THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ECONOMIC DESIGN OF HEAT TRANSFER EQUIPMENT Dale E. Briggs

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ACKNOWLEDGMENTS The Computing Center of the University of Michigan is gratefully acknowledged for generously providing computer time so that this paper might be prepared. The author is also indebted to the Industry Program of the College of Engineering, The Univer sity of Michigan, for providing copies of this paper.

TABLE OF CONTENTS Page ACNWLDMNTS o. o o LIST OF TABLRES....................... v ABSTRACT................V........i ESINTRODUCTION. ooooeooooooooo o1........ SEARCHMETHOS.......................................... 2 EXAMPLE PBLM.,..,,.. 26 tDSos o ~o o o,~o~ooo ooo o.o e oco..* 0. ~e@@*~ o* 0.o*e o o oo oaoooo oo oo ooooooosooeooe oo ~o~o o o o eo o~o e CONCLUSIONSE......... 33 LE o o o e ~e o ~ o e ~ o ~ e e ~ o o ~ o o ~ o o ~ ~~e~oo e ~3 REFERNCES...e eo..,.,.,,.,,....o...oee *eoeeooeeeeoeeceooew-iiioe

LIST OF TABLES Table Page EXTRA FOR ALLOY CONSTRUCTION IN PERCENT OF ALLSTEEL HEAT EXCHANGER PRICE, 450 lb o WORKING PRESSURE................................. 1.....7 1 II COMPUTER VARIABLE TSAB (VTS,U) o EXTRA FOR ALLOY CONSTRUCTION OF TUBE SHEETS AND BAFFLES IN PER CENT OF ALL-STEEL HEAT EXCHANGER COST oo.... 21 III COMPUTER VARIABLE BACST (VTCST, U). COST IN DOLLARS PER SQUARE FOOT FOR TEMA CLASS R ALL-STEEL IEAT EXCHANGERS............................... 22 IV COMPUTER VARIABLE MUSAC (INDEXB, INDEXC) (Multiplying Factors for Mixed Pressures)............... 2 V TRIAL DESIGNS AND CALCULATED RESULTS FOR A FACTORIAL SEARCH PATTERN FOR THE DESIGN OF A n-PROPANOL CDENSER.......................eoooo....... VI TRIAL DESIGNS AND CALCULATED RESULTS FOR AN UNIVARIATE SEARCH PATTERN FOR THE DESIGN OF A n-PROPANOL CONDENSER........................... 15 VII TRIAL DESIGNS AND CALCULATED RESULTS FOR A RANDOM SEARCH PATTERN FOR THlE DESIGN OF A n-PROPANOL iv~~~~~~~~~3

LIST OF FIGURES Figure Page 1 Factorial Search Pattern Where Only Tube Length and Tube Side Water Velocity are Considered as Va iables.....................................e 4 2 Univariate Search - First Level; 3/4 Inch Tubes, 2 Tube Pass, 8 fto/sec. Water Velocity............ 6 3 Univariate Search - Second Level; 3/4 Inch Tubes, 2 Tube Pass, 12 ft. Tube Length. o......... 7... 4 Random Search Pattern Where Only Tube Length and Tube Side Water Velocity are Considered as Variables 8 5 Heat Exchanger Costs as a Function of Heat Transfer Surface for All-Steel 20 Foot Bundles Containing 3/4 in. Tubes on a 15/16 in. Triangular Pitch. The Operating Conditions are 150 lb./sq~in. and 5000F... 10 6 Computer Flow Chart to Determine the Correct Cost Function to be Used in Evaluating the Heat Exchanger 7 Cost Per Square Foot., TEMA Class R, All-Steel Heat Exchangers With 3/4 inch Tubes~.............. 16 8 Computer Routine to Calculate the Economic Information Required in the Search Analysis. The Expressions are in the MAD Language (Michigan Algorithm Decoder), 24

INTRODUCLTON Heat exchangers, condensers, and reboilers, constitute a 1major capital investment and operating expense in most chemical plants and refineries.o Rising costs and increased competition, therefore, are forcing companies to design and select heat transfer equipment in a more judicious manner. The design and selection process has been greatly improved by the availability and use of modern digital com puters Computers have not only relieved engineers from repetitious and laborious calculations but have permitted more elaborate and detailed analysis.o Many trial designs and alternate approaches can now be considered using the computer at a fraction of the cost and time required for an engineer to do a single simple design. The engineer can thus be freed from the routine calculation and can be used more effec tively in preparing data for the computer and in analyzing the computer results to obtain the most economical design. The economic design of heat transfer equipment is complicated by both the large number of independent variables and the nondlinearity of the equations describing heat transfer and pressure drop. A portion of the total number of variables involved. is fixed by the physical limitations of the problem. In a condenser, which will be discussed in this paper, the following variables will be assumed fixed in the problem statement associated with the design- condensing load, inlet pressure, inlet saturated vapor temperature., type of tube side fluid, tube side fluid temperature., fouling rates, allowable pressure drops and. materials

choosableo Such variables are tube side fluid flow rate or tube side fluid velocity, tube dimensions, number of tube passes, tube length, tube arrangement, and tube type. When values are assigned to the free variables, the remaining variables encountered in the heat transfer and pressure drop equations become fixed. Such variables are the number of tubes, total heat transfer surface area, shell size, logarithmic mean temperature difference, tube side liquid flow rate ( if velocity was fixed) or tube side fluid velocity (if flow rate was fixed) and all the heat transfer coefficients and pressure drops. To obtain the most economical condenser, the freely choosable variables must be selected in such a way that the sum of the resulting operating and fixed capital expenses on a yearly basis is a minimum. Finding the set of variables giving the minimum cost can be accomplished by search techniques using a digital computer. SEARCH METHODS Methods for finding the maximum or minimum value of analytic functions are well known. A common technique is the method of steepest ascent(l) or gradient method which assumes that the derivatives of the response function (cost) are continuous and that the independent variables have the same dimensions. For complicated systems procedures utilizing the calculus of variation may be employed. Such methods cannot be used, however, in the heat transfer design problem because the variables cannot be arranged to form an analytic function. The tube dimensions and number of tube passes can only have discrete valueso Furthermore, if the design is in accordance with the "TStandards of

-3Tubular Exchanger Manufacturers Association", (2) the tube length has fixed values and the tube pitch has a minimum value. The search methods used, therefore, must be satisfactory for the types of systems encountered There are three such methods which can be easily used in conjunction with digital computers. They are the factorial method, the univariate method, and the random methodo(394) The factorial method is characterized by selecting the trial values of the free variables at points of a grid in factor space Figure 1 exemplified a 3 x 3 factorial design in whichonly two of the free variables, water velocity and tube length are allowed to vary. The remaining variables are preset at some arbitrarily selected values. The trial values of velocity and tube length are selected to provide an even coverage of the selected region to be searched. The maximum search region or factor space is bounded in Figure 1 by the maximum and minimum tube lengths in one direction and by the minimum and maximum allowable water velocities in the other direction. In the 5 x 5 factorial design nine trial designs are required. The trial design giving the minimum cost is said -co be the optfmx design. As the number of free variables considered is increased, the number of trial designs increases rapidly. For example, if three tube lengths, three water velocities,, two pass arrangements, and two tube diameters are considered, there would be 5 x 5 x 2 x 2 or 56 trials needed. In the univariate method, as the name implies., the effect of changes in one variable at a time are examined. The variables to be

.... I' I...I..... i LARGEST STANDARD TUBE LENGTH: ~~~~~SELECTED /fSEARC H REGIOIN 20o 31 0 w~~~~~~~~~~~~~ W SELECTED u- 3: ~~~~~~~EARCH REGION 3 w w Z(I) 12 ~ <7 8 9 z I.4 tO 4 56 8 I 2 3 SHORTEST STANDARD TUBE LENGTH 0 2 4 6 8 10 12 14 WATER VELOCITY, FT. /SEC. Figure 1. Factorial Search Pattern Where Only Tube Length and. Tube Side Water Velocity are Considered as Variables.

-5are made at several values of the first variable until the best value is obtained with all the other variables being held fixed. Using the best value of the first variable studied, the second variable is studied in a similar fashion until the best value of that variable is obtained The method is extended until all the free variables have been examined individuallyo The process can then be repeated in its entirety or repeated with a reduced number of variables until the desired level of attainment is reached. Figures 2 and. 3 typical univariate schemes. Figure 2 represents the first level of search and Figure represents the second level of search utilizing the informa tion obtained in the first level0 A third procedure is the random method0 This method is characterized by selecting the values of the free variables for each trial design in a random fashion0 The set of randomly selected variables giving the minimum cost heat exchanger is said to be the optimum0 Figure 4 shows the random trial points for a heat transfer system in which only velocity and tube length are allowed to vary0 The random method can be modified by specifying values for certain variables and by selecting the remaining variables in a random fashion0 This is called a stratified random method and is useful when the "tbest"? values for certamn variables are known by prior knowledge or by previous search results. The random methods have one advantage over the other methods. The number of trial designs required for this method is independent of

-61 61000 4~~~~~~~~ 12,1000 Cl) o 8,1000 0 -JL 4~~ 1 21 8ue,2Tb as 10 12/c 14e Vlciy

16,1000 o12,1000 o8,1000 W4,1000 0~~ 0 4 6 8 1O WATER VELOCITY, Fr/SEC. Figure 5.Univariate Search - Secona Level; 3/4 Inch Tubes, 2 Tube Pass, 12 ft. Tube Length.

LARGEST STANDARD TUBE LENGTH 20 8 - z > O w > 3 ~ 3 wwIw oI2 i 2 9 m 0. 0 zS0 -J -J z 8 6 SHORTEST STANDARD TUBE LENGTH I I I I 0 2 4 6 8 10 12 WATER VELOCITY, FT/SEC Figure 4. Random Search Pattern Where Only Tube Length and Tube Side Water Velocity are Considered as Variables.

but it does provide some measure of coverage of the factor space to be searched with a reasonable number of trials. ECONOMICS In the process of the search for the optimum heat exchanger for a particular service by a digital computer or by ordinary means, the cost of each designed unit must be determined This obviously, is one of the most important features of the program, because the resultant optimum is no better than the economic data upon which it is based. Slight errors or differences in the data can drastically change the de signed physical description and operating conditions of the socalled optimum heat exchanger. Cooling water costs are particularly critical Even with small differences in cost per gallon, the final design is greatly altered because of the large amounts of cooling water required on a yearly basis. There appears to be two basic approaches to estimating heat exchanger costs. In the first method the unit is examined in its entirety and placed in a very definite category0(5) The cost of the heat exchanger is then determined by finding a cost graph or plot which is valid for units having the physical description of the designed heat exchanger. For example, suppose that in the course of the search for the optimum design., a floating head exchanger operating at 150 lb./scq~in. and 500'F is selected with 20 ft., 3/4 in. 00D. tubes on a 15/16 in. triangular pitch. In order to determine the cost of the exchanger it

30 _ _ _ _ _ __ _ _ _ _ 20, 7 4) w 0 0 0 0 0 10 0040 V) ~ ~ ~ ~ ~ ~ HA RNFE UFC. QAEFE 1~~iue5.Ha xhagrCssa4aFntono __TanfrSrac rAl Bundles Containing 3/4 in. Tubes on a 15/16 in. Triangular Pitch.~~~~~~~ -J ~ ~ ~ ~ odtosae10 b/qi.ad50O.5

-11it is quite accurate (in that no interpolation or extrapolation is re quired) for the heat exchanger pricingo To utilize the first method of approach with a digital com puter a sufficient number of graphs, similar to Figure 5, are selected for use Each graph is then reduced to a mathematical function which the computer can evaluate. The nature of Figure 5 suggests that a reasonable function might be a simple polynomial in terms of the loga: ritm of heat transfer surface ( in square feet ) as is shown in Equa tion (l). ln($/sqofto) = A(i) + B(i)ln(sqofto) + c(i)[ln(sqft.)] + oo, x(i)[ln(sqofto )]n (1) The total cost of the exchanger is then equal to the product of the cost per square foot times the number of square feet of surface required. cost = (heat transfer surface, sq~ft.) x exp.(ln($/sq~ft,)) (2) The coefficients for Equation (1) can be obtained by any of several techniques for finding the coefficients of a nth order polynomial in a least mean square sense. For the curve in Figure 5, cost/sqioft. =exp4 6.14888 - 0.98756 ln sq~ft. + O0O0t7995(ln sq~ft.)2] (3) Usually a second., third or fourth order polynomial is sufficient, For use with the — -__ —-- _J — _ coptrechfnto and var_ —iabl is assigned.___

-12In the computer program the required physical dimensions are calculated for the proposed operating conditions The computer then searches for the cost function which most nearly matches the designed unit If9 for example9 thirty categories of exchangers are described by equations similar to Equation (5)9 the function whose restrains most nearly match those of the designed unit is selected. The search or selection process for the cost function is accomplished using logical statements. The resulting action taken by the computer in response to a logical statement depends upon the answer to a number of yes-no or true-false questions concerning the input data. Once the proper cost function is selected it is a simple matter to evaluate the cost of the heat exchanger, A modification of the system can provide additional flexi bility and accuracy where a limited number of cost functions are used. Of the restrains specified in Figure 5,, it is not likely that the tube material and tube length. of the designed exchanger will match those specified in Figure 5., whereas the remaining restrains are common to a wide class of applications. Equation (2) is first modified by subtract~ing the cost of the steel tubes from the total heat exchanger cost and then adding the cost of the replacement tubes. This gives., total cost =cost of all steel exchanger -steel tube costs + replacement tube cost (4) The cost of the all steel exch-ange-rl is calculated from Equation (2).

In Equation (5), the exponential term is included to account for the tion in cost with the size of the order. A different equation, based on the cost per square foot9 could, also be used since there is a definite relationship between tube weight and heat transfer surface Frther modification of Equation (2) can be made to account for different length bundles. Equation (4) is then written as, total cost = (cost of all steel exchanger - steel tube costs + replacement tube cost) x length factor (6) These modifications greatly enhance the utility of Figure Having re duced the number of restrains by two9 the number of designs which can be satisfactorily represented by a single cost curve is increased from 1 to 55 (when seven types of alloys and five tube lengths are considered). A typical computer flow chart is shown in Figure 6. The shortcoming of both the unmodif ied and modified forms is the difficulty of adequately covering all designs with a limited number of cost curves., This is especi ally a problem when materials other than steel are used for the construction of the tube sheets, shell, channel 9 and floating head. A cost curve, which is valid for the dimensions of the unit but not for the materials of construction, can be in error by as much as 100 per cent in determining the heat exchanger cost. Further modification of the technique, although possible, is not easily accomplished and is probably not warranted,, The predicament, however, suggests a second method of treating the economic data, In the second method one views the heat exchanger as being built

Calculate Values for all the Required Variables 1.~~~~~~~ _, whenever Tube Length = 12 ft. true OD = 1 in. Admiralty Tubes, etc. false u Tube Length = 12 ft. tue CsOD 1 in. Admiralty Tubes, etc. false etc.. _ V ~~etc. /

-15unit. The cost of each part is figured individually taking into account the materials of construction and size. This method is characteristic of the way a heat exchanger fabricator might price his product or set a bid price on a specified exchanger. A version of this method has been presented by Sieder and Elliott(6) who tabulated the costs of individual component parts as a function of the shell size, working pressures, and materials of construction. The costs of the component parts (ie, tube sheets, shell and cover, and channel and floating head cover) were expressed as a per cent of the cost of an all steel heat exchanger having the same shell size. Table I gives the above cost information for a 450 lb/sqin working pressure, The cost of an allsteel heat exchanger was presented in graphical form with the cost per square foot of heat transfer surface given as a function of the shell size with parameters of working pressure and tube diameter. Figr 7 presents the cost of an all-steel heat exchanger containing 16 foot, 5/4 inch diameter tubes on a square pitch as a function of shell size and TEMAA Class R(2) standard working pressures0- When the standard working pressures of the shell side and tube side are not the same, multipliers are used to correct the cost percentages given in the tables similar to Table I. To account for the extra expense when tubes other than steel tubes are used., Sieder and Elliott gave a table of the extra cost of alloy tubes in dollars per square foot. As an example of this method, suppose the cost of a heat exchanger having the following description is d~esired-.

15 w U. 13 - bL. I.-. 0 0 LUW 9 4 ~450 o a. 0 40 14 18 22 26 30 34 38 42 NOMINAL SHELL SIZE Figure 7.Cost Per Square( ot, TEMA Class R, All-Steel Heat ExchangersWt 3/4 inch tubes.(.

TABLE I EXTRA FOR ALLOY CONSTRUCTION IN PERCENT OF ALL-STEEL HEAT EXCHANGER PRICE, 450 lb. WORKING PRESSURE.(6) Multipliers for Mixed Pressures 12 13 16 18 20 22 24 27 30 33 36 39 42 150 300 All Steel Heat Exchanger 100 100 100 100 100 100 100 100 100 100 100 100 100 Naval Rolled Tube Sheets & Baffles 14 17 19 21 22 22 22 23 23 24 25 26 27 Monel Tube Sheets & Baffles 24 31 35 37 39 39 40 41 41 42 42 42 43 1 Cr-1 Moly Tube Sheets & Baffles 6 7 7 7 8 8 8 9 10 10 11 11 12 4-6 Percent Cr 1 Moly Tube Sheets & Baffles 2 19 22 24 25 26 26 26 25 26 26 26 27 27 11/13 Cr Tube Sheets & Baffles 21 24 26 27 27 27 27 27 27 28 28 28 29 304 Stainless Tube Sheets & Baffles 22 27 29 30 31 31 31 31 30 30 30 31 31 Monel Shell & Cover 51 55 57 57 56 56 55 54 52 50 48 47 46.81.89 1 v Cry Moly Shell & Cover 23 25 26 27 27 27 26 23 22 21 19 19 18.81.89 4_6 Percent Cr 1 Moly Shell & Cover 31 34 37 38 38 37 36 34 32 30 28 27 26 81 89 2 11/13 Chrome Shell & Cover 33 36 37 39 39 39 38 36 34 31 30 28 28.81.89 304 Stainless Shell & Cover 36 39 41 41 41 40 39 37 35 33 31 30 30.81.89 Monel Channel & Floating Head Cover 42 42 43 42 4o 39 37 36 33 32 31 30 30.88.95 1 Cr-i Moly Channel & Floating Head Cover 25 26 26 25 24 23 23 22 22 21 21 20 20.88.95 4-6 Percent Cr — Moly Channel & Floating 2 Head Cover 38 39 39 39 37 35 33 31 29 27 26 25 25.88.95 11/13 Cr Channel & Floating Head Cover 38 40 40 40 37 36 34 32 30 28 27 26 26.88.95 304 Stainless Channel & Floating Head Cover 39 40 40 39 38 36 35 33 31 29 28 27 26.88.95

-18tubes 3/4 inch x 14 BWG shell working pressure 450 lbo/sqn tube working pressure 300 lbo/sqn. area 2255 sqoft. tube material 4 6% chrome tube sheets and baffles 11 - 13% chrome shell and cover 11 13%chrome channel 4 6% chrome From F, the price per square foot for a 32 inch unit with a working pressure of 450 lbo/sqo ino is $5o26 per square foot. From Table In, the component part percentage costs are: all ste1 el heat exchanger 1 00% chrome tube sheets and baffles 28% chrome shell and cover 51% chrome channel 0.95 x 27 26% 185% The extra tube cost is $2.35 per square foot. The heat exchanger cost per square foot is then, ($5.26/sq~ft.)(1.85) + $2055/sqi-ft. = $1210l0sq~ft. The total unit cost is, ($12.10/sq~ft.)(2255 sq~ft.) = 27,300-00 The method is most accurate when the considered heat exchangers

-19an inherent weakness in the procedure. The cost of a 20 foot unit would not cost two-thirds more than a comparable unit 12 feet long This is because the cost of the construction, engineering, channel, floating head, cover, and tube sheets would be nearly identical for both units while the tube and shell expense would be approximately proportional to length For this reason the total unit costs of a 20 foot unit should be only slightly higher than for a comparable 12 foot exchanger. The use of correction factors for tube length and tube pitch or the use of additional cost curves would greatly increase the accuracy of the system. An even greater accuracy could be obtained by calculating the all-steel heat exchanger cost on the basis of the shell size and the square feet of 16 foot steel tubes that could be placed into the calculated shell size on a square pitch. The 16 foot steel tube and shell costs would then be subtracted and the correct length alloy tube and shell costs added. In this manner a minimum of error would be encountered., To utilize the data of Sieder and Elliott in a digital computer program two alternate approaches are possible. The data can either be reduced to a system of nth order polynomials obtained by least square means or be stored as tables in the computer. Use of the first approach, which was discussed earlier, would require numerous equations but would also require less computer storage than storing the data per se. Where the available computer memory is large, as in the IBM 709 digital computer, storage of the data in tabular form is more efficient. The nature of the data suggests that they should be stored as a two dimensional

-20pressure By specifying values of both indices, the desired value of the function can be obtained from memory. Once the percentage costs for component parts are obtained, the method of solving the problem with a digital computer is, therefore, identical to the method used previously in the example problem. In the computer program written to do the actual condenser design a brief routine was included to calculate the cost of the designed unit. When preparing the routine it was found that the data of Sieder and Elliott were most adaptable to computer storage and use if they were first grouped according to component parts rather than working pressure. Table II gives all the data on tube sheets. As can be seen from the table, the index, VTS, accounts for both the tube sheet material and working pressure. When VTS = 1,9 the tube sheets are admiralty and the standard working pressure is 150 lb./sq~in. When VTS = 135, the tube sheets are again admiralty, but the standard working pressure is 450 lb.,/sq~in. The index, U, corresponds to the shell size. When U = 11, the shell size is 241 inches; when U = 17, the shell size = 36 inches. To obtain the cost of the tube sheets as a per cent of an all steel heat exchanger, the indices, VTS and, U, of the assigned storage variable name, TSAB(VIS, U), are specified., The datum~ stored in that location can then be used in calculating the price of the unit. For example TSAB(16117) = 26 In similar manner the shell and cover data were stored in an array

TABLE II COMPUTER VARIABLE TSAB (VTS,i). EXTRA FOR ALLOY CONSTRUCTION OF TUBE SHEETS AND BAFFLES TN PER CENT OF ALL-STEEL HEAT EXCHANGER COST nom. shell size-in. 56 810o12 14 i6 18 20 22 24 26 28 50 52 34 36 38404 45 8 U 1 2 3 4 5 6 7 8 9 10 11 12 15 i4 15 i6 17 i8192 21 2 VTS 150 lb/scpin. acim. 1 5 7 9 11 15 15 18 20 21 21 21 21 21 21 21 21 21 2122 222 2 150 lb/si. in. monel 2 10 13 16 19 23 51 34 36 37 38 59 59 59 59 59 59 59 38 8 858 8 1501lb/sq.in. 11/4 Cr 5 4 56 666b7 77 88 8 899 9999999 150 lb/sq.in. 4 Cr 4 8 11 14 17 19 22 24 25 26 26 26 25 25 25 25 26 24 2424 424 4 150 lb/sq.in. 11-15 Cr 5 9 13 16 19 21 25 25 26 27 27 27 27 27 26 26 26 26 2626 626 6 150 lb/sq.in. 304 6 10 15 i6 19 22 26 28 50 50 51 51 51 50 50 50 50 29 29 92 92 500 lb/scj.in. acim. 7 6 8 10 12 i4 17 19 21 22 22 22 22 22 22 22 25 24 24 2 52 5r 500 lb/si. in. monel 8 11 14 17' 20 24 31 55 57 39 39 40 4o 40 41 41 41 41 4141 242 2H 3001lb/si. in. 1 1/4 Cr 9 4 5 6 6 6 7 7 7 8 8 8 8 8 9 9 1010101 11 11 500 lb/sci.in. 4 Cr 10 8 ii i4 17 19 22 24 25 26 26 26 25 25 25 25 25'26. 2626 626 6 500 lb/sci. in. 11-15 Cr 11 9 13 i6 19 21 24 26 27 27 27 27 27 27 27 27 27 27 2727 828 8 500 lb/sq.in. 304 12 10 13 i6 19 22 27 29 50 51 51 51 31 51 50 50 50 50 5151 151 1 450 lb/sq. in. aclm. 15 6 8 10 12 i4 17 19 21 22 22 22 22 25 25 25 24 25 2526 727 7 45o lb/sq. in. monel 14 11 14 17 20 24 51 55 57 39 39 4o 4o 41 41 42 42 42 42424 s s 4501lb/scp~in. 11/4 Cr 15 4 5 6 6 6 7 7 7 8 8 8 9 9 10.10 10 11 1111 212 2 45o lb/sq.in. 4 Cr 16 8 11 14 17 19 22 24 25 26 26 26 26 25 26 26 26 26 2627 727 6 45o lb/sq_.in. 11-13 Cr 17 9 13 16 19 21 24 26 27 27 27 27 27 27 27 27 28 28 28 82E92 45o 1b/sqi.in. 304 18 10 15 16 19 22 27 29 50 51 51 51 51 51 50 50 50 50 50531 15 6oo lb/sq. in. aclm. 19 6 8 10 12 14 17 19 21 22 22 22 22 22 25 25 24 25 2526 727 7 6oo lb/sq. in. monel 20 11 14 17 20 25 21 24 25 36 37 39 39 4o 41 41 42 43 45 54 54 6001lb/sq_.in. 11/4 Cr 21 4 5 6 6 6 6 7 7 7 7 7 8 8 9 9 10 10 101 2 21 600 1b/sl. in. 4 Cr 22 8 ii 14 17 19 21 25 24 24 24 25 25 25 25 25 25 26 2727 828 8 600 1b/sq. in. 11-15 Cr 25 8 ii 14 17 19 20 22 24 25 26 26 26 26 27 27 27 28 29 95c05 6oo lb/s4. in. 304 24 9 15 16 19 21 25 27 29 29 29 29 29 50 50 50 50 51 52 5555555

TABLE III COMPUTER VARIABLE BACST (VTCST, U). COST IN DOLLARS PER SQUARE FOOT FOR TEMA CLASS R ALL-STEEL HEAT EXCHANGERS nom. shell size-in. 5 6 8 10 12 14 16 i8 20 22 24 26 28 30 52 34 36 38 ho 2 45 8 U 1 2 3 4.5 6 7r 8 9 10 11 12 13 14 15 *i6 17 i8 19 2 21 2 VTCST 3/4 inch tubes 150 lb/sq.in. 1. 24.00 20.00 16.oo 13.00 10.00 8.05 6.70 5.88 5.40 5.10 5.15 4.8o 4.55 4.3o 4.15 4.00 3.90 3.803.0.5 49.0 3/4 inch tubes 300 lb/sq.in. 2 25.00 21.00 17.00 14.oo 10.7o 8.63 7.30 6.42 5.98 5.68 5.51 5.27 4.98 4.75 4.6o 4.46 4.32 3,25 4i ~o.o37 3/4 inch tubes 450 lb/sq.in. 3 26.00 22.00 i8.oo 15.10 11.90 9.68 8.15 7.20 6.68 6.3o 6.00 6.00 5.70 5.50 5.32 5.20 5.05 4.954.0.7 465.4 3/4 inch tubes 6oo lb/sq.in. 4 28.00 24.00 20.00 16.20 12.90 lo.68 9.10 8.03 7.22 6.94 6.90 6.6o 6.3o 6.10 5.90 5.75 5.60 5.505.0.5515.0 1 inch tube 150 lb/sq.in. 5 27.00 23.00 19.00 15.40 11.90 9.30 7.55 6.50 5.8o 5.48 5.40 5.10 4.8o 4.55 4.4o 4.20 4.10 4.oo3.038 37036 1 inch tube 300 lb/sq.in. 6 28.00 24.00 20.00 16.20 12.50 9.85 8.08 6.95 6.3o 5.95 5.95 5.60 5.34 5.10 4.90 4.75 4.63 4.524.7.3 425.1 1 inch tube 450 lb/sq.in. 7 29.00 25-.00 21.00 17.30 14.00 1.1.00 9.05 7.82 7.18 6.80 6.90 6.55 6.25 6.00 5.8o 5.62 5.50 5.40o.751.049 1 inch tube 600 lb/sq.in. 8 30.00 26.00 22.00 18.20 15.10 12.15 10.05 8.67 7.93 7.60 7.70 7.33 7.00 6.70 6.5o 6.3o 6.18 6.oo5.6.7 56o.0

-23in an array called CAFH(VCFH, U). The data for steel heat exchanger costs were also stored, Tube cost data, Table III, were obtained from Figure 7 and a like figure valid for 1 inch diameter tubes, and stored in an array called BACST(VTCST, U). As can be seen in the table when VTCST = 1, the data are for standard 150 lb,/sq.in. units having 4 inch diameter tubes and when VTCST = 6, the data are for standard 00 lbsqin units having 1 inch diameter tubes, The multiplying factors used in heat exchangers having mixed pressures were also stored in arrays. Table IV gives the multiplying factors for the shell and cover. In the array called MUSAC(INDEXB, INDEXC), INDEXB refers to the standard working pressure required for the shell and cover and INDEXC refers to the imaximum standard working pressure required, TABLE IV COMPUJTER VARIABLE MTJSAC (TrDEXB,, INDEXC) (Multi~plying Factors for Mixed Pressures) I1NDEXC 1 2 54 i.NDEXB 1 l.00 0~,92 0.8l 0.75 2 0.00 l1,00 o.89 0o,83 5 0.00 0.00 l,00 0,92 4 0'00 0.00 0.00 LOG0 The completed version of the computer routine used to calculate the heat exchanger cost and the other pertinent economic information is

1. WHENEVER STWPT *G. STWPS 2. MAXWP = STWPT 3. OTHERWISE 4. MAXWP = STWPS 5. END OF CONDI T IONAL 6. INDEX = 6*(MAXWP/150 -1) 7. VTS = INDEX + KTS 8: VSC = INDEX + KSC 9. VCFH = INDEX + KCFH 10. INDEXA = STWPT/150 11. INDEXB = STWPS/150 12. INDEXC = MAXWP/150 13. WHENEVER OD.G. 0.750 14. INDEXD = 4 15. OTHERWISE 16( INDEXD = 0 17. END OF CONDITIONAL 18. VTCST = INDEXD + I'NDEXC 19. CSTPSF = BACST(VTCST,U)*(100Q*O + TSAB(VTSU) + SAC(VSCU)* 1MUSAC(INDEXB-INDEXC) + CAFH(VCFHTU)*MUCFH(INDEXAINDEXC)) 1/100.0 20. AOTST =3o1416*OD*TUBEL*XMAXT/12.0 21. TWTS = XMAXT*TUBEL*WTFTS 22. STUCST = (BASES + 61.0*EXP.(-0.93*TWTS))*TWTS 23. TWT=X*TUBEL*WTFT 24. TUBCST=(BASE+61*EXP.(-0.93*TWT) )*TWT 25. CONCST = CSTPSF*AOTST*CSTIND + (TUBCST -STUCST)*TCSTI 26. INS5TCT =(CONCST*INSTPC)/loU.Q) 27. INCOCT = CONCST + INSTCT 28. YRCAP = INCOCT/AMORD 29. MAINT = AOTST*MCSTF 30* YROPER = POWCST + FLUCST + MAINT 31. YRTOT = YRCAP + YROPER Figure 8. Computer Routine to Calculate the Economic Information Required in the Search Analysis. The Expressions are in the MAID Language (Michigan Algorithm Decoder).

-25programs written in the MAD language into machine-language routines for execution on the IBM-704-709-7090 computers) was developed and is used extensively at The University of Michigan with the IBM 709 digital computer. The names of the variables are, in general, selfexplanatory. The only symbol needing special explanation is the which stands for "greater than". In statement 1. the standard working pressure for the tube side, STWPT, is compared to that on the shell side, STWPSO When the tube side pressure is greater, the maximum working pressure, MAXWP, is set equal to STWPT. Otherwise, MAXWP is set equal to STWPS. Statement 6. is used in calculating the array indices. As seen in Table II, the data for tube sheets are stored according to materials of construction. The first six values of VTS refer to different materials at a working pressure of 150 lb./sqiin, the second six values refer to the same materials at a working pressure of 500 lb./sq~in., etc, The value of INIDEX, therefore, fixes the general location in the array. In statement 7., the exact value of VTS is fixed by adding to the value of INDEX the value of KTS which stands for the type material used. For example, when KTS is 2., monel tube sheets are used and when KTS is 6, stainless steel tube sheets are used. KSC stands for the shell and cover material and KCFH- stands for the channel and floating head material. Statements l0,,, ll,,, and 12. are used to calculate the indices required in the multiplying factors. Statements 15, - 18. are used to determine the index VTCST required in the array containing the all steel heat exchanger costs, Table III. INDlEXD is

-26fixes the general region within the array. The variable INDEXC fixes the working pressure within the general region and therefore fixes the value of the index VTCST in statement 18o In statement l9o the cost per square foot of heat transfer surface is calculated as was done in the example problem. In statement 20. the total area of plain steel tubes is calculated based upon the maximum number of tubes that can be placed in the shell. The steel tube weight is calculated, (statement 21), and the steel tube coat computed (statement 22~)~ The form of the equation in statement 22. has already been discussed, The total tube weight and tube cost for alloy tubes is determined in statements 23. and 24~ In statement 25, the condenser cost is found by first determining the cost of the desired unit having steel tubes, adding the cost of alloy tubes and subtracting the cost of the steel tubes. CSTIND and TCSTI refer to cost indices. The installation cost is calculated in statement 26, and the total installed cost given in statement 27. The yearly capital expense is then found by dividing the installed cost by the amortization rate. Yearly operating costs are calculated in statement 50~ by adding the costs of electric power, tube side fluid (water), and maintenance, and the yearly total cost is calculated by adding the yearly fixed capital expense and the yearly operating costs. EXAMPLE PROBLEM To illustrate the feasibility of the three search methods described, an example problem was solved using each method, The problem considered was to design a 6o,0ooo lbo/hr. n-propanol condenser for a recommended process such that the yearly cost (operating and fixed capital

-27expense) is a minimumo The design conditions were as follows condensing load 60, 0001b./hr. inlet saturated vapor temperature 220~F maximum inlet water temperature 80~F winter inlet water temperature 50~F maximum exit water temperature 140~F allowable shell side pressure drop 3~0 lbsqino/ allowable tube side pressure drop 10.O lbsqin. water cost O. OOl/gal electricity cost $oooo8/kwhr Certain additional restrictions were imposed to reduce the nuber of required trials. They were: tube material admiral ty tube sheets admiralty shell and cover 1 1/4 per cent chrome floating head cover 4-6 per cent chrome tube arrangement square tube diameter 3/4 in. tube pitch 1 irn, All the trial designs were done on an IBM 709 computer at the University of Michigan using a condenser program prepared by the author. After reading in the design conditions, economic data., additional restric-~ tions, tube length, tube passes and water velocity, the computer calculates

-28load under summer conditions. The number of tubes required invariably is less than the number of tubes that could be placed into a standard shell size. The tubes are then fitted into the bottom portion of the shell on the specified pitch until all the required tubes are placed in the shell The shell side pressure drop is calculated based on the vapor flowing down across the tubes. It was assumed that the top portion of the shell would be sufficiently open to permit unrestricted flow of the vapor at the top of the condenser. If the shell side pressure drop affects the condensing temperature, the logarithmic temperature is corrected and is used to obtain a more accurate solution. Finally, the desired economic information is calculated based on the economic input data. The water cost is based on the summer requirements o In the second part of the computer program, the calculated standard shell size is f illed completely with tubes and the shell side pressure drop is determined based on the baffled flow of the vapor. The pressure drop is used to obtain the "exit" vapor temperature which is used in calculating the logaritlimic temperature difference. Since there are more tubes than required, the water velocity is reduced until the sumer operating conditions are satisfied. In similar fashion, the wat~er requirements are found for the winter operating conditions. The yearly water cost is based on an average of the summer and winter requirements. (water costs were based on actual usage rather than on maximum demnand.) The desired economic information is again calculated and printed. The net result is that two complete designs are completed for each set of input dataI

-29Sixteen trials were allotted to each search method. Eight trials were used to evaluate plain tube designs and eight were used to evaluate finned tube designs. The results of the investigation are given in Tables V, VI, and VII Only tube length, water velocity, and tube passes were considered as free variables because of the limited number of trials considered. In the factorial method two values of each variable were tried. As can be seen from Table V, the minimum cost plain tube design is $6886/yr. when a 2 pass, 8 ft. tube bundle is used with a 4 ft./sec. water velocity. A finned tube unit proved to be somewhat better. The best finned tube design would require $6125/yr. and would have a 2 pass, 10 ft. bundle with a water velocity of 10 ft./sec. As can be further noted, many of the trial designs were only partially satisfactory because the maximum allowable exit water temperature -was exceeded. In other cases the design was totally unsuccessful'because either the maximum exit water temperature was exceeded or the maximum allowable tube side pressure drop was exceeded. These results indicate that only a limited region of satisfactory operation exists. This is especially true for finned tube units. Even though they are less expensive to purchase and operate, the factor space of satisfactory operation for a finned tube condenser is even more restricted than for a plain tube condenser. The results of the univariate method are given in Table VI, More successful plain tube designs were obtained with this method but the Dest design was inferior to the best design obtained by the factorial

TABLE V TRIAL DESIGNS AN~D CALCULATED RESULTS FOR A FACTORIAL SEARCH PATTERN FOR TEE DESIGN OF A n-P1ROPANOL C0ONDENSER Plain Tube Trial Designs Trial Trial Trial Trial Trial Trial TrialTra 1 2 3 4 5 6 78 tube length-ft 8.o 8.0 8.o 8.o 12.0 12.0 12.0 1. water velocity-ft./sec. 4.o 8.0 4.o 8.o 4.o 8.o 4.0o. tube passes 2 2 4 4 2 2 44 total yearly cost-$/yr. 6886 8705 -- comments excessive excessive excessive excessive excessive ecesv exit water exit water exit water exit water exit wateluesd temperature temperature temperature temperature temperatue prsuedo cost - $/yr. 9645 14627 7208 9657 8217 11780 ---- (based on partially filled tube bundle and summer water requirements) Finned Tube Trial Designs tube length -ft. 8.o 8.0 8.0 8.0 12.0 12.0 12.0 1. water velocity-ft./sec. 10.0 14.0 10.0 14.0 10.0 14.0 10.0 1. tube passes 2 2 4 4 2 2 44 total yearly-cost-$/yr. 6125 ---- ---- comments excessive excessive excessive excessive excessive excessive ecesv tube side exit water tube side exit water tube side exit water ei ae pressure drop temp. pressure drop temperature pressure drop temperatue teprue (based on partially filled tube bundle and summer water requirements)

TABLE VI TRIAL DESIGNS AND CALCULATED RESULTS FOR AN UNIVARIATE SEARCH PATTERN FOR THE DESIGN OF A n-PROPANOL CONDENSER Plain Tube Trial Designs Trial Trial Trial Trial Trial Trial Trial Trial 1 2 3 4 5 6 7 8 tube length-ft. 10.0 10.0 8.o 12.0 16.0 10.0 10.0 10o.0 water velocity-ft./sec. 6.o 6.0 6.o 6.0 6.o 3.0 4.5 7.5 tube passes 2 4 2 2 2 2 2 2 total yearly cost-$/yr. 7447 - 8753 ---- --- -- 7470 7470 comments excessive excessive excessive excessive exit water exit water exit water exit water temperature temperature temperature temperature cost-$/yr. (based on 10711 7772 11908 9941 8687 7974 9147 12343 H partially filled tube bundle and summer water requirements) Finned Tube Trial Designs tube length-ft. 10.0 10.0 8.o 12.0 16.0 10 1 0. 0 10.0 water velocity-ft./sec. 12.0 12.0 12.0 12.0 12.0 8.o 9.0 lO.0 tube passes 2 4 2 2 2 2 2 2 total yearly cost-$/yr. _ ---- - __ - -- - comments excessive tube side pressure drop excessive exit water temperature excessive exit water temperature cost-$/yr. (based on ---- ---- ---- ---- 6629 7017 partially filled tube bundle and summer water requirements)

TABLE VII TRIAL DESIGNS AND]] CALCULATED RESULTS FOR A RAN'DOM SEARCH PATTERN FOR TEE DESIGN OF A n-P1ROPANOL C0ONDENSER Plain Tube Trial Designs Trial Trial Trial Trial Trial Trial TrialTra 1 2 3 4 5 6 78 tube length-ft. 12.0 8.0 8.o 16.0 16.0 8.o 12.0 1. water velocity-ft./sec. 7.0 7.0 5.0 4.0 10.0 5.0 5.06. tube passes 2 2 4 2 4 2 44 total yearly cost-$/yr. --- 8772 ---- 8726 -- comments excessive excessive excessive excessive excessiveexsiv exit water exit water exit water tube side exit water ei ae temperature temperature temperature drop pressure temperature teprue cost-$/Yr. (based on 10846 13253 7520 8073 ---- 10615 ---- 77 partially filled tube bundle and summer water requirements) Finned Tube Trial Designs tube length-ft. 12.0 10.0 8.o 8.0 16.0 10.0 12.0 1. water velocity-ft./sec. 12.0 6.o 18.0 16.0 6.o 12.0 18.0 6. tube passes 4 2 4 4 2 2 22 total yearly cost-$/yr. ---- ---- -------------- comments excessive excessive excessive excessive excessive excessiveexsiv exit water tube side tube side exit water tube side tube sidetuesd temperature pressure drop pressure drop temperature pressure drop pressure drop prsuedo cost-$/yr. (based on ---- ---- ---- ---- --------- partially filled tube bundle and summer water requirements)

-33The random method had the poorest performance as shown in Table 7. The best plain tube design obtained was approximately $2000/yr., higher than for the best design obtained using the factorial method There were no satisfactory finned tube condenser designs. The total computer time required to process all the trial designs was minutes at a cost of $7000. This is about $150/trial or about $24.00/method if both plain and finned tube condensers are considered. CONCLUSIONS It is difficult to make any definite conclusions from the results of one problem, but it appears obvious that the techniques can be used to advantage with a considerable saving in both time and labor. Certainly more trials should be used once the desired search method. is selected. There seems to be little doubt that a less expensive design could have been achieved if a second level search had been carried out with both. the factorial and univariate methods. There is a limit to the number of trials that can be made, however, before the computer expense exceeds the savings in finding a better design. The best results will probably be obtained by using 8-10 trials to broadly search the factor space. The results of the first level of search would then be used to closely search a more limited region to obtain the best design conditions. Too few initial trials could be uninformative while too many trials would not be economical because of the large number of unsuccessful trials that would be obtained.

_34 - REFERENCES 1. Box, G. E. P., and Wilson, K. Bo, "On the Experimental Attainment of o~p~itimum Condits," J. Roy Stat. Soc. B 13, 1 (1951). 2. Standards of Tubular Exchanger Manufacturers Association," 4th Edition, TEM New York, 1959.. Brooks, S. H., "A discussion of Random Methods for Seeking Maxima," Oper. Res., Vol. 6, No. 2 (April 1958). 4. Brooks., S. H., "A Comparison of Maximum-Seeking Methods," Operu Res., Vol. 6, No. 4 (July-August 1959). 5. Flaxbart, E. W., and Schirmer., D. E., "Economic Considerations in Shell and Tube Heat Exchanger Selection," Chem, Eng. Progr.,, Vol 57, No. 12 (1961), 6, Sieder, E. N., and Elliott, G. H., "How to Check Costs when Selecting Material for Heat Exchangers.," Pet, Ref., Vol. 59., No. 5 (1960).

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