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International Journal of Thermal Sciences 49 (2010) 1427e1436

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

A distributed parameter model and its application in optimizing the plate-finheat exchanger based on the minimum entropy generation

Lina Zhang, Chunxin Yang*, Jianhui ZhouSchool of Astronautics Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China

a r t i c l e i n f o

Article history:Received 25 July 2009Received in revised form14 January 2010Accepted 15 February 2010Available online 2 April 2010

Keywords:Plate-fin heat exchangerDistributed parameter modelOptimizationMinimum entropy generation

* Corresponding author.E-mail address: yangchunxin@sina.com (C. Yang).

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.02.008

a b s t r a c t

A general three-dimensional distributed parameter model (DPM) was developed for designing the plate-fin heat exchanger (PFHE). The proposed model, which allows for the varying local fluid thermophysicalproperties inside the flow path, can be applied for both dry and wet working conditions by using theuniform enthalpy equations. The grids in the DPM were generated to match closely the flow passage ofthe heat exchanger. The classical correlations of the heat transfer and the flow friction were adopted toavoid solving the differential equations. Consequently, the computation burden of DPM becomessignificantly less than that of the Computational Fluid Dynamics method. The optimal design of a PFHEbased on the DPM was performed with the entropy generation minimization taken into consideration.The genetic algorithm was employed to conduct the optimization due to its robustness in dealing withcomplicated problems. The fin type and fin geometry were selected optimally from a customized findatabase. The PFHE included in an environmental control system was designed by using the proposedapproach in this study. The cooling performance of the optimal PFHE under both dry and wet conditionswas then evaluated.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

The plate-fin heat exchanger (PFHE), one type of high-efficiencycompact heat exchangers, has been extensively employed in manyfields [1], such as the aerospace, chemical engineering, and energysystem etc. Performance prediction and optimal design of PFHE aretwo key issues nowadays for the purpose of saving energy andreducing operational cost. There are several challenges regardingthese two issues.

Firstly, the temperature difference between the inlet and outletfluids is considerably large as the result of the high compactness(usually 700e2500 m2/m3) of PFHE. Some thermophysical prop-erties of the fluids vary drastically with the changes in thetemperature (Table 1) [2]. For example, the kinematical viscosity n,and the Prandtl number of the oil reduce by 68 times and 58 timesrespectively when the fluid temperature increases from 0 �C to80 �C. Hence, considering the property variation is crucial to eval-uate precisely the performance of PFHE. Additionally, the phasechange of the fluid also has evident influence on thermophysicalproperties. In a typical environmental control system (ECS)embedded in the aircraft (Fig. 1), there are four main heat

son SAS. All rights reserved.

exchangers (A, B, E, and G in Fig. 1), which occupy around 2/3weight and volume of the entire ECS and affect the performance ofthe ECS significantly. The working conditions for these heatexchangers are always extremely complex. Specifically, the watercondensation, even freeze, may take place inside the passage of thecondenser ‘E’. Such unwanted condensation or freeze likely resultsin blocking the passages and thus reducing the reliability of ECS.Further, water collected by thewater separator ‘F’ is usually sprayedinto the ram air flowing through the secondary and primary heatexchangers, where the water aerosol evaporates gradually. In suchcases, the fluid properties substantially alternate during the heatand mass transfer processes.

Though integrated parametermodel (IPM) has beenwidely usedin designing the heat exchanger, it has inherent difficulty to char-acterize the different thermophysical properties of every point inthe exchanger. Kays and London [1] developed the methods tocorrect the variance of the fluid properties resulted from thetemperature change on the flow section and along the flow direc-tion when the temperature varies pronouncedly large. Neverthe-less, the correction method is not practical to use in modeling thecross-flow heat exchanger, because the fluid in each flow passagebehaves differently.

To overcome such deficiencies of the IPM, the ComputationalFluid Dynamics (CFD)-based methods have been broadly utilized inrecent years. The CFD-based approaches are able to yield accurate

Nomenclature

Ac free flow area (m2)b fin height (m)cp specific heat capacity at constant pressure (J kg�1 K�1)d humidity ratio (kg kg�1 air)Dh hydraulic diameter of passage (m)f friction factorF heat transfer area (m2)h convection heat transfer coefficient (W m�2 K�1)hm mass transfer coefficient (m s�1)H exchanger height (m)i specific enthalpy (J kg�1)ifg latent heat of vapor (J kg�1)j Colburn factorl element length (m)L exchanger length (m)Le Lewis Numberm mass flow rate (kg s�1)MX, MY, MZ grid numbers in the direction of exchanger length,

width and height, respectivelyNfin index number of finP pressure (Pa)Pf fin pitch (m)DP overall pressure drop of fluid (Pa)Dpmax acceptable pressure drop of fluid (Pa)Pr Prandtl numberQ heat duty (kW)R gas constant (J kg�1 K�1)Re Reynolds numberSgen entropy generation (J kg�1 K�1)Sc Schmidt numbert time (s)

T temperature (�C, K)Tex average outlet temperature of exchanger (�C)u velocity (m s�1)W exchanger width (m)X vector of optimization variables

Greek symbolsb Total areas per unit volumes (m2 m�3)d characteristic length of element (m)f fin areas/total areash effectiveness of heat exchangerhd mass transfer effectivenesshfin fin efficiencyl heat conductivity (W m�1 K�1)n kinematic viscosity (m2 s�1)r density (kg m�3)

Subscriptsa airc cold fluidex outlet parameterE current elementf fluidfin finh hot fluidin inlet parameterl liquidmax maximum valuemin minimum valuesat saturation conditionv vaporw wall

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e14361428

results through solving NaviereStokes equation when selectingproper flow models, such as wall function, viscosity model andturbulence model, etc. Many investigators obtained the perfor-mance of the fins through CFD method and the numerical resultswere well validated by the experimental data [3e5]. However, CFDrequires massive computational burden, mainly because of thecomplexity of generating the computational grids and the difficultyof solving the partial differential equations. For instance, the modelin Ref. [6] contained approximately 1,000,000 elements and it took5 h to simulate one operational condition of the compact heatexchanger. Nowadays, CFDmethodwas also applied to simulate the

Table 1Fluid thermophysical properties variations with the change in the temperature [2].

T (�C) r (kg/m3) cp (J/(kg K)

Air 0 1.293 1.00580 1.000 1.009Variation 22.66% 0.40%

Water 0 999.9 4.21280 971.8 4.195Variation 2.81% 0.40%

R134a �30 1385.9 1.26050 1102.0 1.569Variation 20.48% 24.52%

Engine oil 0 905.0 1.83480 857.4 2.148Variation 5.26% 17.12%

performance of the plate heat exchanger [7] and the influence ofthe flow maldistribution on the whole heat exchanger [8]. But thehuge computational cost of the CFD-based method was prohibitiveto design or optimize heat exchanger, especially for the complexPFHE.

The primary purpose of the present study was, therefore, toestablish a feasible three-dimensional (3D) distributed parametermodel (DPM). The proposed approach can rapidly perform thePFHE simulation due to its higher computational efficiency incomparison with the traditional CFD methods, and its strongerability in characterizing the varying fluid thermophysical

) l (W/(m K)) n � 106 (m2/s) Pr

0.0244 13.28 0.7070.0305 21.09 0.69225.00% 58.81% 2.12%

0.551 1.789 13.670.674 0.365 2.2122.32% 79.60% 83.83%

0.1073 0.3106 5.0540.0704 0.1431 3.51534.40% 53.93% 30.45%

0.1449 1336 15,3100.1379 19.7 2634.83% 98.53% 98.28%

BA

E

F

C D

ram air

to cabin

engine bleed

A: primary heat exchangerB: secondary heat exchangerC: compressorD: turbineE: condenserF: water separatorG: regenerator

G

Fig. 1. Aircraft environmental control system with high pressure water separation.

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e1436 1429

properties against the IPM. This novel approach also providesa powerful tool to optimize the PFHE. The distributedmethods havebeen used in the analysis of the finned tube heat exchanger of therefrigeration system [9e12], where usually the thermodynamicperformance along the tube length was concerned. However, thereis little literature in regards of the PFHE designwith DPM due to thecomplexity of the PFHE.

The DPM-based PFHE optimization is a highly-nonlinearproblem involving many design variables. Genetic algorithm (GA),an adaptive searching technique elicited by the natural evolution ofspecies has been widely used in optimizing the heat exchangersdue to its robustness and high computation efficiency [12e17]. Inthese studies, both the IPM [12,13] and CFD method [14e17] wereemployed to compute the optimization objectives.

Another important issue involved in optimization is to establishthe reasonable objective function. Entropy generation is a compre-hensive parameter since it not only describes the energy loss, butalso avoids the conflicts between the heat transfer and fluid fric-tion. Such conflicts are generally encountered in the traditionaloptimal design [18]. In recent few decades, Entropy generationminimization (EGM) has been focusing on the optimization of thetube internal flow, heat exchangers or heat sinks [19e23]. Forexample, Ratts and Raut [19] analytically deduced the solution ofentropy generation for fully developed internal flow. They alsoestablished the relationship between the minimum entropygeneration number and the Reynolds number under constant heatflux. EGM was also applied to determining the best fin arrays andgeometries of heat sinks [17,20,21] and optimizing the heatexchangers included in the ECS [22,23]. The current study woulddetermine the optimal PFHE design by using the DPM with the aidof GA to minimize the entropy generation within the PFHE.

Fig. 2. A typical cross-flow PFHE and the mesh for DPM.

2. Distributed parameter model

An effective DPM, which is capable of accounting for thedetailed thermophysical properties, is primarily characterized bythe special mesh generation method and enthalpy equations aim-ing at the control volumes. The steady state model was developedcomplying with the following assumptions:

a) The outer surface of the heat exchanger is adiabatic;b) The thickness of the metal wall is negligible;c) Heat transfer occurring in the core heat exchanger is

considered;d) The mass flow rate and inlet temperature of each fluid are

constant.

a

b

c

d

Fig. 4. Different cross-sectional fins.

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e14361430

2.1. Mesh generation

The DPM model was based on the grids that coincide with theflow passages. Such grids enable that the current researchachievements (such as heat transfer and friction correlations givenby Kays and London [1]) can be used directly in heat exchangersdesign and lessen the necessary computation time greatly.

Fig. 2(a) shows a cross-flow heat exchanger with the rectangle-plain-fin. The dash lines sketch the mesh generated for the DPM.The overall perspective and top views of the mesh are respectivelylisted in Fig. 2(b) and (c), where that the length and width of thegrid are the fin pitch of hot and cold fluids, respectively. The heightof the grid is the corresponding fin height. The effective heattransfer areas for an element of the grid, Ei,j,k (Fig. 3) taken fromFig. 2(c), can be calculated as follows,

F1 ¼ F3 ¼ Fw þ hfinFfin (1)

F2 ¼ F4 ¼ Ffin (2)

where Ffin and Fw respectively represent the areas of the fin surfaceand the prime (unfinned) surface for single fin passage; hfin is thefin efficiency.

Mesh generation method for the rectangle-plain-fin introducedabove is intuitionistic. It can be extended easily to the heatexchangers with other cross-sectional shape plain-fins. Forinstance, for triangle cross-sectional fin, two adjacent passages arecombined, as the shadowarea exhibited in Fig. 4(b). The grid can begenerated accordingly. The expressions of the effective heattransfer area for the triangle fin are different from the one for therectangle fin:

F1 ¼ F3 ¼ 2ðFw þ hfinFfinÞ (3)

F2 ¼ F4 ¼ Ffin (4)

Trapezoid and other sectional plain fin can be treated similarlyto the triangle fin. The grids of the trapezoid and other sectionalplain fin are described as the shaded region as depicted in Fig. 4(c)and (d), respectively.

With respect to the non-plain surfaces, such as the strip,louvered andwavy fin surfaces, themeshes can be derived similarlyto the plain-fin surfaces with the same cross-sectional shape. Theonly distinctness resides in the concrete values of the heat transferarea and the heat transfer coefficient (see Eq. (6)).

Overall, the mesh generation method of DPM has the followingadvantages against that of CFD method.

a) The grid size is equal to the characteristic dimension used inthe calculation formulas of heat transfer and friction. Asa result, the experimental data or correlations of Colburn factor

x

y

z

o

1

3

4

2

Fig. 3. One of the elements taken from Fig. 2.

j and friction factor f can be used directly in the DPM withoutconsidering the complex turbulent model of the PFHE.

b) The drastically-reduced number of grid leads to significantlyless computation time.

c) The method is flexible to be extended to different kinds of finsin accordance with the fin database (see Section 3.3).

2.2. Governing equations

One aim of current study was to build a universal model forhumid air of PFHE working under both dry and wet conditions inECS, where the phase change of water probably happens. So theenthalpy was used to establish the energy balance equation whichwas involving the simultaneous temperature and humidity.

The energy equation of the fluid element given in Fig. 3 is,

mðiex � iinÞ ¼X4k¼1

hðhFÞEðiTw;k

� iTf ;E Þ.cp;E

i(5)

where, Tf,E represents the fluid temperature of the current element;Tw,k is the wall temperature between the current element and theadjacent element. iTw;k

and iTf ;E indicate the enthalpy at the corre-sponding temperature. The heat transfer coefficient can be calcu-lated as follows [1],

h ¼ jcpmAc

Pr�2=3 (6)

a

b

c

Fig. 5. Reproduction, crossover and mutation techniques in GA.

Initial group in binary

Optimization variablesin decimal

Individual evaluation:record the best one

Reproduction, crossover andmutation process in binary

Next generation group

Arrive the maxoptimization generation

Stop

Performance calculation ofPFHE

Obtain fitness values by overallentropy generation

Yes

No

Decode

Fig. 6. Flowchart to optimize the PFHE by using GA.

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e1436 1431

where, factor j is determined experimentally with constant walltemperature [1]. For a long passage, it represents the average valuealong exchanger. A correction factor for the lack of developmentwould be necessary for a short passage. If there are large transversalgradients, a separate correction for temperature-dependent prop-erties should be made. To be noted, if the data applied to theexchangers along with other types of wall-temperature variation,slightly difference in performance may be obtained [1].

In terms of the balance of the heat transferred through the wall,the following equation could be derived,

ðhFÞE�iTw;k

� iTf ;E�.

cp;E ¼ ðhFÞk�iTf ;k � iTw;k

�.cp;k;

k ¼ 1;2;3;4 ð7Þwhere, Tf,k represents the fluid temperature of the element k, iTf ;k isthe specific enthalpy value at this temperature.

By rewriting the Eq. (7), we obtain,

iTw;k¼

�hF=cp

�EiTf ;E þ

�hF=cp

�kiTf ;k�

hF=cp�Eþ

�hF=cp

�k

; k ¼ 1;2;3;4 (8)

The average enthalpy is then addressed as,

iTf ;E ¼ ðiex þ iinÞ=2 (9)

Substituting Eqs. (8) and (9) into Eq. (5),

iTf ;E ¼

P4k¼1

�1=m�

cp=hF�Eþ

�cp=hF

�k

iTf ;kiin

�þ 2

P4k¼1

"1=m�

cp=hF�Eþ

�cp=hF

�k

#þ 2

iin (10)

The pressure drop in the element can be expressed by thefollowing equation [1],

Table 2A snapshot of the fin database [1].

Nfin Type Pf (mm) b (mm) Dh (mm) d (mm) b (m2/m3) f

11 Plain-fin 1.28 6.35 1.875 0.152 1841.0 0.84931 Louvered-fin 2.29 6.35 3.08 0.152 1204.0 0.75634 Louvered-fin 1.68 9.53 2.68 0.254 1250.0 0.84043 Louvered-fin 2.08 12.3 3.41 0.102 1115.0 0.86248 Strip-fin 1.61 7.72 2.07 0.102 1726.0 0.85954 Wavy fin 2.22 10.49 3.23 0.152 1152.0 0.847

Nfin Fitted coefficient of function j {a3, a2, a1, a0} Fitted coefficient of function f {b3, b2, b1, b0}

11 {�0.1815, 2.1352, �8.4797, 8.8763} {�0.0114, 0.5477, �3.7571, 4.9417}31 {�0.1162, 1.2623, �4.8714, 4.3575} {0.0155, 0.0844, �1.5429, 2.0142}34 {0.0401, �0.3961, 0.9947, �2.2188} {�0.1885, 2.0677, �7.6888, 8.5151}43 {0.0059, �0.0570, �0.2133, �0.8117} {�0.1097, 1.3134, �5.4571, 6.4310}48 {�0.0838, 0.9327, �3.8720, 3.6231} {0.0672, �0.4017, �0.0258, 0.5138}54 {0.1279, �1.3298, 4.2183, �5.9568} {0.0545, �0.5459, 1.4188, �1.9031}

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e14361432

Dp ¼ fru2 4l

(11)

2 Dh

Taking 0 �C as the reference temperature, the specific enthalpyof dry air is [24],

i ¼ cpT (12)

The enthalpy in Eq. (10) of the humid air is formulated as [24],

Fig. 7. Typical plate-fin surface ty

i ¼ cpaT þ dvðifg þ cpvTÞ (13)

In addition, the enthalpy of the saturated humid air containingcondensed water is computed as follows [24],

i ¼ cpaT þ dsatðifg þ cpvTÞ þ ðd� dsatÞcplT (14)

The conservation equation of the water mass for the controlelement is,

pes included in the database.

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e1436 1433

mðdin � dexÞ ¼ �d� dsat;w

�hmrhdF (15)

10-1Colburn factor j a

In light of the analogy between heat andmass transports, if Re isnot severely small, the analogous relationship can be depicted as[25],

h=hm ¼ cprðLeÞ2=3 (16)

where, Le ¼ Sc/Pr. For all gas mixtures, the value of Le is around 1.For the air/water vapor system, ðLeÞ2=3 ¼ 0:95 [26]. Le is approxi-mately taken as 1 to simplify the calculation in this study. There-fore, the mass transfer coefficient hm can be reduced to thefollowing form,

hm ¼ h=rcp (17)

Eqs. (18) and (19) provide the formulas to determine the satu-ration humidity ratio [24].

psat;v ¼ f ðTÞ (18)

dsat ¼ 0:622psat;v

p� psat;v(19)

By using the enthalpy value obtained by Eq. (10), and the abovecorrelations among enthalpy, temperature and humidity, the cor-responding temperature and humidity ratio of every element canbe computed by stepwise iteration.

102 103 10410-4

10-3

10-2

Re

j

100

Friction factor f b

2.3. Entropy generation calculation

From the temperature and pressure distribution of PFHE derivedby DPM above, the entropy change of an element DS, is [18],

DS ¼ mcpln�TexTin

�mRln

�1� Dp

pin

(20)

The total entropy generation of heat exchanger is the sum of theentropy change across all elements,

Sgen ¼XMZ

kZ ¼1

XMY

kY ¼1

XMX

kX ¼1

�mcpln

�TexTin

�mRln

�1� Dp

pin

�kX ;kY ;kZ

(21)

As the amount of water vapor in the air is very tiny in this study,the phase change in the entropy balance was neglected for the sakeof simplifying the calculation.

102 103 10410-4

10-3

10-2

10-1

Re

f

( )

Fig. 8. Comparison between experimental data [1] and fitting curves.

3. Optimization

The PFHE optimization is a multi-objective optimizationproblem with the need of maximizing the heat transfer efficiency,minimizing the weight or the drag, and so on [27], which iscomplicated and time-consuming. To simplify the optimal process,entropy generation was applied as the objective function to predictthe performance of the PFHE in the ECS, which accounts compre-hensively for the confliction resulted from improving the heattransfer efficiency and reducing the pressure drop in the optimi-zation design.

Choosing an effective optimal method is also importantbecause optimal PFHE design is a complicated problem due to thefollowing reasons: (1) The configuration of the PFHE and fins iscomplex and many configuration parameters need to be opti-mized; (2) The type of fin is a discrete variable; (3) The objectivefunction is neither continuous, monotonic nor single-peaked. Inaddition, the gradient of the function is hard to calculate. GA has

strong capacity to solve such complicated problem. So it wasused in our study. A special customized fin database was built tofacilitate the selection of the fin type and fin geometry during theoptimization process.

3.1. Objective function and variables

The optimization variables for EGM included the followingitems:

a) The geometry dimension of exchanger L, W, H;b) The index number of the fin in the customized fin database

Nfin,h, Nfin,c;c) Fin geometric scale coefficient kh, kc.

The objective function for the present optimization problemwas,

4.4

4.6

4.8

5.0

5.2

5.4

5.6

5.8

))K.gk(/J(EHFP fo noitareneg yportnE

mean min

Table 3Initial conditions for the optimization process.

Tin (�C) pin (kPa) m (kg/s) DPmax (kPa)

Hot side 221 425.0 0.169 3.0Cold side 118 96.5 0.338 1.4

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e14361434

min SgenðXÞ ¼ f�L;W;H;Nfin;h;Nfin;c; kh; kc

�(22)

Subject to:8>><>>:

h > hminDPh < DPh;maxDPc < DPc;maxXi;min � Xi � Xi;max i ¼ 1;2;/;7

(23)

0 20 40 60 80 100

Evolutionary generations

Fig. 9. Optimization process.

3.2. Optimization method

GA is a probabilistic search algorithm that uses techniquesinspired by evolutionary biology mainly including natural selectionand survival of the fittest [28,29]. In GA, each solution to theproblem is encoded as a string of symbols named as a chromosome.And there are three basic generic operators used in the evolu-tionary process (Fig. 5) reproduction, crossover and mutation.These operators are applied to an initial generation of chromosomeorderly and repeatedly to find the final optimal solution. Theflowchart for optimizing PFHE by GA is demonstrated in Fig. 6.

3.3. Fin database

In traditional heat exchanger design, the selection of the fin typeis usually depended upon the experience of designers, whichbecomes infeasible for our current study. In order to ensure the GAto determine simultaneously the fin type and sizing, a customizedfin database composing of more than 60 different fins (the standardfin) was established (a snapshot of the database is listed in Table 2).Every standard fin was assigned a unique index number Nfin asso-ciated with fin properties, such as surface type, geometry charac-teristics, factor j and f, etc. The different types of the fin surface aredemonstrated in Fig. 7. During optimization process, once the indexnumber is selected, the detailed parameters of the correspondingfin are decided subsequently.

The original data sets of the fin database were from [1], whereheat transfer factor j and friction factor fwere fitted as the functionsof Re number to simple the coding of the computer program.

lgj ¼ a3ðlgReÞ3þa2ðlgReÞ2þa1lgReþ a0 (24)

lgf ¼ b3ðlgReÞ3þb2ðlgReÞ2þb1lgReþ b0 (25)

These fitting expressions hold high precision as indicated by thecomparison between the experimental data [1] and the fittingcurves in Fig. 8. Using this fin database, the fin type and itsdimensions can be selected automatically by the computerprogram.

Table 4Optimized results.

Nfin;h Nfin;c L (m) W (m) H (m) Pfh (m) Pfc (m) b

34 43 0.190 0.114 0.262 0.0017 0.0021 0

4. Results and discussion

An air heat exchanger utilized in the ECS (‘A’ in Fig. 1) waschosen as the example to illustrate the optimization methodology.The maximum allowable dimension for the heat exchanger was193 � 130 � 280 (mm3). The initial conditions for the optimizationprocess are given in Table 3.

4.1. Optimization

In the GA process, the size of population and the evolutionarygeneration were set as 200 and 100, respectively. The curves ofmean and minimum entropy generation versus evolutionarygeneration are sketched in Fig. 9. Both the mean and minimumentropy generation decrease sharply in the initial optimizationprocess. The objective function approaches a constant valuefollowing thirty generations. At the end of the optimization process,the minimum value of the entropy generation is 4.489 J/(kg K). Thetendency of the curves reflects the diversity of population in earliergeneration and the convergence later. The optimal fins for bothsides are strip-fin surfaces (Table 4).

4.2. Validation of the DPM

The author [30] has conducted heat transfer and thermody-namic analysis of one dimensional internal flow based on DPM, andgot results agreed well with the analytic solution of fully developedlaminar internal flow of air gives a good validation.

Furthermore, comparison of the results between DPM and IPMwere performed with a detailed case, which was the optimal resultof this paper and the parameters were given in Table 4. It can befound that the results obtained by DPM and IPM are in goodagreement (Table 5), which lends some credence to the proposedDPM in this study. On the other hand, some detectable discrep-ancies exist between these two models. This may be because the

h (m) bc (m) h DPh (kPa) DPc (kPa) Sgen (J/(kg K))

.0095 0.0123 0.70 0.545 1.322 4.489

210.3200.9

191.6

182.3

173.0

163.6

154.3

1

11

21

31

41

51

maerts dloc fo noitcerid eht ni xedni dirG

145.0

154.3

163.6

173.0

182.3

191.6

200.9

210.3

219.6

Temperature contour of 12 th

layer hot stream, °°C aTable 5

Comparison between DPM and IPM.

Tex;h (�C) Tex;c (�C) Q (kW) Sgen (J/(kg K))

DPM 148.3 155.2 13.26 4.489IPM 143.4 157.8 14.13 4.853Difference 4.9 2.6 0.75 0.364Relative difference 3.3% 1.7% 6.6% 8.1%

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e1436 1435

DPM instead of the IPM considers the detailed thermophysicalproperties. From this point of view, the results computed from theDPM are more precise than that of the IPM.

4.3. Application

The humidity in the air was ignored under the dry condition andthe heat transfer between metal walls and both sides fluid wereconsidered. The grid number of DPM for the whole PFHE was

210.3201.0

191.7

182.4

173.1

163.8

154.5

1 11 21 31 41 511

11

21

31

41

51

Grid index in the direction of hot stream

maerts dloc fo noitcerid eht ni xedni dirG

145.2

154.5

163.8

173.1

182.4

191.7

201.0

210.3

219.6

124.2

130.2

136.1

142.1148.1

154.1

160.0

1 11 21 31 41 511

11

21

31

41

51

Grid index in the direction of hot stream

maerts dloc fo noitcerid eht ni xedni dirG

118.2

124.2

130.2

136.1

142.1

148.1

154.1

160.0

166.0

a

b

Temperature contour of 12th

layer hot stream, °°C

Temperature contour of 13th

layer cold stream, °°C

Fig. 10. Temperature distributions in dry condition.

1 11 21 31 41 51

Grid index in the direction of hot stream

124.1

129.9

135.8

141.7147.6153.4159.3

1 11 21 31 41 511

11

21

31

41

51

Grid index in the direction of hot stream

maerts dloc fo noitcerid eht ni xedni dirG

118.2

124.1

129.9

135.8

141.7

147.6

153.4

159.3

165.2

Temperature contour of 13th

layer cold stream, °Cb

Fig. 11. Temperature distributions in wet condition.

4.125E-56.500E-5

8.875E-51.125E-4

1.363E-4

1 11 21 31 41 511

11

21

31

41

51

Grid index in the direction of hot stream

maerts dloc fo noitcerid eht ni xedni dirG

1.750E-5

4.125E-5

6.500E-5

8.875E-5

1.125E-4

1.363E-4

1.600E-4

1.838E-4

2.075E-4

Fig. 12. Humidity ratio contour of 13th cold stream, kg/kg, (under wet condition).

L. Zhang et al. / International Journal of Thermal Sciences 49 (2010) 1427e14361436

67,045 (53 � 55 � 23). The temperature distributions within thePFHE in steady state condition are exhibited in Fig. 10.

In the ECS mentioned above, the water gathered by waterseparator is sprayed into the ram air of heat exchanger ‘A’ in orderfor its vaporizing and cooling the hot fluid. As a result, it is neces-sary to consider the phase change of the water. The DPM hasadvantages over IPM to calculate the local properties in every point.The water is sprayed at the rate of 0.00237 kg/s. The temperatureand the humidity ratio distributions of the fluids are exhibited inFigs. 11 and 12, respectively.

It can be noted that the temperatures of the hot and cold fluidsin wet condition are 0.3 �C and 0.6 �C lower than that in drycondition, respectively. This difference is resulted from the evapo-rating of the sprayed water, which absorbs the heat, descends theoutlet temperature of hot fluid, and then decreases the temperatureof the air flowing towards the cabin. Hence, the overall perfor-mance of ECS is improved compared to the dry condition.

5. Conclusions

A universal 3D DPM was developed to evaluate and predict thesteady performance of PFHE. Based on the grids matching the finpassage and enthalpy equations of the control elements, theproposed model is able to unify the calculation for humid air inboth dry and wet conditions. Due to the new mesh generationmethod, the number of grid reduced drastically and the timerequired by the DPM to simulate the PFHE performance wasreduced markedly in comparison with the CFD-based methods.

Considering the capacity of the entropy generation to combinethe heat transfer and fluid friction, an optimization methodologywith the EGM as the objective function for PFHE based on DPMwasstructured. The special fin database built makes it achievable tooptimize the fin surface type and geometry simultaneously. Anoptimal PFHE was designed by means of the proposed method.DPMwas verified by comparing the DPM-derived temperature andhumid ratio distributions of the PFHE with the IPM-derived ones.As mentioned above, the DPM and optimization method proposedhere can be transferred to other heat exchangers.

By applying DPM, it was found that the hot fluid outlettemperature of the heat exchanger under wet condition is lowerthan that in dry condition. This finding implies that spraying waterinto ram air is a useful way to advance the performance of ECS.

The DPM established in this study provide a powerful tool tofind the exact location and the quantity of water condensing andfreezing in heat exchangers. This will assist us to improve the ECSdesign and enhance the reliability of aircraft when encounteringdangerous working conditions.

Acknowledgements

This work was supported by the Natural Science Foundation ofChina (50436010). The authors thank Feng Yang, Ph.D. for editingthe text.

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