A new method to design heat exchangers is proposed, which is based on the process description by Kays and London and lends itself very well to optimization.\ud
The method is described by applying it to the economic optimization of a counter-current exchanger, the extension to other flow configurations being selfexplanatory.\u

Fontein, H.J.

Groot Wassink, J.

English

A new method to design heat exchangers is proposed, which is based on the process description by Kays and London and lends itself very well to optimization. The method is described by applying it to the economic optimization of a counter-current exchanger, the extension to other flow configurations being selfexplanatory

University of Twente Research Information

Engineering and Process Economics 3 (1978) 141-149 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands 141 THE ECONOMICALLY OPTIMAL DESIGN OF HEAT EXCHANGERS H.J. Fontein and J. Groot Wassink Department of Chemical Engineeting, Twente University of Technology, Enschede (The Netherlands) Abstract A new method to design heat exchangers is proposed, which is based on the process description by Kays and London and lends itself very well to optimization. INTRODUCTION THE DESIGN PROCEDURE According to the process description of heat exchangers by Kays and London [ 1 I the number of overall heat transfer units, NTU, can be computed as a function of the effi- ciency E, the ratio between the heat capacity rates of both fluids, and the flow configura- tion. This description offers some advantages over the traditional log mean temperature approach, the most important ones being the thermodynamic significance of E and the greater ease of computation. From the NTU a value for the product of the overall heat transfer coefficient and the required exchanger surface is obtained. Both depend on the lay-out of the exchanger and on the process conditions in a rather com- plicated way. A suitable rearrangement of the pertinent equations and a judicious choice of the design variables result in an acyclic computational scheme, which because of its straightforwardness is well adapted to an optimization program. The method is described by applying it to the economic optimization of a counter- current exchanger, the extension to other flow configurations being selfexplanatory. The problem to be solved is the optimal design of a heat exchanger for a specified duty, i.e. the two flows and their entrance and exit temperatures are known. Dodge [2] introduced the heat transfer effectiveness, which is defined as the ratio between the actual heat transfer rate and the maximum possible rate, as would be realized only in a counterflow exchanger of infinite transfer area: ch (th,in - th,out) Cc @c,out - tc,in) E= = (1) cmin (th,in - tc,in) cmin @h,in - tc,in) Kays and London [ 11 found that it is pos- sible to express E as a function of NTU, Y = CminlCmax and the flow arrangement; in the case of counterflow for instance: 1 - exp (- NTU (1 - 7)) ’ - 1 - yexp (- NTU (1 - 7)) (2) or 142 NTU = 1 -Y One of the more common arrangements for shell and tube construction is multipass cross counterflow; it is realized by baffling the flow around the tubes. The E-NTU relation for this arrangement depends on the number of passes, the above given counterflow relation being its limit if this number tends to infinity [ 1 I. If the number of passes is higher than 3 the counter- flow relations can be used as an acceptable approximation. Eq. 1 can be written as: E= ca QI ,in - in ,out) c2 tr2 ,out - t2 jn) = (la) cmin (t, ,in - t2 ,id cmin @I ,in - t, ,in) where the indices 1 and 2 apply to the flow through and around the tubes respectively. Use of eq. la as compared to eq. 1 sim- plifies the computer program because it now is unnecessary to specify whether the hot or the cold fluid flows through the tubes. With the NTU, which is an overall figure, two partial ntu’s correspond. Combining the definitions UA U,A, NT- = ~ = __ C min Cmin and akA k ntUk = __ kf 0321 with the overall heat transfer equation 1 1 1 2dw 111 u, -=-++---+ +- +- u,A, a,A, Oz ~ww(A+p) A, A, the following relation is obtained: To describe the dependence of ntu on the hydrodynamic conditions and geometrical dimensions, equations are derived as follows. Neglecting the potential and kinetic energy terms reduces the Bernoulli equation to: 2 dp 2 s- + s dW=O 1 p 1 Introducing the Fanning equation and inte- grating: AP 2 _= =f.L_ .1 “2 p 1 s f.f v2 . d” ‘h h 2 or P Eu zi ~ =f!- ‘hpV= f-h Because C=PVA, cp. stzO1_ = &C AC Yh ~ = ntu - = ntu -- VPCp c A L Combining with jh = st Pr 2/3 it is apparent that L ntu ntu - =_=_ P,2/3 ‘h St ih and, from eq. 4 Eu = f !? p,2/3 ih (5) Relations 4 and 5 apply to both flows in the heat exchanger. For the flow through the tubes the follow- ing equations hold: (3) 4rh,, = d, - 2d, (6) 143 ll AC,, = n -(do - 2d,)Z 4 or Re, = 4% n n (d, - 2d,) V$ (7) If the flow is turbulent jh,, = 0.027 Re, -Oi2 (8) and f, = 0.05 Re,-0.2 (9) For the flow around the tubes, which is assumed to be baffled, acyclicity of the computational structure requires a rearrange- ment of the equations: @z * 4rh,l Re, = ~ AC., l Yl 4rh,z = doAx where A, depends on the tube configuration. s-d Ac,z s 2 _ DB = 2 (1 - -:) DB S and consequently Re, = 4~2 4, Ax 2(1-+) DB v2 Because of L/s D and thus: B B= ~C,RP_,~.~L, D d, A,ntu,Pr,2/3 Elimination of respectively B and Re, from the above expression for Re, and B results in the two following equations: Re, = ( _+ )5’3 . (n%)5’3. *1 (l-x)L, . p*, 10/9 B 1 =- L, 2 ( v2 ‘:“’ “‘) 213 . (8y3 . 2 713 . p,,-1019 (11) (12) To compute Eu, the following equation is used f, = Cf Re,-O.l* (13) From the tube configuration two equations can be derived. For triangular pitch for in- stance 2 x J3 A, = -1 (14) n jh,* = CnRe, -“.4 (10) and I31 : eliminating Eu, from eq. 4 and 5 leads to: L, = d, A, ntu Prs213 z 4 Cn Re,-0.4 From the geometry of the exchanger it fol- lows that: D = D (d,, X, n) (15) The coefficients C, and Cf in the eq. 10 and 13 also depend on the tube configura- tion [l]: 144 Cf = Cf (xl (17) It is obvious that: A, = nn(d, -2d,) L, (18) and : A, = rind,,,,, (19) The wall thickness of the tubes depends on the tube diameter and the pressures of the two fluids: d, = d, Cd,, P) (20) The set of eq. 3 through 20 has 4 degrees of freedom. To choose suitable design vari- ables the algorithm proposed by Lee et al. 141 was used. The following two sets result in acyclic computational schemes: I: (0, ntu,, X, d,) II: (L, , ri, X, d,) The scheme using set I is illustrated in Fig. 1. THE OPTIMIZATION To find the economically optimal design an objective function relating the costs to the design variables has to be constructed. Two kinds of costs are considered: depre- ciation on investment and operating costs. The last boundary is handled with a qua- dratic loss penalty function [ 11 I ; if any of the first 2 boundaries happens to be violated no design is possible and an arbitrarily chosen high value is assigned to the objective func- tion. To estimate the investment a number of methods has been proposed [ 5-81. The first three methods relate the investment to the heat transfer area only; the procedure given by Palen [ 81 is a logical supplement to the above design procedure because according to it investment is calculated from the dimensions of the component parts of the exchanger and the manufact- uring costs. To avoid practical difficulties the ratio /3 between tube length and baffle distance should be an integer. Both are dependent variables; introduction of this constraint would result in discontinuities in the ob- jective function. Therefore, the mathe- matically optimal design has to be adapted. This can be done in several ways, the easiest one being as follows. A new baffle distance is computed by rounding off p to the nearest integer value. This leads to new values for Re, , Eu, , NTU and E. Among the operating costs only the costs of pumping the fluids through the exchanger As a consequence the adapted heat ex- changer does not satisfy the original specifica- are dependent on the design variables; ihey can be estimated from the throughputs and the pressure losses, which follow imme- diately from the Eu numbers (eq. 4). To optimize the objective function two optimization procedures were tried: the complex method of Nelder and Mead [ 9 I and a steepest descent method [ 101. Both methods require continuous variables. Tubes being commercially available at discrete values of diameter and wall thickness only, the diameter was not included among the design variables; to find the optimal dia- meter the program is run for different tube diameters. The optimal wall thickness is the smallest standard thickness that is com- patible with eq. 20. The matter is further discussed under the heading “results”. Two of the design variables are bound by con- straints. Too small a value of ntu, results in a negative value of ntu, (eq. 3); if X < 1 the first term at the right hand side of eq. 11 is undefined. An upper boundary on X had to be introduced because literature data on the dependency of C,, and Cf on X are avail- able only for X < 1.65 [ 1 I . 145 18. 'h,l do,D ,X NTU 3 , ntul #do ,dw ntu 2 Legenda : 13 I equation - dependent variable -w- design variable ,,E”2 Fig. 1. Computational scheme in heat exchanger design. tions. The deviations will be smaller at higher values of 0; if this value should prove to be too small another type of flow pattern should be chosen anyway because the phys- ical model would not apply. To change the specifications eq. 1 a are used; inspection of these shows that several ways are open, the easiest one perhaps being re- computing both exit temperatures. Another consequence is that the costs of the final design are no longer mathematically optimal. Extensive experience, however, has taught that the optima are rather flat and the differences in the costs negligible. RESULTS AND DISCUSSION An algol program comprising the above features was implemented on a DEC-I 0 com- 146 puter. For relations that are given in graphical [ 16, 171 or in tabular form [ 151 virial equa- tions were derived from the data by means of the least squares method. As an illustra- tion, the results obtained for a heat exchanger in which two streams of air are treated at atmospheric pressure are discussed. Table 1 gives the input data. The invest- ment costs obtained with the Palen model were multiplied by a factor 3 to allow for inflation and installation; the depreciation rate was put at 10 years. To estimate the pumping costs an energy price of 5 c/kWh was used. TABLE 1 Input data In tubes Around tubes - Gn (“C) 370 40 tout CC) 94 270 @ (m3 /set) 2.5 3 do (mm) 26.9 dw (mm) 2.65 P (kg/m3) 0.596 CP (J/“C kg) 1040 lJ (m* /set) 0.000041 h (J/“C set m) 0.042 hW (J/“C set m) 60 U (J/“C set m2) 0.0003 0.0003 Table 2 gives the mathematical optima reached by the Nelder Mead and steepest descent methods after 2 17 and 95 func- tions calls respectively; in both cases the set I design variables were used. The small difference between the two optimal values strongly suggests that they are reliable TABLE 2 Mathematical optima ~._~ Starting point D ntu, X F 0.8 8.5 1.1 2.15 lo5 estimates of the true minimum, which is further borne out by the decrease of the gradient of the objective function (Table 3) and by the fact that use of the set II decision variables leads to virtually the same results. TABLE 3 Gradients of the objective function 6F/6D 6F/6ntu, 6F/6X Starting point 4.3 X lo5 1.8 X 10’ m-4.0 x lo6 Optimum 3.9 x lo-’ 7.7 x 10-z --1.6 X 10-r Table 4 gives the finzl optimal design. Comparing this table with Table 2 it can be seen that the relative difference between the total costs of the final design and the mathematical optimum is only about 1%. Table 5 illustrates the influence of energy price on the optimal design. As might be TABLE 4 Final optimal design __~ In tubes Around tubes tout c-0 D Cm) L (ml B (ml n X 90.54 272.88 RTIJ 1.219 4.943 0.989 624 1.636 0.847 3.919 lJa (JPC set m*) 29.04 23.31 ntu 6.09 9.61 cz (J/‘C set m*) 45.14 68.58 Re 5723 4068 Eu 8.11 43.39 A (m’) 209 260 v (m/set) 10.9 6.4 AP (atm) 0.0029 0.0013 I) Depreciation _ ($/yr) Pumping costs (WYr) Total $/yr) 0.0055 4301 Mathematical optimum Nelder Mead 1.2191 6.0922 1.6358 5129.9466 940 5241 Steepest Descent 1.2191 6.0926 1.6356 5129.9465 UReferring to inner tube surface. - 147 TABLE 5 Optimal designs for increasing energy prices Energy price ($/kWh) 0.03 0.04 0.05 0.06 0.08 Mathematical optimum Total costs ($/yr) Pumping costs a 4124 4947 5130 5286 5546 15.9 16.2 16.4 16.5 16.8 Final design D (ml L (ml B (ml n X Ub(JpCsecmZ) A b (m’) r1 (m/set) vz (mlsec) r) Depreciation ($/yr) Pumping costs ($/yr) Total costs ($/yr) 1.113 1.172 5.188 5.047 1.038 1.009 517 514 1.633 1.635 32.38 30.47 182 197 13.2 11.9 6.1 6.5 0.0072 0.0061 3963 4148 735 841 4698 4989 1.219 1.259 1.325 4.943 4.859 4.730 0.989 1.215 1.183 623 667 741 1.636 1.637 1.638 29.04 26.25 24.70 209 220 238 10.9 10.2 9.2 6.4 5.0 4.9 0.0055 0.0041 0.0035 4301 4403 4616 940 838 954 5241 5242 5570 St % of total costs. bReferring to inner surface. expected, increasing the energy price in- creases both the investment and the pump- ing costs; rather surprisingly, however, their ratio remains to all practical purposes constant, as follows from the third row of this table. Again the differences and between the total costs of the mathematical optima and of the final designs are satisfactorily small. Table 6 gives optimal designs for a num- ber of standard tubes. From this table it appears that shell diameter hardly depends on tube diameter, that pitch ratio depends on tube diameter if constraints do not inter- fere; and that tube length and number of tubes are strongly dependent. The two last effects, however, cancel each other to a consid- erable degree because heat exchanger surface is rather less dependent on tube diameter. These results also answer the question why tube diameter has not been introduced as a continuous decision variable and the optimal diameter rounded off to its nearest standard values as has been done with num- ber of baffles. This strategy could only have been followed with some confidence if the optimal pitch ratio and heat exchanging surface had been as nearly constant as the shell diameter. As can be seen from Table 6, the optimal designs for tube diameters less then 2 1.3 mm are too optimistic: in these cases the number of baffles is too low to warrant the use of the approximate e-NTU relation. The rigorous relation results in higher values of NTU and as a consequence in more ex- pensive designs. At a diameter of 2 1.3 mm for instance the mathematical optimum changes to 5824 $/yr. The optimal value of the tube diameter therefore, is 26.9 mm. According to Fraas and Ozisik [ 5 1 it has been found that if the ratio n of the pump- ing power requirement to the heat trans- mitted through the heat exchanger has a value between 0.005 and 0.01, the overall cost is usually fairly close to the minimum obtainable. All the minima we found confirm this statement, with the exception of the minima at an energy price of 0.08 $/kWh. 148 TABLE 6 Optimal designs for increasing tube diameters 0.05 17.2 2.35 4977 19.8 Energy price ($/kWh) 0.05 do (mm) 13.5 dw (mm) 2.35 Mathematical optimum Total costs ($/yr) 5507 Pumping costs a 23.0 Final design D (m) 1.281 L (m) 1.762 B (m) 0.881 n 2838 X 1.659 CJ b (JpC set m*) 42.65 A b(m2) 138 v1 (mlsec) 14.5 yz (m/set) 6.8 r) 0.0080 Depreciation ($/yr) 4278 Pumping costs ($/yr) 1358 Total costs ($/yr) 5636 ‘In % of total costs. b Referring to inner surface. 1.247 2.621 1.311 1632 1.650 32.38 168 12.5 4.7 0.0053 3981 897 4878 -__ LIST OF SYMBOLS Latin characters A AC AX B C Cf, Cn CP d D Eu ; ih L NTU n ntu Pr P Re rh St s c u heat exchanging surface (m*) cross sectional area (m*) factor defined in text baffle distance (m) capacity flow (J/“C set) factors in Colburn equations heat capacity (J/oC kg) diameter or thickness (m) shell diameter (m) Euler number objective function ($/yr) friction factor Colburn factor length of flow path (m) number of overall transfer units number of tubes number of partial transfer units Prandtl number pressure (N/m* ) Reynolds number hydraulic radius (m) Stanton number centre-to-centre distance of tubes (m) temperature CC) fouling factor (J/“C set ma) 0.05 0.05 0.05 21.3 26.9 33.7 2.65 2.65 3.25 4929 5130 5542 18.1 16.4 15.4 1.259 1.219 1.211 3.467 4.943 6.562 1.156 0.989 1.094 1068 623 402 1.650 1.636 1.594 30.63 29.04 26.63 186 209 225 11.6 10.9 10.7 5.2 6.4 6.1 0.0050 0.0055 0.0052 4030 4301 4690 854 940 883 4884 5241 5574 u overall heat transfer coefficient (J/“C set m* ) V linear velocity (m/set) W frictional resistance energy (J/kg) X pitch ratio Greek characters partial heat transfer coefficient (J/“C set m2) ratio defined in text ratio of capacity flows heat transfer effectiveness energy ratio defined in text thermal conductivity (JPC set m) kinematic viscosity (m* /set) density (kg/ma) volume flow (m3/sec) Subscripts, not defined above ii in k out min max W 1 2 cold hot in index out minimum maximum wall inside tubes outside tubes REFERENCES 149 BIOGRAPHICAL NOTE 1 2 3 4 5 6 7 8 9 10 11 Kays, W.M. and London, A.L., 1964. Compact Heat Exchangers, McGraw-Hill, New York. Dodge, B.F., 1944. Chemical Engineering Thermo- dynamics, McGraw-Hill, New York. V.D.I., 1963. Warmeatlas, V.D.I.-Verlag, Dusseldorf. Lee, W., Christensen, J.H. and Rudd, D.F., 1966. A.1.Ch.E. Journal, 12(6): 1104. Fraas, A.P. and Ozisik, M.N., 1965. Heat Exchanger Design, John Wiley & Sons, New York. Chilton, C.H. (Ed.), 1960. Cost Engineering in the Process Industries, McGraw -Hill, New York. Guthrey, K.M., 1969. Chem. Eng., 24(3): 114. Palen, J.W., Cham, T.P. and Taborek, J., 1974. A.1.Ch.E. Symp. Series, 70 (138): 205. Nelder, J.A. and Mead, R., 1965. Computer J., 7 : 308. Fontein, H.J. and Groot Wassink, J., 1974. Verfahrens- technik, 8: 200. Lootsma, F.A., 1970. Boundary Properties of Penalty Functions for Constrained Minimizations, Ph.D. Thesis, Technical University of Eindhoven, The Netherlands. H.J. Fontein is a senior research worker in the Chemical Engineering Department at the Twente University of Technology, where he specializes in the application of optimization methods to the design of chemical plants. A chemical engineering graduate of the Delft Technological University, he previously worked in industry for many years. J. Groot Wassink is a professor of process engineering in the same Department at Twente. He holds two master’s degrees, in chemical engineering from Delft and in philosophy from the catholic university of Nijmegen. He has many years industrial experience in chemical process design, pro- duction and management. 

The economically optimal design of heat exchangers

Chemical Engineering Thermodynamics,

Compact Heat Exchangers,

Heat Exchanger Design,

https://research.utwente.nl/en/publications/7b198e4c-ce99-4660-a155-a10c05d209db

The economically optimal design of heat exchangers

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