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    O R IGINA L

    H. M. Soliman M. M. RahmanAnalytical solution of conjugate heat transfer and optimum

    configurations of flat-plate heat exchangers with circular flow channels

    Received: 25 November 2004 / Accepted: 6 September 2005 / Published online: 11 November 2005 Springer-Verlag 2005

    Abstract An analytical solution is developed for conju-gate heat transfer in a flat-plate heat exchanger withcircular embedded channels. The analysis was carriedout for fully-developed conditions in the circular tubeand uniform heat flux at the plate boundary. The resultsare applicable to cooling channels that are 50lm ormore in diameter with a large lengthdiameter ratio. Thethermal characteristics of the heat exchanger have beenexamined for a wide range of the relevant independentparameters and optimum designs for three different setsof constraints have been presented. It was found that theoverall thermal resistance increases with the depth of thetube from the heated surface, as well as the spacingbetween the tubes. For a given combination of tubesdepth and spacing, there is a certain tube diameter atwhich the thermal resistance attains a minimum value.

    List of symbolsAn, Bn, Cn, Dn Coefficients in the series solution, n = 0,

    1, 2, 3,..., NB0, C0 Coefficients in the series solutionCp Specific heat, J/kg Kf Friction coefficientH Depth of the tube from the top surface, mH* Dimensionless tube depthk Thermal conductivity, W/m KL Thickness of the plate, m

    Length of the plate, mM Number of tubes_mT Total mass flow rate, kg/s

    N Number of terms in the seriesP Pressure, PaPT Pumping power, Wq Input heat flux at top surface, W/m2

    qi Heat flux at the solid-fluid interface, W/m2

    R Dimensionless radial coordinater Radial coordinate, mro radius of the tube, mro* Dimensionless tube radius

    Re Reynolds numberT Temperature, KU Dimensionless axial velocity of the fluidu Axial velocity of the fluid, m/s

    um Mean axial velocity of the fluid, m/sW Half of the spacing between tubes cen-

    tre-to-centre, mW* Dimensionless spacing between the

    tubes centre-to-centreWT Overall width of the plate, mX, Y, Z Dimensionless (Cartesian) coordinatesx, y, z Cartesian coordinates, m

    Greek LettersC Overall thermal resistance, K/Wc Geometry-dependent part of thermal

    resistance, K/Wc* Dimensionless thermal resistanceh Dimensionless temperaturel Dynamic viscosity, N s/m2

    q Fluid density, kg/m3

    / Angular coordinate, rad

    Subscripts1 region 1

    H. M. Soliman (&)Department of Mechanical and Manufacturing Engineering,University of Manitoba, Winnipeg, ManitobaR3T 5V6, CanadaE-mail: [email protected].: +1-204-4749307Fax: +1-204-2757507

    M. M. RahmanDepartment of Mechanical Engineering,University of South Florida, Tampa,FL 33620, USA

    Heat Mass Transfer (2006) 42: 596607DOI 10.1007/s00231-005-0031-4

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    2 region 23 region 3ave averageb bulkf fluidi solidfluid interfacein heat-exchanger inletout heat-exchanger outlets solid

    1 Introduction

    A continuous improvement in the methods of heat re-moval is needed to satisfy fast growing circuit integra-tion in electronics equipment. The demand for fastercircuits and increased capacity has led to both increasesin the power dissipation of each circuit, and an increasein the number of circuits per unit volume. The net resulthas been increasing power densities at the chip, module,

    and system level of packaging. Since the expected lifeand reliability of a solid-state device depends verystrongly on its operating temperature, an efficient activecooling system is needed to maintain the required tem-perature limit. Although a number of different tech-niques [1] could potentially provide adequate cooling,microchannels or microtubes embedded in a solid sub-strate offer particular promise due to their simplicity andcomparatively lower thermal resistance.

    A compact, water-cooled, heat sink that was anintegral part of a silicon substrate was first demonstratedby Tuckerman and Pease [2], [3]. Philips [4] reportedmore experimental data using indium phosphide as the

    substrate material for the same kind of heat exchanger.A number of other studies have demonstrated theapplication of microchannel heat sinks.

    Wang and Peng [5] reported experimental data forsingle-phase forced convection of water or methanol inrectangular microchannels. The hydraulic diameter ofthese channels ranged from 311 to 747lm. A fully-developed turbulent convection regime was found tooccur at Reynolds number over 1,0001,500 dependingon liquid temperature, flow rate, and channel size. Intwo later studies, Peng and Peterson [6, 7] did furthermeasurements to characterize the effects of thermo-physical properties and geometrical parameters. Tso and

    Mahulikar [8] studied laminarturbulent transition forconvective heat transfer in microchannels and charac-terized the available test data using Brinkman number inaddition to Reynolds number. Bowers and Mudawar [9]showed that a very high heat transfer rate can beachieved by employing phase change in microchannels.However, the instability of the system when operatingclose to critical heat flux point makes this schemeimpractical for engineering applications.

    In addition to experimental measurements, a numberof theoretical investigations have been performed to

    understand the fundamentals of conjugate heat transferin microchannels of rectangular cross-section. Weisburget al. [10] studied thermal resistance of cooling channelsintegrated into silicon chips. A design procedure for theselection of channel dimensions in conformity withoperational constraints was found for a flat-plate heatexchanger consisting of rectangular channels fabricatedin a silicon wafer and capped with a Pyrex plate. Am-batipudi and Rahman [11] presented a three-dimen-sional numerical simulation model for conjugate heattransfer in rectangular microchannels. The Nusseltnumber variations with Reynolds number, channel as-pect ratio, and spacing between adjacent channels wereinvestigated. Fisher and Torrance [12] numericallystudied conjugate heat transfer in solids with coolingpassages of general, convex cross-section. The effect ofchannel boundary curvature on overall heat transfer wasquantified. Optimum channel shapes for given pressuredrop or pump work were determined.

    Fedorov and Viskanta [13] solved numerically theconventional Navier-Stokes and energy equations forlaminar flow in rectangular micro-channels with conju-

    gate heat transfer in the solid wall of the heat sink. Theydemonstrated that their theoretical results were capableof good agreement with the experimental results ofKawano et al. [14] for pressure drop and heat transfer in aheat exchanger with silicon microchannels of rectangularcross-section with a hydraulic diameter of about 87 lm.

    A number of studies have been reported on fluid flowand heat transfer in circular microtubes. Yu et al. [15]performed an experimental investigation to determineconvective heat transfer characteristics of microtubeswith diameters of 19, 52, and 102 lm at Reynoldsnumber greater than 2,500. It was found that Nusseltnumber compared well with large tube correlation at low

    values but the rate of increase of Nusselt number withReynolds number was found to be significantly largerthan predicted by the correlation. Adams et al. [16]experimentally investigated turbulent convective heattransfer in microtubes with diameters of 760 and1,090 lm. Water was used as the test fluid. Based on thetest data, a new correlation for heat transfer in mi-crotubes was proposed. Adams et al. [17] extended theresearch to non-circular tubes and found that standardturbulent single-phase Nusselt-type correlations can beapplied only when the hydraulic diameter is larger than1,020 lm.

    Mala and Li [18] reported experimental data for the

    pressure drop during laminar flow of water in microtu-bes with diameters ranging from 50 to 254 lm. Theyused fused silica and stainless steel tubes in their tests.For fused silica, the measured pressure gradient agreedwell with the Poiseuille flow theory for tube diameters of101lm or higher up to Re=2,000. With the stainlesssteel tubes, good agreement was obtained betweenmeasurements and the standard theory for tube diame-ters of 152 lm or higher up to Re=2,000.

    The focus of the present study is to analyze conjugateheat transfer in flat-plate heat exchangers with circular

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    flow channels. This kind of heat exchanger is used in awide variety of applications from electronic cooling tosnow-melting systems on pavements [19]. The vastmajority of previous investigations on flat-plate heatsinks have dealt with rectangular channels, while only alimited number of analyses were directed to circularchannels [20,21]. Therefore, a detailed theoretical studyis essential to understand the heat transfer characteris-tics in such a heat exchanger, to examine the effects ofgeometry and property parameters on its performance,and to explore methods for determining the optimumgeometry for different situations.

    Based on the information available in the literature,the conventional equations for the conservation ofmomentum and energy will be applicable provided thatthe channel diameter is of the order of 50lm orhigher. Moreover, if we restrict the analysis to laminarflow, the results will be applicable to tubes of any size,and can be potentially used for a wide range ofapplications. The assumption of fully developed con-ditions necessitates a large length-to-diameter ratio ofthe flow channels.

    2 Analysis

    We consider a flat-plate heat exchanger with circular,longitudinal channels for fluid flow, as shown in Fig. 1.A uniform heat flux is applied at the top surface of theplate and the bottom surface is assumed to be insulated.The overall width of the plate is WT, thickness isL, andlength (in the fluid flow direction) is . There are Mequally-spaced channels within the plate, each with aradiusr0.

    Because of symmetry, we can perform the analysis by

    considering only a cut-section of the heat exchanger, asshown in Fig.2. The depth of the tube from the heatedsurface (top boundary) is H and the width of the cut-section in W. The convective heat transfer is analyzedusing a cylindrical coordinate system attached to the axisof the tube. The conduction within the solid wall of theheat exchanger has been analyzed using a Cartesiancoordinate system attached to the bottom left corner ofthe cut-section, as shown in Fig.2, with z being the

    coordinate in the flow direction. Except for the uniformheating at the top boundary and convection at the tubewall, all other exterior surfaces are at adiabatic condi-tion because of symmetry. The cross-sectional geometryis defined by three dimensionless quantities: H* (= H/L), W* (= W/L), and r0

    * (= r0 /W).

    2.1 The fluid domain

    The following assumptions have been used to simplifythe problem for the analysis: (1) constant fluid and solidproperties, (2) laminar, fully-developed flow (hydrody-namically and thermally) in the tube, and (3) negligibleaxial conduction in the substrate and the fluid. Theapplicable momentum equation in dimensionless form is

    1

    R

    d

    dR R

    dU

    dR

    1

    2fRe; 1

    where, R=r/r0, U=u/um, f=r0(dP/dz)/(q um2 ), and Re

    = 2q umr0/l. In (1), the pressure gradient, (dP/dz), has

    been assumed to be constant because of the fully-developed condition. The solution of (1) is

    U fRe1 R2=8: 2

    Substituting this velocity profile in the conservation-of-mass equation,

    R10URdR 1=2; we get the well-known

    result,fRe=16. Therefore, the velocity profile becomes

    U 21 R2: 3

    The energy equation can be expressed in the follow-ing dimensionless form:

    1

    R

    @

    @R R

    @hf@R 1R2 @

    2hf

    @/2

    2

    p WU; 4where, h=(T Tb)/ (q L/kf). In formulating (4), thefully-developed condition Tf/z =dTb/dz=2q W/(p qCp r0

    2 um) was applied. A solution for (4) can be writtenin the following form:

    hf A0 W

    p

    R2 1

    R2

    4

    XNn1

    AnRn cosn/

    " #: 5

    Fig. 1 Schematic diagram

    of the heat exchanger

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    Solution (5) satisfies the symmetry conditions h f/ /=0 at/=0 and/=p, as well as the requirement thathfis finite at R=0. Imposing the condition that:

    hb 2

    p

    Zp0

    Z10

    RUhfdRd/ 0;

    we get: A0=7 W*/(24 p). Finally, we have

    hf W

    p

    R2

    R4

    4

    7

    24XNn1

    AnRn cosn/

    " #: 6

    2.2 The solid domain

    The energy equation throughout the solid is given by

    @2hs@X2

    @2hs@Y2

    0; 7

    where, X=x/L and Y=y/L. The solid domain was di-vided into three regions, as shown in Fig. 2, in order todevelop solutions that exactly satisfy the energy equa-tion and boundary conditions.

    Within region 1, the solution is given by

    hs;1 B0 kf

    ks

    Y

    XNn1

    Bn cos npX=W

    cosh np1 Y=W =cosh np1 Y1=W ;

    8

    where, Y1=1 H*. Solution (8) satisfies (7) and the

    following set of boundary conditions: @hs;1@X 0 at

    X=0 and X=W*, and@hs;1=@Y kf=ks atY=1.Within region 3, the solution can be written as

    hs;3 C0

    XN

    n1

    Cncos npX=W cosh npY=W =cosh npY2=W

    ;

    9

    where, Y2=1 H* 2 ro

    * W*. This solution satisfies (7)and the following boundary conditions:@hs;3=@X 0 atX=0 and X=W*, and@hs;3=@Y 0 atY=0.

    For region 2, there is only one fixed boundary con-dition;@hs;2=@X 0 at X=0. On the other threeboundaries, the solution must satisfy continuity oftemperature and heat flux. A possible solution for thetemperature distribution in this region can be written as

    hs;2 C0Y1 B0Y2 kf=ks Y1Y2

    Y1 Y2

    B0 C0 kf=ks Y1

    Y1 Y2

    Y

    XNn1

    Dn sin npY Y2Y1 Y2

    cosh npX

    Y1 Y2

    cosh

    npW

    Y1 Y2

    XNn1

    Bn cos npX=W sinh npY Y2=W

    =

    sinh npY1 Y2=W

    XNn1

    Cn cos npX=W sinh npY1 Y=W =

    sinh npY1 Y2=W :

    10

    Solution (10) satisfies (7) and the boundary condition atX=0 stated above. In addition, solution (10) guaranteesthe continuity of temperature at the interfaces betweenregion 2 and regions 1 and 3, i.e.,hs;2 hs;1 at Y=Y1andhs;2 hs;3at Y=Y2. The unknown coefficients in (8)(10), B0, C0, An, Bn, Cn, and Dn (a total of (4N+2)coefficients), can be determined by ensuring continuity ofthe heat flux at Y=Y1 and Y=Y2, as well as continuity of

    temperature and heat flux at the solidfluid interface.

    2.3 Evaluation of the coefficients

    The temperature distributions given by (8)(10) mustsatisfy these conditions:

    Continuity of heat flux at Y=Y1:

    @hs;1@Y

    @hs;2@Y

    11

    Fig. 2 Cross-section of the heat exchanger used for the analysis

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    deviate from this flat shape in the area surrounding theflow channel. It can be seen also that the magnitude oftemperature variation is higher in the part of the plate

    above the channel than it is in the part below thechannel. The direction of the temperature gradient in thesolid indicates that heat enters the fluid channel bothfrom the top and bottom sections. Thus, part of the heatinput at the top surface flows directly into the uppersurface of the channel, while the remainder by-passes thechannel and then returns to enter through the lowersurface of the channel. The isotherms in the fluid deviateslightly from the circular form and this deviation is morenoticeable near the outer radius of the channel.

    Another observation is that the values ofh are higherfor the plate in Fig. 3than the plate in Fig.4. Keepingthe plate thickness L fixed (so that the temperature level

    becomes proportional to h), the results in Figs. 3and4with the corresponding values ofH*,W*, andr0

    * suggestthat a plate with a large number of small-diameterchannels (Fig.4) will experience a lower temperaturelevel than a plate with a small number of large-diameterchannels.

    The temperature distribution at the wall-fluid inter-face (hi) is shown in Figs.57for various geometries andmaterial properties. In Fig. 5, the effect ofW* is inves-tigated keeping the other parameters (H*, r0

    *, and ks/kf)constant. Again, keeping L constant in order to main-

    tain the proportionality between (Ti Tb) andhi, we cansee that decreasing W* at the same r0

    * would amount todecreasing the channel diameter and the distance be-tween the channels. Since the heat source is located atthe top surface of the heat exchanger, it is expected thathiwould decrease gradually as one proceeds around the

    tube periphery from /=0 to /=180 (see Fig.5). Atboth ends of this domain, a zero slope condition ismaintained because of symmetry. The value ofh ican beseen to increase significantly as W* increases. This isconsistent with the results in Figs.3 and 4, where clo-sely-spaced small channels corresponded to a decreasedtemperature level in the heat exchanger. From Eq. (6),the average interfacial temperature can be evaluatedas:hi 11W

    =24p; and the results in Fig. 5appear tobe consistent with this value. Moreover, the change inhibetween /=0 and /=p is seen to increase as W*

    Fig. 4 Isotherms for H*=0.1, W*=0.2, r0*=0.5, and ks/kf=24.6

    Fig. 3 Isotherms for H*=0.3, W*=0.5, r0*=0.4, and ks/kf=24.6

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    increases. That is because the tube occupies a larger partof the block as W* increases.

    Figure6shows the variation ofhiwith r0*, keeping the

    other parameters constant. These results explore theeffect of changing the channel diameter while main-taining the same distance between the channels centre-to-centre. Fig.6 shows a larger temperature variationwith angle as the tube radius is increased. However, theaverage interfacial temperature does not change with r0

    *.

    This graph shows that, for a fixed number of tubes, asmaller tube radius is expected to result in more iso-thermal condition at the solid-fluid interface.

    Figure7 presents the variation of hi with ks/kf.Maintaining a constant kfin order to maintain propor-tionality between (Ti Tb) and hi, the results in Fig. 7show that the interface temperature becomes more uni-form as the solid thermal conductivity is increased dueto the smaller thermal resistance within the solid. Forthe silicon-water system intended for high speed com-puter chips, the heat exchanger provides almost constant

    temperature along the tube periphery. When the thermalconductivity ratio is reduced, the interface temperatureshows significant variation along the tube periphery. A

    larger temperature gradient is needed for a smallerthermal conductivity in order to achieve the giventhermal load of the heat exchanger.

    The effect ofH* onhiwas also explored and found tobe much less significant than the effects of otherparameters.

    3.2 Heat flux at the solidfluid interface

    The local heat flux at solidfluid interface (qi) can beexpressed in the following non-dimensional form:

    qi

    qave 1 X1n1

    nAn cos n/; 16

    where, qave (=q/(p r0*)) is the average heat flux at the

    solid-fluid interface. Results for qi are shown in Figs. 8to10. The effect ofW* is presented in Fig. 8. The value

    Fig. 6 Variation of hi with r0* for H*=0.1, W*=0.1, and ks/

    kf=24.6

    Fig. 7 Variation ofhiwith ks/kffor H*=0.1, W*=0.5, and r0

    * = 0.5

    Fig. 8 Effect ofW* on q ifor H*=0.1, r0

    *=0.4, and ks/kf=24.6

    Fig. 5 Variation of hi with W* for H*=0.1, r0

    *=0.4, and ks/kf=24.6

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    ofqichanges from a maximum at/=0 to a minimum at/=p. The difference between the maximum and theminimum values is not very large and W* does not ap-pear to have much influence on the qi-profile. Fig.9

    demonstrates the effect of tube radius on q i at the sametube spacing centre-to-centre. The circumferential vari-ation ofqidecreases as the tube radius decreases. For thecase of r0

    * =0.9, corresponding to large-diameter tubeswith a small separating distance, the angular profile of qideviates from the smooth shape seen with the lower r0

    *

    and the magnitude of difference between the heat flux at/=0 and /=p is higher. The effect ofks/kfis shown inFig. 10. As expected, a more uniform distribution ofheat flux is seen for a larger solid thermal conductivitybecause of larger temperature uniformity within thesolid.

    3.3 Overall thermal resistance

    Considering a length of the heat exchanger, the overallthermal resistance C is normally defined as

    C Ts;1y L;z Tb;in

    qWT 17

    where, the over-bar indicates average over x.

    Noting thatTs;1y L;z Tb;out Ts;1y L Tb constant;

    we can express C as

    C Ts,l y L Tb

    qWT|ffl{zffl}geometry dependent

    1

    _mTCp;f|fflfflffl{zfflfflffl}flow dependent

    : 18

    Let us focus on the geometry dependent part and define

    c Ts;1y L Tb

    qWT

    L

    WTkf

    hs;1Y 1: 19

    Finally, we can do our computations and geometryoptimizations using a dimensionless thermal resistancec*, defined by

    c WTkf

    L c hs;1Y 1: 20

    Using (8), we can express c* as

    c B0 kf=ks: 21

    In the special case ofkf/ks=0, the plate temperaturebecomes uniform andB0 assumes the value B0=11W

    */(24p). Therefore, based on (21), the minimum value ofthermal resistance is given by cmin

    * =0.1459 W*.

    Values of c *

    for the three values of ks/kf and wideranges of the geometrical parameters are shown inFigs. 1116. In Fig. 11, it can be seen that c* increasesmonotonically with H* for all values of W* for ks/kf=2.3. As the distance between the heated surface andthe top edge of the tube increases, the thermal resistanceis expected to increase because of the larger conductionpath for heat transfer. Again, a geometry with a largenumber of small-diameter tubes (low W*) has less ther-mal resistance than a geometry with a small number oflarge-diameter tubes (high W*). This trend is consistentFig. 10 Effect ofks/kfon q ifor H

    *=0.1, W*=0.5, and r0* = 0.5

    Fig. 9 Effect ofr0* on q ifor H

    *=0.1, W*=0.1, and ks/kf=24.6 Fig. 11 Dependence ofc* onH* andW* forks/kf=2.3 andr0

    *=0.3

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    with the earlier results of the temperaturedistribution.Fig. 12 shows the variation of c * with ro

    * for different

    values ofW*

    atks/kf=2.3. For a given spacing betweenthe tubes (centre-to-centre), the magnitude of c*de-creases with the tube radius, attains a minimum, andthen increases with further increase of tube radius. Theminimum happens around r0

    *=0.3. As the tube radiusincreases, the solid region of the heat exchanger is re-placed by the fluid region. Therefore, there is a decreaseof conduction path between the source and the sink.However, there is less room for the redistribution oftemperature within the solid. The net result of these twoeffects is the variation seen in Fig. 12. As expected, themagnitude of thermal resistance increases with tubespacing.

    The values ofc*

    for ks/kf=24.6 are plotted in Figs. 13and 14. The trends are very similar to those seen inFigs.11and 12. It may be noticed that with an increasein ks/kf, the slopes of the c

    * vs. H* curves decrease and

    the variation of c* with r0* becomes less significant.

    Figures 15 and 16, corresponding to ks/kf=243, show

    that c* is much less dependent on H* and r0*, while thedependence on W* is still very significant. The effect ofW* on c*seen in Figs.11to 16 is completely consistentwith the effect ofW* onh s andh ipresented earlier.

    3.4 Optimum configurations

    Different scenarios are used in design depending on theobjective function that need to be optimized and theconstraints that are relevant to the particular applica-tion. In this section, we will demonstrate how the presentanalysis can be used in three different design scenarios.

    In the first scenario, let us consider a given plate witha known geometry (WT L ) and known solid andfluid properties. The objective is to determine r0 forminimum c * under the constraint of a fixed number of

    Fig. 12 Dependence ofc* onr0* andW* forks/kf=2.3 andH

    *=0.1

    Fig. 13 Dependence of c* on H* and W* for ks/kf=24.6 andr0*=0.3

    Fig. 14 Dependence of c* on r0* and W* for ks/kf=24.6 and

    H*=0.1

    Fig. 15 Dependence ofc* on H* and W* for ks/kf=243 and r0*=0.3

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    channels M. The objective in this case can be achievedby direct application of the results shown in Figs. 1116.Consider for example a concrete slab with WT=1 m,

    = 5 m, and L=20 cm with water flowing in M=10channels. In this case, W=WT/(2M)=5 cm andW*=W/L=0.25. For minimum thermal resistance, as-sume H*=0.1; lower values may cause structural prob-lems. We now search for r0

    * that will produce minimumc* for ks/kf=2.3 and the above values ofH

    * and W* .Direct application of the present analysis producesr0*=0.28, r0=1.4 cm, and c

    *=0.1184.In the second scenario, let us consider a given plate

    with known geometry (WT L ) and known solid andfluid properties. The objective is to determine Mand r0for minimum pumping power PTunder the constraint offixed overall thermal resistanceC. The pumping power is

    given byPT _mTDP=qf; 22

    where, DP is the pressure drop across the heat ex-changer, which (under the present conditions of laminarfully-developed flow) can be expressed as

    DP 8lfpr40qfM

    _mT: 23

    Knowing thatC c _mTCp;f 1

    ; and using definition(20) ofc*, we can write

    _mTCp;f

    C Lc

    WTkf 1

    24

    Substituting (23) and (24) into (22) and rearranging, weget

    bPT 16

    p WT= r0 4

    W 3 1 Lc

    WTkfC

    h i2; 25where,

    b qfCP;fC 2

    L3=lf:

    The procedure is illustrated through the followingexample: Consider a stainless-steel plate (WT=10 cm,=50 cm, and L=2 cm) receiving a uniform heat flux at

    the top surface and cooled by water flowing in channelsinside the block. It is desired to achieve an overallthermal resistance of C=0.05 K/W. Determine thechannels layout (Mand ro) that will result in the mini-mum pumping power PT.

    Taking water properties at 300 K, we get ks/kf=24.6.The value ofH* was assumed to be 0.1. A search pro-cedure is required to find the solution. Select a value ofM, from which W and W* can be determined. For afixed M, use different values ofr0

    * and determine c * foreach combination of H*, W*, r0

    *, and ks/kf from thepresent analysis. The value of b PT can then be easilydetermined from (25). This search procedure was con-

    ducted for different values of M and a sample of theresults is shown in Fig. 17. The minimumPTwas foundto correspond to M=14 and r0

    *=0.88.

    Fig. 16 Dependence ofc* on r0* and W* for ks/kf=243 and H

    *=0.1 Fig. 17 Example on the optimization of pumping power

    Fig. 18 Example on the optimization of thermal resistance

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    In the third optimization scenario, we consider aplate heat exchanger with a known geometry (WT L ) and known solid and fluid properties. The objective isto determine M and r0 for minimum C under the con-straint of fixed PT. (25) can be reformulated to read

    C Lc

    WTkf

    4

    qfr20CP;f

    lf2pMPT

    1=2: 26

    For example, consider a silicon wafer (WT=1 cm,=5 cm, andL=2 mm) receiving a uniform heat flux atthe top surface and cooled by water flowing in circularchannels. It is desired to limit the frictional pumpingpower to PT=2 10

    4 W. Determine the channelslayout (Mandr0) that will result in a minimum C .

    The search procedure is demonstrated in Fig. 18. Foreach value of r0

    *, there is a value of Mthat produces aminimum C. The value ofMfor minimumC can be seento increase as r0

    * increases. The limit of diminishing re-turns is probably reached at r0

    *=0.9. A reasonablesolution corresponds toM=9 andr0

    *=0.9. All results inFig. 18correspond to H*=0.1 and ks/kf=243.

    4 Conclusions

    Flat plate heat exchangers with circular longitudinalchannels have been analyzed for the conditions ofuniform heat flux at the top surface of the plate andfully-developed flow and heat transfer in the tubes. Ananalytical series solution has been developed for thetemperature distributions in the fluid and solid regionsand the series solution is shown to converge to highaccuracy with a small number (N=15) terms. Calcu-

    lations have been carried out for a wide range of thegeometry parameters (H*, W*, and r0

    *) and the solidfluid thermal conductivity ratio ks/kf, thus coveringtypical applications of this heat exchanger for thermalmanagement of electronics, process equipment, andpavement slabs.

    It was found that the temperature level in the solidand at the solidfluid interface increases with an increasein tube spacing (centre-to-centre) for a given depth oftubes from the heated surface. For a given tube spacing,the interface temperature shows wider variation alongthe tube periphery when the tube radius is increased;however, the average value of temperature does not

    change with tube radius. When the thermal conductivityratio is increased, the interface temperature becomesmore uniform because of the decrease in thermal resis-tance within the solid. The peripheral heat flux distri-bution at the solidfluid interface became more uniformas the tube spacing (centre-to-centre) and/or the thermalconductivity ratio were increased.

    The overall thermal resistance of the heat exchangerincreased with increases in the tubes depth from theheated surface and/or the spacing between tubes (centre-to-centre). However, for given depth and spacing, the

    thermal resistance achieved a minimum value at anintermediate tube diameter indicating the importance oftube size in the design of this heat exchanger. A largerthermal conductivity ratio resulted in lower thermalresistance within the solid and heat transfer in that sit-uation is heavily controlled by the convective resistanceat the solid-fluid interface.

    The objective function for design optimization of thisheat exchanger has been derived for three different sce-narios including minimum pumping power and mini-mum thermal resistance. The optimization procedurehas been illustrated by an example in each scenario.

    Acknowledgements The financial assistance provided by the Natu-ral Sciences and Engineering Research Council of Canada isgratefully acknowledged.

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