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Available online at www.sciencedirect.com

Chemical Engineering and Processing 47 (2008) 906–913

Design and optimization of multipass heat exchangers

J.M. Ponce-Ortega a,b, M. Serna-Gonzalez a, A. Jimenez-Gutierrez b,∗a Facultad de Ingenierıa Quımica, Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Mexico

b Departamento de Ingenierıa Quımica, Instituto Tecnologico de Celaya, Celaya, Mexico

Received 28 April 2006; received in revised form 11 February 2007; accepted 12 February 2007Available online 23 February 2007

bstract

In this paper, a simple algorithm is developed for the design and economic optimization of multiple-pass 1–2 shell-and-tube heat exchangers in

eries. The design model is formulated using the FT design method and inequality constraints that ensure feasible and practical heat exchangers.imple expressions are obtained for the minimum real (non-integer) number of 1–2 shells in series. A graphical method is also presented to developome insight into the nature of the optimization problem. The proposed algorithm enables engineers to design optimum multipass heat exchangersuickly and easily. It is also shown how the method can be applied for optimal design of multipass process utility exchangers.

2007 Published by Elsevier B.V.

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eywords: Multipass heat exchangers; Design; Optimization

. Introduction

Multiple-pass shell-and-tube heat exchangers allow thermalxpansion and easy mechanical cleaning, as well as longerowpaths for a given exchanger length. In addition, the highelocities achieved for the tube fluid help increase heat transferoefficients and reduce surface fouling [1].

The design of multipass exchanger involves the determina-ion of the heat transfer area and number of shells in series for apecified heat duty. A typical solution to this problem involvestrial-and-error graphical method to determine the number of

hells in series for values of the FT correction factor equal tor greater than 0.75 (see for instance, Kern [2] and Ahmad etl. [3]). The size of the exchanger is then found from the basicesign equation. This graphical method is often tedious and timeonsuming. Furthermore, the FT correction factor may be diffi-ult to compute, particularly in the steep regions of the FT charts.akheri [4] has presented some explicit expressions that avoid

he difficulties associated with the use of the FT charts.Ahmad et al. [3] introduced a new parameter, Xp, and derived

imple equations to calculate the number of shells in seriesxplicitly, in terms of the heat capacity ratio, R, the heatxchanger thermal effectiveness, P, and the parameter Xp. Thus,

∗ Corresponding author. Tel.: +52 461 611 7575x139; fax: +52 461 611 7744.E-mail address: arturo@iqcelaya.itc.mx (A. Jimenez-Gutierrez).

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255-2701/$ – see front matter © 2007 Published by Elsevier B.V.oi:10.1016/j.cep.2007.02.004

f the values of Xp and the terminal temperatures are speci-ed, these expressions allow a more straightforward solutionor multiple-pass heat exchangers problems than the graphicalethod. In addition, these equations are useful for targeting and

ynthesis of multipass heat exchanger networks.As stated by Ahmad et al. [3] the Xp parameter must be spec-

fied by the designer such that 0 < Xp < 1. Since the Xp valuef 0.9 guarantees that FT ≥ 0.75 almost over the full range of, Ahmad et al. [3] suggest the general use of that value toimplify calculations. As an alternative approach, Shenoy [5]eported some equations to estimate Xp for different values of. However, these approximations can predict values of Xp thatield unfeasible exchangers. To avoid this limitation, Moita et al.6] attempted to quantify a trade-off between large heat-transferrea at high Xp and large number of shells in series at low Xp

alues, through the minimization of the investment cost of thexchanger. As a constraint, a lower bound on the FT correc-ion factor of 0.75 was used. Similarly and based on the logicalequirement that Ns must be integer, Ponce-Ortega et al. [7] pro-osed a mixed integer non-linear programming formulation forhis optimization problem. As an additional constraint in thatpproach, an upper value for the area per shell was specified.

In this paper, we present an optimization method for the

esign of multiple-pass shell-and-tube heat exchangers. Theethod is simpler and more efficient than previous algorithms.imple analytical expressions for minimum real (non-integer)umber of 1–2 shells in series are derived. A simple design

J.M. Ponce-Ortega et al. / Chemical Engineering and Processing 47 (2008) 906–913 907

Table 1Basic design equations

Description Equations

Energy balance hot fluid Q = CPh(Thi − Tho) (1)Energy balance cold fluid Q = CPc(Tco − Tci) (2)Design equation Q = UAFT �TLM (3)

Thermal effectiveness P = Tco − Tci

Thi − Tci(4)

Heat capacity ratio R = CPc

CPh= Thi − Tho

Tco − Tci(5)

log-mean temperature �TLM = (Thi − Tco) − (Tho − Tci)

ln((Thi − Tco)/(Tho − Tci))for R �= 1 (6)

log-mean temperature �TLM = Thi − Tco = Tho − Tci for R = 1 (7)

FT correction factor FT =√

R2 + 1 ln[(1 − P1,2)/(1 − RP1,2)]

(R − 1) ln ((2 − P1,2(R + 1 − √R2 + 1))/(2 − P1,2(R + 1 + √

R2 + 1)))for R �= 1 (8)

Effective P1,2 P1,2 =(

1 −(

1 − RP

1 − P

)1/Ns) (

R −(

1 − RP

1 − P

)1/Ns)−1

for R �= 1 (9)

FT correction factor FT =√

2P1,2

(1 − P ) ln ((2 − P (2 − √2))/(2 − P (2 + √

2)))for R = 1 (10)

E + Ns

aouefloltg

2

tob

w

o2wtRb

P

atwI

1,2

ffective P1,2 P1,2 = P

P − NsP

lgorithm is developed taking advantage of the fact that Ns cannly take integer values. It should be noted that the design modelses directly the FT correction factor instead of the Xp param-ter. First, we consider the problem in which the heat capacityow rates of both fluids are known and one wishes to find theptimal number of shells in series. Then, we consider the prob-em of finding both the optimal number of shells in series andhe optimal mass flow rate of a process utility exchanger for aiven heat duty.

. Mathematical model and constraints

In the design of multiple-pass shell-and-tube heat exchangers,he major items are the total heat transfer area and the number

f shells in series. The basic equations for such problems haveeen reported elsewhere [2,8], and are summarized in Table 1.

An examination of Eq. (8) from Table 1 shows thathen P1,2 = 2/(R + 1 + (R2 + 1)1/2), a division by zero will

otAf

Fig. 1. FT correction factor for shell-and-tube heat exchan

1,2 1,2

for R = 1 (11)

ccur in the logarithm term of the denominator. Also, when− P1,2(R + 1 + (R2 + 1)1/2) < 0, the argument of the logarithmill be negative. Thus, these conditions impose limitations on

he values one can assign to P1,2. Specifically, for any value of, the maximum theoretically attainable value of P1,2 is giveny (see Ahmad et al. [3] and Smith [1]):

max = 2

1 + R + √R2 + 1

(12)

Fig. 1 shows the relationship between the FT correction factornd P1,2 for different values of R. It shows how FT is a mono-onically decreasing function of P1,2. Note that, in the limit,here P1,2 → Pmax, the correction factor FT approaches zero.

t can be seen from Eq. (3) that, in this limiting case, the area

f the heat exchanger is infinite. It can also be seen from Fig. 1hat FT = 1 as P1,2 → 0. Thus, the minimum area requirement= Amin = Q/(U �TLM) is obtained when P1,2 → 0. It follows

rom this analysis that the region, where Eq. (8) has positive

gers as a function of P1,2 for different values of R.

908 J.M. Ponce-Ortega et al. / Chemical Engineer

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Fig. 2. Relationship between R and P1,2 for R = 0.8333 and P = 0.878.

nd, therefore, physical solutions for the FT correction factor is:

< P1,2 < Pmax (13)

Eqs. (12) and (13) also apply to Eq. (10) when R = 1.Also, for all real multiple-pass shell-and-tube heat exchang-

rs the range of the heat-transfer area satisfies:

Q

U �TLM

< A < ∞ (14)

Eqs. (9) and (11) can be rewritten to yield expressions for theeal (non-integer) number of 1–2 shells in series in terms of R,

and P1,2 (see Ahmad et al. [3]):

s = ln[(1 − RP)/(1 − P)]

ln((1 − RP1,2)/(1 − P1,2)), R �= 1 (15)

nd

s =(

P

1 − P

)(1 − P1,2

P1,2

), R = 1 (16)

For given values of R and P, the relation between Ns and1,2 may be generated from Eqs. (15) or (16). Fig. 2 showsne such graph for R = 0.8333 and P = 0.878. One can noticehat ∂Ns/∂P1,2 < 0 (i.e. Ns decreases when P1,2 increases). Onean also observe there are two asymptotes. The first one is ver-ical, and it defines the limit of infinite number of 1–2 shellseeded at P1,2 → 0. The second asymptote is horizontal (Ns,minn Fig. 2) and is obtained when P1,2 → Pmax. Ns,min repre-ents the minimum real (non-integer) number of 1–2 shells ineries theoretically needed for multiple-pass shell-and-tube heatxchangers. One can obtain similar results, for any values of Rnd P, for either limiting case of P1,2 → Pmax or P1,2 → 0.

The minimum asymptotic values of Ns are found fromhe combination of Eqs. (12), (15) or (16), and the condition1,2 = Pmax:

s, min = ln((1 − RP)/(1 − P))

ln((1 − R + √R2 + 1)/(R − 1 + √

R2 + 1)), R �= 1

(17)

and

s, min =(√

2

2

)(P

1 − P

), R = 1 (18)

FNvr

ing and Processing 47 (2008) 906–913

Therefore, we conclude that the physically allowable rangef the number of 1–2 shells in series, Ns, satisfies:

s, min < Ns < ∞ (19)

As shown in Section 3, the specification of Ns is sufficiento fix P1,2 and A. Thus, the bounds given by conditions (13)nd (14) cannot be treated as independent constraints since onlyxplicit bounds on Ns are necessary.

The above analysis shows that an increase on the FT valueeads to a decrease of the exchanger area A and P1,2, whichncreases Ns. It should be noted that, for any value of R, the lowestalue of FT corresponds to Ns = Ns,min. This value increases withhe number of shells.

Fig. 1 shows that there is a point FT = F∗T that divides the

urves into two regions. When 0 < FT < F∗T , the curves have

teep slopes. This implies that a heat exchanger design locatedn this region will not be able to handle fluctuations in processonditions. On the other hand, in the region F∗

T ≤ FT ≤ 1, theependence of FT on the thermal effectiveness becomes veryeak when P1,2 moves close to 0. Thus, a proper design shoulde such that the FT curves provide values of the correction factorbove F∗

T .It is convenient, therefore, to find the point FT = F∗

T thativides the graphical correlations of FT into two differentegions. According to Shah and Skiepko [9] it is a commonndustrial practice to use a F∗

T value of 0.8. Below this point, asay be seen in Fig. 1, FT decreases sharply because it approachesvertical slope at given values of R and P1,2. In addition, this

alue avoids the condition of temperature cross in each shell2]. Thus, for designing feasible multiple-pass shell-and-tubeeat exchangers, the following constraint must be satisfied (seehah and Skiepko [9]):

T ≥ 0.8 (20)

. Degrees of freedom and design algorithm

The heat exchanger model has 15 variables and eight inde-endent equations (Eqs. (1)–(5), (6) or (7), (8) or (10) and (9) or11), as needed). For design problems, the input parameters arehe inlet and outlet temperatures of the hot (or cold) fluid, thenlet temperature of the cold (or hot) fluid, and the heat capacityow rates of both fluids. We assume that the overall heat trans-er coefficient is also specified. Hence, there are nine remainingariables in the heat exchanger model [Q, Tco (or Tho), A, �TLM,, R, FT, P1,2 and Ns], providing one degree of freedom.

We can first notice that Eqs. (1), (2), (4), (5) and (6) or7) allow a sequential calculation of Q, Tco, P, R, and �TLM,espectively. Thus, these five variables provide the first set to beomputed.

The second subset is composed of three equations (Eqs. (3),8) or (10) and (9) or (11), as appropriate) and four variables (A,

T, P1,2, and Ns). A or Ns can be used as the best design variable;s is selected by convenience because it can take only integeralues. Such a selection provides a sequential solution for theemaining variables.

ineering and Processing 47 (2008) 906–913 909

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J.M. Ponce-Ortega et al. / Chemical Eng

It is important to notice that the design has to satisfy the con-traints given by Eqs. (19) and (20). Thus, not all integer valuesf Ns yield feasible designs. However, one can also notice thats,min sets a lower limit on meaningful values of Ns. Conse-uently, the design algorithm can be simplified and initializedy choosing an integer value for Ns equal to or higher than Ns,min.hen, FT is calculated from Eqs. (8) or (10). A search over Ns

s conducted until Eq. (20) is satisfied.Therefore, the following algorithm provides the initial feasi-

le design for the optimization problem.

1) Set input parameters: inlet and outlet temperatures of hotfluid, inlet temperature of cold fluid, heat capacity flow ratesof both the hot fluid and the cold fluid, and the overall heattransfer coefficient.

2) Calculate Q, Tco, P, R, and �TLM from Eqs. (1), (2), (4),(5) and (6) or (7).

3) Calculate Ns,min from Eqs. (17) or (18).4) Choose the smallest integer value for Ns satisfying

Ns > Ns,min to ensure that the initial design specification ismeaningful.

5) Calculate P1,2 from Eqs. (9) or (11) for the current value ofNs.

6) Find the FT correction factor from Eqs. (8) or (10).7) Check whether the condition given by Eq. (20) is satisfied. If

so, a feasible solution has been detected; calculate the heatexchanger area from Eq. (3). Otherwise, set Ns = Ns + 1 andgo to step 5.

. Formulation of the optimization problem

The optimization problem uses the minimization of thenvestment cost as the objective function. The optimal designhould satisfy constraints (19) and (20), with Ns integer. Thenvestment cost for heat exchangers may be expressed as a func-ion of the equipment area through a power law expression.or multiple-pass heat exchangers, the investment cost can bestimated by (see for instance Smith [1]):

= a + bNs

(A)c

(21)

Ns

here a, b, and c are cost law constants which depend on con-truction materials, pressure rating and the type of exchanger.alues for c are typically between 0.6 and 0.9 [1,10,11], which

atsl

able 2ata for the examples

xample Ex1 Ex2 Ex3

hi (◦C) 410 500 500

ho (◦C) 110 270 130

ci (◦C) 0 40 40

co (◦C) 360 195 180(kW) 2000 2000 2000(kW/m2 K) 0.1 0.1 0

8600 0 0670 7000 7000

0.83 0.65 0

ig. 3. The investment cost C as a function of the number of shells in series Ns

or example 1. The asterisk denotes the location of the unconstrained optimumolution.

eflect the economies of scale generally observed in chemicalrocess equipment. It should be noted that for these fractionalalues of c this optimization process is different to that of min-mizing the total area; the minimum area would be obtained inhe limit when Ns → ∞.

The solution to this optimization problem is first shownhrough a graphical method. The idea is that the knowledgeained through this method will be helpful for the numericalolution of the optimization problem.

.1. Graphical method

Example 1, taken from Moita et al. [6] and shown in Table 2,s used to illustrate the graphical method. Fig. 3 shows the invest-

ent cost as a function of Ns. The curve was obtained by directolution of the model equations, for several values of Ns. Theonstraint imposed by the correction factor FT is shown as aashed line. The feasibility region lies on the right hand side ofhe FT constraint.

For nonlinear cost functions with 0 < c < 1 (which reflectshe cases of practical interest), Eq. (21) has a unique global

inimum in the feasible region given by condition (19). The fol-owing analysis shows this property. When Ns → Ns,min, FT → 0,

nd A → ∞ as discussed in Section 2. Thus, in this limiting case,he objective function goes to infinite. When Ns → ∞, the oppo-ite occurs: FT → 1, and A has the smallest value. Hence, in thisimit, the function objective will also approach infinity. There-

Ex4 Ex5 Ex6

570 570 400150 150 320

50 0 120150 200 330

2000 2000 2,000.1 0.1 0.1 0.1

0 0 40,0007000 7000 2,300

.65 0.65 0.65 1.0

910 J.M. Ponce-Ortega et al. / Chemical Engineer

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fv

emmetuf

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((

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ig. 4. Investment cost C as a function of the number of shells in series Ns forxample 1. Dark circles indicate integer values of number of shells in series.

ore, a minimum investment cost must exist at some intermediatealue of Ns.

It should be noted that the mathematical behavior for lin-ar cost functions is such that the objective function decreasesonotonically with A, providing the trivial and not useful opti-um solution at the limit A → Amin for Ns → ∞ (which is

quivalent to the problem of minimizing the total area). To avoidhis impractical result, it is needed in these cases to include anpper limit lower than 1 for the FT correction factor in the modelormulation.

It is apparent from Fig. 3 that the minimum investment costs obtained by using a number of 1–2 shells in series of approx-mately 6.25. The minimum value of C is D 109,037. However,easible multiple-pass shell-and-tube heat exchangers must pro-ide FT correction factors greater than 0.8, with an integerumber of 1–2 shells in series. Therefore, the optimization prob-em is to find the best integer value of Ns, with the constraintsiven by Eqs. (19) and (20), that yields the minimum investment.his situation is shown in Fig. 4 for example 1, where circles

ndicate the acceptable integer values of Ns. The optimal integerolution is Ns = 6, for a minimum investment cost of D 109,064.n this case, restricting Ns to integer values has only increasedhe investment cost by D 27.

In general, depending on the economic data and process con-itions, the relaxed optimum solution may lie on either side ofhe FT constraint. However the best integer solution is alwayst or to the right hand side of the FT constraint. These observa-ions allow the development of a simple numerical optimizationlgorithm.

.2. Numerical algorithm

The optimization algorithm is based on a single variableearch, in this case over the number of shells in series. The

esign algorithm discussed above provides a starting feasibleesign. Then, the design process is simply repeated for highernteger values of Ns until a minimum investment cost is detected.he optimization procedure is summarized as follows:

t

pm

ing and Processing 47 (2008) 906–913

1) Specify Thi, Tho, Tci, CPc, CPh, U, a, b, and c.2) Use the design algorithm to generate an initial feasible

design.3) Compute the investment cost C associated with the initial

design using Eq. (21).4) Fix Ns = Ns + 1 and generate the corresponding feasible

design.5) Evaluate the cost function of the new design. If a better

objective function is found, go to step 4; otherwise, theformer design is the optimum solution.

It can be noted that the algorithm is based on three basiceatures. First, the number of 1–2 shells in series must be integer.econd, the optimum integer solution must be at or to the rightand side of the FT constraint. And third, the objective functions unimodal.

.3. Optimization of multipass coolers

The analysis shown above is extended for optimization prob-ems of multipass coolers (or heaters). In the previous analysis,he heat capacity flow rates of both fluids are fixed; now theroblem is to find the optimal mass flow rate or outlet tempera-ure for the cold (hot) utility as well. Therefore, this problem isormulated so as to minimize the total annual cost (TAC):

AC = Af

[a + bNs

(A

Ns

)c]+ HYCumu (22)

here Af is the annualization factor for capital investment, mus the flowrate of the process utility with a unit cost Cu, and HYs the annual operating time.

We outline below the analysis of a multipass cooler that usesooling water as the process utility. The hot fluid inlet and outletemperatures as well as the heat duty are usually fixed by therocess, and the water inlet temperature Tci is determined by siteonditions. Thus, the cooling water flowrate can be calculatedrom:

u = Q

CPu(Tco − Tci)(23)

This problem has two degrees of freedom, and two suitableptimization variables for this case are Ns and Tco.

To minimize the total annual cost, an iterative procedure issed. The basic idea of this algorithm is to choose the waterutlet temperature, Tco, and calculate the corresponding optimalalue of Ns using the single-variable optimization algorithm ofection 4.2; then, for the assumed value of Tco, the coolingater flow rate and the total annual cost can be determined fromqs. (23) and (22), respectively. This procedure can be applied

or varying water outlet temperatures to determine the minimumotal annual cost. As shown in Fig. 5, for nonlinear cost functionsith 0 < c < 1, the total annual cost is unimodal with respect to

he water outlet temperature. We use the golden section method

o determine the optimal value of Tco.

Many designers [2] limit the maximum water outlet tem-erature to 49 ◦C because of the rapid deposit formation ofineral salts over the heat transfer surface at greater temper-

J.M. Ponce-Ortega et al. / Chemical Engineering and Processing 47 (2008) 906–913 911

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Table 3Results obtained with the proposed optimization algorithm

Example Ns Xp FT A (m2) C (kD )

Ex1 6 0.717 0.907 289.888 109.06Ex2 1 0.720 0.909 82.807 123.55Ex3 2 0.719 0.929 118.747 199.05Ex4 1 0.915 0.814 110.159 148.73EEE

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tItsFbt2

oT92s

ig. 5. Finding the optimum outlet temperature for cooling water in multipassoolers for 0 < c < 1. Each point implies a local optimum solution for the optimumumber of shells (solution to example of Table 4).

tures. We use this value as a constraint. A similar applicationf the algorithm can be used for heaters with no phase change.

. Illustrative examples

Example 1 is now used to show the numerical applicationsf the optimization algorithm.

.1. Initial design

The constraint for the correction factor is taken as FT ≥ 0.8.he thermal effectiveness is 0.878, and the heat capacity ratio is.8333, from Eqs. (4) and (5). The minimum real (non-integer)umber of 1–2 shells in series may be obtained by solving Eq.17) as Ns,min = 3.062. Thus, the design algorithm begins withs = 4. For this value, the correction factor is found from Eq.

8) to be 0.7594. Therefore, the number of shells in series muste increased by 1. When Ns = 5, the correction factor provideshe feasible value of 0.8599. From Eqs. (3) and (21), the corre-ponding heat exchange area and cost values are A = 305.7 m2,nd C = D 110,375.

.2. Optimization

We now increase the initial integer value of the number ofhells in series to search for a better design; thus, for Ns = 6he following results are obtained: FT = 0.9066, A = 289.9 m2,nd C = D 109,064. Since C is lower than the initial design,he search continues. For Ns = 7: FT = 0.9329, A = 281.7 m2, and

= D 109,312. C is greater than the previous solution; therefore,he optimum number of shells in series is 6, with an objectiveunction value of D 109,064.

Table 3 shows the results obtained for this example. Moitat al. [6] reported an optimum number of shells in series of, with an investment cost value of D 110,375. We have shownhat this is not the optimum solution to this problem. Althoughhe difference in the objective function value is small, it can be

stablished for theoretical purposes that the optimization algo-ithm by Moita et al. [6] fails to provide the global optimumolution. The algorithm of this work provides therefore a betterand simpler) design tool.

afto

x5-a 2 0.6736 0.937 87.571 163.30x5-b 1 0.9520 0.661 124.205 160.80x6 2 0.646 0.952 169.683 430.27

A sensitivity analysis with respect to the change in the num-er of shells in series can be easily conducted. For this example,he investment cost C is within 1.2% of the optimum so longs Ns lies between 5 and 7. Therefore, we can notice that the Cersus Ns curve is fairly flat in the neighborhood of the mini-um investment cost. Although the C values are very close for

hose two values of Ns, it is observed that the heat-transfer areaequired is rather different (5.45% higher for Ns = 5 and 2.83%ower for Ns = 7).

The results to the examples Ex2–Ex4 shown in Table 2, alsoiscussed by Moita et al. [6], are given in Table 3. The resultsre identical to the ones obtained by Moita et al. [6], but with these of a simpler and easier to implement optimization method.

For examples Ex1–Ex4, one can notice that the implemen-ation of the constraint FT ≥ 0.8 does not affect the optimalolution (see Table 3). This is because the FT constraint lies,n all cases, at the left hand side of the optimal constrainedesigns. For example Ex5, however, the optimization procedureives different solutions for the constrained and unconstrainedroblems. These solutions are given in Table 3, where Ex5-a andx5-b represent the solutions for the constrained and uncon-trained optimization problem, respectively. The results showhat case Ex5-b has one less shell, which gives a 1.55% savingn the capital cost and a FT value of 0.661. Note, therefore, thatf the optimization problem is solved without the constraint onhe FT value, the solution would probably be close in economicerms, but would not meet practical design criteria.

Example 6, taken from Moita et al. [6], serves also to illustratehe application of the model to a case with a linear cost function.n this case, the problem is unbounded with the optimal solu-ion A = Amin when Ns = ∞, as described above. The reportedolution to this problem was obtained by setting the limit forT,max of 0.96 (which corresponds to Xp,min = 0.6), as suggestedy Moita et al. [6] The model therefore finds the closest solu-ion to this limit as the optimal solution, which corresponds toshells with an FT factor of 0.952 (see Table 3).To illustrate the application of the optimization procedure

f multipass coolers, we consider the problem data given inable 4; the basic problem consists of cooling a stream from3 to 32 ◦C using cooling water at an assumed temperature of9 ◦C. The outlet cooling water temperature and the number ofhells were used as the search variables. The results from the

pplication of the algorithm are reported in Table 5. The searchor the optimal solution is displayed in Fig. 5. It should be noticedhat each search point reflects a local optimum solution for theptimal number of shells, obtained through a direct application

912 J.M. Ponce-Ortega et al. / Chemical Engineer

Table 4Data for the cooler problem

Variable Value

Thi (◦C) 93Tho (◦C) 32Tci (◦C) 29Q (kW) 348.167U (kW/m2 K) 0.0443a 0b 5.405c 0.99Af (1/y) 1Cu (D /kg) 3.97 × 10−7

HY (h/y) 8400

Table 5Optimal solution—cooler design

Variable Value

Tco (◦C) 35.29A (m2) 427.11FT 0.9947Ns 6mT

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f the original design algorithm for cases with one degree ofreedom. The optimal values for the search variables are an outletemperature of cooling water of 35.3 ◦C and a number of shellsf 6, which provide a total annual cost of 2396 D /yr, based on thessumed economic environment. It can be noted that, as in theases of problems with process streams, the design algorithmor utility process exchangers with no phase changes is veryffective, showing no convergence problems.

. Conclusions

The problem of designing and optimizing 1–2 shell-and-tubeeat exchangers in series has been addressed, and a simple designnd optimization algorithm has been developed based on the FT

orrection factor model.The optimization algorithm uses the number of shells as

convenient search variable to provide an effective designethod. The procedure guarantees the global constrained opti-um solution since the objective function is unimodal with

espect to the number of shells in series when the coefficientfor the cost function take values between 0 and 1. A graphicalethod that allows a proper insight into the optimization prob-

em has also been presented. In addition, it has been shown howlower bound for the number of shells in series Ns as a functionf R can be used to provide an initial feasible design quickly.

We have also shown how the design method can be usedor the optimization of multipass utility exchangers, coolers oreaters, without phase change.

Because the proposed algorithms are simple and reliable, theyan be used within the solution of larger industrial problems,uch as the synthesis of multipass heat exchanger networks andooling water networks.

ing and Processing 47 (2008) 906–913

cknowledgement

Partial financial support from the Consejo de Investigacionientıfica of the UMSNH, Mexico, is gratefully acknowledged.

ppendix A. Nomenclature

, b, c constants of heat exchanger cost functionheat exchanger area (m2)

f annualization factor (1/y)heat exchanger capital cost (D )

P heat capacity flow rate (kW/◦C)Pu heat capacity for utility (W/kg ◦C)u unitary cost for utility (D /kg)T correction factor for logarithmic mean temperature dif-

ference∗T minimum allowed correction factor for logarithmic

mean temperature differenceY operation hours per year (h/y)u utility mass flow rate (kg/h)s number of 1–2 shells in series

heat exchanger thermal effectiveness1,2 thermal effectiveness of each 1–2 shell in the series

heat exchanger duty (kW)heat capacity ratiostream temperature (◦C)

AC total annual cost (D /y)TLM logarithm mean temperature difference (◦C)

overall heat exchanger transfer coefficient (kW/m2 K)P Ahmad’s parameter, dimensionless

ubscriptscold streamhot streaminlet

ax maximumoutlet

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