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Internal heat generation in a discrete heat source: Conjugate heat transfer analysis
Applied Thermal Engineering 26 (2006) 2201–2208 www.elsevier.com/locate/apthermeng
Internal heat generation in a discrete heat source: Conjugate heat transfer analysis O. Bautista a, F. Me´ndez a
b,*
Seccio´n de Estudios de Posgrado e Investigacio´n, IPN 02550, Me´xico, DF, Mexico b Facultad de Ingenierı´a, UNAM, 04510 Me´xico, DF, Mexico Received 8 December 2005; accepted 28 March 2006 Available online 5 June 2006
Abstract In the present work, we conduct an asymptotic and numerical analysis for the cooling process of a discrete heat source, which is placed in a rectangular-channel laminar cooling flow. In our physical model, the heated strip is embedded in a substrate, generating continuously a uniform volumetric heat rate. We assume that this heat-generation mechanism is due to an electrical current in the heat source. Hence, heat losses to the cooling fluid and to the substrate material during this process are presented. The governing equations of the cooling flow and the participating solid are reduced to an integro-differential equation that predicts the temperature variations of the heat source. We show that the conjugate heat transfer process is controlled by a conjugate nondimensional parameter, here denoted by a, which determines the basic heat transfer regimes between the cooling flow and the discrete heat source. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Channel flow; Discrete heat source; Regular perturbation techniques
1. Introduction The electronic cooling analysis of small heat-generating sources has been recognized in the specialized literature as an active and fundamental research’s area, due to the influence of this factor to control the electric efficiency of different types of board circuitries. Yun et al. [1] showed that in order to maintain the device junction temperature below of a maximum limit, the increase of volumetric heat production rates must be carefully controlled. Otherwise, the temperature differences for conventional systems affect the component’s efficiency and these overestimated chip temperature gradients can introduce a thermal failure between the elements of the board circuitry, changing drastically the electronic performance. In consequence, these thermal failures in some cases generate irreversible mechanical fractures. Therefore, the physical influence that thermal conditions have on the *
Corresponding author. Tel.: +52 55 56 22 81 03; fax: +52 55 56 22 81
06. E-mail address: [email protected] (F. Me´ndez). 1359-4311/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.03.018
electronic package surfaces is very important. In general, these conditions are unknown and for a given heat-generation rate, the prediction of temperature profiles at the electronic chip, is of primordial importance to obtain a high performance of the involved electronic components. The foregoing fundamental and practical aspects offer an excellent opportunity to explore systematically this class of conjugated heat transfer models. Here, we accept that the volumetric heat production rate drives the optimal cooling conditions and consequently the conjugate heat transfer formulation is inevitable. In the past, Cole [2], Incropera [3] and Jaluria [4] emphasized the fundamental importance of this type of thermal interactions between forced and natural convection flows and thermal sources on surfaces. Later, Sathe and Joshi [5] showed the importance of the coupled heat transfer process between a heat-generating substrate-mounted protrusion and a liquid-filled twodimensional enclosure. In the above works, the flush heaters operated as uniform heat sources. Me´ndez and Trevin˜o [6] analyzed the conjugated heat transfer process between a natural convection flow and an embedded vertical strip in
O. Bautista, F. Me´ndez / Applied Thermal Engineering 26 (2006) 2201–2208
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Nomenclature B(n, l) C h H L Nu Pr Re Pe Tw TF Tw DTwm T1 U q_ q0 q_ 000 x, y z
beta function specific heat of the laminar cooling flow thickness of the discrete source channel width length of the discrete source Nusselt number Prandtl number of the laminar cooling flow Reynolds number of the laminar cooling flow Peclet number of the laminar cooling flow chip temperature fluid temperature average temperature on the discrete source maximum difference temperature, Twjx=L Twjx=0 free stream temperature of the laminar cooling flow velocity profile of the laminar cooling flow volumetric heat production heat rate to the substrate heat flux to the substrate Cartesian coordinates nondimensional normal coordinate of the discrete source
d c e k kw m q qw hw hw v n
thickness of the thermal boundary layer in the laminar cooling flow nondimensional parameter aspect ratio of the discrete source thermal conductivity of the fluid thermal conductivity of the discrete source kinematics viscosity of the fluid density of the fluid density of the discrete source nondimensional temperature of the discrete source nondimensional average temperature of the discrete source nondimensional longitudinal coordinate nondimensional longitudinal coordinate
Subscripts l conditions at the leading edge of the discrete source w conditions at the discrete source 1 conditions in the laminar cooling flow
Greek symbols a nondimensional heat conduction or conjugate parameter
a substrate with nonuniform generation heat rate. They used numerical and analytical perturbation techniques in order to clarify the role of the longitudinal heat transfer effects on a vertical thin plate in a natural convective cooling process. Sometimes, the passive cooling mechanism by natural convection is sufficient, due to the simplicity of design, absence of noise and high reliability. However, we require other cooling techniques for increasing volumetric heatgeneration rates, as was pointed opportunely out by Tout et al. [7] and Sun et al. [8]. Ramadhyani et al. [9] theoretically and Incropera et al. [10] experimentally, considered the problem of conjugate heat transfer from discrete heat sources mounted on a wall of a channel exposed to fully developed laminar flow. Following a simpler physical model, Rizk et al. [11] derived an analytical solution for the conjugate heat transfer problem of a flow past a heated block. Recently, Chuang et al. [12] considered a more complex situation: the heat transfer between a three-dimensional rectangular duct and heat-generating chips, showing that the higher inlet velocity leads to heat transfer enhancement in the internal region of the duct. In this work, with the aid of perturbation as well as numerical techniques, we obtain the temperature profile in a thin chip with uniform internal heat generation. The heat-generation strip is embedding in a substrate and hence
heat losses to the cooling flow and to the substrate material during this process, are presented. For simplicity, we assume a fully developed velocity profile in the rectangular channel and with the aid of thermal boundary layer theory; we evaluate the heat transfer to the laminar cooling flow. For simplicity, we assume a uniform heat transfer to the substrate, which in the present model appears as a nondimensional parameter given lines below. Therefore, the simultaneous participation of both heat transfer effects has a profound influence on the process, since the temperature of the chip becomes regulate by this thermal interaction. 2. Formulation In Fig. 1 we show the physical model under study. In the Cartesian coordinate system, the upper left corner of the strip coincides with the origin, whose y axis points out in the normal direction to the discrete heat source or strip and its x axis points out in the longitudinal direction of the source. The heat-generating source of length L and thickness h, embeds in a rectangular channel of width H and a depth M H, so that we can established a bi-dimensional flow into the rectangular channel. Due to the internal heat generation of the strip (with a uniform volu_ an important fraction of the heat transfer metric rate q),
O. Bautista, F. Me´ndez / Applied Thermal Engineering 26 (2006) 2201–2208
net heat transferred to the cooling fluid and to the substrate. Eqs. (2) and (3) can be combined to yield 2 1=3 _ _ 2 H L qh q_ 00 qh 1 0 : ð4Þ and DT w DT F c _ k 6Pe kw qh
u (y) y H
8
T 0
internal heat generation q
h
x
q
L
Fig. 1. Schematic diagram of the physical model.
occurs between the discrete source and the laminar cooling flow. The other fraction transfers to the substrate. Because in practical cases, the ratio h/L is very small in comparison with unity, we assume that the right and left faces of the source are practically adiabatic. Therefore, the heat loss to the substrate, here denoted q0, is only present through the lower face of the chip. It is really important to note that we assume, for simplicity, a uniform heat loss to the substrate. It is indispensable to introduce a more realistic model like a nonuniform variation of heat to the substrate. However, the present analysis can be readily generalized to include this condition. Following the above considerations, the upper face of the strip contacts a laminar cooling flow with a well-known fully developed velocity profile given by y y 2 1 dP 2 u ¼ 6 u H ; ð1Þ ; with u¼ H H 12l dx where u represents the mean value of the velocity. In order to obtain the appropriate scales of the problem, we use an order of magnitude analysis on the energy equation of the cooling fluid in order to show that the ratio of the thickness of the thermal boundary layer to the length of the strip can write as " #1=3 2 d H 1 ; ð2Þ L L 6Pe where Pe = Re Pr and represents the well-known Peclet number, Re = u H/m denotes the Reynolds number and Pr is the Prandtl number. q and l are the density and the dynamic viscosity, respectively. On the other hand, a global balance of thermal energy establishes that the heat flux from the strip to the cooling flow can write as _ kw qh
DT w DT F k þ q_ 000 ; h d
2203
ð3Þ
where k and kw are the thermal conductivities of the discrete source and fluid, respectively. DTw is, in the transverse direction, the characteristic temperature drop of the strip and DTF is the characteristic temperature drop in the cooling fluid. q_ 000 denotes heat flux to the substrate. On the other hand, the first term in Eq. (1) corresponds to the thermal energy generated in the strip, while the third one is the
Using these relationships, we also obtain that " #1=3 2 DT FM a kw h H 1 2 with a ¼ c ; e L 6Pe DT w k L
ð5Þ
e is the aspect ratio of the strip, e = h/L, which is assumed to be very small compared with unity and c a constant of order unity to be given later. In the above relationship (5), we define DTFM as the maximum value of the characteristic temperature drop in the cooling fluid and replacing q_ 000 ¼ the relationship (4), we obtain that DT FM ¼ 0 into _ c qh k
H 2L 6Pe
1=3
.
We will use the above relationship to adimensionalize the temperature distribution. Furthermore, parameter a is the nondimensional longitudinal heat conductance of the discrete source and is a measure of the longitudinal heat conduction effects in this electronic device on the cooling process. This parameter can assume any value depending on the geometrical and physical properties of the heat-conducting element and the fluid. For instance, from the order relationship, Eq. (5) with a/e2 1, the transverse temperature variations of the strip are very small compared with the temperature differences in the fluid, that is DTw DTF. In a previous work of Me´ndez and Trevin˜o [13], we called this limit as the thermally thin wall regime. For values of a/e2 1, the temperature variations in both directions of the strip now are very important and are of the same order of magnitude of that temperature differences in the fluid. This is the thermally thick wall regime. In this limit and with e 1, the longitudinal heat conduction through the strip is very small, neglecting in a first approximation. Both limits will be explored in the next sections. With the above discussion, we can introduce the following nondimensional variables for the discrete source x yþh Tw T1 v¼ ; z¼ ; hw ¼ with L h DT FM 2 1=3 _ qh H L DT FM ¼ c ð6Þ k 6Pe and for the fluid y T T1 n¼ ; h¼ : d DT FM
ð7Þ
Therefore, the nondimensional energy balance equations for the strip and fluid take the form a
o2 hw a o2 hw þ 2 þ1¼0 e oz2 ov2
ð8Þ
and n
oh o2 h ¼ : ov on2
ð9Þ
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We derive Eq. (9) taking into account that Pe 1, which means that the longitudinal heat conduction effects in the cooling flow are negligible, approximation widely used for laminar flow in ducts and channels. Furthermore, the presence of a thermal boundary layer adjacent to the upper surface of the strip, permit us to linearize the velocity profile. The details are given lines below. The corresponding boundary conditions for the strip are,
proportional to the transverse distance y, when the temperature field is confined inside of the zone of the velocity field, which occurs frequently for large values of the Prandtl number. However, this approximation gives excellent results even for Prandtl numbers of order unity. In particular, the Kernel in Eq. (18) is given by
ohw ¼0 ov
where, hwl is the value of the nondimensional temperature at the leading edge of the heated strip. The assumed value for the constant c in the definition of a is c = 31/3/C(1/3). In general, Eqs. (8), (10), (11) and (18) can numerically integrate. With this purpose, in the following sections, we obtain asymptotic solutions for the thermally thin and thick wall regimes. We also include the numerical solution for the thermally thin wall regime.
at v ¼ 0; 1
ð10Þ
and ohw e2 ¼c at z ¼ 0; ð11Þ oz a where the nondimensional parameter c is defined as c¼
q_ 000 : _ qh
ð12Þ
This nondimensional parameter represents the competition between the loosed thermal energy to the substrate and the generated thermal energy by the discrete heat source. Obviously, this nondimensional parameter is always less than unity. In order to complete the boundary conditions, we add that hðv; n ! 1Þ ¼ hðv ¼ 0; nÞ ¼ 0
ð13Þ
and moreover we require an additional condition at z = 1 (n = 0), which corresponds to accept the continuity of the temperatures and heat transfer rates ohw e2 oh hðv; n ¼ 0Þ ¼ hw ðvÞ and ¼ ; ð14Þ oz z¼1 a onn¼0 where the temperature gradient oh/onjn=0 represents the nondimensional heat flux from the upper surface to the cooling fluid. In this form, the relationships (14) define the matching conditions between both zones. In order to obtain the solution of the energy equation, Eq. (9), we propose the quasi-similarity variables given by h ¼ hw u
and f ¼
n v1=3
;
ð15Þ
ð16Þ
with the corresponding boundary conditions uð0Þ ¼ 1
and uðf ! 1Þ ¼ 0:
In addition, Eq. (14) can be written as
Z hw ohw e2 c 0 0 ¼ hwl þ Kðv; v Þdhw : oz z¼1 a v1=3 hwl
1=3
;
ð19Þ
3. Thermally thin wall regime (a/e2 1) For very large values of a/e2 compared with unity, the relationship (5) dictates that the temperature variations in the normal direction of the strip are insignificant. Therefore, in a first approximation, hw only depends on the coordinate v. In this regime, the energy equation for the chip, Eq. (8), can be integrated along the normal coordinate and after applying the appropriate boundary conditions (10), (11) and (18), we obtain
Z hw d2 hw 1 0 0 a 2 þ 1 ¼ 1=3 hwl þ Kðv; v Þdhw þ c: ð20Þ v dv hwl This equation must be solving with the adiabatic conditions for the lateral surfaces of the strip given by Eq. (10). In the following subsections we present asymptotic solutions for very large values of a, that is a 1 and for values of a ! 0, respectively. In both cases, we accept that the limit a/e2 1 is valid. In addition, we obtain numerical solutions for these limits in order to validate the asymptotic perturbation scheme. On the other hand, we solve Eq. (20) using numerical techniques reported elsewhere (see for instance, Me´ndez and Trevin˜o [13]). 3.1. Asymptotic limit a 1
reducing Eq. (9) to o2 u 1 2 ou ou ¼ vf ; þ f ov of2 3 of
Kðv; v0 Þ ¼ ð1 v0 =vÞ
ð17Þ
ð18Þ
The right-hand side of this last boundary condition, Eq. (18), can easily derive with the aid of the Lighthill’s approximation, [14]. This integral approach takes into account that the longitudinal velocity component, u, is still
In order to study this regime, the temperature of the chip is obtaining with the aid of a regular expansion technique, using the inverse of a as a small perturbation parameter. For very large values of the parameter a, the nondimensional temperature of the discrete source, hw, changes very little in the longitudinal direction (of order a1). Therefore, we assume that the nondimensional temperature of the strip can be expanding as 1 X 1 hwj ðvÞ: ð21Þ hw ¼ aj j¼0 Introducing the above relationship (21) into the nondimensional governing equation for the chip (20), we obtain after
O. Bautista, F. Me´ndez / Applied Thermal Engineering 26 (2006) 2201–2208
collecting terms of the same power of a, the following set of equations d2 hw0 ¼ 0; dv2
ð22Þ
Z h0 d2 hw1 1 0 0 þ 1 ¼ 1=3 h0l þ Kðv; v Þdhw0 þ c; v dv2 h0l Z hi d2 hwðiþ1Þ 1 0 0 ¼ h þ Kðv; v Þdh for i > 1; il wi v1=3 dv2 hil with the following adiabatic boundary conditions dhwj ¼ 0; for all j: dv
ð23Þ ð24Þ
3.2. Asymptotic limit a ! 0
Integration of Eq. (22) with the corresponding boundary conditions (25) gives hw0 = C0, where C0 is an integration constant and must be determined by solving the next R 1 higher order Eq. (23). Integrating Eq. (23) in the form 0 dv and using the adiabatic conditions at both edges of the strip, yields that C0 = 2 (1 + c)/3. This procedure can use to obtain the solution to higher orders. Therefore, the solution to Eq. (23) is v2 9 v2 þ v5=3 C 0 þ c þ C 1 ; 2 10 2
where C1 is a constant given by 2 2 18 5 2 2 2 C 1 ¼ B 2; B ; C 0 B 2; c; 8 3 42 3 3 8 3
in addition, the averaged nondimensional temperature hw , up to terms of order 1/a2, can write as Z 1 1 1 hw1 þ 2 hw ¼ hw2 þ hw dv ’ hw0 þ a a 0
2 1 9 1 C 0 ð1 cÞ þ C 1 ¼ ð1 cÞ þ 3 a 10 2 1 27 5 2 9 9 2 þ 2 B ; C 0 þ C 1 B 2; ð1 cÞ þ C 2 . a 1400 3 3 10 88 3 ð31Þ
ð25Þ
v¼0;1
hw1 ðvÞ ¼
2205
ð26Þ
ð27Þ
where B(l, n) represents the complete beta function. Following the same procedure, the second order solution is 9 9 2 11=3 27 5 2 5=3 B ; C 0 v10=3 h2 ðvÞ ¼ C 1 v B 2; v þ 10 88 3 140 3 3 9 2 þ B 2; cv11=3 þ C 2 ; ð28Þ 88 3 with C2 given by 18 5 2 1 2 3 5 2 B 2; B ; C0 C2 ¼ B ; 42 3 3 4 3 7 3 3 1 2 3 2 11 2 ; B 2; c þ B 2; B 4 3 52 3 3 3 3 5 2 10 2 3 2 11 2 ; C 0 B 2; B ; c: B ; B 28 3 3 3 3 52 3 3 3 ð29Þ Therefore, up to terms of second order, the nondimensional discrete source temperature becomes
2 1 9 v2 5=3 C 0 v ð1 cÞ þ C 1 hw ¼ ð1 cÞ þ 3 a 10 2 1 9 27 5 2 10=3 5=3 C1v þ C0B ; v þ 2 a 10 140 3 3
9 2 11=3 B 2; v ð1 cÞ þ C 2 ; ð30Þ 88 3
In the thermally thin wall regime, we have the limiting case of a ! 0, but with a/e2 1. Here, the longitudinal heat conduction in the strip is very small and can neglect except in regions very near to the edges of the strip, where the presence of local thermal boundary layers are indispensable to understand it. However, the structure of these regions has only a local influence. Therefore, in the present analysis we exclude to consider these thermal boundary layers. Hence, Eq. (20) with a = 0 transforms to Z v dh0 1=3 ð1 cÞv ¼ Kðv; v0 Þ wa0 dv0 : ð32Þ dv0 0 The solution of foregoing equation can obtain with the aid of the well-known Abel’s Integral transform. The nondimensional temperature of the strip, hw(v), is then hwa0 ¼
3ð1 cÞ 1 v3 ; B 13 ; 23
ð33Þ
moreover, the nondimensional average temperature is hwa0 ¼ 9ð1 cÞ : 4B 13 ; 23
ð34Þ
4. Thermally thick wall limit (a/e2 1) In this other case, the longitudinal heat conduction is also very small and therefore can neglect. The energy balance Eq. (8) for the strip then reduces to o2 hw e2 ¼ : 2 oz a
ð35Þ
Eq. (35) has to be solving with the boundary conditions ohw e2 ð36Þ ¼ c oz z¼0 a and ohw e2 1 ¼ 1=3 oz z¼1 av Z hwl þ 0
hw
0 e2 1=3 dhw 0 dv ¼ ð1 cÞ: 0 dv a
ð1 v0 =vÞ
ð37Þ
Integrating Eq. (35) in the normal direction and using both conditions (36) and (37), we obtain that
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Z v ohw e2 1 dh0 ¼ 1 1=3 ð1 v0 =vÞ1=3 w0 dv0 z : v oz a dv 0 Therefore, the nondimensional temperature is
e2 1 hw ¼ hwu þ ð1 z2 Þ cð1 zÞ ; a 2
ð38Þ
ð39Þ
where hwu is the nondimensional temperature at the upper surface of the strip, given by hwu ¼
3ð1 cÞ 1 v3 ; B 13 ; 23
ð40Þ
The nondimensional average temperature is then 9ð1 cÞ 1 e2 hw ¼ 1 2 þ : 3 a 4B 3 ; 3
ð41Þ
It is interesting to note that in the limit of e2/a ! 0, the total thermal energy of the strip in this regime is the same as for the case of a ! 0, for the thermally thin wall regime given by Eq. (34).
Fig. 3. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.01 and different values of the nondimensional parameter a.
5. Results and concluding remarks In Figs. 2–7, we present the most relevant numerical and asymptotic results derived from the present analysis. In all calculations carried out in this work, we use the following data: T1 = 300 K, kw/k = 100 and Pr = 0.7. Also as illustration, we assume representative values for the characteristic geometric scales of the system: the horizontal length of the discrete heat source was 5 cm, its thickness 1 cm and the vertical distance of the channel was H = 2 cm. In order to validate the regular perturbation scheme to study the thermally thin wall regime, Eq. (20) was numerically integrated together with the adiabatic boundary conditions, Eq. (10); using the well-known Runge–Kutta technique. In order to make it, the boundary value problem is transFig. 4. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.05 and different values of the nondimensional parameter a.
Fig. 2. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.0 and different values of the nondimensional parameter a.
formed to an initial value problem with the given initial conditions at the leading edge of the chip for the nondimensional temperature and its gradient. Together with the above procedure, a conventional shooting-iteration method was applied, due to the unknown value of hwl for given values of the nondimensional parameters a and c. For the thermally thin regime, Figs. 2–5 show the comparison between the asymptotic solutions—given by relationship (30), and the numerical results based on the above iteration scheme for the nondimensional temperature distribution, hw, as a function of the nondimensional coordinate v, different values for the parameter a (=3.0, 5.0, 10.0 and 100) and including the variation of the nondimensional parameter c (=0.0, 0.01, 0.05 and 0.1). In all figures, we found that for a’s smaller than 3, the differences
O. Bautista, F. Me´ndez / Applied Thermal Engineering 26 (2006) 2201–2208
Fig. 5. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.1 and different values of the nondimensional parameter a.
Fig. 6. Nondimensional average temperature differences, ðT w ðvÞ T 1 Þ, as a function of the Peclet number, Pe, and for different values of the nondimensional parameter c.
between both solutions are increasing and are practically independents of the assumed values of the parameter c. Also, it is interesting to note that the asymptotic solutions offer good results even for values of a of order 3. On the other hand, the deviations between both solutions can not necessarily assign to an error of the perturbation scheme. In this domain of a O(1) or smaller values than unity, the influence of the temperature gradients in the transverse direction begins to be important and the first corrections of the thermally thick regime should be taking into account. In these figures, the numerical and asymptotic solutions show that for decreasing values of a, the nondimensional temperature hw decreases strongly at the leading edge of the strip and increases at the trailing edge. The influence of c does not change the above behavior of
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Fig. 7. Maximum difference of the nondimensional strip temperature, DTwm, as a function of the Peclet number, Pe, and different values of the nondimensional parameter c.
the temperature; however, for increasing values of this parameter, the temperature hw decreases, because the parameter c measures the influence of the loosed heat to the substrate. The above results confirm the observation made by Bejan [1]: higher temperatures lead to higher internal heat generations, which derive again to higher temperatures. This thermally unstable limit behavior must be avoided in this type of applications. Hence, the set of Figs. 2–5 clearly reveal the fundamental role of the parameters a and c on the temperature hw and their corresponding longitudinal gradients. Thus, the criterion to operate with a better electrical performance depends on which tolerance is accepted: a strip working with large nondimensional longitudinal temperature differences (decreasing values of a) and a low average temperature hw or a strip that can attenuate the nondimensional longitudinal temperature gradients with high average temperature hw. Both cases controlled by the influence of the parameter c. In order to clarify the above aspect, we present lines below other numerical results to complement these observations. Following the above comments, in Fig. 6 we show the corresponding nondimensional average temperature differ_ =kÞ, plotted as a function of the ences, ðT w ðvÞ T 1 Þ=ðqH Peclet number, Pe, and three different values of the parameter c. The influence of Peclet number is clear: for increasing values of the cooling effect the average temperature of the strip is drastically reduced. In addition, this reduction also increases for increasing values of c. In this case, the presence of the loosed heat to the substrate operates as a favorable effect. However, the heating of the substrate can yield undesirable effects on the electric performance of the system. In this figure, we show also analytical results and the corresponding comparison between both solutions provides a very good agreement.
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Finally, in Fig. 7 we show for the thermally thin wall regime, the maximum difference of the nondimensional strip temperature between both adiabatic edges, DTwm = DTFMDhw, as a function of the Peclet number Pe, three different values of the parameter c (=0.01, 0.2 and 0.3) and also, three different values of the internal heat generation _ For increasing values of the Peclet number, DTFMDhw q. decreases, being more notable this effect for increasing values of c. On the other hand, the influence of the internal heat generation changes drastically the levels of these temperature differences in the strip. Because the maximum characteristic temperature DTFM depends sensibly on Peclet number, we can conclude that the numerical calculations show that DTFM is the dominant factor that yields the continuous decrement of the nondimensional function DTwm. In the present work, we have carried out an analytical and numerical analysis to study the cooling mechanism of a discrete heat source embedded in a rectangular channel. Inside this channel is circulating a laminar flow, which permits to transfer heat from the heating-strip to the fluid. Because the substrate is not properly adiabatic, the direct contact between the heated strip and the substrate permits a finite heat transfer rate. Here, this mechanism is modulated by the introduction of the nondimensional parameter c. Therefore, in terms basically of the nondimensional parameters c and a, the thermal performance and characteristics of this typical device are well defined. Acknowledgements This work has been supported by a research grant no. 43010-Y of Consejo Nacional de Ciencia y Tecnologı´a at Mexico.
References [1] Y.-S. Yun, Z.-Y. Guo, Heat wave phenomena in IC chips, Int. J. Heat Mass Transfer 38 (1995) 2919–2922. [2] K.D. Cole, Conjugate heat transfer from a small heated strip, Int. J. Heat Mass Transfer 40 (1997) 2709–2719. [3] F.P. Incropera, Convection heat transfer in electronic equipment cooling, ASME J. Heat Transfer 110 (1988) 1097–1111. [4] Y. Jaluria, Interaction of natural convection wakes arising from thermal sources on a vertical surface, ASME J. Heat Transfer 107 (1985) 883–892. [5] S.B. Sathe, Y. Joshi, Natural convection arising from a heatgenerating substrate-mounted protrusion in a liquid-filled twodimensional enclosure, Int J. Heat Mass Transfer 34 (1991) 2149–2163. [6] F. Me´ndez, C. Trevin˜o, The conjugate conduction-natural convection heat transfer along a thin vertical plate with nonuniform internal heat generation, Int. J. Heat Mass Transfer 43 (2000) 2739–2748. [7] K.W. Tou, G.P. Xu, C.P. Tso, Direct liquid cooling of electronic chips by single-phase forced convection of FC-72, Exp. Heat Transfer 11 (1998) 121–128. [8] H. Sun, C.F. Ma, W. Nakayama, Local characteristics of convective heat transfer from simulated microelectronic chips to impinging submerged round water jets, J. Electronic Package 115 (1993) 71–79. [9] S. Ramadhyani, D.F. Moffatt, F.P. Incropera, Conjugate heat transfer from small isothermal heat sources embedded in a large substrate, Int. J. Heat Mass Transfer 28 (1985) 1945–1952. [10] F.P. Incropera, J.S. Kerby, D.F. Moffatt, S. Ramadhyani, Convection heat transfer from discrete heat sources in a rectangular channel, Int. J. Heat Mass Transfer 29 (1986) 1051–1058. ¨ zisik, Analytical solution to the [11] T.A. Rizk, C. Kleinstreuer, M.N. O conjugate heat transfer problem of flow past a heated block, Int. J. Heat Mass Transfer 35 (1992) 1519–1525. [12] S.H. Chuang, J.S. Chiang, T.M. Yan, Numerical heat transfer analysis in a three-dimensional rectangular vent duct with heatgenerating chips, Int. Commun. Heat Mass Transfer 30 (2003) 835– 844. [13] F. Me´ndez, C. Trevin˜o, Film condensation generated by a forced cooling fluid, Eur. J. Mech. B/Fluids 15 (1996) 217–240. [14] M.J. Lighthill, Contributions to the theory of heat transfer through a laminar boundary layer, Proc. R. Soc London A 202 (1950) 359–377.
Internal heat generation in a discrete heat source: Conjugate heat transfer analysis O. Bautista a, F. Me´ndez a
b,*
Seccio´n de Estudios de Posgrado e Investigacio´n, IPN 02550, Me´xico, DF, Mexico b Facultad de Ingenierı´a, UNAM, 04510 Me´xico, DF, Mexico Received 8 December 2005; accepted 28 March 2006 Available online 5 June 2006
Abstract In the present work, we conduct an asymptotic and numerical analysis for the cooling process of a discrete heat source, which is placed in a rectangular-channel laminar cooling flow. In our physical model, the heated strip is embedded in a substrate, generating continuously a uniform volumetric heat rate. We assume that this heat-generation mechanism is due to an electrical current in the heat source. Hence, heat losses to the cooling fluid and to the substrate material during this process are presented. The governing equations of the cooling flow and the participating solid are reduced to an integro-differential equation that predicts the temperature variations of the heat source. We show that the conjugate heat transfer process is controlled by a conjugate nondimensional parameter, here denoted by a, which determines the basic heat transfer regimes between the cooling flow and the discrete heat source. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Channel flow; Discrete heat source; Regular perturbation techniques
1. Introduction The electronic cooling analysis of small heat-generating sources has been recognized in the specialized literature as an active and fundamental research’s area, due to the influence of this factor to control the electric efficiency of different types of board circuitries. Yun et al. [1] showed that in order to maintain the device junction temperature below of a maximum limit, the increase of volumetric heat production rates must be carefully controlled. Otherwise, the temperature differences for conventional systems affect the component’s efficiency and these overestimated chip temperature gradients can introduce a thermal failure between the elements of the board circuitry, changing drastically the electronic performance. In consequence, these thermal failures in some cases generate irreversible mechanical fractures. Therefore, the physical influence that thermal conditions have on the *
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06. E-mail address: [email protected] (F. Me´ndez). 1359-4311/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.03.018
electronic package surfaces is very important. In general, these conditions are unknown and for a given heat-generation rate, the prediction of temperature profiles at the electronic chip, is of primordial importance to obtain a high performance of the involved electronic components. The foregoing fundamental and practical aspects offer an excellent opportunity to explore systematically this class of conjugated heat transfer models. Here, we accept that the volumetric heat production rate drives the optimal cooling conditions and consequently the conjugate heat transfer formulation is inevitable. In the past, Cole [2], Incropera [3] and Jaluria [4] emphasized the fundamental importance of this type of thermal interactions between forced and natural convection flows and thermal sources on surfaces. Later, Sathe and Joshi [5] showed the importance of the coupled heat transfer process between a heat-generating substrate-mounted protrusion and a liquid-filled twodimensional enclosure. In the above works, the flush heaters operated as uniform heat sources. Me´ndez and Trevin˜o [6] analyzed the conjugated heat transfer process between a natural convection flow and an embedded vertical strip in
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Nomenclature B(n, l) C h H L Nu Pr Re Pe Tw TF Tw DTwm T1 U q_ q0 q_ 000 x, y z
beta function specific heat of the laminar cooling flow thickness of the discrete source channel width length of the discrete source Nusselt number Prandtl number of the laminar cooling flow Reynolds number of the laminar cooling flow Peclet number of the laminar cooling flow chip temperature fluid temperature average temperature on the discrete source maximum difference temperature, Twjx=L Twjx=0 free stream temperature of the laminar cooling flow velocity profile of the laminar cooling flow volumetric heat production heat rate to the substrate heat flux to the substrate Cartesian coordinates nondimensional normal coordinate of the discrete source
d c e k kw m q qw hw hw v n
thickness of the thermal boundary layer in the laminar cooling flow nondimensional parameter aspect ratio of the discrete source thermal conductivity of the fluid thermal conductivity of the discrete source kinematics viscosity of the fluid density of the fluid density of the discrete source nondimensional temperature of the discrete source nondimensional average temperature of the discrete source nondimensional longitudinal coordinate nondimensional longitudinal coordinate
Subscripts l conditions at the leading edge of the discrete source w conditions at the discrete source 1 conditions in the laminar cooling flow
Greek symbols a nondimensional heat conduction or conjugate parameter
a substrate with nonuniform generation heat rate. They used numerical and analytical perturbation techniques in order to clarify the role of the longitudinal heat transfer effects on a vertical thin plate in a natural convective cooling process. Sometimes, the passive cooling mechanism by natural convection is sufficient, due to the simplicity of design, absence of noise and high reliability. However, we require other cooling techniques for increasing volumetric heatgeneration rates, as was pointed opportunely out by Tout et al. [7] and Sun et al. [8]. Ramadhyani et al. [9] theoretically and Incropera et al. [10] experimentally, considered the problem of conjugate heat transfer from discrete heat sources mounted on a wall of a channel exposed to fully developed laminar flow. Following a simpler physical model, Rizk et al. [11] derived an analytical solution for the conjugate heat transfer problem of a flow past a heated block. Recently, Chuang et al. [12] considered a more complex situation: the heat transfer between a three-dimensional rectangular duct and heat-generating chips, showing that the higher inlet velocity leads to heat transfer enhancement in the internal region of the duct. In this work, with the aid of perturbation as well as numerical techniques, we obtain the temperature profile in a thin chip with uniform internal heat generation. The heat-generation strip is embedding in a substrate and hence
heat losses to the cooling flow and to the substrate material during this process, are presented. For simplicity, we assume a fully developed velocity profile in the rectangular channel and with the aid of thermal boundary layer theory; we evaluate the heat transfer to the laminar cooling flow. For simplicity, we assume a uniform heat transfer to the substrate, which in the present model appears as a nondimensional parameter given lines below. Therefore, the simultaneous participation of both heat transfer effects has a profound influence on the process, since the temperature of the chip becomes regulate by this thermal interaction. 2. Formulation In Fig. 1 we show the physical model under study. In the Cartesian coordinate system, the upper left corner of the strip coincides with the origin, whose y axis points out in the normal direction to the discrete heat source or strip and its x axis points out in the longitudinal direction of the source. The heat-generating source of length L and thickness h, embeds in a rectangular channel of width H and a depth M H, so that we can established a bi-dimensional flow into the rectangular channel. Due to the internal heat generation of the strip (with a uniform volu_ an important fraction of the heat transfer metric rate q),
O. Bautista, F. Me´ndez / Applied Thermal Engineering 26 (2006) 2201–2208
net heat transferred to the cooling fluid and to the substrate. Eqs. (2) and (3) can be combined to yield 2 1=3 _ _ 2 H L qh q_ 00 qh 1 0 : ð4Þ and DT w DT F c _ k 6Pe kw qh
u (y) y H
8
T 0
internal heat generation q
h
x
q
L
Fig. 1. Schematic diagram of the physical model.
occurs between the discrete source and the laminar cooling flow. The other fraction transfers to the substrate. Because in practical cases, the ratio h/L is very small in comparison with unity, we assume that the right and left faces of the source are practically adiabatic. Therefore, the heat loss to the substrate, here denoted q0, is only present through the lower face of the chip. It is really important to note that we assume, for simplicity, a uniform heat loss to the substrate. It is indispensable to introduce a more realistic model like a nonuniform variation of heat to the substrate. However, the present analysis can be readily generalized to include this condition. Following the above considerations, the upper face of the strip contacts a laminar cooling flow with a well-known fully developed velocity profile given by y y 2 1 dP 2 u ¼ 6 u H ; ð1Þ ; with u¼ H H 12l dx where u represents the mean value of the velocity. In order to obtain the appropriate scales of the problem, we use an order of magnitude analysis on the energy equation of the cooling fluid in order to show that the ratio of the thickness of the thermal boundary layer to the length of the strip can write as " #1=3 2 d H 1 ; ð2Þ L L 6Pe where Pe = Re Pr and represents the well-known Peclet number, Re = u H/m denotes the Reynolds number and Pr is the Prandtl number. q and l are the density and the dynamic viscosity, respectively. On the other hand, a global balance of thermal energy establishes that the heat flux from the strip to the cooling flow can write as _ kw qh
DT w DT F k þ q_ 000 ; h d
2203
ð3Þ
where k and kw are the thermal conductivities of the discrete source and fluid, respectively. DTw is, in the transverse direction, the characteristic temperature drop of the strip and DTF is the characteristic temperature drop in the cooling fluid. q_ 000 denotes heat flux to the substrate. On the other hand, the first term in Eq. (1) corresponds to the thermal energy generated in the strip, while the third one is the
Using these relationships, we also obtain that " #1=3 2 DT FM a kw h H 1 2 with a ¼ c ; e L 6Pe DT w k L
ð5Þ
e is the aspect ratio of the strip, e = h/L, which is assumed to be very small compared with unity and c a constant of order unity to be given later. In the above relationship (5), we define DTFM as the maximum value of the characteristic temperature drop in the cooling fluid and replacing q_ 000 ¼ the relationship (4), we obtain that DT FM ¼ 0 into _ c qh k
H 2L 6Pe
1=3
.
We will use the above relationship to adimensionalize the temperature distribution. Furthermore, parameter a is the nondimensional longitudinal heat conductance of the discrete source and is a measure of the longitudinal heat conduction effects in this electronic device on the cooling process. This parameter can assume any value depending on the geometrical and physical properties of the heat-conducting element and the fluid. For instance, from the order relationship, Eq. (5) with a/e2 1, the transverse temperature variations of the strip are very small compared with the temperature differences in the fluid, that is DTw DTF. In a previous work of Me´ndez and Trevin˜o [13], we called this limit as the thermally thin wall regime. For values of a/e2 1, the temperature variations in both directions of the strip now are very important and are of the same order of magnitude of that temperature differences in the fluid. This is the thermally thick wall regime. In this limit and with e 1, the longitudinal heat conduction through the strip is very small, neglecting in a first approximation. Both limits will be explored in the next sections. With the above discussion, we can introduce the following nondimensional variables for the discrete source x yþh Tw T1 v¼ ; z¼ ; hw ¼ with L h DT FM 2 1=3 _ qh H L DT FM ¼ c ð6Þ k 6Pe and for the fluid y T T1 n¼ ; h¼ : d DT FM
ð7Þ
Therefore, the nondimensional energy balance equations for the strip and fluid take the form a
o2 hw a o2 hw þ 2 þ1¼0 e oz2 ov2
ð8Þ
and n
oh o2 h ¼ : ov on2
ð9Þ
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We derive Eq. (9) taking into account that Pe 1, which means that the longitudinal heat conduction effects in the cooling flow are negligible, approximation widely used for laminar flow in ducts and channels. Furthermore, the presence of a thermal boundary layer adjacent to the upper surface of the strip, permit us to linearize the velocity profile. The details are given lines below. The corresponding boundary conditions for the strip are,
proportional to the transverse distance y, when the temperature field is confined inside of the zone of the velocity field, which occurs frequently for large values of the Prandtl number. However, this approximation gives excellent results even for Prandtl numbers of order unity. In particular, the Kernel in Eq. (18) is given by
ohw ¼0 ov
where, hwl is the value of the nondimensional temperature at the leading edge of the heated strip. The assumed value for the constant c in the definition of a is c = 31/3/C(1/3). In general, Eqs. (8), (10), (11) and (18) can numerically integrate. With this purpose, in the following sections, we obtain asymptotic solutions for the thermally thin and thick wall regimes. We also include the numerical solution for the thermally thin wall regime.
at v ¼ 0; 1
ð10Þ
and ohw e2 ¼c at z ¼ 0; ð11Þ oz a where the nondimensional parameter c is defined as c¼
q_ 000 : _ qh
ð12Þ
This nondimensional parameter represents the competition between the loosed thermal energy to the substrate and the generated thermal energy by the discrete heat source. Obviously, this nondimensional parameter is always less than unity. In order to complete the boundary conditions, we add that hðv; n ! 1Þ ¼ hðv ¼ 0; nÞ ¼ 0
ð13Þ
and moreover we require an additional condition at z = 1 (n = 0), which corresponds to accept the continuity of the temperatures and heat transfer rates ohw e2 oh hðv; n ¼ 0Þ ¼ hw ðvÞ and ¼ ; ð14Þ oz z¼1 a onn¼0 where the temperature gradient oh/onjn=0 represents the nondimensional heat flux from the upper surface to the cooling fluid. In this form, the relationships (14) define the matching conditions between both zones. In order to obtain the solution of the energy equation, Eq. (9), we propose the quasi-similarity variables given by h ¼ hw u
and f ¼
n v1=3
;
ð15Þ
ð16Þ
with the corresponding boundary conditions uð0Þ ¼ 1
and uðf ! 1Þ ¼ 0:
In addition, Eq. (14) can be written as
Z hw ohw e2 c 0 0 ¼ hwl þ Kðv; v Þdhw : oz z¼1 a v1=3 hwl
1=3
;
ð19Þ
3. Thermally thin wall regime (a/e2 1) For very large values of a/e2 compared with unity, the relationship (5) dictates that the temperature variations in the normal direction of the strip are insignificant. Therefore, in a first approximation, hw only depends on the coordinate v. In this regime, the energy equation for the chip, Eq. (8), can be integrated along the normal coordinate and after applying the appropriate boundary conditions (10), (11) and (18), we obtain
Z hw d2 hw 1 0 0 a 2 þ 1 ¼ 1=3 hwl þ Kðv; v Þdhw þ c: ð20Þ v dv hwl This equation must be solving with the adiabatic conditions for the lateral surfaces of the strip given by Eq. (10). In the following subsections we present asymptotic solutions for very large values of a, that is a 1 and for values of a ! 0, respectively. In both cases, we accept that the limit a/e2 1 is valid. In addition, we obtain numerical solutions for these limits in order to validate the asymptotic perturbation scheme. On the other hand, we solve Eq. (20) using numerical techniques reported elsewhere (see for instance, Me´ndez and Trevin˜o [13]). 3.1. Asymptotic limit a 1
reducing Eq. (9) to o2 u 1 2 ou ou ¼ vf ; þ f ov of2 3 of
Kðv; v0 Þ ¼ ð1 v0 =vÞ
ð17Þ
ð18Þ
The right-hand side of this last boundary condition, Eq. (18), can easily derive with the aid of the Lighthill’s approximation, [14]. This integral approach takes into account that the longitudinal velocity component, u, is still
In order to study this regime, the temperature of the chip is obtaining with the aid of a regular expansion technique, using the inverse of a as a small perturbation parameter. For very large values of the parameter a, the nondimensional temperature of the discrete source, hw, changes very little in the longitudinal direction (of order a1). Therefore, we assume that the nondimensional temperature of the strip can be expanding as 1 X 1 hwj ðvÞ: ð21Þ hw ¼ aj j¼0 Introducing the above relationship (21) into the nondimensional governing equation for the chip (20), we obtain after
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collecting terms of the same power of a, the following set of equations d2 hw0 ¼ 0; dv2
ð22Þ
Z h0 d2 hw1 1 0 0 þ 1 ¼ 1=3 h0l þ Kðv; v Þdhw0 þ c; v dv2 h0l Z hi d2 hwðiþ1Þ 1 0 0 ¼ h þ Kðv; v Þdh for i > 1; il wi v1=3 dv2 hil with the following adiabatic boundary conditions dhwj ¼ 0; for all j: dv
ð23Þ ð24Þ
3.2. Asymptotic limit a ! 0
Integration of Eq. (22) with the corresponding boundary conditions (25) gives hw0 = C0, where C0 is an integration constant and must be determined by solving the next R 1 higher order Eq. (23). Integrating Eq. (23) in the form 0 dv and using the adiabatic conditions at both edges of the strip, yields that C0 = 2 (1 + c)/3. This procedure can use to obtain the solution to higher orders. Therefore, the solution to Eq. (23) is v2 9 v2 þ v5=3 C 0 þ c þ C 1 ; 2 10 2
where C1 is a constant given by 2 2 18 5 2 2 2 C 1 ¼ B 2; B ; C 0 B 2; c; 8 3 42 3 3 8 3
in addition, the averaged nondimensional temperature hw , up to terms of order 1/a2, can write as Z 1 1 1 hw1 þ 2 hw ¼ hw2 þ hw dv ’ hw0 þ a a 0
2 1 9 1 C 0 ð1 cÞ þ C 1 ¼ ð1 cÞ þ 3 a 10 2 1 27 5 2 9 9 2 þ 2 B ; C 0 þ C 1 B 2; ð1 cÞ þ C 2 . a 1400 3 3 10 88 3 ð31Þ
ð25Þ
v¼0;1
hw1 ðvÞ ¼
2205
ð26Þ
ð27Þ
where B(l, n) represents the complete beta function. Following the same procedure, the second order solution is 9 9 2 11=3 27 5 2 5=3 B ; C 0 v10=3 h2 ðvÞ ¼ C 1 v B 2; v þ 10 88 3 140 3 3 9 2 þ B 2; cv11=3 þ C 2 ; ð28Þ 88 3 with C2 given by 18 5 2 1 2 3 5 2 B 2; B ; C0 C2 ¼ B ; 42 3 3 4 3 7 3 3 1 2 3 2 11 2 ; B 2; c þ B 2; B 4 3 52 3 3 3 3 5 2 10 2 3 2 11 2 ; C 0 B 2; B ; c: B ; B 28 3 3 3 3 52 3 3 3 ð29Þ Therefore, up to terms of second order, the nondimensional discrete source temperature becomes
2 1 9 v2 5=3 C 0 v ð1 cÞ þ C 1 hw ¼ ð1 cÞ þ 3 a 10 2 1 9 27 5 2 10=3 5=3 C1v þ C0B ; v þ 2 a 10 140 3 3
9 2 11=3 B 2; v ð1 cÞ þ C 2 ; ð30Þ 88 3
In the thermally thin wall regime, we have the limiting case of a ! 0, but with a/e2 1. Here, the longitudinal heat conduction in the strip is very small and can neglect except in regions very near to the edges of the strip, where the presence of local thermal boundary layers are indispensable to understand it. However, the structure of these regions has only a local influence. Therefore, in the present analysis we exclude to consider these thermal boundary layers. Hence, Eq. (20) with a = 0 transforms to Z v dh0 1=3 ð1 cÞv ¼ Kðv; v0 Þ wa0 dv0 : ð32Þ dv0 0 The solution of foregoing equation can obtain with the aid of the well-known Abel’s Integral transform. The nondimensional temperature of the strip, hw(v), is then hwa0 ¼
3ð1 cÞ 1 v3 ; B 13 ; 23
ð33Þ
moreover, the nondimensional average temperature is hwa0 ¼ 9ð1 cÞ : 4B 13 ; 23
ð34Þ
4. Thermally thick wall limit (a/e2 1) In this other case, the longitudinal heat conduction is also very small and therefore can neglect. The energy balance Eq. (8) for the strip then reduces to o2 hw e2 ¼ : 2 oz a
ð35Þ
Eq. (35) has to be solving with the boundary conditions ohw e2 ð36Þ ¼ c oz z¼0 a and ohw e2 1 ¼ 1=3 oz z¼1 av Z hwl þ 0
hw
0 e2 1=3 dhw 0 dv ¼ ð1 cÞ: 0 dv a
ð1 v0 =vÞ
ð37Þ
Integrating Eq. (35) in the normal direction and using both conditions (36) and (37), we obtain that
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Z v ohw e2 1 dh0 ¼ 1 1=3 ð1 v0 =vÞ1=3 w0 dv0 z : v oz a dv 0 Therefore, the nondimensional temperature is
e2 1 hw ¼ hwu þ ð1 z2 Þ cð1 zÞ ; a 2
ð38Þ
ð39Þ
where hwu is the nondimensional temperature at the upper surface of the strip, given by hwu ¼
3ð1 cÞ 1 v3 ; B 13 ; 23
ð40Þ
The nondimensional average temperature is then 9ð1 cÞ 1 e2 hw ¼ 1 2 þ : 3 a 4B 3 ; 3
ð41Þ
It is interesting to note that in the limit of e2/a ! 0, the total thermal energy of the strip in this regime is the same as for the case of a ! 0, for the thermally thin wall regime given by Eq. (34).
Fig. 3. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.01 and different values of the nondimensional parameter a.
5. Results and concluding remarks In Figs. 2–7, we present the most relevant numerical and asymptotic results derived from the present analysis. In all calculations carried out in this work, we use the following data: T1 = 300 K, kw/k = 100 and Pr = 0.7. Also as illustration, we assume representative values for the characteristic geometric scales of the system: the horizontal length of the discrete heat source was 5 cm, its thickness 1 cm and the vertical distance of the channel was H = 2 cm. In order to validate the regular perturbation scheme to study the thermally thin wall regime, Eq. (20) was numerically integrated together with the adiabatic boundary conditions, Eq. (10); using the well-known Runge–Kutta technique. In order to make it, the boundary value problem is transFig. 4. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.05 and different values of the nondimensional parameter a.
Fig. 2. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.0 and different values of the nondimensional parameter a.
formed to an initial value problem with the given initial conditions at the leading edge of the chip for the nondimensional temperature and its gradient. Together with the above procedure, a conventional shooting-iteration method was applied, due to the unknown value of hwl for given values of the nondimensional parameters a and c. For the thermally thin regime, Figs. 2–5 show the comparison between the asymptotic solutions—given by relationship (30), and the numerical results based on the above iteration scheme for the nondimensional temperature distribution, hw, as a function of the nondimensional coordinate v, different values for the parameter a (=3.0, 5.0, 10.0 and 100) and including the variation of the nondimensional parameter c (=0.0, 0.01, 0.05 and 0.1). In all figures, we found that for a’s smaller than 3, the differences
O. Bautista, F. Me´ndez / Applied Thermal Engineering 26 (2006) 2201–2208
Fig. 5. Numerical solution of the strip temperature hw as a function of the longitudinal coordinate v for c = 0.1 and different values of the nondimensional parameter a.
Fig. 6. Nondimensional average temperature differences, ðT w ðvÞ T 1 Þ, as a function of the Peclet number, Pe, and for different values of the nondimensional parameter c.
between both solutions are increasing and are practically independents of the assumed values of the parameter c. Also, it is interesting to note that the asymptotic solutions offer good results even for values of a of order 3. On the other hand, the deviations between both solutions can not necessarily assign to an error of the perturbation scheme. In this domain of a O(1) or smaller values than unity, the influence of the temperature gradients in the transverse direction begins to be important and the first corrections of the thermally thick regime should be taking into account. In these figures, the numerical and asymptotic solutions show that for decreasing values of a, the nondimensional temperature hw decreases strongly at the leading edge of the strip and increases at the trailing edge. The influence of c does not change the above behavior of
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Fig. 7. Maximum difference of the nondimensional strip temperature, DTwm, as a function of the Peclet number, Pe, and different values of the nondimensional parameter c.
the temperature; however, for increasing values of this parameter, the temperature hw decreases, because the parameter c measures the influence of the loosed heat to the substrate. The above results confirm the observation made by Bejan [1]: higher temperatures lead to higher internal heat generations, which derive again to higher temperatures. This thermally unstable limit behavior must be avoided in this type of applications. Hence, the set of Figs. 2–5 clearly reveal the fundamental role of the parameters a and c on the temperature hw and their corresponding longitudinal gradients. Thus, the criterion to operate with a better electrical performance depends on which tolerance is accepted: a strip working with large nondimensional longitudinal temperature differences (decreasing values of a) and a low average temperature hw or a strip that can attenuate the nondimensional longitudinal temperature gradients with high average temperature hw. Both cases controlled by the influence of the parameter c. In order to clarify the above aspect, we present lines below other numerical results to complement these observations. Following the above comments, in Fig. 6 we show the corresponding nondimensional average temperature differ_ =kÞ, plotted as a function of the ences, ðT w ðvÞ T 1 Þ=ðqH Peclet number, Pe, and three different values of the parameter c. The influence of Peclet number is clear: for increasing values of the cooling effect the average temperature of the strip is drastically reduced. In addition, this reduction also increases for increasing values of c. In this case, the presence of the loosed heat to the substrate operates as a favorable effect. However, the heating of the substrate can yield undesirable effects on the electric performance of the system. In this figure, we show also analytical results and the corresponding comparison between both solutions provides a very good agreement.
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Finally, in Fig. 7 we show for the thermally thin wall regime, the maximum difference of the nondimensional strip temperature between both adiabatic edges, DTwm = DTFMDhw, as a function of the Peclet number Pe, three different values of the parameter c (=0.01, 0.2 and 0.3) and also, three different values of the internal heat generation _ For increasing values of the Peclet number, DTFMDhw q. decreases, being more notable this effect for increasing values of c. On the other hand, the influence of the internal heat generation changes drastically the levels of these temperature differences in the strip. Because the maximum characteristic temperature DTFM depends sensibly on Peclet number, we can conclude that the numerical calculations show that DTFM is the dominant factor that yields the continuous decrement of the nondimensional function DTwm. In the present work, we have carried out an analytical and numerical analysis to study the cooling mechanism of a discrete heat source embedded in a rectangular channel. Inside this channel is circulating a laminar flow, which permits to transfer heat from the heating-strip to the fluid. Because the substrate is not properly adiabatic, the direct contact between the heated strip and the substrate permits a finite heat transfer rate. Here, this mechanism is modulated by the introduction of the nondimensional parameter c. Therefore, in terms basically of the nondimensional parameters c and a, the thermal performance and characteristics of this typical device are well defined. Acknowledgements This work has been supported by a research grant no. 43010-Y of Consejo Nacional de Ciencia y Tecnologı´a at Mexico.
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