Selecting Optimal Configurations for Multisection Plate Heat

6112

Ind. Eng. Chem. Res. 2003, 42, 6112-6124

Selecting Optimal Configurations for Multisection Plate Heat Exchangers in Pasteurization Processes Jorge A. W. Gut† and Jose´ M. Pinto*,†,‡ Department of Chemical Engineering, University of Sa˜ o Paulo, Avenida Prof. Luciano Gualberto, trav. 3, 380/Sa˜ o Paulo, SP/05508-900 Brazil, and Department of Chemical and Biological Sciences and Engineering, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201

An optimization method for configuring the regeneration, heating, and cooling sections of a plate and frame heat exchanger (PHE) in pasteurization processes is presented. The objective is to select the configuration with minimum operational and capital costs that satisfies constraints on fluid pressure drops, channel flow velocities, and pasteurizer thermal performance. The configuration of each section of the PHE is defined by its number of channels, pass arrangement, feed connection locations, and type of flow in the channels. Because the steady-state PHE simulation model for generalized configurations is in algorithmic form, it is not possible to represent the optimization problem in the form of a mixed-integer nonlinear programming model. Therefore, a branching method is proposed for its solution. Using a structured search procedure, the proposed method is able to successfully determine the optimal and near-optimal configurations with a very reduced number of exchanger evaluations. An example of optimization for a milk pasteurization process is presented, and it is verified that only 154 thermal simulations were required to obtain the optimal configuration from a feasible region containing 2.36 × 108 elements. 1. Introduction The plate heat exchanger (PHE) consists of a pack of gasketed corrugated metal plates pressed together in a frame, as presented in Figure 1. The fluids flow in alternate channels that are formed between adjacent plates and exchange heat through the thin metal plates. A large number of different flow arrangements are possible for a PHE, and one frame can as well hold more than one heat-exchange section using special intermediate connector plates that allow fluids to be introduced or removed from the plate pack. PHEs are largely used in food, dairy, and pharmaceutical industries for pasteurization and sterilization processes because of their compactness and flexibility for disassembling and cleaning. The design of PHEs is highly specialized because of the large variety of plates and possible arrangements. The PHE manufacturers have developed their own exclusive design methods, and despite the large number of industrial applications, rigorous PHE design methods are not as easily available as those for tubular exchangers. The available methods often have limited application to certain types of configurations or utilize simplified mathematical models of the PHE. To the best of the authors’ knowledge, there are no design methods for PHEs with multiple sections and generalized configurations in the open literature. According to Jarzebski and Wardas-Koziel,1 designers find it difficult to determine operating conditions and unit dimensions for PHEs due to the complexity of the optimization of costs for this type of exchanger. There are several possible configurations for a PHE and a poor choice can lead to significantly higher operational costs. * To whom correspondence should be addressed. Tel.: 718 260-3569. Fax: 718 260-3125. E-mail: [email protected]. † University of Saõ Paulo. ‡ Polytechnic University.

Figure 1. Plate heat-exchanger assemblage and parts: a, opened plate pack; b, fixed end cover; c, moveable end cover; d, upper carrying bar; e, lower carrying bar; f, support column; g, tightening bolts; h, corrugated chevron plate; i, plate gasket; j, plate port.

Jarzebski and Wardas-Koziel1 present simple expressions for dimensioning a PHE; however, the simplifications of the PHE modeling may compromise the optimization results. Focke2 worked on the optimization of the plate pattern of the PHE for minimizing the heat-transfer area of exchangers with single-pass arrangements. Pure countercurrent-flow conditions are assumed for deriving the model. Shah and Focke3 presented a detailed stepby-step design procedure for rating and sizing a PHE, which is however also restricted to single-pass arrangements. Kandlikar and Shah4 analyzed several usual configurations and presented guidelines for selecting the appropriate flow arrangement among those considered. It was verified that symmetric configurations with countercurrent-flow yield the highest thermal effectiveness. However, some applications require asymmetrical configurations because of differences in fluid heat capacities, and a more careful analysis is required for the configuration selection from the heat-exchange and

10.1021/ie0303810 CCC: $25.00 © 2003 American Chemical Society Published on Web 10/30/2003

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6113

Figure 2. Schematic of a three-section PHE used for pasteurization.

pressure-drop viewpoints. With this concern, Pinto and Gut5 developed a screening procedure for selecting the optimal configuration of a PHE targeting the minimum number of plates necessary for the desired process conditions. This procedure is based on a mathematical model that accounts for deviations from the pure countercurrent conditions and can represent any configuration.6 Although the screening method can optimize the configuration of a single-section PHE, it is not suitable to optimize multisection PHEs such as the ones used in pasteurization processes because the operational costs and the interdependence of the sections play an important role. Wang and Sundeń7 recently presented an optimization method for determining the best pass arrangements and plate patterns for PHEs. The full utilization of the available pressure drops is used as a design objective. However, the statement that the minimum heattransfer area is achieved when the available pressure drops are fully utilized is only valid when the configuration of the PHE is fixed. In this work, an optimization method to select a detailed configuration with minimum capital and operational costs for a three-section PHE in a pasteurization process is presented. The PHE simulation model developed by Gut and Pinto6 for generalized configurations is used for the exchanger evaluation, and a branching procedure is proposed to solve the optimization problem. The branching method is a structured and efficient search procedure, which enumerates the feasible region of the problem with a very reduced number of exchanger evaluations. The structure of this paper is as follows: First, the fundamentals of the pasteurization process are introduced. Further, in sections 3-5, the parametrization of the different configurations and the mathematical modeling of the PHE are presented. The configuration optimization problem is described in section 6, and the branching method for its solution is presented in section 7. Finally, an example of optimization for a milk pasteurization process is presented in section 8 to illustrate the efficiency of the proposed method.

2. Pasteurization Process Pasteurization is the process of heating a product to a specific temperature for a given period of time to destroy and/or inactivate pathogenic/spoilage microorganisms and undesirable enzymes. Because the extent of inactivation depends on the combination of time and temperature, these process parameters must be carefully selected to ensure that all pathogenic microorganisms are destroyed, while preventing the thermal degradation or overprocessing of the product. The PHEs are widely used for the continuous hightemperature short-time (HTST) pasteurization of liquid foods such as milk, fruit juices, and beer.8 The schematic diagram of the process is presented in Figure 2 for a PHE with three heat-exchange sections, namely, regeneration, heating, and cooling. The cold raw product is preheated at the regeneration section (path p1 f p2 in Figure 2) before being heated to 70-75 °C at the heating section (path p2 f p3). Immediately after this section, there is an insulated holding tube that is designed for a residence time of approximately 15 s. If the temperature of the product at the exit of the holding tube (marked p4) does not reach the desired pasteurization temperature, Tpstr, the product stream must be diverted back to the raw product tank (the diversion line is not shown in Figure 2). At the regeneration section, the hot pasteurized product (path p4 f p5) is used to preheat the incoming raw product before being cooled to a suitable storage temperature at the cooling section (path p5 f p6). For the HTST pasteurization processes, the circulating heating fluid is usually hot water (indirectly or directly heated with steam) and the cooling fluid is usually chilled water or a food-grade water/glycol mixture. The regeneration, heating, and cooling sections can be assembled side by side in the same frame using connector plates, as shown in the box of Figure 2. In a pasteurizer, in addition to the three sections that are considered in this work, it is also possible to have a supplementary cooling section or to split the regeneration section in two if the raw product must be removed for homogenization before further heating.

6114 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003

Figure 3. Representation of the configuration parameters for one section of the PHE.

3. Characterization of the PHE Configuration The configuration of a PHE (or of one section of a multisection PHE) defines the flow distribution of the hot and cold fluids inside the plate pack and the terminal connections at the covers and/or connector plates. For the characterization of such a configuration, six parameters are used: NC, PI, PII, φ, Yh, and Yf, which are described as follows.5,6 Number of Channels (NC). The space comprised between two plates is a channel, and the PHE section is represented as a row of channels numbered from 1 to NC. By definition, the odd-numbered channels belong to side I and the even-numbered channels belong to side II (see Figure 3a). When NC is even, both sides have the same number of channels (NIC ) NII C ); otherwise, side I has one more channel (NIC ) NII C + 1). Allowable values are NC g 2. Number of Passes at Sides I and II (PI and PII). A pass is a set of channels where the main fluid stream is split and distributed (see Figure 3d for an example). The parameters NC, PI, and PII are related as follows: I II II II I NIC ) PINI, NII C ) P N , NC ) NC + NC , where N and II N are the number of channels per pass for sides I and II, respectively. The pass arrangement of the PHE section is represented by PI × NI/PII × NII or simply by PI/PII. Allowable values for PI and PII are all integer divisors of NIC and NII C , respectively. Feed Connection Relative Location (O). The feed connection of side I is set arbitrarily to channel 1. The relative location of the feed of side II is given by the parameter φ, as shown in Figure 3a.9 Note that the configuration represented by these parameters can be freely rotated or mirrored to fit the PHE frame. Allowable values for φ are 1-4. Hot-Fluid Location (Yh). This binary parameter assigns the fluids to the exchanger sides, as shown in Figure 3b. If Yh ) 1, the hot fluid is at side I and the cold fluid at side II; otherwise, Yh ) 0.

Type of Flow in Channels (Yf). This binary parameter defines the type of flow inside the channels, which can be vertical or diagonal depending on the gasket type (see Figure 3c). If Yf ) 1, the flow is diagonal in all channels; otherwise, Yf ) 0. It is not possible to use both types in the same plate pack. The configuration of a three-section PHE can be thus represented by 3 × 6 ) 18 parameters, as shown in Figure 4, where the relative location of the sections is also represented. This characterization can be easily extended to an n-section PHE, which would be defined by n × 6 parameters. More details on the configuration of the three-section PHE are given in section 5.1. 4. Pasteurizer Thermal Modeling The fundamental structure of the pasteurizer comprises the three sections of the PHE, the heating and cooling circuits, and the holding tube, as shown in Figure 2. The flow rates of the product, of the heating fluid, and of the cooling fluid (Wprod, Whot, and Wcold), as well as their inlet temperatures (Tp1, Th1, and Tc1, according to the notation in Figure 2), are assumed to be known for the steady-state operation of the pasteurizer. The heat load at the regeneration section, QR, at the heating section, QH, and at the cooling section, QC, are defined in eqs 1a-c, assuming that there are no heat losses or phase changes at the PHE. Note that the index p1-2 denotes the average value for the product path p1 f p2 in Figure 2, and so forth.

QR ) WprodCpp1-2(Tp2 - Tp1) ) WprodCpp4-5(Tp4 Tp5) (1a) QH ) WprodCpp2-3(Tp3 - Tp2) ) WhotCph1-2(Th1 Th2) (1b) QC ) WcoldCpc1-2(Tc2 - Tc1) ) WprodCpp5-6(Tp5 - Tp6) (1c)


Figure 4. Configuration parameters for a PHE with three sections.

The heat loads can also be represented in terms of the thermal effectiveness of the sections, R, H, and C, as shown in eqs 2a-c.

QR ) R min(WprodCpp1-2,WprodCpp4-5)(Tp4 - Tp1) (2a)

is considered for the estimation of the average physical properties of each one of the six streams in Figure 2 (p1 f p2, p2 f p3, p4 f p5, p5 f p6, c1 f c2, and h1 f h2). Therefore, the convective heat-exchange coefficients (hI and hII) can be calculated using typical correlations.3,10,11

QH ) H min(WhotCph1-2,WprodCpp2-3)(Th1 - Tp2) (2b)

1 1 1 e ) + + + RI + RII U hI hII kP

QC ) C min(WcoldCpc1-2,WprodCpp5-6)(Tp5 - Tc1) (2c)

The regeneration ratio, RR in eq 5, represents the fraction of the heat load employed for the heating of the raw product up to the pasteurization temperature that is recovered at the regeneration section. Because of the PHE high thermal effectiveness, it is possible to achieve regeneration ratios as high as 0.9.

The thermal effectiveness of a section is obtained using the steady-state thermal model of the PHE, which is presented by Gut and Pinto6 in the form ) (NC,PI,PII,φ,Yh,Yf,RI,RII), where RI and RII are the dimensionless thermal coefficients for sides I and II of the exchanger (see eqs 3a,b).

RI )

AplateUNI

RII )

AplateUNII

WICpI

WIICpII

(3a)

(3b)

The main assumptions of the PHE model are (1) constant overall heat-exchange coefficient along the PHE, (2) uniform distribution of flow throughout the channels of a pass, (3) plug flow inside the channels, (4) no heat exchanged in the direction of the channel flow, and (5) non-Newtonian viscosity obeying the power-law rheological model. Because it is not possible to represent the PHE model explicitly as a function of the six configuration parameters, an “assembling algorithm” was developed. For a given configuration, the algorithm builds the corresponding system of differential equations (with the proper set of boundary conditions), which is solved analytically or numerically to obtain the channel temperature profiles and consequently the thermal effectiveness, . In eqs 3a,b, U is the average overall heat-exchange coefficient for the PHE section. To calculate U using eq 4, the desired thermal performance of the pasteurizer

RR )

QR ) Q + QH Cpp1-2(Tp2 - Tp1)

(4)

R

Cpp1-2(Tp2 - Tp1) + Cpp2-3(Tp3 - Tp2)

=

Tp2 - Tp1 Tp3 - Tp1 (5)

To account for a possible temperature drop in the holding tube, the parameter Tdrop is required for the thermal modeling, as shown in eq 6. If the holding tube is assumed to be adiabatic, then ∆Tdrop ) 0.

Tp3 - Tp4 ) ∆Tdrop

(6)

After using the PHE thermal model to obtain R, H, and C, the system of eqs 1a-c, 2a-c, 5, and 6 can be solved to obtain Tp2, Tp3, Tp4, Tp5, Tp6, Th2, Tc2, and RR. This process can be simplified by using eqs 7-9 to automatically obtain the three most important variables for the evaluation of the pasteurization process: the temperature of the product at the end of the holding tube with eq 7, the regeneration ratio with eq 8, and the temperature of the cooled product with eq 9. The auxiliary coefficients ωR1, ωR2, ωH, and ωC are defined in eqs 10a-d.

Tp4 )

Tp1(1 - ωR1)(1 - ωH) + ωHTh1 - Tdrop 1 - ωR1(1 - ωH)

(7)


RR ) ωR1Cpp1-2(Tp4 - Tp1) [ωR1Cpp1-2 + (1 - ωR1)Cpp2-3](Tp4 - Tp1) + Cpp2-3Tdrop

(8) Tp6 ) ωCTc1 + (1 - ωC)[(1 - ωR2)Tp4 + ωR2Tp1] (9) ωR1 ) R

min(Cpp1-2,Cpp4-5) Cpp1-2

(10a)

ωR2 ) R

min(Cpp1-2,Cpp4-5) Cpp4-5

(10b)

min(WhotCph1-2,WprodCpp2-3) WprodCpp2-3

(10c)

ωH ) H

min(WcoldCpc1-2,WprodCpp5-6) ωC ) WprodCpp5-6 C

(10d)

5. Pasteurizer Hydraulic Modeling The pressure drop of a fluid flowing through a side of the PHE can be calculated with eq 11 using suitable correlations for the evaluation of the friction factor in the channel flow, f.3,11 In eq 11, LP is the plate length measured between the centers of the orifices, Achannel is the cross-sectional area for channel flow, and Aport is the area of the plate orifice. The first term on the righthand side of eq 11 accounts for the friction loss, and the second term represents the pressure drop at the plate orifices.

∆P )

(

4fLPP W 2FDe NAchannel

)

2

( )

P W + 1.4 2F Aport

2

(11)

The pressure drop of the product (for the path p1 f p6 in Figure 2), ∆Pprod in eq 12a, is represented by the sum of the pressure drops in the three PHE sections (calculated with eq 11) and in the holding tube (∆Ptube, known a priori) and due to a level change (this last term is always considered positive for a conservative estimate). A supplementary booster pump for the product is not accounted for in the pressure drop calculation in this work, but the nominal head of the pump could be included in the left-hand side of eq 12a for the calculation of ∆Pprod.

∆Pprod ) ∆Pp1-2 + ∆Pp2-3 + ∆Pp4-5 + ∆Pp5-6 + ∆Ptube + Fp1-6gLP (12a) ∆Pheat ) ∆Ph1-2 + Fh1-2gLP

(12b)

∆Pcool ) ∆Pc1-2 + Fc1-2gLP

(12c)

The fluid velocity inside a channel, v, can be calculated with eq 13 for a given side of a PHE section. The calculation of v is important for further verification of the minimum fluid velocity constraint that is imposed in the configuration optimization problem. The channel flow velocity is also useful for determining the apparent viscosity for non-Newtonian fluids.

v ) W/NFAchannel

(13)

6. Configuration Optimization Problem The goal of the optimization problem is to configure the three-section PHE for achieving the minimal annual

Figure 5. Cumulative number of configurations for different problems.

pasteurization cost. The configuration of each one of the three sections could be, in principle, independently optimized using the “screening method” developed by Pinto and Gut5 for the minimization of the capital costs of a single-section PHE. However, this procedure would generate a poor solution because the operational costs of pasteurization would not be accounted for and unrealistic assumptions would be required, such as defining the local pressure drop constraints for the product flow throughout the pasteurizer. Moreover, because the sections would be evaluated separately, an iterative procedure would be necessary to converge the temperature of the product flow that leaves a section and enters the following one. Therefore, it is desired to optimize all of the three sections simultaneously, targeting the operational and capital costs of the process. The optimization variables for this problem are the 18 parameters shown in Figure 4. The objective function suggested for this problem is the fraction of the annual pasteurization cost that depends on the PHE configuration, PC ($/yr) defined in eq 14. The first term on the right-hand side of eq 14 accounts for the depreciated capital costs of the plate pack, where Ntotal ) NRC + NH C C C + NC is the total number of channels. The second term represents the overall pumping costs for the product and the heating and cooling fluids, where PP ) W∆P/F is the pumping power. The last two terms represent the utilities/power consumption for maintaining the heating and cooling circuits.

PC ) cc1Ntotal + cc2(PPprod + PPhot + PPcold) + C cc3QH + cc4QC (14) It is important to mention that the optimization method presented in this work can accept any objective function because the elements of the feasible region are enumerated and the objective function is evaluated a posteriori to locate the optimal and near-optimal elements. Consequently, the usage of eq 14 is not mandatory. Nevertheless, this feature of the method will be discussed further in section 7. The constraints for the configuration optimization problem are grouped into three distinct sets: design constraints, hydraulic performance constraints, and thermal performance constraints. Each set is detailed as follows.


6.1. Design Constraints. The three sections of the PHE could be configured independently a priori using the 18 parameters in Figure 4. In this case, it can be e 100 there are 2.92 × 104 shown that for 0 e Ntotal C possible configurations for a single-section PHE and 9.23 × 1011 configurations for a three-section PHE, as shown in Figure 5. Nevertheless, it is possible to reduce the dimension of this optimization problem by applying the set of design constraints presented in eqs 15a-j.

Nmin e Ntotal e Nmax C C C

(15a)

NRC is even

(15b)

PI,R ) PII,R

(15c)

YRh ) 0

(15d)

YH h ) 0

(15e)

YCh ) 1

(15f)

{

3 if PII,R is even φ ) 4 if PII,R is odd R

{ {

Figure 6. Different flow arrangements for the heating fluid when PII,H ) 1.

(15g)

3 if PII,H is odd ∧ * 1 φ ) 4 if PII,H is even 3 or 4 if PII,H ) 1

(15h)

3 if PII,C is odd ∧ * 1 φC ) 4 if PII,C is even 3 or 4 if PII,C ) 1

(15i)

C YRf ) YH f ) Yf ) 0 ∨ 1 (given)

(15j)

H

The upper bound of the total number of channels in eq 15a is associated with the capacity of the PHE frame and with the number of available plates. The smallest ) 6 (with two channels per possible bound is Nmin C section), but a more realistic value for Nmin C can be used when a larger number of plates are expected for the PHE. The constraints in eqs 15b,c,g impose that the regeneration section has a symmetrical configuration with the passes arranged countercurrently. When the heat capacities of the hot and cold fluids are very similar, as in the case of the regeneration section, this type of configuration yields the highest thermal effectiveness.4,12 Therefore, these constraints prevent the evaluation of low-effectiveness configurations for the regeneration section. Under these conditions of symmetry, YRh ) 0 and YRh ) 1 lead to equivalent configurations, and thus eq 15d is used to remove one unnecessary degree of freedom of the problem.6 C Configuration parameters YH h and Yh are also set to simplify the optimization problem because their influence on the pressure drop and thermal effectiveness is rather small. With eqs 15e,f, side I of the heating and cooling sections receives the product whereas side II receives the utility fluid. Note that the first channel of

these sections is assumed to be the one adjacent to the neighboring connector plate, as shown in Figure 4. The design constraints in eqs 15h,i ensure the potential connection between the three sections of the PHE by preventing conflicts at the nozzles of the connector plates. These constraints are necessary because the corner of a connector plate can handle only one stream. Because the use of φ ) 1 or 2 with multipass arrangements reduces the thermal effectiveness of the section, these values of φ are not considered. Note that φ has no influence on the fluid pressure drops or channel fluid velocities (see eqs 11-13). A heuristic rule is applied for the heating and cooling sections when there is a single pass at the utility side, i.e., when PII,H ) 1 and/or PII,C ) 1. In this case, the U-type arrangement is preferred instead of the Z-type because all of the connections for the utility are made at the end cover of the PHE without using the nozzles of the connector plate (see the example in Figure 6 for the heating section). Therefore, φH and φC may vary in eqs 15h,i when there is a single pass at side II. The last design constraint, eq 15j, is required because all of the sections must have the same type of channel flow. Because Yf is associated with the convective coefficient and friction factor correlations, this parameter is set to simplify the solution of the optimization problem. By application of the set of design constraints (presented in eqs 15a-j), the dimension of the problem is reduced by several orders of magnitude, as shown in Figure 5. For example, the number of possible configurations for 0 e Ntotal e 100 drops from 9.23 × 1011 to C 5.49 × 107. 6.2. Thermal Performance Constraints. The set of constraints related to the thermal performance of the pasteurizer is presented in eqs 16a-d, where Tpstr is the minimum required pasteurization temperature and Tstrg


is the maximum storage temperature of the pasteurized product.

pressure drop: eq 11 for calculating ∆Pp1-2, ∆Pp2-3, ∆Pp4-5, ∆Pp5-6, ∆Ph1-2, and ∆Pc1-2

Tp4 g Tpstr

(16a)

Tp6 e Tstrg

(16b)

PHE thermal model (regeneration): R ) R(NRC,PI,R,PII,R,φR,YRh ,YRf ,RI,R,RII,R)

R

R,min

g

RR g RRmin

(16c) (16d)

6.3. Hydraulic Performance Constraints. Equations 17a-c define bounds for the fluid pressure drops. The lower bounds in eqs 17b,c are used to prevent the folding of the plates due to a high-pressure differential between product and utilities. The bounds on the channel-flow velocities for the product and utilities in eqs 17d-i are required to prevent the formation of stagnation areas, air bubbles, or burnouts that may occur when the fluid velocity is too low inside the channel.

∆Pprod e ∆Pmax prod

(17a)

max ∆Pmin heat e ∆Pheat e ∆Pheat

(17b)

max ∆Pmin cool e ∆Pcool e ∆Pcool

(17c)

vp1-2 g vmin prod

(17d)

vp2-3 g vmin prod

(17e)

vp4-5 g vmin prod

(17f)

vp5-6 g vmin prod

(17g)

vh1-2 g vmin heat

(17h)

vc1-2 g vmin cool

(17i)

7. Branching Method In section 6, the problem of configuration optimization for a three-section PHE used for pasteurization was detailed. The problem of minimization of the annual pasteurization costs can be summarized as follows:

min annual pasteurization costs: eq 14 subject to set of design constraints: eqs 15a-j set of thermal performance constraints: eqs 16a-d set of hydraulic performance constraints: eqs 17a-i pasteurizer model: eqs 7-9, 10a-d, and 12a-c channel-flow velocity: eq 13 for calculating vp1-2, vp2-3, vp4-5, vp5-6, vheat, and vcool

PHE thermal model (heating): I,H II,H H H H I,H II,H H ) H(NH ,P ,φ ,Yh ,Yf ,R ,R ) C ,P PHE thermal model (cooling): C ) C(NCC,PI,C,PII,C,φC,YCh ,YCf ,RI,C,RII,C) where the optimization variables are the 18 configuration parameters shown in Figure 4. The main difficulty in solving this problem lies on the algorithmic form of the PHE thermal model. Because the thermal model for obtaining cannot be represented in algebraic form, it is not possible to use mixed-integer nonlinear programming (MINLP) techniques. Because the optimization variables are discrete, an enumeration procedure could be used to locate the optimal solution. However, as shown in Figure 5, the number of possible configurations for this problem can be very large. Thus, this procedure would be prohibitive because of the large computational time required. Instead, a branching procedure is proposed to solve the optimization problem. In this procedure, the constraints are successively used to remove infeasible elements until the feasible region of the problem and the optimal configuration(s) are obtained. A structured and efficient search algorithm is presented to obtain the problem solution with very reduced computational effort. The tree structure in Figure 7 illustrates the branching sequence. The first three pairs of nodes define the number of channels and pass arrangements for the regeneration, heating, and cooling sections, respectively. The last three nodes define the relative location of the inlet and outlet connections at the connector plates and C R F C end covers. Binary variables YRh , YH h , Yh , Yf , Yf , and Yf are not represented in the tree structure because they are specified in eqs 15d-f,j. The expansion of branches and nodes of the tree is restricted by the design constraints in eqs 15a-c,g-i. e 7, there For example, for the simple case of 6 e Ntotal C are 28 672 possible configurations. By application of the design constraints, the total amount drops to 20 configurations (as previously shown in Figure 5), which are represented in the expanded tree of Figure 8. If the sets of hydraulic and thermal performance constraints are applied to each element of the expanded tree, the infeasible elements can be discarded and the feasible region of the problem is enumerated. However, the branching method can enumerate the feasible solutions with a reduced number of performance evaluations by applying the following principles: (A1) The hydraulic performance constraints should be applied prior to the ones on thermal performance because the latter require the solution of a system of differential equations to obtain while the former are calculated using simple algebraic equations. (A2) The parameter φ has no influence on the calculation of ∆P and v because a uniform flow distribution throughout the channels is assumed. Consequently, the constraints on hydraulic performance can be applied prior to level 7 in the tree in Figure 7.


Figure 7. Structure of the tree for the branching method.

Figure 8. Expansion of the tree for 6 e Ntotal e 7 (without performance verifications). C

(A3) ∆P and v are independent for sides I and II of a PHE section. Therefore, for a given number of channels, the calculation of ∆P and v is made only once for each allowable number of passes, rather than evaluating all of the possible pass-arrangement combinations PI/PII. (A4) For a given number of channels at side I or II, ∆P is proportional to the number of passes; therefore, if ∆P > ∆Pmax is verified, any larger number of passes will also lead to an infeasible solution and does not need to be considered. It is important to note that the verification of the maximum pressure drop for the product stream can be done with only the known terms in the right-hand side of eq 12a because all of them are positive. (A5) Before simulation of a section of the PHE, its thermal effectiveness assuming a pure countercurrentflow condition, CC, can be calculated using eqs 18a-c.4 Because e CC, the pure countercurrent effectiveness represents a rigorous upper bound for that section. Therefore, if the countercurrent condition is not sufficient to achieve the desired thermal performance, the section configuration under analysis can be discarded without further simulation. Note that the verification of eq 16b assuming a pure countercurrent condition for the cooling section can only be done if the effectiveness of the regeneration and heating sections, R and H, was previously obtained and used for the calculation of Tp6,CC

with eq 9. However, the verification of eqs 16a,d, using RRCC and Tp4,CC, is possible by assuming the pure countercurrent condition for both the regeneration and heating sections.

{

1 - e-NTU(1-C*)

if C* e 1 -NTU(1-C*) CC ) 1 - C*e NTU if C* ) 1 NTU + 1

( )

NTU ) (NC - 1) max

C* )

RI RII , NI NII

min(WICpI,WIICpII) max(WICpI,WIICpII)

(18a)

(18b)

(18c)

(A6) A given configuration of a PHE section may appear more than once when the tree in Figure 7 is expanded. Storing and tracking the obtained thermal effectiveness values avoids repeated evaluations of (NC,PI,PII,φ,Yh,Yf,RI,RII) for the considered section. For example, a cooling section with two channels, pass arrangement 1/1, and φC ) 4 appears six times in the tree of Figure 8, and thus C needs to be obtained only once for this section configuration.

6120 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 Table 1. Structure of the Screening Algorithm for the Configuration Optimization Problem level

variable

variable bounds

0 1 2

NRC PI,R/PII,R

2 e NRC e Nmax -4 C I,R I,R 1 e PI,R e NI,R is an integer) C (NC /P II,R II,R II,R is an integer) 1eP e NC (NII,R C /P

3 4

NH C PI,H/PII,H

max 2 e NH - NRC - 2 C e NC I,H I,H I,H is an integer) 1 e P e NC (NI,H C /P II,H II,H 1 e PII,H e NII,H (N /P is an integer) C C

5 6

NCC PI,C/PII,C

C max max(2, Nmin - NRC - NH - NRC - NH C C ) e NC e NC C I,C I,C I,C I,C 1 e P e NC (NC /P is an integer) II,C II,C 1 e PII,C e NC (NC /PII,C is an integer)

7

φR

3 e φR e 4

8

φH

3 e φH e 4

9

φC

3 e φC e 4

10

criteria for node creation root node eq 15b: NRC is even eq 15c: PI,R ) PII,R eq 17a: constraint on ∆Pproda eq 17d: constraint on vp1-2 eq 17f: constraint on vp4-5 eq 16c: constraint on R b eq 17a: constraint on ∆Pproda eq 17b: constraint on ∆Pheat eq 17e: constraint on vp2-3 eq 17h: constraint on vh1-2 eq 16a: constraint on Tp4c eq 16d: constraint on RRc eq 17a: constraint on ∆Pprod eq 17c: constraint on ∆Pcool eq 17g: constraint on vp5-6 eq 17i: constraint on vc1-2 eq 15g: constraint on φR eq 16c: constraint on R (simulate R) eq 15h: constraint on φH eq 16a: constraint on Tp4 (simulate H) eq 16d: constraint on RR (simulate H) eq 15i: constraint on φC eq 16b: constraint on Tp6d eq 16b: constraint on Tp6 (simulate C) terminal node

∆Pprod is calculated partially only with the known terms of eq 12a. b Using RCC instead of R (see eq 18a). c Using RCC and H CC instead of R and H (see eq 18a). d Using CCC instead of C (see eq 18a). a

Considering the aforementioned principles, the proposed structure of the branching procedure is presented in Table 1. Starting from the lowest level of the tree, a new node is created only if a certain group of performance constraints is satisfied, respecting the bounds on the optimization variables. The structure of the tree in Figure 7 is determined based on principles A1, A2, A3, and A5 to minimize the number of section evaluations when applying the performance constraints. The tree should be explored in a “depth first” strategy instead of “breadth first” because variables calculated at the lowest levels may be required at the highest levels of the same branch. For example, the value of ∆Pp1-2 is calculated at level 2 and used at levels 2, 4, and 6 of the same branch for the verification of eq 17a. Each time the search reaches the terminal node successfully, a feasible configuration is obtained and stored. After the tree is fully explored, the feasible region of the optimization problem is enumerated and the objective function can be used to obtain the optimal solution. It is important to note that (1) objective functions other than eq 14 can be used because it only evaluates the solutions a posteriori and (2) because the elements of the feasible region are ordered, it is possible to obtain more than one optimal configuration as well as the near-optimal elements. It is possible to use the branching method assuming pure countercurrent flow for all sections of the PHE. In this case, the thermal performance verifications that require simulations are bypassed (see levels 7-9 in Table 1). This assumption simplifies largely the mathematical modeling of the PHE and consequently the application of the branching method. For a large number of passes and/or channels per pass with φ ) 3 or 4, the PHE behavior approaches the ideal pure countercurrent-flow conditions.4 Thus, solving this simplified

branching method may be useful for rapidly obtaining an estimate of the optimal solution. It is important to mention that the branching method is not to be confused with the branch-and-bound method, which is used for solving optimization problems with discrete variables.13 Despite some similarities such as the tree structure to represent integer variables, the branching method is essentially a structured enumeration procedure to obtain the feasible solutions of the problem, while in the branch-and-bound method, a relaxed optimization problem is solved at each node of the tree. The branch-and-bound method could not be used to solve the problem of configuration optimization because it is not possible to mathematically represent the pasteurizer model explicitly on the configuration parameters. 8. Optimization Example: Milk Pasteurization The proposed branching method is applied to select the optimal configuration for a three-section PHE used for the continuous HTST pasteurization of cow milk with 13 wt % solids content at a mass flow rate of Wprod ) 3000 kg/h (2900 L/h) and at Tp1 ) 5 °C. The heating fluid is hot water at Wheat ) 4500 kg/h and Th1 ) 80 °C. Pasteurized milk is cooled with Wcool ) 5500 kg/h of chilled water at Tc1 ) 2 °C. The stainless steel plates have chevron corrugations with an inclination angle of 45° and an effective heat-transfer area of Aplate ) 0.30 m2/plate. The main dimensions of the PHE are similar to those of the Quasar Q030 RKS-10 PHE:14 e ) 0.07 cm, De ) 0.643 cm, LP ) 720 cm, Aport ) 78.5 cm2, and Achannel ) 15.5 cm2. The holding tube (2 in.) was designed for a holding time of 16 s and has ∆Ptube ) 0.13 psi and ∆Tdrop ) 2 °C. A maximum number of 140 channels is considered for this example, with a lower bound of six channels (the


Figure 9. Structure of the applied branching method.

Figure 10. Performance of the branching method in the optimization example.

Table 2. Constraint Bounds for the Optimization Example pressure drop

channel-flow velocity

thermal performance

∆Pmax prod ) 45 psi ∆Pmax heat ) 17 psi ∆Pmax cool ) 20 psi ∆Pmin heat ) 5 psi ∆Pmin heat ) 5 psi

vmin prod ) 0.1 m/s vmin heat ) 0.1 m/s vmin cool ) 0.1 m/s

R,min ) 80% RRmin ) 0.80 Tpstr ) 72 °C Tstrg ) 5 °C

lowest attainable value for three sections). The bounds on pressure drops, channel-flow velocities (recommended minimum velocity by Kho and Mu¨ller-Steinhagen15), and thermal performance are presented in Table 2. For determining the objective function in eq 19 (PC, $/yr; PP, W; Q, W), the cost coefficients suggested by Wang and Sundeń7 were considered. This function evaluates the annual pasteurization costs comprising the capital costs of the PHE and pumps (assuming an amortization period of 10 years) and the electric power consumption for the operation of pumps and for the maintenance of the heating and cooling circuits.

PC ) 110(Ntotal - 1)0.85 + 0.17(PPprod + PPhot + C PPcold) + 0.13(QH + QC) (19)

Figure 11. Relative location of the inlet and outlet connections at the plate pack for the optimal configuration.

Average physical properties for the streams are calculated (considering the desired thermal performance in Table 2) through the equations provided by Choi and Okos16 for milk and the equations in Perry et al.17 for water. The correlations for calculating the convective heat-transfer coefficients and friction factors that are presented by Saunders10 and the fouling factors suggested by Marriott18 for water and by Lalande et al.19 for milk were used in this example. Adequate fouling factors are very important when dimensioning a PHE for a pasteurization process because fouling plays a major role in these processes. The PHE must be designed to satisfy the required thermal inactivation under fouled conditions. Process control is required to prevent overprocessing of the product when running under clean conditions, and regular cleaning of the PHE is indispensable. Figure 9 shows the structure for the application of the branching method. The search method presented in Table 1 was implemented in C language, and the thermal simulations were solved separately with the software gPROMS20 with an appropriate finite-differences method.21 The assembling algorithm6 is applied for the automatic generation of the gPROMS input files, and the simulation results in the form )

(NC,PI,PII,φ,Yh,Yf,RI,RII) are stored in a spreadsheet. After the feasible region is obtained, the objective function is applied for sequencing its elements and locating the optimal one(s). The total computational time for the problem solution was approximately 3 min in a personal computer with 128 MB of RAM and a 450 MHz processor. The performance of the branching method in solving this example is illustrated in Figure 10. Note that only 154 thermal simulations were required for obtaining the unique optimal configuration within the universe of 235 699 024 possibilities. The optimal configuration obtained and its characteristics are presented in Tables 3 and 4. The relative locations of the inlet and outlet connections on the plate pack are represented in Figure 11. The elements of the feasible region are represented as a function of Ntotal in Figure 12. It is possible to C observe that there are six distinct configurations in the range of 16 200 e PC e 12 800 $/yr, while PCmin ) 12 660 $/yr. One of the main advantages of the branching method is that the near-optimal elements are also obtained. The third best configuration (PC ) 12 697 $/yr), also with 124 channels, can be seen very close to

6122 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 Table 3. Optimal Configuration of the PHE Sections for the Example no. of channels pass arrangement (I/II) other parameters U (W/m2‚K) (%) Q (kW) performance

regeneration

heating

cooling

96 24 × 2/24 × 2 Yh ) 0, φ ) 3, Yf ) 1 1014 90.1 183.7 RR ) 0.875

16 2 × 4/4 × 2 Yh ) 0, φ ) 4, Yf ) 1 811 59.5 26.2 Tp4 ) 72.1 °C

12 3 × 2/3 × 2 Yh ) 1, φ ) 3, Yf ) 1 1444 69.0 20.2 Tp6 ) 5.0 °C

Table 4. Process Conditions for the Optimal Configuration in the Example

pressure drop (psi) channel flow velocity (m/s)

milk

hot water

chilled water

∆Pprod ) 41.8

∆Pheat ) 6.0

∆Pcool ) 8.1

vp1-2 ) 0.26 vp2-3 ) 0.13 vp4-5 ) 0.26 vp5-6 ) 0.26

vheat ) 0.41

vcool ) 0.49

Figure 12. Representation of the feasible region elements for the optimization example.

the optimal point in Figure 12. This configuration has an important feature: the pass arrangements of the three sections are symmetrical (R, 24/24; H, 4/4; C, 3/3), which simplifies largely the assembling of the plate pack. The influence of the maximum allowed number of channels for the solution of this problem is analyzed through the charts in Figure 13. It can be observed in Figure 13a that, by increasing Nmax C , the feasible region expands almost exponentially while the required number of simulations (proportional to the computational time) increases almost linearly, which represents a major advantage of the branching method. In Figure 13b, it can be seen that the constraint on the total number of channels is active for the cases in which e 124, although the optimal solution does not Nmax C ) Nmax necessarily have Ntotal C C . When the simplified branching method (assuming pure countercurrent-flow conditions) was solved, a configuration similar to the optimal one in Table 3 was obtained. The only difference is in the pass arrangement of the cooling section: 2 × 3/3 × 2 instead of 3 × 2/3 × 2. By simulating this configuration, one can see that it does not satisfy the constraint in eq 16b; however, it is a very good estimate of the optimal solution obtained

Figure 13. Influence of Nmax in the optimization example. C

by the branching method, and the required computational time was only 8 s. This optimization problem was also solved using the screening method presented by Pinto and Gut5 for minimizing the number of channels of each section individually. The sections were optimized in the following order: regeneration, heating, and cooling. An iterative procedure was required for converging Tp2 and Tp4. The maximum pressure drop for the product in the PHE (∆Pmax prod - ∆Ptube) was equally divided among the four product streams (paths p1 f p2, p2 f p3, p4 f p5, and p5 f p6 in Figure 2) for defining the local pressuredrop upper bounds, and the design constraints in eqs 15a-j were included in the screening search procedure. A configuration with 100 channels was obtained (re-


generation, 60 channels with 10 × 3/10 × 3; heating, 20 channels with 5 × 2/5 × 2; cooling, 20 channels with 5 × 2/5 × 2) with an annual cost of 15 317 $/yr, while the optimal configuration obtained through the branching for minium costs has 126 channels and PCmin ) 12 660 $/yr. It is interesting to note that when the screening method individually minimizes the number of channels of the sections, the global optimum is not obtained. Nevertheless, when the branching method minimizes the total number of channels of the pasteurizer, a configuration with 92 channels is obtained (regeneration, 56 with 14 × 2/14 × 2; heating, 20 with 5 × 2/5 × 2; cooling, 16 with 4 × 2/4 × 2), which is the global optimum for this objective. 9. Conclusions The problem of configuration optimization of a threesection PHE used in pasteurization processes was formulated as the minimization of capital and operational costs, subject to constraints on the PHE assembly, on the fluid pressure drops, on the channel-flow velocities, and on the pasteurizer thermal performance. A steady-state simulation model for the PHE with generalized configurations was used. Because configuration parameters cannot be explicitly included in the model, it is available only in algorithmic form. Consequently, a MINLP approach could not be used. A branching procedure was proposed to solve the optimization problem. In this procedure, the constraints are successively applied to eliminate infeasible elements from the set defined by the bounds on the total number of channels, thus enumerating the feasible region of the problem and locating the optimal configuration. The structure of the branching method was developed to solve the problem with a very reduced number of exchanger evaluations. An optimization example was presented to show the efficiency of the proposed method in finding the best configuration for a milk pasteurization process. In this example, the constraints on the equipment assembly define a set of 2.36 × 108 distinct configurations, and only 154 thermal simulations were required to find the optimal solution with a computational time of approximately 3 min. Because the feasible region is known, it is possible to obtain multiple optimal configurations and the nearoptimal elements can also be further analyzed. For example, in the milk pasteurization problem presented, the third best configuration offers an advantage for the plate pack assembling with little increase in the annual pasteurization costs. Moreover, it is possible to use the branching method assuming pure countercurrent conditions for the PHE. In this case, the thermal simulation model is not required and an estimate of the optimal configuration can be rapidly obtained.

cc2 ) cost coefficient for pumping power ($/W‚yr) cc3 ) cost coefficient for the heating utility ($/W‚yr) cc4 ) cost coefficient for the cooling utility ($/W‚yr) C* ) heat capacity ratio (C* e 1) Cp ) specific heat at constant pressure (J/kg‚°C) De ) equivalent diameter of the channel (m) e ) thickness of the metal plate (m) f ) Fanning friction factor g ) gravitational acceleration (g ) 9.8 m/s2) h ) convective heat-transfer coefficient (W/m2‚°C) kP ) plate thermal conductivity (W/m‚K) LP ) plate length measured between the center of the orifices (m) N ) number of channels per pass NC ) number of channels Ntotal ) total number of channels of the multisection PHE C NTU ) number of heat-transfer units P ) number of passes PC ) pasteurization annual cost ($/yr) PP ) pumping power (W) Q ) heat load (W) R ) fouling factor (m2‚°C/W) RR ) heat regeneration ratio T ) temperature (°C) Tpstr ) pasteurization temperature (°C) Tstrg ) pasteurized product storage temperature (°C) U ) overall heat-transfer coefficient (W/m2‚°C) v ) channel flow velocity (m/s) W ) mass flow rate (kg/s) Yf ) binary parameter for the type of channel flow Yh ) binary parameter for the hot fluid location Greek Letters R ) dimensionless heat-transfer parameter defined in eqs 3a,b ∆P ) fluid pressure drop (Pa) ∆Tdrop ) temperature drop of the product at the holding tube (°C) ) exchanger thermal effectiveness (%) µ ) viscosity (Pa‚s) F ) density (kg/m3) φ ) parameter for the feed connection relative location ωi ) auxiliary coefficients defined in eqs 10a-d Subscripts CC ) pure countercurrent-flow conditions cool ) cooling fluid heat ) heating fluid i ) ith element p*, c*, h* ) mark * in Figure 2 p*-*, c*-*, h*-* ) average for the path *-* in Figure 2 prod ) product, process fluid tube ) holding tube Superscripts

The authors thank FAPESP for its financial support (Grant 00/13635-4).

C ) cooling section of the PHE H ) heating section of the PHE I ) side I of the PHE section II ) side II of the PHE section max ) maximum min ) minimum R ) regeneration section of the PHE

Nomenclature

Literature Cited

Achannel ) cross-sectional area for channel flow (m2) Aplate ) effective plate heat-transfer area (m2) Aport ) area of the plate orifice (m2) cc1 ) cost coefficient for plate pack ($/plate‚yr)

(1) Jarzebski, A. B.; Wardas-Koziel, E. Dimensioning of Plate Heat-Exchangers to Give Minimum Annual Operating Costs. Chem. Eng. Res. Des. 1985, 63 (4), 211. (2) Focke, W. W. Selecting Optimum Plate Heat-Exchanger Surface Patterns. J. Heat Transfer 1986, 108 (1), 153.

Acknowledgment

6124 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 (3) Shah, R. K.; Focke, W. W. Plate Heat Exchangers and their Design Theory. In Heat Transfer Equipment Design; Shah, R. K., Subbarao, E. C., Mashelkar, R. A., Eds.; Hemisphere: New York, 1988. (4) Kandlikar, S. G.; Shah, R. K. Multipass Plate Heat ExchangerssEffectiveness, NTU Results and Guidelines for Selecting Pass Arrangements. J. Heat Transfer 1989, 111, 300. (5) Pinto, J. M.; Gut, J. A. W. A Screening Method for the Optimal Selection of Plate Heat Exchanger Configurations. Braz. J. Chem. Eng. 2002, 19 (4), 433. (6) Gut, J. A. W.; Pinto, J. M. Modeling of Plate Heat Exchangers with Generalized Configurations. Int. J. Heat Mass Tranfer 2003, 46 (14), 2571. (7) Wang, L.; Sundeń, B. Optimal Design of Plate Heat Exchangers with and without Pressure Drop Specifications. Appl. Therm. Eng. 2003, 23, 295. (8) Lewis, M. J. Heat Treatment of Milk. In Modern Dairy Technology; Robinson, R. K., Ed.; Aspen Publishers: Gaithersburg, MD, 1999; Vol. 1. (9) Pignotti, A.; Tamborenea, P. I. Thermal Effectiveness of Multipass Plate Exchangers. Int. J. Heat Mass Transfer 1988, 31 (10), 1983. (10) Saunders, E. A. D. Heat Exchangers: Selection, Design & Construction; Longman: Harlow, U.K., 1988. (11) Kakaç , S.; Liu, H. Heat Exchangers: Selection, Rating and Thermal Design; CRC Press: Boca Raton, FL, 1998. (12) Zaleski, T.; Klepacka, K. Plate Heat-ExchangerssMethod of Calculation, Charts and Guidelines for Selecting Plate HeatExchangers Configurations. Chem. Eng. Process. 1992, 31 (1), 45.

(13) Nemhauser, G. L.; Wolsey, L. A. Integer and Combinatorial Optimization; Wiley: New York, 1988. (14) APV. Data Sheet: Quasar-Plate Heat Exchanger Q030 RKS-10; APV: Kolding, Denmark, 2000. (15) Kho, T.; Mu¨ller-Steinhagen, H. An Experimental and Numerical Investigation of Heat Transfer Fouling and Fluid Flow in Flat Plate Heat Exchangers. Chem. Eng. Res. Des. 1999, 77A, 124. (16) Choi, Y.; Okos, M. R. Thermal Properties of Liquid Foodss Review. In Physical and Chemical Properties of Food; Okos, M. R., Ed.; ASAE: St. Joseph, MI, 1986. (17) Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s Chemical Engineers’ Handbook, 7th ed.; McGraw-Hill: New York, 1997. (18) Marriott, J. Where and How to Use Plate Heat Exchangers. Chem. Eng. 1971, 5 (4), 127. (19) Lalande, M.; Corrieu, G.; Tissier, J. P.; Ferret, R. E Ä tude du Comportement d’un E Ä changeur a` Plaques Vicarb Utilise´ pour la Pasteurisation du Lait. Lait 1979, 59 (581), 13. (20) Process Systems Enterprise. gPROMS Introductory User Guide, release 2.1.1; Process Systems Enterprise: London, 2002. (21) Georgiadis, M. C.; Macchietto, S. Dynamic Modeling and Simulation of Plate Heat Exchangers under Milk Fouling. Chem. Eng. Sci. 2000, 55, 1605.

Received for review May 5, 2003 Revised manuscript received September 10, 2003 Accepted September 24, 2003 IE0303810

Selecting Optimal Configurations for Multisection Plate Heat

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