Ind. Eng. Chem. Res. 2009, 48, 2927–2935
2927
PROCESS DESIGN AND CONTROL Optimal Design of Shell-and-Tube Heat Exchangers Using Particle Swarm Optimization Mauro A. S. S. Ravagnani,*,† Aline P. Silva,†,‡ Evaristo C. Biscaia, Jr.,‡ and Jose A. Caballero§ Departamento de Engenharia Quı´mica, UniVersidade Estadual de Maringa´, AV. Colombo 5790, Maringa-PR, 87020900, Brasil, Programa de Engenharia Quı´mica, PEQ/COPPE/UFRJ, UniVersidade Federal do Rio de Janeiro-RJ, 21949900 Brasil, and Departamento de Ingenierı´a Quı´mica, UniVersidad de Alicante, Carretera de San Vicente s/n, Alicante, 03690, Spain
In this paper, the shell-and-tube heat exchangers design is formulated as an optimization problem and solved with particle swarm optimization (PSO). The objective is to minimize the global cost including area cost and pumping cost or just area minimization, depending on data availability, rigorously following the standards of the Tubular Exchanger Manufacturers Association and respecting pressure drops and fouling limits. Given fluids temperatures, flow rates, physical properties (density, heat capacity, viscosity, and thermal conductivity), pressure drop and fouling limits, and area cost data, the proposed methodology calculates the optimal mechanical and thermal-hydraulic variables. The Bell-Delaware method is used for the shell-side calculations. Some literature cases are studied and results show that in this type of problem, with a very large number of nonlinear equations, the PSO algorithm presents better results, avoiding local minima. Introduction Due to their resistant manufacturing features and design flexibility, shell-and-tube heat exchangers are the most used heat transfer equipment in industrial processes. They are also easy adaptable to operational conditions. In this way, the design of shell-and-tube heat exchangers is a very important subject in industrial processes. Nevertheless, some difficulties are found, especially in the shell-side design, because of the complex characteristics of heat transfer and pressure drop. As pointed out by Taborek,1 some methods were proposed in the literature to calculate the heat exchange area, the individual and overall heat transfer coefficients, and pressure drops for the shell side. Kern2 published the first and most known method of thermal-hydraulic heat exchangers design. The method allows one to design heat exchangers or to rate existent equipment with respect to pressure drop and fouling. For the shell side, the correlations were proposed on the basis of the equivalent diameter. Although it overestimates the parameters design, it is, to this day, the most used method. Also according to Taborek,1 the method of Bell-Delaware is a more complete shell-and-tube heat exchanger design method. It is based on mechanical shell-side details and presents more realistic and accurate results for the shell-side film heat transfer coefficient and pressure drop. The method flow model considers five different streams: leakages between tubes and baffles, bypass of the tube bundle without cross-flow, leakages between shell and baffles, leakages due to more than one tube pass and the main stream, and tube bundle cross-flow. These streams do not occur in so well defined regions but interact with each other, needing a complex mathematical treatment to represent the real shell-side flow. These streams are not considered in the work of Kern.2 * To whom correspondence should be addressed. E-mail:
[email protected]. † Universidade Estadual de Maringa´. ‡ Universidade Federal do Rio de Janeiro. § Universidad de Alicante.
In spite of this, both the Kern and Bell-Delaware methods are used in practice. The real problem, however, is that in the majority of published papers and in industrial applications, heat transfer coefficients are estimated on the basis of, generally, literature tables. These values have always a large degree of uncertainty, so more realistic values can be obtained if these coefficients are not estimated but calculated during the design task. A few papers present shell-and-tube heat exchanger design including overall heat transfer coefficient calculations. The first authors that considered this situation were Polley et al.,3 and Panjeh Shah,4 Jegede and Polley,5 and Panjeh Shah.6 These authors, using both the Kern and Bell-Delaware methods, proposed an algorithm for shell-and-tube heat exchangers design to be incorporated into a heat exchanger network synthesis algorithm. The method consists of solving simultaneously three equations. In these equations, the authors proposed three constants: one depends on the mass flow rate and the tube-side fluid physical properties, and the other two are complex functions involving shell-side fluid physical properties, shell geometry, and the Fanning and Colburn ideal factors of the Bell-Delaware method. With the simultaneous solution of these equations, film coefficients and pressure drops can be obtained. The authors were not worried about industrial international parameters standards, like the Tubular Exchanger Manufacturers Association (TEMA)7 standards. In the work of Ravagnani,8 a systematic procedure was proposed for the design of shell-and-tube heat exchangers using the Bell-Delaware method. Overall and individual heat transfer coefficients are calculated on the basis of a TEMA tube counting table, beginning with the smallest heat exchanger with the biggest number of tube passes, to use all the pressure drop and fouling limits, fixed before the design and that must be satisfied. If the pressure drops or fouling factor is not satisfied, a new heat exchanger is tested, with a lower number of tube passes or larger shell diameter, until the pressure drops and fouling are under the fixed limits. Using a trial and error systematic, the
10.1021/ie800728n CCC: $40.75 2009 American Chemical Society Published on Web 02/13/2009
2928 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
final equipment is the one that presents the minimum heat exchanger area for fixed tube length and baffle cut, for a counting tube TEMA table including 21 types of shell-and-tube bundle diameter, two types of external tube diameter, three types of tube pitch, two types of tube arrangement, and five types of number of tube passes. This systematic procedure was also incorporated in a heat exchanger network synthesis method, as it is shown by Ravagnani et al.9 Mizutani et al.10 presented a mathematical programming model for heat exchangers design. The optimization model uses the Bell-Delaware correlations to calculate the heat transfer coefficient and pressure drop in the shell-side flow. The model is based on GDP (generalized disjunctive programming) and is optimized with a MINLP reformulation to determine the heat exchanger design that minimizes the total annual cost accounting for area and pumping expenses. The model follows some of the TEMA standards. Some other features, however, such as the shell diameter and the number of tubes, are calculated and optimized during the algorithm use and are not always in accordance with the standards. The model finds the best value to optimize the objective function, so the number of tubes can be very different from the tube counting TEMA standards. Also, tube length must be fixed before the design. Serna and Jime´nez11 presented an analytical expression that relates the pressure drop, the heat exchanger area, and the film heat transfer coefficient for the shell side of a shell-and-tube heat exchanger. The equation was based on the Bell-Delaware method and the use of the compact formulation within design and optimization algorithms was illustrated. Selbas et al.12 presented a method for the area optimization of shell-and-tube heat exchangers based on genetic algorithms. The authors applied the genetic algorithms to vary some design variables to determine the heat transfer area for a given design configuration. Babu and Munawar13 applied the differential evolution (DE) and its various strategies for the optimal shell-and-tube heat exchangers design. DE is an improved version of genetic algorithms and was applied by the authors with different strategies using the Bell-Delaware method to find the best heat transfer area. Ravagnani and Caballero14 used the Bell-Delaware method to formulate the mathematical model that involves discrete and continuous variables for the selection of the configuration and operating levels, respectively. A tube counting table, similar to the one used by Ravagnani et al.,9 is proposed. Following the TEMA standards, it allows one to find the shell diameter, tube bundle diameter, external tube diameter, tube pitch, tube arrangement pattern, number of tube passes, and number of tubes. By using a GDP formulation, these optimization variables are easy and rapidly obtained. Other variables are obtained by using some of the Mizutani et al.10 GDP model disjunctions. Furthermore, some complementing features are proposed. Besides the table counting, shell-and-tube side pressure drops and fouling factor are calculated, and the model has as constraints operational limits, previously fixed, as in industrial applications. The objective function can be considered as the minimization of area and pumping expenses or just heat exchange area minimization, depending on data availability. This formulation was incorporated to a heat exchanger network synthesis method in Ravagnani and Caballero.15 In the present paper, the problem of shell-and-tube heat exchangers design is formulated as an optimization problem. An objective function is proposed considering area and pumping cost or just heat exchange area minimization, depending on data
Figure 1. Heat exchanger with one pass at the tube side.
availability. The design rigorously follows the TEMA standards and respects shell- and tube-side pressure drops and fouling limits. The Bell-Delaware method is used for the shell-side calculations and the counting table presented in Ravagnani and Caballero14 for the mechanical parameters is used to avoid some nonlinearities in the model. A particle swarm optimization (PSO) algorithm is proposed to solve the optimization problem. Three cases extracted from the literature were studied and the results showed that the PSO algorithm for this type of problems, with a very large number of nonlinear equations, being a global optimum heuristic method, can avoid local minima and present better results than mathematical programming MINLP models. Model Formulation The problem of shell-and-tube heat exchanger design is formulated as an optimization problem. The main objective is to find the equipment that presents the minimum cost including heat exchange area cost and/or pumping cost while rigorously following the standards of TEMA and respecting pressure drop and fouling for shell-and-tube side limits. The inlet data for hot and cold fluids, Tin (inlet temperature), Tout (outlet temperature), m (mass flow rate), F (density), Cp (heat capacity), µ (viscosity), κ (thermal conductivity), ∆ (pressure drop limits), rd (fouling factor limits), and cost data (area and/or pump) are given. The mechanical variables to be calculated for the proposed methodology are tube inside diameter (din), tube outside diameter (dex), tube arrangement, tube pitch (pt), tube length (L), number of tube passes (Ntp), and number of tubes (nt), for the tube side. For the shell side, the variables to be optimized are the external diameter (Ds), the tube bundle diameter (Dft), the number of baffles (Nb), the baffles cut (lc), and the baffles spacing (ls). The thermal-hydraulic variables to be calculated are heat duty (Q), heat exchange area (A), tubeside and shell-side film coefficients (ht and hs), dirty and clean global heat transfer coefficient (Ud and Uc), pressure drops (∆Pt and ∆Ps), fouling factor (rd), log mean temperature difference (LMTD), the correction factor of LMTD (Ft), and the fluids location (shell or tube side) in the heat exchanger. Figure 1 shows an example of a 1-1 heat exchanger (one pass on the tube side and one pass on the shell side). The model equations were presented in Ravagnani and Caballero14 and can be seen also in Appendix A. The optimization problem (object function and constrains) is shown in eq 1. Minimize Ctotal ) Carea + Cpump Subject to
(1)
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 2929 t
max
∆P e ∆P ∆Ps e ∆Pmax rd g rddesign Ft g 0.75 1 e Vt (m/s) e 3 0.5 e Vs (m/s) e 2 where
(
Carea ) aa1
Q U × Ft × MLDT
(
Cpump ) aa3
)
aa2
∆Ptmt ∆Psms + Ft Fs
)
(2) (3)
For the optimization problem it is possible to consider heat exchange area cost and pumping cost or just heat exchange area, depending on the data available. If the cost parameters are not available, the methodology will find the heat exchanger with minimum heat exchange area. For these cases, one can consider aa1 ) aa2 ) 1, aa3 ) 0
(4)
Some Comments about PSO
process and are modified in each iteration by the eqs 5 and 6. Each particle is formed by the following variables: tube length, hot fluid allocation, and position in the TEMA table (that automatically defines the shell diameter, tube bundle diameter, internal and external tube diameter, tube arrangement, tube pitch, number of tube passes, and number of tubes). After the particle generation, the heat exchanger parameters and area are calculated using eqs 7-48 in Appendix A. This is done to all particles even when they are not a problem solution. The objective function value is obtained, and if the particle is not a solution of the problem (any restriction is violated), the objective function is penalized. The cases studied in this paper were tested with various sets of different parameters and the influence of each case was evaluated in the algorithm performance. The final parameters set was the set that was better adapted to this kind of problem. The parameters used in all the cases studied in the present paper are shown below. c1
c2
w
Npt
1.3
1.3
0.75
30
Proposed PSO Algorithm
16
Kennedy and Elberhart, on the basis of some animal groups’ social behavior, introduced the particle swarm optimization (PSO) algorithm. In the last years, PSO has been successfully applied in many research and application areas. One of the reasons that PSO is attractive is that there are few parameters to adjust. An interesting characteristic is its global search character in the beginning of the procedure. In some iteration, it becomes a local search method when the convergence of the final particles occurs. This characteristic, besides increasing the possibility of finding the global optimum, assures a very good precision in the obtained value and a good exploration of the region near to the optimum. It also assures a good representation of the parameters by using the method evaluations of the objective function during the optimization procedure. In the PSO, each candidate to the solution of the problem corresponds to one point in the search space. These solutions are called particles. Each particle also has an associated velocity that defines the direction of its movement. At each iteration, each one of the particles changes its velocity and direction taking into account its best position and the group best position, bringing the group to achieve the final objective. This work used the PSO proposed by Vieira and Biscaia.17 The particles and the velocity that define the direction of the movement of each particle are actualized according to eqs 5 and 6 Vik+1 ) wVik + c1r1(pik - xik) + c2r2(pglobalk - xik)
(5)
xik+1 ) xik + Vik+1
(6)
where xk(i) and Vk(i) are vectors that represent, respectively, position and velocity of the particle i, ωk is the inertia weight, c1 and c2 are constants, r1 and r2 are two random vectors with uniform distribution in the interval [0, 1], pk(i) is the position with the best result of particle i, and pkglobal is the position with the best result of the group. In above equations, subscript k refers to the iteration number. In this problem, the variables considered independent are randomly generated in the beginning of the optimization
A PSO algorithm is proposed to solve the optimization problem. The algorithm is based on the following steps: (i) input data maximum number of iterations number of particles of the population (Npt) c1, c2, and w maximum and minimum values of the variables (lines in TEMA table) streams, area and cost data (if available) (ii) random generation of the initial particles (There are no criteria to generate the particles. The generation is totally randomly done.) tube length (just the values recommended by TEMA) hot fluid allocation (shell or tube) position in the TEMA table (that automatically defines the shell diameter, the tube bundle diameter, the internal and the external tube diameter, the tube arrangement, the tube pitch, the number of tube passes and the number of tubes) (iii) objective function evaluation in a subroutine with the design mathematical model; with the variables generated at the previous step, it is possible to calculate parameters for the tube side (eqs 7-13) parameters for the shell side (eqs 14-35) heat exchanger general aspects (eqs 36-48) objective function (eqs 1-3) All the initial particles must be checked. If any constraint is not in accordance with the fixed limits, the particle is penalized. (iv) begin the PSO Actualize the particle variables with the PSO equations (5 and 6), re-evaluate the objective function value for the actualized particles (step iii), and verify which is the particle with the optimum value (v) Repeat step iv until the stop criteria (the number of iterations) be satisfied. During this PSO algorithm implementation is important to note that all the constraints are activated and they are always tested. When a constraint is not satisfied, the objective function is weighted and the particle is automatically discharged. This
2930 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 Table 1. Example 1 Data stream
Tin (K)
Tout (K)
m (kg/s)
µ (kg/m s)
F (kg/m3)
Cp (J/kg K)
κ (W/m K)
rd (W/m K)
kerosene crude oil
371.15 288.15
338.15 298.15
14.9 31.58
0.00023 0.00100
777 998
2684 4180
0.11 0.60
1.5 × 10-4 1.5 × 10-4
proceeding is very usual in treating constraints in the deterministic optimization methods. When discrete variables are considered, if the variable can be an integer, it is automatically rounded to the closest integer number at the level of objective function calculation, but it is maintained at its original value at the level of PSO; in this way we keep the capacity of changing from one integer value to another. Case Studies Three examples from the literature are studied, considering different situations. In all of the cases the computational time for a Pentium 2.8 GHz computer was about 18 min for 100 iterations. For each case studied, the program was executed 10 times and the optima values reported are the average optima between the 10 program executions. The same occurs with the PSO success rate (how many times the minimum value of the objective function is achieved in 100 iterations). Example 1. The first example was extracted from the work of Shenoy.18 The problem can be described the design of a shelland-tube heat exchanger to cool kerosene by heating crude oil. Temperature and flow rate data as well as fluids physical properties and limits for pressure drop and fouling are in Table 1. In Shenoy’s report18 there is no available area and pumping cost data, and in this case, the objective function will consist of the heat exchange area minimization, assuming the cost parameters presented in eq 4’/. It is assumed that the tube wall thermal conductivity is 50 W m K-1. Pressure drop limits are 42 kPa for the tube side and 7 kPa for the shell side. A fouling factor of 0.00015 m2 K W-1 should be provided on each side. Shenoy18 uses three different methods for the heat exchanger design: the method of Kern,2 the method of Bell-Delaware (Taborek1), and the rapid design algorithm developed in the papers of Polley et al.,3 Polley and Panjeh Shah,4 Jegede and Polley,5 and Panjeh Shah6 that fixes the pressure drop in both tube side and shell side before the design. Because of the fouling tendency, the author fixed the cold fluid allocation on the tube side. The tube outlet and inlet diameters and the tube pitch are fixed. Table 2 presents the heat exchanger configuration of Shenoy18 and the designed equipment, by using the best solution obtained with the proposed MINLP model of Ravagnani and Caballero14 and the proposed PSO algorithm in the present paper. Shenoy18 did not take the standards of TEMA into account. This type of approach provides just a preliminary specification for the equipment. The final heat exchanger will be constrained by standard parameters, such as tube lengths, tube layouts, and shell size. This preliminary design must be adjusted to meet the standard specifications. For example, the tube length used is 1.286 m and the minimum tube length recommended by TEMA is 8 ft or 2.438 m. As can be seen in Table 2, the proposed methodology with the PSO algorithm in the present paper provides the best results. Area is 19.83 m2, smaller than 28.40 and 28.31 m2, the values obtained by Shenoy18 and Ravagnani and Caballero,14 respectively, as is the number of tubes (102 vs 194 and 368). The shell diameter is the same as presented in Ravagnani and Caballero,14 i.e., 0.438 m, as is the tube length, although with a higher tube length the heat exchanger would have a smaller
Table 2. Results for the Example 1 Shenoy18 area (m2) Ds (m) tube length (mm) doutt (mm) dint (mm) tubes arrangement baffle spacing (mm) number of baffles number of tubes tube passes shell passes ∆Ps (kPa) ∆Pt (kPa) hs (kW/m2 °C) ht (kW/m2 °C) U (W/m2 °C) rd (m2 °C/W) Ft factor hot fluid allocation Vt (m/s) Vs (m/s) a
28.40 0.549 1286 19.10 15.40 square 0.192 6 368 6 1 3.60 42.00 8649.6 1364.5 1000.7 0.00041 0.9 shell a a
Ravagnani and Caballero14 best solution
present paper
28.31 0.438 2438 19.10 17.00 triangular 0.105 6 194 4 1 7.00 26.92 3831.38 2759.84 1017.88 0.00030 0.9 tube 1.827 0.935
19.83 0.438 2438 25.40 21.2 square 0.263 8 102 4 1 4.24 23.11 5799.43 1965.13 865.06 0.00032 0.9 tube 2.034 0.949
Not available.
diameter. Fouling and shell-side pressure drops are in accordance with the fixed limits. The PSO success rate (how many times the minimum value of the objective function is achieved in 100 executions) for this example was 78%. Example 2. The second example was extracted from the work of Serna and Jime´nez.19 In this example, the objective function also considers the minimization of the heat exchange area. Area and pressure drop costs are not considered, assuming the cost parameters as eq 4. Table 3 the fluids properties, temperatures, flow rates, fouling, and pressure drop limits. It is assumed also that the tube thermal conductivity is 45 W/m K. With these fluid temperatures, the LMTD correction factor will be 0.9165 (greater than 0.75), and one shell is necessary to satisfy the thermal balance. Table 4 shows three heat exchanger configurations. The first column presents the Serna and Jime´nez19 best solution, the second one presents the results of Ravagnani and Caballero,14 and the third column presents the equipment designed by using the proposed PSO algorithm in the present paper. The heat transfer areas obtained are 165.3, 148.56, and 131.27 m2, respectively. It is interesting to note that just the two last columns in the table take in account the standards of TEMA. As the final heat exchanger will be restricted to standard parameters, such as tube lengths, tube layouts, and shell size, this preliminary design must be adjusted to meet the standard specifications. It can be seen that the third column presents the best results, with smallest area, shell diameter, and number of tubes, although with a higher tube length. The tube length used in the optimum design is 6.096 m, in accordance with the standards of TEMA. The smaller heat transfer area can be explained by the use of the different tube length as well as the shell diameter, smaller than the used by the authors (0.771, 0.737, and 0.635 m, respectively), that affects the number of tubes (529, 509, and 360, respectively). The other differences are relative to the square tube arrangement and the inside tube diameter, in
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 2931 Table 3. Data for Example 2 fluid A B
Tin (K) 483.15 324.81
Tout (K) 377.59 355.37
m (kg/s)
µ (kg/m s) -4
1.2 × 10 2.9 × 10-4
19.15 75.22
F (kg/m3)
Cp (J/kg K)
κ (W/m K)
∆Pmax (kPa)
rd (W/m K)
789.72 820.12
2428 2135
0.106 0.123
78.81 83.63
3.5 × 10-4 3.5 × 10-4
Table 4. Results for Example 2 Serna and Jime´nez19 Ravagnani and best solution Caballero14 area (m2) Ds (m) tube length (mm) doutt (mm) dint (mm) tubes arrangement baffle cut (%) baffle spacing (mm) no. of baffles no. of tubes tube passes ∆Ps (kPa) ∆Pt (kPa) hs (kW/m2 °C) ht (kW/m2 °C) U (W/m2 °C) rd (m2 °C/W) hot fluid allocation Vt (m/s) Vs (m/s) a
165.3 0.771 5422 19.05 14.83 triangular 25.4 258.4 18 529 6 83.63 78.805 1336.73 1267.23 381.86
148.56 0.737 4880 19.05 17.00 triangular 25 0.305 15 509 6 43.69 76.74 928.60 1,174.36 425.10
a
a
tube
tube
a
a
a
a
present paper 131.27 0.635 6096 19.05 15.40 square 25.4 0.381 14 360 6 10.58 60.37 2638.64 3479.08 642.48 7.091 × 10-4 tube 2.171 0.520
Not available.
the PSO solution (square arrangement and 15.40 mm, respectively). The success rate for the PSO algorithm was 76%. Example 3. The third example was first used for Mizutani et al.10 and is divided in three different situations. Part A. In this case, the authors proposed an objective function composed of the sum of area and pumping cost. Table 5 presents the fluid properties, the inlet and outlet temperatures, and pressure drop and fouling limits, as well as area and pumping costs. The optimization problem consists of minimizing the global cost function subject to the constraints in eq 45. As all the temperatures and flow rates are specified, the heat load is also a known parameter. Part B. In this case, it is desired to design a heat exchanger for the same two fluids as those used in part A, but it is assumed that the cold fluid target temperature and its mass flow rate are both unknown. Also, it is considered a refrigerant to achieve the hot fluid target temperature. The refrigerant has a cost of $7.93/1000 tons, and this cost is added to the objective function. Part C. In this case, it is supposed that the cold fluid target temperature and its mass flow rate are unknowns, and the same refrigerant used in part B is used. Besides, the hot fluid target temperature is also unknown and the exchanger heat load may vary, assuming a cost of $20/kW yr to the hot fluid energy not exchanged in the designed heat exchanged, in order to achieve the same heat duty achieved in parts A and B. All of the three situations were solved with the PSO algorithm proposed in the present paper, and the results are presented in Table 6. Also presented in this table are the results of Mizutani et al.10 and the result obtained by the MINLP proposition presented in Ravagnani and Caballero14 for part A. It can be observed that in all cases the PSO algorithm presented better results for the global annual cost. In part A, the area cost is higher than the presented by Mizutani et al.10 but inferior to that presented by Ravagnani and Caballero.14 Pumping cost, however, is always lower. Combining both area and pumping costs, the global cost is lower.
In part B, the area cost is higher than that presented by Mizutani et al.,10 but the pumping and the cold fluid cost are lower. So, the global cost is lower (11 572.56 vs 19 641). The outlet temperature of the cold fluid is 335.73 K, higher than 316 K, the value obtained by Mizutani et al.10 In part C, the area cost is higher but pumping, cold fluid, and auxiliary cooling service costs are lower, and because of this combination, the global annual cost is lower than the presented by Mizutani et al.10 The outlet cold fluid temperature is 338.66 K, higher than the value obtained by the authors, and the outlet hot fluid temperature is 316 K, lower than the value obtained by Mizutani et al.10 The PSO success rates were 74%, 69%, and 65% for parts A, B, and C, respectively. Conclusions In the present paper, the design of shell-and-tube heat exchangers is formulated as an optimization problem. The main objective is to design the heat exchanger with the minimum cost including heat exchange area cost and pumping cost or just heat exchange area minimization, depending on data availability, rigorously following the standards of TEMA and respecting shell-and-tube sides pressure drops and fouling limits. Given a set of fluids data (physical properties, pressure drop and fouling limits, and flow rate and inlet and outlet temperatures) and area and pumping cost data, the proposed methodology allows one to design the shell-and-tube heat exchanger and to calculate the mechanical variables for the tube and shell sides, tube inside diameter (din), tube outside diameter (dex), tube arrangement, tube pitch (pt), tube length (L), number of tube passes (ntp), number of tubes (nt), the external shell diameter (Ds), the tube bundle diameter (Dft), the number of baffles (Nb), the baffles cut (lc), and the baffle spacing (ls). Also the thermal-hydraulic variables are calculated, including heat duty (Q), heat exchange area (A), tube-side and shell-side film coefficients (ht and hs), dirty and clean overall heat transfer coefficients (Ud and Uc), pressure drops (∆Ps), fouling factor (rd), log mean temperature difference (LMTD), the correction factor of LMTD (Ft), and the fluids location inside the heat exchanger. The Bell-Delaware method is used for the shellside calculations and a counting table presented in earlier papers for mechanical parameters is used in the model. The optimization problem is solved using a particle swarm optimization (PSO) algorithm. Three cases from the literature cases are studied. In two of them, the objective function was the area optimization. For the third one, the objective function was composed of the sum of the area and pumping costs. In this case, three different situations were studied. In the first one, all the fluids temperatures are known and, because of this, the heat load is also a known parameter. In the second situation, the outlet cold fluid temperature is unknown, and in the third one, the outlet hot and cold fluids are unknown. In this way, the optimization model considers these new variables. All of the cases are complex nonlinear programming problems. Results showed that in all cases the values obtained for the objective function using the proposed PSO algorithm in the present paper are better than the values presented in the literature. This can be explained because all the optimization models used in the
2932 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 Table 5. Data for Example 3a fluid A B a
Tin (K) 368.15 298.15
Tout (K) 313.75 313.15
µ (kg/m s)
m (kg/s)
F (kg/m3)
Cp (J/kg K)
κ (W/m K)
∆Pmax (k Pa)
rd (W/m K)
750 995
2840 4200
0.19 0.59
68.95 68.95
1.7 × 10-44 1.7 × 10-44
-4
3.4 × 10 4 8.0 × 10-44
27.78 68.88
Acost ) 123A0.59; pumpcost ) 1.31(∆Ptmt/Ft + ∆Psms/Fs) ($/year) (A is in m2, ∆P is in Pa, m is in kg/s, F is in kg/m3).
Table 6. Results for Example 3 part A
part B
part C
Mizutani et al.10 Ravagnani and Caballero14 present paper Mizutani et al.10 present paper Mizutani et al.10 present paper total cost ($/year) area cost ($/year) pumping cost ($/year) cold fluid cost ($/year) aux cool. cost ($/year) mc (kg/s) Tcout (K) Thout (K) area (m2) Ds (mm) tube length (mm) doutt (mm) dint (mm) tubes arrangement baffle cut baffle spacing (mm) baffles no. of tubes tube passes no. of shell passes ∆Ps (kPa) ∆Pt (kPa) hs (kW/m2 °C) ht (kW/m2 °C) U (W/m2 °C) rd (m2 °C/W) Ft factor hot fluid allocation Vt (m/s) Vs (m/s) a
5250 2826 2424
5028.29 3495.36 1532.93
a
a
a
a
3944.32 3200.46 743.86 -
a
a
a
264.63 1.067 4.88 25.04 23.00 square 25% 0.610 7 680 8 1 4431 23312 3240.48 1986.49 655.29 3.46 × 10-4 0.812 tube 1.058 0.500
250.51 0.8382 6.09 19.05 15.75 square 25% 0.503 11 687 4 1 4398.82 7109.17 5009.83 1322.21 700.05 3.42 × 10-4 0.812 tube 1.951 0.566
a a
202 0.687 4.88 15.19 12.6 square b
0.542 8 832 2 b
7494 22676 1829 6480 860 b
0.812 shell b b
19641 3023 1638 14980 a
58 316 a
227 0.854 4.88 19.05 14.83 square b
0.610 7 777 4 b
719 18335 4,110 2632 857 b
0.750 tube b b
11572.56 4563.18 1355.61 5653.77 -
21180 2943 2868 11409 3960 46 319 316 217 0.754 4.88 19.05 14.83 triangular
335.73 386.42 1.219 3.66 19.05 14.20 triangular 25% 0.732 4 1766 8 3 5097.04 15095.91 3102.73 1495.49 598.36 3.40 × 10-4 0.797 tube 1.060 0.508
b
0.610 7 746 4 b
5814 42955 1627 6577 803 b
0.750 shell b b
15151.52 4000.38 1103.176 6095.52 3952.45 338.61 315.66 365.63 1.219 4.88 25.40 18.60 square 25% 0.732 5 940 8 2 2818.69 17467.39 3173.352 1523.59 591.83 3.40 × 10-4 0.801 tube 1.161 0.507
Not applicable. b Not available.
literature that presented the best solutions in the cases studied are based on MINLP and they were solved using mathematical programming. When used for the detailed design of heat exchangers, MINLP (or disjunctive approaches) is fast, assures at least a local minimum, and presents all the theoretical advantages of deterministic problems. The major drawback is that the resulting problems are highly nonlinear and nonconvex and therefore only a local solution is guaranteed and a good initialization technique is mandatory, which is not always possible. GA or PSO have the great advantage that do not need any special structure in the model and tend to produce near global optimal solutions, although only in an “infinitely large” number of iterations. In general, PSO tends to produce better results than GA. Using PSO it is possible to initially favor the global search (using a one-best strategy or using a low velocity to avoid premature convergence) and later the local search, so it is possible to account for the tradeoff local vs global search. Considering the cases studied in the present paper, it can be observed that all of the solutions obtained with MINLP were trapped in local minima. By using the PSO algorithm, a metaheuristic method, because of its random nature, the possibility of finding the global optima in this kind of nonlinear problem is higher. The percentage of success is also higher, depending on the complexity of the problem. Computational time (about 18 min for all cases) is another problem and the user must work with the possibility of a trade off between the computational effort and the optimum value of the objective function. However, for small-scale
problems the PSO algorithm proposed in the present paper presents the best results without excessive computational effort. Appendix A The following model equations were already presented by Ravagnani and Caballero.14 Tube Side 1. Reynolds number: Ret )
4mtnpt πµtdintnt
(7)
2. Prandl number: Prt )
µtCpt κt
(8)
3. Nusselt number: Nut ) 0.072(Ret)0.8(Prt)1/3 4. Individual heat transfer coefficient: ht ) 5. Fanning friction factor:
(Nu)tκt
dint
(9)
(10)
fit )
0.079 (Ret)0.25
(11)
6. Velocity: Vt )
(Re)tµt
(12)
Ftdint
[
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 2933
λ)
(21) 1 Fc ) [π + 2λ sin(arccos(λ)) - 2 arccos(λ)] π 16. Fraction of cross-flow area available for bypass flow: Fsbp )
7. Pressure drop: ∆Pt )
(
t t t 2 1 2fi ntpL (V ) + 1.25ntp(Vt)2 Ft dint
)
8. Sn, cross-flow area at or near centerline for one cross-flow section:
[
[ ( [ (
)] )]
(pt - dex )(Dft - dex ) triangular w Sn ) ls Ds - Dft + pt (pt - dext)(Dft - dext) square w Sn ) ls Ds - Dft + pn
[
δsb )
1.038(Ds)2 π(dext)2nt(1 - Fc) (25) 4 8 20. Shell-side heat transfer coefficient for an ideal tube bank: Sw )
t
µsSn
(15)
(24)
19. Area for flow through the windows:
hoi ) m dex
(23)
Stb ) 0.0006223dextnt(1 + Fc)
9. Reynolds number: Res )
(22)
3.1 + 0.004(Ds1000) 1000 (π - a cos(0.5))Dsδsb Ssb ) 2 18. Tube-to-baffle leakage area for one baffle:
t
(14)
s
ls (D - Dft) Sn s
17. Shell-to-baffle leakage area for one baffle: (13)
Shell Side
t
0.5Ds Dft
( )
jiCpsms κs Sn Cpsµs
2/3
(26)
21. Correction factor for baffle configuration effects: Jc ) Fc + 0.54(1 - Fc)0.345
10. Velocity:
[
(27)
22. Correction factor for baffle-leakage effects: Vs )
ms/Fs (Ds/pt)(pt - dext)ls
(16)
11. Colburn factor:
[
a3
a)
1 + 0.14(Res)a4 ji ) a11.064a(Res)a2
(17)
where a1, a2, a3, and a4 assume different values depending on the tube arrangement and the range of the Reynolds number flowing in the shell side. These values are presented in Appendix B. 12. Fanning friction factor:
[
1 + 0.14(Res)b4 fi ) b11.064b(Res)b2
(18)
)
Ssb Stb + Ssb Ssb + Stb Jl ) R + (1 - R) exp -2.2 Sn
(
)
(28)
23. Correction factor for bundle-bypassing effects: Jb ) exp(-0.3833Fsbp)
(29)
24. Shell-side heat transfer coefficient: hs ) hoiJcJlJb
(30)
25. Pressure drop for an ideal cross-flow section: ∆Pb )
b3
b)
(
R ) 0.44 1 -
2fisNc(ms)2
(31)
FsSn2
26. Pressure drop for an ideal window section:
c
where b1, b2, b3, and b4 depend also on the tube arrangement and on the range of the Reynolds number flowing in the shell side. They are also presented here. 13. Number of tube rows crossed by the ideal cross-flow: 0.5Ds Nc ) pp
(19)
14. Number of effective cross-flow tube rows in each window: Ncw )
0.2Ds pp
15. Fraction of total tubes in cross-flow:
∆Pw ) (2 + 0.6Ncw)
(ms)2 2SnSwFs
(32)
27. Correction factor for the effect of baffle leakage on pressure drop:
[
( [ (
) )(
Ssb + 0.8 Stb + Ssb Ssb Stb + Ssb Rl ) exp -1.33 1 + Stb + Ssb Sn K ) -0.15 1 +
)] κ
(33)
28. Correction factor for bundle bypass: (20)
Rb ) exp(-1.3456Fsbp) 29. Pressure drop across the shell side:
(34)
2934 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
(
∆Ps ) 2∆Pb 1 +
)
Ncw R + (Nb + 1)∆PbRbRl + Nb∆PwRl Nc b (35)
General Aspects of the Heat Exchanger 30. Heat exchanged: Q ) mhCph(Tinh - Touth) ) mcCpc(Toutc - Tinc)
(36)
31. LMTD: ∆T1 ) Tinh - Toutc
LMTD )
(∆T1 - ∆T2) (37) ∆T1 ln ∆T2
( )
32. Correction factor for the LMTD: c
(39)
(Toutc - Tinc)
RS - 1 [ S-1 ] P ) RS - 1 R-[ S-1 ]
1/NS
x
Ft )
(40)
1/NS
(√ )
[
R +1 × R-1 2
(
ln
( ) ( )
1 - Px 1 - RPx
)
2 - 1 - R + √R2 + 1 Px ln 2 - 1 - R - √R2 - 1 Px
33. Tube pitch:
]
(41)
pt ) 1.25dext
(42)
Lt (Nb + 1)
(43)
34. Baffles spacing: ls )
[
35. Definition of the tube arrangement (pn and pp) variables:
[ [
pn ) 0.5pt pn ) 0.5pt pn ) pt square w pp ) pt
triangular w
]
]
(44)
36. Heat exchange area: A ) ntπdextLt
(45)
37. Clean overall heat transfer coefficient: Uc )
(
t
dex
htdint
t
+
rindex dint
t
+
dex
1 log(dext/dint) 1 + rex + 2kw hs
)
(46)
38. Dirty overall heat transfer coefficient: Ud )
Q A × LMTD
10 -10 104-103 103-102 102-10
