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193

MINLP Heat Exchanger Network Design incorporating Pressure Drop Effects S. Frausto-Hernandez, V. Rico-Ramfrez, S. Hernandez-Castro^ and A. Jimenez Institute Tecnologico de Celaya, Av. Tecnologico y Garcfa Cubas S/N, Celaya, Gto. C.P. 38010, Mexico ^Universidad de Guanajuato, Facultad de Quimica, Col. Noria Alta S/N, C.P. 36000, Mexico

Abstract Successful work has been done on heat exchanger network synthesis (HEN) using both Pinch Technology and MINLP techniques. However, most of the design procedures reported to date assume constant stream heat transfer coefficients. Motivated by the fact that detailed heat exchanger design is based on pressure drop of the streams, in this paper we extend the simultaneous MINLP model for the design of heat exchanger networks (Yee and Grossmann, 1990) by removing the assumption of constant film heat transfer coefficients and incorporating instead the effect of allowable pressure drop. An illustrative example is used to show the relevance of the approach. Keywords: Heat Exchanger Network, Pressure Drop

1. Introduction Recent approaches for the synthesis of heat exchanger networks with both Pinch Technology and MINLP techniques have shown to be capable of synthesizing near optimal networks for real industrial problems. Some of the design procedures can be reviewed, for instance, in Linhoff and Flower (1978), Papoulias and Grossmann (1983) and Yee and Grossmann (1990). However, although most of the current synthesis techniques are based on the assumption of constant film heat transfer coefficients, detailed heat exchanger design is based on the satisfaction of three major objectives (Polley and Panjeh Shahi, 1991): i) Transfer of required heat duty, ii) Tube side pressure drop below a maximum allowed value, and Hi) Shell side pressure drop below a maximum allowed value. Hence, since synthesis and detailed design are not conducted on the same basis, there is no guarantee that the values assumed for the heat transfer coefficients in the synthesis stage are the same as those actually achieved in equipment design. Polley and Panjeh Shahi (1991) proposed that one way of making consistent network synthesis and detailed exchanger design is to base network synthesis on allowable stream pressure drop rather than on constant film transfer coefficients. Since then, a number of applications have been reported which suggest the incorporation of pressure drop effects into the synthesis stage based on pinch technology (Serna, 1999). Further applications of this approach have also been considered for network retrofit based on MINLP techniques (Nie and Zhu, 1999). In this paper, the simultaneous MINLP model

194 for heat exchanger design proposed by Yee and Grossmann (1990) has been extended to incorporate pressure drop considerations into the synthesis stage. So, the network is synthesized based on allowable pressure drop rather than on constant film heat transfer coefficients and it is, therefore, made closer to industrial reality.

2. MINLP Synthesis Incorporating Pressure Drop Effects In this section we present the equations that have been incorporated to the simultaneous MINLP formulation of Yee and Grossmann (1990) in order to consider pressure drop effects during the synthesis stage. The main issue here is that the heat exchanger network to be synthesized must satisfy not only a specified heat recovery target but also pressure drop constraints. 2.1 Simultaneous MINLP approach to heat exchanger network design The MINLP model proposed by Yee and Grossmann (1990) is intended to provide an appropriate trade-off between utility consumption, number of units and exchanger areas. Such a model is based on a stage-wise superstructure with the temperature driving forces as optimization variables. The superstructure is constructed so that each cold stream can be potentially matched with each hot stream in each of the stages. In this model, all the constraints but the objective function are linear. The objective function consists of the annual cost for the network and is nonlinear because of the heat exchanger area calculation (the Chen approximation is used, (Chen, 1987)). Also, the overall heat transfer coefficients required by the area calculation are obtained in terms of the constant (assumed as given) film heat transfer coefficients of the streams. 2.2 Pressure drop considerations Policy et al. (1990) developed a general relationship between frictional pressure drop and convective film heat transfer coefficients as follows: AP = KAh'"

(1)

where AP is the exchanger pressure drop, A is the heat transfer area and h is the film heat transfer coefficient. Serna (1999) developed equations of the type of Equation (1) based on the detailed Bell-Delaware method for the design of heat exchangers. Hence, for turbulent flow in shell and tube exchangers, the equation for the tube side is: APj^ = Kp^Ah'/

(2)

and for the shell side is: AP,=K,,Ahr

(3)

where S and T stand for shell and tube, correspondingly. The parameters KPT and Kps depend on the physical properties of the streams and on the geometry. The equations derived by Serna (1999) are used in this wok. Expressions such as (2) and (3) apply for

195 those streams which do not experiment change of phase. Hence, it will be assumed that the heat transfer coefficients for heating and cooling utilities are still given constants. Furthermore, in order to be consistent with the assumptions of the simultaneous MINLP approach described above, only shell and tube heat exchangers will be considered here. 2.2.1 Pressure drop calculations for heat exchanger networks When a process stream exchanges heat in more than one unit, individual and overall pressure drops must be calculated. We use a simplified approach presented by Shenoy (1995) for the calculation of the pressure drop in each of the stages of a network superstructure. That approach assumes that the pressure drop of a stream is linearly distributed according to the surface area of the exchange units. 2.2.2 Calculation of Cost of Power In order to incorporate the cost of power to the objective function of the synthesis problem, the expression proposed by Serna (1999) can be applied: (4)

Cost of Power = CW Q^P

where CW is a cost coefficient (unit cost of power, $/KW-year), Q is the volumetric flow rate and AP is the pressure drop of the stream. 2.3 The extended model equations Seeking concreteness, this section presents only the equations that have been added to the simultaneous MINLP formulation for HEN synthesis presented by Yee and Grossmann (1990). It will be assumed that hot streams flow in the shell side and the cold streams flow in the tube side of the heat exchangers of the network. Hot streams are represented by index / (iG HP) and cold streams by index j (jeCP)- Index k denotes the k-th stage of the superstructure (keST). cu stands for cooling utility (cooling water) and hu for hot utility. The equations are presented next.

Vie HPj£CP,ke Ajk ~

{^T,,,l^T,^,^

' ^^icu "^ 'oiITi ^INcul

~ IV^hlUjA^lNhu

^OUTjl

hNcu

1 1 —+ h. K,.

\Jie HP

(6)

(7)

^hu \ui

(5)

h,^ h,

A7;,,+A7;,,„

A™ = K^^icuA'oUTi

ST

ry

196

'4o=ZEAy.+A.;

VyeCF

(9)

A/^ = /:pA,''f'°'

Vie//P

(10)

A/', = A:p,/l,,/if

VyeC/'

(H)

A/^

V/e///'

(12)

APj

yjeCP

(13)

Af;, = -^

AP.

V; € HP,k€ ST

(14)

^r/

l^

ijk

(15)

+ ESEcf;,^,, +Ec/^,.„^„, +XcF,..^,.; '•

y

'

7

k

i

J

(16) '

J

k

Equations (5) through (7) correspond to the explicit calculation of heat exchanger area of each unit. Evaluation of the total contact area for hot and cold streams are provided by equations (8) and (9). Equations (10) and (11) relate the film heat transfer coefficients to the pressure drop for each hot and cold stream. Besides, these values of pressure drop are related to the maximum allowable values for the pressure drop of each stream in equations (12) and (13). Pressure drop for hot and cold streams in each stage of the superstructure is determined by using the Shenoy (1995) approximation, equations (14) and (15). Equation (16) is the objective function which consists of the annualized cost of the network. Compared to the objective provided by Yee and Grossmann (1990), there are two extra terms which are incorporated to calculate the cost of power for hot and cold streams.

3. Illustrative Example An example has been developed in order to show the effect of pressure drop in the synthesis stage of heat exchanger networks. The example corresponds to the problem described by Policy and Panjeh Shahi (1991) for an HRAT of 20 °C. Specifications of the problem are given in Table 1. Two designs obtained with the proposed MINLP approach are compared. Figure 1 shows the network configuration, which is the same for both designs. The network consists of 7 units with minimum utility consumption (1075 KW and 400 KW for heating and cooling, correspondingly). Hence, the utility cost is $122,249.999/year. The designs, however, are different from each other with respect to the values of heat duty, exchanger areas, heat transfer coefficients and pressure drops. Such values are shown in Table 2. For the design 1, the total exchanger

197 area is 567.36 m^, with an investment cost of $236,474. For this case the cost of power is $23,310/year. Therefore, the total cost of the network is $382,034/year. For design 2, the total exchanger area is 618.24 m^, with an investment cost of $249,937, and the cost of power is $9,044/year. Therefore, the total cost of the network is $381,231/year. Note that in Table 2 the summation of the values corresponding to the contact area is not equal to the total area of the network. That is because the exchange area of a given match is considered as contact area for both the hot and the cold streams of the match. Observe that, although the total heat exchanger area for design 1 is smaller than that of design 2, the total cost of design 1 is larger that that of design 2. As a matter of fact, this example has been selected to show that the cost of power might be important in the determination of an optimal design. Also, since pressure drop has been considered into the synthesis stage, it can be expected that these designs are consistent with what is finally achieved in terms of industrial hardware.

4. Discussion In this work, the simultaneous MINLP formulation developed by Yee and Grossmann (1990) has been extended by incorporating equations that relate the overall pressure drop of a stream to its film heat transfer coefficient. Hence, the optimization of the problem not only minimizes capital but also satisfies both the transfer of required heat duty and the pressure drop constraints imposed by practical considerations. Also, the cost of power has been incorporated into the objective function of the optimization problem. The numerical disadvantages of the proposed approach can be easily identified (non-convexities) but the results obtained so far have shown the potential of the approach. As described with the illustrative example, the networks obtained by the proposed approach are expected to be consistent with what is finally achieved in terms of industrial hardware.

5. Acknowledgements Financial support provided by CONACYT (Mexico) is gratefully acknowledged.

^-^

90

HI

— V — r > 90

C2

100

^.^82

JL

70

o

70

r 60

65

-o

H2

125

60

M

^ 20

O

o-

Figure J ResuUing HEN for the Example

25

198 Table 1 Temperature Specifications for the Example Stream HI H2 CI C2 Steam Cooling Water

Tin (°C) 150 90 20 25 180 10

Tou, (°C) 60 60 125 100 180 15

F CP (KW/°C) 20 80 25 30

Table 2 Film Heat Transfer Coefficients for Two Designs of the Example

Stream HI H2 CI C2

Design 1 Contact Area h (m') (WW°C) 139.42 734 259.24 893 308.91 572 169.49 1049

AP (Kpa) 20 30 10 60

Design 2 h Contact Area (m') (wW°C) 721 154.71 680 293.52 565 325.39 814 200.08

AP (Kpa) 20 6.71 10 23.17

References Chen, J. J. J. (1987), Letter to Editors: Comments on Improvement on a Replacement for the Logarithmic Mean, Chem. Eng. Sci., 42, 2488. Linhoff, B. and J. R. Flower (1978), Synthesis of Heat Exchanger Networks, AlChE Journal, 24, 633. Nie, X. R. and X. X. Zhu (1999), Heat Exchanger Network Retrofit Considering Pressure Drop and Heat-Transfer Enhancement, AlChE Journal, 45, 1239. Papoulias S. A. and I. E. Grossmann (1983), A Structural Optimization Approach in Process Synthesis-II. Heat Recovery Networks. Comp. Chem. Eng. 7, 707. Policy, G. T., Panjeh Shahi, M. H. and Jegede, F. O. (1990), Pressure Drop Considerations in the Retrofit of Heat Exchanger Networks, Trans IChemE, 68, Part A, 211. Policy, G. T. and M. H. Panjeh Shahi (1991), Interfacing Heat Exchanger Network Synthesis and Detailed Heat Exchanger Design, Trans IchemE, 69, 445. Shenoy, U. V., (1995), Heat Exchanger Network Synthesis. Process Optimization by Energy and Resource Analysis, Gulf Publishing Company. Serna, G. M., (1999) Development of Rigorous Methods for Heat Integration of Chemical Processes, Ph.D. Thesis, Department of Chemical Engineering, Instituto Tecnologico de Celaya, Mexico. Yee, T. F. and I. E. Grossmann (1990), Simultaneous Optimization Models for Heat Integration - II. Heat Exchanger Network Synthesis, Comp. Chem. Eng., 14, 1165.

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