THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING TRANSIENT RESPONSE STUDY OF GAS FLOWING THROUGH IRRIGATED PACKING Fran'ces c? De Maria A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1958 Mvay, 1958 IP-293

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Doctoral Committee: Professor Robert R. White, Chairman Professor Julius T. Banchero Assistant Professor Kenneth F. Gordon Professor Victor L. Streeter Assistant Professor Charles M. Thatcher ii

ACKNOWLEDGEMENTS Appreciation is hereby expressed to all those persons who offered their valuable time and availed me of their experience and advice to the preparation of this work. I wish to express my gratitude to all the members of my Doctoral Committee for their friendly advice, expert guidance and encouragement given; and above all, to my Chairman, Professor Robert R. White, for his experienced advice, for his patient, constructive criticism of the manuscript and his untiring moral support which enabled me to complete this research. Moreover, I am grateful to Mr. Gordon Colpitts for the exchange of ideas and frequent discussions of various phases of this work; to Messrs. John D. Janicek and Muhammad T. Tayyabkhan for their friendly advice and interest; to the secretaries and shop members of the Chemical and Metallurgical Engineering Department who did their best to make my tas~d a pleasant one. Lastly, I am indebted to the Visking Corporation and Hercules Powder Company who made this work possible with financial assistance through fellowship grants; the U. S. Stoneware Company who made available the ceramic packings; and to Mr. Patrick Cummiskey of the Friden Calculating Machine Company who provided the use of a desk calculator for the extensive computations required. iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii LIST OF TABLES vii LIST OF FIGURES ix CHAPTER I. INTRODUCTION...........o.o..o.......,. 1 CHAPTER II. LITERATURE SURVEY ON GAS-LIQUID PACKED COLIMNS.... 7 CHAPTER III. THEORYo.....00o.o....................... 11 Transient Response..o....................... 11 Impulse Input, Distribution of Residence-Times Function................................, 12 The Average Residence-Time, Moments of the Distribution of Residence-Times............. 14 Comparison Between the Response of Two Systems, The Variance....................... 16 The Skewness, Correlation of Distribution Curves...................................... 20 Step Input...... 22 Comparison of the Two Methods of Signal Introduction................................ 27 Frequency Response..............................31 Axial Mixing in Packed Beds, Correlation Models... 33 The Diffusion Model............................. 33 Solutions of the Diffusion Equation, Boundary Conditions..................................35 A Packed Bed as a Series of Perfectly Mixed Stages.......46 Comparison of the Two Models................... 56 CHAPTER IV. EXPERIMENTAL..ooo...........65 Equipment........................................5 Flow System................................ 65 Continuous Gas Analyzer.....*...... 70 iv

TABLE OF CONTENTS (Cont'd) Page CHAPTER IV Procedure.O.o.a8................ 7 Preliminary Tests.............................. 78 Standard Experimental Technique................ 81 Experimental Data................................. 85 Dry Packing.......................o......... Irrigated Packing............................ 91 Calculated Results.........................99 CHAPTER V. DISCUSSION........................... 113 Dry Packing............................... 113 Material Balance...oooo.o000 oo*000000........ 113 Effect of Tracer Composition......... 114 Relationship Between Number of Stages and Diffusion Coefficient..ooooo........O...... 116 Effect of Bed Height, Correction for End Effectss.. lo o 117 Comparison with Previous Investigators........ 119 Effect of Particle Diameter and Packing Orientation..... 00 0 0 * 000906 00... 122 Summary...........Q.................... 125 Irrigated Packing............................. 129 Porosity of Irrigated Packing................. 129 Correlation of Experimental Response Curves... 137 End Effects................................... 142 Axial Mixing Correlation...................... 143 Interpretation of Mixing Data................150 Comparison with Previous Investigators........150 Significance of Time Distribution Data as Applied to Mass Transfer... 0. 158 Recommendations....1............................ 161 Conclusions............................... 163 V~~~~~~~6

TABLE OF CONTENTS (Cont'd) Page APPENDIX I Solution "A" atT 0 0 167 Solution "B"A "..........a.................. 176 The Normal Distribution........................... 184 APPENDIX II A Number of Mixed Stages in Series Having Equal Volumes... + a * a a a 0 a v.. 191 A Number of Mixed Stages in Series Having Unequal Volumes.................... 0...0.....0 199 APPENDIX III Typical Irrigated Packing Experiment............. 206 a) Bed Properties............................ 206 b) Air Properties............................. 207 c) Tracer Properties.. 208 d) Response Curve, Air Displacing Tracer... 208 e) Response Curve, Tracer Displacing Air. 214 f) Water Properties.......................... 219 APPENDIX IV Response to a Step Function for Solution "A" of the Diffusion Equation..................... 221 Rotameter Calibration Curves...................... 222 Viscosities of Gases. 226 Water Properties................................. 227 NOMENCLATURE.................................................... 229 BIBLIOGRAPHY.............................................. 235 vi

LIST OF TABLES Table Page I: Quantities Characterizing a System in Terms of its Transient Response............................... 28 II Solutions of Diffusion Equation..................... 45 III Response of a Number of Mixed Stages in Series and Moments of the Time Distribution.................... 55 IV Comparison Between Series of' Mixed Stages and the Diffusion Models................................. 60 V Experimental Conditions and Results for Gas Flowing Through Dry Packing........................ 102 VI Experimental Conditions and Results for Gas Flowing Through Irrigated Packing.......................... 103 VII Number of Mixed Stages for Dry Packing Section Above Liquid Distributor......................... 108 VIII Effect of Tracer Composition....................... 115 IX Effect of Bed Height................................ 118 X GCaracteristic Lengths for Peclet Number Correlation. 124 XI Packing Properties....,.............. 126 XII Coefficients for Porosity Correlation............... 132 XIII Agreement Between Liquid Holdup and Porosity for Irrigated Packing................................. 136 XIV Effect of Irrigated Bed Depth on Longitudinal Mixing. 144 XV Coefficients for Mixing Correlation................... 147 XVI Determination of Mixing Parameters for Response CurvAir Displacing Tracer..................... 210 vii

LIST OF TABLES CONT'D Table Page XVII Determination of Mixing Parameters for Response Curve, Tracer Displacing Air..................... 215 XVIII Response to a Step Function for Solution "A" of Diffusion Equation C/co as Function of t/8 for Values of uL from 80 to 200...................... 221 2DL viii

LIST OF FIGURES Figure Page 1 Response to a Tracer Impulse.......o.0.00000 e..00.... 13 2 Distribution of Residence-Times Plot............... 15 3 Distribution of Residence-Times Plot for Two Systems Having Different Average Residence-Time......... 17 4 Dimensionless Residence-Times Plot Showing Various Degrees of Mixing.................Oo... 19 5 Dimensionless Residence-Times Plot Showing Asymmetry About the t/G = 1 Axis............................... 21 6 Response to a Step In Tracer Concentration........... 23 7 Dimensionless Step Function Response.....o........ 24 8 Transient Response........................ 31 9 Conditions at Inlet of Packed Bed.................... 38 10 Conditions at Outlet of Packed Bed................... 41 11 Time Distribution Curves for Solutions "A" and "B" of the Diffusion Equation............................ 47 12 Time Distribution Curves for Solution "B" and a Series of Mixers....................................... 0.62 13 Flow Diagram......................................... 66 14 Timer and Solenoids Wiring........................... 68 15 Gas Analyzer Assembly............v.......,.,.... 72 16 Assembly of Gas Analyzer and Accessories.......... 73 17 Current Output of Gas Analyzer With Impressed Potential............................................ 74 18 High Voltage D.C. Supply......................... 76 19 Current output of Gas Analyzer With Helium Concentration..........,~...O.........O............ 77 ix

LIST OF FIGURES CONT'D Figure Page 20 Dry Packing, Run No. D-8............................. 86 21 Dry Packing, Run No. D-12..oo.........oooo........ 87 22 Dry Packing, Run No. D-16............................ 88 23 Dry Packing, Run No. D-23............................ 89 24 Dry Packing, Run No. D-27....S....................90 25 Irrigated Packing, Run No. W-1....................... 92 26 Irrigated Packing, Run No. W-5....................... 93 27 Irrigated Packing, Run No. W-8.................... 94 28 Irrigated Packing, Run No. W-ll..................... 95 29 Irrigated Packing, Run No. W-17.... o......o......... 30 Irrigated Packing, Run No. W-26..................... 97 31 Irrigated Packing, Run No. W-30.................... 98 32 Irrigated Packing, Run No. W-32....................99 33 Irrigated Packing, Run No. W-39.................... 100 34 Estimation of the Dimensionless Group UL for Dry Packing Runs............................L..........109 35 Axial Mixing Through Beds of Spheres and Raschig Rings (Peclet No. vs. Reynolds No.)...*.............0900~. 121 36 Axial Mixing for Gas Flowing Through Beds of Spheres and Raschig Rings (Average Value of Modified Peclet No. vs. Reynolds No.)....................... 127 37 Effect of Liquid Flow Rate on Porosity of Irrigated Bed (Porosity vs. Liquid Reynolds No.).............. 130 38 Effect of Gas Flow Rate on Wet Porosity for 1/4" Raschig Rings..... eoooooooooo..... 133 39 Effect of Particle Diameter on Wet Porosity........ 134 X~~~~~~3

LIST OF FIGURES CONT'D Figure Page 40 Response Curve Showing Departure from Uniform Flow (v - 24.0003) for Run No. W-8, Air Displacing Tracer............0..........OOOOO. 138 41 Response Curve Showing Departure from Uniform Flow (v 2a4 =.0011) for Run No. W-18, Air Displacing Tracer..................................... 139 42 Response Curve Showing Departure from Uniform Flow (v - 2=4 =.0050) for Run No. W-19, Tracer Displacing Air............................o...... 140 43 Effect of Liquid Flow Rate on Peclet No. for Irrigated Bed (Peclet No. vs. Liquid Reynolds No.)... 145 44 Effect oi Particle Diameter on Longitudinal Mixing... 148 45 Effect of Gas Flow Rate on Longitudinal Mixing....... 149 46 Longitudinal Mixing for Gas Flowing Through Irrigated Beds of Raschig Rings............................... 152 47 Total Bed Area Exposed to Flowing Gas and Effective Mass Transfer Area as Function of Liquid Flow Rate... 155 48 Peclet Number for the Liquid ard G"-s Phases Flowing Countercurrently in a Packed Bed of Raschig Rings.... 157 49 A Perfectly Mixed Tank............................... 189 50 Two Perfectly Mixed Tanks in Series................. 191 51 Series of Perfectly Mixed Tanks Having Unequal Volumes. 9........................... 197 52 Graphical Integration of Response Curve, Air Displacing Tracer.....,...................... 211 53 Graphical Integration of Response Curve, Tracer Displacing Air................................. 2]6 54 Rotameter No. 1, Tube No. 2F-1/4-16-5/70 Calibration for Air at 60~F and 14.7 psia....................... 222 55 Rotameter No. 2) Tube No. B4-27-10/70-G Calibration for Water at 64.4~F............................ 223 xi

LIST OF FIGURES CONT'D Figure Page 56 Rotameter No. 3, Tube No. B4-27-10/27 Calibration for Air at 600 F and 14.7 psia....................... 224 57 Rotameter No. 4, Tube. No. B5-27-10/77 Calibration for Air at O0~ F and 14.7 psia....................... 225 58 Viscosity of Gases................................ 226 59 Properties of Water............................... 227 xii

C HAPTER I INTRODUCTION Chemical engineering is concerned with complex and interconnected physical structures (systems) within which a number of physical and chemical changes (processes) take place either simultaneously or consecutively, often at competing rates and with intricate interactions. If an accurate formulation of these systems is attempted in terms of the microscopic events taking place therein, the problem in many cases defies mathematical formulation. Moreover, research tools and methods designed to probe the "fine structure" of interrelated phenomena such as those taking place in process equipment have not yet been developed. As a result, the engineer must often resort to the study of a system of process equipment as a whole. This is accomplished by simulation of large commercial scale units by smaller experimental systems. Inlet streams and experimental conditions are set to a predetermined value and once the experimental equipment has reached "steady state", the properties of the outlet stream are measured and recorded. Empirical correlations of overall performance in terms of the independent variables investigated are the end result. Such correlations are very useful in determining optimum operating conditions and for a limited amount of scaleup. Recently "unsteady state" experimental methods have gained increased attention in chemical engineering research, mainly because statistical information concerning the "time-distribution" of the response can be readily obtained. Such information has significance in that it permits the characterization of the succession of events _1

-2occurring in a system in terms of physical models (or which is the same in terms of probability laws). It may also provide a more adequate similarity criterion in the scaleup of equipment (i.e., two systems of different size in which the statistical distribution of a certain succession of events is the same, ought to be considered similar to one another). Unsteady state experiments are based on the measurements of the response of a system to a known disturbance in experimental conditions. Work of this type has already been reported for relatively simple systems of chemical engineering, such as on single phase flow in pipes and packed beds /13/, /17/, /18/, /23/, /47/ reactors /9/, /15/, /27/, /28/, /55/, /64/, and heat exchangers /8/. An unsteady state investigation of packed gas-liquid absorption equipment appeared to be both useful and interesting for the following reasons: 1. Simplified equipment such as wetted wall columns and disc absorption columns yield data which is not easily applicable to predictions on the performance of packed absorbers /11/. Mass transfer data as obtained from experimental columns operated at steady state conditions appears to be in wide disagreement /51/. These difficulties are apparently due to the fact that very little is known about the flow of layers of liquid and gas over discontinuous surfaces and the fundamental concepts of hydrodynamics are not sufficient in furnishing valid similarity criteria. It is possible that statistical information on the "distribution of residence-times" of the gas and liquid stream flowing through the column

-3may provide a valid similarity criterion and thereby provide a more satisfactory parameter for the correlation of absorption equipment performance. 2. As the knowledge of absorption processes becomes more complete, a greater and greater number of these processes appear to involve chemical reactions between components of the gas and liquid phase in addition to physical absorption /51/. For such processes, data on distribution of residence-times of both phases as function of their volumetric rates of flow and column geometry are of direct use in design calculations. 3. Unsteady state experimental techniques applied to gasliquid flow systems present unusual problems concerning the introduction of a predetermined disturbance in one of the entering phases and the separation of the effluent phases for rapid analysis. The existing theory on the analysis of unsteady state measurements /13/, /38/, is not general enough to make possible the analysis of more complex equipment and conditions of flow other than uniform. Once the reliability and generality of unsteady state measurements could be demonstrated, such a technique would certainly find increased application in chemical engineering research. For the foregoing reasons, we elected to study a typical gasliquid packed absorber (a column packed with Raschig rings) under unsteady state conditions.

-4The problem was limited Io June study of the gas phase response to a sudden disturbance in tracer concentration in the inlet stream. Complicating effects such as absorption or chemical reaction between the liquid and gas streams were eliminated. The method of introduction of the transitory disturbance permitted the measurement of the total volume of the column occupied by the gas. Such gas holdup data, for a packed absorption column, not previously available in the literature was one of the specific objectives of this work. The theory of unsteady state experimentation has been reorganized and presented stressing the interpretation of transient response in terms of its statistical significance. Various models proposed for the correlation of mixing data in packed beds have been presented and discussed in the theory. The concept of a packed bed as a series of perfectly mixed stages was generalized in order to adequately formulate the complications presented by the experimental equipment required to handle a gas and a liquid phase. The experimental equipment involved the development of a continuous gas analyzer based on an alpha particle ionization chamber already used for the analysis of gas mixtures /16/ and as a leak detector. Analyzers capable of furnishing a continuous output signal of this type are strongly recommended for unsteady state work since they allow direct and automatic computations on the output signal to furnish the desired characteristics of the response. The reported data consists primarily of gas porosities and Peclet numbers, a dimensionless mixing parameter calculated from the recorded response of the gas stream, as function of gas and liquid flow rates and packing sizes. A discussion of these data is given.

-5It is recommended that a study of the behavior of the liquid phase be made in a manner analogous to the sne followed here for the gas. It is hoped that such a course of action may lead to a more complete picture of the mechanics of heterogeneous flow in packed beds.

CHAPTER II LITERATURE SURVEY ON GAS-LIQUID PACKED COLUMNS Packed columns are widely used for the contacting of two fluid phases to effect mass transfer and/or chemical reaction between them. A packed column consists of a vertical shell, constructed of metal, ceramics or glass filled with packing materials such as Raschig rings or other regular or irregular shapes made out of a wide variety of materials. Two types of columns find use in chemical engineering operations; one in which the gas and liquid flow countercurrently to each other, the other in which both phases flow down the column concurrently. The first type finds applications in most separations in which the absorption of a gaseous component by the liquid or the Stripping of a liquid component by the gas is desired. In this case, the countercurrent operation is advantageous in obtaining the maximum possible driving force between the two contacting streams, the packing producing a large surface of contact between the two phases. The second type is generally used in hydrogenation and hydrodesulfurization of heavy petroleum feed stocks and in other catalytic conversion processes. Here, in general, the packing is the catalyst for the reaction. The concurrent flow technique is used to favor greater capacity rather than greater driving force between the flowing phases. When the two fluid phases are flowing through a porous medium such as a packed tower, the liquid normally wets the solid and flows over it, thus changing the volume of the voids available to the gas flow in absence of liquid. We shall, from now on, refer to the fraction -7

of the total tower volume occupied by gas, as porosity and the fractional volume occupied by the liquid as holdup. The extent of our knowledge on the nature of such flow is limited to pressure drop and liquid holdup measurements for countercurrent operation which we shall presently review. The pressure drop in packed columns increases with an increase in gas rate, liquid rate, or both. The pressure drop for a constant liquid flow rate is proportional to the mass flow rate of the gas raised to a power between 1.8 and 2 until a critical flow velocity is reached where the curve increases in slope /51/. This point is called "flooding point". Another less clearly defined change in slope is sometimes detected before the flooding point is reached. This is called loading point. In recent times, however, a more generally accepted belief is that loading and flooding phenomena develop gradually and not suddenly at a definite flow point. In general, it has been found difficult to obtain correlations of the pressure drop from consideration of the proportions of the voids occupied by the two fluids. Therefore, most of the reported data /44/, /57/, /50/, /60/ is given as function of gas and liquid superficial flow rates for various packing independently of the actual flow conditions in the column. The total quantity of liquid present in an absorption column at any time during operation has been called liquid holdup. In general, this quantity has been arbitrarily defined to consist of a static and a dynamic or operating holdup. The first is the amount of liquid in the packing which does not drain when the flow to the column is stopped. The latter represents the liquid flowing through the packing and which drains when the flow is interrupted. The total holdup has been found

-9to increase with liquid flow rates. Below the flooding gas velocity, holdup was found almost independent of gas rate until, as the flooding was approached, the holdup increased exponentially /58/, /19/, /34/, /54/. Holdup information has been used to shed some light on the flow mechanics of the column. Theoretical discussions of flooding in packed equipment and generalized correlations of limiting velocities with fluid properties and packing dimensions are given by Elgin and Weiss /19/, Sherwood, Shipley and Holloway /52/, and Lobo, Friend, Hashmall and Zenz /43/. In all cases, fluid velocities based on the empty cross section of the column, the superficial area and the void fraction based on the dry packing were the correlation parameters. Recently, the pressure drop for coke packing was correlated in terms of the true gas velocity by Gardner /26/, this being the first effort to relate quantitatively holdup and pressure drop. No data of this nature are available for concurrent flow except for a recent study on mixing and flow distribution on the two phases flowing concurrently over glass spheres and porous pellets published by Lapidus /40/ while the present work was in progress. The performance of packed absorption towers is also dependent on the extent of interfacial area available for the transfer of mass between the two phases. The effective value of the wetted area is, in general, less than the geometrical surface of the packing and there are reasons to believe that it changes with the hydrodynamic conditions in the column. Wetted surface area has been determined either directly in

-10various ingenious ways /45/, /29/, or by using gas film controlled absorption data which is amenable to analysis resulting in indirect estimates of effective wetted surface areas /22/, /51/, /53/. These data are in agreement only in that for the operating range below the loading zone, gas rates had no effect on wetted area but extent of wetted area was liquid rate dependent. F. R. Whitt /62/, in a recent review on this subject points out that the directly determined wetted surface values for 3/8" and 1/2" Raschig rings appear to be considerably higher than the indirectly estimated effective values for the same size rings and liquid rates. For a given liquid rate, the effective wetted surface per unit volume is surprisingly similar for Raschig rings from 1/2" to 2" in diameter in spite of a 114/29 or approximately fourfold range of total available surface. The prevalent uncertainty on the actual value of this quantity is believed to be partly responsible for the wide scatter of mass transfer data /51/. Recent work on correlation of mass transfer data /25/, /20/, /46/, /5/, has shown that gas porosity is also an important correlation parameter. No direct measurements of gas porosities in countercurrent packed towers have been available to date.

CHAPTER III THEORY Experiments in which unsteady state response of a system is to be studied involve the use of an input signal and measurement of the output as function of time. For process systems one is more or less restricted to use as a signal some alteration of the intensive property of the fluid being processed. Of these, the most readily measurable properties are temperature and concentration. If the disturbance is introduced in a transitory manner, we obtain the so-called transient response of the system. The signal can also be introduced in a cyclic manner and in this case the system response is called frequency response. Transient Response An early application of transient response in chemical engineering research is reported by F. C. Fowler and G. G. Brown /23/ in a study of the successive flow of fluids through pipes. Gilliland and co-workers used it in a pioneering study of fluidization /27/, /28/. Later, transient response found increased application in the study of fluidized reactors /15/, /55/, heat exchangers /8/, packed beds /18/, and stirred pot reactors /9/, /64/. Danckwerts /13/ gives a detailed discussion of this technique as a tool for equipment and process studies. We shall restrict our attention to the case of a flow system in which the signal is generated by changes in concentration of a given component of the fluid stream designated as the tracer. There are two methods of introduction of the concentration signal which lead to a simple mathematical representation. These are: -11

-12 - 1. Injection in the inlet stream of a quantity of tracer virtually instantaneously or within a period of time very short compared to the average residence-time. Such an input is designated mathematically as "impulse function" or "delta function". 2. Introduction of a discontinuity in the concentration of the tracer in the inlet stream by displacing the fluid flowing through the equipment with a fluid of different tracer concentration. If the transition from the leading fluid to the follow-up fluid takes place sharply, the input may be represented mathematically by a "step function". Impulse Input, Distribution of Residence-Times Function We consider a flow system having a total volume V and through which the volumetric rate of flow is v. If the quantity (tQ of tracer is injected instantaneously at the inlet, a continuous analysis of the effluent fluid will yield a plot of tracer concentration versus time as shown in Figure 1. No matter what the shape of the concentration-time plot the material balance for the tracer: C tdt G ~o~~~~~~~ ~~~(1) must hold. Here t is the time measured from the introduction of the impulse signal and c is the tracer concentration.

Figure 1 Response To A Tracer Impulse By transposing Q in equation (1) the material balance can also be written in the form: -13 - o (2) Equation (2) suggests that the quotient - may be regarded as 0 interval of time, tsum of all such fractions over all times must of course be equal to unity. We realize then, that if one could tag each molecule which is introduced in the system at any time t over a very small interval of time dt the same relationship would exist between all the tagged mole

cules at the outlet so that quite independently from the injection of an impulse of tracer all the molecules flowing through the system obey a "probability" law governing their time of residence in the system. The pulse experiment offers a convenient method to measure such a probability so that with the quantity VC_ we may associate the function E(t)* such that: o () 0 E(t) may thus be defined as the fraction of the entering feed, flowing in the effluent per unit time, which has been in the reactor between times t and t+dt. We shall designate E(t) as the distribution of residence-time function or simply time distribution. The Average Residence-Time, Moments of the Distribution of ResidenceTimes A plot of E(t) versus t as shown in Figure 2 gives a graphical representation of the mass fraction of those elements of fluid which enter the system at the same moment and which are flowing out per unit of time as a function of the time spent in the system. The time spent on the average by the fluid flowing through the system may be readily found by the well-known method of calculating the average of a continuous function. aO tE(t)it IE(t) Jt * In probability theory E(t) is called a probability frequency function /10/. Equation (3) gives a basic property of the frequency function.

-15 - t Time, t Figure 2 Distribution of Residence-Times Plot O is the average or mean residence-time and is a useful quantity to characterize the system, since no matter what the flow conditions are: VV (5) where e is the fraction of the total volume occupied by the fluid. Thus, if the average residence-time 0 and the volumetric flow rate of the fluid v are known, the volume of the system occupied by the fluid eV may be readily calculated. Vice versa, if both eV and v are known one may check the consistency of the residence-time distribution curve as determined in the tracer experiment by the use of equations (4) and (5). Combining equations (3) and (4) we may write simply: (6) Referring to Figure 2 we see that on the basis of equation (6), the

average residence-time @ can also be regarded as the first moment of the distributin function about the ordinate axis since it is the sulmnation of all the differential elements of the area E(t)dt multiplied by the abs issa t. In general, we define the nth moment of the distribution function as the summation of the differential elements of area multiplied by the abscissa raised to the nth power: hth moment t (t (7) Comparison Between The Response Of Two Systems, The Variance When it is required to compare the pulse response for two systems having different mean residence-times the time scale for the two systems will be different since in the one for which the residence-time is larger, the tracer elements * will spend a longer time in flowing through even though their relative distribution with respect to the average time spent in the system may be the same as Figure 3 shows. In this case, it is useful to use as the abscissa the dimensionless time t/o and as the ordinate the dimensionless function GE(t)so that equation (3) becomes: (3-a) * The word "elements" is used to identify small but discrete portions of the flowing fluid. Depending upon the scale of interest (smoothing out of the response curves) or the nature of the detecting apparatus one may identify such elements with molecules or segments of the flowing stream of various dimensions down to a molecular scale.

-17I \ t = 1 t =2 Figure 3 Distribution of Residence-Times Plot For Two Systems Having Different Average Residence-Time and equation (6) gives: Io the imenionlss pot E~t)verss t/ is he (6-a) If the dimensionless plot GE(t) versus t/@ is the same for the two systems being compared, then the two systems exhibit the same transient behavior. It is apparent that the shape of the residence-times plot will vary with the nature of the flow taking place in the column depending upon the flow path followed by each tracer element. Two limiting possibilities may be visualized: One is the piston flow case in which tracer

-18elements which enter the vessel at the same moment proceed through it with constant and equal velocity in parallel flow paths; the other is the case of perfect mixing in which the fraction of tracer elements which escape the system immediately after injection is a maximum. This condition is achieved when the concentration throughout the vessel is the same as the concentration of the effluent. Consequently, at the moment of injection: t-o and Q=:Vc so that:;J E t) = N J therefore, the dimensionless time-distribution curve has a value of 1 at t/@ = 0 and as seen in Appendix II, pp. 185 it decays at a rate equal to 1/G. Between these two limiting cases as illustrated in Figure 4, there are various degrees of mixing depending upon the manner of propagation of the impulse signal through the system. Figure 4 clearly shows that the major difference between the various degrees of mixing is the degree of dispersion of the impulse signal about the mean residence-time. For piston flow the tracer signal at the outlet is concentrated about t/G = 1 and the dispersion is zero. The dispersion increases as the mixing increases until for perfect mixing the dispersion is greatest and equal to 1 (See pp. 49). The second moment of the dimensionless distribution of residence-times curve about t/g = 1 gives a suitable measure of the degree of dispersion.

-19r~ ~~~~~~d ] FE Piston Flow WO | XPerfect M 1 o Mixing 0 r. 0 From our definition of the nth moment (see page 15 ) we have for the second moment of GE(t) about the origin: ~2 lrt ~2e scnmo ent atbot9) (8) | (t / 0_ ) oE ) d (~/8) 02gure 4 H iialtescn omn bu / s 0~~jb' EtdtB 9

-20by use of equation (3a) and (6a), this reduces to: Jo 2_ r~tl) s E~t) d (TV) - i(lo) Equation (10) gives the desired quantity which we shall call "variance" and designate it by the symbol a because of its similarity with an analogous quantity used in statistics /10/, /21/, /24/. The Skewness, Correlation of Distribution Curves While the variance of the residence-times distribution typifies the extent of mixing taking place in a flow system by giving a measure of the dispersion of the fluid elements about the average residence-time, no clue is obtained as to the "asymmetry" of the dispersion. In fact, two impulse response curves may have the same average residencetime and variance as shown in Figure 5 and yet they are not completely congruent because one is symmetric about the axis t/@ = 1 and the other is not. If the distribution curve were symmetric about the t/e = 1 axis, all central moments of odd order would vanish. It is practice in statistics and probability theory /10/ to use the departure from zero of the third central moment as a measure of the asymmetry of the distribution. We shall call this quantity "skewness" and designate it by the symbol v. By analogy with equation (9) the third central moment * In some hypothetical models of flow systems, the average residencetime G as calculated from the computed response of the model to an impulse may not be equal to the true residence-time eV/v. In this case, if one desires to calculate the variance about the true residence-time eV/v equation (10) becomes: =(")5t'Ett)dt- (')[ tEt) (10-a)

-210 $4'Q a)4 I t/Q = 1 Dimensionless Time, t/Q Figure 5 Dimensionless Residence-Times Plot Showing Asymmetry About The t/@ = 1 Axis is: By simplifying: The skewness of the distribution function is used in the exd ~ ~~~~~~~~~~~~~ 1 ~ —--—' n-__ 1 __-x

-22tion model. The fourth moment of the distribution function is also needed to completely define a distribution curve /24/; however, in practical analysis the evaluation of the variance and skewness may already prove to be a tedious task. Empirical methods of correlation based on the various moments of the distribution curve have been developed to fit all sorts of statistical distributions. The Gram-Charlier series, for example, is capable of fitting a distribution curve. The coefficient of each term of the series is a moment of the distribution curve in question. The degree of accuracy of the series increases as terms containing moments of higher and higher order are added. For a description of this series and other methods of correlation, reference is made to Fry's excellent work on probability and its engineering uses /24/. Step Input The foregoing information about the residence-time distribution function can also be obtained by the introduction in the flow system of a step (or discontinuity) in tracer concentration. For a step function input, a continuous analysis of the effluent fluid will yield a plot of tracer concentration versus time. When the displacing fluid contains no tracer while the fluid being displaced has a tracer concentration equal to co, the response curve will have the general shape shown by the solid line in Figure 6.

-23 - o C- o Curve A h | Cle N..r. Fluid \ / Curve B Clear Fluid 4-) ~ a Tracer Displacing Displacing Tracer Clear Fluid O / O~~~~~~ ~~~Time, t Figure 6 Response To A Step In Tracer Concentration No matter what the shape of the curve, the material balance: (13) must hold. Here t is the time from the moment of discontinuing the tracer flow, co is the initial tracer concentration, c is the concentration at any time t, eV is the volume of the system occupied by the fluid, v is the volumetric fluid velocity. If the displacing fluid has a tracer concentration equal to co while the leading fluid does not, the response curve would have the form as shown by Curve B in Figure 6. The material balance for the tracer is now: j(c.-c)dt= c0CeV) o (14)

-24On rearranging equations (13) and (14) and substituting for EV/v the average residence-time 0, we obtain respectively: CO (13-a) and 0CO)dt) (14-a) A plot of c/co versus t/O will thus be independent of the mean residence-time. In addition, equations (13-a) and (14-a) clearly show that Curves A and B are one the complement of the other since for any value of t/o as shown in Figure 7, the sum of the ordinate of Curves A and B is equal to unity. Curve A Curve B o Clear Fluid \ / Tracer Displacing I0 Displacing Tracer Clear Fluid' // \ o t/Q Figure 7 Dimensionless Step Function Response

-25 - For the sake of mathematical consistency in the treatment that follows, we shall be concerned with response curves of the type represented by Curve B, i.e., with experiments in which the step function input is produced by a sudden change in concentration from 0 to co. The conclusions which we shall obtain will be equally applicable to the complementary case as presented by Cairve A. The step function response must be in some simple way related to the distribution of residence-times function since for the same system, the only difference in the two response curves lies in the input signal. By intuitive reasoning we may argue that since a step in tracer concentration results from the summation of an infinite nLunber of impulses introduced in the system over a continuous interval of time, the response to a step function ought to be obtained by integration of the distribution of residence-times curve or vice versa, the distribution of residence-times curve should be obtained by differentiation of the step input response. A verification of this statement is obtained by the following reasoning: For a step input in tracer concentration at time zero, the flowing stream at the inlet of the system is suddenly changed from O to co. At any time t thereafter, the rate of flow of tracer in the effluent stream is cv and the molecules of tracer in the effluent stream have been in the system for a period of time anywhere from O to t. At time t + dt the rate of outflow of tracer is cv + d(vc)dt and dt the tracer molecules in the effluent have been in the system for a

*26period of time anywhere from 0 to t + dt. The increment in tracer outflow d()dtis due to molecules of feed which have been in the system for an interval of time between t and t + dt. Therefore, the fraction of entering feed emerging at the outlet in the interval of time t + dt and which has been in the system for an interval of time from t to t + dt is: d,vc:t d I dt = Co t V co dt The fraction of entering feed which has been in the reactor for an amount of time between t and t + dt emerging at the outlet for unit of time, i.e.,.dLt/ corresponds to the distribution of residence-times function E(t) as defined on page 13 From the foregoing we have established the fact that the response to a step function is the integral of the distribution of residence-times curve. If it were required to obtain information about the distribution from a step function experiment a differentiation of the response curve would be necessary. Numerical methods of differentiation are rather inaccurate, however, so that one may recur to the evaluation of the moments of the distribution of residence-times curve which are obtainable from the step function response by an integration procedure.

-27Thus, the material balance from a step function experiment as given in equation (14) is the first moment of the distribution of residence-times: tE(tt =t ( c dt (i6) this equality can be verified on integration by parts of the right hand side of equation (16) by using the relationship between E(t) and CO given in equation (15). In an analogous manner, one may prove that: <fit o |Cb~~d~ f/ EfA=t_-1 (17) and also: v -| (t/SE(t) d (t/9)-3 3 (t,4)z&(v) 8a 2 ft )t _____ __ _ ___, _ __ _ __ _ __ _ (18) A summary of the relationships between impulse, and step input response with the distribution of residence-times function E(t) is given in Table I, page Z7. Comparison of The Two Methods of Signal Introduction Whenever it is desired tO correlate the distribution of residence-times in terms of mechanistic models, one is confronted with

TABLE I QUANTITIES CHARACTERIZING A SYSTEM IN TERMS OF ITS TRANSIENT RESPONSE Moments of the Distribution of Relationships in Ti-rms of Outflow Coincentration Residence-Times Function E(t) Step Input from 0 to c0 Impulse Input (Amount Q of Tracer Injected Instantaneously) Zero Moment of E(t) vs. t cc Go 00 About t = Oaxis f E(t)dt = 1 i dc/c =1 fcdt=Q 0 0 0 v First Moment of E(t) vs. t Cc Co Co About t = 0 axis f tE(t)dt = 0 = f (1 - c/c )dt = eEV v I tcdt = 0 = EX 0 v 0~~~~o v Q 0 Variance Second Moment of 2c 2 t022= 0f 0 -l _/ d d OE(t) vs. t/O2 = (t/O) OE(t)d(t/) - 2f t0i /c ) About t/6 = 1 axis 0 22 Skewness Third Momeint of c 3fct2(l c/c )dt f 3cdt eE(t) vs. t/O = f (t/e)3eE(t)d~t/o) = 0 - v = About t/O = 1 axis 0 3 3 00 0 2otcdt - 31 (t/e) GE(t)d(t/0) + 2 -6f t(i - c/c0)dt 03Q)Q + 2 0 0 +2 02 02

-29the solution of "systems" of differential equations which simulate the behavior of the experimental system. In this case, the step signal lends itself more readily to the mathematical solution of the problem by elementary methods /35/ and, in most cases, the solutions are reported in terms of step function response. The impulse response can be readily obtained by differentiating such solutions. Another important advantage of the response to a step input experiment is that the volume of the system occupied by the fluid stream which carries the tracer is readily obtained from the value of the area under the response curve as equations (13) and (14) clearly show. When a response to an impulse is available instead, one must evaluate the first moment of the response curve. When this is done by graphical means, the procedure is more laborious and less accurate*. In many cases, especially in the study of large equipment, the impulse signal may prove to be more advantageous because it requires much smaller quantities of tracer which may be injected just ahead of that portion of equipment under study thus avoiding entrance problems and the problem of obtaining a sharp transition from tracer to ordinary process stream as required in the step input method. On the passive side, however, is the high sensitivity required by the detection equipment since the amount of tracer injected in a pulse will be small in comparison to the volumetric flow rate of the clear fluid so that the tracer concentration at the outlet will be relatively small even at its peak * In the event that a continuous analysis of the effluent is feasible, one may feed the output signal of the analyzer to suitable electronic integrators to obtain the various moments of the distribution to any desired degree of accuracy.

-30 - value. Instruments of high sensitivity and suitable for this purpose, such as a mass spectrograph, are very costly and make the impulse technique less accessible to the experimenter. Lastly, it is worth mentioning that in special systems it may be advantageous to use both methods of signal introduction. Lapidus, in a recent article /39/ reported a transient study of a reactor packed with porous catalyst pellets and pointed out that impulse response did not detect the intraparticle voids because the rate of tracer transfer in and out of the pores of the catalyst occurred at a much lower rate than the speed. of propagation of the impulse throughout the system. In this case, comparison between impulse and step experiments may lead to the evaluation of intraparticle void volume.

-31Frequency Response This unsteady state method of experimentation consists of using as input a purely sinusoidal signal. No direct information is obtained, however, about the volume of the system occupied by the fluid or about the distribution of residence-times. Frequency response is valuable when a mathematical model which approximates the system is already available. In this case, the mathematical analysis is greatly simplified since the phase shift and amplitude attenuation of the response, as exemplified in Figure 8, permit a direct evaluation of the constants involved in the "system" of differential equations which simulates the physical situation. PHASE SHIFT 0 OUTPUT AMPLITUDE INPUT SINE WAVE ftat.~ MEMO C OUTPUT et i SINE WAVE 2 - -AMPLITUDE ATTENUATION tE / -INPUT MPUEUI TIME, t Figure 8 Frequency Response * This is generally the case in electric networks and instrument loops where frequency response finds its widest application.

-32Frequency response has found some application in chemical engineering investigations, notably in the study of longitudinal diffusion occurring in one phase flow through packed beds /17/, /18/, /36/ /47/, /38/, and in the study of heat transfer /8/. However, in addition to the one mentioned above, three other disadvantages to this method of analysis as applied to process systems are: 1. A great deal of specialized equipment is necessary to obtain an experimental frequency response on a particular process since the equipment must be capable of applying a sinusoidal disturbance upon one of the intensive properties of the process stream. 2. If the signal is not purely sinusoidal, the mathematical analysis is greatly complicated, thus offsetting the primary advantage of frequency response /18/. 3. If the differential equations which simulate the dynamics of the system are not linear, this method of mathematical analysis breaks down /35/. Because of the disadvantages enumerated above, the experimental work undertaken in the present investigation was limited to the use of transient response and we shall no longer be concerned with frequency response analysis.

-33 - Axial Mixing In Packed Beds, Correlation Models In this study we are concerned with the mechanism by which mass is dispersed in a packed bed through which it is flowing in the absence of absorption or chemical reaction. In all cases of practical interest, the dispersion mechanism, which is usually referred to as mixing, involves turbulence. The intermingling of turbulent eddies is apparently responsible for a "coarse scale" of mixing while the ultimate homogenization to a molecular scale is effected by molecular diffusion.* The Diffusion Model Numerous authors /2/, /3/, /13/, /38/, /42/, have treated the mixing taking place in a packed bed as a diffusional process in which the rate of mass transfer is described by the relation: (19) where: D = Effective diffusivity sq. ft/sec. grad c =- o gd = i + ~ j + -A - moles/cu.ft./ft. r = Mass transfer rate, moles/sec.sq.ft. Assuming that the bed can be treated as a continuum, we obtain the following expression for the accumulation in a differential element: * This point is just mentioned here to render the reader aware that the "concentration at the point" has a significance only upon the scale of interest. And, the mixing rate may be dependent upon how fine a mixing is detectable by available instruments /12/.

-34-.c _ _ d.. CD SvacJ c +uc)'at (20) where u is the average velocity through the interstices of the bed. The effective diffusivity has been found to be anisotropic, i.e., it is sensibly greater in the longitudinal than in the radial direction. Bernard and Wilhelm /3/ and Singer and Wilhelm /56/ have shown experimentally that at sufficiently high Reynolds numbers and tube-to-particle-diameter ratio, the radial Peclet number (the dimensionless quantity du ) is approximately 11. Baron /2/ from a statistical treatment of the random side-stepping deflections of the fluid in the bed estimated that the radial Peclet number is between 5 and 13. The average length of each step was assumed by Baron to be dependent on the characteristics of the packing, the Reynolds number and the kinematic viscosity. Ranz /49/ calculated that the radial Peclet number inrhomboehedrally packed beds of spheres is 11.2. More recently, measurements of longitudinal mixing by Kramers and Alberda /38/ and McHenry and Wilhelm /47/ showed that the longitudinal Peclet du number (the dimensionless quantity DL ) is between 1 and 2 for gases. Ebach's /18/ measurements for liquids gave values of 0.5 to 1. Despite the discrepancy in reported results, it can be stated with some certainty that the Peclet number in the radial direction is significantly higher. We shall discuss the possible reasons for the anisotropy of the effective diffusivity later (see page 55 ).

-35 - It suffices now to say that we restrict our discussion to longitudinal mixing alone since we assume that the transient concentration signal is introduced in such a manner that the concentration in the direction normal to flow is uniform. For this particular case, equation /20/ reduces to: XC, -_DL C. LC (21) The solution of equation (21) involves a suitable choice of boundary conditions which will completely specify the problem. The choice depends on how detailed or complex a physical model is desired; of course, the resulting solutions become correspondingly more complex and consequently more difficult to use. Solutions of The Diffusion Equation, Boundary Conditions Here we shall briefly review the solutions of the diffusion equation (21) as reported in the literature /1/, /6/, /40/, /64/, in order of complexity of the physical model and thereby discuss their shortcomings for the correlation of the experimental data. The simplest choice of boundary conditions is obtained by assuming continuity of fluid concentration at the inlet of a packed bed of infinite extent. The solution to this problem /6/, /40/, shall be referred to as solution "A". The conditions of continuity at the bed entrance, i.e., at x = 0 are:

-36 - also for a bed of infinite extent: lim c -o (23) A step input in tracer concentration from 0 to c0 may be described as follows: C-o at Lt<o c.O cat t co C~OC at to O (24) c c. at t>o The set of boundary conditions which satisfies equations (22), (23) and (24) is, therefore: 1-a) cCot>) Co 2-a) im c(,t) (25) 3-a) C( b t) o The solution of the diffusion equation with this set of boundary conditions evaluated at the outlet where x = L is: uL C.O 2 -I+cF "t- utL e~L er~es )_MQ P f-~- 1 (26) By differentiation of the step input response as given by equation (26) above, one obtains the impulse response or distribution of residencetimes function: ut L dt 2z-~ti: F (27) See deriv tion details in Apendix I, * See derivation details in Appendix I9 pp. i64

-37It may be shown that the zero moment of the distribution of residence-times as given by equation (27) is equal to unity. Moreover, the second moment of the distribution is equal to L/u* or the average residence-time. Therefore, solution "A" appears to be consistent with the conditions required by the time distribution of an actual system as specified in equation (3), page 13 and equation (6), page 14 Aris and Amundson have recently pointed out /1/ however, that the space distribution ** of this solution does not behave correctly because its zero moment is greater than one, due to the simplicity of the model which does not take into account the diffusion flux at the inlet boundary. For this reason, solution "A" should not be used when the extent of mixing is large (large values of the effective diffusivity). A somewhat more complex model results when such a diffusion flix is considered to be present immediately at the entrance of a packed bed of infinite extent. We shall call the solution to this problem, recently given by Aris and Amundson /1/, solution "B". * See derivation details in Appendix I, pp. 166 and 167 N* Note that equation (27) is really a function of space (x) and time (t) and that the space variable was kept constant by letting x = L thus obtaining a distribution in time (Dr residence-times distribution as we have called it). By maintaining the time variable constant it is possible to study the space distribution as it was done by Aris and Amundson.

-38 - C(O-) C(o+) Fe Secti n Y)I Bed of Infinite Extent Effective Longitudina ffetive Longitudina ffusivity Dif /usivity DOj x x =0 X — C Figure 9 Conditions At Inlet Of Packed Bed The physical situation at the inlet to the semi-infinite bed is shown schematically in Figure 9. To the left of the boundary x = 0, we have: aC = C(-)-DL oC wO-) 3t X to the right:'ac Duco+}. Dc 2+'at ta a~ m a so that a material balance gives the required inlet condition:.cCo-)- Co-)_,c(oc)_DL-u (28)

-39For a semi-infinite column as for solution "A", we have: lim C = (23) and for a step function in tracer concentration from 0 to cO as before we can write: C=o at t o C=-Co ct t -o CCo at t>o (24) If the step function is applied immediately at the boundary of the packed bed as we would obviously like to assume for simplicity, then D; (o-) in equation (28) must be equal to zero as dictated by the discontinuity in the step function. This essentially means that in order to obtain a sharp step input, the diffusivity in the foresection D' must be zero. Thus the set of boundary conditions which satisfies equations (28), (23) and (24), is: 1-b) CO:.C (oCt) -)DL j(o, t) 2-b) im C(X)QO (29) C — oo 3-b) C The solution of the diffusion equation evaluated at the

-40 - outlet, i.e., at x = L is:* (Lvt)e ArL C- = Vt dt (30) and the impulse response* ~L- t)~ uL &() 4 u UZ C e+f t (31) dt eoe,'.e The zero moment of the residence-times distribution as given by equation (31) above is equal to unity*. However, for the first moment we find*: (.-ut) L f!E(t)dt- __e _e cO Therefore the distribution of residence-times given by this model Therefore, the distribution of residence-times given by this model does not correspond to that of an actual system. Normalizing equation (32) i.e., dividing throughout by the residence-time 9 or L/u we obtain: *jSe d atio)nd E(ta n/Apeni I, pp 17DL (33) * See derivation details in Appendix I, pp. 172 and 175

-41A comparison of this result with equation (6-a) clearly shows that the first moment of the time distribution as given by solution "B" will approximate that of an actual system within 1% when: DL.ol DAL 1 O (34) A third solution which we shall refer to as "C" has been reported by Yagi and Miyauchi /64/. The boundary value problem not only takes into account the flux due to diffusion at the entrance but also at the exit from the bed which is now considered to be of finite extent. The physical conditions at the bed outlet are schematically shown in Figure 10. c(L- ) c(L+) EffctiZe Longitudinal Effective Longitudinal Diffusivity DL Diffusivity D X _ O0 x A L Figure 10 Conditions At Outlet Of Packed Bed

-42At the boundary between the aftersection and the packed bed we have to the left: XC _ EC(L-)_ XSC and to the right: = ucC(L~)-L aL t' the exchange of tracer across the boundary is, therefore: UC(L-)-...... CL+) it (35) If continuity of concentration is assumed in the outlet stream: u CL- = QL-) so that equation (35) reduces to: D -OC( —) = D" aZC.t (36) L Ly s Equation (36) is considered a perfectly general representation of the outlet boundary for a transient problem such as this /59/, Yagi and Miyauchi, however, used at the outlet a more restrictive condition, that is: DL C (L-) =o (37)

-43 - As equation (36) shows, this condition amounts to saying that the flow past the bed exit is perfectly piston-like, (D" = 0), which L is certainly quite arbitrary. Had one assumed that: DL = D"L the model would be reduced to that of a semi-infinite columm and solution "B" would have again resulted. At any rate, the set of boundary conditions which satisfies the outlet condition as given by equation (37), the inlet condition of equations (28), and the step function input, equation (24) is: 1-c) WCo = UC(o,t) - DL(,Lt) 2-C) 3cCLot) =L )2-c) ac.(L-tX0 (38) 3-c) C(,O)A O The solution of the diffusion equation evaluated at x = L is: m uo (U!Fiat,,,,I cot l cO n=l (I + +A is (39) where: ^ and cot7a 2C(A U and the impulse response: i( At - 2_ L (13r Z3 22( ) 40

It is easy to see that the resulting expressions are complex and the fact that the eigenvalues An cannot be expressed in an explicit form makes the study of the behavior of this solution difficult. It is possible that this model may be capable of correlating more accurately the performance of an actual packed bed, since the authors, Yagi and Miyauchi /64/ claim solution "C" reduces to the uL perfect mixing case when the group uL tends to zero, a behavior which DL is not exhibited by solutions "A" and "B". However, to state this with certainty, one should check the moments of its space and time distributions. This proves to be a tedious task and again underlines the fact that equations (39) and (40) are devoid of practical significance in the correlation of experimental data because of their complexity. The boundary conditions, step and impulse response for solutions "A", "B", and "C" are summarized in Table II. The zero, first moment and variance of the respective distributions are also listed for solutions "A" and "B". Table II includes the Normal Distribution which also satisfies equation (21). The Normal Distribution, ordinary designation in statistics and probability for the Gaussian error curve /24/, is directly obtained from the random walk theory (see Chandrasekhar /4/, pp. 3) using a suitable bias on each random step to account for the superimposed longitudinal velocity and letting the number of steps approach infinity. It can be shown that both solutions "A" and "B" approach the Normal Distribution* as the dimensionless group DL becomes uL small (DL <.01). uL On this basis, one may conclude that solutions "A", "B" and the normal distribution are all usable correlation models when the value * See Appendix I, pp. 177

TABLE II SOLUTIONS OF DIFFUSION EQUATION Soluti on "A" "B" "C" Normal Distribution - l1-a) c(o,t) = cO l-b) uc0= uc(O,t)-DL c (O,t) l-c) uc0 = uc(0,t) - DL c (O,t) Asymptotic Approximation of x ax Solution "A" and "B" as | 2S-a) lin c(x,t) = 0 2-b) lim c(x,t) 0 2-c) ac (L,t) = o DL O V0o X- x.-. x. 0 o 3-a) c(x,O) = O 3-b) c(x,O) = 0 3-c) c(x,O) = 0 Also Arises from Random Walk 0 0 t -(L-ut) co c/c0 = 1erfcL - ut'\ c/c0 =f ue 4Dlt dt + c/cO0= U o(U silin +,n cos on) 2 0? \ s' — -) - (L jt) 2o 20 \|DI J tO n=l (u2 + 2U + 25)(1U2+ n2) t Dt o Di t ut [~J-(US+~~ut] c/c0f4D n+ eerfu + f u2eDL erfe + u dt e L \ 2U e L PIPi —t 7 +,ID I]n I U uL cots: 1(1-U oA R 2DT 2 U 1S 3 < 20 g=L+ U+2+ut )-L (L-ut)2 /- d c/cSo= 4DLtS ue + d c/cO = 2 u Uen(U sin d + n-, cos Pn) 2 dt a d jdt tEt3L di t dt n=l L( + 2 U+ n2) -(L-ut) 055~an 4Dt out u= d/c0 u e dt s ed 0 +_ _ ut. Varianee 21 2DL + 3D | 2DL + 8D u +For derivation details see f.(t)dt 1 0 f tE(t)dt L/u L/u + DL/u2 L/u + 2DL 0 u R. Aris and N. R. Amundson R. V. Churchill "Mod. Op. Math. fcr S. Yag and T. Myauchs, Chem. Eng. (Japan) 17, 0. Levenspiel and W. K. Smith AIChE Journal 3, 280(1957). Eng." 38251953). Chem. Eng. Sci. 6, 227(1957) For derivation details see Appendix I,

-46D D of the dimensionless group DL is small. However, when L is greater UE uL than.01, an appreciable disagreement is to be expected as exemplified by the dimensionless distributions of residence-times shown in Figure 11. Moreover, we note from the variance for solutions "A", "B" and the normal distribution that as DL tends to infinity (perfect mixing) the limit for the variance is zero rather than one as would be expected for an actual system under conditions of complete mixing. A Packed Bed As A Series of Perfectly Mixed Stages It is useful to think of a packed bed as a series of stages or cells. We imagine that in each stage perfect mixing exists. As defined in page 17 this requires that the concentration in each stage be uniform and equal to the concentration in the effluent. Consequently, the differential equation which characterizes each cell is: am% _0 it (c;,-c;) (41) where ci 1 and ci are the concentrations at the inlet and outlet respectively, vi is the volumetric velocity of the fluid through the ith cell and Vi' is its volume. For n such stages in series the effluent concentration from the nth stage is obtained from the simultaneous solution of n equations analogous to equation (41)*. More specifically, if all stages have the same volume V', the response * See details of these derivations in Appendix II, pp. 187

3.2 SOLUTION "A" 2.8- -- SOLUTION "B" 2.4 / 2.0.8 uLDL 10l.8 uL.4 0 ~2.4 8 1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 t/i Figure 11. Time Distribution Curves For Solutions "A" and "B" Of The Diffusion Equation

to a step input in concentration is: cO = oI (V t(42) e-, CZ-') W The corresponding response to an impulse may be obtained by differentiating equation (42) with respect to t: == (43) Equation (43) is the distribution of residence-times for the fluid flowing through n stages in series. We shall now show that, under certain conditions this distribution is consistent with the material balance requirements and mixing behavior of an actual system. Taking the integral from zero to infinity of equation (43) we find that:* F.~t)Jt- Mt, t \Pdt=l Thus, the zero moment of the time distribution fulfills the material balance of an impulse signal response for an actual system (see eq. (3) pp. 13 )provided that the number of stages n be greater than zero. We note further that the integral in equation (44) is none other than the Laplace transform of the function tn-1 / (n-l)!, thus the number n * See derivation details in Appendix II, pp. 189

-49need not be restricted to a whole number greater than zero since (n - 1)t can be expressed as the Gamma function of n, any real number greater than zero /63/. Since it appears meaningless to speak of less than one mixing stage we may arbitrarily restrict the number n to any real number from one to infinity. In an analogous manner we find that the first moment of the function E(t) of equation (43) is * 0C+t IT I ot t (e (45) From the above equality we establish, therefore, the obvious fact that if the cells are all equal the overall residence-time of the system is given by: 9-s~~~ "~Be~ ~(46) If we visualize a packed bed as made up of m mixing cells, the overall residence-time of the bed is given by equation (45) only m if the cells are disposed in n parallel banks, each bank having n cells in series such that: (47) * See derivation details in Appendix II, pp. 190

-50In other words, in order to make use of equations (45) and (46), we must assume that the flow of fluid through the bed is uniformly distributed. In this case the height of a mixed stage is: HV _J (48) where A is the cross sectional area of the empty column. Using the same procedure, we obtain the variance of the dimensionless distribution function QE(t) about the mean residence-time V' * n -, v Ali _____ID'IV It (49) Consequently, the dispersion about the mean residence-time is equal to the reciprocal of the number of mixing stages and it is found to vary from 0 for the case of piston flow where n = Xc to the value of one for the case of perfect mixing when n = 1. The variance, therefore, gives a convenient method for the evaluation of the number of mixing stages in series that can be associated with a packed bed from the response of the bed to a step function or an impulse signal input. Finally, to verify the correctness of the assumption that the flow is distributed uniformly across the bed, we can calculate the skewness of the dimensionless residence-times distribution curve OE(t) * See derivation details in Appendix II, pp. 190

-51vs. t/O. This is*:' t t (50) [s0v~n4)! CAt] | n 0) V Thus, if n t is the true residence-time, the skewness will be twice the square of the variance. In the affirmative case, the proposed model of m mixing cells with _ banks of cells in parallel each n bank having n cells, is adequate in simulating the response of the packed. bed. We note further that the transient response expressions we have discussed would only give the number of stages in series and the residence-time for each stage or the height of each stage but no information can be obtained on the actual volume of each stage unless the total number of cells m is determined independently. In the event that the variance and skewness are in disagreement one might postulate that the flow in the packed bed is not uniformly distributed, i.e., a certain degree of channeling occurs. The difference between the skewness and twice the square of the variance would be a measure of such channeling. Theoretically, one could set up alternate models for the packed bed in which the flow rate v' through parallel banks of cells is different in each bank. The additional relationships necessary may be furnished by calculating higher moments of the distribution of residence-times curves. * See derivation details in Appendix II, pp. 191

-52By referring to Appendix II, pp. 184, it is apparent that one of the outstanding advantages of perfectly mixed stages as building blocks of packed bed models is the ease by which the moments of the resulting response functions can be predicted by the use of the Laplace transform because of the presence of an exponential term. The moments of the experimental response curves could also be readily obtained if the response of a continuous analyzer at the outlet of the system were fed to a number of integrators, each giving higher moments of the distribution of residence-times. So far in this discussion we have concerned ourselves with series of perfectly mixed stages all having the same volume such that the overall contact time of the column is given by the product of n with the contact time of a single stage. If the stages are unequal, one can show* that the first moment of the resulting distribution of residence-times function gives the relation: j9; C 8 (51) and that the variance of the distribution yields: (52) In this case the variance a2 is not equal to the reciprocal of the total number of stages in series except when the volume of each stage * See derivation details in Appendix II, pp. 196

-53 - is not widely different. In fact, by letting: equation (52) can be rewritten in the form: 1 multiplying both sides by n and taking the square root: 9 = | 3 ( ) (i53) It follows from equation (53) that n is not equal tothe total number of stages present in the system but rather it is equal to a number such that by dividing the total contact time by n the mean square average of the contact time for each of the n stages is obtained. If on the other hand, the volume of each stage in series is not widely different: we have: so that: substituting these last two equalities in (53) we find that: Comparison with equation (46) shows that this is the ordinary case of an equal number of stages.

-54When a column is made up of packed sections having different properties or large empty sections interposed with the packing, we may conclude that: 2 ZZ or in terms of mixed stages: n tS, n,, where n as before refers to the apparent number of stages calculated as the reciprocal of the variance and the subscripts 1 to n refer to different sections making up the total system. But rather the variance contribution for each section to the variance of the residence-times distribution of the whole system is: 2 2 z a 1( )% The same relationship in terms of number of mixing stages is: 4,n, &6 ) n ( E )~ ~ n ( (54-a) Table III gives a summary of the various relationships discussed in the previous pages.

TABLE III RESPONSE OF A NUMBER OF MIXED STAGES IN SERIES AND MOMENTS OF TEE TIME DISTRIBUTION n1 STAGES HAVING VOLUME V; QUANTITY SERIES OF n EYolUAL STAGES IN SERIES WITH n2 STAGES HAVING VOLUME VT -V T II v'(7h t -V~V T ~ -V' Step Input / T n-i e Vt dT t T n ( V4 2 n (n2-i) eV Re s/o0 ne (Vi)1 n1ne.i 2v-X e 2 (T-X)ldrdX sp on2 sC e -vI t Impulse Response VI n tn-i e VI T V (t-T) (Residence-Times V I n (n1IV (n-l)! f inVt 1 T n1-i) e Vj (t - T) e V dT Distribution) (V)(-)0 V Till_____I_ ( -Tn2dl)! cv = E(t) f E(t)dt 1 1 0 00 f tE(t)dt nvI l + n2v I 0 VF IV' VI i 2 C2 1 2 02 1 1 + 1 2 n n, e2 n2602 v 2 n7

-56Comparison Of The Two Models The similarity between the frequency responses of a series of perfect mixers and of a system which combines plug flow and axial diffusion has already been considered by Kramers and Alberda /38/. On the basis of this similarity and by observing that the longitudinal Peclet Number for gas flowing through beds of spheres is about 2, McHenry and Wilhelm j47/ concluded that the height of a mixed stage is approximately equal to the diameter of each sphere. Aris and Amundson /1/I investigated this point further and found both mechanisms to give comparable results at least in the neighborhood of t = 0 and more generally for low values of the effective diffusivity. Ranz /49/ in predicting the radial Peclet number for a bed of spheres considered a series of perfectly mixed stages in the radial direction even though not stating so. His calculations show that for a bed of spheres of infinite extent the ratio of the mean residence-time of the fluid proceeding through a cell interconnected longitudinally with that of a cell interconnected by radial openings is.179. Thus, for generally accepted values of longitudinal and radial Peclet numbers, we have: PeL 2 Pe 11.183 (55) r The agreement between these two ratios may be fortuitous but not devoid of significance since the Peclet number, consistent with the series of mixed stages mechanism, is inversely proportional to the mean residencetime through each cell mixer. Consequently, one can consider a packed

-57bed as a matrix of mixing cells connected some in the radial and some in the longitudinal direction through which the flowing medium undergoes complete mixing. This mechanism is equivalent to assuming a mean value of the concentration and of the mass transfer rate within the cells. Even though this assumption may not be exact the resulting mathematical expressions are greatly simplified. While we have limited ourselves to the treatment of this model in the longitudinal direction, it should be possible to extend it to two and three dimensional flow with the presence of absorption and chemical reaction. Such an approach is exemplified in a recent article by Churchill, Abbrecht and Chu /7/ on a heat transfer problem through porous media. The authors have shown that such a model yields a solution of striking simplicity. From equation (43) we see that the dimensionless residencetimes distribution function for a single stage @'E(t) is:'E) (<''' e- (56) Equation (56) is the familiar Poisson distribution of probability (See Feller /21/ page 110.) It expresses the fraction of those tracer molecules introduced at the inlet at time t over a time interval dt which are in the nth cell at time t or, alternatively it expresses the probability that a tracer molecule introduced at the inlet at time t over a time interval dt may be in the nth cell at time t. The Poisson distribution in probability theory arises in two ways (see Fry /24/ page 214 through 227):

-581. As an approximation of the Binomial law governing the distribution of random numbers or steps as arising for instance in the random walk theory /4/, we note that the Normal distribution is also an approximation of the Binomial law. However, probability theory tells V' us that when V,t of equation (56) is small only the Poisson distribution is a good approximation. On the V' other hand, if'r t is large both the Poisson and the Normal distribution can be used. 2. The second way in which the Poisson distribution arises is as an exact solution of a system of differential equations of the same form as the one in equation (41). Many other physical systems such as the splitting of physical particles and the arriving of telephone calls at an exchange behave in the same manner as the series of mixed stages discussed in the previous paragraph. In this case, however, no information whatever is rev' quired about the number t or V, and no knowledge is Vt required of the number of steps in any interval large or small in contrast with the random walk theory. The only two necessary requirements of the physical processes which obey the Poisson law are that the events described by the basic differential equation (41) be all of the same nature and their occurrence be in no way related to previous or past ones. Thus, the Poisson law possesses a large degree of flexibility in fitting various physical situations.

-59It is well known that the Poisson distribution converges to the Normal distribution at least for a large number of steps (see Chandrasekhar /4/ pp. 81). Equating the two distributions, one may arrive at some relationship between the number of mixed stages and the effective diffusivity coefficient. For a more direct comparison, with the solutions of the diffusion equations mentioned in previous paragraphs, however, we have chosen to look at the moments of the time distribution function which are listed on Table IV. As we have already noted, the zero moment is in all cases equal to one. Since the sum of all probabilities for a system must equal unity, it is no surprise that the two models are in agreement on this point. As far as the first moment is concerned, this should be equal to the residence-time. From Table IV we note that this is so in all cases provided that the dimensionless group L is small enough. The UL reason for the discrepancy presented by solution "B" and the Normal distribution is probably due to the fact that the Normal distribution and, therefore, the diffusion mechanism, does not give an adequate approximation of the probability unless there is an appreciable number of random steps in a given interval of time. Solution "A" gives perhaps fortuitously the proper residence-time because the approximation error is compensated by neglecting the diffusion flux at the entrance of the bed. Passing to a comparison of the variance of the time distribution we see that the relationship sought between the number of stages

TABLE IV COMPARISON BETWEEN SERIES OF MIXED STAGES AND THE DIFFUSION MODELS Moments n Mixed Of The Time Stages Solution Solution Normal Distribution In "A" "B" Distribution Function E(t) Series ft E(t)dt 1 1 1 1 0 DL 2DL f t2 E(t)dt 0 0+ -`L 4+ 0 0 ~~~~~~~~~~~~~~u 1 2DL 2DL 3D2 2DL 8D2 uL L2u2

-61and the effective longitudinal diffusion coefficient is: do - (57) 2 D. uL This relationship is valid within one per cent for D- greater than ten as far as the variances for the distributions of solution "B" and the Normal approximation are concerned. Graphical comparison between the diffusional model and the mixing stage model is furnished by Figure 12 where distribution curves for a series of mixed stages and the solution "B" of the diffusion equation are shown. As one would expect, the agreement is not very good for a small number of mixed stages but it rapidly improves as the number of mixed stages is increased. One may also note that for values of t/@ larger than one the agreement is generally better even for small values of uL/DL as the theory of probability predicts (/21/,pp. 144 ) From this point of view, the series of mixed stage mechanism, i.e., the Poisson distribution should furnish, in any case, a better approximation for values of t/G less than one. In absence of data on mixing for shallow beds or in which the number of mixing stages is small, we are really not justified in any statement concerning which model gives a more precise representation of the mixing process over the whole range of conditions from plug flow to complete mixing. While the diffusion model considers the overall effects produced by a very fine mixing scale, the series of mixed stages model

3.2 SERIES OF MIXERS 2.8 SOLUTION "A" 2.4 2.0 1.6- uL w DL, t// 1.2, 0n =50.8 uL 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 24 tie Figure 12. Time Distribution Curves For Solution "B" And A Series Of Mixers

-63amounts to assuming average conditions over discrete volume fractions of the bed. In addition, the former model gives rise to fairly cumbersome equations to describe the system. The latter instead, yields much simpler relationships which may be easily modified to fit possible anomalies in the behavior of the bed.

I I I I I

CHAPTER IV EXPERIMENTAL The prime objective of the experimental work was the collection of data on the gas porosity of an irrigated packed column and on the extent of mixing of the gas phase through the irrigated bed as function of liquid and gas flow rates and column geometry. The transient response method was chosen because, as discussed in the theory, it readily yields such information. The experimental apparatus was designed with the intent of introducing a signal of helium tracer in the inlet gas stream of an absorption column of suitable dimension and recording the composition of the outlet gas stream continuously and with minimum delay. The equipment, therefore, consists of a liquid supply system, a gas and tracer supply system, a packed test column and a gas analyzer. The details on the various pieces of apparatus are not essential to the presentation of the results except where pertinent data and dimensions are needed. The reader may, therefore, omit the section entitled: Equipment which follows and continue on page 76. Equipment Flow System A flow diagram showing liquid, gas and tracer supply systems and the packed test column is given in Figure 13. The liquid supply system handles water flow rates from.1 to 1.1 gal/min and is arranged as to make possible the introduction of a step function or a pulse of dye in the water stream even though the

PRESSURE REDUCER SET AT 50 PI VENT I- ANALYSIS CELL TRACER WATER SUPPLY SOLENOID us OII LOW PRESSU ) IPRE URE - _~~~~~~~~~~~~~~~~~~ L THERM. — ~,,SPRTR/( A DR a WATERAIN T-wRTOR ~~~~FOCOTHOLERM.a'FLOW PRE~SSURE oC~~arru /\ Y D(XXX ~ ~ I I r WATE, PRSSR PRESSUREREDU D1 GAUGES rm ~ - r O 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ w ~ ~ ~ ~ ~ ~ ~~~ DRIN Figur 13 Flw Darm W~~~~~~~~~ W ~~~~~~~~~~~~~~~PUMP w ~~~~~~~~~~~~~THERM ~~~~~~ ~~~~~DRAIN x N04 41 N03 NOI -4 —-cTTO DRL\I ROTAMETERS AIR ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ GS SOLENOID I LEVELING TUBE SUPPLY TEM PRESSURE REDUICER SETT AT 50 PS.1 DRAIN Figure 13. Flow Diagram.

-67experimental work is only limited to the tracer study of the gas stream. City water is softened and purified in a Bruner 60,000 grain capacity water softener and transferred to a 100 gallon storage tank where it is allowed to come to equilibrium with the temperature of the room. The gas system allows air flow rates from.1 to 4 SCF/minute. Compressed air at 50 psi pressure is humidified in a brass column 4 inches in diameter and 4' 8" in height. The column is heated by a 2 Kwatt wire coil wound on the outer surface of the column. A Fenwall temperature controller maintains the temperature of the water such that the gas leaving the humidifier has a dry bulb temperature approximately equal to the temperature of the water flowing down the packed column. The air rate is measured through a bank of three rotameters in parallel. The air rate is adjusted and maintained constant by a pancake low pressure regulator ahead of the rotameters. Pure helium or a mixture of helium in N2 (15% He - 85% N2) are used as tracers and are available in pressurized tanks. The tracer flow after a primary reduction in pressure is maintained at a constant rate by means of a second pancake low pressure regulator. A three-way solenoid valve immediately before the inlet to the test column permits replacing the air flow to the column with tracer or the introduction of a pulse of tracer as slugs of short duration in the air stream. An automatic timer and switching assembly controls the operation of the solenoid valve and supplies an electric impulse marker to the recorder. Details of the electric wiring for the timer and manual switches are shown in Figure 14.

-68EAGLE SIGNAL MICROFLEX TIMER WATER SOLENOID VALVE SIGNAL SOLENOID 41B t SWITCH SWITCH A- V.I BATERR GAS A SOLENOID LIN VALVE CHANNEL *2 1/2 V. BATTERY A. C. Figure 14. Timer and Solenoids Wiring.

-69The test column is made out of 4" I.D. flanged Pyrex glass pipe. The mid-section of the column is interchangeable in lengths of 3' and 4'. The bottom section of the column is conical and a 3/4" I.D. liquid outlet is located at the vertex of the cone. An adjustable liquid level device is used to maintain an adequate liquid seal to allow the outgoing liquid to disengage any gas bubbles before going to the drain. A 3/4" I.D. gas inlet is located at right angles with the bottom section and it is surmounted by a hood to protect the gas line from any splashing liquid. A packing support screen is fastened between gaskets and flanges which connect the column's bottom with the midsection. The screen mesh is of the same dimension as the nominal diameter of the packing used. The total volume of the bottom section from the three-way solenoid valve to the beginning of the packing is.0065 cu.ft. and it constituted 1.5 to 2% of the total volume of the column depending on the height of the mid-section. The upper extremity of the column is reduced in diameter down to 2" I.D. for connection with the gas analyzer. The liquid is introduced on a side inlet and is distributed by six jets made of 1-1/2 mm. capillary tubing evenly spaced on a 3" radius. The volume of the upper section is.0437 cu.ft. The mid and upper column sections are completely filled with packing up to the gas analysis cell, thus that portion of packing which is in the upper section remains dry during the gasliquid runs because it is above the liquid distributor. The column is packed with Raschig rings of 1/4", 3/8" and 1/2" nominal diameter. The rings, made of unglazed porcelain, were supplied through the courtesy of the United States Stoneware Co.

-70Continuous Gas Analyzer A rapid monitoring system capable of producing a continuous and accurate recording of the tracer concentration in the gas stream is the most important component of the experimental equipment. For this purpose an alpha particle ionization cell was constructed. This device is frequently used in vacuum practice as a vacuum gauge and leak detector. Deisler, McHenry and Wilhelm /16/ first used it as an analyzer for binary and ternary gas mixtures at pressures above atmospheric. The ionization cell utilizes the alpha particles emitted by a radium source to ionize the gas passing through it. The number of ions produced per unit time depends on the nature of the gas which is being analyzed. The ions can be collected between two electrodes immersed in the flowing stream to produce a current. For a binary mixture of gases at constant temperature and pressure, the rate of production of ions by the alpha particles is function of the gas composition only. If the voltage impressed upon the electrodes is maintained constant, then the current produced is also a function of gas composition only, and measurement and recording of this current makes possible the measurement and recording of the gas composition. Further details on the theory and design considerations for this instrument may be found in the paper on this subject by Deisler, McHenry and Wilhelm /16/. The electrodes of the cell are two rectangular stainless steel plates 1-1/4" x 1-11/16", 1/32" thick. The one used as the alpha particle emitter has on one side an active area of 1-1/8" x 1-1/8"

-71covered with 100 micrograms of Ra226 rolled in gold foil. The other plate is the ion collector. The plates are spaced at a distance of one inch within a 2" O.D. Teflon mount specially streamlined to minimize extraneous mixing of the gas entering the cell. The assembly is housed in a specially fabricated glass flanged tube 3" long which is inserted between two 2" Pyrex brand pipe flanges forming the connection between the top of the test column and the exhaust pipe as shown in Figure 15. Electric leads are clamped on the electrodes by means of copper clips and sealed through the glass housing by means of rubber vial caps. The glass housing and high impedance lead are shielded with brass foil. The accessory equipment to the analyzer consists of a high voltage D.C. Supply, a 187 megohm resistor circuit between the collecting plate and the ground, a Keithley Model 210 Electrometer which matches the high impedance of the cell resistor combination and a two channel Brush amplifier and direct writing oscillograph. Details of the analyzer and recorder cell circuit are shown in Figure 16. The rectangular cell geometry was chosen to give most favorable current stability and noise characteristics. In fact, when using pure helium as the tracer, the output is exceedingly stable and completely free of stray noise. The output current of the ionization chamber for air and helium as functions of the applied voltage at the electrodes is shown in Figure 17. It can be seen here that a potential of about 1000 volts across the electrodes is necessary to collect all the ions formed by the alpha particles. Thus, this voltage was selected as standard throughout the experimental work. The variable high voltage D.C. power supply

-72 - U) to co:

-73 - HIGH VOLTAGE VARIABLE POWER a- EMITTER CELL SUPPLY -.- a| I V-l 187 MEG oo LO. II o MEG I o o 6o G0 + z m C)0 C[A II Z a ~| z I ll LL D~OCHANNEL# I Fiur 1.Asemlyo Gs nayeran AcNsris

0 100 - ZONE A ZONE B a: ZONE A-RECOMBINATION REGION n;'"0 B-APPROXIMATELY ALL IONS w | PRODUCED REACH ELECTRODES N C-ADDITIONAL IONS FORMED BY <60 I EXISTING IONS 0L I SELECTED POTENTIAL 0 40 ZONE ZONE B ZONE C Q. 0 a: S 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 POTENTIAL ACROSS ANALYZER ELECTRODES; VOLTS Figure 17. Current Output of Gas Analyzer with Impressed Potential.

-75 - was built using a high voltage transformer and athermionic rectifier. The power supply circuit is shown in Figure 18. The output current of the electrodes at 1000 volts potential -8 varies linearly with helium concentration from a value of 1.9 x 10 amps for pure helium to 9.1 x 10-8 for air. The output current vs. concentration shown in Figure 19 was obtained by passing through the cell known gas mixtures. The overall precision of the measurement is better than one per cent. The low output current from the electrode is boosted by a 187 megohm resistor across the ion collector plate and the ground to a maximum potential drop of 17 volts. To allow the use of 15% helium — 85% N2 mixture as a tracer a current blocking circuit is inserted in series with the resistor. This consists of a 15 volt mercury battery connected with inverted polarity so that this constant voltage is subtracted from the output voltage of the resistor. Thus, when the battery is inserted in the high resistance circuit, the maximum output voltage corresponding to 100% air is two volts. The high sensitivity circuit causes a certain amount of noise to be detected in the cell output, thus a 250 C4lfarad capacitor was added across the high sensitivity circuit to damp some of it out (see circuit details in Figure 16, page 71 The cell-resistor combination results in an impedance of the order of 108 ohms so that a matching circuit is provided to measure the output voltage. This purpose is served by the Keithley Model 210 Electrometer which is capable of measuring input voltages having an impedance as high as 1014 ohms. The electrometer features five scales from 0 to.8, 2, 8, 20 and 80 volts. The 0 to 2 volt scale is used with the high sensitivity circuit while the 0 to 20 volt scale is

/ — FILAMENT SWITCH 1.9 MEGR 3/8 AMP. | FFUSE 03 MEG~ ~~~~~~ o< > wE~s rl2.5 V. T | | I Ll 0 F~~IL | 35/8 AMP. FUSE 879 PRIMARY SWITCH I I1OKQ I1O0K,: f 0 0 i~~= o CT )c; ~~~~~~~~~~N~~~~ll~~~~ 2~ IpIII~, ~f C: Figure C.T. Spl 2~~~ w 0 0" POWERSTAT 6.3 V. FIL. UNUSED HIGH VOLT. D.C. SUPPLY Figure 18. High Voltage D.C. Supply.

I00 CD 0 o 80 Q_ w 60 N 2 00 H D >00 <[ z 0 W III I I Ir r 0 tO0 0 10 20 30 40 50 60 70 80 90 100 o HELIUM CONCENTRATION VOL % Figure 19. Current Output of Gas Analyzer with Helium Concentration.

-78used for the normal sensitivity. The electrometer contains an output amplifier which develops 10 volts in a balanced connection for full scale reading of the meter and a 1.0 milliampere current to drive the recorder. The output of the electrometer is fed to a Brush dual channel direct-writing oscillograph model BL 202 through a Brush model BL530 amplifier by means of a balanced connection to the electrometer. Procedure Preliminary Tests A number of test runs were performed to give preliminary estimates of the limitations of the equipment and to establish a consistent and reproducible experimental procedure. The manner of introduction of the tracer at the column inlet was soon limited to step changes in concentration since it was found impossible to inject an appreciable amount of tracer in an interval of time short compared with the contact time of the gas flowing through the column in order to approximate an impulse signal. The contact time through the column was varied from 2 to 40 seconds, thus a time of injection as short as.02 to.04 seconds was required. In the runs at low gas velocity some impulse responses were recorded and the first moments and variance of the resulting distributions were found to be within 1% of those obtained by a step change in tracer concentration. Pure helium was first used as a tracer in preliminary work. It was found, however, that when the helium displaced the air through the column, the recorder traces (which we shall also call response curves) were severely distorted, especially at low flow rates. This phenomenon

-79is clearly detectable in the reproduction of the Brush recording for Run D-23 shown on Figure 23, pp. 87 where the gas flow rate was.482 cu.ft./second. No distortion is visible when air displaces helium however. As the gas flow rate was increased, the distortion became less pronounced. These observations led to the belief that convective currents were produced by the eightfold difference in density between helium and air and disturbed the smooth displacement of the air by the helium. Since the flow was upward, such currents would not be present when air displaced helium; in addition when the average gas velocity was presumably well above the convective velocity, the disturbing effect was much less pronounced. On further experiments, carbon dioxide was used as tracer and for this gas, the distortion appeared in the response curves recorded when air displaced the carbon dioxide. Since this gas is about 1.5 times heavier than air, the convective currents set up by the difference in density were, indeed, responsible for the observed phenomenon. It was, therefore, decided to carry out the experimental work using as tracer a mixture of 15 volumes of helium and 85 volumes of nitrogen. This mixture was prepared in standard gas cylinders at a total pressure of 1000 psi. The difference in density with air is about 16% and the resulting response curves for the tracer displacing the air and for the air displacing the tracer were in reasonable agreement as discussed in Chapter V. The high sensitivity circuit for the analyzer described previously (see pp. 73 ) was added at this time to permit both the use of 15% helium or pure helium as tracer whenever desired. When helium

-0 - was used, the output from the analyzer through the normal sensitivity circuit was practically free of noise as Figure 23, pp. 87 shows. With 15o helium used as a tracer, the high sensitivity circuit was required to allow full deflection of the meter whenever 100% tracer was present in the analysis cell. Due to the sevenfold magnification, the cell output was more sensitive to slight pressure fluctuations and/or stray currents so that a certain amount of random noise was registered at the recorder. To filter out some of this noise, a 250 x 10-1244 farad capacitor was added across the high sensitivity circuit, a slight amount of random noise was, however, still detectable as exemplified by a typical run record reproduced in Figure 21, pp. 85 The increased capacitance of the high sensitivity circuit resulted in an increase in the time constant* of the analyzer recorder system from.005 to.05 seconds. To insure that this increase of the time constant would not disturb the respors e curves, the data was checked for consistency as discussed on page 110, Chapter V. The analyzer cell, especially at higher sensitivity was found to be somewhat susceptible to pressure fluctuations, thus changes in flow rate would alter the analyzer calibration. This factor was eliminated by calibrating the analyzer-recorder system each time a different flow rate was used. Once the system was properly warmed up, no appreciable drift was noticed for the entire duration of any one run. The pressure sensitivity of the analyzer was found useful as a check to insure that the air and tracer would produce the same flow conditions through the system since for unequal flow conditions at the moment of * The time constant referred to here is the time required for the pen to travel.632% of the total width of the paper.

switching the flow from air to tracer or vice versa, the recording would register slight deviations from normal setting due to a pressure change at the analyzer cell. During the experiments performed with irrigated packing or humidified gas, some moisture would deposit on the electrodes of the analysis cell and the output of the cell would gradually be shorted out. This problem was completely corrected by maintaining the analyzer at a temperature of approximately 100~F with a heating tape wound around the glass housing and shield. To investigate the effect of the degree of humidification of the air and tracer upon the response curves, test runs were made by alternately routing the air stream or the gas stream through the humidifier with or without liquid descending over the packing. No detectable effect of the degree of humidification of either stream on the response curves was noticed as long as the temperatures of the flowing streams were kept approximately the same. It was noticed that the temperature of the tracer was usually within a degree of room temperature. The liquid was kept in the storage tank and allowed to come to equilibrium with the temperature of the room. Consequently, only the air temperature needed occasional adjustment and this was done by conditioning it through the humidifier. Standard Experimental Technique Raschig rings of the desired size were introduced into the column by partially filling the column with water, then slowly pouring the rings from the top. The water prevented breakage and provided slow and even settling of the rings. The column was then reassembled.

-82Column and packing were dried out by allowing dry air to flow through the column overnight. The porosity of the dry bed was measured by back filling the column with water to a level two inches above the packing support screen. 1600 cc. of water were then added and the increment of the water level in the column measured by means of a cathetometer. This procedure was repeated by successively adding 1600cc. of water until the top of the column was reached. The porosity was calculated for each increment knowing the net amount of water added to the column and the resulting increase of the liquid level. The porosity of the bed was taken as the average of the total number of determinations. The deviation from the average was for all packing sizes within 0.5% indicating reasonable uniformity of the porosity throughout. Column and packing were again dried prior to a series of runs. A typical run was started by turning on the high voltage generator, amplifier, recorder, and the heaters for the analysis cell and humidifier. Warmup of the analyzer-recorder system required about two hours. If a series of dry packing runs were executed, the required gas and tracer flow rates were preset by adjusting the low pressure regulators and measuring the flow rates through the bank of rotameters. The flow rates of the two streams -were adjusted so as to give approximately the same pressure drop through the column. The temperatures of the flowing streams were adjusted within one or two degrees Fahrenheit. The liquid seal level at the bottom of the column was set at a reference mark by means of the leveling tube. The electrometer was set to zero and the sensitivity and proper voltage scale selected. The recorder was calibrated by allowing air to flow through the column and setting the

-83recorder pen to coincide with the zero line of the chart. Tracer was then allowed to flow through the column and the full deflection of the pen was adjusted by proper setting of the Brush amplifier controls. The chart speed was selected and air and tracer flow rates were again checked. A run was started by turning on the switch for the three-way solenoid valve which interrupts the air flow and replaces it with tracer. When the response curve for the tracer displacing the air was completed at the recorder, the flow was switched back to air and a response curve for the air displacing the tracer was recorded. The air and tracer flow rates, all the temperatures and the pressures were then rechecked and recorded. For an irrigated packing run, the packing was first thoroughly wetted by turning the water to 6000 lb/hr.sq.ft. for about one hour. Then a series of runs was performed by adjusting the liquid flow to the desired rate and following the same procedure described above for the gas. For each liquid flow rate, the column was allowed to reach equilibrium for about a half-hour, then response curves were recorded at 15minute intervals and compared by superposition to note if the curves were exactly congruent. In the negative case the gas and liquid flow rates were checked and the procedure repeated until two successive sets of curves were in agreement. When flooding was reached, the flow conditions through the column were too erratic to obtain consistent response curves so that no data was obtained on or about this range of conditions. Only for Run W-19 was a reasonable agreement for successive response curves obtained even though the column was flooded. Whenever a second person was available, the dynamic liquid holdup was measured at the end of an irrigated packing run by interrupting simultaneously the gas and liquid

84flow and collecting the total amount of liquid drained from the column. When a series of runs was completed and a changeover of packing was necessary, the porosity was again checked. Then the packing was removed and weighed in such a manner as to determine the total number of particles.

-85 - Experi,-ental Data Dry Packing The experimental effort was first directed to the study of the gas phase mixing through the dry bed in order to verify the suitability of the equipment and method of analysis against existing data obtained by the frequency response /47/. Since it was possible to measure the void volume of the dry packing independently, the precision of the gas holdup information furnished by the transient response could also be determined. A total of 28 experiments were performed with gas flowing over dry Raschig rings. Figures 20 through 24 are typical examples of data collected during these experiments. The variables and the ranges covered are: Variables Ranges 1. Column Length 3, 4 Ft. 2. Tracer Pure helium and 15% He-85% N2 3. Average gas interstitial flow rate 0.06 - 1.0 ft./sec. 4. Nominal Particle diameter 1/4, 3/8, 1/2 in. 5. Packing Arrangement Random and Oriented Other variables which were held constant are: 6. Column Diameter 0.334 Ft. 7. Packing Ty~je Unglazed Porcelain Raschig Rings 8. Method of Signal Introduction Step Function

-86Figure 20. Dry Packing, Run No. D-8 Date: 7/15/57 Room Temp-rature: 78.8 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 3 Reading 70 Height of Packed Bed 3.42 ft. Temp. of Flowing Steaam 79.4~F Porosity of Dry Bed 0.633 Pressure at Rotameter 15.7 psia Air: Rotameter No. 3 Reading 70 Heater Temperature 79.2 ~F Dry Bulb Temperature 78.3 F Wet Bulb Temperature 78.5 ~F Pressure at Rotameter 15.7 psia RECORDER TRACES: T__ONI —.S C.. / /......- 9'. E L -- EC.ONICS CO PRINTED N U.S... CATN.BL 909 BRUSH E_-,:RONICS COMPANY PRINTED NN U.S.. _BL. B90 - 9: EL

-87Figure 21. Dry Packing, Run No. D-12 Date: 8/26/57 Room Temperature: 73.4 ~F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 1 Reading 9.6 Height of Packed Bed 4.45 ft. Temp. of Flowing Stream 73.2 F Porosity of Dry Bed ~685 Pressure at Rotameter 14.7 psia Air: Rotameter No. 1 Readi-ng 10 Heater Temperature 73. 0~F Dry Bulb Temperature 72. 8 ~F Wet Bulb Temperature 72.3 0F Pressure at Rotameter 14.7 psia RECORDER TRACES: = = =; I " 7"'7i'7- t i? r: -:? i-::: - i /: t:' j t t tI t r -e 7'-~-~- i i i i:- I i i i- c _ I+ v t \ i-V _1 ou~-~: s ~ ~~ ~ ~~Ct4~T r~~. j: ~Tt

-88 - Figure 22. Dry Packing, Run No. D-16 Date: 8/22/57 Room Temperature: 77.4 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 3 Reading 68 Height of Packed Bed 3.42 ft. Temp. of Flowing Stream 77.2~F Porosity of Dry Bed.666 Pressure at Rotameter 15.8 psia Air: Rotameter No. 3 Reading 70 Heater Temperature 77.2 ~F Dry Bulb Temperature 76.5 ~F Wet Bulb Temperature 76.0~F Pressure at Rotameter 15.9 psia RECORDER TRACES::~~~~~-l l l \ l —~I~~ ~ ~ ~ —:1 W g X W t t t - I A. _ CHART NO. B!_ 909 iI-Ut"H.! t!l. I-:(':T'()N'NC COMPAN' PRNTO IN U.S A........ LL LL I L __L~~~~~~~~~~~~~~IL

Figure 23. Dry Packing, Run No. D-23 Date: 9/7/57 Room Temperature: 70.6 F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 3/8 in. Rotameter No. 1 Reading 18.0 Height of Packed Bed 4.45 ft. Temp. of Flowing Stream 69.8~F Porosity of Dry Bed.660 Pressure at Rotameter 14.7 psia Air: Rotameter No. 1 Reading 15.3 Heater Temperature 71.0 ~F Dry Bulb Temperature 69.5 ~F Wet Bulb Temperature 69.0 ~F Pressure at Rotameter 14.7 psia RECORDER TRACES: _',:~~~~~ ~~ _: t~ — _t-=- i-._i_4-.— <;=< _ i -t R $ - r FX fL W' = W S; i= i'A A IL —-i _ -777T -

-90Figure 24. Dry Packing, Run No. D-27 Date: 9/17/57 Room Temperature: 71.70F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 1/2 in. Rotameter No. 1 Reading 15 Height of Packed Bed 4.45 ft. Temp. of Flowing Stream 71.50F Porosity of Dry Bed.703 Pressure at Rotameter 14.7 psia Air: Rotameter No. 1 Reading 15 Heater Temperature 72.0 ~F Dry Bulb Temperature 69.7 ~F Wet Bulb Temperature 69.2 ~F Pressure at Rotameter 14.7 psia RECORDER TRACES:,-://:~,/:f~ _ F/!:/I/ / f////_ -/, -I- t - 4 [ i: j' - - fl= t7t:; tt t X t WW-t \t - if t.: i t -t~t4L t: F fi:fff; t | I I t f | g |E'; Jf t I I I, t I, I I | f r | f f ff t i j I

-91 - Irrigated Packing Having determined the effect of column geometry, manner of signal introduction and tracer composition by means of experiments on the dry packing, the porosity of the wet packing and the extent of mixing taking place in the gas stream were measured in a series of 42 runs employing a step input of tracer having a composition of 15% He and 85~% N2. Figures 25 through 33 are typical examples of the data collected. The variables considered and the ranges investigated are: Variables Range 1. Gas rate lb/hr ft2 19.2 - 198 2. Liquid rate lb/hr ft2 0 - 6200 3. Particle diameter 1/4, 3/8, 1/2 in. Beds of 1/4" Raschig rings, three and four feet deep, were used to ascertain the effect of a section of -acking which remained dry at the top of the column. The remaining variables which were ke;t constant are: 4. Column Diameter.334 ft. 5. Packing Type Unglazed Porcelain Raschig Rings 6. Surface Tension of the Liquid' 70 dynes/cm.

-92Figure 25. Irrigated Packing, Run No. W-l Date: 7/25/57 Room Temperature: 78.80F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 3 Reading 21.5 Height of Irrigated Temp. of Flowing Stream 79.20F Section 3.0 ft. Pressure at Rotameter 14.7 psia Porosity of Dry Bed.693 Water: Air: Rotameter No. 2 Reading.30 Rotameter No. 1 Reading 15 Temperature In 78.90F Heater Temperature 79.5 F Temperature Out 78.50F Dry Bulb Temperature 79.20F Wet Bulb Temperature 78.7 OF Pressure at Rotameter 14.7 psia RECORDER TRACES: WONICS CO]MIPANY P~R!N N u s CHART NO. BL 90' BRS4L-'RNC OMAY~NhOI:::SH ELECTRONICS COMPAN- -V A-:: -:D -— = - t.: ~USH~~~~~~~~~~~- J.TOI$~~~ ~.'o,~u s ^ H

-93 - Figure 26. Irrigated Packing, Run No. W-5 Date: 7/29/57 Room Temperature: 85.20F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 1 Reading 13.7 Height of Irrigated Temp. of Flowing Stream 84.2~F Section 3.0 ft. Pressure at Rotameter 14.7 psia Porosity of Dry Bed.693 Water: Air: Rotameter No. 2 Reading.60 Rotameter No. 1 Reading 15 Temperature In 84.90F Heater Temperature 86.O ~F Temperature Out 84.7 ~F Dry Bulb Temperature 85.3 ~F Wet Bulb Temperature 84.7 0F Pressure at Rotameter 14.7 psia RECORDER TRACES: CHART NO. BL 909 BRUSH ELECTRONICS COMPANY PRINTED IN U.S. CHART NO. BL 909,, BRUSH ELECTRONICS COMPANY PRINTED IN U.S.A. Dot~ ~~~~~~~~~~~~F 4 — ='fF{H z +Ih {i{8a+ _

-94 - Figure 27. Irrigated Packing, Run No. W-8 Date: 7/19/57 Room Temperature: 87.8 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 3 Reading 29 Height of Irrigated Temp. of Flowing Stream 38.0~F Section 3.0 ft. Pressure at Rotameter 14i 5 pZia Porosity of Dry Bed.693 Water: Air: Rotameter No. 2 Reading.40 Rotameter No. 3 Reading 30 Temperature In 88.50F Heater Temperature 8. 0CF Temperature Out 88.2 F Dry Bulb Temperature F7. F Wet Bulb Temperature $6. 9 ~F Pressure at Rotameter 14.9 psia RECORDER TRACES: ECTRONICS COMPAN Y PRNTED lN U S C CHART NO. B L 909 BRUSH ELECTRONICS COMPANY 0;1 - V CS COMPANY pR,IrTE IN u s A C HART N O. BL 909 BRUSH ELECTRONICS COMPANY PD -: - 4 -:-4 - _; _ I - 1

-95 - Figure 28. Irrigated Packing, Run No. W-ll Date: 7/30/57 Room Temperature: 86.5 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 3 Reading 28 Height of Irrigated Temp. of Flowing Stream 85.6 Section 3.0 ft. Pressure at Rotameter 14.9 psia Porosity of Dry Bed.693 Water: Air: Rotameter No. 2 Reading.30 Rotameter No. 3 Reading 30 Temperature In 86.2 ~F Heater Temperature 87.8 ~F Temperature Out 85.8 F Dry Bulb Temperature 86.7 OF Volume Drained From Wet Bulb Temperature 86.3 OF Column.0118 cu.ft. Pressure at Rotameter 14.9 psia RECORDER TRACES: ONICS COMPANY *PR, U..^. CHART NO. BL 909 RSELECTRONICS COMPANY.

Figure 29. Irrigated Packing, Run No. W-17 Date: 8/3/57 Room Temperature: 81.2 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 3 Reading 28 Height of Irrigated Section Temp. of Flowing Stream 81.9~F Section 3.0 ft. Pressure at Rotameter 14.9 psia Porosity of Dry Bed.693 Water: Air: Rotameter No. 2 Reading.90 Rotameter No. 3 Reading 30 Temperature In 81.70F Heater Temperature 82.O ~F Temperature Out 81.40F Dry Bulb Temperature 81.8 ~F Volume Drained From Wet Bulb Temperature 81.5 ~F Column.0385 cu. ft. Pressure at Rotameter 14.9 psia RECORDER TRACES: I i,TRONICS COMPANY PR.TE. U A CHART NO. BL 909 BRUSH ELECTRONICS COMPANY i'j-~~~~~~~~~ _i tt:: t i, i-4: t: -4 ~ANYn~~~~ PRINTED~SA C~~HART -NO BL 909'-BRUSH ELECTRONICS COMPANY "RINTED IN U S - ---- +-::f; - IA- I< -1 -

-97Figure 30. Irrigated Packing, Run No. W-26 Date: 8/31/57 Room Temperature: 76.2~F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 1/4 in. Rotameter No. 1 Reading 12 Height of Irrigated Temp. of Flowing Stream 76.40F Section 4.0 ft. Pressure at Rotameter 14.7 psia Porosity of Dry Bed.685 Water: Air: Rotameter No. 2 Reading.90 Rotameter No. 1 Reading 15 Temperature In 76.0~F Heater Temperature 76.60F Temperature Out 75.6~F Dry Bulb Temperature 76.2~F Wet Bulb Temperature 75.70F Pressure at Rotameter 14.7 psia RECORDER TRACES: "PINTF0 IIJ U Sf 1,A1

-98Figure 31. Irrigated Packing, Run No. W-30 Date: 10/29/57 Room Temperature: 77.0 ~F Chart Speed: 1 division/second Column Properties: Tracer: Nominal Particle Size 3/8 in. Rotameter No. 1 Reading 13 Height of Irrigated Temp. of Flowing Stream 77.4~F Section 4.o ft. Pressure at Rotameter 14.7 psia Porosity of Dry Bed.658 Water: Air: Rotameter No. 2 Reading 0.0 Rotameter No. 1 Reading 15 Temperature In --- Heater Temperature 78.O~F Temperature Out --- Dry Bulb Temperature 77.6 ~F Wet Bulb Temperature 77.1 ~F Pressure at Rotameter 14.7 psia RECORDER TRACES::.~:' —-:~- -__ —~ - - ~t 1 t- T f iT CHART NO. 8 L'909. I I.St. E. FCTIO()\NIC S COM PANY P',NT~ 1N U.v A ~..... — —:.... f - --- - - — I " ~~~~~~~~~~~~~Ti J_= f W W X X W W W g iWW S H~~~~~~~~~f S m S v7 0~~~~~~~~~~~~~~~~~4 J 4LgfT:]V- Jg~W

-99Figure 32. Irrigated Packing, Run No. W-32 Date: 10/30/57 Room Temperature: 75.6 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 3/8 in. Rotameter No. 3 Reading 64 Height of Irrigated Temp. of Flowing Stream 75.0~F Section 4.0 ft. Pressure at Rotameter 15.8 psia Porosity of Dry Bed.658 Water: Air: Rotameter No. 2 Reading.60 Rotameter No. 3 Reading 70 Temperature In 75.80F Heater Temperature 76.2 ~F Temperature Out 75.3 F Dry Bulb Temperature 75.6 ~F Wet Bulb Temperature 75.2 ~F Pressure at Rotameter 15.9 psia RECORDER TRACES: o k,'~'~.'~ ~'~ "~-~~ ~iil~: l~! I 1!,T-F i i f- I-W ( NI ft t 7 ~~~~-L-T 7-~~~~~~~~~~~~~~~

-100Figure 33. Irrigated Packing, Run No. W-39 Date: 11/4/57 Room Temperature: 77.0 ~F Chart Speed: 5 divisions/second Column Properties: Tracer: Nominal Particle Size 1/2 in. Rotameter No. 3 Reading 66 Height of Irrigated Temp. of Flowing Stream 77.1~F Section 4.0 ft. Pressure at Rotameter 15.8 psia Porosity of Dry Bed.705 Water: Air: Rotameter No. 2 Reading 1.10 Rotameter No. 3 Reading 70 Temperature In 77.4 ~F Heater Temperature 78. 0 F Temperature Out 76.8~F Dry Bulb Temperature 77.7 F Wet Bulb Temperature 77.3 OF Pressure at Rotameter 15.9 psia RECORDER TRACES: I~~~~~~~~ ~~~~~~~~T- ~ -~-t t-: I, I- - 4_: 1 i:";. ~ I::::: 4ART NO. BL 909 BRUSH I (ECTTRONICS COMIAN -:o - ~.4 arT N. F3L 909 I.~-..L —~lfr. --.~-~-:i-'-.... —:~.._f~_~.P... -4=- 4 ~ ~ ~ ~ ~ ~ i = t -VA-IARTNO. L 90 BHU)-1 EE~:RONIS COi2.4 Y,,,F3;Ui T~ —i: I- I -~~ --— 1 —L4 ___i~~~~~~~~~~~~~~~~~t

-101Calculated Results Calculated results and experimental conditions for the 28 runs _performed with dry packing are listed in Table V, pp. 100. Table VI, pp. 101 gives an analogous tabulation for the 42 experiments performed with wet packing. Standard run calculations consisted of obtaining liquid and gas flow rates and respective Reynolds numbers, also in estimating various moments and appropriate mixing parameters from the recorder traces as described below. Gas and Liquid Reynolds Numbers were calculated from the measured flow rates converted to superficial mass velocities in the unpacked column, nominal diameter of Raschig rings and viscosities. The Residence-Time was evaluated by integration of the area bounded by the response curve from the time at which the step input in concentration is introduced. This quantity is: for tracer displacing air: (See Eq. 14, pp. 22 ) for air displacing tracer: A dt (See Eq. 13, pp. 22) ateale The ratio of the residence-time calculated as above with that calculated from the total void free volume and the gas flow rate was designated as Material Balance, i.e.

TABLE V EXPERIMENTAL CONDITIONS AND RESU~LTS FOR GAS FLOWING THROUG11 DRY PACKING 2~~~~~~~~~~~~~~~~~~~~7 3 4 8 19 227 30 319 2 ~ ~ ~ ~ ~ ~ ~ ~ 1- 3 42 9 3 -4 5 BED PROTERTIES A i?~ AIP. DISPLACING TRACER TDLACR TRCRDRPAIN I T~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ a~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, CD~~~~~~~~~ j El >: 0 >~~~~~~~~: " 0 Id U~~~~~~~~~~~~~~~~~ >.2.1 478 ~06 7o.4o80 8o].9 o731910.o1 o4.11 473 0791. 8'.2. 61o6.07 2 ~ o77i.2217.0208 J.63 3.42 J.17 114.7 6 J. 0058 17.0.040 8.05 3.60 1.01!.00903 11 110 --!......100 1!~.[ 56.0051 2.1).059.98 4.80 1.002.029(1 42.........028 J.69 ].2 J.21 j 4.777.0034 8.5.045 1.6 2.8 i 984i.076 13 13.070.042.72 1~ 4.7 7].001626. j 043 11.5 2.6 1.54 0076 19 10.072.048 16 Q) Q, a, 0 —~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~0208.6933.42 J.217 14.7 7 ~.0934 285.045 1~.622.86:.986.0075 133 10.0099 J.O39 1.4 15 1.FJ 7.0101 26.1i.043 J12. 23.071.078.007461to E5L.0064.02I 1.7.0208.693 3.42.217 J 14.7 78. 00786 24.0.045 11.5 27.28 j.92.00602 166 15.00541:.0185 225 15 14.7J 78 J0817 21.0 I.043 10.10 25.15 0.97.00660 151 150.00604.020[ I.,.008J.933.2.27 1. 7 J 066510 045 41~1351 103.074 4010 0061[ 026.8 1 4.J 7.0~0 4.0Ij.03 2.212761.16.071 4014.069 025.>.0208~~~~~~~~~I.6]34.271.77.0165..032.41.1.65:051010 ---...101.J7.1O.2 05.01.4.9 ~.63 3.42.217 J 14.7 86 J.058 70 o4 4o005 80 9 07 129 io.072 044 i7 5 47 36.0436 1. o3 6. 2 4.26 03 5.007 28 1306.00 76.00750 134 14o. oo6g O~9 174 i 14 7 0 i( -2.1 o433 12.75.00 8.983.2.179 6. 73J 04 3 1 3.2.04617. 5194 97: 009011 1000-93.... 00859J59j 03804.1.0547.3.1.922027946...... ~-.0208 j.693 3,42!.217 14~.7 78 1.00716 12.6. o435 6 1.0 27.408 1.992 oc6O20 665 150.00541 ~.0208 2.0 25 1~ 4.7 78 /.0487 1 1.71.0433 5 0.6 25.615.947. 0066o 156 150.0054.20:0 D6.0208.693 3.42 i.217 17.2 77 oi664 523.0.0435 24, 103.391 1,6i.033 o7l42 135 35 oc,6690.0226 1.77 15. 14 7 -11 "9 1.63 199.0o.o4,34 21.2 12.1 ~.o1.0695134 1 40). 04o.02 0 19.D8.0208 i.68_ 12.4 27 15779 ~c 12.60 431 62.0 16 5.96.00763 il-071 jD-9.028.6934.42.2176 16.0 79.0040 133.2.04363 63.07 55.29'.984.00908 112 1200.0047',) l.2131 17.19 o4514.7 783 [-1.0922.00478 29 20.445. 09 D10.0208.6985 3.42.2761..026 7 53.4o.051.022.976.053 189 1950 o o.02508 2.002 18 15 14.97 78 oo69 4.58 i.71 433 2 5.63 1 56. 1.005.004252 5 ~~~~~~~~~~~~~1 4- 8 o~6 1.06 0.0435 4.6 9i.27 15.9.0401 133.0.0434 63.7 6.951 1.009 ~.00507 197 200.00476.0212 1.97 15 15.81 75 I.0390236 110.6y.0432 57.1 7957663 9 5 o434059951593171.9516400695244.65 ~~~D-11 ~ 3 42.02082 1.6 ~93 3.425 1.27 177217.065 2075.0:3 o436 i26.39 1.006.00742 135 135.0c776.9966 1.57 1-12 35 4.45 ~.276 i2.1.0433 2 -7 55'.966 1.008 oo7.02812.7 20 13 -049 12 o31600- o 75 1 C9.90 ~ 0208 I~~~~~~~ ~~~~~.7 6 o.o2 o.0402 13..0977 7.3 1 5 7 i.72.973 9 654 4 5 1 4.7 543 D ~~~ ~ ~~~~~~~.002 4 5.02 3 4.29 5215 o 7!.433.0973 [.033 1.255 D1 o8.666.260082 2- 435 6 3.4 25.28 1.031.00979 1202 1301O75 ~06 4.7'2 1.97 T.010077 19. 043 0.6 6 1.00 082 12 13 D-I~~~~~.020 66 3.42 06 132.5 o3 34 53.3.00954 10 115..0097424.3 11 5d 7 4, i5.6 o3 55.6 474 93 0 13 9 li D17.03208.66 206 o4 o4358 6. Io2.266~~~~~~~~~~~~~~~~~~~~~~~~~~~53 1..043 7.0947.97 10. 15 944.00323 1529 155.0 0 6 0 7 -43 5.069 2.315 3.42 ~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~132.5 73.04 115.6.043.4 60.45.9715 7. 006011 160.05C. 2423 D-8.0312 i.660 4:45 ~17 26 8 140o793.0058 i1o.043 6.4 739 96.00630 1659 5 oo 29 23 5 1. 3 -02 11 o3.4 6.5 -7 o.o 1i io 18. 0 1 ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4. 68 ~ 007(96 2o. 8.0428 1 5. 16, 1.030.06 go04 D-19 i.66o 4.45.266 47 68 096 24.8.o430 18. 55 -6 oc 6 6o. 00 579.0258 2.42 15. 10 io..0312 67.0005!33. 15. 6 7. 0349 c27 55 54 (5 co9 4 130.8 B-20 C312.66o 4.45.266 15-~~~ -03: o. o429 952 6.63.970.00723 1-38 130.00 01.0 1 i~~~~~~~~~~1. 67.0069052. 0427 95.4 5. 1 46,i~.06915 4.452 66 17.236 2.09.o42 0 657 145 14o.00665.0296 2.1i 15 1.01 5 197.0oI 4.07 194: 9 D-21. C1.66o7 0 i,,9 2 1029 oo 77 16549 -22 66o ~~4.45 7. 2 0 o670 516j3 i.o c64 103 --- 2 --. 0o 4.0479 56.53 142 D ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1. 66 7 37.407 5:D-23.031-2.660 4.5.266 14.7 370.000 16.c 3259 94.021 — -- --- 4~ ~ ~~~ ~~~~~~~~~~~~~~~~.1 432oo4 r43 17 5 14.9 7I.016702 3.8 404381 i.00654 15 16018 ) lc.... D-2~~ ~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~4.77.048651 7-97 04 5.3 4.45228 1~.900.016513 66 I.. 376.041 9-.84..........7,,6 00 4045.283 6 6 c42.1.9934085 9 85.083.38 21 5 62( 0161 2 g 160 oo848 ii8 120.0834 15 ~~~14.7 7 ~95 107090.0 9: D-26.046 6703 4.45 I28 14.7 71 1.008034 l4.-, o9 1.043-31 18-,1.037 2.2494((

TABLE VT EXPERIMENTAL CONDITIONS AND RESULTS FOE GAS FLOWING THROUGH IRRIGATED PACKING 1 0 1 1 1 6 7 8 9 00 J1j1j 0~ E i E Ih 1 10 19 20 21 2221 2"E6 2 1~O J1 ~ 7 138 19 40 1E1216 BED PROPERTIES AIR OIR DISPLACING TRACER TRACER TRACER DISPLACING AIR WATER d k jl t~~~~~~~~~~~ i%,ro, a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~,4~~~~I I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I EL 2 I 2.. PI 0 0 Id 10 1. 10 1~~~~~~~~~~~~. ~~~~j.7 1. NR1 K l L EE> EN E NE RN > 140 N KEN El I IN N N N N 1E.21 NI W-1 0208 -93 N. D EE7 16 03 o1i 0G 24 o4 198K N.5 CE ENoN 100 K K W-7 020.63 3co 259.03"1 014.7 19.00780 56.49 1. 2-5 2.5.2 o11167 OOIRI,.V245 E.6E.0002 VOCE1 14.7 09.0096 19 8 09 GO 9 15 665 1 5.5R2 O.0119 oB1496.0440 9.46 OOV.0001 79 42 01. N-8 o2VE V69 3o 29 069 014.7 79 co1E5 ANN. 04141 26 5 lbC.011110.01070 o~o.050 oVE O.000 VOVO 11.7 09.007 45 o5 11 1.2 88.594.00124.01650.000 1R14 VE00l.0002 CS 02 I W-9.O~B.93.oo.25 3 EE1.o 09 oV98E 1290 o441 11. 45 8 35.9.011216.10215 oE481 0.4. 0010 o.V025 04.9 79.0096 il o49 5. 4 -.501 012EE.V2VB4 V075EE7 I oo.10001 79 61 5. N1.0200.691 3o 29 069 117 El 8.005E4 20( 1 00419211 2.6 261.7.01079.00054.E25o 1.02.0000.0000 11.7 05.000o 120 2 0459 965 5II E 2.945.60E Vo1598.020E09 V626 1.6 OC 0VE7.0002 Es 62.4 47 5-1 EAE8 0695 3 C.29 069 14.9 E6.0V165 1 01 11 o4o 25 22 29.4 17i 51 o.01106.010178.01531 0.19.0005.0002 49 8 l, 42N6 05.022 9 15 22 90 E10.590.01912.10002.05414 710 V0002 EOVoo E6 (lE 09 0 1. W-12 0208.693 3.00.259 o569 11.7 E6s 30 15 4o 6.5 46 2 ooE57 109.0 2211Oq.o..1 0.010.0oo0.1021 15.9 E6s o.VV7 94 152 21 V 1. 0 15.6.14.011452 o1E90 V465 E89 011 001.E0E2 86 021 0 1 E 1 6. W-15.028.95 303 259 -o59 lE7. 86 oE084 216 E94 0409 119 2 251E 21 56 610,.01071.020011.0245 E.6B6.01020 COVE EG.9 86 Volo 9 1543 8. 26 10' 2E 5f.511 VOlE5.0C 024 9.O19 10 7G5 V0lE.OV Eso 02.. w1 02E08 0693 5oo -59 o69 10.9 OE.0165 50.0 E441 21 6 loOlEOR.58 015 o011.V.o40E 0.038.000 o.00014.9 08.0170 45 o5 2.. (6.595.01098.01220 0106.1(, 14 000.0002 40 62.I 49 W159 E02ER 693 3 00 C 259.0169 16.0 00.E3980 2 41 6. 4.2 E 285.592.01867.01602 oG~l o.621.01013.0000 15.9 88.03741o39 4..6 6.594.01224.01570 06 72.0004.,0001 40, 02 49 N1 102E08.691 5o 29 069 17.0 42 05698.03140.9 2.4.9.010054.00457.0103 0.601.0020 o0015 06.9 80.061'1 1( 046 79 5 2180 2.45.511.01590,.02002.8 5 0017.0012 82 02 19 W11.0100 069 3.00.259 0169 14.9 80.0165 508 0 0G438 21. 7.0101148 CB4.01176.0758 1.17 ocEi4.0001( iG.9 86.016'( 4. 46 2..1'.4.51( o00992. 0114 c69 64 0002.0000 06 0.6 19 7 5 1. W 12.0208 0691 1 00.259 0109 10.0 86 oE198 10 l o040o 61.5 4 65 1.( 72 450.01175.01710.0511 V.011 oo0E9.0005 15.9 86.020712 o48 59 34( 40 169.567.012529.01549 10405 0953 0005.0002 00 02 1E. N 19 E02EE 0691..29 o6 11.0 80.0566 094.0440 919 2.7 18 39.018551 o04421.1326 0.600.00203.00 5' 10.9 00.001 7(.4 4.C 256 17.512.018552 )54 64 02460 0015.ooo0 00 62107. 142.0208 1691 1 00 7259 0109 14.9 40.0105 50. 0 0441 23 6 1 19.84 5 510 46 o01650.0291 016 051 0.4000.0010.1006 14.9 88.0161 4. 48 2. 7 69.521.01495.03191 091 48 0006.0002 86 020 9 58 101. W15.0279 6953.0 0.259 E169 10.0 RE.0198 1B0 B44o 610 9.3 290.4.01964 oE2203.1651 E.621.00213.BEE7 15.9 ER o7377 12E 7481 W9 9 4 40 3 42.515.01454.01877 05618 5429 ool4 ToB6 86 E6S.9 E 1 1. W-2 16 0200 0691 1 00 259 0109 17. 82 7.0566 198 0455 94o 0.5 402.1.020N8.01457.101510.21.0055.0044 16.9 7 02 0 14 04 91 95 2 028 2 10.616 VEERS6.01269 -0408 5154 VE0l.0021 02 624018 W-217 0200.691 1.o. 29 0109 14.9 82.0057 50 84. 0078 24. 2 18 72 1.001047 65c.0256 6o5 0.549.E014.0002 14.9 026.0160 1. 43.65. ) 81.677.01650.0210 024 15 0009.000402 022720606 54 5: W-248.02108.695 4 00 549 0169 14.7 80.00190 1102 04340 611 03 575.02072~q,.0292.0879 0.41.0019.0011 15.9 80. o5007. 44.63.( 68.167.02195.01153 o45.3 0022.0011 86 02 05. 41 W-19.0208.685 1.00.2549.0169 147. 86.05669 1942 o0434 91.9 28.47 1.82.199.02115.0441.11259 0.113.0011.0001 10.9 06.0610 1973.018 81.0 29.160 1.76.124.00552.01164. o,2.82. 0071.0050 86 01E 03. W-26.0210.695 4lEoc.259.066 16.0 76.E079 1502.0440 61.5 16.81 22.90.450.02968 oE4503.1151 0.500.0021 EIOI 15.9( 86.0475 10.01 19.44 9.2 18.20 25.42.144.02081.02959.0880.469.0014.0006 06 62.6 19.0 1 6. W-27.0320.691 1.00.259.0169 16.0 75.09078 22 155 5.01 1764. 4.951 24.021.648.02070.00818.015 1.20.0002.0001 15.9( 71.00065 1147.0411 55.2 4.84 15.91.61 1 019.01924923 g7 1.042 IE1.CooE6 71 022.8 000. W-23.0320.6585 6.00.49.E366 14.7 76.E0788 24.2.0435 11.5 21.85 27.21.616.00612.00050.0260 1.40 EVE29 oE0l4 14.7 77.0E725 19.5.044 95.26 29.98 28.10.o60.00219.006599.0244 1.44.0007.IEVO 70 625 2.9 08 44.2 W-29.0320.658 4Eoo.749.0355 14.7 76.00789 24.2 E43G 11.5 10.19 25.75.582.01252. ol44.0941 1.00 ooEo6 OIEO5 14.7 761.007250 19.1.0111 95.26 11.10 286.0.593.00950.01115.0455.914.0000.0000 76 625 2.9 27 14052 N-I0.0320.6585 4.Eo.549.006 14.7 70.00709 24.2.E435 11.5 20.22 27.58.655.01110.502.0515 49 E.32.0009.0007 14.7 76.00750 19.1.0434 9.26 529.60 27.00.592 o00999.01179.62 o3 1.402.4002.OOEO 76 625 2.9 54 14018 0.26.0321.68 5 4.00.149.0166 14.975y.04070 24.2.0434 11.5 26.82 22.10.501.01557.01701.1000 0.611.0009.0005 14.'8 76.04070 19.1.0134 90.26 528.1 24.50.49N.011291.052.1584.711.0003.0000 76 623 2.9.6 50081 W-327.0312.6s85 4.00.349.0155 14.7 75.0o402 24.2.0434 17.4 20.71 24.21.517.02253.02814.0050 0.015.0021.00121 14.7 75.00725 18.7.0432 14.5 10.88 25.99.529.0124.02115.09474.642.40119.0026 75 626 2.2.4 14070 0-28.0112.658 6.00.349.0355 14.7 77.00402 24.0.0435 17.2 27.4 22.96.5710.02446.037618.147 0.42.0029.0011 14.7 775 075 1..04 11o3 11.1 29.11 21.42.506.021194.01599.440.44.0050.0041 77 621 2.6 ls 60098 W-29.0312.68 4.00.349.0155 14.7 77.00402 24.0.0435 17.2 590.2 25.74.581.01202.1404 o6524 1.0.000o6.0003 14.7 775 775 1..04011 o3 10.2 611.0 520.1.591.01175.01379.0552 1.11.0003.0000 77 62.1 21 0 1764 8-30.0312.6058 6.00.549.0554 14.7 77.003718 23.5.04154 16.9 71.99 27.1.5. 0103.01 103.0 477 89 1.32.o7l4.0002 i4.7 77.00742 19.0.0431 11.7 12.60 25.82.5892 o.0966.01092.0436 1.45.40102.09 7 6.4 21.0007 05.0 w-31.0312.605 4.00.349.0554 15.9- 75.00028 175 o0454 95.6 5312 4.6-5.599.026387.02304.1417 0.441.0045.00231 15.0 75.04002 1811.0132 00.2 52.21241.31.596.025200.01136.1174 0.758.054.0047 75 62.6 22.0 000. 0-12.0312.708 4.01.349.03554 15.9 75.04028 133.0434 95.6 52.42 47.48.651.02285O1 7.0292 o.16 0.535.0023.0012 14.8 75.0400 1116.0432 00.2 5.495 48.40.529.02400.03110.06N70 1.501.0039.0020 75 622 2.2 54 14070 0-38.04112.6058 4.00.349.0355 15.9 75.0402 153 o0435 95.1. 5.79 4.918.657.01456.0069i.0708 0.83.0010.o006 14.0 75.04002 1116.0132 00.2 56.04 5.155.505.01394.01679.1672 1.291.4010.0006 75 62.6 22 4762 0-39.0412.705 4.00.549.0155 15.9 75.0402 1532 E455 95.6 5.91 5.02.581.018307.01561.0647 1.00,.0C26.00010 15.0 75.0000 111.0432 00.26.1 5. 21.9.591.012606.10.421.050o 160.1.00.04o5 622622.0002 07.0 W-4o.o416.705 4.00.349.0374 15.7 77.00402 124..041 21.0 3.42659.50.599.02132.02950.1173 0.711.0034.0027 14.7 77.0072 18.6.0411 17.9 2.91 21.42.575.02200.03757.1102.794.0024.oco4 77 624 2.6 ol 600 0 0-41.0416.705 4.00.349.0371 15.9 77.0402 132.0435 126 0.42 5.49.653 T.0155.02257.0903 0.924.E017.0010 15.0 77.0410 113.0431 108.6 6.22 5.31.624.00875.02288.0915.911.0001.0004 77 62.4 21 00 4786 0-42.C416.705 4.00.349.0374 15.9 77.0002 132.o435 126 6.78 5.45.628.01700 o00045.0018 1.02.EOEE A00E2 15.0 77(.0410 111.0431 108.6 6.26 5.35.629.01940.02376.0950.878.0005.0000 77 62.4 21 000.

f or tracer displacing air: for air displacing tracer: O where cVT is the volume available to gas flow in the packed section, VE is the volume of the empty bottom section and v is the volumetric f low rate of the follow-up gas. The above ratios must be equal to unity for a perfect material balance. Since the free volume was known independently only for the dry bed, the material balance was calculated for the dry packing runs only. The porosity of the irrigated packing can be calculated froN the mean residence-time of the gas flowing through the irrigated portion of the bed (i.e., 9 -GD-GE) divided by the residence-time which the gas would require to flow through the same section of empty column VW. This quantity is., therefore: v for tracer displacing air: 4F... a. —--------- (5 8) IV dVI. \VW for air displacing tracer: 4...Z........ (58a)~.MMMMMOM Nr ~~~~~~~~~~~IT

-105 - where ew is the porosity of the irrigated packing., EV is the free volume of that p~ortion of the column which remains dry and V is the empty volume of the irrigated p~ortion. Again v refers to the volumetric flow rate of the displacing gas. The Variance was evaluated as: 2 2o for tracer displacing air: oCJo (See Eq. (i7), pp. 26 ) for air displacing tracer: 4t ( ompemetr The variance as calculated by the above method measures the mixing produced by the entire system. For the dry packing experiment,, this includes a packed section and an empty section. According to the mixing stage mLodel (see!pp. 53 eq. (54) and (54a) the variance of the total system is: where ax, n and G are the variance', number of mixed stages and average residence-time for the total system and the subscripts P and E refer to the packed and empty portions respectively. Since we are interested in the variance or number of mixed stages for the packed section alone and we assume that the empty section is perfectly mixed,,

-106 - the corrected variance for dry packing runs is therefore: alp( )2 ((59) Since the volumetric flow rate is the same in each sect ion, equation (59) can be rewritten in terms of the free volume of the packed section eV and the volume of the empty bottom section V For the irrigated packing experiments the packed bed was in turn made up of two sections: an irrigated portion and a portion remaining dry above the liquid distributor so that according to the mixing stage model (see pp.53 eq. (54) and (54a) where the sbscripts w, D and E refer to those portions of the column

-107.. The number of mixed stages contributed by the dry section of packing above the liquid distributor nD is estimated from the results of previous experiments on dry packing. Values of nD used for various groups of runs are listed in Table VII. The height of a perfectly mixed stage is obtained from the variance which after appropriate correction as described above is multiplied by the effective height of the packed section being considered, (S ee Eq. (4 8), pp.- 49 ). The effective height of the bed was taken equal to the empty volume of the column occupied by packing, divided by the average cross sectional area of the column. The dimensionsless group uLwas estimated for dry packing 2DL ruqns by graphical comparison of actual recorder traces with the solution "A" of the diffusion equation giving the response to a step input (eq. (26), pp. 35 ). This was done by superimposing a graph of c/c 0 Vs t/@ as obtained from a given recorder trace over prvously prepared graphs of solution "A"T evaluated for increments of the group uL u -of 10 units from u - 80 to U 200*. An example of this 2DL 2DL 2DL procedure is shown in Figure 34 where the points Plotted were taken from the response curve for Run D-4 when tracer displaces air. The uL superimposed curves were calculated for - eqato1o 4o 6o uL DL qa o10 4,10 The estimated value of ~TLfor the experimental points was chosen to be 140. * A tabulation of this data can be found in Appendix IV, Table

TABLE VII NUMBER OF MIXCED STAGES FOR DRY PACKING SECTION ABOVE LIQUID DISTTRIBUTOR Run No. Nominal Modified Packing Hydraulic Height of Equil~avent Number ofMie Packing Peclet No. Porosity Mean Diam. Mixed Height of Stages ForPce Diameter From Eq. e d Stag-e Dry Packed Section Abv In. (63b) m 2d Section Liquid Distiuo m pp. 120H Pell F.N L Pe" _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _L_ _ _ _ _ _ _ _ _ _ _ _ _ W-23 through ~W-26 1/4.338.685.00362.0214.45 21.0 W-23 through W-26 1/4.338.693.00371.0220.4520 W-27 through W-34 3/8.338.658.00512.0303.45 14.8 W-35 through W-42 1/2.33-~8.705.00748 o0442.45 10.2

1.0 >Z%. ~~~~~~~~~UL12 2DL UL:140.8 2DL U:160 2DL 0 POINTS DETERMINED FROM RECORDER TRACE FOR o 6 RUN D-4 WHEN TRACER DISPLACED AIR. z 0 ~ 4H z0 0.2.70.80.90 1.00 1.10 1.20 1.3 DIMENSIONLESS TIME, "e8 Figure 34i. Estimation of the Dimensionless Group- U For Dry Packing Runs. 2D L

The Peclet Number is generally defined as: DLI (61) using the relationship found earlier (Eq. (57), PP. 60 ) between the number of stages and the effective longitudinal diffusivity the Peclet number can be expressed as the ratio of the particle diameter with the height of a m-,ixed stage. Pe~~~ck 2c;1_ (62) The equation above is used to calculate the basic value of the longitudinal Peclet number. A modif ied Peclet number is also used in Chapter V for correlation purposes where the nominal particle diameter is replaced by other characteristic lengths. The skewness of the ti me distribution function was evaluated for wet packing runs as a measure of the departure from uniform flow. This quantity is: for tracer displacing air:'i)-.. -4.2 (18) tpt for air displacing tracer: COC- LMM (complementary to above) [00 cCtC f#44t 0.A

-111If the flow of gas through the column is not uniform., then: 2z07z.-/ For an illustration of detailed calculations f or the various quantities., a sample calculation for Run No. W-8 is given in Appendix IIIY p-p. 199

-113 - CHAPTER V DISCUSSION Dry Packing We examine first the reported results to seek a confirmation of the conclusions drawn from theoretical considerations and the vaLidity of the experimental technique. Material Balance The first requirement of the step input technique is that the area bounded by the response curve from the time at which the step input in concentration is introduced must coincide with the time required for the gas to replenish itself through the system. The degree of agreement is given by the material balance listed in columns 13 and 28 of Table V, pp. 100. It may be observed that this quantity fluctuates about unity with a maximum deviation of + 8%. The uncertainty of the flow rate measurements (the precision of the rotameter calibrations was about 3%) and the precision of reading the recorder traces (the recorder chart can be estimated within + 1/2 line or 2.5%) with subsequent graphical integration give ample justification for the reported fluctuations of the material balance. We may thus conclude that the step input used gives the true response of the system. It may further be noted that if the tine lag of the analyzer-recorder system did affect the resoonse curves then the material balances would have to -be Soesis mti; greate-r than one because of thle additional delay of the ou u- sis,na -aused' c; - analyzer seystem and not due to the gas flow tl rom, -v.) sysse-;:. TIlus totI' thle normal sensitivity o eration using helium as Fracer and the high sensitivity oTVeration are consistent on

-114this point. Effect of Tracer Composition The second point concerning the validity of the experimental technique was to discover whether the signal output was affected by changes in tracer composition or by the manner in which the step signal in tracer concentration was introduced. From the preliminary experiments, some doubt was created on whether the use of pure helium would alter the shape of the response curves not only for the tracer displacing the air which was quite evident in Figure 23j pp. 87 but also for the air displacing the helium. Table VIII gives a comparison for analogous runs carried out with both 15% helium and pure helium as tracer. From the variance for helium displacing air, we see that the discrepancy in the response curves is quite large and this can be explained by the convective effect already discussed on page 77. For the air displacing the helium, this effect should be non-existent however. Nevertheless, a comparison of the variance calculated from these response curves shows a significant difference for Runs D-2, D-7, D-9, D-20, and D-23 as compared with runs executed at similar volumetric flow rates but with 15% helium. Thus, some other effect concurs to increase the variance of the distribution. A plausible cause of the increased mixing may be found in the difference in line pressure necessary to obtain comparable volumetric flow rates for air and pure helium. Because of this difference it is possible that a certain amount of extraneous mixing takes place at the solenoid valve when the tracer flow is interrupted and replaced with air. At any rate, this point was

TABLE VIII EFFECT OF TRACER COMPOSITION Calculated Calculated Nominal Variance Variance Packing Tracer Composition Flow Rate For Air For Tracer Run No. Diameter, in. Vol. % H cu.ft./sec. Displacing Tracer Displacing Air D-1 1/4 15.00519.00773.00778 D-2 1/4 100.00521.00903.02971 D-6 1/4 15.01703.00714.00713 1D-7 1/4 100.01501.00835.01824 D-8 1/4 15.0435.00763.00746 D-9 1/4 100.0458.00890.02179 D-18 3/8 15.00429.00630.00620 3-21 3/8 15.0652.oo6o6.00716 D-20 3/8 100.01865.00654.01513 D-23 3/8 100.00970.00620.03998

-116 - not investigated further. Having noted the undesirable effect of the helium air system on both air and tracer response curves, it was decided to base all the subsequent conclusions on results obtained with 15% helium as tracer. For the runs made using 15% helium, Table VIII shows no consistent difference between the value of the variance for air displacing helium and. that for helium displacing air. Thus, both values of the variance were used in the subsequent treatment. Relationship Between Number of Stages and Diffusion Coefficient A third point which was verified experimentally is the relationship between the number of stages and the longitudinal diffusion coefficient. The average number of stages in series can be obtained directly as the inverse of the variance and is listed in Columns 15 and 30 of Table V, pp. 100 for displacing air and helium respectively. The group uL was estimated by means of solution "A" as described on page 2DL 105 and is listed in Columns 16 and 31 of the table mentioned above for air and helium respectively as the displacing gases. By comparing column 15 with 16, and 30 with 31, it can be seen that the agreement between these two quantities is within 10%. On considering that the unuL certainty in the estimation of the group 2DL is probably not better than 2DL 10%, one may conclude that the relationship: -V L- L a(57) i2. is valid at least for the range investigated. This fact lends addition

-117al proof to the validity of the step function technique for measuring longitudinal mixing because it is in substantial agreement with the theory (see pp. 55-62 ). Because the variance as calculated from the response curves was considered more precise than the estimation of the uL group 2D it was decided to adopt the number of mixing stages or the inverse of the variance as a measure of the extent of mixing taking place in the column. Effect of Bed Height, Correction For End Effects Lastly, we may consider the consistency of the data with respect to bed height and end effects. According to the correlation model in which the packed bed is considered as made up of a series of perfectly mixed stages, one might expect that the height of a mixed' stage: W _X = L (48) LA n be independent of bed height. A tabulation of the data showing the effect of the packing height on the variance and on the height of a mixed stage is given on the following page. The variance was corrected as described on page 104 to account for entrance effects produced by the bottom empty section. Table IX shows the variance and height of a mixed stage before and after correction. Assuming that there is no effect of flow rate or packing porosity, the height of a mixed stage appears in fair agreement for both bed heights. Before correction the average of all the runs for

TABLE IX EFFECT OF BED HEIGHT AIR DISPLACING TRACER TRACER DISPLACING AIR Effective Height of a Actual Height of a Actual Bed Mixed Stage Height of A Mixed Stage Height of A Height Calculated Before Cor- Corrected Mixed Stage Calculated Before Cor- Corrected Mixed Stage Run No. Ft. Variance rection Ft. Variance Ft. Variance rection Ft. Variance Ft. D-1 3.42.00773.0265.00712.0244.00778.0267.00707.0242 D-3 3.42.oo760.0260.00(09.0242.00776.0266.00726.0248 D-4 3.42.00750.0257.00699.0239.00746.0259.oo00694.0238 D-5 3.42.00602.0206.00541.0185.oo660.0226.00604.0207 D-6 3.42.00714.0244 00661.0 220661.00226.oo13.0244.oo659.0225 D-8 3.42.00763.0262.00713.0244.00746.0256.oo694.0238 D-10 3.42.00665.0228.oo609.0208.00642.0220.00584.0200 D-ll 3.42.00742.0254.oo690.0236.oo695.0238.00641.0220 D-12 4.45.005o8.0226.00477.0212.oo478.0213.00445.0198 D-13 4.45.00530.0236.00500.0223.00452.0202.00419.0187 D-14 4.45.00507.0226.oo476.0212.00591.0264.00564.0252 ~~~~~~~~~~~~~~~~~~~.0252

-119 - the shorter bed gives a height of a mixed stage equal to.0247 ft. as compared with.0228 ft. for the longer bed. After correction, the averages are.0228 and.0214 for the shorter and longer bed respectively. Comparison of the above data shows that the correction for the bottom empty section amounts to about 7-8% for the shorter bed and 5-6% for the longer bed. Consequently, while the height of a mixed stage for the two bed lengths before correction differs by 8%, the difference is only 6% after the end correction was applied. This type of behavior is expected if the correction is real. On this basis, we may conclude, therefore, that the height of a mixed stage is independent of column height. Moreover, the correction for the extraneous mixing introduced by the empty portion of the column, although small, appears justified. Having established in the above discussion that the equipment and experimental technique yield reliable and consistent results, we wish to compare the present results with analogous data of previous authors. The discussion that follows also offers the basis for the overall correlation of the data. Comparison with Previous Investigators An extensive study of axial mixing for gases flowing through packed beds of spheres is reported by McHenry and Wilhelm /47/, who used a frequency response technique. Their findings can be summarized as follows: du 1. The Peclet number calculated as D L where d is L the spherical diameter and u is the interstitial velocity, is essentially independent of Reynolds number. 2. For 21 determinations of Peclet number in the Reynolds number range from 10 to 400, the mean Peclet number

-120was found to be 1.88. Expressed in statistical language, their reported result is: Pr (2.03 > Pe > 1.73) = 95% where Pr is read, "the probability of" and Pe is the true mean of 21 determination. 3. No detectable effect of gas density was found for the two gas systems studied. These were, methane-nitrogen and hydrogen-nitrogen. The data of these authors is represented in Figure 35 as a shaded band indicating the 95% confidence limits for the Peclet number. The data for the Raschig rings as obtained in the present work are listed in Table V, columns 19 and 34, and are also shown graphically in Figure 35. In this case, the Peclet number is calculated as the ratio of twice the nominal particle diameter to the height of a perfectly mixed stage and it is entirely analogous to the Peclet number used by McHenry and Wilhelm except that the nominal ring diameter is used in place of the diameter of spheres. The gross agreement between the two sets of data as shown by Figure 35 may be quite fortuitous because clearly there is little hydrodynamic relation between Raschig rings and spheres having the same diameter. However, Figure 35 serves to point out that the Peclet number is of the same order of magnitude in both sets of experimental data for both techniques of signal input, i.e., step function response and frequency response. An analogous conclusion was also obtained by Ebach /18/ who has extensively studied axial mixing of liquids, packed beds of spheres and other packing shapes. The dotted line in the lower region of Figure

4 95% confidence interval for data on gas. AA| O Flowing thru spheres /47/. 2,e o * - I I _.1.8 CL.6.6 Data on liquids flowing HH.5 thru spheres/18/ 3 4 5 6 7 8 9 10 20 30 40 50 60 70 8090100 200 300 400 Reynolds No.;Gd/L o 1/4" Raschig Rings, air displ. tracer ~ 1/4",1, tracer displ. air A 3/8" ",, airdispl. tracer A 3/8" 1' tracer displ. air o 1/2" " 2 air displ. tracer * 1/2 1",, tracer displ. air Figure 35. Axial Mixing through Beds of Spheres and Raschig Rings. (Peclet No. vs. Reynolds No.

-12235 is representative of data obtained in his experiments using two types of unsteady state techniques, i.e., transient and frequency response. Other data for liquids obtained by Danckwerts /13/ and Kramers and Alberda /38/ also fall in the neighborhood of the dotted line. A statistical analysis of the data obtained on Raschig rings with the exception of Runs D-15, D-16 and D-17 on oriented packing, shows that a least squares straight line through the points plotted on Figure 35 has no significant slope. Thus, no apparent effect of Reynolds number in the investigated range is detectable. Therefore, the Peclet number can be expressed as the true mean of forty points and is found to be equal to 1.94 +.07. In statistical language the foregoing can be expressed as: Pr (2.01 > e > 1.87) =95% (63) These results compare well with those of McHenry and Wilhelm, thus the principal objective of the first part of this work concerning the dry packing was reached. Effect of Particle Diameter and Packing Orientation In the preceding paragraph it was tacitly assumed that the Peclet number comiputed as the ratio of the nominal Raschig rings diameter to the height of the mixing stage was independent of particle diameter. * In terms of height of a mixed stage we have: H = 1.03d +.07 (63a)

-123 - However, this dimension is seldom used to characterize Raschig rings. More frequently, the equivalent spherical diameter dp (diameter of spheres having the same volume as the Raschig rings) is used /41/. Ebach /18/ employed this quantity in the study of axial mixing of liquid flowing through a bed of Raschig rings. For pressure drop through granular beds Kozeny /37/1 suggested as hydraulic mean diameter the quantity: d -e e-.)~ = (64) where e is the void fraction, S is the specific surface of the material and a is the surface area of the material in the bed. For spheres, this quantity is: d_ ed 6(1-a) (65) Since the porosity for a packed bed of spheres varies about a value of 0.4, the value of dm is approximately 1/10 of the actual diameter of the sphere. To compare the suitability of each of these quantities in the correlation of the data, Table X was prepared. On the basis of the foregoing results, it is apparent that the hydraulic mean diameter dm is the best correlation parameter for the effect of particle size and porosity on Peclet number. Using the modified Peclet number computed with the above characteristic length, the results for 40 determinations can be summarized as follows: Pr (.349 > p-, > 327) = 95% (63b)

TABLE X CHARACTERISTIC LENGTHS FOR PECLET NUMBER CORREIATION Particle MEAN PECLET NUMBER Nominal to Bed Particle Diameter Number of Runs PeL pi p2 Diameter Ratio Using Random H HL - H In. ccdtit Packing 1/4.0625 22 1.87 1.73.331 3/8.0938 8 2.18 2.04.557 1/2.1250 10 1.89 1.82.335 Mean Value of Above Numbers.............. 1.98 1.86.540 Mean Square Deviation.........OOOQQOQQOOO.05006.02545.00043

-125 - Since no trend is detectable in the Peclet numbers of Table X with particle size irrespective of characteristic length used, one may assume that the variation of particle diameter to bed diameter ratio from.0625 to.1250 does not appreciably affect the longitudinal mixing. The packing properties for all the dry runs listed in Table V are summarized in Table XI. As seen here, Runs D-15, D-16 and D-17 were performed with oriented packing. The orientation,visually very noticeable, was produced by flooding the column so that the packing was lifted by the severe pulsation in the gas flow and rearranged anew. The longitudinal mixing observed in this type of bed is appreciably greater than that for a bed packed as described on page 79. Figure 36 gives a graphical comparison of the mean values of the Peclet number as determined for the random Raschig rings, the oriented Raschig rings and Wilhelm and McHenry's data for spheres. Summary 1. The material balance for the step function technique employed in these experiments appears consistent with the precision of the measurements. The area bounded by the response curves can thus safely be used in the irrigated packing experiments as a measure of the volume of the system occupied by the gas. 2. Because of extraneous mixing effects detected when pure helium was used as the tracer stream, all subsequent work was limited to the use of 15% helium. 3. The relationship between number of mixing stages and diffusion coefficient is as predicted by the theory

TABLE XI PACKING PROPERTIES Surface Area Nominal Surface of Total Per Unit Hydraulic Particle Particle Packing No. of Bed Porosity Vol. of Bed Mean Diameter Runs d, In. In.2 Arrangement Particles Q, Sq.Ft/Cu.Ft. dm = a D-1 through 1/4.375 random 21500.693 187.00371 D-ll D-12 through 1/4.375 random 28300.685 189.00362 D-14 D-15 through 1/4.375 oriented 22800.666 199.00335 D-17 D-18 through 3/8.767 random 9440.660 129.00512 D-23 D-24 through 1/2 1.38 random 3880.703 95.4.00736 D-28

.9.8.7.6 E. 4 6 Random Raschig Rings.3 C) 1/4 Oriented Raschig Rings Spheres /47/.2 0 4 5 6 7 8 9 10 15 20 25 30 40 50 60 70 80 90 100 150 200 Reynolds No.; Gd/u Figure 36. Axial Mixing for Gas Flowing through Beds of Spheres and Raschig Rings. (Average value of Modified Peclet No. vs. Reynolds No.)

-128 - (eq. (57), pp.60 ) at least in the range investigated when values of 2L were fairly large. 2DL 4. The corrections for end effects introduced by the presence of an empty portion of column at the entrance of the bed appear adequate. On this basis, a similar correction will be used for irrigated packing experiments. 5. The longitudinal mixing for beds of Raschig rings as measured by the present data can be expressed as follows: Pr (2.01. Pe > 1.87) = 95% In order of magnitude of the Peclet number and in the precision of the data, this result is comparable with data of previous investigators. 6. The effect of particle diameter and packing porosity appears best accounted for by the use of the hydraulic mean diameter (d = e ) as the characteristic length m a in the Peclet number. On this basis, the result of the overall correlation is: Pr = (.349 > Pe" ~.327) = 95%

-129Irrigated Packing Porosity of Irrigated Packing As described on page 102, the fractional volume of the wet bed occupied by the gas or the porosity of the irrigated packing was calculated as the ratio of the actual residence-time of the flowing gas to the residence-time which the gas would have required to flow through the same section of empty column. The wet porosity is presumably dependent on the mass flow rates and viscosities of the flowing streams, on the size and shape of the packing as well as on the surface tension of the liquid. The liquid used was water so that variations in surface tension do not enter into consideration for the correlation of the data. Raschig rings were the only packing shape investigated, thus: ew = F (G, W, % G` Aw, d) where Ew is the porosity of the irrigated packing, G is the superficial mass flow rate for the gas, W is the superficial mass flow rate for the liquid, 4G and yw are the viscosities for the liquid and gas respectively and d is the diameter of the Raschig rings. We choose for convenience to express the independent variables in the form of dimensionless Reynolds number and particle to bed diameter ratio. The influence of the variables on the data is shown in Figure 37 where the wet porosity is plotted versus the Reynolds number of the liquid. This Figure indicates that the liquid rate is the most important variable and that the relationship between the liquid rate and porosity may be

1/2 RASCHIG RINGS;I~~~~~~~~~~~~~/ ACI IG 0 18.6-24.2 lb/hr sq.fl GAS R 18.6-126 GAS RArE E1113 -126.6 _ _ _ _ __ _ __13_ _ _ _ _ _ _ _ _ _ _ _ _ _.5 o< 0.8 3/8"1 RASCHIG RINGS.7 0 01~~~~~~~~~~~~~~~~~~~~~~~~~018.6-24.2 Ib/hr sq~l't GAS L i I I I I I i 1 I.6 133 EAS RAT 0 4 __ aI-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1/40' RASCHIG RINGS 19.2 - 50.8 GAS RATE 0 19-2-824.2 lb/hr sq. GAS RA' ~~~~~~~... ~ ~ ~ ~ ~ ~ ~ ~ 1~A42.6-50.8.5 _ _`~~~~ c` a 1~~~~~~~3101 —133 e-z ~~~~101 -133 GAS 0167 -198 RAT.4~~~~~~~~~~~~~ 167- 8GAS RT 0 10 20 30 40 50 60 00 80 90 100 110 120 LIQUID REYNOLDS NO. Wd/,u FIGURE 37. EFFECT OF LIQUID FLOW RATE ON POROSITY OF IRRIGATED BED. (POROSITY vs LIQUID REYNOLDS NO.)

-131represented by the equation: -MRe = 710 w (66) where Ewo is the intercept, M the slope of the lines shown in Figure 37 and Rew is the Reynolds number for the liquid. The constants Ewo and M were determined by the method of least squares for the group of runs listed in Table XII. Each run in a group was performed with packing of the same size; also a statistical test showed that the effect of gas rate for each run in the same group could not be distinguished from the random scatter of the data. On this basis, the effect of gas rate was only noticeable for beds packed with 1/4" Raschig rings and for gas rates greater than 50 lb/hr ft. It can be noticed from Figure 37 that for 1/4" Raschig rings, an increase in gas rate above 50 lb/hr.ft2 does not affect the value of the intercept wo, whereas the value of the slope M increases rapidly at gas flow rates in the loading zone range (167 to 198 lb/hr. ft2). Considering the effect of particle diameter, it can be noted from the last column of Table XII that the ratio E is independent of particle diameter. Moreover, the slope M varies proportionally with the 2.31 power of the particle diameter as is clearly shown in Figure 39. On this basis, equation (66) was extended to comprise the effect of particle diameter as follows: w 9 x.900 x -343 x 10- 31Rew (67)

TABLE XII COEFFICIENTS FOR POROSITY CORRELATION CI IJ 0 co C ~ ~|o ~. W | COEFFICIENTS a) o (1 -E o.) o) o.H d CH rA~'cd 0 Cd _ __4 00 4-P Uz1 I RUN NUMBER j c) o h ~ a) r c o 95% Con-I 95* conW!-l7 and W-20 19.2-50.8 -H — q h)'_ 1 w bo Xi M O U 4-~ r4 4- rd L0 ~d 4-4 fidence fidence E-4 0 Interval Interval W-1 through W-17 W-8; W-11, w-14, 1/4.0625 0-6190.068-.693 32.623 +.002.00200 +.00003.899-.909 W-17 and W-20 19.2-50.8 W-23 through W-26 W-9, W-12, W-15,.693 W-18, W-21, and 1/4.0625 0-6190 101-133 12.616 +.004.00218 +.00015.890 W-22. W-10, W-13.693 W-16, W-19 1/4.0625 446-5050 167-198 8.612 +.012.0015 +.000o64.883 W-27.658 through 3/8.0938 0-6200 18.6-133 16.589 +.005.000783 +.00013.896 W-34 W-55.705 through 1/2.125 0-6200 18.6-132 16.636 +.oo6.000402 +.00013.902 W-42

.006.005 0.004.003 _ 0 c 0 0.002 o E.. LA tII (0 a..001 I I I i 3I to 15 20 25 30 40 50 60 70 80 90 lOO 150 200 250 300 G,Superficiol Mass flow rate, lb./hr. sq. ft. Figure 38. Effect of Gas Flow Rate on Wet Porosity for 1/4" Raschig Rings.

-134-.004.003.002 LL) -J z z.001.0009 z.0008 o.0007 L. uL.0006 L.\.0005 z 0..0004 -J rL o.0003 o o.0002.0001 I *.I0o.15.20.25.30.40.50.60.70.80.90 1.00 d, PARTICLE DIAMETER, INCHES Figure 39. Effect of Particle Diameter on Wet Porosity.

-135The intercept e of the wet porosity versus liquid Reynolds wo number plot (Figure 37) represents the porosity of the wet packing at zero liquid flow rate, thus the difference Ewo - EW should correspond to the dynamic holdup. This quantity is compared in Table XIII with the amount of liquid drained from the column when both liquid and gas rates were suddenly interrupted. As seen here, such data was obtained only for nine runs in which 1/4" packing was used. However, the agreement is as good as could be expected from the precision of the experimental technique (see pagellO ). Since the measurements were made at the most drastic flow conditions, least favorable to experimental precision (Runs W-16, W-18, W-19 were conducted in the loading region), the agreement can be considered quite general. The values of wo - ~w for 3/8" and 1/2" rings are also in fair agreement with operating holdup data of Shulman, Ullrich, and Wells /54/, and Jesser and Elgin /34/ considering slightly different packing dimensions, differences in porosities of the dry packing and differences in drainage time.

TABLE XIII AGREEMENT BETWEEN LIQUID HOLD UP AND POROSITY OF IRRIGATED PACKING Superficial Volume Of Dynamic AIR DISPLACING TRACER TRACER DISPLACING AIR Liquid Liquid Liquid Flow Rate Collected Holdup Superficial et wo- SuperficiaZ Wet W From Drain- Air Mass Poro- Tracer Mas PoroLb./Hr. Sq.Ft. ing Column Flow Rate G sity Flow Rate sity Cu.Ft. Lb./Hr. Sq. Ft. iLb./Hr. W w __....w _Ft Sq.Ft. __. W-ll 1590.0118.045 50.1.585.038 i 42.6,590.042 W-12 1590.0118.045 130.573.050 111,.580.043 W-13 1590.0146.055 194.560 o.063 177 557.66 W-14! 3340.0286.110 50.0.518.098 44.5 520.096 W-15 3340 1.0279 i.108 129.504.112 101.515 i.101 W-16 3340.0336.130 198.491.125 167 [480.136 W-17 5060.0385.149 50.8.484.128 42.9 i477.135 W-18 5050.0380.147 130.462.150 126 467.145 W-19 5050.0540.208 194.399.213 177 412.200

-137Correlation of Experimental Response Curves In the study of the gas mixing through the irrigated packing, it was noted that as the liquid and gas flow rates were increased, the response of the system could no longer be correlated in terms of any of the models discussed in the theory in which uniformity of flow is assumed. In absence of any information on the actual flow distribution in the column, the variance of the distribution of residence-times is still to be considered the most appropriate measure of the mixing process since it expresses the dispersion of the gas elements as they flow through the column. The variance, however, says nothing about the relative weight that the various fractions of feed have on the total dispersion. Therefore, the skewness as defined by equation (12), page 20 was introduced. In accordance with the series of mixing stage mechanism, as 2 discussed previously (see page 50), the skewness is equal to -7-, pron vided that the flow through the column is uniformly distributed. It follows, therefore, that the difference between the skewness and twice the square of the variance should be a measure of the degree of departure from uniform flow. Figures 40, 41 and 42 give a graphical representation of the distortion in the response curves by comparing a theoretical curve for which the quantity v-2a is zero with an experimental curve having the same variance but for which v-2ac is increasingly important. The distortion as produced in the experimental curve appears to show that the "younger" elements of gas (i.e., those elements which have been in the system for a time less than the average) replace them

1.0.8 -- Calculated curve,o =.0111, v =.0002 o- -.oExperimental curve,=.0118,v =.0007 o C).6 ().0.70.80.90 1.00 1.10 1.20 1.30 1.40 Dimensionless time t10. 4=, Figure 40. Response Curve Showing Departure from Uniform Flow (v - 264.0003) for Run No. W-8 Air Displacing Tracer.

1.0.8 ~~~~~~~~~~~- Calculated curve,o =.20,v.00 o- -oExper imental curve,oa, 0207, v C01 06 0 00 60 7.80.90 1.00 1.10 1.20 1.30 IAO~~~~ o ~ ~ ~ ~~~~~~Dmesols iet/ Fiue4.Rsos uv hwn eatr rmUiomFo v 2r 01 o u o C8A U~~lcn rcr

1.0 _____ Calculated curve,.0370 v =.0027 o —--- Experimental curve, a=. 0355, v =. 0075.8.6.0\.2 ) 0 o 0_.2 \.o.60.70.80.90 1.00 1.10 1.20 1.30 1.40 1.50 Dimensionless time t/8 Figure 42. Response Curve Showing Departure from Uniform Flow (v - 2a4 =.0050) for Run No. W-19 Tracer Displacing Air.

-141selves at a greater rate, or in other words, have a greater probability of leaving the system than the "older" elements of gas. This, of course, is essentially what happens when we speak of "channeling" or "by-passing". What is generally meant by these terms is that a substantial fraction of the fluid entering the system finds its way out through inner channels or by-passes much before the average residence time has elapsed while the remaining feed is held back and appears at the effluent at a much slower rate. By referring to Table VI, pp. 101 columns 21 and 36 where the quantity v-2c4 is listed, one can make the following observations: 1. The magnitude of this quantity is somewhat erratic. Although this is partly due to the difference of two numbers a4 and v of the same order of magnitude both subject to inaccuracy in their numerical evaluation, it is evident that the departure from uniform flow may fluctuate greatly for any one run as evidences by large variations in the value of v-2t4 when air displaces helium from corresponding values when helium displaces air. 2. The departure from uniform flow appears to be most affected by increase in liquid flow rates. 3. Gas flow rates appear also to have an effect but while this is not detectable for liquid loadings below 3,000 lb/hr.sq.ft. it becomes more pronounced above this flow rate. 4. The departure from uniform flow is greatest for Run No. W-19 where flooding was observed.

-142The above qualitative information gives some indication that the distortion in the response curve is perhaps due to the random occlusion of the gas flow channels by the liquid resulting in maldistribution of the gas through the bed. Consistent with this hypothesis, the departure from uniform flow would increase with liquid flow rates until a point is reached in which the liquid forms a continuous phase throughout the bed (flooding). When this occurs, the gas is compelled to lift the liquid from the pores in a pulsating manner and will proceed through the column in the form of bubbles rather than a continuous stream. Consequently, uniformity of gas distribution can hardly be expected. The skewness of the response curve and the departure from uniform flow as measured by the difference v-2a have only been introduced here to give a qualitative indication of the nature of the flow taking place in the bed. Having done so, we shall continue to use the variance of the distribution of residence-times curve as the only practical measure of the mixing process in the discussion that follows. End Effects The variance of the residence-times distribution was calculated from the recorded response curves as outlined on page 104. Because of the presence of end effects due to a section of dry packing at the upper extremity of the column and of an empty section at the bottom, the calculated variance was corrected as discussed on page according to equation (60). The significance of the correction as applied to the variance is best understood when data on beds of different depths are compared.

-1143 - This is done in Table XIV where data obtained from 1/4" Raschig ring beds having irrigated depths of 3 and 4 feet are compared under similar operating conditions. For each run, the height of a perfectly mixed stage as calculated from the variance before and after correction is reported. It may be readily seen that a more satisfactory agreement is obtained after the correction is applied. On this basis, the end effects are adequately accounted for and the bed depth is no longer considered as a pertinent variable. Axial Mixing Correlation In a manner analogous to the previous correlation for mixing of the gas through the dry packing, we choose to express the axial mixing in terms of the Peclet number defined as the ratio of twice the nominal particle diameter to the height of a mixed stage. Having concluded from the previous discussion that the height of a mixed stage is independent of bed depth, the remaining variables, the effects of which need to be considered in the correlation of the experimental data, are liquid and gas flow rates and particle diameter. Gas and liquid flow rates are expressed as before, in terms of the respective Reynolds number and particle-to-bed-diameter ratio. The data in question are shown in Figure 43 where the Peclet number is plotted versus the Reynolds number of the liquid at constant gas rate for all three different packing sizes investigated. The lines drawn through the points are calculated by the least squares method

TABLE XIV EFFECT OF IRRIGATED BED DEPTH ON LONGITUDINAL MIXING AIR DISPLACING TRACER TRACER DISPLACING AIR,O to0 O 0 0 cd cd 0 c0 I I 1 I 1-) 1 I, I -) I ) ~ 0 0 ~-.. 0 I 0 ~2 a) ~2 Cl C) D 0 CD C. oO cii C 0 cii C| O ci h H 0 2 * d rd ~2 o ~ e0 ~ H 0 r o C +), d 04 d 4- 3 |; hh a, t *Hi *H XU k o ci $- h; CH Pi h | p C ~ ~2p > d ci rHd c......~c. Q0'., > 8 J) w-4 1 c/4 H3 0 446 23.6 ~. 00793d.0238.0084.026 20.2.00803.024 ) I H~ *r 0 "C) rd a)O r) H 0 "C) rd W-1 1/4 3 1600 24.0 1144.0343.01383.045 24.6.01199.360 x.0146.04400 W-24 1/4 4 160 0 24.2. 086.358.01038.045 19.3.00950.380.0113.0445 W-2 1/4 3 3340 24.0 1330 0399.01670.05 1 24.6.01324 0397.0165.048 0 ~2cd H 0 C) bflu ti u od. O t H Oi *H.x *Hii' CC'J 0 E-4 r\ 04 0 u bC 00 w-4 1/4 3 446 23.6.00793.0238.00854.0256 20.2.00803.0241.00869.0261 W-23 1/4 4 447 24.2.00612.0245.00650.0260 19.3.00620.0248.00659.0264 w-' 1/4 3 i600 24.0.01144.0343.01383.0415 24.6.01199.0360.o1466.0440 W-24 1/4 4 i600 24.2.00896.0358.01038.0415 19.3.00950.0380.01113.0445 W-2 1/4 3 3340 24.0.01330.0399.01670.0501 24.6.01324.0397.01659.0498 W-25 1/4 4 3340 24.2.01133.0453.01372.0549 19.3.00999.0400.01179.0472 W-3 1/4 3 5070 24.0.01625.0488.02156.0647 24.6.01581.04743.02080.0624 W-26 1/4 4 5070 24.2.01357.0543.01701.0680 19.3.01193.0477.01462.0585

1/2 " RASCHIG RINGS LO ---- - ----- ~~~18.6-24.2 GAS RAT 1 16GSRT.6 0 18.6-24.2 lb/hr sq. t GAS R E 13 113- 126 t.2 3/8"1 RASCHIG RINGS 0 j.8 W..6 W L Q _.4 __ _ _ __ _ _ III1- 133 GAS RATEI 0 18.6 -24.2 lb/hr sq. I GAS RAFE 0 III- 133 i" I~r II1~n 1/4" RASCHIG RINGS 19.2-24.6 GAS RATE C 3.00 IRATE _ to.8 8 "Ina 2. 6- 8 GAS -RI XT.6.4 _ _ _ _ _ _ _ _ _ _ _ _ _ 167- 198 G 4S RATE __ _ _ __ _ ___ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 LIQUID REYNOLDS NO. Wd/w FIGURE 43. EFFECT OF LIQUID FLOW RATE ON PECLET NO. FOR IRRIGATED BED. (PECLET NO. vs LIQUID REYNOLDS NO.)

-146 - according to the equation: (68) PP 1-NRew Pew OPewo where Pew is the longitudinal Peclet number for the irrigated packing. The constants Pewo and N were evaluated separately for nine groups of runs during each of which the bed properties, i.e., particle diameter and bed porosity were the same and the gas flow rate varied over a narrow range so that its effect could be assumed negligible. Table XV gives a list of the constants and the groups of runs for which they were evaluated. From an inspection of Figure 43, it appears that due to the scatter of the data, the slope of the least squares line, i.e., the constant N in Table XV can be considered independent of particle diameter. A statistical test of this hypothesis shows that the average value of N for all runs having the same particle size is no different than the least squares value for individual groups of runs reported on Table XV. Figure 44 shows the effect of particle size on the constant N of equation (68). A similar test indicates that the intercept Pewo of the Peclet number vs. Reynolds number plot can be considered a function of gas Reynolds number only. Figure 45 gives the relationship between Pewo and the gas Reynolds number. From the foregoing assumptions, the data can be represented

-147TABLE XV COEFFICIENTS FOR MIXING CORRELATION c P tI 0 Z | c; | COEFFICIENTS RUN NO. | H ) ~,'~ b4 hO 5 FW P Po 95% Con- N 95% Cono.,-4 Cc 0~ - o o fidence fidence,,z ^ | ); D H, 1 < oInterval Interval W-1 through 1/4 0-6190 19.2 - 24.0 10.8 14 1.528 +.030.0069 +.0013 W-7 W-23 through 1/4 447-5070 19.3 - 24.2 10.4 8 1.500 +.o64.0079 +.00oo38 W-26 W-8, W-11 W-14,W-17 1/4 446-6190 42.6 - 50.8 22.2 10 1.334 +.o038.0068 +.0017 and W-20 W-9,W-12 W-15,W-18 1/4 0-6190 101 - 133 58.2 12 1.072 +.035.0067 +.0014 W-21,W-22 W-10, W-13 W-16,W-19 1/4 446-5050 167- 198 87.7 8.895 +.053.0079 +.0027 W-27 through 3/8 0_6200 18.6 - 24.2 15.3 8 1.265 +.052.0053 +.0009 W-30 W-31 through 3/8 0-6200 111 - 133 87.9 8.979 +.045.0041 +.0013 W-34 w-35 through 1/2 0-6200 18.6 - 24.2 20.5 8 1.254 +.021.0020 +.0004 w-38 w-38 through 1/2 o-6200 113 - 126 117.5 8.928 +.060.0014 +.0012 W-42

.0008.0006 ~ (t).0004- ~ X c.0002. 0.0000 I 0.20 0.30 0.40 0.50 d, Particle Diameter, Inches Figure 44. Effect of Particle Diameter on Longitudinal Mixing.

CO o 2.0 c 0 o 1/4" Raschig Rings._ A~~ 3/8" Raschig Rings C 0 1/2" Raschig Rings 0.9 0.6 C@.5 C ~ 4 5 6 7 8 9 10 15 20 25 30 40 50 60 70 80 90 100 150 200 250 Gos Reynolds No; Gd/A Figure 45. Effect of Gas Flow Rate on Longitudinal ixing. Figure 45. Effect of Gas Flow Rate on Longitudinal Mixing.

-150by the equation: -.20 -(.013 -.088 d )Re (69) Pe = 2. 4 (ReG) 10 dt w Equation (67) can also be written in terms of the height of perfectly mixed stages: 3d (Re 20 10 (.013 -.088 A. )ReW (69a) ~ ~ =.83d (ReG) dt w Interpretation of Mixing Data If one uses the series of perfectly mixed stages model to interpret the results pertaining to the mixing taking place in the bed, one observes that the height of a mixed stage is smaller for dry packing than it is for wet packing. Since the height of a mixed stage is, as seen in the dry packing correlation, proportional to the particle diameter, one may postulate that as the liquid descends on the packing, it envelops several packing elements to form larger aggregates, thus

-151increasing the apparent diameter Df the particles or in an equivalent manner, increasing the height _f a xiixed stage H. Consistent with this interpretation, equation (69a) shows that H increases with liquid flow rate. An idea of the validity range of equation (69) is given by Figure 46 where the average Peclet number for dry Raschig rings is compared with data on irrigated packing. The line for wet packing calculated from equation (69) for a liquid flow rate equal to zero is representative of all three packing sizes used in the experiments. It is apparent that equation (69) cannot be used to extrapolate mixing data for Reynolds number much less than ten since from what we remarked before, one would expect the Peclet number for wet packing to remain always below that for dry packing. The lower line shows 1/4" packing data for a liquid flow rate from 5050 to 5070 lb/hr.sq.ft. The points on the extreme right of this line are for Run W-19 which was conducted under flooding conditions. One may infer, therefore, that in the flooding region, equation (69) yields higher values of the Peclet number than those observed. A more interesting but highly speculative interpretation of the Peclet number may be given by assuming that the ratio of hydraulic mean diameter to height of a mixed stage is a constant. As seen from Table X page 123, this is quite true generally for dry packing. We may assume, therefore: 2 2 w Constant Pe Pe" Constant L w H H_

395% CONFIDENCE INTERVAL FOR DATA ON GAS FLOWING THROUGH DRY RASCHIG RINGS 2 _ _ _ \\ _ __X_ _ _ _ _ __\\\\\\\_ __\X_ _\_\\\\\\\\\ \\\\\\\\___\ _\\\\\\\\\\\\\\\\' 0 -I|1/4, 3/8,~ 1/2.,; RASCHIG RINGS THOROUGHLY WET, Z.9 Em.8.7 06 ~ o1/4" RASCHIG RINGS, LIQUID RATE.6: 5050 LB/HR. SQ.FT. -..5 c'J o.4 0 Z - II -.3IJ FLOODING a. 00.2 0 1/4' RASCHIG RINGS A 3/8 RASCHIG RINGS O 1/2" RASCHIG RINGS.1 I A i I I I111 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 8090100 200 300 400 GAS REYNOLDS NO. Gd//G Figure 46. Longitudinal Mixing for Gas Flowing through Irrigated Beds of Raschig Rings.

-153 - Accordingly, by dividing equation (69a) by H as given by equation (63a) pp. 119we find that: 0.20 (0.013 - o.o88 d ) Rew Hw _w = X a = 0.81 (ReG) 10 dt(70) H c aw so that the ratio of total wet packing area to dry area is obtained by dividing equation (67) pp. 128 by (70): - [0.013 - 0.088 d -3.43 x 10-6(d) -2.31] Re dt W 8w = 1.11 (ReG) 0.20 10 a (71) Equation (71) although highly speculative, indicates that the wetted area would decrease rapidly with liquid flow rate and at faster rate, in fact, than the porosity of the wet packing. In addition, the rate of decrease in area is much more pronounced for the smaller size packing than it is for larger sizes. The wetted area could also be affected, but to a lesser extent by the gas Reynolds number. This may be due to the fact that higher gas flow rates tend to make the liquid-packing aggregate more compact so that it presents less surface area to gas flow.

-154The cumulative result of these effects is to reduce the wetted surface area in inverse proportion of the particle size so that the area of the packed bed exposed to gas flow may, at high liquid flow rates, be larger for large packing sizes than for the smaller packing. Figure 47 shows the relationship between the total area of the bed in contact with flowing gas as estimated by equation (71) together with estimates of effective mass transfer area /22/, /45/, /62/. The uncertainty of data on effective mass transfer area is illustrated by the low and high values reported for 1/2" Raschig rings. The data of Mayo, Hunter and Nash /45/ obtained by measuring the colored surface area of Raschig rings made of paper over which water containing a red dye was circulated appear in line with the total surface area as given by equation (71). In fact, one would expect the mass transfer area to be only a fraction of the total surface area exposed to gas flow and that the former increases with increasing liquid flow rates so that the two quantities approach each other asymptotically. It is possible that the area which the bed presents to liquid flow is more directly related to the effective mass transfer area. If the reasoning followed in the derivation of equation (71) is valid, (this could be verified by obtaining mixing data on other packing shapes) then the surface area which the bed presents to gas flow could be evaluated by a study analogous to the present one for the liquid phase. Comparison With Previous Investigators In connection with the previous discussion, it is interesting to compare the present data on the gas phase mixing with the results reported in graphical form by Kramers and Alberda /38/ on a number of

-155 - 200 - 8 EQ. 71 100 LL 80 -- w, -p~~- _TOTAL BED AREA EXPOSED TO FLOWING U 6 GAS ESTIMATED FROM EQUATION 71 4g~~~ ~EFFECTIVE MASS TRANSFER AREA 4 —-- ESTIMATED FROM MASS TRANSFER MEASUREMENTS /22/ AND /62/ AVERAGE VALUE OF EFFECTIVE MASS ----- TRANSFER AREA AS ESTIMATED 2 FROM DIRECT MEASUREMENTS /45/ O 1000 2000 3000 4000 5000 6000 LIQUID SUPERFICIAL MASS RATE Ib/hr sq.ft. Figure 47. Total Bed Area Exposed to Flowing Gas and Effective Mass Transfer Area as Function of Liquid Flow Rate.

-156 - experiments in which the mixing of water running over 10mm. Raschig rings in an absorption column with a countercurrent air flow was measured by frequency response. Figure 48 shows the Peclet number for the gas as measured in the present work for 3/8" rings (closest dimension to 10mm) with analogous Peclet numbers for the liquid as estimated roughly from the graphical data reported by the above authors. It is apparent from this Figure that while mixing in the gas phase increases both with liquid and gas rates, in the liquid phase instead, mixing decreases with increasing liquid rates. The results on the liquid phase have to be accepted with some reservation because of their scarcity and because Kramers and Alberda do not give details about the experimental technique used. However, they seem to show that according to the discussion of the previous paragraph, the area which the bed presents to the flow of liquid does, indeed, behave in a similar manner to the effective mass transfer area. Another interesting aspect of this comparison is that according to Figure 48, the Peclet number and, thus the height of the mixing stage for both phases is of the same order of magnitude in the practical range of liquid flow rates. This fact may have some importance in greatly simplifying the interpretation of mass transfer data as we briefly discuss in the next section. Lapidus' data on countercurrent flow of liquid and gas over spheres and porous pellets /39/ is not sufficient to be used for comparisons of the sort shown previously.

3 GAS PHASE MIXING 0 18.6-24.2 Ib/hr.sq. ft.,GAS RATE. 3/e RASCHIG RINGS A III - 133 "',,, it,, 2p LIQUID PHASE X ESTIMATED RANGE OF DATA BY KRAMERS E ALBERDA /38/ MIXING IOmm I HIGH POINT REFERS TO GAS RATE=O Ib/hr.sq.ft. RASCHIG RINGS L A LOW POINT REFERS TO GAS RATE: 236- 280Ib/hrsq. ftt. I ~ 18.6-24.2 3t O.Q GAS RATE-X._ 2, 0. N 0.7 | 236-280, GAS RATEo 3 0.1 0 1000 2000 3000 4000 5000 6000 7000 0 I 000 2000 3000 4000 5000 6000 7000 Figure 48. SUPERFICIAL LIQUID FLOW RATE, LB/HR. SQ. FT. Peclet Number for the Liquid and Gas Phases Flowing Counter Currently through a Packed Bed of Raschig Rings.

-158Significance Of Time Distribution Data As Applied To Mass Transfer It is interesting to speculate on the significance of the mixing stage concept in interpreting the exchange of mass taking place between the liquid and gas stream in the packed absorber. If the gas and liquid phase effectively undergo a series of complete mixing through discrete intervals of bed (a perfectly mixed stage), the rate of renewal of the fluid through the stage might have some relationship with the rate of surface renewal as visualized in the modern penetration theory of mass transfer. The penetration theory originally conceived by Higbie /32/ and later developed by Danckwerts /11/, /14/, supposes that turbulence generated in both flowing streams extends to the mass transfer surface so that there is a constant renewal of both surfaces. Consequently, the rate of absorption in each phase would depend upon the time distribution of the elements of surface. Depending on what distribution function one assumes for the liquid and gas surface, one obtains various expressions for the liquid and gas film transfer coefficients respectively. Danckwerts suggests that the time distribution function for elements of surface is of the form: E (t) = te-*t where V is the fractional rate of surface renewal. This equation is identical in form to the rate of change of concentration in a perfectly mixed stage where * is now the rate of renewal of the fluid contained in the stage V' Hanratty /31/ in a recent study of mass and momentum transfer of fluid with a solid boundary found that an analogous time distribution for the elements of fluid in contact with the wall

-159permitted an accurate prediction of the concentration profile in the fluid. Danckwerts has also pointed out/ll/ that mass transfer data from actual packed beds of Raschig rings may be rationalized by supposing that the time distribution of the elements of surface area is distorted to favor relatively higher rates of replacement of the "younger" elements of surface. A phenomenon of this kind (see page 134) has actually been observed in the course of this research. It is possible, therefore, that the residence-time distribution data of the kind presented in this work may be of value in the interpretation of mass transfer data. In this case, and in the event that the penetration theory gives a sufficiently accurate representation of the mass transfer process in packed beds, simple measurements of mass transfer and chemical reaction rates as occurring through a stagnant surface and distribution of residence-time data would be all that is required in the prediction of gas and liquid film coefficients.

SUMMARY 1. The porosity of the irrigated bed for which no other method of measurement is presently known is calculated from a knowledge of the mean residence-time of the gas and the corresponding volumetric flow rate. For flow conditions up to loading, the porosity of the irrigated bed is correlated by equation (67) PP. 128 For liquid loadings above 5000 lb/hr. sq. ft. the experimental response curves can no longer be satisfactorily correlated by assuming uniform flow through the column. This condition is apparently brought about by "channeling" or "by-passing" of the gas through the irrigated bed. While the variance is still the most.ppropriate measure of the extent of mixing, the difference between the skewness and twice the square of the variance gives a measure of the deviation from uniform flow. 3. The presence of an empty portion of column at the entrance of the bed and of a dry packing portion above the liquid distributor is accounted for by a correction analogous to the one used for dry packing. On this basis the depth of the packed bed does not affect the height of a perfectly mixed stage. 4. The Laxial mixing through the bed as expressed by the Peclet number is dependent on liquid and gas flow rates as Well as particle diameter as expressed by equation (69) pp. 147 -160

5. Since the height of a mixed stage (see eq. (69a) pp. 147) increases markedly with liquid flow rates, one may postulate that "s the liquid descends on the packing, it envelops several packing elements to form li. rger a-nd larger obstacles to the flow of gas. 6. If one assumes that the ratio of hydraulic mean diameter to the height of a mixed stage is constant, one may obtain information on the area of the wet packing exposed to the gags flow as given by eq. (71) pp. 150. 7. Comparison of the present data with results reported by Kramers and Alberda/38/ shovs that while mixing in the gas phase increases both with liquid and gas rates, in the liquid phase insteld, mixing decreases with increasing liquid ra.tes. In the practical range of flow rates the height of a mixed stage for both phases is of the same order of magnitude. 8. It is possible that residence-time distribution data of the kind presented in this,ork may be of value in the Lapplication of the Higbie - Danckwerts penetration theory of mass transfer.

Re comendat ions In the previous discussion we have seen that there is an appreciable amount of mixing occurring in the longitudinal direction for the gas phase flowing through irrigated packing. This may have considerable effect on processes of simultaneous absorption and chemical reaction especially for the final exposure of the gas where presumably the conversion for the gas phase is fairly large. In addition, the large dispersion and the poor distribution of the gas in the loading and flooding region as revealed by the experimental response curves are perhaps the major factors responsible for the poor performance of packed absorbers in this operating range. The brief account given of the consequences which are derivable from a knowledge about residence-times distribution of the flowing streams has served to point out the desirability of an extensive study of the liquid running over the packing. The existing evidence seems to show that the extent of mixing in both phases is of the same order of magnitude. Since minor changes in the magnitude of the mixing parameter (i.e., variance, height of mixing stage or diffusivity) do not greatly affect the overall behavior of the system, one may hope to simply apply the penetration theory of mass transfer to a system as complex as a packed absorber. Moreover, the mixing in the liquid phase appears to increase with decreasing flow rates. This is rather interesting since, accordingly, it would imply a restricted flow region for most efficient operation of an absorber.

.164Lastly, we wish to recall that the characterization of the residence-times distribution by means of the moments of the distribution is believed to be most general since it is not restricted to any one specific correlation model. While the computation of these moments from experimental concentration-time records is rather tedious since it involves numerical methods of integration, we strongly suggest that automatic methods of computation be applied to the output of the continuous analyzer. If this suggestion is workable, it would enable the experimenter to analyze a large number of data and to obtain all the desired moments of the distribution with the same uniform degree of accuracy. It is hoped that the present work has furnished the student of "rate processes" a glimpse of the numerous possibilities offered by transient response as a method of chemical engineering experimentation, and that the increased interest in this field may serve in the long range to adequately resolve scale-up and automation problems in the chemical industry.

Conclusions By the use of unsteady state experiments, the present research has resulted in the evaluation of gas holdup and distribution of residencetimes spent by the elements of the gas stream flowing through an irrigated bed of Raschig rings. Dry packing experiments served to verify that the mean residencetime, as obtained from recorded concentration-time response curves can be used to calculate the volume of the bed occupied by the gas. On this basis, the porosity of' the irrigated bed for which no other method of measurement is presently known is calculated from a knowledge of the mean residencetime of the gas and the corresponding volumetric flow rate. For flow conditions up to leading, the porosity of the irrigated bed is correlated by equation (67), pp 128. The variance of the residence-time distribution was used as a measure of the logitudinal mixing of the gas phase flowing through the bed. This quantity obtained from the recorded response curves for dry packing was compared with various mixing models. The relationship between the variance a2, the number of perfectly mixed stages n and the effective diffusivity coefficient DL is:.2 =,1 uL n 2DL Two bed depths were studied to evaluate the magnitude of end effects on the overall response. On the basis of the series of perfectly mixed state model, an adequate correction for end effects was obtained. For a dry bed of Raschig rings, the ratio of the height of' a mixed stage to the nominal packing diameter is essentially a constant at least in the investigated range of gas flow rates. For irrigated packing, the height of' a mixed stage is both affected by liquid and gas flow as expressed by equation.(69a) pp. 147. -165

I

-167APPENDIX I Solution "A" The problem in terms of longitudinal diffusion and superimposed plug flow is given by eq. (21) pp. 34 which is transcribed here: at'x ax The boundary conditions for a step input in concentration from 0 to co for a packed bed of infinite extent assuming continuity of fluid concentration at the inlet are (see Chapter III pp. 34-38 ): (i2a) C f m o (3a) C( )) =o taking the Laplace transform of eq. (21) (1-AI) SC.S Ct7C DO)DCAC X,5)_ U XS)

-168and the boundary conditions become: l-al) C (oS)= Co 3-a') C(~, ) = O applying 3-a') to eq. (1-AI) we get: ICsr TAXIS- _ c (JS)= (2-AI) Eq. (2-AI) is an ordinary differential equation, the general solution of which is: C(~,)= $ " +e <_(3-AI) where cp and X are arbitrary functions of s since c (x,s) is bounded,(condition 2-a') cP must be zero also to satisfy condition 1-a' ): At.. Co (4-AI)

-169 - thus, the solution of equation (2-AI) is: co e D )D C. ) = - C e~ (5-AI) to find the inverse transform we write eq. (5-AI) in the form: =k 1 e.e V(6-AI) Ca, S the inverse transforms are: A'M 2 1e 2 2D, (7-AIT) -AS u~ Wi_,.t Uz Wt * (8-AI) * This latter transform is obtained from /6/, pp. 299 eq. (82)

-170Thus, by the convolution theorem (see /6/ pp. 36 ): C. _ Co:Z tVt us uzt 16L =.~q_ i tA t (9-AI) In order to integrate eq. (9-AI) above, we write it in the form: MOut us, +e \t t t ttg, Now we let: a ast - - and_ OX 2- 1A_ at

-171thus we can write eq. (10-AI) in the form: zeS c2c ( iP -i 2 0tC ZP) (l-AI) and substituting back the values of UC and f we get: C 2 [Fzz W' J)s2 t) (12-AI) We are interested in the response of the system at the outlet, i.e., x = L thus, equation (12-AI) becomes: C I Ir(L. t _____ [ (\2Yv/ WLD (13-AI) The response to an impulse signal or the distribution of residence-times function at the outlet is: _- L- 4DNt 4t. 2\lGrt3 (14-AI)

-172We thus proceed in evaluating the zero and first moment of E(t) and the central second moment or the variance of @E(t) vs. t/o curve. For the zero moment we have: _(L-ut) uL _ Lt E (t)lJt)t Let e t O V J YPz, OYD~t4 (15-AI) We may evaluate the integral on the right hand side of equation (15-AI) by regarding it as a Laplace transform in which the parameter s is equal to u Accordingly we write formally: ~L -LZ _ L LI L 5f By substituting this relation in equation (16-AI): | IECtt) t = e 2~~7rP~~tt c(17-AI) Frmatal ftrnfrs ~ p ~9e.(8)w i0 ht

-173 - and substituting for s the value u2 we obtain: 4DL E(t)t) (18-AI) as required for the zero moment of the residence-times distribution (see eq. (3) pp. 13 ). The first moment of the distribution function is according to the definition given on page 14: ~tE~t)dt 4|\,e.n t (19-AI) Here again we may evaluate the integral by use of the Laplace transform. Since multiplication of a function by t corresponds to differentiating and changing the sign of the transform of said function we obtain from equations (16-AI) and (17-AI):,L _13 L -uL e4 $ t IDL 3 LCt)( t) l. ) e (20-AI) and by carrying out the differentiation with respect to s: uL _L tECt 2t =\.( L) (21-AI) 0tE~t~dt R

-174u2 Hence, by substituting s = bDL E.t) -t = (22-AI) Since L/u is equal to the true mean residence-time equation (22-AI) satisfies the requirements of an actual system (see eq. (r) pp. 14, and eq. (6) pp. 14) The variance of the dimensionless distribution GE(t) vs. t/O is according to equation (10) pp. 19: (L-u 40 e = | (to aElt)82tl6) - l -| 052 It a 4A co 40,t We let s = u2 and as before we use the Laplace transform to evaluate the integral on the right hand side of equation (23-AI): ~L uL uL ~ Pt~t c L 0 tD.L?. L.Te 4.. (24-AI) LZ L2.'~1~"~1 I Dtd

-175 - by evaluating the differential, we get: UL LIS 2. D 6 4L. I _- (25-AI) by substituting s = u2 and simplifying, we find that the variance 4DL is: ~~6'z~2 2 1)' Z~~~(26-AI) u L

-176Solution "B" Equation (21) pp. 34 can also be solved for another set of boundary conditions which take into account the flux due to diffusion at the inlet of a semi-infinite bed (see Chapter III pp. 37 to 38 ) These are: l-b) A CO= - t o) t) 9ID Cot) 2-b) iM 3-b) C,4o) A o The Laplace transform of eq. (21) pp. 34 is again: i- c. C, )- C. C S)-XX (1-AI) and the boundary conditions become: 2-b') X- ~oo 3-b') C.. ) = 0 applying condition 3-b') to eq. (1-AI): CX;( R 5 U C X, 7Z5 _5C()9 (2-AI)

-177the general solution for equation (2-AI) is as before: /(SS)=f+ +,e (3-AI) where p and X are arbitrary functions of s. In order that condition 2-b') be satisfied cp must be zero, also to satisfy condition 1-b'): Ucc. (27-AI) Hence, the solution of equation (3-AI) for the new set of boundary conditions is: I. C. \s (U /4+ ) (28-AI) To find the inverse transform, we write eq. (28-AI) in the form: 2L& Z(2DL c~~~~~~~~~wb~~~~~~~ i i

-178where in the second factor on the right hand side we have made the substitutions: d\ ij X' as >E,4DL (30-AI) the inverse transforms are: -4 1 An- ~- L Mal=e tk 2 fx c'tz'\T~~~ -3t[~~~~~~ e Pe er(33-AI) At e CL (32-AT) By the convolution theorem (/6/, pp.36 ) we find that the inverse transform of equation (6-AT) is: C*411 I f'w~... -Tr

-179substituting back the values of a, and 7 and simplifying we find: (X-ut)L Co [ s__ -!413 zrcQC 1 Dt(34-AI) Since we are interested in the resuponse of the system at the outlet, we can rewrite the above equation, substituting x = L: (L-t)z uL C J| [ u _ ___ at [ t.t z (35-AI) the above integral is rather difficult to evaluate analytically. One can resort to the graphical integration of the impulse response function E(t) (L-dt)2 uL Wemywd C to f the e \f tiut3 (36-AI) We may now proceed to find the moments of the distribution of res idence -t imes.

-180For the zero moment we have: L ZQ:\C Et)| _ )7m __c(37-AI) we can rewrite the equation above in the form: ~ Flt~dt-, rr.4tQ 4 /0( s e,__dt -(38-AI) L where now / D and y remaining as defined in (30-AI). We note that the integral on the right side of equation (15-AI) is the Laplace transform of the function enclosed in square brackets having argument 7. Hrence, we may write formally: uL - p1Z 2 Dt. 4t ~ECL)At: = Yk~z 4. tl U (39-AI) Ihe transform can be evaluated Fwith the help of (32-AI), replacing s with:

-181substituting back the values of CZ', 3 and 7 we find: E~t) At 1 (41-AI) as is required for the response to an impulse signal (see eq. (3) pp. 13). For the first moment of the distribution function E(t), we may write: |b a) % |t a t t (42( \) since multiplication of the function by -t corresponds to differentiation of the transform of the function, we may write formally: LO i_ 4t i W uL av'6; (43-AI) _L.

-182 - substituting the values of G', a and y we find: 00 tA L#2.At(44-AI) which indicates that the first moment approximates the residence-time only when DL is small. u2 Since the average residence-time as given by equation (44-AI) is not equal to the true residence-time L/u or which is the same ev/V. The variance may be evaluated by using Eq. (10-a) (footnote on page 19) thus: 00 z ( t2E.t)clt (45-AI) In a manner analogous to the procedure used in obtaining eq. (38-AI) and by substitution of equation (44-AI) in (45-AI) we obtain: uL 64J+2 )t~e ar * (46-AI) L a a & /. I A I.It I f

-183 - By use of the Laplace transform we may write formally: 6|__2:, < 4)2(C0O_ ~ (47-AI) Evaluating the second derivative in gamma, we find: UL 3 32DL- L 2, (49-^I) (49-AI)

The Normal Distribution Considering the impulse response of solution "B":.(t-~)z uL C 4A0,t P_ E(t)- = - 2D~ /d 2 (36-AI) and the asymptotic expansion for large values of the argument of the complementary error function: = ~ (1 2 1.5 (50-AI) we can write equation (36-AI) in the form: Eqt) = a2 jzcct oa at'P E~t)=-^) (51-A) we see that for large values of L + ut and especially for L in the neighborhood ut, that is ut = t = equation (51-AI) simplifies to: CL-. Ut)' Et)= a t _A_dt liro~t (52-AI)

-185This is the normal or Gaussian distribution. It is worth noting that for large values of uL the response to a step function for solution DL "A" reduces to: 4C 2. L'~i (%.ZYJ (53-AI) This expression, the approximate response of a packed bed to a step function, was the result of a simplified derivation by Danckwerts /13/. The derivative of eq. (53-AI) with respect to t or the response to an impulse function for large values of u is: Car it) E~t) -d c, = (54-AI) which again reduces to the normal distribution of equation (52-AI) in the neighborhood of t 1. We should like to find the moments of the time distribution function E(t). Zero Moment: (L-%kit uL U u.t -t4D t 94L. ___t__ dt o PevnslandSmth / in heir dicussionof packd beds(55-AI) * Levenspiel and Smith /42/ in their discussion of packed beds use equation (52-AT) as the response to the impulse function.

-186 - If we let u = s we can evaluate the last integral on the right vDL hand side of eq. (55-AI) by finding the Laplace transform of the function -Le/4DLt this is: 1 et.' i\r~I ~~ ~~~; ~(56-AI) Consequently, eq. (55-AI) becomes: ECt)At = -.) (57-AI) substituting for s the value u we find that: 4DL | Ct) att (58-AI) * This transform is obtained from /6/ pp. 299, eq. (84)

-187First Moment: (L- uL _L 00 0AI 4Dtte 2. 4 1~S ItE t)t U ___ X} e t ~Lt= ~:-e emf~ -l-} uL ~ 3L (59-AI) ZOL __ td _ tJd evaluating the differential: 4 L Lf ttt)dtiLe i o ) (60-Ai) and substituting the value of s = U2 we get: DL Z I Z~~4t L Ot+ (61-AT) i, U~~~

-188 - Variance of the dimensionless time distribution: 400 2(L Ltd)L (62-AI) utilizing again the principle of differentiation of the Laplace trans-s uL Lfi _A) _ -, - e - ---- (63-AI) 4t IT PLt DL hence, eq. (62-AT) becomes: - L2UZ LA LA )

-189by simplifying and substituting s = u2 we obtain the variance: 2. 2 8D I-am, (68D5.4L M L! (65-AI)

I

-191APPENDIX II A Number of Mixed Stages in Series Having Equal Volumes Let us consider a perfectly mixed tank such that the concentration at the outlet is at all times equal to the concentration in the bulk of the tank as shown in Figure 49: V' cu.ftAec, V' cu.ft./sec. Co MOLES/C.ft. C MOLES/CU.ft. VOLUME V Figure 49 A Perfectly Mixed Tank If the concentration at the inlet of the tank is suddenly changed from O to co, we can write that the rate of change of concentration in the bulk of the tank or at the outlet is: _4s d&c. c=a s(1-AII) cit

-192 - Also the boundary condition at the outlet is: Gc= at t=o (2-AII) which expresses the fact that at the outlet the concentration remains O at least up to time t = O. Taking the Laplace transform of (1-AII) we have: -C,Cs)- C.o)= r C, aa Ot,(5) (3-AII) II, since c(o) 0 by (2-AII) we can solve for c(s): (4-AII) the inverse transform of which is: Cz-CO(_ z2 ) (5-AII) * Note that in this limiting case the distribution function is: ct thus it is an exponential curve decaying at the rate v'/V' or 1/0.

-193Suppose now that the outlet stream from this tank is admitted to a second tank of equal volume as shown in the Figure below: V' cu.ft/sec. V' cu.ft/sec. V' cuft/sec. Co MOLES/Cuft. C, MOLESkuft C2 MOLES/Cu.ft. =tl~~ ~t"2 VOLUME VI VOLUME V' Figure 50 Two Perfectly Mixed Tanks in Series The rate of change in concentration for tank No. 2 is: _' V _ ='cz _ 4'c (6-AII) with the boundary condition: CX= o at (7A=o (7-^zz)

-194Taking the Laplace transform of (6-AII) we find: s~S, C _ C2 ) _ - (5). Ct.i ) (8-AII) Since by eq. (7-AII) c2(0) = 0 we can solve for c2(s): C. CS);L Z (Q-AII) but cl(s) is given by eq. (4-AII) which substituted in (9-AII) gives: (C? (___ (10-AII) For the nth tank we have: V C1 4C c-C,,tl (11-AII) d t hence, the Laplace transform of the concentration at the outlet of the nth tank can be written immediately by analogy with eq. (10-AII): C s> (r)\ ) C (12-AII) s(~+s+t >

-195The inverse transform of eq. (12-AII) can be found with the aid of the convolution integral from the following inverse transforms: 1A 1 Y'- l Ai t t (13-AII) Hence, the inverse transform of (12-AII) is: V' _ CO ( A> t-''1c II )! t } (it (14-AII) Integration of equation (14-AII) for any finite number of stages n greater than 1 yields the equation: (15-AII) This form of the response of a number of mixers in series has already been reported by McMullin and Weber /48/j; however, eq. (14-AII) is more

-196general since n need not'be restricted to an integral number greater than one but may be any real positive number. The time distribution function for this model is given directly by differentiation of eq. (14-AII): t dE~t)- 1o N (r)(16-AII) This equation written in the dimensionless form: ut Ai,~E~t= <sta = 0% eat) (17-AII) V' is the well known Poisson probability distribution. Many of the results derived on the following pages can be derived directly from probabilistic considerations, see Feller /21/. The zero moment of the time distribution function is obtained by integration of eq. (16-AII) between the limits of zero to infinity: t (18-A

-197By letting v = s we see that the right hand side is the Laplace VI transform of the function tn-1 Thus we have immediately: (n-l),. 5:~ ct) at ( r 4l) =1' l (19-AII) The first moment of the distribution can be found in an analogous manner: 00 t IA t t=' )O ("H e)!V'} sawn: Ad (20-AII) This is, therefore, the overall residence-time 0 through the series of n stages. The variance of the distribution is: (,t ),2 I (ttt,5,,I r (t - q\ Ot (21-AII) ~ ~= _ Zr~l \1 00 Again with the aid of the Laplace transform, we have: d 1N i/\ VIY1)

-198 - Finally, we may also find the skewness of the distribution function by writing according to Eq. (12) pp. 20: 1)_ ( ) BE lt)( )3 + 2t E (tt +2 ) v 3 )~ ~oot (v ) i (23-AII) we can then write the integrals in Eq. (23-AII) in terms of Laplace transforms: I)1 —~~~~~ _(. ) -(s ) nZ(n" i ) A (24-AII) substituting s L and simplifying we find: VI (-AII) Vt"

-199A Number of Mixed Stages in Series Having Unequal Volumes Let us suppose that we have nl perfectly mixed stages, each having a volume V1' and n2 stages, each having volume V2' connected in series as shown in Figure 51. "I #n, I n,-,- n nn2 v' cu.ft/sec v v V, Cu.ftec V V V2 Figure 51 Series of Perfectly Mixed Tanks Having Unequal Volumes Then if each stage i* perfectly mixed, we can immediately write the Laplace transform of the concentration of the stream emerging at the end of the series of tanks by comparison with eq. (12-AII) pp. 187:,') s(26-AII) \VI'~ ~~ + )V' (; 2.'~

-200the inverse transforms are: (27-AII) v., by the convolution theorem (see ref. /6/ pp. 36) we see that: VIb I)" ~~c = t. )' 3 dZ7 2 (28-AII) Therefore, the response to a step function in concentration for a series of stages having different volumes V1' and V2' is: C FI(wY An'r") J (a1t~X't iA(2 All)

-201We find the response to the impulse input by differentiating the equation above with respect to time: (n1A" - _Ntt)d Al Ctt)(30-AII) We can now calculate the zero moment of the time distribution E(t): (31-AII) ( 1 12V ck~t [, By use of the Laplace transform and by letting s = v' and s2 v' V1I V2' we may write the above equation in the form: E E~rl - (V7 5A, xV7) j($2 ) (. ^ A(32-AII) By substituting back the values of sl and s2 we get: $.t-) o I\ (33-AII) 0,

-202The overall contact time through the series of tanks is obtained from the first moment of the time distribution function: 0E a t t(34-AII) making use of the fact that multiplication of the function by -t is equivalent to differentiating the corresponding transform with respect to s we may write formally: &~ E)(t t -— + [ r lti l (t) (Fiji;) >; ] (35-AII) v:)'v A IsA - 2~~~~~~,

-203 - Substituting the values of s1 and s2 and simplifying, we get: S t Ett)J~+ "~I= a + tant (36-AII) thus, the overall contact time is the sum of the contact time for each stage as would be expected. We can thus write the general result: (37-AII) Passing to the variance of the time distribution function: 00 (38-AII) =M Nf 2 ( ( =~,~V,',~/ t4,,-).',,,.

-204Again, with the use of the Laplace transform and the chain rule for differentiation we may write eq. (38-AII) in the form: (39-AII) _',''o l 1 A At \ hence, by evaluation of the differentials in sl and s2: 2 /< f >"|1 n, \nz nlh,(fI) v 2n" + flLt ) 1) (40-AII) 81 +VI, J2 419 l1n'*t54t j +Z2 substituting for sl and s2, vT and v' respectively, and simpliV' V2' fying: 6~ (a )(41-AII) (+ VI -6?t' 3'~~~~~~~%

-205since:.tV, t- ~ and 4W iT' W we may write eq. (41-AII) in the form: 2 -_. _ _ _ (42-AII) F9Z Generalizing these results for a large number of unequal mixing stages, we have: 6Z - ffi ( (43-AII) LZ= L g-_p

-206 - APPENDIX III Typical Irrigated Packing Experiment For experimental record, see Figure 27, pp. 92 a) Bed Properties 1. Average I.D. of column, dt.334 ft. 2. Average cross-sectional area, d =.0874 sq.ft. 3. Volume of Empty Column Volume of Bottom Section, VE.0065 cu.ft. Empty Volume of Mid-Section, Vw =.2590 cu.ft. Empty Volume of Top Section (above liquid distributor), VD =.0437 cu.ft. Total Empty Volume, VE + Vw + VD.3092 cu.ft. 4. Free Volume of Packed Column Volume of Bottom Section, VE =.0065 cu.ft. Volume of Packed Mid-Section (Wetted), eVw.693 x.2590 =.180 cu.ft. Volume of Top Section, eVD.0303 cu.ft. Total Free Volume, VE + EVw + eVD.217 cu.ft. Volume Not Wetted, VE + eVD.oo65 +.0304.0369 cu.ft. 5. Nominal Diameter of Raschig Rings.0208 cu.ft. 6. Surface Area of One Raschig Ring Outer Cylindrical Surface Area: i x 6.5 mm. x 6mm. = 122.6 mm2 Inner Cylindrical Surface Area: 2 t x 4.5mm. x 6mm. - 84.9 mm Bottom Ring Surface: T x (6.52 x 4.52) mm2 - 17.3 mm2 Top Ring Surface' n x (6.52 x 4.52) mm2 = 17.3 mm2 Total Surface of Particle = 242.1 mm2 or.375 sq.in.

-207a) Bed Properties (conttd) 7. Total Surface Area of Packing Total Weight of Raschig Rings, Loaded to Column 6380 gr. Average Weight of 4 Samples of 200 Rings Each 59.4 gr. Total Number of Rings _ 21 500 Total Surface Area Per Unit Volume of Bed: a = 21,500 x.375 sq. in. 187 sq.ft./cu.ft. 144 sq.in./sq.ft. x.3027 —cu.ft. 8. Hydraulic Mean Diameter: d =.693 a i87 sq.ft./cu.ft. b) Air Properties 1. Average Temperature of All Flowing Streams is taken to nearest degree 88~F 2. Air Flow Rate: (from No. 3 Rotameter calibration, Figure 56, pp. 218 ) - 1.01 cu.ft./min. at 600F and 14.7 psia 3. True Air Flow Rate: 1/2 v = 1.01 cu.ft./min (520 x 14.9 =.0165 cu.ft./sec. 60 min/sec \548 x 14.7 4. Air Density: P = 29 lb./mole 492 x 14.9.0735 lb/cu.ft. 359 cu.ft./mole K548 x 14' 7 5. Superficial Mass Flow Rate: G =.0735 lb./cu.ft. x.0165 cu.ft./sec. x 3600 sec./hr. = 50.0 lb/hr..0874 sq.ft. sq.ft. 6. Viscosity: (From Figure 58, pp.220 at 88~F), G =.0441 lb/ft.hr.

-208b) Air Properties (cont'd) 7. Reynolds No.: Re = Gd = 50.0 lb.Ar.sq.ft. x.0208 ft. = 23.6 G G.0441 lb/ft.hr. c) Tracer Properties 1. Average Temperature 88~F 2. Tracer Flow Rate: (From No. 3 Rotameter calibration, Figure 56, pp. 218 ).98 cu.ft/min at 60~F and 14.7 psia 3. True Tracer Flow Rate: 2 v.98 cu.ft./min. 20 x 14.9 x 29 =.0175 60 min.sec. 548 14.) cu.ft./sec. 4. Tracer Density: G = 24.4 lb/mole (492 x 14.9 =.0619 359 cu.ft./mole 5 7 lb/cu.ft. 5. Superficial Mass Flow Rate: G -.0619 lb/cu.ft. x.0175 cu.ft./sec. x 3600 sec./hr. = 44.5.0874 sq. ft. lb/hr.sq.ft. 6. Viscosity: (From Figure 58, pp. 220 at 880F),,G =.0439 lb/ft. hr. 7. Reynolds No.: ReG = Gd = 44.5 lb./hr.sq.ft. x.0208 ft. = 21.1 G 1uG * ~.0439 d) Response Curve, Air Displacing Tracer 1. Residence-Time As defined in page 99, the residence-time of the gas phase is evaluated by integration of the area

-209d) Response Curve, Air Displacing Tracer (cont'd) bounded by the response curve. Table XVI gives the necessary quantities as obtained from the recorder traces shown in Figure 27, pp. 92. Figure 52 shows the graphical integration of that part of the response curve which has a significant slope. Values oft c. are obtained graphically av. as the average values for small time increments from Figure 52. The value of c =1 for t _? 8.67, also, c = / for t > 15.47, therefore: 00 t=8.67 t=15.47 c f c dt - f dt+ Z (c) LAt o t=O t=8.67 0av ~ 8.67 + 2.992 = 11.67 seconds 2. Porosity of Irrigated Packing Residence-time in Empty Section (that which is usually occupied by irrigated packing): @ =.259 cu.ft. 15.70 seconds.0165 cu.ft./sec Residence-time in Sections Not Wetted: VE + eVp.0369 cu.ft. = 2.24 seconds v.0165 cu.ft./sec. Porosity of Irrigated Packing: w = 11.67 - 2.24.601 15.7

-210TABLE XVI DETERMINATION OF MIXING PARAMETERS FOR RESPONSE CURVE, AIR DISPLACING TRACER t Chart t t (t)av (CO)avAt Se conds Divisions At 8.67 40.0 1.000.40 8.87 78.6769.996.3984 9.07 39.7.992.40 9.27 85.9329.984.3936 9.47 39.0.975.40 9.67 93.5089.961.3844 9.87 37.8.945.40 10.07 101.4049.920.3680 10.27 35.6.890.40 10.47 109.6209.850.3400 10.67 32.2.805.40 10.87 118.1569.750.3000 11.07 27.4.685.40 11.27 127.0129.607.2428 11.47 21.0.525.40 11.67 136.1889.447.1788 11.87 15.2.380.40 12.07 145.6849.320.1280 12.27 10.9.272.40 12.47 155.5009.225.0900 12.67 7.5.187.40 12.87 165.6369.158.0632 13.07 5.3.132.40 13.27 176.0929.108.0432 13.47 3.5.0875.40 13.67 186.8689.074.0296 13.87 2.3.0590.40 14.07 197.9649.0o48.0192 14.27 1.5.0375.40 14.47 209.3809.030.0120 14.67 1.o.0250.40 14.87 221.1169.015.00oo60 15.07 0.5.0125.4o 15.27 233.1729.005.0020 15.47 0.0.000

-211 - 1.00.90.80.70.60.50 -.40.30.20.10.00 8.67 9.07 9.47 9.87 1027 10.67 11.07 11.47 11.87 12.27 12.67 13.07 13.47 13.87 14.27 14.67 15.07 15.47 Time from signal introduction, t, sec. Figure 52. Graphical Integration of Response Curve, Air Displacing Tracer.

-212 - d) Response Curve, Air Displacing Tracer (cont'd) 3. Calculated Variance The variance is calculated as indicated on page 103. By reference to d)l, we assume that: CZ t=8.67 t=15.47 f tO dt _. f tdt+ Z ta c) At o t=O t=8.67 av where tav = t + At 2 The foregoing formula for graphical integration involves the assumption that for small intervals of time: tav(Co)av (t co)av This was verified by detailed graphical integration of the curve t c vs. t and found to hold co for all cases encountered in the experimental work. c dt 8672 3 t oo dt' 2 + 31227122 _ 68.85567 sec2 Variance: 00 2 f t - dt a2 0 O 00 0 co 2 x 68.85567 7ii 67 <- 1 ~7=.01118 (11.67)

-213 - a) Response Curve, Air Displacing Tracer (cont'd) 4. Skewness In a similar manner, as for the calculation of the variance, we may write: t2 C ~t=8.67 t=15.47 f t2 dt t2dt + t2 (c )av At o o t=O t=8.67 av Oj (8 67)3 (8.) + 330.69345 = 547.9326 3 And the skewness is: v = 3 x 547.9326 -3(1.01118) + 2 =.00073 (11.67)5 5. Variance Correction For End Effects Modified Peclet Number From pp. 120 eq. (63b): =.338 Hydraulic Mean Diameter From a) 8 =.00371 ft. Height of Mixed Stage: H = 2 x.00371 ft. =.0220 ft..338 Equivalent Height of Section Above Liquid Distributor -.45 ft. Number of Stages in Section Above Distributor: nD =.45 ft. = 20.4.022 ft.

-214d) Response Curve, Air Displacing Tracer (cont'd) Residence-time in Section Above Distributor: VD.0303 cu.ft. v _ 03 luf.=1.4 sec. OD = =.0165 cu.ft./sec..4 sec. Residence-time in Empty Section: V.oo66 cu. ft. E v E -.0165 cu.ft./sec..40 sec. Residence-time in Irrigated Section: Ow = 1 - QD - = 11.67 - 1.85 - 0.39 = 9.43 sec. Corrected Variance: g =.01118 (1167 2 1 (1.842 )2 (119.) 2o- -9.43 9.43 0 6. Height of Mixed Stage Height of Irrigated Section = 3.00 ft. Height of Mixed Stage: Hw =.01346 x 3.00 ft. =.0404 7. Peclet Number: 2d 2 x.0208 Pe -= 0404 1.03 ft. e) Response Curve, Tracer Displacing Air 1. Residence-time In manner analogous to d)l (see Table XVII and Figure 53): X00l t=8.12 t=14.52 f (l-c)dt — f dt + Z (l-~) At o 0 t=O t=8.12 o av. 8.12 + 2.8004 = 10.92 sec.

-215 - TABLE XVII DETERMINATION OF MIXING PARAMETERS FOR RESPONSE CURVE, TRACER DISPLACING AIR C C t Chart cO 1 —c. t t (1L) (l_2)aVAt co av av aav Seconds Divisions | At 0 l 0 8.12.0.0000.000.40 8.32 69.2224.997.3988 8.52. 3.0075.993.40 8.72 76.0384.984.3936 8.92.9.0225.978.40 9.12 83.1744.963.3852 9.32 2.3.0575.943.40 9.52 90.6304.910.3640 9.72 5.5.137.863.40 9.92 98.4064.822.3288 10.12 9.6.240.760.4o 10.32 106. 5024.697.2788 10.52 15.0.375.625.4o 10.72 114.9184.555.2220 10.92 20.4.510.490.40 11.12 123.6544.417.1668 11.32 26.0.650.350.40 11.52 132.7104.280.1120 11.72 31.1.777.223.40o 11.92 142.0864.165.o660 12.12 35.0.875.125.40 12.35 151.7824.100.0400 12.52 37.0.925.075.40 12.72 161.7984.o6o.0240 12.92 38.0.950.050.4o 13.12 172.1344.038.0152 13.32 39.0.975.025.40 13.52 182.7904.022.oo88 13.72 39.3.983.017.40 13.92 193.7664.011.oo44 14.12 39.7.993.007.40 14.32 205.0624.005.0020 14.52 4o.0 1.000.000

-2161.00.90 -.80.70.60 Ul0.50.40.30.20.10.00 8.12 8.52 8.92 9.32 9.72 10.12 10.52 10.92 11.32 11.72 12.12 12.52 12.92 13.32 13.72 14.12 14.52 Time from signal introduction,t,sec. Figure 53. Graphical Integration of Response Curve, Tracer Displacing Air.

-217e) Response Curve, Tracer Displacing Air (cont'd) 2. Porosity of Irrigated Packing Residence-time in empty section occupied by irrigated packing only:.259 cu. ft. = 14.8 sec..0175 cu.ft./sec. Residence-time in Section Not Wetted: VE+EVD _.0369 cu.ft. v.0175 cu.ft./sec. Porosity of Irrigated Packing: = 10.92 - 2.11 595 14.8 3. Calculated Variance In manner analogous to d)3 (See Table XVII and Figure 53)' t=8.12 t=14.52 f t(1-co)dt tdt + Z tav(1-C)av At o t=O t=8.12 0 av. "- 2 (8.12) + 27.38541 = 60.35261 sec2 Variance, 2f t(l-c)dt o a2 1-0 (l- c)dt o o 2 x 60.35261 sec2 2 X- 1 -.01038 (10.92)2sec2

-218 - e) Response Curve, Tracer Displacing Air (cont'd) 4. Skewness 0 2 t=8.12 t=14.52 f t2(1C —)dt-0 f t2dt + t2 (1 — ) At o c0 t=O t=8.12 av 0 av. (8.12)3 + 270.43762 = 448.9000 3 And the skewness is: v = 3 x 448.9000 -3 (1.01038) +2 =.00022 (10.92)3 5. Variance Correction for End Effects From d)5, the number of stages above distributor are: = 20.4 Residence-time in section above distributor V -V.0304 cu.ft. = 174 sec D D 0175 cu.ft./sec. 1.74 sec. V Residence-time in empty section: V.oo65 cu.ft. E = v E.0175 cu.ft./sec. =.37 sec. Residence-time in irrigated section: Qw = - D - GE = 10.92 - 1.59 - 0.52 = 8.81 sec. Corrected Variance: -.01038(I.Z. - (39 =.1220 6. Height of Mixed Stage Height of Irrigated Section: = 3.00 ft. Hw =.1220 x 3.00 ft. =.3660 ft. 7. Peclet Number 2d 2 x.0208 Pe = -.03660 = 1.14

f) Water Properties 1. Average Temperature = 88 OF 2. Density (see Figure 59, pp. 221), p = 62.13 lb/cu.ft. 3. Viscosity (see Figure 59, pp. 221), pw = 1.89 lb/ft.hr. 4. Water Flow Rate: (From No. 2 Rotameter Calibration, Figure 55, pp.217 ) =.078 gal/min. 5. Superficial Mass Flow Rate: W = 62.13 lb/cu.ft. x.078 gal/min x.1337 cu.ft/gal x 60 min/hr.0874 sq. ft. - 446 lb/hr.ft.2 6. Reynolds Number Re = Wd = 446 lb/hr.ft.2 x.0208 ft. 4.91 L 1Uw 1. lb./ft.hr.

I

APPENDIX IV TABLE XVIII RESPONSE TO A STEP FUNCTION FOR SOLUTION "All OF DIFFUSION EQUATION C t FRM8 t 0.c- AS FUNCTION OF - FOR VALUES OF FROM 8o to 200 =0 2DL uL I L 80 90 100 110 120 130 i4o 150 i6o 170 180 190 20.6 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000.7.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000.8.974.981.985.988.994.994.995.997.998.998.999.999.999.85.920.932.943.956.959.965.970.975.978.981.984.987.988 H.90.813.829.843.856.867.877.886.896.902.909.916.922 927.95.656.669.674.688.698.706.717.721.729.736.743.749.755 1.00.477.479.48o.481.482..488.483.484.484.484.485.486.486 1.05.311.303.294.288.281.274.268.261.256.250.445.244.254 1.10.i81.169.157.447.138.129.120.114.io6.100.094.o88.083 1.15.095.o84.069.o66.055.047.045.o4o.035.031.028.025.022 1.20.046.037.030.025.020.017.oi4.011.009.008.oo6.005.ook 1.25 -- -- -- -- -- -- -- --.002.002.001.001.000 1.30.oo8.oo6.004.003.002.001.001.000.000.000.000.000.000 1.4o.001.000.000.000.000.000.000.000.000.000.000.000.000

-222-.5.4.3- cL w Ii 3: /.2,.1 0,, J,,,, I 0 2 4 6 8 10 12 14 16 ROTAMETER SETTING Figure 54. Rotameter No. 1 Tube No. 2F-1/4-16-5/70 Calibration for Air at 60~F and 14.7 Psia.

1.2 I0 C w.8 0 -J w H.4.2 0A.1.2.3.4.5.6.7.8.9 1.0. ROTAMETER REAUING GALS /min Figure 55 - Rotameter No. 2 Tube No. B4-27-10/70-U Calibration for Water at 64.49F.

3.0 2.5 2.0 1.5.LL I0 0 20 30 40 50 60 70 80 I 0 I0 20 30 40 50 60 70 80 90 ROTAMETER SETTING Figure 56. Rotameter No. 3 Tube No. B4-27-10/27 Calibration for Air at 600F and 14.7 Psia.

12 10 8 Ij06 w I 0 U2,,, X,,, 0 10 20 30 40 50 60 70 80 90 00 ROTAMETER SETTING Figure 57. Rotameter No. 4 Tube No. B5-27-10/77 Calibration for Air at 60~F and 14.7 Psia.

.046 IHELIUM ILL 0 j.044 50 55 60 65 70 75 80 85 TEMPERATURE 0 F Figure 58. Viscosity of Gases.

-22762.50 34 62.40 3.2 3.0 62.30 _ 2.8 2.6 62.20 _ 7 24. _ 2.2 O 0 z O 6210 _ 2.0 1.8 62.00 1.6 50 60 70 80 90 100 TEMPERATURE F Figure 59. Properties of Water.

NO4ENCLATURE A = cross sectional area of empty column, sqo ft. a = total surface area of bed, sq. ft./cu. ft. aw = total surface area of irrigated bed, sq. ft./cu. ft. c = tracer concentration in effluent at time t, moles/unit vol. c = tracer concentration in entering stream, moles/unit vol. c (s) = Laplace tranisform of c D =:ffective diffusivity, sq. ft./sec. DL = effective diffusivity in the direction parallel to flow in the packed bed, sq. ft./sec. D' = effective diffusivity in the direction parallel to flow L in the foresection, sq. ft./sec. D = effective diffusivity in the direction parallel to flow in the aftersection, sq. ft./sec. Dr = effective diffusivity in the direction normal to flow, sq. ft./sec. d = nominal diameter of packing element, ft. dm hydraulic mean diameter, cE, ft. a dp = diameter of sphere having equal volume as packing element, ft. dt = diameter of packed column, ft. E(t) = distribution of residence-times function, seconds-1 = d c/co for a step input dt vc for an impulse input Q -229

-230NOMENCLATURE (Cont'd) erf(X) = 2 f e dX X2 erfc(X) = 2 f e dX f{s} = Laplace transform of any function f1{s} = inverse Laplace transform of any function f'{s} = derivative of Laplace transform with respect to the arg-unent G = superficial mass flow rate of gas, lb./hr., sq. ft. H = height of a perfectly mixed stage L/n, ft. Hw = height of a perfectly mixed stage for irrigated bed, ft. i = unit vector in the x direction 3 = unit vector in y direction k = unit vector in z direction L = height of packed bed, ft. M = coefficient for porosity correlation, dimensionless m = total number of perfectly mixed cells in packed bed N = coefficient for mixing correlation for irrigated packing, dimensionless n = number of perfectly mixed cells in series in longitudinal direction n = apparent number of perfectly mixed stages in series or inverse of the variance PeL = longitudinal Peclet number, du/DL, dimensionless

-231NOMENCLATURE (Cont'd) Pe' = modified longitudinal Peclet number, dpu, dimensionless L DL Pe" = modified longitudinal Peclet number, dmU dimensionless L D DL Pe = radial Peclet number, du, dimensionless Dr Pe = longitudinal Feclet number for irrigated packing, 2d' Hw dimensionless Pe' = modified longitudinal Peclet number for irrigated packing, w 2dm, dimensionless Pew = coefficient for mixing correlation for irrigated packing, dimensionless Q = quantity of tracer material injected, moles r = mass transfer rate, moles/sec., sq. ft. Re = Reynolds number for gas, Gd, dimensionless GFG Re = Reynolds number for liquid, Wd, dimensionless w Aw S = specific surface of the material, sq. ft./cu. ft. s = argument of the Laplace transform t = time measured from introduction of signal, sec. u = average superficial velocity through interstices, ft./sec. U = dimensionless group uL used in solution "C" of diffusion 2DL equation V' = volume of a perfectly mixed stage, cu. ft.

-232 - NOMENCLATURE (Cont'da) V = volume of empty column, cu. ft. v = volumetric flow rate through column, cu. ft./sec. v' = volumetric flow rate through a perfectly mixed stage, cu. ft./sec. W = superficial mass flow rate of liquid, lb./hr., sq. ft. x = distance along coordinate axis parallel to flow, ft. y = distance along coordinate axis normal to x and z, ft. z = distance along coordinate axis normal to x and y, ft.,a = variable defined in Appendix I a-' = variable defined in Appendix I D = variable defined in Appendix I Y = variable defined in Appendix I Lt = finite increment of time, sec. e = porosity of dry bed; volume fraction of empty column Ew = porosity of wet bed; volume fraction of empty column w = coefficient for porosity correlation, dimensionless wo O _= mean residence-time; EV or L, sec. v u 0' = mean residence-time for a single perfectly mixed cell; V', sec. VI X = variable used for a generic designation of the argument uf the error function or as dummy variable of integration 1G _ viscosity of the gas, lb./ft., hr. -w = viscosity of the liquid, lb./ft., hr.

-233 - NOMENCLATURE (Cont' d) Ln = eigenvalues of solution "C" of diffusion equation v = skewness of the dimensionless time distribution function about t = e; dimensionless < = 3.1416 radians PG = density of gas; lb./cu. ft. P, = density of liquid; lb./cu. ft. 2 a2 = variance of the dimensionless time distribution function, dimensionless T = dummy variable of integration expressing time p= arbitrary function of s =X arbitrary function of s = time distribution function for element of surface, seconds,'V= fractional rate of surface renewal, seconds1 Subscripts D, E, P and w for the quantities n, V, e and a refer to portion of packed bed above liquid distributor, empty section below packed bed, total packed bed and irrigated portion of packed bed respectively Subscripts 1, 2, i and n for quantities V', e' refer to mixed stages number 1, 2, i and n respectively. These subscripts may also refer to several mixers of types 1, 2, i and n when used for quantities V, e and n.

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BIBLIOGRAPHY 1. Aris, R., Amundson, N. R., A.I. Ch. E. Journal 3 280 (1957) 2. Baron, Thomas, Chem. Eng. Progress 48 118 (1952) 3. Bernard, R. A., Wilhelm, R. H., Chem. Eng. Prog. 46 233 (1950) 4. Chandrasekhar, S., Rev. of Modern Physics 15 1 (1943) 5. Chu, J. C., Kalil, J. and Welteworth, W. A., Chem. Eng. Progress 49 141 (1953) 6. Churchill, R. V., "Modern Operational Math. In Engineering", pp. 117, McGraw Hill Book Co., New York 1944 7. Churchill, S. W., Abbrecht, P. H., Chu, Chiao-Min, Ind. Eng. Chem. 49 1007 (1957) 8. Cohen, W. C., and Johnson, E. F., Ind. Eng. Chem. 48 1031 (1956) 9. Colpitts, G. P., Ph.D. Thesis in Progress, University of Michigan 10. Cramer, H. "The Elements of Probability Theory", John Wiley and Son, New York (1955) 11. Danckwerts, P. V., A.I. Ch. E. Journal 1 456 (1955) 12. Danckwerts, P. V., Appl. Sci. Research A3 279 (1953) 13. Danckwerts, P. V., Chem. Eng. Sci. 2 1 (1953) 14. Danckwerts, P. V., Ind. Eng. Chem., 43 1460 (1951) 15. Danckwerts, P. V., Jenkins, J. W., and Place, G., Chem. Eng. Sci. 3 26 (1954) 16. Deisler, P. F., McHenry, K. W., and Wilhelm, R. H., Anal. Chem. 27 1366 (1955) 17. Deisler, P. F., and Wilhelm, R. H., Ind. Eng. Chem. 45 1219 (1953) 18. Ebach, E. A., Ph.D. Thesis, University of Michigan, May, 1957 19. Elgin, J. C., and Weiss, F. B., Ind. Eng. Chem. 31 435 (1939) 20. Evans, G. C., and Gerald, C. F., Chem. Eng. Progress 49 135 (1953) -235

-236 - BIBLIOGRAPHY (CONT' D) 21. Feller, Wm., "Probability Theory and its Applications", Volume 1, John Wiley and Sons, Inc., New York (1950) 22. Fellinger, L., ScD. Thesis, Mass. Inst. Technology (1941) 23. Fowler, F. C., and Brown, G.G., Trans. Am. Inst. Chem. Engrs. 39 491 (1943) 24. Fry, T. C., "Probability and Its Engineering Uses", D. Van Nostrand Co., Inc., New York (1928) 25. Gamson, B. W., Chem. Eng. Progress, 47 19 (1951) 26. Gardner, G. C., Chem. Eng. Science 5 101 (1956) 27. Gilliland, E. R., and Mason, E. A., Ind. Eng. Chem. 41 1191 (1949) 28. Gilliland, E. R., and Mason, E. A., Ind. Eng. Chem. 44 218 (1952) 29. Grimley, S. S., Trans. Inst. Chem. Engrs. (London) 23 228-35 (1945) 30. Handlos, A. E., Kunstman R. W., Schisler, D. O., Ind. Eng. Chem. 49 25 (1957) 31. Hanratty, T. J., A.I.Ch.E. Journal 2 359 (1956) 32. Higbie, R., Trans. Am. Inst. Chem. Engrs. 31 365 (1935) 33. Hughes, R. R., Ind. Eng. Chem. 49 947 (1957) 34. Jesser, B. W., and Elgin, J. C., Trans. Am. Inst. Chem. Engrs. 34 277 (1943) 35. Kaplan, W., Operational Mathematics for System Analysis, Notes for Math. 148 Univ. of Michigan (1956) 36. Keyes, J. J., Jr., A.I.Ch.E. Journal 1 305 (1955) 37. Kozeny, J., Ber. Wien. Akad., 136A 271 (1927) 38. Kramers, H. and Alberda, G., Chem. Eng. Sci. 2 173 (1953) 39. Lapidus, Leon, Ind. Eng. Chem. 49 1000 (1957) 40. Lapidus, L., and Amundson, N., J. of Phys. Chem. 56 984 (1952)

-237BIBLIOGRAPHY (CONT'D) 41. Leva, M., "Tower Packings and Packed Tower Design" U. S. Stoneware Co., Akron, Ohio (1953Y 42. Levenspiel, 0., and Smith, W. K., Chem. Eng. Sci. 6 227 (1957) 43. Lobo, W. E., Friend, L., Hashmall, F., and Zenz, F., Trans. Am. Inst. Chem. Eng. 41 693 (1945) 44. Mach, E.)Dechema Monograph 6 38 (1933); Z. Ver. Deut. Ing. Forsh. 375, (1935) 45. Mayo, F., Hunter, T. G., and Nash, A. W., J. Soc. Chem. Ind. (London) 44 375T (1935) 46. McCune, L. K., and Wilhelm, R. H., Ind. Eng. Chem. 41 1124 (1949) 47. McHenry, K. W., Jr., Wilhelm, R. H., A.I.Ch.E. Journal 3 83 (1957) 48. McMullin, R. B., and Weber, M., Jr., Trans. Am. Inst. Chem. Engrs 31 409 (1934-35) 49. Ranz, W. E., Chem. Eng. Progress 48 247 (1952) 50. Sarchet, B. R., Trans. Am. Inst. Chem. Engrs. 38 283 (1942) 51. Sherwood, T. K., and Pigford, R. L., Absorption and Extraction, McGraw-Hill Book Co., Inc., New York, 1952 52. Sherwood, T. K., Shipley, G. H., and Holloway, F. A. L., Ind. Eng. Chem. 30 765 (1938) 53. Shulman, H. L., Ullrich, C. F., Proulx, A. Z., and Zimmerman, J. 0., A.I.Ch.E. Journal 1 253 (1955) 54. Shulman, H. L., Ullrich, C. F., and Wells, N., A.I.Ch.E. Journal 1 247 (1955) 55. Singer, E., D. B. Todd, Guin, V. P., Ind. Eng. Chem. 49 11 (1957) 56. Singer, E., Wilhelm, R. H., Chem. Eng. Progress 48 247 (1952) 57. Tillson, P., SM Thesis MIT (1939) 58. Uchida, S., and Fujita,S., Soc. Chem. Ind. (Japan) 39 432 (1936); 40 238 (1937)

-238BIBLIOGRAPHY (CONT' D) 59. Wehner, J. F., and Wilhelm, R H., Chem. Eng. Sci. 6 89 (1956) 60. White, R. R., and Othmer, D. F., Trans. Am-. Inst. Chem. Engrs. 38 1067 (1942) 61. Whitman, W. G., Chem. and Met. Eng. 29 147 (1923) 62. Whitt, F. R., Brit. Chem. Eng. 1 43-37 (1957) 63. Wylie, C. R., Jr. "Advanced Engineering Math" McGraw-Hill Book Co. (1951) 64. Yagi, S., Miyauchi, T., Chem. Eng. (Japan) 17 382 (1953)

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