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Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm
Anton A. Kiss, Edwin Zondervan, Richard Lakerveld, Leyla Özkan (Eds.) Proceedings of the 29th European Symposium on Computer Aided Process Engineering June 16th to 19th , 2019, Eindhoven, The Netherlands. © 2019 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/B978-0-12-818634-3.50096-5
Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm Oscar D. Lara-Montañoa and Fernando I. Gómez-Castroa a Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Campus Guanajuato, Universidad de Guanajuato, Noria Alta S/N, Guanajuato, Guanajuato 36050, Mexico [email protected]
Abstract Most chemical processes require heat exchangers to modify the temperatures of the streams involved in the production steps. Due to their many applications, the rigorous design of such exchangers is of interest, aiming to determine the physical characteristics required to obtain the desired variation in temperature. The mathematical models representing such devices have several degrees of freedom and non-lineal equations, thus robust optimization algorithms are required to obtain the optimal solution in a short time. In this work, the use of the grey wolf algorithm is proposed for the optimization of a shell-and-tube heat exchanger, modeled by the Bell-Delaware equations. With the proposed method, a minimum total annual cost of 3,978.2 USD was obtained, in a mean time of 0.7976 seconds. Keywords: meta-heuristic optimization, shell-and-tube heat exchanger, grey wolf algorithm.
1. Introduction Heat exchangers are among the most used auxiliary equipment in chemical industry, since cooling and heating are common operations which allow modifying the temperature, or even the phase, of process streams, according to the operational requirements. This implies that heat exchangers are necessary for almost any chemical process. Shell-and-tube exchangers are widely used in industry to obtain wide ranges of temperature variations for streams from low to high flow rates, with high heat transfer efficiency due to the turbulence occurring on the shell. The design of a shell-and-tube heat exchanger involves several variables, which are usually related through energy balances and empirical correlations. One of the most common strategies to design heat exchangers in the industry is the Bell-Delaware method, which considers the variations of the convective heat transfer coefficient with the baffle configuration and phenomena occurring in the baffles, as leakage, pass partition bypass, among others. Nevertheless, the equations which model the heat exchanger are highly non-lineal and non-convex, mainly due to the presence of fractional and logarithmic terms in the mathematical model. Moreover, some of the variables are integer, as the number of tubes, and other variables can be taken as continuous, but in practice they take discrete values, as the internal and external diameter of the tubes, which are constrained by the standard dimensions for commercial tubes. Thus, a robust method is necessary to obtain proper solutions to the optimal design problem of the heat exchanger. The use of meta-heuristic optimization strategies offers a viable alternative to the solution of such problem. In this work, a recently reported stochastic optimization method, namely, the grey wolf optimization algorithm, is proposed to solve the optimization problem for a shell-and-tube heat exchanger, represented by the Bell-Delaware model, aiming to the minimization of the total annual cost. The grey wolf optimization algorithm emulates the hunting mechanisms of grey wolves, and it has been reported that it possesses a superior
O.D. Lara-Montaño and F.I. Gómez-Castro
572
exploitation capacity than other meta-heuristic algorithms, together with a high capacity to avoid local optima (Mirjalili et al., 2014)
2. Grey Wolf Optimizer Grey Wolf Optimizer (GWO) emulates the hunting technic of wolves. It was modeled by Mirjalili et al. (2014). This optimization algorithm considers the wolves hierarchy; alpha (α ), is not necessarily the strongest but is the one who makes important decisions that affect the whole group; beta (β ) wolves help the alpha in decision making activity, these wolves respect the alpha but command the other lower level wolves; omega (ω ) are the lowest ranked wolves in the hierarchy, although they are the least important, they help to maintain the structure of the pack; delta (δ ) wolves are those that are not in any of the previous categories, they have a lower hierarchy than alpha and betas but higher than the omega. The GWO algorithm takes into account different actions within the hunting process: • Tracking, chasing, and approaching the prey. • Pursuing, encircling, and harassing the prey until it stops moving. • Attack towards the prey. 2.1. Mathematical model The GWO algorithm defines the fitness solution as alpha α , the second and third best solutions are beta β and gamma γ , respectively. Other solutions are named omega ω . The optimization process is guided by α , β and γ . As mentioned before, wolves tend to encircle its prey. This behaviour is modeled with the equations 1 and 2. ·X D = |C p (t) − X (t)| X (t + 1) = X p (t) − A · D
(1) (2)
are coefficient p is the position vector of the prey, X is the position vector of a grey wolf, A and C X are calculated according to the equations vectors , and t is the current iteration. The vectors A and C 3 and 4 A = 2a · r1 −a = 2 · r2 C
(3) (4)
where components of the vector a are linearly decreased form 2 to 0 as the iterations pass, r1 and r2 are random vector with elements between 0 and 1. In nature, grey wolves know where the prey is, however, in the mathematical framework we don’t know where the optimum is (prey). To overcome this problem, it’s supposed that the alpha, beta and delta wolves have a better idea about the position of the prey. Assumed this, the best three results are saved and oblige the other search agents to update their position. To simulate this, the equations 5-7 are used. 1 · Xα − X, D 2 · Xβ − X, D 3 · Xδ − X| β = |C δ = |C Dα = |C 1 · Dα , X 2 · D 3 · D 1 = Xα − A 2 = Xβ − A 3 = Xδ − A β , X δ X X (t + 1) = X1 + X2 + X3 3
(5) (6) (7)
Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm
573
In summary, the population of wolves begins randomly within the search space. Each position of the wolves is a possible solution. The parameter a decreases in each iteration from 2 to 0, this parameter indicates the exploration and exploitation capacity of the GWO. If the value of |A ≥ 1|, the wolf diverges from the prey, if |A ≤ 1| the wolf converges towards the prey.
3. Heat Exchanger Mathematical Model 3.1. Heat transfer of shell side The Bell-Delaware mathematical model is used to predict the shell-side convective heat transfer coefficient hs . This model considers different parameters and geometrical variables (Shah and Sekulic, 2003). An ideal convective heat transfer coefficient hi is calculated and corrected using five parameters j according to the equation 8. hs = hi jc jl jb js jt
(8)
hi is evaluated as −2/3
hi = j
C ps Prs Ao,cr
(9)
where C ps is the fluid heat capacity, Prs is the Prandlt number, and jc , jl , jb , js , jt are correction factors for baffle configuration, shell to baffle and tube to baffle leakage effects, bundle pass and partition bypass, baffle spacing, and temperature gradient, respectively. Ao,cr is the flow area near the shell center-line for one cross-flow section and j is the Colburn factor, it is calculated with the equation 10 (Wildi-Tremblay and Gosselin, 2007). The variable a is determined with 11. The parameters a1 , a2 , a3 and a4 are reported in Wildi-Tremblay and Gosselin (2007). 1.33 a (Res )a2 (10) j = a1 Pt/do a3 (11) a= 1 + 0.14Reas 4 (12) The Reynolds number Res is evaluated with equation 13, μs is fluid viscosity, ms is the mass flow rate in shell-side and do is the tube external diameter. Res =
ms do μs Ao,cr
(13)
The area of the heat exchanger area is calculated with the Equation 14. A=
Q UΔTml Ft
(14)
where Q is the heat transfer rate, U is the global heat transfer coefficient, TLMDT is the logarithmic mean temperature difference and Ft is a correction factor. The U coefficient depends of the convective heat transfer of shell and tube side as well as fouling resistance of both sides, it is calculated with the Equation 15 1
U= 1 hs
+Rfs +
do ln ddo i
2kw
(15) + R f t ddoi + h1t
do di
ht is convective heat transfer coefficients for tube side, kw is the material thermal conductivity, R f s and R f t are the fouling factors for shell and tubes. The tube internal diameter, di , is calculates as di = 0.8do .
O.D. Lara-Montaño and F.I. Gómez-Castro
574 3.2. Heat transfer in tube side
To calculate ht , the equation 16 is used, this equation only works when the fluid in tubes is water (Sinnott and Towler, 2009). TCm is the medium fluid temperature and vt is the fluid velocity in tubes that is calculated with the equation 17. Where ρt fluid density, mt is the mass flow rate and Nt is the total number of tubes, is is calculated according the equation 4200 (1.35 + 0.02TCm ) vt0.8 di0.8 s mt vt = Nt π di2i ρ ht =
(16) (17)
t
4
3.3. Pressure drop calculations The pressure drop in the tubes side is calculated with the equation 18, the friction factor, f , is evaluated as f = 0.046Ret−0.2 , s is the the pass number and L the length of tubes that is calculated using 19. ρt vt 4fL (18) + 2.5 ΔPt = s di 2 A L= (19) π do Nt The expression 20 is used to calculate the pressure drop in shell-side Nr,cw ζb ζs ΔPs = (Nb − 1) ΔPb,id ζb + Nb ΔPw,id ζl + 2ΔPb,id 1 + Nr,cc
(20)
where Nb is the number of baffles, ΔPb,id is the ideal pressure drop in the central section, Nr,cw is the effective tubes en the cross flow, Nr,cc is the effective number of tubes rows crossed, ζ is a correction factor. 3.4. Cost estimation In this study, the total annual cost (TAC) is used, this is the sum of fixed, C f and operation cost Cop . To calculate the fixed cost the equation 22 is used (Smith, 2005). The purchase price C p is corrected using the factors fm , f p and fc for construction material, operating pressure and operation temperature, respectively. While the Cop is evaluated with equation 23. C p = 32800
A 80
0.68
C f = Cp fm f p fc (Es + Et ) EcHr Cop = 1000 r (1 + r)n ) +C f TAC = C f (1 + r)n ) − 1
(21) (22) (23) (24)
where Es and Et is the pump power (Watts) used in shell and tube side, respectively. Ec is the energy cost ($/kWh), Hr is the operational hours per year, n is the projected life time and r the interest rate per year. The optimization is performed using eleven parameters that can have a value between a given range, those are: • Diameter of shell (Ds ): 300mm -1000mm.
Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm
575
• Outer diameter of tube (do ): 15.87mm 63.5mm. • Tube pitch Pt : [1.25do, 1.5do]. • Tube layout angle (T L): [30°, 45°, 90°] • Baffle spacing at center (Lbc ): 0.2Ds − 0.55Ds • Baffle spacing at the inlet and outlet (Lbo = Lbi ): Lbc − 1.6Lbc • Baffle cut (Bc ): [25%, 30%, 40%, 45%] • Number of tube passes (s): [1, 2, 4] • Tube-to-baffle diametrical clearance (δtb ): 0.01do − 0.1do • Diametrical clearance of shell-to-baffle (δsb ): 0.01Ds − 0.1Ds • Outer diameter of tube bundle (Dotl): 0.8 (Ds − dsb ) - 0.95 (Ds − dsb ) The optimization process is subject to constrains for maximum allowed pressure drop in tube-side and shell-side, fluid velocity, and length to shell-diameter ratio, as follows: ΔPs , ΔPt ≤ ΔPa 1m/s ≤ vt ≤ 3m/s
(25) (26)
L/Ds < 15
(27)
The equation 28 is the fitness function. The constants r1 , r2 , r3 and r4 are penalty values, if a constrain is violated a penalty value is activated. FF = TAC + r1 max [(ΔPs − ΔPa ) , 0] + r2 max [(ΔPt − ΔPa ) , 0] + r3 [max {(1 − vt )} + max {(3 − vt )} , 0] + r4 max [(L/Ds − 15) , 0]
(28)
4. Case study The case study is taken from Wildi-Tremblay and Gosselin (2007). It is required to design a heat exchanger with a flow rate of 18.8 kg/s of cooling water, the inlet temperature is 33 °C and the outlet temperature is 37.2 °C. The hot fluid is nafta, the inlet temperature is 114 °C and the outlet temperature is 40 °C. Water is placed in tube-side and nafta in shell-side. The construction material for tube and shell side is stainless steel and carbon steel, respectively. Calculations are performed using a 5 % interest rate, 20 years life time period, 5000 operation hours per year, a pump efficiency of 0.85 and an electricity cost of 0.1 $/kWh. The maximum allowed pressure drop is 70,000 Pa. The factors to modify the purchase cost are fm = 1.7, ft = 1.0 and f p = 1.0.
5. Results MATLAB was used as codification environment for the heat exchanger mathematical model and GWO model. A population of 50 individuals was used with 50 iterations. To generate statistical information 30 runs where made. An average cost of 4,095.36 USD was found with a standard deviation of 103.57 USD. The minimum value was 3,977.76 USD. Particle swarm optimization (PSO) algorithm was also used to solve the problem as a comparison. With PSO an average cost of 4,197.56 USD with a standard deviation of 220.07 USD, the minimum value was 3,976.38 USD. The Table 1 shows the parameters value for both optimization methods. It can be seen that both method reached a similar solution. The computing time was 1.0161s for GWO and 2.1881s for PSO.
O.D. Lara-Montaño and F.I. Gómez-Castro
576
Table 1: Parameters for optimal heat exchanger configuration. Parameter Unit
Ds mm
do mm
Nt
A m2
L m
Pt mm
TL °
s
Lbc mm
Lbo mm
Bc %
302.36 301.90
15.87 15.87
144 143
32.35 32.30
4.50 4.53
19.83 19.83
90 90
1 1
60.47 60.38
60.47 60.38
25 25
Parameter Unit
δtb mm
δsb mm
Dotl mm
Nb
vt m/s
L/Ds
ΔPt Pa
ΔPs Pa
Cop $/year
Cf $/year
TAC $/year
GWO PSO
0.15 0.15
3.02 3.02
284.37 283.93
74 74
1.64 1.66
14.90 15.00
8938.76 9028.80
12843.52 12889.25
189.20 190.91
3789.08 3784.94
3978.29 3975.85
GWO PSO
6. Conclusion The Grey Wolf Optimizer has been proposed as a technique to solve the design and optimization of a shell-and-tube heat exchanger, modeled through the Bell-Delaware method. The optimization model is a MINLP, which is codified and solved in a MATLAB routine. The method has been proved to be efficient, reaching the optimal solution in a relatively low computing time. Moreover, the standard deviation between the obtained solutions in different tests is low, implying that the method tends to reach very similar optimal points for a number of runs.
References S. Mirjalili, S. M. Mirjalili, A. Lewis, 2014. Grey wolf optimizer. Advances in engineering software 69, 46–61. R. K. Shah, D. P. Sekulic, 2003. Fundamentals of heat exchanger design. John Wiley & Sons. R. K. Sinnott, G. Towler, 2009. Chemical engineering design: SI Edition. Elsevier. R. Smith, 2005. Chemical process: design and integration. John Wiley & Sons. P. Wildi-Tremblay, L. Gosselin, 2007. Minimizing shell-and-tube heat exchanger cost with genetic algorithms and considering maintenance. International Journal of Energy Research 31 (9), 867–885.
Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm Oscar D. Lara-Montañoa and Fernando I. Gómez-Castroa a Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Campus Guanajuato, Universidad de Guanajuato, Noria Alta S/N, Guanajuato, Guanajuato 36050, Mexico [email protected]
Abstract Most chemical processes require heat exchangers to modify the temperatures of the streams involved in the production steps. Due to their many applications, the rigorous design of such exchangers is of interest, aiming to determine the physical characteristics required to obtain the desired variation in temperature. The mathematical models representing such devices have several degrees of freedom and non-lineal equations, thus robust optimization algorithms are required to obtain the optimal solution in a short time. In this work, the use of the grey wolf algorithm is proposed for the optimization of a shell-and-tube heat exchanger, modeled by the Bell-Delaware equations. With the proposed method, a minimum total annual cost of 3,978.2 USD was obtained, in a mean time of 0.7976 seconds. Keywords: meta-heuristic optimization, shell-and-tube heat exchanger, grey wolf algorithm.
1. Introduction Heat exchangers are among the most used auxiliary equipment in chemical industry, since cooling and heating are common operations which allow modifying the temperature, or even the phase, of process streams, according to the operational requirements. This implies that heat exchangers are necessary for almost any chemical process. Shell-and-tube exchangers are widely used in industry to obtain wide ranges of temperature variations for streams from low to high flow rates, with high heat transfer efficiency due to the turbulence occurring on the shell. The design of a shell-and-tube heat exchanger involves several variables, which are usually related through energy balances and empirical correlations. One of the most common strategies to design heat exchangers in the industry is the Bell-Delaware method, which considers the variations of the convective heat transfer coefficient with the baffle configuration and phenomena occurring in the baffles, as leakage, pass partition bypass, among others. Nevertheless, the equations which model the heat exchanger are highly non-lineal and non-convex, mainly due to the presence of fractional and logarithmic terms in the mathematical model. Moreover, some of the variables are integer, as the number of tubes, and other variables can be taken as continuous, but in practice they take discrete values, as the internal and external diameter of the tubes, which are constrained by the standard dimensions for commercial tubes. Thus, a robust method is necessary to obtain proper solutions to the optimal design problem of the heat exchanger. The use of meta-heuristic optimization strategies offers a viable alternative to the solution of such problem. In this work, a recently reported stochastic optimization method, namely, the grey wolf optimization algorithm, is proposed to solve the optimization problem for a shell-and-tube heat exchanger, represented by the Bell-Delaware model, aiming to the minimization of the total annual cost. The grey wolf optimization algorithm emulates the hunting mechanisms of grey wolves, and it has been reported that it possesses a superior
O.D. Lara-Montaño and F.I. Gómez-Castro
572
exploitation capacity than other meta-heuristic algorithms, together with a high capacity to avoid local optima (Mirjalili et al., 2014)
2. Grey Wolf Optimizer Grey Wolf Optimizer (GWO) emulates the hunting technic of wolves. It was modeled by Mirjalili et al. (2014). This optimization algorithm considers the wolves hierarchy; alpha (α ), is not necessarily the strongest but is the one who makes important decisions that affect the whole group; beta (β ) wolves help the alpha in decision making activity, these wolves respect the alpha but command the other lower level wolves; omega (ω ) are the lowest ranked wolves in the hierarchy, although they are the least important, they help to maintain the structure of the pack; delta (δ ) wolves are those that are not in any of the previous categories, they have a lower hierarchy than alpha and betas but higher than the omega. The GWO algorithm takes into account different actions within the hunting process: • Tracking, chasing, and approaching the prey. • Pursuing, encircling, and harassing the prey until it stops moving. • Attack towards the prey. 2.1. Mathematical model The GWO algorithm defines the fitness solution as alpha α , the second and third best solutions are beta β and gamma γ , respectively. Other solutions are named omega ω . The optimization process is guided by α , β and γ . As mentioned before, wolves tend to encircle its prey. This behaviour is modeled with the equations 1 and 2. ·X D = |C p (t) − X (t)| X (t + 1) = X p (t) − A · D
(1) (2)
are coefficient p is the position vector of the prey, X is the position vector of a grey wolf, A and C X are calculated according to the equations vectors , and t is the current iteration. The vectors A and C 3 and 4 A = 2a · r1 −a = 2 · r2 C
(3) (4)
where components of the vector a are linearly decreased form 2 to 0 as the iterations pass, r1 and r2 are random vector with elements between 0 and 1. In nature, grey wolves know where the prey is, however, in the mathematical framework we don’t know where the optimum is (prey). To overcome this problem, it’s supposed that the alpha, beta and delta wolves have a better idea about the position of the prey. Assumed this, the best three results are saved and oblige the other search agents to update their position. To simulate this, the equations 5-7 are used. 1 · Xα − X, D 2 · Xβ − X, D 3 · Xδ − X| β = |C δ = |C Dα = |C 1 · Dα , X 2 · D 3 · D 1 = Xα − A 2 = Xβ − A 3 = Xδ − A β , X δ X X (t + 1) = X1 + X2 + X3 3
(5) (6) (7)
Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm
573
In summary, the population of wolves begins randomly within the search space. Each position of the wolves is a possible solution. The parameter a decreases in each iteration from 2 to 0, this parameter indicates the exploration and exploitation capacity of the GWO. If the value of |A ≥ 1|, the wolf diverges from the prey, if |A ≤ 1| the wolf converges towards the prey.
3. Heat Exchanger Mathematical Model 3.1. Heat transfer of shell side The Bell-Delaware mathematical model is used to predict the shell-side convective heat transfer coefficient hs . This model considers different parameters and geometrical variables (Shah and Sekulic, 2003). An ideal convective heat transfer coefficient hi is calculated and corrected using five parameters j according to the equation 8. hs = hi jc jl jb js jt
(8)
hi is evaluated as −2/3
hi = j
C ps Prs Ao,cr
(9)
where C ps is the fluid heat capacity, Prs is the Prandlt number, and jc , jl , jb , js , jt are correction factors for baffle configuration, shell to baffle and tube to baffle leakage effects, bundle pass and partition bypass, baffle spacing, and temperature gradient, respectively. Ao,cr is the flow area near the shell center-line for one cross-flow section and j is the Colburn factor, it is calculated with the equation 10 (Wildi-Tremblay and Gosselin, 2007). The variable a is determined with 11. The parameters a1 , a2 , a3 and a4 are reported in Wildi-Tremblay and Gosselin (2007). 1.33 a (Res )a2 (10) j = a1 Pt/do a3 (11) a= 1 + 0.14Reas 4 (12) The Reynolds number Res is evaluated with equation 13, μs is fluid viscosity, ms is the mass flow rate in shell-side and do is the tube external diameter. Res =
ms do μs Ao,cr
(13)
The area of the heat exchanger area is calculated with the Equation 14. A=
Q UΔTml Ft
(14)
where Q is the heat transfer rate, U is the global heat transfer coefficient, TLMDT is the logarithmic mean temperature difference and Ft is a correction factor. The U coefficient depends of the convective heat transfer of shell and tube side as well as fouling resistance of both sides, it is calculated with the Equation 15 1
U= 1 hs
+Rfs +
do ln ddo i
2kw
(15) + R f t ddoi + h1t
do di
ht is convective heat transfer coefficients for tube side, kw is the material thermal conductivity, R f s and R f t are the fouling factors for shell and tubes. The tube internal diameter, di , is calculates as di = 0.8do .
O.D. Lara-Montaño and F.I. Gómez-Castro
574 3.2. Heat transfer in tube side
To calculate ht , the equation 16 is used, this equation only works when the fluid in tubes is water (Sinnott and Towler, 2009). TCm is the medium fluid temperature and vt is the fluid velocity in tubes that is calculated with the equation 17. Where ρt fluid density, mt is the mass flow rate and Nt is the total number of tubes, is is calculated according the equation 4200 (1.35 + 0.02TCm ) vt0.8 di0.8 s mt vt = Nt π di2i ρ ht =
(16) (17)
t
4
3.3. Pressure drop calculations The pressure drop in the tubes side is calculated with the equation 18, the friction factor, f , is evaluated as f = 0.046Ret−0.2 , s is the the pass number and L the length of tubes that is calculated using 19. ρt vt 4fL (18) + 2.5 ΔPt = s di 2 A L= (19) π do Nt The expression 20 is used to calculate the pressure drop in shell-side Nr,cw ζb ζs ΔPs = (Nb − 1) ΔPb,id ζb + Nb ΔPw,id ζl + 2ΔPb,id 1 + Nr,cc
(20)
where Nb is the number of baffles, ΔPb,id is the ideal pressure drop in the central section, Nr,cw is the effective tubes en the cross flow, Nr,cc is the effective number of tubes rows crossed, ζ is a correction factor. 3.4. Cost estimation In this study, the total annual cost (TAC) is used, this is the sum of fixed, C f and operation cost Cop . To calculate the fixed cost the equation 22 is used (Smith, 2005). The purchase price C p is corrected using the factors fm , f p and fc for construction material, operating pressure and operation temperature, respectively. While the Cop is evaluated with equation 23. C p = 32800
A 80
0.68
C f = Cp fm f p fc (Es + Et ) EcHr Cop = 1000 r (1 + r)n ) +C f TAC = C f (1 + r)n ) − 1
(21) (22) (23) (24)
where Es and Et is the pump power (Watts) used in shell and tube side, respectively. Ec is the energy cost ($/kWh), Hr is the operational hours per year, n is the projected life time and r the interest rate per year. The optimization is performed using eleven parameters that can have a value between a given range, those are: • Diameter of shell (Ds ): 300mm -1000mm.
Optimization of a shell-and-tube heat exchanger using the grey wolf algorithm
575
• Outer diameter of tube (do ): 15.87mm 63.5mm. • Tube pitch Pt : [1.25do, 1.5do]. • Tube layout angle (T L): [30°, 45°, 90°] • Baffle spacing at center (Lbc ): 0.2Ds − 0.55Ds • Baffle spacing at the inlet and outlet (Lbo = Lbi ): Lbc − 1.6Lbc • Baffle cut (Bc ): [25%, 30%, 40%, 45%] • Number of tube passes (s): [1, 2, 4] • Tube-to-baffle diametrical clearance (δtb ): 0.01do − 0.1do • Diametrical clearance of shell-to-baffle (δsb ): 0.01Ds − 0.1Ds • Outer diameter of tube bundle (Dotl): 0.8 (Ds − dsb ) - 0.95 (Ds − dsb ) The optimization process is subject to constrains for maximum allowed pressure drop in tube-side and shell-side, fluid velocity, and length to shell-diameter ratio, as follows: ΔPs , ΔPt ≤ ΔPa 1m/s ≤ vt ≤ 3m/s
(25) (26)
L/Ds < 15
(27)
The equation 28 is the fitness function. The constants r1 , r2 , r3 and r4 are penalty values, if a constrain is violated a penalty value is activated. FF = TAC + r1 max [(ΔPs − ΔPa ) , 0] + r2 max [(ΔPt − ΔPa ) , 0] + r3 [max {(1 − vt )} + max {(3 − vt )} , 0] + r4 max [(L/Ds − 15) , 0]
(28)
4. Case study The case study is taken from Wildi-Tremblay and Gosselin (2007). It is required to design a heat exchanger with a flow rate of 18.8 kg/s of cooling water, the inlet temperature is 33 °C and the outlet temperature is 37.2 °C. The hot fluid is nafta, the inlet temperature is 114 °C and the outlet temperature is 40 °C. Water is placed in tube-side and nafta in shell-side. The construction material for tube and shell side is stainless steel and carbon steel, respectively. Calculations are performed using a 5 % interest rate, 20 years life time period, 5000 operation hours per year, a pump efficiency of 0.85 and an electricity cost of 0.1 $/kWh. The maximum allowed pressure drop is 70,000 Pa. The factors to modify the purchase cost are fm = 1.7, ft = 1.0 and f p = 1.0.
5. Results MATLAB was used as codification environment for the heat exchanger mathematical model and GWO model. A population of 50 individuals was used with 50 iterations. To generate statistical information 30 runs where made. An average cost of 4,095.36 USD was found with a standard deviation of 103.57 USD. The minimum value was 3,977.76 USD. Particle swarm optimization (PSO) algorithm was also used to solve the problem as a comparison. With PSO an average cost of 4,197.56 USD with a standard deviation of 220.07 USD, the minimum value was 3,976.38 USD. The Table 1 shows the parameters value for both optimization methods. It can be seen that both method reached a similar solution. The computing time was 1.0161s for GWO and 2.1881s for PSO.
O.D. Lara-Montaño and F.I. Gómez-Castro
576
Table 1: Parameters for optimal heat exchanger configuration. Parameter Unit
Ds mm
do mm
Nt
A m2
L m
Pt mm
TL °
s
Lbc mm
Lbo mm
Bc %
302.36 301.90
15.87 15.87
144 143
32.35 32.30
4.50 4.53
19.83 19.83
90 90
1 1
60.47 60.38
60.47 60.38
25 25
Parameter Unit
δtb mm
δsb mm
Dotl mm
Nb
vt m/s
L/Ds
ΔPt Pa
ΔPs Pa
Cop $/year
Cf $/year
TAC $/year
GWO PSO
0.15 0.15
3.02 3.02
284.37 283.93
74 74
1.64 1.66
14.90 15.00
8938.76 9028.80
12843.52 12889.25
189.20 190.91
3789.08 3784.94
3978.29 3975.85
GWO PSO
6. Conclusion The Grey Wolf Optimizer has been proposed as a technique to solve the design and optimization of a shell-and-tube heat exchanger, modeled through the Bell-Delaware method. The optimization model is a MINLP, which is codified and solved in a MATLAB routine. The method has been proved to be efficient, reaching the optimal solution in a relatively low computing time. Moreover, the standard deviation between the obtained solutions in different tests is low, implying that the method tends to reach very similar optimal points for a number of runs.
References S. Mirjalili, S. M. Mirjalili, A. Lewis, 2014. Grey wolf optimizer. Advances in engineering software 69, 46–61. R. K. Shah, D. P. Sekulic, 2003. Fundamentals of heat exchanger design. John Wiley & Sons. R. K. Sinnott, G. Towler, 2009. Chemical engineering design: SI Edition. Elsevier. R. Smith, 2005. Chemical process: design and integration. John Wiley & Sons. P. Wildi-Tremblay, L. Gosselin, 2007. Minimizing shell-and-tube heat exchanger cost with genetic algorithms and considering maintenance. International Journal of Energy Research 31 (9), 867–885.