THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Chemical and Metallurgical Engineering Multi-phase Fluids Laboratory Progress Report No. 2 A MATHEMATICAL MODEL FOR THE POPPET NOZZLE K. H. Coats J. L. York M. R. Tek ~" ^' -.;....'-UMIrI:roject'291 -,''''''~.. - -' ~''~..'..,.. under contract with: DELAVAN MANUFACTURING COMPANY WEST DES MOINES, IOWA administered by: THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR February 1960

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TABLE OF CONTENTS Page LIST OF FIGURES v NOMENCLATURE vii I. INTRODUCTION 1 II. SUMMARY OF RESULTS 3 III. REVIEW OF PROGRESS AND RECOMMENDATIONS FOR FUTURE WORK 11 IV. PREDICTED PERFORMANCE OF NOZZLE WITH DASHPOT 13 V. EXPERIMENTAL DETERMINATION OF ORIFICE COEFFICIENTS 15 VI. COMPARISON BETWEEN PREDICTED AND OBSERVED NOZZLE PERFORMANCE 17 APPENDIX I. EQUATIONS GOVERNING PERFORMANCE OF A POPPET NOZZLE WITH A DASHPOT 19 APPENDIX II. VISCOUS DAMPING COEFFICIENT FOR DASHPOT 21 iii

LIST OF FIGURES No. Page 1. Pintle opening and nozzle flow rate vs. time for nozzle with dashpot. 4 2. Pintle opening and nozzle flow rate vs. time for nozzle with dashpot. 3. Flow rate vs. Ap for spring-loaded poppet nozzle with dashpot. 6 4. Sketch of dashpot. 7 5. Sketch of test nozzle. 7 6. Orifice coefficient Co vs. Reynolds number Re. 9 7. Orifice coefficient Co vs. Reynolds number Re. 9 8. Flow rate vs. pressure drop for Delavan nozzle DMC 2360. 10 9. Sketch of test nozzle. 10 v

NOMENCLATURE ao radius of pintle A area bo radius of sleeve Co orifice coefficient c viscous damping coefficient for dashpot D diameter F force on pintle Fo force exerted by spring when nozzle is closed f frequency of pintle vibration k spring constant lw lost work L length of contact between dashpot piston and cylinder m mass of pintle plus moving parts p pressure Ap PI - P2 or P2 - P1 q nozzle flow rate qe average nozzle flow rate Re Reynolds number at cross section 2 t time u fluid velocity x pintle opening Xe equilibrium or average position of pintle Subscripts 1 cross section 1 (see Fig. 1 of report 2931-1-P) 2 cross section 2 (see Fig. 1 of report 2931-1-P) i inner o outer Greek 9 angle of conical pintle head p density of flowing liquid p. viscosity of flowing liquid vii

I. INTRODUCTION A mathematical model for the poppet nozzle was formulated and presented in the previous progress report 2931-1-P of September, 1959. This report presents results pertinent to the extension, refinement, and application of this model. First, the model has been adapted to treat the case of a nozzle with a dashpot. Second, a test nozzle with two different pintles has been built and tested to determine the orifice coefficient (or lost work) as a function of pintle opening and Reynolds number. The manner in which the orifice coefficient is incorporated into the equations governing nozzle performance is shown. Finally, a comparison is given between the actual performance of a nozzle as determined experimentally by Mr. Richard Wilcox of Delavan and the performance predicted by the mathematical model. Throughtout this report the word "poppet" is used in a descriptive sense, referring to the type of nozzle. The word "pintle" is employed to designate the central component of the nozzle, the moving rod and its integral head piece. 1

II. SUMMARY OF RESULTS Calculations have been carried out on the IBM 704 digital computer to determine the performance of a spring-loaded poppet nozzle with a dashpot. The results show that a stable, nonoscillatory pintle position is attained almost instantaneously for a sufficiently large dashpot viscous damping coefficient. Figure 1 shows pintle position and flow rate as a function of time for a nozzle operating at an upstream pressure of 480 psia and having a dashpot damping coefficient of.05 pound force-second per inch. An essentially stable pintle opening and flow rate is attained within.04 sec after start-up. Figure 2 presents the calculated performance of the same nozzle with the exception of the viscous damping coefficient, which has been increased to.3 pound force-second per inch. The same equilibrium values of pintle opening and flow rate are attained considerably faster due to the larger damping coefficient. The relationship between equilibrium flow rate and pressure drop across the nozzle is presented.in Fig. 3. This plot shows that the flow rate in gallons per minute increases with Ap at a rate slightly greater than that corresponding to a straight-line relationship. All the results plotted on Figs. 1, 2, and 3 were calculated with the assumption of negligible lost work. The important conclusion to be drawn from these results is that a nonoscillatory pintle position can be attained for each upstream pressure pi if a dashpot with a sufficiently large damping coefficient is built into the nozzle. The stable pintle position means, of course, that wear due to collision between the pintle head and seat:or sleeve is avoided. A relation has been derived giving the viscous damping coefficient as a function of design dimensions of the dashpot. This relation 4 8 p. L Di g (Do-Di)2 (D-D) should provide a basis for design of an effective dashpot. The terms in Eq. (1) are defined in the nomenclature and in Fig. 4, a sketch of the dashpot. If a dashpot piston with a rounded edge is to be employed to avoid sticking, Eq. (1) will not be strictly accurate and the value of c should in that case be determined experimentally. A test nozzle, sketched in Fig. 5, has been built and tested to determine the orifice coefficient for poppet nozzles as a function of Reynolds number. This coefficient is defined by the relation P P l —C -, (2) p

c =-.05 lbf -sec/in., P=480 psia x = pintle opening (axial displacement.014 of pintle from closed position) q nozzle flow rate, gpm 0Joo6- -3,008 0 [ l I I I I X I I I I o 10 20 30 40 50 120 130 140 150 160 NUMBER OF.0003-SECOND TIME INCREMENTS Fig. 1. Pintle opening and nozzle flow rate vs. time for nozzle with dashpot.

c =.3 /lbf - sec/in..015 p,= 480psia.010 xe~~~~~~~~~~~~,.00966 in. C-) \J1 Z q.005^ /, q*1.4861 -21 0 I~~~~~III I I 0 10 20 30 40 50 60 70 8090 NUMBER OF.0003-SECOND TIME INCREMENTS Fig. 2. Pintle opening and nozzle flow rate vs. time for nozzle with dashpot.

2. 6 2.4 2.2 2.0 1.8 1.6 1.4E E < 1.2 C0 1.0 0 100 200 300 400 500 600 Ap, psia Fig. 3. Flow rate vs. Ap for spring-loaded poppet nozzle with dashpot. 6

CYLINDER Fig. 4. Sketch of dashpot. WATER IN 7/ v 9= A C /CROSS SECTION 1 SLEEVE,^ XX,X~ CROSS SECTION 2 PINTLE / -bo=.570 IN. a=.448 IN. Fig. 5. Sketch of test nozzle. 7 Fig. 5. Sketch of test nozzle.

where lw is the energy lost by friction and Ap is the pressure drop through the orifice. The experimentally determined values of Co are plotted in Figs. 6 and 7. The plots show that Co is a function of pintle opening x and pintle head angle 0 as well as Reynolds number. More data need to be taken in the low Reynolds number region to determine whether Co tends toward unity or zero (as does Co for a fixed area orifice) at small Re. The results given in Figs. 6 and 7 indicate that Co decreases (i.e., lost work increases) as the angle of the pintle head increases (x and Re held constant) and that Co decreases as the pintle opening x increases. The data, although incomplete, yield the important fact that lost work in the poppet nozzle orifice is not negligible, i.e., the orifice coefficient Co is significantly less than 1.0. Delavan nozzle DMC 2360 has been tested by-.. Richard Wilcox and the flow rate obtained as a function of pressure drop. The design dimensions and other necessary data were employed along with the equations developed in report 2931-1-P to calculate the performance of this nozzle. The predicted and observed flow rates are compared in Fig. 8. The results show that, when lost work is assumed to be negligible (Co = 1.0), the predicted flow rates are considerably above the observed values. The experimental data on lost work discussed above show that a relation Co = a - bx gives a qualitatively correct relationship for CO. When the assumed relation C =.61 - 12x is employed in the mathematical model, good agreement is obtained between the observed and predicted flow rates, as shown in Fig. 8. A frequency of 380-390 cps was recorded on the sonic analyzer during the testing of the nozzle. The frequency of pintle vibration can be predicted from Eq. (3) of report 2931-1-P. The slope of the F vs x curve must be calculated from the design dimensions of the nozzle. The slope for nozzle DMC 2360 was calculated by the 704 computer as 113 pounds force per inch; this slope yields a calculated frequency of 390 cps, in good agreement with the observed 380-390 cps. During the testing of the nozzle a loud noise and erratic spray was reported at a flow rate of 288 lb/hr and upstream pressure of about 300 psig. The computer calculations for the nozzle showed that the conical pintle head would become completely exposed (beyond the seating or sleeve) at a pressure between 250 and 300 psig. At this large pintle opening, the bearing or positioning ring (ridge) on the pintle will come in line with the fuel entry ports or holes, as is evident from Fig. 9. Thus the noise and erratic spray at the upper pressure range of operation could be due to pintle oscillations induced by partial blocking of the fuel entry holes by the pintle positioning ring. If this is the cause of the trouble, then moving the positioning ridge back farther on the pintle should remedy the situation. A nozzle of second design was tested by Mr. Wilcox and within a certain flowrate range the conical pintle head was observed sliding around the periphery of the orifice, yielding a nonuniform spray. This situation could be remedied either by reducing the clearance at the pintle positioning surfaces (with attending problem of inducing "sticking")arbyincressing theaxial distance between the positioning surfaces (see Fig. 9). The latter is suggested as a feasible method of keeping the pintle centered during operation. 8

1.0 8=Angle of conical pint/e head = /50.9 x=.053 in. Co.8 ~~ x=./0 /in. 7 -x =.150 in. 103 104 105 Re Fig. 6. Orifice coefficient Co vs. Reynolds number Re. 6 = Angle of conical pintle head= 45".8 x =. 051 in..6 Co.4 x =.105 in..2 x =.157 in. 103 104 105 Re Fig. 7. Orifice coefficient Co vs. Reynolds number Re. 9

_ Data o Co assumed =1.0 4 ao Co assumed =.61 - /2x E 3 o J LL 2 0 50 100 150 200 250 300 350 P =p,-po, psi Fig. 8. Flow rate vs. pressure drop for Delavan nozzle DMC 2360. FFUEL ENTRY HOLE SLEEVE PINTLE _///./__///////// / //// /// /. __ PINTLE POSITIONING RIDGE Fig. 9. Sketch of test nozzle. 10

III. REVIEW OF PROGRESS AND RECOMMENDATIONS FOR FUTURE WORK A mathematical model has been formulated for the poppet nozzle which allows prediction of amplitude and frequency of pintle vibration and of flow rate as a function of pressure drop. The model shows that pintle vibration is inevitable for a spring-loaded poppet nozzle operating without a dashpot. The model has been adapted to treat the case of a nozzle with a dashpot and shows that a stable nonoscillatory pintle position is attained very quickly for a sufficiently large damping coefficient. A relation has been derived as a guide in designing an effective dashpot. Calculated curves of flow rate versus pressure drop yield a flow rate which increases with Ap at a greater rate than that corresponding to a straight-line relationship. This agrees well qualitatively with Delavan data for various nozzles. Experimental data taken at Michigan show that the orifice coefficient Co is significantly less than unity for the poppet nozzle in the operating range of Reynolds number. This necessity of accounting for the lost work is confirmed by application of the mathematical model to an actual test case: ignoring the lost work results in poor agreement between observed and predicted performance while choice of a realistic Co less than unity produces close agreement between observed and predicted nozzle flow rates. Concerning recommended future work, the major difficulty remaining in the development of an accurate mathematical method of predicting nozzle performance is the relationship between orifice coefficient CO and Reynolds number. At present, insufficient data have been taken in the low Reynolds number range. Since Co seems to be a function of pintle opening and conical head angle as well as Re, a correlative problem of significant magnitude exists. Once Co has been correlated, a step-by-step outline, aided to maximum extent by generalized tables and charts, of the method of calculation for prediction of nozzle performance should be formulated. This method of calculation would allow determination of a nozzle's performance without building and testing it and would allow selection of one of several proposed nozzles on a basis of optimum performance. The very important intermediate step between determination of Co and formulation of the general method of calculation is of course the testing of the mathematical method by application to actual test cases. The excellent data already gathered by Mr. Wilcox on the DMC 2360 nozzle are sufficient for partial evaluation of the calculational method. The feasibility of designing an effective dashpot to obtain a stable pintle opening and of lengthening the distance between pintle positioning surfaces should be investigated. In the formulation of a long-range research and development program concerning variable-area devices, radical departures from the current design such 11

as pneumatic control of pintle position, hydraulic centering through swirl flow, and magnetic pintle-centering devices should be entertained. 12

IV. PREDICTED PERFORMANCE OF NOZZ.TE WITH DASHPOT The predicted performance of a spring-loaded poppet nozzle without a dashpot was presented in the previous report 2931-1-P. The calculated results showed that this type of nozzle will operate with an oscillatory pintle vibration of definite amplitude and frequency. The question arises as to whether introduction of a dashpot into the nozzle assembly will eliminate the oscillatory pintle movement and, if so, how such a dashpot should be designed. The dashpot has been taken into account mathematically by adding a force on the pintle proportional to its velocity; that is, dx Forcedashpot = - where c is the dashpot viscous damping coefficient and dx/dt is the velocity of the pintle. The values of design dimensions and parameters* used in all calculations reported in this section are: ao =.055 in. bo =.07 in. Fo =.36 pound force m =.0003 slug c =.05,.3, or.6 lb force-sec/in., as noted P2 = 14.7 psia k = 120 lb force/in. 9 = 25~ pi = as noted Figure 1 presents the calculated pintle displacement and nozzle flow rate as a function of time for a damping coefficient of c =.05 pound force-sec/in. and pi = 480 psia. The results were calculated by a numerical solution of the equation governing the pintle motion. This equation (I-3)** and its finite difference equivalents (I-5) and (I-6) are derived in Appendix I. The oscillatory pintle motion is seen to damp out rapidly, establishing a steady-state pintle opening of xe =.0096 in. The term xe denotes the pintle opening at which the fluid pressure forces exactly counterbalance the spring force so that the total force on the pintle is zero. A steady state is attained essentially at about *Nomenclature and geometry of the nozzle are the same in this report as in report 2931-1-P. The reader is referred to page 1 and Fig. 1 of that report for this information, as well as to page vii of this report. **(I-3) means Appendix I, Eq. (3). 13

time t =.04 sec. The nozzle flow rate attains a steady value of 1486 gallons per minute. The results plotted in Fig. 2 are for a nozzle identical to that for which the results in Fig. 1 were calculated, except that the viscous damping coefficient has been increased from.05 to.3. The steady-state pintle opening and flow-rate values are identical, but the oscillatory motion is damped much more rapidly. Figure 3 presents the equilibrium or steady-state values of flow rate q as a function of pressure drop Ap = Pi - P2. These results are valid for any nonzero damping coefficient c since the magnitude of c affects only the rate of damping of the oscillatory motion or rate of approach to steady state. Actually the digital computer results showed no change in the fourth decimal place of x after t = about.012 sec for c =.3 and t = about.006 sec for c =.6. Figure 3 shows that flow rate increases with Ap at a slightly greater rate than that corresponding to a straight-line relationship. A derivation given in Appendix II relates the viscous damping coefficient c to dashpot design dimensions as 8t A. L Di c = where p. is the liquid viscosity and Do, Di, and L are as noted in Fig. 4. This relation assumes laminar liquid flow through the annular region between the dashpot piston and cylinder wall; it should give an accurate estimation of c for known L, Do, and Di, and thus give some basis for design of an effective dashpot. 14

V. EXPERIMENTAL DETERMINATION OF ORIFICE COEFFICIENTS Whenever a viscous fluid flows through a pipe, some energy is lost from the fluid due to friction at the pipe wall. The increased fluid velocity in the region of a constriction in the pipe amplifies the lost energy or lost work. Such a constriction theoretically results in a conversion from pressure energy to kinetic energy; in actual practice a portion of the pressure energy is lost through friction, the remainder appearing as increased kinetic energy of the fluid. The lost work or, energy is'usually accounted for by a coefficient in the case of orifices, as w + A C2 AP (2 P 0 P where lw is the energy lost by friction, Ap is the pressure drop through the orifice, and Co is the "orifice coefficient." This coefficient appears in the equation relating velocity and pressure drop as /C 2gc(-Ap)/p ul = CO JV(AC/A2)2_ 1' (3) ~ ^V(Al/A2)l-1 where ul is the fluid velocity at the entrance to the constriction, p is fluid density, and Al and A2 are the flow areas at the entrance and exit, respectively, of the constriction. Since ul can be directly related to the flow rate in gallons per minute through the orifice, Co can be easily calculated if Ap and the flow rate are measured and the dimensions necessary to calculate Al and A2 are known. A nozzle, sketched in Fig. 5, has been built and tested for the purpose of determining CO for converging annular orifices pe.culiar to poppet nozzles. Two pintles were employed with the same sleeve, one with 0 = 15~, the other with 0 = 45~. The ends of the pintles were threaded so that the pintle could be set at any desired opening. The body of the nozzle was threaded to fit a 2-in. pipe which began at a water line and connected to a gear pump and rotameter. A pressure gage was fitted to the nozzle as shown in Fig. 5. The data were taken in the following manner. The pintle was screwed out to an opening which was measured with a depth gage and recorded as x. Valves in the water line were set to give a range of flow rates. At each valve setting, the flow rate was read from the rotameter and the pressure at the nozzle was read and recorded. The pintle opening was then changed and recorded and flow rates and corresponding nozzle pressures were again recorded. The pressure gage and rotameter were calibrated prior to insertion into the apparatus, and depth-gage 15

readings on the pintle position were taken both before and after each run. For low-pressure runs, the pressure gage was replaced by a mercury manometer. The values of Co for each run were calculated from Eq. (3), where Al = ( - a2), A2 = t x sin 0 (2bo - x tan 9), and u = q(gpm) (8.33/Aip). The orifice coefficient Co is plotted in Figs. 6 and 7 versus Reynolds number where Reynolds number is evaluated at cross section 2 (see Fig. 2 of report 2931-1-P) as Re = D u2 = C ___ 1q 2t bo - X x sin 9 cos 9 p where Ci is a conversion constant to obtain a dimensionless Reynolds number, q is in gpm, bo and x in inches and p in centipoises. 16

VI. COMPARISON BETWEEN PREDICTED AND OBSERVED NOZZLE PERFORMANCE The force on the pintle can be calculated from the equation F = pA1 - p2A2 cos - 2A1 CO (P1 - P2) (A1/A2)Co -1 - P2 (bo - x sin 9 cos )2 + P T a - F - kx (4) where CO is the orifice coefficient accounting for lost work. The areas A1 and A2 are calculated as A1 = t (b - a) A2 = t x sin 0 (2bo - x tan 0). The mean or average value of x (pintle opening) can be calculated by setting F = 0 in Eq. (4); this mean value of x, denoted by xe, can then be employed to calculate the average flow -ate qe through the nozzle as 311A C /288 go (Pl-P2) = -oV p (A1/A2) - -1 where A2 is calculated for x = xe. The value of Fo, the initial spring loading (force exerted by spring when x = 0 or nozzle is closed), is determined by setting x equal to 0 in Eq. (4). The total force F is also equal to 0 at the critical value of upstream pressure pi which is sufficient just to crack the nozzle open. Thus setting x and F equal to 0 and P1 equal to Pc in Eq. (4), one obtains 2 2 0 = Pc Al - t P2 bo + Pc t ao -Fo or 2 2 2 2 2 Fo = Pc i (bo - ao) + Pc ao - I P2 bo = bo (Pc P2) Thus F0 is determined by the value of pi necessary to initiate flow through the nozzle. The data given on DMC 2360 nozzle are as follows: 17

a0 =.055 in. bo =.07025 in. k = spring constant = 114.2 pounds force/inch Pc = 3 psig m = 3.2822 grams =.000225 slug 9 = 25~ p = 47.5 lb/ft3 = 1.145 centistokes f = frequency = 380-390 cps, from sonic analyzer Pressure Drop, Pi-P2 Flow Rate, lb/hr 100 psig 110 200 502 250 713 300 898 A computer program was written to employ the design dimensions listed above in solving Eq. (4) for F as a function of x at each of several pi values. The value of x for which F = 0 (xe) was employed in Eq. (5) to calculate the average flow rate qe. This calculation process was carried out twice, once with Co assumed equal to 1.0 (equivalent to assumption of no lost work) and again with CO set equal to.61 - 12x. The observed and calculated flow rates are plotted versus Ap on Fig. 8. The assumption of no lost work is seen to yield excessively high flow rates, while the relation Co =.61 - 12x results in close agreement between observed and predicted flow rates. The assumption of a relation of the form Co = a - bx is qualitatively correct if the Reynolds number at the orifice is sufficiently high so that at a given value of x the operating point in Fig. 6 or 7 is located on the flat portion of the curve. Clearly, more data on Co are necessary and a correlative relationship must be derived so that C0 may be expressed as a function of x in Eqs. (4) and. (5). The frequency of pintle vibration may be predicted from Eq. (3) of report 2931-1-P, f = 1/2g \/-s/m, where s is the slope of the F versus x curve in pounds force per foot. The computer yielded this slope as minus 1358 so that f = 1/2i'1358/.000225 = 390 cps which compares well with the 380-390 cps noted on the sonic analyzer during testing of the nozzle. 18

APPENDIX I EQUATIONS GOVERNING PERFORMANCE OF A POPPET NOZZLE WITH A DASHPOT Progress Report No. 2931-1-P presented a mathematical model for the poppet nozzle. The nomenclature, equations, and geometrical sketches included in that report are referred to below but are not repeated here. The basic equation in the mathematical model is simply Newton's law, FT = ma, (I-l) where FT is the total force on the pintle, m ins the mass of the pintle, and a is the acceleration. When the poppet nozzle includes a dashpot, a viscous damping force acts on the pintle. Thus the total force on the pintle is given by FT(x) F(x) - c(dx/dt), (1-2) where F(x) is given by Eq. (13P)* and c is the viscous damping coefficient. Equation (I-1) may now be written x + c x= 1 F(x). (-3) dt2 m dt m The variables whose values must be known to solve this equation for x as a function of time are ao, bo, 9, c, m, Fo, and the spring constant k. The damping coefficient c is given by 87i UL L Di C )2 ( -(I-4) gc (Do-Di) (D-i)' where. is the fluid viscosity, L the length of contact between the dashpot piston and cylinder, and Do and Di are the diameters of the cylinder and piston, respectively. This expression is derived below with the assumption of laminar liquid flow through the annular region between cylinder and piston. Equation (1-3) can be solved by numerical techniques in which the derivative d2x/dt2 is replaced by a finite difference form and x(t) is calculated stepwise at time t = At, 2At, 3At,.... An approximate solution to (1-3) can be obtained by substituting a linear function of x for F(x). The numerical technique involves solution of the equations *All equation numbers followed by letter P refer to equations in Progress Report No. 2931-1-P. 19

Xl =12(At)2 F(xo) and (I-5) m M+2 Xn+l. 4M+2 - 1 + 12(At)2 F(xn) (i-6) Xn+l = M+l xn M+l Xn- + M where xn = value of x at time t = nAt, and M = cAt/m. The initial conditions for which (I-5) is valid are o = (X)t=o = 0 (I-7) (dtt= o (-8) that is, the displacement x and velocity dx/dt are both initially zero. The analytical solution to (1-3) for the same initial conditions appears as x = xe 1- e (/m)t (osh dt + - sinh d t), (1-9) where the force F(x) [from Eq. (13P)] has been represented by F(x) = a (1 - x/xe). (I-10) In Eqs. (I-9) and (I-10), 1 1; 4a d 1A and 2 2 mxe and Xe = value of x for which F(x) =. The flow rate in gallons per minute through the nozzle is a function of x and is given by Eq. (27P). The area A2 in that equation is a function of x as given in Eq. (15P). 20

APPENDIX II VISCOUS DAMPING COEFFICIENT FOR DASHPOT A relation is derived below for the viscous damping coefficient of a dashpot. Figure 4 is a sketch of the dashpot mechanism considered. The piston of diameter Di slides in a cylinder of diameter Do. The length of contact or width of the piston is L. The liquid pressures on the two sides of the piston are denoted by pi and P2 as shown. The displacement of the piston from some reference position is denoted by x. The damping coefficient c is calculated from the equation dx 2 Fdashpot = - c t P 1 ao, (II-1) where Fdashpot denotes the force on the piston due to the difference in pressures pi and P2; thus Fdashpot = (P2-P1)4 P - Pi ao (II-2) The pressure difference P2-P1 can be related to the lost work occurring due to friction in the annulus. 1w = _ p = _ (Pip2) = v2 (II-3) P P 2gc D For laminar flow through the annulus, 64 64 (i-4) f - - --. (11-4) Re Dv p Combining Eqs. (11-3) and (II-4), one obtains P2-P1 fpLv2 32 LLv P2'-P1 2g D - g D(-) If the fluid is assumed incompressible, the velocity v through the annulus can be related to the velocity dx/dt of the piston as This is Eq. (63) of Unit Operations, Brown, G. G., et al., Wiley, 1953. 21

it 2 2 dx ~ 2 v (Do - Di) = D 4 dt 4 or 2 dx Di = ~cix D~ (ii-6) dt Do - Di Also, the term D in Eq. (11-5) should be replaced by (Do-Di), for an annulus. Combination of (11-5) and (II-6) now yields 2 32 U L i dx P2-Pi = = (Do-Di) Di - dt so that Fdashpot, from Eq. (11-2), becomes 8F p.~ L Di dx 2 Fdashpt = g (D-Di) (Do-D) - a(II-7) or dx Fdashpo =-t d - P-1 Pi a2 where 8T- L Di c = g -(D D)(D^ ) ~ c =22 22

UNIVERSITY OF MICHIGAN 3 9015 02826 7584