An analytical approach for a Hiemenz flow in a porous medium with heat exchange

International Journal of Heat and Mass Transfer 54 (2011) 3613–3621

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An analytical approach for a Hiemenz flow in a porous medium with heat exchange Max A.E. Kokubun, Fernando F. Fachini ⇑ Combustion and Propulsion Laboratory, National Institute for Space Research, Cachoeira Paulista, São Paulo 12630-000, Brazil

a r t i c l e

i n f o

Article history: Received 20 November 2010 Received in revised form 5 March 2011 Available online 12 April 2011 Keywords: Hiemenz flow Stagnation-point flow Porous medium Heat transfer

a b s t r a c t In this work, a Hiemenz flow established inside a semi-infinite low porosity medium with heat exchange is analyzed analytically. Heat is supplied to the system through two different wall conditions: a constant wall temperature, and a constant wall heat flux. Local thermal non-equilibrium is considered, and the two-equation model is used to consider heat exchange between gas and solid phases. The flow is analyzed through a non-Darcian model, in which viscous and convective terms are considered in the Darcy pressure equation. The results obtained point out the importance of the inner zone (close to the wall) analysis in the case of a constant wall heat flux. From the analysis, a new dimensionless parameter, j, emerges, which gathers information of the transport phenomena in gas and solid phases, and it is responsible to determine the flow field. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The interest on flows inside porous media has grown considerably in the last decades, since this geometry arises in many different systems, ranging from natural to man-manufactured technological ones. Oil wells, underground aquifers, chemical catalysts and wood are some examples of porous media [1,2]. Therefore, the fluid flow and heat transfer study in those systems is a subject of great interest in many branches of engineering and science. The governing equations of a fluid flow in a porous medium are non-linear and difficult to handle analytically. So, problems concerning flows in porous medium are usually solved by numerical methods. Many numerical analysis related to isothermal flows in porous media have been conducted. Nevertheless, analytical analysis dealing with such problems are not extensive in the current literature. Siddiqui et al. [3] used inverse methods to obtain exact solutions of several types of two-dimensional steady, viscous and incompressible fluid flows through porous medium. They compared the obtained solutions with known solutions in the absence of the porous medium. Wu et al. [4] studied an impinging flow configuration to analyze transition in flow behavior, from the classical Hiemenz flow to the local solution of the Brinkman equation [5]. Also, by considering small permeability values, Wu et al. [4] utilized the perturbation technique to obtain asymptotic solutions for velocity and pressure fields. They found a new dimensionless parameter relating the square of two lengths, the classical boundary layer thickness for a high Reynolds number flow and the boundary layer ⇑ Corresponding author. Tel.: +55 12 3186 9266; fax: +55 12 3101 1992. E-mail addresses: [email protected] (M.A.E. Kokubun), [email protected] (F.F. Fachini). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.03.021

thickness associated with the viscous attachment at the pore size. Kumaran et al. [6] also analyzed an impinging flow. By using an implicit perturbation technique, they extended Wu et al. [4] previous analytic results by considering large permeability. Heat transfer in porous media has also been a subject of interest, as it can be used as heat sinks for heated impermeable solids. The large contact surface between solid and fluid phases enhances considerably the heat transfer. This feature allows heat removal from heated bodies to be more efficient than free jet impingement systems [7,8]. Under the consideration of heat transfer, numerical studies are vastly found in the literature. Attia [9] studied the effect of the porosity in a stagnation-point flow impinging on a permeable surface by using a porosity parameter (inversely proportional to the porosity). Results indicated that by increasing the porosity parameter, and hence, reducing the medium porosity, causes a decrease on the thickness of both thermal and velocity layers and an increasing in the heat transfer at the permeable surface. Jiang and Ren [10] numerically investigated the forced convection heat transfer inside a porous medium by considering a thermal nonequilibrium model, known as two-equations model. They also analyzed the effects of viscous dissipation, appropriate boundary conditions, thermal dispersion and geometric properties of the medium, and compared the results with experimental data. The analysis showed that it is possible to predict numerically the convection heat transfer in porous medium by using the thermal nonequilibrium model with the ideal constant wall heat flux as a boundary condition. Jiang and Lu [11] analyzed thermal boundary characteristics at the contact interface between a porous medium and an impermeable wall subject to a constant heat flux on the upper surface, with and without the consideration of a thermal contact resistance. They considered both a finite-thickness and a zero-thickness wall. Alazmi and Vafai [12] investigated different

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forms of the constant wall heat flux boundary conditions. Effects of variable porosity and thermal dispersion were also studied. A comparison between different models of heat transfer at the wall was performed by the authors. The cooling of heated elements in a parallel-plate channel with porous inserts at the adiabatic walls (upper and lower plates) and under a laminar forced convection flow was studied numerically by Yucel and Guven [13]. Their results showed that heat transfer can be enhanced by using highthermal-conductivity porous inserts and that the insertion of heated elements and a porous matrix cause a rapid increase in the pressure drop along the channel with increasing the Reynolds number. Marafie et al. [14] studied numerically non-Darcian effects on the mixed convection heat transfer in a metallic porous block with a confined slot jet. The effectiveness of the presence of the porous block, in the scope of heat transfer performance, was confirmed. Also, the effects of the Richardson number, the Darcy number, heights of the channel and of the porous layer, medium porosity and thermal conductivity ratio on the heat transfer characteristics were extensively analyzed. Several applications to the jet impingement system were also exhibited. Numerical studies on stagnation-point flow in porous medium are also found extensively in the literature. Jeng and Tzeng [7] investigated numerically the impinging cooling of porous metallic foam heat sink. Their simulation results revealed that when the Reynolds number is low, the maximum Nusselt number occurs at the stagnation point. However, when the Reynolds number increases, the maximum Nusselt number moves downward, to the narrowest part between the recirculation zone and the heating surface. Calculations were performed for a highly porous medium. Dórea and de Lemos [15] performed simulations of a laminar jet impinging on a flat plate covered by a porous layer at the wall. They considered two macroscopic models for analyzing energy conservation, the one-energy equation model, which considers a local thermal equilibrium hypothesis, and the two-energy equation model, where distinct conservation equations for the fluid phase and for the porous matrix follow the local non-thermal equilibrium assumption. Thermal physical properties were varied and their influences on the energy transport were obtained. The results showed that for low porosities (low permeabilities), for thin porous layers and for high values of solid-to-fluid thermal conductivity ratio, a different distribution of local Nusselt number at the wall is calculated depending on the energy model applied. Differently from numerical studies, analytical results concerning heat transfer are scarce in the open literature. Lee and Vafai [16] provided an extensive analytical characterization of forced convective flow through a channel filled with a porous material. A heated wall was considered to provide heat flux transversely to the flow. They obtained exact solutions for the temperature profiles of solid and fluid phases, and classified heat transfer characteristics into three regimes, each of them dominated by a physical heat transfer mechanism: fluid conduction, solid conduction and internal heat exchange between solid and fluid phases. In the present work, an analytical study is performed on a stagnation-point flow inside a porous medium with heat exchange between phases and with heat provided by the wall at the stagnation-point. Heat supply is considered in two similar situations: a prescribed wall temperature, and a prescribed constant wall heat flux. A procedure using asymptotic theory is proposed in order to solve the problem by assuming a very large value for the interphase heat exchange and a low porosity medium. Under these assumptions, the problem presents two different length scales, ls and lg, associated with solid and gas phase thermal conductivities, respectively. These characteristic length scales are identified [17,18] and the results for temperatures and flow field are obtained for each scale, providing an analytical solutions for the problem.

2. Length scales Based on the hypotheses mentioned before, the proposed problem presents two different length scales. These length scales are illustrated in a schematic representation of the problem in Fig. 1.  1 Þ, where  In the region of order ls ¼  ks =ðq1 cp v ks is the solid phase thermal conductivity, q1 is the injected gas mass density, cp is the  1 is the injection velocgas specific heat at constant pressure and v ity of the gas, thermal equilibrium between solid and gas phases occurs, due to the high value of the interphase heat exchange. This region corresponds to the solid phase thermal diffusivity (modified) length scale. In the region near the wall, thermal non-equilibrium between phases occurs. This region of non-equilibrium is of  1 Þ and it corresponds to the gas phase thermal order lg ¼  kg =ðq1 cp v diffusivity length scale. The ratio between these length scales is ls =lg ¼  ks = kg  C  1. This feature ensures separation of scales and allow us to perform an analysis based on asymptotic theory. More detailed discussion of the importance of the length scales is found elsewhere [17,18]. 3. Model and mathematical formulation An impinging flow configuration inside a porous medium is assumed for the studied system in this work, considering z the normal coordinate and  x the tangential coordinate to the wall. Conservation of mass, momentum, gas-phase energy and solid-phase energy are considered and boundary-layer approximation for temperatures are assumed, leading to a z dependent model. Since we are assuming a negligible small gas thermal expansion (constant density) a state equation for the impinging gas is not necessary. The porous medium acts as a barrier for the flowing gas. Such resistance is quantified by the Darcy term, which is dependent on the medium permeability K and on the flow velocity. Additionally, we must consider the interphase heat exchange between gas and solid phases, linearly related with the local temperature difference between phases. The thermodynamic and transport properties are assumed to be constant in the whole domain. The steady-state volume-averaged mass, momentum and energy conservation equations (omitting the volume-averaging notation) under the above assumptions are given by

 @ v @u þ ¼ 0; @ x @z      @u @u @p @2u u qu  þ qv  ¼ e  þ l 2  el ; @x @z @x @z K  @ v @ v @p @ 2 v v qu  þ qv  ¼ e  þ l 2  el ; @z @x @z @z K @T @2T eqv cp g ¼ ekg 2g þ hv ðT s  T g Þ; @z @z 2 d Ts  0 ¼ ð1  eÞks 2  hv ðT s  T g Þ; dz

ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ

ls

lg

z x

T0

Q

Fig. 1. Schematic diagram at the vicinity of the stagnation-point.

M.A.E. Kokubun, F.F. Fachini / International Journal of Heat and Mass Transfer 54 (2011) 3613–3621

where e is the medium porosity and hv is the volumetric surfaceconvection heat exchange coefficient. The boundary conditions far from the wall ðz ! 1Þ are given by

  du  ¼ x  ; u dx 1

v ¼ v 1 ;

Tg ¼ Ts ¼ T1

and at the wall, z ¼ 0:

v ¼ u ¼ 0;

¼p 0 ; p

Tg ¼ Ts ¼ T0:

The condition at the wall ðz ¼ 0Þ for the energy problem depends on whether we are considering a constant temperature value of the wall or a constant heat flux. 3.1. Non-dimensionalization  =v 1; v  The non-dimensional variables .  q=q1 ¼ 1; u  u =ðq1 v  21 Þ; hg  T g =T 1 ; hs  T s =T 1 ; x   pp x=ls ; z  z=ls ,  1 Þdu  =d and the non-dimensional strain-rate a  ðls =v xj1 are used in the present analysis, where ls is a characteristic length scale  1 Þ, and a modified variable change [19] is given by ls   ks =ðq1 cp v performed in order to analyze the region near the stagnation-point in a scale ls:

v =v 1 ;

u ¼ axUðzÞ; p0  p ¼

v ¼ a1=2 f ;

   1 1 2FðzÞ x2 þ Pr a2 1 þ ; 2 a jC

g ¼ a1=2 z 2

in which Pr is the Prandtl number, j  aK=ls is the non-dimensional porous medium permeability and C   ks = kg  1. The pressure expression is based on the classical expression [19] with an extra term, 1/(jC), to consider the Darcy effect. In this work, the medium in which the flow is established is considered to be of low porosity, satisfying e2  1, such that the permeability is also low. Under this assumption, it is assumed that j  1/C2, or 1/(jC) = bC, where b is a unitary order parameter that determines the type of porous medium used. In addition, the non 1 Þ2 , is asdimensional interphase heat exchange, N v ¼  ks hv =ðq1 cp v sumed to be of order C, being re-scaled by Nv/a = nvC, where nv is a unitary order parameter that collects information about the thermal interaction between solid phase and gas phase. Eqs. (1)–(5) become

df U¼ ; dg

ð6Þ

 2 3 2 Pr d f d f df df þ f   CebPr ¼ ePrð1 þ bCÞ; dg2 dg dg C dg3

ð7Þ

2

Pr d f df dF þf  CebPrf ¼ ePrð1 þ bCÞ ; dg dg C dg2

ð8Þ

e d 2 hg dhg þ ef ¼ Cnv ðhs  hg Þ; C dg2 dg

ð9Þ

flow at the porous walls and a thermal boundary layer. The first boundary layer is the classical viscous boundary layer, and its effects are accounted by the first term in the left side of Eqs. (7) and (8). It has an order of magnitude of C1, as pointed by Eqs. (7) and (8). The second boundary layer is related to the fluid attachment at the porous wall, and its effect are accounted by the choice of j. The thickness of the thermal boundary layer is determined by the order of C1. At the same time that the viscous and thermal boundary layers are considered, it is necessary to take into account the thermal equilibrium or thermal non-equilibrium between the gas and solid phases. The thermal condition between the phases is controlled by the parameter Nv. The choice of Nv determines the thickness of the thermal equilibrium and non-equilibrium zones. In the present work, the non-dimensional parameters j and Nv are chosen in such a way that the three distinct boundary layers and the thermal nonequilibrium zone have the same thickness, C1, hence, these effects are coupled in the inner zone. If a lower porosity medium was considered, by using, for instance, j  C3 the Darcy flow would be observed practically in the whole inner zone. Therefore, the (impermeable wall) viscous boundary layer would not have the same thickness of the thermal boundary layer (inner zone). The viscous effects of the porous medium internal area would not permit the establishment of the (impermeable wall) viscous boundary layer in the inner zone (thickness of order of the C1). The viscous boundary layer would occur in a zone of order of the C3/2 attached to the impermeable wall. If, on the other hand, a higher porosity medium was considered, by using j  C1, the flow in the outer zone would be governed by the Darcy equation with the inertia terms. In this situation, the porous–fluid viscous interaction would be minimized, and in the scale of C1 the viscous boundary-layer due to the impermeable wall would be observed. The choice of the value of j (a non-dimensional parameter that relates the permeability of the medium with the thermal-physical properties of gas and solid phases) determines the length scale of the viscous boundarylayers. 4. The constant wall temperature case If we consider a wall with a prescribed temperature h0, the energy equations must obey a boundary condition given by hs = hg = h0 at g = 0, together with the boundary conditions mentioned in the previous section. The problem of unitary order, corresponding to the solid-phase thermal diffusivity length scale, is described first, then the problem of order C1, corresponding to the gas thermal diffusivity length scale. 4.1. Outer zone: problem of order unity

2

d hs ð1  eÞ 2 ¼ Cnv ðhs  hg Þ: dg

3615

ð10Þ

The boundary conditions are hs = hg = 1 and df/dg = U1 for g ? 1, and f = df/dg = 0 at g = 0. Two different energy boundary conditions at the wall are analyzed: a constant wall temperature and a constant wall heat flux. In a region of the order of unity, corresponding to ls, the viscous effects and the thermal non-equilibrium among flowing gas and porous medium are not observed. These effects occur only in a region of order C1 near the wall, corresponding to lg, and another boundary-layer expansion is necessary to describe them. It is important to point that the studied system present three different boundary layers: one due to viscous attachment of the flow at the impermeable wall, one due to the attachment of the

In the outer zone, corresponding to the solid-phase thermal diffusivity [17,18], the flow is basically governed by the pressure gradient (Darcy flow). The equations to be analyzed to obtain the momentum variations and the pressure field in this region are given by

 2 3 2 Pr d f d f df df þ f   CebPr ¼ ePrð1 þ bCÞ; dg2 dg dg C dg3

ð11Þ

2

Pr d f df dF þf  CebPrf ¼ ePrð1 þ bCÞ : dg dg C dg2

ð12Þ

The solution for the momentum function f is obtained from Eq. (11) and may be expressed as f = f(0) + C1f(1) + O(C2). Substituting it in Eq. (11) and collecting equal powers of C, two differential equations are found for the first two terms:

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0 fð0Þ ¼ 1; 0 ebPr fð1Þ

ð13Þ  2 00 0 ¼ ePr þ fð0Þ fð0Þ  fð0Þ

The solutions of Eqs. (17) and (18) give the pressure field in the outer zone as

ð14Þ

in which the prime denotes differentiation with respect to g. The 0 boundary conditions are f(0) = f(1) = 0 for g = 0 and fð0Þ ¼ 1 and 0 fð1Þ ¼ U 1 for g ? 1. The value U1 comes from the expansion U1 = 1 + C1U1 + O(C2). Solving Eqs. (13) and (14) with the appropriate boundary conditions, we obtain

FðgÞ ¼ 

g2 2

f ðgÞ ¼ g  C

1  ePr g þ OðC2 Þ: ebPr

ð15Þ

If one evaluates all terms of the proposed series solution of Eq. (11), then it will be seen that all of them have a linear form, and hence, the first and the second terms in the left side of Eq. (11) are null for every evaluation performed. If one notes that those terms are the viscous terms, then the analysis of the problem in the outer zone is conducted to the Darcy equation with corrections given by the third term on the left side of Eq. (11). The horizontal component of the velocity in the outer zone is given by U, and is obtained by deriving Eq. (15):

UðgÞ ¼ 1  C1

1  ePr þ OðC2 Þ: ebPr

F 0ð0Þ ¼ fð0Þ ; F 0ð1Þ ¼

0 fð0Þ fð0Þ

ebPr

ð17Þ  fð1Þ 

F 0ð0Þ b

ð18Þ

which must satisfy the boundary conditions F(0) = F(1) = 0 at g = 0.

ð19Þ

2

1 d h d h dh þc 2þf ¼0 dg dg C dg2

ð20Þ

which was obtained by summing Eqs. (9) and (10) and using the definition c  (1  e)/e. If one express the solution as an expansion of the form h = h(0) + C1h(1) + O(C2), the following differential equations are obtained, when equal powers of C are collected:

ch00ð0Þ þ fð0Þ h0ð0Þ ¼ 0; ch00ð1Þ þ fð0Þ h0ð1Þ ¼ h00ð0Þ  fð1Þ h0ð0Þ

ð21Þ ð22Þ

which must satisfy the boundary conditions h(0) = h0 and h(1) = 0, at g = 0, and h(0) = 1 and h(1) = 0 for g ? 1. The solutions of Eqs. (21) and (22) give the temperature in the outer zone:

ð16Þ

The result exhibited by Eq. (16) points out to a constant value of U in the whole region of order of unity (solid characteristic length scale) as shown in Fig. 2. This value is below one, U1 < 0, which indicates the influence of the solid-phase on the strain-rate of the flow. Most of the analysis on impinging flow configurations established in porous media usually considers f0 j+1 = 1 as a boundary condition. However, this condition comes from the classical Hiemenz solution of an impinging flow without the porous medium. However, if one considers the porous medium, the resistance exerted by the tortuous channels requires a correction term in velocity fields, as pointed by Eqs. (15) and (16). The pressure field is obtained from Eq. (12) and considered as F = F(0) + C1F(1) + O(C2). Substituting the proposed solution in Eq. (12) and collecting equal powers of C, two differential equations are found for the first two terms:

g2 þ OðC2 Þ: ebPr

In the outer zone, gas phase and solid phase are in thermal equilibrium due to the high value of the interphase heat exchange. Under this condition, it is assumed hg  hs = h in that region, and one must solve the following equation: 2

1

þ C1

g

!

H

hðgÞ ¼ h0  Herf pffiffiffiffiffiffi þ C1 2c 2c sffiffiffiffiffiffi  2 1  ePr 2 1þ  c geg =2c þ OðC2 Þ; pc ebPr

ð23Þ

where for the sake of compactness was defined by H  (h0  1) > 0. In the outer zone, both gas and solid phases experience an exponential increase in the temperature, due to the heated wall. Since the interphase heat exchange is assumed to be high, thermal equilibrium is observed in the outer zone. In the present case, the temperature in the wall is assumed to be known. Plots for three different temperatures in the wall are presented in Fig. 3. The exponential increase in the temperature downward to the wall is observed clearly. The analysis of the limit cases of a very high porosity medium and of a very low one for the leading order term of Eq. (23) shows

hðgÞ ¼



1

for c  1;

h0

for c  1:

The first solution represents the limit in which the porous medium is eliminated. In this limit, c  1, the temperature in the outer zone is the same as the temperature far from the region. The temperature variations will occur in the inner zone (gas phase characteristic length scale, lg). The second solution corresponds to the limit c  1, a solid with no void spaces, the temperature is of that of the wall, h0. The analyzed cases in the present work are not in these limits, i.e., c is of order of unity. 4.2. Inner zone: problem of the order C1 In this region, due to viscous effects, variations in the momentum are of order of C1, as already pointed by Eq. (15) with g  C1. Then, to capture those variations it is necessary to re-scale ~f ¼ Cf and also to express it as ~f ¼ ~f ð0Þ þ C1 ~f ð1Þ þ OðC2 Þ. With this in mind, after performing a boundary-layer expansion ~ ¼ Cg, the first two governing equations in the spatial coordinate, g for the momentum are

  ~f 000  eb ~f 0  1 ¼ 0; ð0Þ ð0Þ  2 ~ ~00 ~0 Pr~f 000  ebPr~f 0ð1Þ ¼ ePr ð1Þ þ f ð0Þ f ð0Þ  f ð0Þ Fig. 2. Velocity in the outer zone.

ð24Þ ð25Þ

M.A.E. Kokubun, F.F. Fachini / International Journal of Heat and Mass Transfer 54 (2011) 3613–3621

3617

Fig. 3. Temperature in the outer zone. Fig. 4. Velocity in the inner zone.

~. in which the prime now denotes differentiation with respect to g ~ ¼ ~f ð1Þ ¼ The boundary conditions are given by ~f ð0Þ ¼ d~f ð0Þ =dg ~ ¼ 0 at g ~ ¼ 0, and account for the non-slip condition at d~f ð1Þ =dg the wall for both vertical and horizontal velocities. The remaining boundary conditions are given by matching this solution with the solution provided by the outer zone analysis:

If Eqs. (28) and (29) with boundary conditions given by e F ð1Þ ¼ 0 at g = 0 are solved, the pressure field in the inner F ð0Þ ¼ e zone is found:

  g~ g~ 2 e ~ Þ ¼ C1 pffiffiffiffiffi  F ðg þ OðC2 Þ: eb 2

ð30Þ

  d~f ð0Þ  dfð0Þ  ¼ 1;  ¼ ~  dg dg 0 1  d~f ð1Þ  dfð1Þ  1  ePr ¼ :  ¼ ~  dg dg 0 ebPr

In the inner zone, the problem analyzed is in a region very near the stagnation-point. As a result, the variable e F is nearly null, because the pressure is very close to that of stagnation, and variations in e F become relevant in higher order terms. It is possible to write Eq. (30) as

Solving Eqs. (24) and (25) with the boundary conditions, one obtains the momentum in the inner zone:

  ge 2 2 e ~ Þ ¼  C1 1  pffiffiffiffiffi þ OðC2 Þ: F ðg 2 g~ eb

1

 1  pffiffiffiffi ~f ðg ~Þ ¼ g ~ þ pffiffiffiffiffi e ebg~  1 eb " pffiffiffiffiffi 2

pffiffiffiffi 10g ~ þ 2 ebg ~ 1  ePr g~ þ e ebg~  C1 8ebPr ebPr  pffiffiffiffi  # ~  eb g e  1 ð18  8ePr Þ pffiffiffiffiffi þ OðC2 Þ: þ : 8ebPr eb

~ ! 1 is taken, the pressure field in the inner zone is If the limit g given by 2

g~ e ~ ! 1Þ  C1 : F ðg 2 ð26Þ

From derivative of Eq. (26), the horizontal velocity profile for the inner zone is obtained as

pffiffiffiffi

e g ~ Þ ¼ 1  e ebg~ Uð pffiffiffiffi "  pffiffiffiffiffi 2  e ebg~ 1 1  ePr ~ ~ þ 2 ebg C  pffiffiffiffiffi  10g ebPr 8 ebPr pffiffiffiffi #  e ebg~  pffiffiffiffiffi ~  8ð1  ePrÞ þ OðC2 Þ: þ 4 ebg 8ebPr

ð27Þ

In the inner zone, the viscous effects become relevant due to the fluid attachment at the wall. Such behavior is observed in Fig. 4. The pressure field is described by the same transformation, given by e F ¼ CF ¼ e F ð0Þ þ C1 e F ð1Þ þ OðC2 Þ. Performing the same spatial coordinate change used before for the momentum, and after collecting equal powers of C, Eq. (8) provides the following differential equations:

e F 0ð0Þ ¼ 0; e F 0ð1Þ ¼

~f 00 ð0Þ

eb

ð28Þ  ~f ð0Þ :

ð29Þ

ð31Þ

e, Eq. (31) has to be compared with Eq. (19), written in terms of g which leads to

Fðg ! C1 Þ ¼ C1

g~ 2 2

þ OðC2 Þ:

ð32Þ

Note that taking appropriate limits, the coupling between Eqs. (30) and (19) is obeyed. In the inner zone the temperature profiles detach one from another because the thermal equilibrium is no longer satisfied. Then, the governing equations must be analyzed separately. The energy flux from the outer zone must match the energy flux from the inner zone in all orders, a condition that accounts for the continuity of the first derivative. This mandatory matching conditions is expressed by

   dh  dhg  dhs  ¼ C  ¼ C  :  ~ þ1 ~ þ1 dg 0 dg dg

ð33Þ

The temperature variation is not abrupt in the inner zone, unlike those in combustion problems [17,18]. Then, differently from the momentum variable f, in the inner zone, one does not re-scale temperature variables hs and hg. Both temperature solutions are expressed in a general form as h = h(0) + C1h(1) + O(C2), but to respect the order of magnitude in the matching condition ~ ¼ dhsð0Þ =dg ~ ¼ 0 for every given by Eq. (33), the conditions dhgð0Þ =dg

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g~ must be imposed. From the boundary condition hg(0) = hs(0) = h0 at g~ ¼ 0; hgð0Þ ¼ hsð0Þ ¼ h0 for every g~ is obtained, and the temperature solutions in the inner zone are expressed as

hg ¼ h0 þ C1 hgð1Þ þ C2 hgð2Þ þ OðC3 Þ; hs ¼ h0 þ C1 hsð1Þ þ C2 hsð2Þ þ OðC3 Þ: Substituting these expressions into Eqs. (9) and (10) with the ~ ¼ Cg and collecting equal powers re-scaled spatial coordinate g of C, a set of governing equations is obtained:





eh00gð1Þ ¼ nv hsð1Þ  hgð1Þ ; h00gð2Þ

e

ð34Þ



þ e~f ið0Þ h0gð1Þ ¼ nv hsð2Þ  hgð2Þ ;

ð1  eÞh00sð1Þ ¼ 0; Þh00sð2Þ

ð1  e

ð35Þ ð36Þ



¼ nv hsð1Þ  hgð1Þ :

ð37Þ

The boundary conditions are hg(1) = hs(1) = hg(2) = hs(2) = 0 at

g~ ¼ 0. Eqs. (34)–(37) must match the temperature solution from

Fig. 5. Temperature in the inner zone.

the outer zone as given by Eq. (33), thus the following conditions must be obeyed:

sffiffiffiffiffiffi    dhgð1Þ  dhsð1Þ  dhð0Þ  2 ; ¼ ¼ ¼ H ~ þ1 ~ þ1 dg dg dg 0 pc sffiffiffiffiffiffi     dhgð2Þ  dhsð2Þ  dhð1Þ  H 2 1  ePr 1 þ ¼ ¼ ¼ c ~ þ1 ~ þ1 dg dg dg 0 2c pc ebPr If Eqs. (36) and (37) are solved, then the solid phase temperature profile in the inner zone is obtained:

sffiffiffiffiffiffi 2

~ Þ ¼ h0  C1 H hs ðg

sffiffiffiffiffiffi



H 2 1  ePr 1þ g~ þ C2 c g~ þ OðC3 Þ pc 2c p c ebPr ð38Þ

The gas phase temperature profile in the inner zone is given by solving Eqs. (34) and (35). In the process of solving Eq. (35) it is assumed that e2  1, so one treats a simplified form of such equation when all variables are substituted in Eq. (35). Under such conditions, the gas phase temperature profile in the inner zone is given by

sffiffiffiffiffiffi " sffiffiffiffiffiffiffiffiffi 2e 1 2 H ~ Þ ¼ h0  C H hg ðg g~ þ C H þ pc pcb nv p c 2c sffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffi 1  ePr 2 e c g~ þ H e ebg~  1þ 2 ebPr pceb e b  nv sffiffiffiffiffiffiffiffi # p ffiffiffi ffi 2 nv 2eb e  g~ e þ OðC3 Þ: H e pc nv ðe2 b  nv Þ 1

sffiffiffiffiffiffi 2

2

ð39Þ

In the inner zone, temperature profiles detach one from another due to differences in solid phase and gas phase thermal conductivities. However, in the inner zone one is analyzing a region very near the stagnation-point, such that the velocity field is near zero – from the re-scaling of f, the velocity field in the inner zone is of order C1. Because of the very low velocity of the gas, there is enough time to solid phase and gas phase to reach an equilibrium temperature in lower orders, and the thermal non-equilibrium is only observed in the terms of order C2, as can be seen from Eqs. (38) and (39). It is possible to observe such nearly-equilibrium situation in Fig. 5. In Fig. 6 the corrections of order C2 are presented. The thermal non-equilibrium between solid and gas phase appears only on this order and higher. The convective heat transport is a process that occurs in the outer zone, while the heat removal, mainly by the solid phase conductive heat transport, is a process that occurs in the inner zone if

Fig. 6. Corrections of temperatures.

the heat flux is not too high. If the convective heat transport is balanced with the conductive heat transport, a plane of thermal stagnation is found. An estimative of the position of this plane may be performed, by balancing the convective heat transport from the outer zone, with the solid phase conductive heat transport, from the inner zone. In a non-dimensional form, the following approximated relation for the position of the plane of thermal stagnation is found:

ef hjgts  Cð1  eÞ

 dhs  ~ g~ ts dg

ð40Þ

~ ts are the position of the thermal stagnation plane in which gts and g observed from the outer zone and from the inner zone, respectively. Collecting similar power terms:

dh

 

 : efð0Þ hð0Þ jgts  ð1  eÞ sð1Þ ~ g~ ts dg

ð41Þ

From Eq. (41) the position of the thermal stagnation plane is found by solving the following transcendental expression for gts:

g

"

ts pffiffiffiffiffi ffi 1 2c

!# ðh0  1Þ gts ðh0  1Þ 1 ffi pffiffiffiffi :  erf pffiffiffiffiffi h0 h0 p 2c

ð42Þ

M.A.E. Kokubun, F.F. Fachini / International Journal of Heat and Mass Transfer 54 (2011) 3613–3621

3619

5.1. Outer zone: problem of order unity In the outer zone, velocity and pressure fields are given by Eqs. (15) and (19), and energy conservation is given by Eq. (20). The same type of solution for h is performed as before, and then one must solve the already obtained set of two equations given by Eqs. (21) and (22). The boundary condition at g = 0, when observed from the outer zone, is a temperature h0 and its correction of order C1 given by h1, but, differently from the previous case, this temperature is unknown and dependent on the value of the heat flux Q. Boundary conditions for Eqs. (21) and (22) are then h(0) = h0 and h(1) = h1 for g = 0, and h(0) = 1 and h(1) = 0 for g ? 1. The thermal solution in the outer zone is then given by solving Eqs. (21) and (22) with the appropriate boundary conditions, obtaining: !

g

hðgÞ ¼ h0  ðh0  1Þerf pffiffiffiffiffiffi 2c sffiffiffiffiffiffi  " !#  g2 h0  1 2 1  ePr g þ C1 h1 þ  1þ c ge 2c  h1 erf pffiffiffiffiffiffi 2c pc ebPr 2c

Fig. 7. Position of the thermal stagnation plane.

Even though Eq. (42) is an approximation, some features can be observed from it. It is possible to observe that as the wall temperature increases, the thermal stagnation plane goes farther from the wall. Since the inner zone is a very small region, if the wall temperature is high enough (if compared to the injection temperature), the heat flux is high, leading the heat removal from the wall to be a process that occur in the outer zone. Hence, the thermal stagnation plane pffiffiffiffiffiffi 1=2 = pffiffiffiffiffiffi the wall, as shown by gts = 2c  p goes far from 1  erf ðgts = 2cÞ for (h0  1)/h0 ? 1. Contrarily, if the wall temperature is close to the air injected temperature, the thermal pffiffiffiffiffiffi stagnation plane is close to the wall, according to gts = 2c  p1=2 ðh0  1Þ=h0 for (h0  1)/h0 ? 0. Fig. 7 presents the position of the thermal stagnation plane as a function of the wall temperature. The horizontal axis have been normalized and the vertical axis have been re-scaled. Cooling devices do not present significant temperature differences between the injected gas and the heated component (impermeable wall), and hence, below the limit (h0  1)/h0 < 0.2, or h0 < 1.25 (a case of interest), the plane of thermal stagnation is approximately located in a region of the order of C1/2. 5. The constant wall heat flux case If the interest relies not on a prescribed temperature at z ¼ 0, but instead on removing an amount Q of heat through the wall, the development considered before may be performed by only changing the boundary condition at z ¼ 0 to

  dT s  dT g  ð1  eÞks þ ekg ¼ Q  dz 0 dz 0

ð43Þ

or, in its non-dimensional form, at g = 0:

  dhs  e dhg  ð1  eÞ  þ ¼ q dg 0 C dg 0

ð44Þ

 in which is considered that q  Q =ðcp T 1 Þ ½1=ðq1 v 1 Þ½1=a1=2  is of order of unity. The problem in the outer zone, corresponding to the solid-phase thermal diffusivity length scale, is described first. Then the problem in the inner zone, corresponding to the gas thermal diffusivity length scale, is described. In both regions, solutions will be obtained using the perturbation method, as in the previous case. The velocity and the pressure fields will be the same as those obtained for the case considering a constant wall temperature, but expressions for the temperature profiles will be different.

þ OðC2 Þ:

ð45Þ

5.2. Inner zone: problem of order C1 The same re-scaling performed previously is made, and velocity and pressure fields are also the same as those obtained for the case of a constant wall temperature, and are given by Eqs. (26) and (30), respectively. Considerations made previously about matching the heat flux from the outer zone with the inner zone are also valid in the present case. The difference appears on the boundary condi~ ¼ 0, that now is given by tion at g

  dhs  dhg  þ e ¼ q: ~ 0 ~ 0 dg dg

Cð1  eÞ

ð46Þ

Expressing temperature profiles as before one must solve the same set of equations obtained previously, given by Eqs. (34)–(37), and imposing the boundary conditions given ~ ¼ Q =ð1  eÞ and by hsð1Þ ¼ hgð1Þ ¼ h1 ; hsð2Þ ¼ hgð2Þ ¼ h2 ; dhsð1Þ =dg ~ þ edhgð1Þ =dg ~ ¼ 0, at g ~ ¼ 0. The matching conditions ð1  eÞdhsð2Þ =dg impose that

sffiffiffiffiffiffi    dhgð1Þ  dhsð1Þ  dhð0Þ  2 ; ¼ ¼ ¼ ðh0  1Þ ~ þ1 ~ þ1 dg dg dg 0 pc    dhgð2Þ  dhsð2Þ  dhð1Þ  ¼ ¼ ~ þ1 ~ þ1 dg dg dg 0 sffiffiffiffiffiffi  pffiffiffiffiffiffi pffiffiffi! pc  2 ðh0  1Þ 2 1  ePr : 1þ ¼ c þ h1 pffiffiffiffiffiffi 2c pc ebPr pc

ð47Þ

ð48Þ

It is worth to note that, since the heat removed from the wall is assumed to be of order of unity, the leading order term for the temperature expressions must be a constant value. This condition reinforces the consistence of the proposed model. Solid phase temperature profile is obtained solving Eqs. (36) and (37) with boundary and heat flux matching conditions. The result is

  q ~ Þ ¼ h0 þ C1 h1  hs ðg g~ 1  e sffiffiffiffiffiffi "   h 2 1  ePr 0 1  1þ c þ C2 h2 þ 2c pc ebPr pffiffiffiffiffiffi pffiffiffi!! # pc  2 ~ g þ OðC3 Þ: þ h1 pffiffiffiffiffiffi

pc

ð49Þ

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M.A.E. Kokubun, F.F. Fachini / International Journal of Heat and Mass Transfer 54 (2011) 3613–3621

The same simplifying assumption made before, e2  1, is performed here, and similarly from the previous case, a simplified form of Eq. (35) is solved, obtaining a gas phase solution in the inner zone:

  q ~ Þ ¼ h0 þ C1 h1  hg ðg g~ 1  e pffiffiffi

  pffiffiffiffi nv q e e q 1 2 pffiffiffiffiffi e e g~ þ h2 þ þ C  h2 þ c nv e2 b  nv c nv eb ! ! p ffiffiffi   pffiffiffiffiffiffi pc  2 ~ q 1 1  ePr þ c þ h1 g 1þ pffiffiffiffiffiffi c 2ð1  eÞ ebPr pc pffiffiffiffi # q e ebg~ þ pffiffiffiffiffi 2 þ OðC3 Þ: c ebðe b  nv Þ

ð50Þ

From the matching conditions expressed by Eqs. (47) and (48) the temperature at the stagnation-point as a function of the removed heat q, is obtained:

h0 ¼ 1 þ q

Fig. 8. Temperature at the wall.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

; 2eð1  eÞ   p ffiffiffiffiffiffi pc q  e 2 1  ePr pffiffiffi 1  c : h1 ¼ p ffiffiffiffiffiffi 2e 1  e ebPr pc  2

ð51Þ ð52Þ

Eq. (51) points that the temperature at the stagnation-point increases as the heat delivered from the wall to the system increases. It also points out an increase temperature at the wall by reducing the medium porosity. Lowering the porosity of the medium, the volume of solid increases – the gas fills the volume given by (1  e) – and consequently the gas phase convective heat transfer decreases. The result is that the heat transfer becomes predominantly conductive by the solid phase. Since the length of the porous material is considered infinite with a prescribed temperature on one ‘‘side’’ (g ? 1), and a heat flux on the other side, the stationary solution for this material leads to an infinite temperature at the wall as e ? 0. This behavior is captured by Eqs. (51) and (52). Eq. (51) also shows that the temperature at the wall increases, as the porosity increases. This behavior is not physically correct by two reasons. First, as the porosity is made large, the heat transfer becomes predominantly convective and conductive through the gas phase. Then, under this condition, the wall temperature does not go to infinite. Second, and the stronger reason, the model is not valid for high porosity. The limit of this model is determined by the temperature correction h1, as will be seen subsequently. The first correction to the stagnation-point temperature is given by Eq. (52), and one must note that h1 is negative, since the last term in the parentheses is larger than one. The heat supplied from the wall appears as a boundary condition, as can be seen in Eq. (46), and its value q is assumed to be of order of unity and having none corrections, since its value is assumed to be known. Under this feature, we observe that Eq. (46) points out that all heat provided by the wall goes to the solid phase, in a first approximation. In this scenario, we are overestimating the temperature at the stagnationpoint, since we are not considering the existence of a small heat flux from the wall to the gas phase, in a first approximation. The negative value for h1 compensates this fact. Eq. (52) is a correction of order C1 for Eq. (51), except in the limits of e  1, e  0 and pffiffiffiffiffiffi pffiffiffi pc  2. At these limits, h1 diverges and becomes as significant as the leading order term h0. The first two limits are compatible with the limits presented before for the leadingpffiffiffi order term h0, pffiffiffiffiffiffi but the third are not. With the condition pc  2 it is possible to obtain an estimative for the porosity value at which the proposed model fails. This limit leads to an estimated limiting value of porosity, given by e < p/(2 + p)  0.6, at which the proposed

model is valid. The dependence of the temperature with porosity is presented in Fig. 8. In order to obtain the value for h2, appearing in Eqs. (49) and (50), one must evaluate the temperature solution of order C3. This procedure results in a new constant h3. This constant is obtained by the term of order C4, and so on. The main goal in this analysis was to show that the non-equilibrium between solid phase and gas phase occurs only in higher order terms. In face of that, the constant h2 will not be obtained, although we reinforce that if one wishes to obtain it, one should evaluate the result for hs(3) and hg(3) and apply the boundary condition for the heat flux at the wall in these lower order terms.

6. Conclusions A Hiemenz flow established inside a low porosity medium with high heat exchange between phases was analyzed in this work. Solutions for temperature, momentum and pressure fields were obtained analytically. When one considers the existence of a heat exchange between phases, the parameter C ¼  ks = kg emerges. Two different regions must be considered in this situation: an outer zone and an inner zone. Since the value of C is high, the perturbation method is applied to detail profiles in both zones. The model studied in this work may be thought as a porous heat dissipator for cooling devices, e.g. electronic components. Two boundary condition cases are analyzed: a constant wall temperature and a constant wall heat flux. The first case is useful if the electronic components have a maximum operating temperature, and the second is useful when one is interested on removing a certain amount of heat from the components. The present results show that the porous medium is the main responsible from removing heat from the heated components, as pointed out by Eq. (46), but it is also shown that the flowing gas have a key role in dissipating the removed heat. It is also shown that since in the inner zone f ? 0, the solution in the outer zone is sufficient to describe the problem considering a prescribed wall temperature. However, when one considers a constant wall heat flux, it is necessary to treat the thermal process in the inner zone, since heat release occurs in this small region. Calculations were performed assuming a low porosity medium and a high value for the interphase heat flux. The modeling fails when one increases the medium porosity, close to e < p/(2 + p), but the proposed model opened an opportunity to explore a wider range of the space parameter by only modifying the term 1/(jC). The analysis presents a modification to include the Darcy term

M.A.E. Kokubun, F.F. Fachini / International Journal of Heat and Mass Transfer 54 (2011) 3613–3621

on the classical pressure expression that is able to describe the influence of the linear Darcy term. It is shown the importance of the description of the inner zone process for systems in which the heat flux at the wall is known. It must be pointed that if one wishes to analyze a gas cooling system by means of a porous matrix, would only be necessary to change the sign of the term in the right-side in Eq. (43). Acknowledgment This work was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). References [1] J. Bear, Dynamics of Fluids in Porous Media, Dover, New York, 1988. pp. 1–26. [2] A.A.M. Oliveira, M. Kaviany, Non-equilibrium in the transport of heat and reactants in combustion in porous media, Prog. Eng. Combust. Sci. 27 (2001) 523–545. [3] A.M. Siddiqui, A. Zeb, Q.K. Ghori, Some exact solutions of 2d steady flow of an incompressible viscous fluid through a porous medium, J. Porous Media 9 (2006) 491–502. [4] Q. Wu, S. Wienbaum, Y. Andreopoulos, Stagnation-point flows in a porous medium, Chem. Eng. Sci. 60 (2005) 123–134. [5] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid in a dense swarm of particles, Appl. Sci. Res. A 1 (1947) 27–34. [6] V. Kumaran, R. Tamizharasi, K. Vajravelo, Approximate analytic solutions of stagnation point flow in a porous medium, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 2677–2688.

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