Heat Mass Transfer (2006) 42: 596–607 DOI 10.1007/s00231-005-0031-4
O R I GI N A L
H. M. Soliman Æ M. M. Rahman
Analytical 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 conjugate heat transfer in a flat-plate heat exchanger with circular embedded channels. The analysis was carried out for fully-developed conditions in the circular tube and uniform heat flux at the plate boundary. The results are applicable to cooling channels that are 50 lm or more in diameter with a large length–diameter ratio. The thermal characteristics of the heat exchanger have been examined for a wide range of the relevant independent parameters and optimum designs for three different sets of constraints have been presented. It was found that the overall thermal resistance increases with the depth of the tube from the heated surface, as well as the spacing between the tubes. For a given combination of tubes’ depth and spacing, there is a certain tube diameter at which the thermal resistance attains a minimum value. List of symbols An, Bn, Cn, Dn Coefficients in the series solution, n = 0, 1, 2, 3,..., N B0, C0 Coefficients in the series solution Cp Specific heat, J/kg K f Friction coefficient H Depth of the tube from the top surface, m H* Dimensionless tube depth k Thermal conductivity, W/m K L Thickness of the plate, m
H. M. Soliman (&) Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Manitoba R3T 5V6, Canada E-mail:
[email protected] Tel.: +1-204-4749307 Fax: +1-204-2757507 M. M. Rahman Department of Mechanical Engineering, University of South Florida, Tampa, FL 33620, USA
‘ M m_ T N P PT q qi R r ro r*o Re T U u um W W* WT X, Y, Z x, y, z
Length of the plate, m Number of tubes Total mass flow rate, kg/s Number of terms in the series Pressure, Pa Pumping power, W Input heat flux at top surface, W/m2 Heat flux at the solid-fluid interface, W/ m2 Dimensionless radial coordinate Radial coordinate, m radius of the tube, m Dimensionless tube radius Reynolds number Temperature, K Dimensionless axial velocity of the fluid Axial velocity of the fluid, m/s Mean axial velocity of the fluid, m/s Half of the spacing between tubes centre-to-centre, m Dimensionless spacing between the tubes centre-to-centre Overall width of the plate, m Dimensionless (Cartesian) coordinates Cartesian coordinates, m
Greek Letters C Overall thermal resistance, K/W c Geometry-dependent part of thermal resistance, K/W c* Dimensionless thermal resistance h Dimensionless temperature l Dynamic viscosity, N s/m2 q Fluid density, kg/m3 / Angular coordinate, rad
Subscripts 1
region 1
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2 3 ave b f i in out s
region 2 region 3 average bulk fluid solid–fluid interface heat-exchanger inlet heat-exchanger outlet solid
1 Introduction A continuous improvement in the methods of heat removal is needed to satisfy fast growing circuit integration in electronics equipment. The demand for faster circuits and increased capacity has led to both increases in the power dissipation of each circuit, and an increase in the number of circuits per unit volume. The net result has been increasing power densities at the chip, module, and system level of packaging. Since the expected life and reliability of a solid-state device depends very strongly on its operating temperature, an efficient active cooling system is needed to maintain the required temperature limit. Although a number of different techniques [1] could potentially provide adequate cooling, microchannels or microtubes embedded in a solid substrate offer particular promise due to their simplicity and comparatively lower thermal resistance. A compact, water-cooled, heat sink that was an integral part of a silicon substrate was first demonstrated by Tuckerman and Pease [2], [3]. Philips [4] reported more experimental data using indium phosphide as the substrate material for the same kind of heat exchanger. A number of other studies have demonstrated the application of microchannel heat sinks. Wang and Peng [5] reported experimental data for single-phase forced convection of water or methanol in rectangular microchannels. The hydraulic diameter of these channels ranged from 311 to 747 lm. A fullydeveloped turbulent convection regime was found to occur at Reynolds number over 1,000–1,500 depending on liquid temperature, flow rate, and channel size. In two later studies, Peng and Peterson [6, 7] did further measurements to characterize the effects of thermophysical properties and geometrical parameters. Tso and Mahulikar [8] studied laminar–turbulent transition for convective heat transfer in microchannels and characterized the available test data using Brinkman number in addition to Reynolds number. Bowers and Mudawar [9] showed that a very high heat transfer rate can be achieved by employing phase change in microchannels. However, the instability of the system when operating close to critical heat flux point makes this scheme impractical for engineering applications. In addition to experimental measurements, a number of theoretical investigations have been performed to
understand the fundamentals of conjugate heat transfer in microchannels of rectangular cross-section. Weisburg et al. [10] studied thermal resistance of cooling channels integrated into silicon chips. A design procedure for the selection of channel dimensions in conformity with operational constraints was found for a flat-plate heat exchanger consisting of rectangular channels fabricated in a silicon wafer and capped with a Pyrex plate. Ambatipudi and Rahman [11] presented a three-dimensional numerical simulation model for conjugate heat transfer in rectangular microchannels. The Nusselt number variations with Reynolds number, channel aspect ratio, and spacing between adjacent channels were investigated. Fisher and Torrance [12] numerically studied conjugate heat transfer in solids with cooling passages of general, convex cross-section. The effect of channel boundary curvature on overall heat transfer was quantified. Optimum channel shapes for given pressure drop or pump work were determined. Fedorov and Viskanta [13] solved numerically the conventional Navier-Stokes and energy equations for laminar flow in rectangular micro-channels with conjugate heat transfer in the solid wall of the heat sink. They demonstrated that their theoretical results were capable of good agreement with the experimental results of Kawano et al. [14] for pressure drop and heat transfer in a heat exchanger with silicon microchannels of rectangular cross-section with a hydraulic diameter of about 87 lm. A number of studies have been reported on fluid flow and heat transfer in circular microtubes. Yu et al. [15] performed an experimental investigation to determine convective heat transfer characteristics of microtubes with diameters of 19, 52, and 102 lm at Reynolds number greater than 2,500. It was found that Nusselt number compared well with large tube correlation at low values but the rate of increase of Nusselt number with Reynolds number was found to be significantly larger than predicted by the correlation. Adams et al. [16] experimentally investigated turbulent convective heat transfer in microtubes with diameters of 760 and 1,090 lm. Water was used as the test fluid. Based on the test data, a new correlation for heat transfer in microtubes was proposed. Adams et al. [17] extended the research to non-circular tubes and found that standard turbulent single-phase Nusselt-type correlations can be applied only when the hydraulic diameter is larger than 1,020 lm. Mala and Li [18] reported experimental data for the pressure drop during laminar flow of water in microtubes with diameters ranging from 50 to 254 lm. They used fused silica and stainless steel tubes in their tests. For fused silica, the measured pressure gradient agreed well with the Poiseuille flow theory for tube diameters of 101 lm or higher up to Re=2,000. With the stainless steel tubes, good agreement was obtained between measurements and the standard theory for tube diameters of 152 lm or higher up to Re=2,000. The focus of the present study is to analyze conjugate heat transfer in flat-plate heat exchangers with circular
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flow channels. This kind of heat exchanger is used in a wide variety of applications from electronic cooling to snow-melting systems on pavements [19]. The vast majority of previous investigations on flat-plate heat sinks have dealt with rectangular channels, while only a limited number of analyses were directed to circular channels [20, 21]. Therefore, a detailed theoretical study is essential to understand the heat transfer characteristics in such a heat exchanger, to examine the effects of geometry and property parameters on its performance, and to explore methods for determining the optimum geometry for different situations. Based on the information available in the literature, the conventional equations for the conservation of momentum and energy will be applicable provided that the channel diameter is of the order of 50 lm or higher. Moreover, if we restrict the analysis to laminar flow, the results will be applicable to tubes of any size, and can be potentially used for a wide range of applications. The assumption of fully developed conditions necessitates a large length-to-diameter ratio of the flow channels.
coordinate in the flow direction. Except for the uniform heating at the top boundary and convection at the tube wall, all other exterior surfaces are at adiabatic condition because of symmetry. The cross-sectional geometry is defined by three dimensionless quantities: H* (= H/ L), W* (= W/L), and r*0 (= r0 /W). 2.1 The fluid domain The following assumptions have been used to simplify the problem for the analysis: (1) constant fluid and solid properties, (2) laminar, fully-developed flow (hydrodynamically and thermally) in the tube, and (3) negligible axial conduction in the substrate and the fluid. The applicable momentum equation in dimensionless form is 1 d dU 1 R ¼ f Re; ð1Þ R dR dR 2 where, R=r/r0, U=u/um, f=r0(dP/dz)/(q u2m), and Re = 2q um r0/l. In (1), the pressure gradient, (dP/dz), has been assumed to be constant because of the fullydeveloped condition. The solution of (1) is
2 Analysis
U ¼ f Reð1 R2 Þ=8:
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 the plate and the bottom surface is assumed to be insulated. The overall width of the plate is WT, thickness is L, and length (in the fluid flow direction) is ‘. There are M equally-spaced channels within the plate, each with a radius r0. Because of symmetry, we can perform the analysis by considering only a cut-section of the heat exchanger, as shown in Fig. 2. The depth of the tube from the heated surface (top boundary) is H and the width of the cutsection in W. The convective heat transfer is analyzed using a cylindrical coordinate system attached to the axis of the tube. The conduction within the solid wall of the heat exchanger has been analyzed using a Cartesian coordinate system attached to the bottom left corner of the cut-section, as shown in Fig. 2, with z being the
Substituting thisR velocity profile in the conservation-of1 mass equation, 0 URdR ¼ 1=2; we get the well-known result, fRe=16. Therefore, the velocity profile becomes
Fig. 1 Schematic diagram of the heat exchanger
U ¼ 2ð1 R2 Þ:
ð2Þ
ð3Þ
The energy equation can be expressed in the following dimensionless form: 1 @ @hf 1 @ 2 hf 2 R W U ; þ 2 ¼ ð4Þ 2 R @R R @/ p @R where, h=(T Tb)/ (q L/kf). In formulating (4), the fully-developed condition ¶Tf/¶z =dTb/dz=2 q W/(p q Cp r20 um) was applied. A solution for (4) can be written in the following form: # " X N W R2 2 n hf ¼ A0 þ R 1 þ An R cosðn/Þ : ð5Þ p 4 n¼1
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where, Y1=1 H*. Solution (8) satisfies (7) and the following set of boundary conditions: @hs;1 @X ¼ 0 at X=0 and X=W*, and@hs;1 =@Y ¼ kf =ks at Y=1. Within region 3, the solution can be written as hs;3 ¼C0 N X þ Cn cos ðnpX =W Þcosh ðnpY =W Þ=cosh ðnpY2 =W Þ; n¼1
ð9Þ *
Fig. 2 Cross-section of the heat exchanger used for the analysis
Solution (5) satisfies the symmetry conditions ¶h f/ ¶/ =0 at /=0 and /=p, as well as the requirement that hf is finite at R=0. Imposing the condition that: 2 hb ¼ p
Zp Z 1 0
RU hf dR d/ ¼ 0;
r*o
*
where, Y2=1 H 2 W . This solution satisfies (7) and the following boundary conditions:@hs;3 =@X ¼ 0 at X=0 and X=W*, and@hs;3 =@Y ¼ 0 at Y=0. For region 2, there is only one fixed boundary condition;@hs;2 =@X ¼ 0 at X=0. On the other three boundaries, the solution must satisfy continuity of temperature and heat flux. A possible solution for the temperature distribution in this region can be written as C0 Y1 B0 Y2 ðkf =ks ÞY1 Y2 hs;2 ¼ Y1 Y2 B0 C0 þ ðkf =ks ÞY1 þ Y Y1 Y2 N X Y Y2 Dn sin np þ Y1 Y2 n¼1 npX npW cosh cosh Y1 Y2 Y1 Y2 N X Bn cos ðnpX =W Þsinh ðnpðY Y2 Þ=W Þ= þ n¼1
0
sinh ðnpðY1 Y2 Þ=W Þ *
we get: A0=7 W /(24 p). Finally, we have # " N W R4 7 X 2 n R þ hf ¼ An R cosðn/Þ : p 4 24 n¼1
þ ð6Þ
N X
Cn cos ðnpX =W Þsinh ðnpðY1 Y Þ=W Þ=
n¼1
sinh ðnpðY1 Y2 Þ=W Þ: ð10Þ
2.2 The solid domain The energy equation throughout the solid is given by @ 2 hs @ 2 hs þ ¼ 0; @X 2 @Y 2
ð7Þ
where, X=x/L and Y=y/L. The solid domain was divided into three regions, as shown in Fig. 2, in order to develop solutions that exactly satisfy the energy equation and boundary conditions. Within region 1, the solution is given by N X kf hs;1 ¼B0 þ Bn cos ðnpX =W Þ Yþ ks n¼1 cosh ðnpð1 Y Þ=W Þ=cosh ðnpð1 Y1 Þ=W Þ; ð8Þ
Solution (10) satisfies (7) and the boundary condition at X=0 stated above. In addition, solution (10) guarantees the continuity of temperature at the interfaces between region 2 and regions 1 and 3, i.e.,hs;2 ¼ hs;1 at Y=Y1 andhs;2 ¼ hs;3 at 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 of the heat flux at Y=Y1 and Y=Y2, as well as continuity of temperature and heat flux at the solid–fluid interface. 2.3 Evaluation of the coefficients The temperature distributions given by (8)–(10) must satisfy these conditions: – Continuity of heat flux at Y=Y1: @hs;1 @hs;2 ¼ @Y @Y
ð11Þ
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– Continuity of heat flux at Y=Y2: @hs;3 @hs;2 ¼ @Y @Y
ð12Þ
– Continuity of temperature at the solid–fluid interface: hs;2 ¼ hf
ð13Þ
where, the solid–fluid interface (tube surface) is defined by X ¼ W 1 ro sin / ; Y ¼ Y1 ro W ð1 cos /Þ ð14Þ – Continuity of heat flux at the solid-fluid interface, which can be expressed as kf @hf @hs;2 @hs;2 sin / cos / ¼ 0: ð15Þ þr W ks @R R¼1 0 @X @Y
Each of (11)–(13), and (15) was applied at a number of equally-spaced points, NP, along the respective interface. This resulted in (4NP) linear algebraic equations for the (4N+2) unknown coefficients with 4NP > (4N+2). This linear system of equations was solved by the well-known least-squares method. Once the unknown coefficients are determined, values of h anywhere in the solution domain can be easily determined. The input parameters necessary for the solution are: H*, W*, r*0, and kf/ks. For the special case of kf/ks=0, (11)–(15) are satisfied by the expected condition of uniform temperature throughout the solid with An=Bn=Cn=Dn=0 and B0=C0=11 W*/(24 p). The effects of N and NP on the accuracy of the results were investigated for various combinations of the independent parameters. The assessment of the accuracy was based on the values of B0 and C0. Keeping in mind that the average temperature at Y=0 ishs;3 ¼ C0 and the
average temperature at Y=1 ishs;1 ¼ B0 þ kf =ks ; we can see that these two coefficients are of significant importance for the accuracy of the temperature field. In addition, the value of B0 is the only coefficient used in calculating the thermal resistance of the heat exchanger, as will be seen later. Table 1 summarizes the effects of N and NP on B0 and C0 for various geometries (H*, W*, r*0) and material properties (ks/kf). These results demonstrate that the series solution converges very fast and reasonably accurate solutions can be obtained with N=10. In order to insure better than 0.5% accuracy in the results, it was decided to perform all computations with N=20 and NP=400.
3 Results and discussion Results were generated for three different values of ks/kf, namely ks/kf=243 (representing silicon-water), ks/ kf=24.6 (representing stainless steel-water), and ks/ kf=2.3 (representing concrete-water). These values of ks/kf and the materials they represent cover typical applications of this heat exchanger for electronic cooling, process equipment heat transfer, as well as pavement slab thermal control. In addition, these values provide two orders-of-magnitude variation in the thermal-conductivity ratio. 3.1 Temperature distribution Plots of the isotherms within the computational domain are shown in Figs. 3 and 4 for two heat exchangers with different values of H*, W*, and r*0; both cases correspond to ks/kf=24.6. The value of h is indicated on some isotherms and the magnitude of D h between successive isotherms is uniform in both plots. These results show that the temperature variation in the X-direction is small near the upper heated surface; however, the isotherms
Table 1 Effects of N and NP on the values of B0 and C0 for various geometries and material properties ks/kf=243
ks/kf=24.6
B0
C0
H*=0.1, W*=0.5, and r*0=0.5 N=5, NP=100 N=10, NP=200 N=20, NP=400
0.070465 0.070500 0.070500
0.072173 0.072144 0.072144
H* = 0.3, W* = 0.1, and ro*= 0.8 N=5, NP=100 N=10, NP=200 N=20, NP=400
0.012198 0.012371 0.012378
H*=0.5, W*=1.0, and r0*=0.2 N=5, NP=100 N=10, NP=200 N=20, NP=400
0.14569 0.14577 0.14577
B0
ks/kf=2.3 C0
B0
0.048045 0.048319 0.048319
0.065704 0.065487 0.065488
0.22229 0.22187 0.22187
0.024534 0.024459 0.024459
0.014217 0.014057 0.014052
0.0086828 0.0078937 0.0078837
0.010568 0.0098799 0.0098714
0.25719 0.25604 0.25603
0.0082010 0.0082370 0.0082320
0.14661 0.14661 0.14661
0.14376 0.14445 0.14447
0.15309 0.15310 0.15310
0.11214 0.11502 0.11505
C0
0.23070 0.23036 0.23037
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Fig. 3 Isotherms for H*=0.3, W*=0.5, r*0=0.4, and ks/kf=24.6
deviate from this flat shape in the area surrounding the flow channel. It can be seen also that the magnitude of temperature variation is higher in the part of the plate above the channel than it is in the part below the channel. The direction of the temperature gradient in the solid indicates that heat enters the fluid channel both from the top and bottom sections. Thus, part of the heat input at the top surface flows directly into the upper surface of the channel, while the remainder by-passes the channel and then returns to enter through the lower surface of the channel. The isotherms in the fluid deviate slightly from the circular form and this deviation is more noticeable near the outer radius of the channel. Another observation is that the values of h are higher for the plate in Fig. 3 than the plate in Fig. 4. Keeping the plate thickness L fixed (so that the temperature level becomes proportional to h), the results in Figs. 3 and 4 with the corresponding values of H*, W*, and r*0 suggest that a plate with a large number of small-diameter channels (Fig. 4) will experience a lower temperature level than a plate with a small number of large-diameter channels. The temperature distribution at the wall-fluid interface (hi) is shown in Figs. 5–7 for various geometries and material properties. In Fig. 5, the effect of W* is investigated keeping the other parameters (H*, r*0, and ks/kf) constant. Again, keeping L constant in order to main-
Fig. 4 Isotherms for H*=0.1, W*=0.2, r*0=0.5, and ks/kf=24.6
tain the proportionality between (Ti Tb) and hi, we can see that decreasing W* at the same r*0 would amount to decreasing the channel diameter and the distance between the channels. Since the heat source is located at the top surface of the heat exchanger, it is expected that hi would decrease gradually as one proceeds around the tube periphery from /=0 to /=180 (see Fig. 5). At both ends of this domain, a zero slope condition is maintained because of symmetry. The value of h i can be seen to increase significantly as W* increases. This is consistent with the results in Figs. 3 and 4, where closely-spaced small channels corresponded to a decreased temperature level in the heat exchanger. From Eq. (6), the average interfacial temperature can be evaluated as:hi ¼ 11W =ð24pÞ; and the results in Fig. 5 appear to be consistent with this value. Moreover, the change in hi between /=0 and /=p is seen to increase as W*
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Fig. 5 Variation of hi with W* for H*=0.1, r*0=0.4, and ks/ kf=24.6
Fig. 7 Variation of hi with ks/kf for H*=0.1, W*=0.5, and r*0 = 0.5
temperature along the tube periphery. When the thermal conductivity ratio is reduced, the interface temperature shows significant variation along the tube periphery. A larger temperature gradient is needed for a smaller thermal conductivity in order to achieve the given thermal load of the heat exchanger. The effect of H* on hi was also explored and found to be much less significant than the effects of other parameters. 3.2 Heat flux at the solid–fluid interface The local heat flux at solid–fluid interface (qi) can be expressed in the following non-dimensional form: Fig. 6 Variation of hi with r*0 for H*=0.1, W*=0.1, and ks/ kf=24.6
increases. That is because the tube occupies a larger part of the block as W* increases. Figure 6 shows the variation of hi with r*0, keeping the other parameters constant. These results explore the effect of changing the channel diameter while maintaining the same distance between the channels centreto-centre. Fig. 6 shows a larger temperature variation with angle as the tube radius is increased. However, the average interfacial temperature does not change with r*0. This graph shows that, for a fixed number of tubes, a smaller tube radius is expected to result in more isothermal condition at the solid-fluid interface. Figure 7 presents the variation of hi with ks/kf. Maintaining a constant kf in order to maintain proportionality between (Ti Tb) and hi, the results in Fig. 7 show that the interface temperature becomes more uniform as the solid thermal conductivity is increased due to the smaller thermal resistance within the solid. For the silicon-water system intended for high speed computer chips, the heat exchanger provides almost constant
1 X qi ¼1þ nAn cos ðn/Þ; qave n¼1
ð16Þ
where, qave (=q/(p r*0)) is the average heat flux at the solid-fluid interface. Results for qi are shown in Figs. 8 to 10. The effect of W* is presented in Fig. 8. The value
Fig. 8 Effect of W* on qi for H*=0.1, r*0=0.4, and ks/kf=24.6
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Fig. 9 Effect of r*0 on qi for H*=0.1, W*=0.1, and ks/kf=24.6
of qi changes from a maximum at /=0 to a minimum at /=p. The difference between the maximum and the minimum values is not very large and W* does not appear to have much influence on the qi-profile. Fig. 9 demonstrates the effect of tube radius on qi at the same tube spacing centre-to-centre. The circumferential variation of qi decreases as the tube radius decreases. For the case of r*0 =0.9, corresponding to large-diameter tubes with a small separating distance, the angular profile of qi deviates from the smooth shape seen with the lower r*0 and the magnitude of difference between the heat flux at /=0 and /=p is higher. The effect of ks/kf is shown in Fig. 10. As expected, a more uniform distribution of heat flux is seen for a larger solid thermal conductivity because of larger temperature uniformity within the solid.
Fig. 11 Dependence of c* on H* and W* for ks/kf=2.3 and r*0=0.3
C¼
Ts;1 ðy ¼ L; z ¼ ‘Þ Tb;in qWT ‘
ð17Þ
where, the over-bar indicates average over x. Noting that Ts;1 ðy ¼ L; z ¼ ‘Þ Tb;out ¼ Ts;1 ðy ¼ LÞ Tb ¼ constant; we can express C as C¼
Ts,l ðy ¼ LÞ Tb þ qWT ‘ |ffl{zffl} geometry dependent
1 m_ T Cp;f |fflfflffl{zfflfflffl}
:
ð18Þ
flow dependent
Let us focus on the geometry dependent part and define Ts;1 ðy ¼ LÞ Tb L ¼ c¼ hs;1 ðY ¼ 1Þ: ð19Þ WT ‘kf qWT ‘
3.3 Overall thermal resistance
Finally, we can do our computations and geometry optimizations using a dimensionless thermal resistance c*, defined by
Considering a length ‘ of the heat exchanger, the overall thermal resistance C is normally defined as
c ¼
WT kf ‘ c ¼ hs;1 ðY ¼ 1Þ: L
ð20Þ
Using (8), we can express c* as c ¼ B0 þ kf =ks :
Fig. 10 Effect of ks/kf on qi for H*=0.1, W*=0.5, and r*0 = 0.5
ð21Þ
In the special case of kf/ks=0, the plate temperature becomes uniform and B0 assumes the value B0=11 W*/ (24p). Therefore, based on (21), the minimum value of thermal resistance is given by c*min=0.1459 W*. Values of c *for the three values of ks/kf and wide ranges of the geometrical parameters are shown in Figs. 11–16. In Fig. 11, it can be seen that c* increases monotonically with H* for all values of W* for ks/ kf=2.3. As the distance between the heated surface and the top edge of the tube increases, the thermal resistance is expected to increase because of the larger conduction path for heat transfer. Again, a geometry with a large number of small-diameter tubes (low W*) has less thermal resistance than a geometry with a small number of large-diameter tubes (high W*). This trend is consistent
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Fig. 12 Dependence of c* on r*0 and W* for ks/kf=2.3 and H*=0.1
Fig. 14 Dependence of c* on r*0 and W* for ks/kf=24.6 and H*=0.1
with the earlier results of the temperature distribution. Fig. 12 shows the variation of c * with r*o for different values of W* at ks/kf=2.3. For a given spacing between the tubes (centre-to-centre), the magnitude of c*decreases with the tube radius, attains a minimum, and then increases with further increase of tube radius. The minimum happens around r*0=0.3. As the tube radius increases, the solid region of the heat exchanger is replaced by the fluid region. Therefore, there is a decrease of conduction path between the source and the sink. However, there is less room for the redistribution of temperature within the solid. The net result of these two effects is the variation seen in Fig. 12. As expected, the magnitude of thermal resistance increases with tube spacing. The values of c* for ks/kf=24.6 are plotted in Figs. 13 and 14. The trends are very similar to those seen in Figs. 11 and 12. It may be noticed that with an increase in ks/kf, the slopes of the c* vs. H* curves decrease and
the variation of c* with r*0 becomes less significant. Figures 15 and 16, corresponding to ks/kf=243, show that c* is much less dependent on H* and r*0, while the dependence on W* is still very significant. The effect of W* on c*seen in Figs. 11 to 16 is completely consistent with the effect of W* on h s and h i presented earlier.
Fig. 13 Dependence of c* on H* and W* for ks/kf=24.6 and r*0=0.3
3.4 Optimum configurations Different scenarios are used in design depending on the objective function that need to be optimized and the constraints that are relevant to the particular application. In this section, we will demonstrate how the present analysis can be used in three different design scenarios. In the first scenario, let us consider a given plate with a known geometry (WT · L · ‘) and known solid and fluid properties. The objective is to determine r0 for minimum c * under the constraint of a fixed number of
Fig. 15 Dependence of c* on H* and W* for ks/kf=243 and r*0=0.3
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Fig. 16 Dependence of c* on r*0 and W* for ks/kf=243 and H*=0.1
Fig. 17 Example on the optimization of pumping power
channels M. The objective in this case can be achieved by direct application of the results shown in Figs. 11–16. Consider for example a concrete slab with WT=1 m, ‘ = 5 m, and L=20 cm with water flowing in M=10 channels. In this case, W=WT/(2M)=5 cm and W*=W/L=0.25. For minimum thermal resistance, assume H*=0.1; lower values may cause structural problems. We now search for r*0 that will produce minimum c* for ks/kf=2.3 and the above values of H* and W* . Direct application of the present analysis produces r*0=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 and fluid properties. The objective is to determine M and r0 for minimum pumping power PT under the constraint of fixed overall thermal resistance C. The pumping power is given by
The procedure is illustrated through the following example: 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 channels inside the block. It is desired to achieve an overall thermal resistance of C=0.05 K/W. Determine the channels layout (M and ro) that will result in the minimum pumping power PT. Taking water properties at 300 K, we get ks/kf=24.6. The value of H* was assumed to be 0.1. A search procedure is required to find the solution. Select a value of M, from which W and W* can be determined. For a fixed M, use different values of r*0 and determine c * for each combination of H*, W*, r*0, and ks/kf from the present analysis. The value of b PT can then be easily determined from (25). This search procedure was conducted for different values of M and a sample of the results is shown in Fig. 17. The minimum PT was found to correspond to M=14 and r*0=0.88.
PT ¼ m_ T DP =qf ;
ð22Þ
where, DP is the pressure drop across the heat exchanger, which (under the present conditions of laminar fully-developed flow) can be expressed as 8‘lf DP ¼ ð23Þ m_ T : pr04 qf M 1 Knowing thatC ¼ c þ m_ T Cp;f ; and using definition (20) of c*, we can write Lc 1 m_ T Cp;f ¼ C ð24Þ WT kf ‘ Substituting (23) and (24) into (22) and rearranging, we get bPT ¼
16 h i2 ; 4 pðWT =‘Þ r0 ðW Þ3 1 WTLckf ‘C
where, 2 b ¼ qf CP ;f C L3 =lf :
ð25Þ
Fig. 18 Example on the optimization of thermal resistance
606
In the third optimization scenario, we consider a plate heat exchanger with a known geometry (WT · L · ‘) and known solid and fluid properties. The objective is to determine M and r0 for minimum C under the constraint of fixed PT. (25) can be reformulated to read Lc 4 ‘lf 1=2 þ 2 C¼ : ð26Þ WT kf ‘ qf r0 CP ;f 2pMPT For example, consider a silicon wafer (WT=1 cm, ‘=5 cm, and L=2 mm) receiving a uniform heat flux at the top surface and cooled by water flowing in circular channels. It is desired to limit the frictional pumping power to PT=2 · 10 4 W. Determine the channels layout (M and r0) that will result in a minimum C . The search procedure is demonstrated in Fig. 18. For each value of r*0, there is a value of M that produces a minimum C. The value of M for minimum C can be seen to increase as r*0 increases. The limit of diminishing returns is probably reached at r*0=0.9. A reasonable solution corresponds to M=9 and r*0=0.9. All results in Fig. 18 correspond to H*=0.1 and ks/kf=243.
4 Conclusions Flat plate heat exchangers with circular longitudinal channels have been analyzed for the conditions of uniform heat flux at the top surface of the plate and fully-developed flow and heat transfer in the tubes. An analytical series solution has been developed for the temperature distributions in the fluid and solid regions and the series solution is shown to converge to high accuracy with a small number (N=15) terms. Calculations have been carried out for a wide range of the geometry parameters (H*, W*, and r*0) and the solid– fluid thermal conductivity ratio ks/kf, thus covering typical applications of this heat exchanger for thermal management of electronics, process equipment, and pavement slabs. It was found that the temperature level in the solid and at the solid–fluid interface increases with an increase in tube spacing (centre-to-centre) for a given depth of tubes from the heated surface. For a given tube spacing, the interface temperature shows wider variation along the tube periphery when the tube radius is increased; however, the average value of temperature does not change with tube radius. When the thermal conductivity ratio is increased, the interface temperature becomes more uniform because of the decrease in thermal resistance within the solid. The peripheral heat flux distribution at the solid–fluid interface became more uniform as the tube spacing (centre-to-centre) and/or the thermal conductivity ratio were increased. The overall thermal resistance of the heat exchanger increased with increases in the tubes’ depth from the heated surface and/or the spacing between tubes (centreto-centre). However, for given depth and spacing, the
thermal resistance achieved a minimum value at an intermediate tube diameter indicating the importance of tube size in the design of this heat exchanger. A larger thermal conductivity ratio resulted in lower thermal resistance within the solid and heat transfer in that situation is heavily controlled by the convective resistance at the solid-fluid interface. The objective function for design optimization of this heat exchanger has been derived for three different scenarios including minimum pumping power and minimum thermal resistance. The optimization procedure has been illustrated by an example in each scenario. Acknowledgements The financial assistance provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
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