heat transfer, heat leakage and mixing in context to a heat exchanger. Bejan (1977) presented the design of a gas-to-gas counterflow heat exchanger for minimum irreversibility ..... Free flow areas on the two fluid-sides are a,L b a,fa,f a,f a,f a.
THERMOECONOMIC DESIGN-OPTIMISATION OF CROSSFLOW PLATE-FIN HEAT EXCHANGER USING GENETIC ALGORITHM Manish Mishra and P.K. Das 1. Introduction Compact heat exchangers are characterised by a high rate of heat exchange within a small volume. Amongst different varieties of compact heat exchanger, crossflow plate-fin heat exchangers are widely used in aerospace, automobile, cryogenic and chemical process plants for their low weight and volume, high efficiency and ability to handle many streams. However, the superior thermal performance of compact heat exchangers is in general associated with high pressure-drops. Therefore, it often becomes necessary to find a trade-off between the increased rate of heat exchange and the power consumption due to higher pressure-drop within the constraints of specified performance requirements and available resources. Analysis based on second law of thermodynamics is applied for this purpose. Second-law based optimisation by entropy generation minimisation (EGM) is the method of thermodynamic optimisation of real systems that owe thermodynamic imperfection to heat transfer, fluid flow and mass transfer irreversibilities. The thermodynamic irreversibility or number of entropy generation units (Ns) indicates the amount of lost useful power, which is not available due to system irreversibilities. In the heat exchangers irreversibilities are generated due to the finite temperature difference heat transfer between the fluid streams and the pressure drops along them. Optimising heat exchanger or any other system on this basis means minimising the amount of lost or unavailable power by accounting for the finite size constraints of actual devices and finite time constraints of actual process (Bejan, 1996; Bejan et al., 1996). Poulikakos and Bejan (1982) established a theoretical framework for the minimisation of entropy generation in
1
extended surfaces. London (1982) has discussed in detail about the entropy generation, irreversibility evaluation and the relationship between irreversibility and the economics by taking an example of a condenser. An operationally convenient methodology is presented by London and Shah (1983) for relating economic costs to entropy generation. This methodology allows the designer to determine the trade-offs between the individual irreversibilities due to flow friction, heat transfer, heat leakage and mixing in context to a heat exchanger. Bejan (1977) presented the design of a gas-to-gas counterflow heat exchanger for minimum irreversibility and the design of a regenerative heat exchanger for minimum heat transfer area with fixed irreversibility. Seculic and Herman (1986) have presented the optimisation of a compact crossflow heat exchanger for the minimum enthalpy exchange irreversibility (EEI) using numerical ‘Fibonacci method’. Instead of optimising single component, global performance of the installation is used by Vargas et al. (2001) for optimisation by taking an example of crossflow heat exchanger used in the environmental control system of an aircraft for total component volume, and wall material volume. Vargas and Bejan (2001) further used the concept of optimising global performance by selecting finned and /or smooth parallel plate type crossflow heat exchanger of the environmental control system of aircraft. Apart from minimising the lost power or increasing the efficiency, there are other aspects also in designing a thermal system. Many of them are basically economic aspects and the decision-making is usually based on cost considerations. The total cost consists of initial or capital cost of the equipment, and the operating and maintenance cost of driving energy systems over a period of time. The second law based thermoeconomic optimisation can be explained as the minimisation of the combined cost of entropy generation with the annualised capital cost of
2
the thermal component. Due to its perceptible attraction thermoeconomics tends to obscure the thermodynamics portion of its foundation. A closed form analytical method for the second law based thermoeconomic optimisation of two-phase counterflow heat exchangers used as condensers or evaporators are developed by Zubair et al. (1987). Adding economic aspects to pure thermodynamic optimisation models, von Spakovsky and Evans (1989) have discussed a model using feed water heater as an example. The basic component of a heat exchanger, ‘fin’, is thermodynamically designed and optimised by Shuja and Zubair (1997) by taking an example of constant cross-section area pin fin and plate fin.
A
combination
of
entropy
generation
minimisation method and thermoeconomics is presented by Mahanta and Saha (2002) for a solar heater. A compact plate-fin type crossflow heat exchanger possesses a large number of design variables. The performance parameters of the heat exchanger bear complex functional relationships with these variables. Further, some of these variables are often discrete in nature. These render the optimisation of such equipments a rather difficult task. Though there are a few classical techniques of optimisation, which can handle a combination of continuous and discrete variables, the solution procedure becomes rather complex (Tayal et al., 1999). On the other hand, Genetic Algorithm (GA) based on evolutionary global search technique is particularly suitable for such problems (Schmit et al, 1996; Pacheco-Vega et al., 2001). Genetic algorithm has been applied successfully for the optimum design of different thermal systems and components namely convectively cooled electronic components (Queipo et al., 1994) and cooling channels (Wolfersdorf et al., 1997), fin profiles (Fabbri, 1997; Younes and Potiron, 2001), finned surface and finned annular ducts (Fabbri, 1998), compact high performance coolers (Schmit et al., 1996) and shell and tube heat exchangers (Tayal et al. 1999).
3
Further, crossflow plate-fin heat exchanger has also been optimised for total annual cost using GA by Mishra et al. (2004) In the present work, a GA based optimisation technique for crossflow plate-fin heat exchangers has been developed, which minimises total thermoeconomic cost (Zubair et al., 1987; Mahanta and Saha, 2002) for a specified heat duty under given space restrictions. The result compares well with that obtained by the graphical technique. The selection of optimum GA parameters (Grefenstette, 1986; Wolfersdorf et al., 1997) has also been done to achieve the faster result with better accuracy.
2. Outline of the Scheme of Optimisation Genetic algorithm is an evolutionary search procedure based on the principles of genetics and natural selection. An elaborate description of this technique is available in a number of references for example, Goldberg (2000) and others. The genetic search is started with an initial set of population termed as ‘Np’. The members of population can conveniently be represented by a binary coding consisting of 0’s and 1’s. The value of the objective function for a particular member decides its merit (competitiveness) in comparison with its counterparts. In GA language this is termed as fitness function. After creating an initial population, a simple GA works with three operators: reproduction, crossover and mutation. Reproduction, which constitutes a selection procedure whereby individual strings are selected for mating, based on their fitness values relative to the fitness of the other members. Individuals with higher fitness values have a higher probability of being selected for mating and for subsequent genetic production of offsprings. This operator, which weakly mimics the Darwinian principal of survival of the fittest, is an artificial version of
4
natural selection, where the selection is done stochastically. The probability for selecting the i th string is
pi
fi
,
Np
f j 1
where Np is the population size.
(1)
i
After reproduction, the crossover operator alters the composition of the offspring by exchanging part of strings from the parents and hence creates new strings. Crossover is also achieved stochastically using a suitable crossover probability, pc . The need for mutation is to create point in the vicinity of the current point, thereby achieving a local search around the current solution, which sometimes is not possible by reproduction and crossover. Mutation increases the variability of the population. For a GA using binary alphabet to represent a chromosome, mutation provides variation to the population by changing a bit of the string from 0 to 1 or vice versa with a small mutation probability p m . GA does not guarantee convergence to global optimum solution, and so requires suitable stopping criteria. The GA can be terminated when there is no improvement in the objective function (fitness) for a defined number of consecutive generations within a prescribed tolerance range, or when it covers a pre-specified maximum number of generations. In the simplest form GA can be formulated as unconstrained maximisation (Goldberg, 2000). For the present problem GA is used for constrained minimisation. If there are number of constraint conditions and the objective function needs to be minimised the problem can be stated as follows: Minimise f(X), X=[x1,……xk]
(2)
gj(X) 0, j=1,……,m
(3)
Where,
5
and xi, min xi xi,max, i=1,……,k.
(4)
For implementation in GA, the first step is to convert the constrained optimisation problem into an unconstrained one by adding a penalty function term. Minimise f(X) Mj1 ( g j ( X )) ,
(5)
subject to xi, min xi xi,max, i =1,……, k.
(6)
Where is a penalty function defined as,
(g(X)) = R1.g(X) 2.
(7)
Here R is the penalty parameter having an arbitrary large value. The second step is to convert the minimisation problem to a maximisation one. This is done redefining the objective function such that the optimum point remains unchanged. The conversion used in the present work is as follows Maximise F(X), where, F(X) = 1 / { f(X) +
(8)
m j 1
( g j ( X )) }.
(9)
More details regarding the scheme and the algorithm are given by Mishra et al. (2004).
3. Thermoeconomic Optimisation Fig. 1 depicts a schematic view of a crossflow plate-fin heat exchanger with offset-strip fins. Following assumptions are made for the analysis. 1. The steady state condition is assumed to be prevailing.
6
2. Offset-strip fins having the same geometrical specifications are used for both sides of the exchanger. 3. Distribution of primary and secondary areas and the heat transfer coefficients are assumed to be uniform and constant. 4. Physical property variation of the fluids with temperature is neglected. 5. Number of fin layers for fluid b is assumed to be one more than that of fluid a (Nb=Na+1). Rate of entropy generation for the two fluid streams is S ma ( S a ) mb ( Sb )
(10)
Further, number of entropy generation units can be given by,
Ns
S C max
(11)
Optimisation based on second law of thermodynamics gives the optimum design by minimising the number of entropy generation units, Ns. By incorporating the economic aspect to the thermodynamic optimisation problem, the thermoeconomic cost (Mahanta and Saha, 2002) of a plate-fin heat exchanger can be defined as follows. Total thermo-economic cost (TTEC) = Annual capital cost (ACC) + Cost involved with lost work due to heat transfer + Cost involved with lost work due to friction pressure drop = ACC + HT0 SH + pT0 Sp
(12)
Where, Annual capital cost (ACC) = Capital recovery factor (CRF) x Capital investment (CI) + Running cost + Maintenance cost – Annual salvage value (ASV), (13)
7
Capital recovery factor (CRF) =
i( i 1 ) N {( i 1 ) N 1 }
,
(14)
Annual salvage value (ASV) = Salvage fund factor (SFF) x Salvage value (SV), Salvage fund factor (SFF) =
i {( i 1 ) N 1 }
,
(15) (16)
Salvage value (SV) = 0.1 x Capital investment,
(17)
Maintenance cost = 0.1 x Capital recovery factor x Capital investment,
(18)
Capital investment (CI) = Initial cost of heat exchanger = HX. A,
(19)
Total area of heat transfer A = La Lb (2Na+1)[1+ 2na(Ha - ta)],
(20)
Running cost = [
ma
a
Pa
mb
b
Pb ]
C powTime / year
pump
.
(21)
Entropy change due to heat transfer between the two fluids can be given as,
S H ma Cpa ln
Ta ,2 Tb ,2 mb Cpb ln . Ta ,in Tb ,in
(22)
Now, the effectiveness of the exchanger is
So,
Ca ( Ta ,in Ta ,2 ) C (T T ) b b ,2 b ,in Cmin ( Ta ,in Tb ,in ) Cmin ( Ta ,in Tb ,in ) Cmin ( Ta ,in Tb ,in ) Ca
(24)
Cmin ( Ta ,in Tb ,in ) Cb
(25)
Ta ,out Ta ,in
Tb ,out Tb ,in
(23)
S H Ca ln 1
Cmin Tb ,in Cmin Ta ,in 1 C ln 1 1 b Ca Ta ,in Cb Tb ,in
(26)
For crossflow heat exchanger with both fluids unmixed, effectiveness (Incropera and Dewitt, 2001) is given by
8
1 0.22 exp[ Cr .NTU 0.78 ] 1 , NTU Cr
1 exp
and
1 C 1 1 min Cmin NTU UA ( hA )a ( hA )b
(27)
(28)
Heat transfer areas for the two sides can be obtained as given below. Aa = La.Lb.Na[1+ 2.na.(Ha-ta)]
(22)
Ab = La.Lb.Nb[1+ 2.nb.(Hb-tb)]
(23)
Introducing the expressions for heat transfer coefficients, 1 1 Affa 1 Affb Cmin . . . 2 / 3 2 / 3 NT U ma A a jbCp b Prb mb A b ja Cp a Pra
(24)
In the similar fashion the entropy change due to friction pressure drop is given as,
S p ma Ra ln
Pa ,2 P mb Rb ln b ,2 Pa ,in Pb ,in
(25)
where, Pa,2 = Pa,in-( Pa,in- Pa,2) = Pa,in - Pa, and
(26)
Pb,2 = Pb, in -( Pb, in - Pb,2) = Pb, in - Pb So,
S p ma Ra ln( 1
ma Ra
ma
Pa Pa ,in
Pa Pa ,in
mb Rb
Pa
a ,inTa ,in
mb
) mb Rb ln( 1
(27)
Pb Pb ,in
)
Pb Pb ,in
Pb
b ,inTb ,in
(28)
Pressure drop for the two fluid streams can be calculated as
Pa
4.f a .La .Ga 2. a .Dh ,a
2
,
(29)
9
2
4.f .L .G Pb b b b . 2. b .Dh ,b
(30)
Free flow areas on the two fluid-sides are Aff a (H f , t f ,a ).( 1 n f ,a t f ,a ).Lb .N L ,a
(31)
Aff b (H f,b t f ,b ).( 1 n f ,b t f ,b ).La .N L ,b
(32)
a
Now, Colburn j factor and friction coefficient f used in eq. (24), (29) and (30) may be evaluated from available correlations (Joshi and Webb, 1987). For laminar flow (Re1500)
j 0.53(Re) 0.5 ( l f / Dh )0.15 { s f /( H f t f )}0.14
(33)
f 8.12(Re) 0.74 ( l f / Dh )0.41{ s f /( H f t f )}0.02
(34)
and for turbulent flow (Re>1500)
j 0.21(Re) 0.4 ( l f / Dh )0.24 ( t f / Dh )0.02
(35)
f 1.12(Re) 0.36 ( l f / Dh )0.65( t f / Dh )0.17
(36)
where, Re
GDh
m.Dh Aff .
(37)
The statement of the optimisation problem is as follows. Minimise f(X)=TTEC, Subjected to the constraints: g1(X) 0.1 ≤ La ≤ 1; g2(X) 0.1 ≤ Lb ≤ 1;
10
(38)
g3(X) 0.002 ≤ H ≤ 0.01; g4(X) 100 ≤ n ≤ 1000; g5(X) 0.0001 ≤ t ≤ 0.0002; g6(X) 0.001 ≤ l ≤ 0.010; g7(X) 1 ≤ Na ≤ 10.
(39)
The heat duty of the exchanger is given by the equality constraint g8(X) 1 – [ .Cmin (Ta,in-Tb,in)]/ Q = 0,
(40)
where Q is the minimum heat duty mentioned in the example as 160 kW.
Air is assumed to be the working fluid for both hot side and the cold side of the heat exchanger separated by separating sheet and fins made of aluminium. The operating and geometrical constraints and other additional operating variables, selected for the present example are given as follows (Shah, 1980).
ma = 0.8962 kg/s
mb = 0.8296 kg/s
Ta,in = 513 K
Tb,in = 277 K
Cpa = 1017.7 J/kg-K
Cpb = 1011.8 J/kg-K
a = 0.8196 kg/m3
b = 0.9385 kg/m3
a = 241.0 N-s/m2
b = 218.2 N-s/m2
Pra = 0.6878
Prb = 0.6954
pump = 0.7
Operational time/year = 8000 Hrs
Number of years, N = 10
Q = 160 kW
11
Cpow = 0.00005 $ / kW-Hr
H = $ 0.1 / kW
HX = $ 270 / m2 of heat transfer area
p = $ 0.04 / kW
4. Results and Discussion Though the designer has some independence in selecting the GA parameters, it has been observed that selection of proper GA parameters renders a quick convergence of the algorithm and the proper GA parameters are problem specific (Wolfersdorf et al., 1997; Grefenstette, 1986). Therefore initially an exercise has been made following the methodology of Wolfersdorf et al. (1997) to select the optimum GA parameters for the present problem. The effect of population size, crossover and mutation probabilities and penalty parameter on TTEC, fmax and Ns is depicted in Fig. 2. It is observed the population size 60, p c = 0.6 and p m = 0.01 gives the minimum TTEC, however, the points in some cases may not be the points for maximum fmax and minimum Ns. For the range tested, penalty function parameter does not show any effect on TTEC, fmax and Ns, so, an arbitrary value 100 is selected for the solution. The optimum solution using the above values of selected parameters is given in the table 1. To check the solution, the variation of Ns, Q and TTEC are plotted in the form of contours for different combinations of La and Lb keeping other parameters constant, and are as shown in Fig. 3. Though the solution obtained through GA is not the global solution for the minimum TTEC, but it corresponds to the minimum solution with the specified heat duty. It can also be observed from the contours that for the specified heat duty, the solution for the minimum TTEC and that for minimum total entropy generation units are close to each other and also the
12
value of Ns corresponds to the optimum TTEC matches with that (as shown in Table 1). This shows that the thermoeconomic optimisation also satisfies the thermodynamic optimisation. The design parameters (La, Lb and Na) are varied individually keeping other parameters fixed at that of optimum TTEC and the variations of TTEC and Ns, and the pressure drops on the two sides are plotted in Fig. 4 and 5 respectively. Due to restriction in heat duty, the domain of feasible design becomes very small, but the values of La, Lb and Na obtained for minimum TTEC are same as shown in Table 1 and also as the graphical solution obtained in Fig. 3. The variation can be taken up as the effect of putting additional constraints (La, Lb and Na) along with heat duty Q. The variation of the pressure drops can be very useful for the designers as the prior information to start with.
5. Conclusion An optimisation model for crossflow plate-fin heat exchanger having large number of design variables of both discrete and continuous type has been developed using genetic algorithm. The case of multilayer plate-fin heat exchanger has been solved for minimum thermoeconomic cost. The parametric study of selected input variables and their effects on solution/ performance result is anticipated. It has also been shown that the solution for minimum thermoeconomic cost also satisfies the solution for minimum entropy generation units in the same environment. The results can be very useful for the designers to start with such complex thermal system
Nomenclature A –heat transfer area, m2
Aff – free flow area, m2
13
C – heat capacity rate (mCp), W/K
Na, Nb – number of fin-layers
Cp – specific heat of fluid, J/kg-K
Np – population size
Cpow - cost of power, $/unit
Ns – number of entropy generation units,
Cr – Cmin/Cmax
dimensionless
CHX - cost of heat exchanger per unit
NTU – number of transfer units
heat transfer area, $/m2
P – pressure, N/m2
Dh – hydraulic diameter, m
p c – crossover probability
f – Fanning friction factor
p m – mutation probability
f(X) - objective function Pr - Prandtl number fmax – GA fitness parameter
P – pressure drop, N/ m2
g(X) - constraint G – mass flux velocity ( m/Aff), kg/ m2-s h – heat transfer coefficient, W/m2-K
Q – rate of heat transfer, W R – specific gas constant, J/kg-K R1– penalty parameters
H - height of the fin, m Re – Reynolds number I – rate of interest annually s – fin spacing (1/n-t), m j - Colburn factor l – lance length of the fin, m L - heat exchanger length, m m – mass flow rate of fluid, kg/s M - number of constraints. n - fin frequency, fins per meter n’ – number of generations
N – number of years
S - rate of entropy generation, W/K
S – entropy difference, W/K St – Stanton number [h/(GCp)] t – fin thickness, m T-Temperature, K TTEC – total thermoeconomic cost, $ Time/year – yearly operational time, hours
14
T0 – ambient temperature, K
- viscosity, N/ m2-s
U– overall heat transfer coefficient, W/
- overall efficiency of heat transfer
m2 K
pump - pump efficiency
xi - variable
(.) – penalty function
X – (x1, x2,……xk) Greek letters - effectiveness
Subscripts a, b – fluid a and b i - variable number
– density, kg/ m3
in – inlet
H – unit cost of heat transfer, $ / kW
max -maximum
P – unit cost of pressure drop, $ / kW
min – minimum
HX – unit cost of heat exchanger, $/m2
out - exit
References Bejan, A. (1977). The Concept of Irreversibility in Heat Exchanger Design: Counterflow Heat Exchangers for Gas-to-Gas applications. ASME J. Heat Transfer, 99, 374-380. Bejan, A. (1996). Entropy Generation Minimization. New York: CRC Press. Bejan, A., Tsatsaronis, G., & Moran, M. (1996). Thermal Design and optimisation. New York: John-Wiley & Sons, Inc. Fabbri, G. (1997). A Genetic Algorithm for Fin Profile Optimisation. Int. J. Heat Mass Transfer, 40(9), 2165-2172. Fabbri, G. (1998). Heat Transfer Optimisation in finned Annular Ducts Under Laminar Flow Conditions. Heat Transfer Engineering, 19(4), 42-54.
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Goldberg, D.E. (2000). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Longman, Inc. Grefenstette, J.J. (1986). Optimization of Control Parameters for Genetic Algorithms. IEEE Trans. of Systems, Man and Cybernatics, 16(1), 122-128. Incropera, F.P., & DeWitt, D.P. (2001). Fundamentals of Heat and Mass Transfer. John Wiley and Sons, Inc. Joshi, H.M., & Webb, R.L. (1987). Heat Transfer and Friction in the Offset Strip-fin Heat Exchanger. Int. J. Heat Mass Transfer, 30(1), 69-84. London, A.L. (1982). Economics and Second Law: An Engineering View and Methodology. Int. J. Heat Mass Transfer, 25(6), 743-751. London, A.L., & Shah, R.K. (1983). Cost of Irreversibilities in Heat Exchanger Design. Heat Transfer Engineering, 4(2), 50-73. Mahanta, D.K., & Saha, S.K. (2002). Second Law Analysis of a Solar Air Heater. Proc. Of 5th ISHMT-ASME Heat and Mass Transfer Conference and 16th National Heat and Mass Transfer Conference, Calcutta, India, 1352-1357. Mishra, M., Das, P.K., & Sarangi, S. (2004). Optimum Design of Crossflow Plate-Fin Heat Exchangers Through Genetic Algorithm. Int. J. Heat Exchangers, 5 (2), 379-402. Pacheco-Vega, A., Sen, M., Yang, K.T., & McClain, R.L. (2001). Correlations of FinTube Heat exchanger Performance data Using Genetic algorithms, Simulated annealing and interval Methods. Proc. of ASME International Mechanical Engineering congress and Exposition, 1-9. Poulikakos, D., & Bejan, A. (1982). Fin Geometry For Minimum Entropy Generation in Forced Convection. ASME J. Heat Transfer, 104, 616-623.
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Queipo, N., Devarakonda, R., & Humphery, J.A.C. (1994). Genetic Algorithms For Thermosciences Research: Application to the Optimized Cooling of Electronic Components. Int. J. Heat Mass Transfer, 37(6), 893-908. Schmit, T.S., Dhingra, A.K., Landis, F. & Kojasoy, G. (1996). A Genetic Algorithm Optimization Technique for Compact high Intensity Cooler Design. J. Enhanced Heat Transfer, 3(4), 281-290. Seculic, D.P., & Herman, C.V. (1986). One Approach to Irreversibility Minimization in Compact Crossflow Heat Exchanger. Int. Commn. Heat Mass Transfer, 13, 23-32. Shah, R.K. (1980), Compact Heat Exchanger Design Procedure,
Heat Exchangers,
Thermal Hydraulic Fundamentals and Design, eds. S. Kakac, A.E. Bergles and F. Mayinger, Hemisphere Publishing Corporation, 495-536. Shuja, S.Z., & Zubair, S.M. (1997). Thermoeconomic Optimization of Constant CrossSectional Area Fins. ASME J. Heat Transfer, 119, 860-863. Tayal, M.C., Fu, Y. & Diwekar, U.M. (1999). Optimum Design of Heat Exchangers: A Genetic Algorithm Framework. Ind. Eng. Chem. Res., 38, 456-467. Vargas, Jose V.C., & Bejan, A. (2001). Thermodynamic Optimization of Finned Crossflow Heat Exchangers for Aircraft Environment Control System. Int. J. Heat and Fluid Flow, 22, 657-665. Vargas, Jose V.C., Bejan, A., & Siems, D.L. (2001). Integrative Thermodynamic Optimization of of the Crossflow Heat Exchanger for an Aircraft Environment Control System. ASME J. Heat Transfer, 123, 760-769. von Spakovsky, M.R., & Evans, R.B. (1989). The Design and Performance Optimization of Thermal System Components. J. Energy Resources Technology, 111, 231-238.
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Wolfersdorf, J.V., Achermann, E., & Weigand, B. (1997). Shape Optimization of Cooling Channels Using Genetic Algorithms. ASME J. Heat Transfer, 119, 380-388. Younes, M., & Potiron, A. (2001). A Genetic Algorithm for the Shape Optimisation of Parts Subjected to Thermal Loading. Numerical heat Transfer (Part A), 39, 449-470. Zubair, S.M., Kadaba, P.V., & Evans, R.B. (1987). Second-Law-Based Thermoeconomic Optimization of Two-Phase Heat Exchangers. ASME J. Heat Transfer, 109, 287-294.
Figure Captions Fig. 1. Schematic representation of (a) crossflow plate-fin heat exchanger, and (b) offsetstrip fin. Fig. 2. Effect of different GA parameters, (a) population size, (b) crossover probability, (c) mutation probability, and (d) penalty parameter, on maximum fitness, number of entropy generation units and TTEC. Fig. 3. Total thermoeconomic cost TTEC, number of entropy generation units Ns and heat duty contours in the design space. Fig. 4. Effect of variation of (a) La, (b) Lb, and Na on number of entropy generation unitsNs and total thermoeconomic cost TTEC. Fig. 5. Effect of variation of (a) La, (b) Lb, and Na on optimum pressure drops on the two sides.
18
Table Captions Table 1. The optimum solution using the selected values of parameters from Fig. 2.
Fluid b
(a)
Fluid a
Lb La
1/n
l H
h t
19
Fig. 1. Schematic representation of (a) crossflow plate-fin heat exchanger, and (b) offsetstrip fin.
(b) Maximum fitness, fmax No of entropy generation units, Ns TTEC
0.0635 0.0634
0.0638 0.0636
0.0633
10900
TTEC
0.000094
fmax
TTEC
0.0628
11200 0.00009
fmax
fmax 0.00007
0.000090
60
80
10800
0.00006
10700
40
0.0630
0.00008
10800
fmax0.000092
Ns
0.0632
Ns
0.0632 0.0631
11600
Ns 0.0634
11000
TTEC - $
Ns
11100
0.2
100
0.4
(c) Ns
60000
0.062 0.060
TTEC
40000
fmax
fmax 20000 1E-5 1E-3
1
Ns
TTEC - $
0.064
Ns
(d)
Maximum fitness, fmax 80000 No of entropy generation units, Ns TTEC
0.066
0.6
0.8
1.0
Crossover Probability, p c
Population Size
Maximum fitness, fmax No of entropy generation units, Ns Total thermoeconomic cost, TTEC 12000
0.1
Ns
11500
TTEC
11000
0.01
1E-3
TTEC - $
0.0636
Maximum fitness, fmax No of entropy generation units, Ns TTEC
0.0640
TTEC - $
(a)
10500
fmax 1E-4
fmax 10000
0.01
0.1
Mutation Probability, pm
1
1
10
100
1000
Penalty Function Parameter, R1
20
Fig. 2. Effect of different GA parameters, (a) population size, (b) crossover probability, (c) mutation probability, and (d) penalty parameter, on maximum fitness, number of entropy generation units and TTEC.
1.0
Ns
Lb - m
0.9
0.053 0.058
0.068
0.8
1.5E4
0.7
1.3E4
0.063
0.6
Q=160 kW
1.2E4
0.5 0.4
TTEC
8E3 9E3
7E3
1.4E4
1E4 1.1E4 1.2E4
0.3 0.2 0.1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
La - m
Fig. 3. Total thermoeconomic cost TTEC, number of entropy generation units Ns and heat duty contours in the design space.
21
11600
No of Entropy Generation Units, Ns
0.06336 11600
0.063385
0.06334 11200
TTEC
11400
TTEC - $
Ns
11000
10800
0.063360 0.063355
Ns
11000
0.063350 0.063345 0.063340
10800
0.063335
0.06331 0.8
0.063330
1.0
0.6
0.8
1.0
Lb - m
La - m
(c) 0.11
10800 0.10
0.09
10600
Ns 0.08
TTEC 10400
0.07
10200
0.06
6
7
8
9
No of Entropy Generation Units, Ns
0.6
0.063375
0.063365
11200
0.06332
TTEC
0.063380
0.063370
0.06333
TTEC - $
TTEC - $
0.06335 11400
0.063390
No of Entropy Generation Units, Ns
(b)
(a)
10
Na
22
Fig. 4. Effect of variation of (a) La, (b) Lb, and Na on number of entropy generation units-
(a) Pressure Drop, Pa & Pb- Pa
(b)
6000
5000 Pb
4000
3000 Pa
2000
1000 0.6
0.8
1.0
Pa
5000
Pb
0.6
0.8
1.0
Lb- m
La- m
(c) Pressure Drop, Pa & Pb- Pa
Pressure Drop, Pa & Pb- Pa
Ns and total thermoeconomic cost TTEC.
8000 7500 7000 6500 6000 5500 5000 4500 4000 3500 3000 2500 2000
Pb
Pa
6
7
8
9
10
Na
23
Fig. 5. Effect of variation of (a) La, (b) Lb, and Na on optimum pressure drops on the two sides.
Table 1 The optimum solution using the selected values of parameters from Fig. 2.
La, m Lb, m 0.618
0.987
H, mm 9.9
n,
t,
fins/m
mm
654.5
0.101
l, mm
Na
TTEC, $
7.551
9
10801.1
Ns
Q, kW
0.06335 159.96
24
