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On the thermally developing forced convection through a porous material under the local thermal non-equilibrium condition: An analytical study
International Journal of Heat and Mass Transfer 92 (2016) 815–823
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On the thermally developing forced convection through a porous material under the local thermal non-equilibrium condition: An analytical study Maziar Dehghan a,⇑, M.S. Valipour a, Amir Keshmiri b,c, S. Saedodin a, Nima Shokri d,⇑ a
Faculty of Mechanical Engineering, Semnan University, Semnan, Iran School of Engineering, Manchester Metropolitan University, Manchester, UK School of Mechanical Aerospace and Civil Engineering, , University of Manchester, Manchester, UK d School of Chemical Engineering and Analytical Science, University of Manchester, Manchester, UK b c
a r t i c l e
i n f o
Article history: Received 1 June 2015 Accepted 30 August 2015 Available online 1 October 2015 Keywords: Thermally developing forced convection Analytical study Iso-flux thermal boundary condition modeling Porous medium Local thermal non-equilibrium
a b s t r a c t The aim of the present analytical study is to investigate a thermally developing forced convective heat transfer inside a channel filled with a porous medium whose walls are imposed to a constant heat flux (i.e. the iso-flux thermal boundary condition). The Darcy’s law of motion and the two-energy equation (i.e. local thermal non-equilibrium, LTNE) model are considered. A perturbation analysis is conducted to avoid using any model for the iso-flux thermal boundary condition. Thermally developing forced convection inside a porous-filled channel has previously not been analyzed without implementing a heuristic model for the iso-flux thermal boundary condition. The temperature difference between the fluid and solid phases (called the LTNE intensity) is analytically obtained. Results concerning the LTNE intensity is compared with the available models of the iso-flux thermal boundary condition. In addition, the Nusselt number of a thermally developing convection through a porous material is obtained for the LTNE condition. Also, an expression representing the developing length, i.e. the dimensionless axial location at which the flow becomes fully developed, for a porous-filled channel is proposed for the first time. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Local thermal non-equilibrium (LTNE) may be observed in various engineering applications including microchannel heat sinks, heat pipes, fluidized beds, dryers, and catalytic reactors, thus, motivating numerous researchers to study the LTNE condition with an aim of accurately predicting the thermal behavior of porous media [1–10]. Amiri et al. [11] numerically focused on the inertial as well as viscous effects of the equation of motion of a fluid saturated porous medium. For the first time, they discussed on the iso-flux thermal boundary condition modeling. Kuznetsov [12] used the Brinkman–Forchheimer-extended Darcy equation to analytically study the LTNE condition for the first time. Kuznetsov [12] discussed the thermal non-equilibrium effects in a channel filled with a fluid saturated porous medium based on a perturbation analysis. Kuznetsov used the results of the previous studies (Vafai and Kim [13,14], and Nield et al. [15]) for the flow ⇑ Corresponding authors. E-mail addresses: [email protected], [email protected] (M. Dehghan), [email protected] (N. Shokri). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.091 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
field and a two-equation energy model [16,17] for the temperature field and proposed a simplified relation for measuring the LTNE. Using the Darcy’s law of motion, Nield and Kuznetsov [16] analytically investigated the conjugate heat transfer problem of a saturated porous channel. Lee and Vafai [17] analytically characterized the forced convective flow through a channel filled with a porous material and obtained the dimensionless temperature for the fluid and solid phases as well as the Nusselt number. Alazmi and Vafai [18] investigated four important categories in modeling the fluid flow and heat transfer through porous media, namely constant and variable porosity, thermal dispersion, and local thermal non-equilibrium phenomena. They numerical experiments showed that the above-mentioned variants have stronger influences on the flow field than the heat transfer. Alazmi and Vafai [19] numerically studied two primary assumptions for the iso-flux thermal boundary condition with seven different sub-models. They found the Nusselt number obtained by each model. Nield et al. [20] applied a modified Graetz methodology to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial
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Nomenclature asf specific surface area (m1) b a constant value Bi Biot number cp specific heat at constant pressure (J kg1 K1) d0, dn constant values di, i = 1–4 constant values pore diameter (m) dp f an intermediate function g an intermediate function H half of the channel gap (m) hsf fluid–solid heat transfer coefficient (W m2 K1) k conductivity ratio (kf,eff /ks,eff) kf conductivity of fluid phase (W m1 K1) kf,eff effective conductivity of fluid phase (W m1 K1) effective conductivity of the medium (kf,eff + ks,eff) km (W m1 K1) ks conductivity of solid phase (W m1 K1) effective conductivity of solid phase (W m1 K1) ks,eff MA Model A MB Model B MC Model C n counter Nu Nusselt number O order of magnitude Pe Peclet number Pr Prandtl number q00 w heat flux at the wall (W m2)
conduction and viscous dissipation included. Their analysis based on the LTNE assumption led to expressions for the local Nusselt number. Kuznetsov et al. [21] followed a similar methodology to the one used in Nield et al. [20] to investigate the thermally developing forced convection inside a fluid saturated pipe with an isothermal wall. Later, Nield and Kuznetsov [22] applied the classical Graetz methodology to study the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at constant temperature, for the case of a non-Newtonian fluid of power-law type. Nield and Kuznetsov [22] obtained the Nusselt number at LTNE condition. Based on a pore-scale numerical simulation, Jiang and Lu [23] explored effects of the thermal contact resistance between a porous medium and an impermeable wall. They showed that the first approach of the iso-flux thermal boundary condition models (i.e. Model A of Ouyang et al. [7]) gives the closest results to the experiments when the thermal contact resistance is negligible and the wall has a finite thickness. Yang and Vafai [4] analytically investigated the effects of internal heat generations within the fluid and solid phases on the thermal behavior of a channel filled with a porous material. Later on, Yang and Vafai [5] analytically solved the problem of transient forced convection inside a saturated porous medium incorporating the LTNE model. Ouyang et al. [7] derived analytical solutions for thermally developing flows in porous media under LTNE condition for the constant wall heat flux boundary condition. They used three different models to obtain the local Nusselt number and to predict the dimensionless thermal entry length. Recently based on a perturbation analysis, Dehghan et al. [24,25] investigated the LTNE situation in a fluid saturated channel or pipe filled with a saturated porous material. They proposed a dimensionless number representing the LTNE intensity in terms of pertinent parameters. Following the work of Dehghan et al. [26] who analyzed the heat transfer rate in a solar heat exchanger filled with porous media for the case of combined conduction– convection–radiation heat transfer, Dehghan et al. [27]
Re T Ti u x⁄, y⁄ x, y
Reynolds number temperature (K) inlet temperature (K) Darcian velocity (m s1) dimensional coordinates (m) dimensionless coordinates
Greek letters
DNE
e
h
q rn
/
LTNE intensity (hs hf) small parameter (1/hsfasf, W1 m3 K) dimensionless temperature fluid density (kg m3) parameter in the obtained dimensionless temperature profile porosity of the medium
Subscripts dev f fd i m n s w
developing fluid phase fully developed component inlet bulk-mean value counter solid phase wall
investigated effects of the slip-flow regime on the combined convective–radiative heat transfer rate. Dehghan et al. [27] showed that the heat transfer rate is mainly controlled by the temperature jump phenomenon. Nield and Kuznetsov [28] analytically obtained the Nusselt number for forced convection heat transfer of nanofluids flow through a clear or porous-filled channel at LTE condition. Dehghan et al. [29] solved the non-linear problem of forced convection inside a porous-filled channel with a temperaturedependent conductivity using the perturbation technique. They found that the Nusselt number increases with a variable conductivity approach. In the present study, thermally developing forced convection in a fluid saturated porous medium bounded by two infinite parallelplates held at a constant heat flux is analytically investigated allowing the temperature of fluid and solid phases to be different (LTNE). It is assumed that the Darcy’s law governs the fluid motion in the porous medium. The energy equation has been solved by the method of Fourier series. The temperature difference between the solid and fluid phases, called the ‘LTNE intensity’, is found without the need for implementing any model for the iso-flux thermal boundary condition. In addition, the Nusselt number and thermally developing length are also obtained. To the best of authors’ knowledge, no other study has investigated the thermally developing forced convection heat transfer in a porous-filled channel without implementing a model for the iso-flux thermal boundary condition. 2. Mathematical modeling The schematic diagram of the problem considered in this study is shown in Fig. 1. The following assumptions are invoked in the formulation of the model: The flow in the porous medium is incompressible and uniform along the cross-sectional area (Darcy’s law).
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ks;eff @ 2 T f þ Oðe2 Þ hsf asf @y2
ðT s T f Þ ¼
ð8Þ
To define the temperature difference (or the intensity of LTNE condition), the value of
@2 T f @y2
should be defined. From Eqs. (1), (2)
and (7) it can be shown that:
qcp u
@T f @2T f ¼ km 2 þ OðeÞ @x @y
ð9Þ
where km is the effective thermal conductivity of the medium and equals to ks;eff þ kf ;eff . For the sake of brevity, the subscript ‘‘f” will be dropped hereafter. Now, the solution reduces to solve a fluid saturated porous medium under the LTE condition (Eq. (9)). The boundary conditions for the LTE model under the situation of a constant heat flux imposed at the walls are:
km
Fig. 1. Schematic diagram of the present problem.
The thermally developing forced convection is desired. The porous medium is isotropic and homogenous. Walls are impermeable with infinite dimension perpendicular to the plane of view to ensure the two-dimensionality of the problem. The temperature of the solid and fluid phases could be different (i.e. LTNE condition).
@T ¼0 @y
qcp u
0 ¼ ks;eff
@2T s hsf asf ðT s T f Þ @y2
ð1Þ
ð2Þ
ks;eff ¼ ð1 /Þks
ð3Þ
kf ;eff ¼ /kf
ð4Þ
where / is the porosity of the medium, and hsf and asf are the fluid– solid heat transfer coefficient and specific surface area (surface per unit volume) respectively, and are defined as [30]:
"
hsf ¼
ð5Þ
1 kf qudp 2 þ 1:1Pr 3 dp l
0:6 # ð6Þ
where dp is particle diameter and Pr is the Prandtl number. In Eqs. (1) and (2) it is assumed that the axial heat conduction term is negligible [7]. 3. Analysis and solution
Ts ¼ Tf þ O
e¼
1 hsf asf
ð11Þ
x¼
ðT T 0 Þks;eff q00w H x k f
¼
qcp H u 2
x =H ; Pe
ð12Þ
y¼
y ; H
k¼
kf ;eff ks;eff
ð7Þ
where e is a positive parameter with a small value, i.e. OðeÞ < 1. It is chosen for the perturbation analysis. Combining Eqs. (2) and (7) yields:
ð13Þ
where h is the dimensionless temperature, x is the dimensionless axial coordinate, Pe is the Peclet number, y is the dimensionless vertical coordinate, and k is the effective conductivity ratio. Combining Eqs. (7), (9), (12) and (13) results in:
ð70 Þ
hs ¼ hf þ OðBi Þ k
@h @2h 1 ¼ ðk þ 1Þ 2 þ OðBi Þ @x @y
ð14Þ
and the dimensionless boundary conditions for the LTE model are:
@h 1 ¼ y¼0 @y kþ1 @h ¼0 y¼1 @y
ð15Þ
x¼0
h¼0
ð16Þ
Combining Eqs. (8) and (14) yields:
ðT s T f Þ ¼
k q00w @h þ Oðe2 Þ ðk þ 1Þ hsf asf H @x
ð17Þ
Eq. (17) reveals that the temperature difference between the . Now in fluid and solid phases is proportional to the value of @h @x order to find the dimensionless temperature distribution (h), one can solve Eq. (14) with the boundary conditions (15) and (16) through:
hðx; yÞ ¼ gðx; yÞ þ f ðyÞ
Using the scale analysis and noting that the product of hsfasf is a large value, one can show from Eq. (2) that [2,12,24,25]:
x ¼ 0
1
where subscripts ‘s’ and ‘f’ denote solid and fluid phases, respectively, T is the temperature, cp is the specific heat of the fluid phase, u is the Darcy velocity and, ks,eff and kf,eff are the effective thermal conductivities of the solid and fluid phases given by [30]:
asf ¼ 6ð1 /Þ=dp
ð10Þ
where Ti is the inlet temperature of the fluid phase. To solve Eq. (9), the following dimensionless variables and parameters are defined:
2
@T f @ Tf ¼ kf ;eff 2 þ hsf asf ðT s T f Þ @x @y
y ¼ 0 y ¼ 1
T ¼ Ti
h¼ According to the above assumptions, the steady-state energy conservation equations of the solid and fluid phases are given as [7,24,30]:
@T ¼ q00w @y
ð18Þ
Combining Eqs. (14) and (18) results in:
kg x ¼ ðk þ 1Þg yy þ ðk þ 1Þf
00
ð19Þ
with the following updated boundary conditions:
1 y¼0 kþ1 0 gy þ f ¼ 0 y¼1 0
gy þ f ¼
ð20Þ
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gð0; yÞ þ f ðyÞ ¼ 0
x¼0
ð21Þ
Now, the solution procedure is reduced to the solution of two individual differential equations. Since the boundary conditions in the transverse direction are of Neumann type (the first order of differentiation), an additional step is required to satisfy the boundary conditions:
ðhs hf Þ ¼ DNE ¼
00
f ¼
1 ; kþ1
8 > > <
1 Biðk þ 1Þ > > :fully
0
f ð0Þ ¼
1 0 ; f ð1Þ ¼ 0 kþ1
DNE ¼ ðhs hf Þ ¼ ð23Þ
where b is an unknown constant. Subsequently, combining Eqs. (19) and (22) results in the second differential equation expressed as:
kg x ¼ ðk þ 1Þg yy þ 1
ð24Þ
with the following boundary conditions:
y¼0 y¼1
ð25Þ ð26Þ
Based on the boundary conditions (25), the solution of Eq. (24) is expected to have the following form:
gðx; yÞ ¼ c0 ðxÞ þ
1 X
½cn ðxÞ cosðnpyÞ
ð27Þ
n¼1
Substituting Eq. (27) into Eq. (24) and using the orthogonality relations of the trigonometric functions results in:
c0 ðxÞ ¼
x þ d0 k
cn ðxÞ ¼ dn expðrn xÞ;
ð28Þ n ¼ 1; 2; . . .
ð29Þ
where d0 and dn are constants and rn ¼ pÞ .The inlet condition, Eq. (26), should be used to obtain the constants (d0 and dn). One can derive the following solution by combining Eqs. (23) and (26)–(29): ðkþ1Þ ðn k
gðx; yÞ ¼
2
x 1 þ b k 3ðk þ 1Þ " # 1 X 2 expð r xÞ cosðn p yÞ þ n 2 n¼1 ðk þ 1ÞðnpÞ
ð30Þ
Combining Eqs. (18), (23) and (30) yields the dimensionless temperature distribution for the LTE model under the condition of constant heat flux imposed at the walls:
hðx; yÞ ¼
ðT s T f Þks;eff q00w H
ð34Þ
DNE consists of two components (see Eq. (33)): developing and fully developed components. The fully developed component of the LTNE intensity (DNEfd) is:
DNEfd ¼ ðhs hf Þfd ¼
1 2 þ OðBi Þ Biðk þ 1Þ
ð35Þ
where the Biot number, Bi ¼ hsf asf H2 =ks;eff . Eq. (35) shows that DNEfd is the same as what was proposed by Kuznetsov [12] and Dehghan et al. [25] for thermally fully developed flow under the Darcy’s law and under the constant heat flux thermal boundary condition. Also, one can show that OðeÞ OðBi Þ. In summary, a fundamental analytical relation, Eq. (33), representing the intensity of LTNE condition (DNE) has been obtained. Eq. (33) is a simple and general relation that can be used for the estimation of the LTNE intensity (DNE) as well as validation of numerical simulations. Further parametric studies on DNE and comparison with available models are presented in the next section. 1
x¼0
gð0; yÞ ¼ f ðyÞ
ð33Þ
The LTNE intensity, DNE, represents:
ð22Þ
y 2y þb 2ðk þ 1Þ
gy ¼ 0
And the dimensionless temperature difference between the fluid and solid phases (called LTNE intensity) is:
dev eloping component
2
gy ¼ 0
ð32Þ
9 > > = X1 2 1 þ ½2 expð r xÞ cosðn p yÞ þ OðBi Þ n |{z} n¼1 > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl } > ; dev eloped component
The solution can be written as:
f ðyÞ ¼
( ) 1 X q00w ðT s T f Þ ¼ ½2 expðrn xÞ cosðnpyÞ þ Oðe2 Þ 1þ hsf asf Hðk þ 1Þ n¼1
1 x 3ðy 1Þ2 1 X 2 þ expðrn xÞ cosðnpyÞ k 6ðk þ 1Þ krn n¼1
ð31Þ
The temperature difference between the fluid and solid phases can be found by combining Eqs. (17) and (31):
4. Results and discussions 4.1. The LTNE intensity at the wall Fig. 2 shows the LTNE intensity (hs hf) versus dimensionless axial coordinate for Bi = 10 and different conductivity ratios (k) at the channel wall (y = 0). The figure compares the present dimensionless temperature difference between the fluid and solid phases obtained analytically with those obtained by Ouyang et al. [7] at the channel wall. Ouyang et al. [7] used three different models (i. e. models A, B, and C) to investigate the thermally developing heat transfer in a fluid saturated porous medium imbedded in a channel. These models were used to model the iso-flux thermal boundary condition at the walls [7]; Model A:
hs ¼ hf ;
k
@hf @hs ¼1 @y @y
y¼0
ð36Þ
Model B:
k
@hf @hs ¼ ¼1 @y @y
y¼0
ð37Þ
M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
0.6
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0.4
(θs-θf )
0.2 0 -0.2 -0.4 -0.6 -0.8 0.001
0.01
x
0.1
1
(a) 0.7
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0.6 0.5
(θs-θf )
0.4 0.3 0.2 0.1 0 -0.1 -0.2 0.001
0.01
x
0.1
1
(b) 0.45 0.4 0.35
(θs-θf )
0.3 0.25 Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0.2
0.15 0.1
Meanwhile, the fully developed component of Model A is shown in the figures. In the following figures ‘‘MA”, ‘‘MB”, and ‘‘MC” represent ‘‘Model A”, ‘‘Model B”, and ‘‘Model C” of Ref. [7], respectively. Fig. 2 shows that the LTNE intensity (hs hf) obtained by the present study decreases in the streamwise direction and with increasing the conductivity ratio (k), which is in agreement with the findings of Dehghan et al. [24,25]. The conductivity ratio (k) increases with increasing the porosity (/). At higher porosities, the occupying solid phase has less thermal inertia and would follow the fluid phase more easily and vice versa [24,25,31,32], which lead to lower LTNE intensities. In addition, by increasing the conductivity ratio (k), the ability of heat transfer in the fluid phase compared to the solid phase increases and results in a lower degree of LTNE intensity. Based on the assumptions made for Model A, the absolute value of the LTNE intensity is zero for this model (i.e. Model A) at y = 0. Comparing Fig. 2(b) with Fig. 2(c) reveals that the LTNE intensity obtained by Model B increases with increased conductivity ratio (k). Model B returns negative LTNE intensity for k < 1 but becomes positive for k P 1. Meanwhile, Model C gives negative values for k 6 1. It should be noted that a negative value for the LTNE intensity (hs hf) on the wall (y = 0) is unphysical as there is no internal heat generation in the fluid phase. Since the heat is transferred from the wall to the porous medium and is cooled via entering the fluid phase with a cold temperature at the inlet (Ti), the heat is transmitted through the solid phase to the fluid phase. Consequently, based on the second law of thermodynamics the fluid phase cannot have a higher temperature than the solid phase. Fig. 3 shows the effects of the Biot number on the LTNE intensity for k = 1. It is seen that the LTNE intensity decreases by increasing the Biot number (Bi). A higher Biot number represents an enhanced internal convective heat transfer between the solid and fluid phases and consequently, a lower temperature difference between the two phases (i.e. lower LTNE intensity). For example, the LTNE intensity corresponding to Bi = 10 is about one order of magnitude larger than that corresponding to Bi = 100. Moreover, the LTNE intensity at the wall (y = 0) decreases along the streamwise direction since the fluid phase moves in contact with the solid phase to reach a fully developed LTNE intensity. Another finding to emerge implicitly from Fig. 3 is that the portion of the wall which is in direct contact with the fluid phase has a lower dimensionless temperature (equal to zero) due to the ‘no-jump condition’ which says that the fluid should have the same temperature as the adjacent wall temperature [27,31,32]. On the other hand, the other portion of the wall, which is in direct contact with the solid phase,
0.05 0 0.001
0.01
x
0.1
1
(c) Fig. 2. LTNE intensity versus dimensionless axial direction (Bi = 10 and y = 0); (a) k = 0.1, (b) k = 1, (c) k = 10.
Model C:
k
@hf ¼ /; @y
@hs ¼1/ @y
y¼0
ð38Þ
Since Model C depends on the porosity of the medium and this dependency is not the focus of the present study, a constant value for / = equal to 0.9 was used in plotting the results of Model C.
819
Fig. 3. Effects of the Biot number on the LTNE intensity (k = 1 and y = 0).
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has a dimensionless temperature equal to the value of the LTNE intensity. This matter indicates the need for introducing a model for the iso-flux thermal boundary condition. This has led to, for example the work of Ouyang et al. [7], where three different models for the iso-flux thermal boundary condition have been used which have resulted in different LTNE intensities.
0.2
(θs-θf )
0.15 0.1
0.05 Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0 -0.05 0.001
0.01
0.1
x
1
(a) 0.1
(θs-θf )
0.05
0
-0.05 0.001
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC 0.01
x
0.1
1
(b) 0.05 0.04
In the previous subsection, the LTNE intensity at the channel wall (y = 0) was discussed for different pertinent parameters. In this subsection, the LTNE intensity at the channel centerline (y = 1) is discussed. The LTNE intensity at the centerline for Bi = 10 is shown in Fig. 4. Similar to Figs. 2 and 3, the LTNE intensity obtained by the present analysis decreases by increasing the conductivity ratio (k). However, in contrast to the results reported earlier at the wall, it is now seen that the LTNE intensity at the channel centerline increases in the streamwise direction (x). Moreover, it is worth noting that in Fig. 4 the present analysis shows that the LTNE intensity is zero at the entrance for different values of conductivity ratio, while the models of Ouyang et al. [7] (i.e. models A, B, and C) return non-zero LTNE intensity at the entrance. Another point to highlight about the results in Fig. 4 is that further downstream, values of the LTNE intensity of the present analysis approach to those of models A and C. Based on the observations presented in Figs. 2 and 4 (particularly in relation to the sign and magnitude of the LTNE), it is concluded that the present analytical results are in agreement with the physics of the problem and the ‘scale analysis’ prediction [11,24,25]. The reason for the discrepancies between the present analysis and the models of Ouyang et al. [7] is that no heuristic model was used to describe the iso-flux thermal boundary condition at the wall (i.e. a constant heat flux imposed at the walls). This matter shows the importance of the modeling of the iso- flux thermal boundary condition. In Fig. 5, the effects of the Biot number on the LTNE intensity are investigated at the centerline for k = 1. It shows that the temperature difference between the solid and fluid phases (LTNE intensity) is negligible in the entrance region due to a high convective heat transfer coefficient of the fluid phase in this region. The LTNE intensity increases along the channel length since the convective heat transfer coefficient of the fluid phase decreases in the streamwise direction. In addition, Fig. 5 indicates that the LTNE intensity decreases with increased Biot number due to a lower internal convective thermal resistance between the two phases. Finally, similar to the argument of Ref. [17], when there is no heat generation within the solid and fluid phases and the heat is imposed from the wall, the dimensionless temperature of the solid phase should be greater than the one of the fluid phase. This occurs since the solid phase acts as an extended surface propagating the heat flux throughout the fluid volume. Consequently, the heat flux must flow through the solid phase in order to be convected via the
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
(θs-θf )
0.03
4.2. The LTNE intensity at the centerline
0.02 0.01 0
-0.01 0.001
0.01
x
0.1
1
(c) Fig. 4. LTNE intensity versus dimensionless axial direction (Bi = 10 and y = 1); (a) k = 0.1, (b) k = 1, (c) k = 10.
Fig. 5. Effects of the Biot number on the LTNE intensity (k = 1 and y = 1).
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M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
fluid phase. This matter imposes a limit on the validity of the models in which the dimensionless temperature of the solid phase must be greater than that of the fluid phase resulting in positive LTNE intensities (hs hf > 0). Therefore, the magnitude of the LTNE intensity plays an important role in determining the validity of the models used for the thermal boundary condition. It should be noted that the above-mentioned discussion is valid when there is no internal heat generation/absorption within the porous material. Effects of heat generations/absorptions within the solid and fluid phases of a porous material on the thermal response have been studied by Refs. [33–37] in details. 4.3. The Nusselt number The Nusselt number for a thermally developing flow based on the LTNE approach is defined as [30]:
Nu ¼
q00w 4H km ðT w T f ;m Þ
ð39Þ
where hf ;m ðxÞ is the bulk-mean temperature of the fluid phase given by:
Z hf ;m ðxÞ ¼
1
hf ðx; yÞdy
ð40Þ
0
Based on the present solution, the local Nusselt number is obtained as follows:
NuðxÞ ¼ 1
3
P1
4
1 rx þ n¼0 ðnpÞ e Biðkþ1Þ 2
2
þ OðBi Þ
ð41Þ
2
The terms in the denominator of Eq. (41) represents the following three phenomena involved in a thermally developing force convection through a porous material: (1) the fully developed component (i.e. 1/3) of the Nusselt number of a fluid convected through a Darcian porous material based on the LTE approach, (2) the developing component of the Nusselt number, and (3) the LTNE intensity of the convective heat transfer. It should be noted that such a simple expression for the local Nusselt number has previously not been proposed, which does not use any model for the iso-flux thermal boundary condition. Fig. 6 shows the developing Nusselt number along the dimensionless flow direction (x = x⁄/(H.Pe), where x⁄ is the dimensional axial-direction) for LTE condition (Bi ! 1). As one would expect, the Nusselt number starts off with a high value in the entrance region because of the inlet condition and thin thermal boundary layer thickness. As the fluid moves downstream along the channel,
the Nusselt number decreases until reaches a fully developed value. The fully developed value of Nusselt number for a channel under the iso-flux thermal boundary condition is equal to 12 [30], which is indicated in Fig. 6. Comparison has also been made with the developing Nusselt number obtained analytically by Haji-Sheikh et al. [38], where good agreement can be found with the present analysis. In addition, Fig. 6 shows the effects of the effective conductivity ratio (k) on the developing Nusselt number, where it can be seen that the Nusselt number increases with increasing the conductivity ratio (k). In a porous medium with a higher conductivity ratio, which translates to a medium having a fluid phase with a relatively higher effective conductivity than that of the solid phase, the fluid phase plays the key-role in heat transferring. In other words, the ‘convective’ heat transfer, which is governed by the fluid phase, is enhanced by increasing k. The dimensionless thermally developing length (xdev.) is defined as a dimensionless axial location where:
DNu ¼
NuðxÞ Nufd ¼ 0:01 Nufd
ð42Þ
Combining Eqs. (41) and (42) yields the location at which the flow becomes thermally fully developed. Fig. 7 shows the relationship between thermally developing length, xdev., against the effective conductivity ratio, k. Similar to the observations in Fig. 6, by reducing k, the flow reaches its fully developed condition at a smaller dimensionless axial location. In a porous medium with a lower effective conductivity ratio, the convective heat transfer plays a relatively weaker role in comparison to the conductive heat transfer in the solid phase. It should be noted that in the present study the axial conduction is ignored and the thermal development phenomenon is only associated with the convective heat transfer. The foregoing discussion on the dependency of developing length (xdev.) on k explains why the value of the Nusselt number in the developing region is a function of k, while in the fully developed region it is independent of k as seen in Fig. 6 for LTE situation. Following a curve fitting based on the exponential nature of the present analytical results, the following expression is proposed for the first time to calculate the dimensionless thermally developing length. This expression reveals the validity region of numerous convection heat transfer studies, both analytical and numerical, with the simplified fully-developed assumption.
xdev : ¼ d1 ed2 k þ d3 ed4 k d1 ¼ 0:3608; d3 ¼ 0:3526;
100
d4 ¼ 0:8357
Nu
1
xdev.
10 k=10
0.1
Analytic results Fitted curve
k=1
xdev.= d1*exp(d2*k)+d3*exp(d4*k) d1 = 0.3608, d2 = 0.001339, d3 = -0.3526, d4 = -0.8357 R-square = 0.9978
k=0.1 Haji-Sheikh et al. (2006)
1 0.001
ð43Þ
d2 ¼ 0:001339
Fully developed Nu=12
0.01
x
0.1
1
Fig. 6. Thermally developing Nusselt number for the LTE condition (Bi ! 1).
0.01
0.1
1
k
10
100
Fig. 7. The thermal entrance length (xdev.) versus the effective conductivity ratio (k).
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100
Nu
5. Conclusion
10
The thermally developing forced convection of a Newtonian fluid through a Darcian porous material has been analytically studied based on the local thermal non-equilibrium (LTNE) approach. A constant heat flux is prescribed at the wall boundaries of the channel studied here (i.e. the iso-flux thermal boundary condition). A perturbation analysis has been performed to avoid using any model for the iso-flux thermal boundary condition. The main findings to emerge from the present study are as follows:
Present analysis Yang and Vafai (2010) Ouyang et al. (2013) Model A Ouyang et al. (2013) Model B Ouyang et al. (2013) Model C
1 0.001
0.01
x
0.1
1
Fig. 8. Validation of the local Nusselt number for a LTNE condition (Bi = 10, k = 1).
Nu
100
10
k=10 k=1
1 0.001
k=0.1
0.01
x
0.1
1
The LTNE intensity obtained by the existing models of the isoflux thermal boundary condition showed neither a unique behavior nor a similar trend for different pertinent parameters. While, the LTNE intensity obtained in the present analytical study is in agreement with the physical interpretations. It was shown that the LTNE intensity (hs hf) decreases by increasing the conductivity ratio (k) and the Biot number (Bi). The present analysis revealed that the LTNE intensity increases in the streamwise direction (x) at the centerline (y = 1) and decreases at the channel walls (y = 0). Also, it was shown that the LTNE intensity of the centerline is zero at the entrance (x 0). The validity limit of the iso-flux thermal boundary condition models in the absence of internal heat generation was discussed and based on the physics of the problem it was found that the results of any model would not be valid if the obtained LTNE intensity was negative. The Nusselt number of a thermally developing flow through a Darcian porous material was obtained based on the LTNE approach. The obtained developing Nusselt number has a simple form without the need of using any heuristic model. Finally, an expression representing the thermal entrance region, i.e. a dimensionless axial location where the flow becomes thermally developed, was proposed based on the present analytical solution. This expression reveals the validity region of the simplified fully developed heat transfer studies. Conflict of interest
Fig. 9. Thermally developing Nusselt number for a LTNE condition (Bi = 10).
There is no conflict of interest. To further study the thermally developing Nusselt number for a LTNE situation (Bi = 10), Figs. 8 and 9 are plotted. Fig. 8 depicts the present analysis local Nusselt number for k = 1 and Bi = 10 in comparison with those of Refs. [3,7]. A good agreement is seen between the results. Fig. 9 shows the distributions of the Nusselt number in the streamwise direction for three different effective conductivity ratios. As expected, it is seen that the Nusselt number increases with increasing the effective conductivity ratio. Also, it exhibits that the Nusselt number decreases along the flow direction, similar to the LTE situation. However, each curve, corresponding to a different effective conductivity ratio, reaches a slightly different fully developed value equal to 9.43, 10.43, and 11.68 for k = 0.1, 1, and 10 when Bi = 10, respectively. For LTNE situations, fluid and solid phases of the porous material have their independent roles in heat transferring due to the imperfect inter-phase heat exchange 1
between the solid and fluid phases of the porous material. k 1
and Bi represent the importance of the solid-phase conduction and inter-phase thermal resistances, respectively. So that, the Nusselt number is a function of both k and Bi for LTNE situations. Furthermore, the foregoing statement reveals why the Nusselt number of a thermally developing flow, as well as the fully developed one, for the LTNE condition (Bi = 10) is lower than that for the LTE condition (Bi ! 1).
References [1] P.X. Jiang, R.N. Xu, W. Gong, Particle-to-fluid heat transfer coefficients in miniporous media, Chem. Eng. Sci. 61 (2006) 7213–7222. [2] B. Deng, Y. Qiu, C.N. Kim, An improved porous medium model for microchannel heat sinks, Appl. Therm. Eng. 30 (2010) 2512–2517. [3] K. Yang, K. Vafai, Analysis of temperature gradient bifurcation in porous media – an exact solution, Int. J. Heat Mass Transfer 53 (2010) 4316–4325. [4] K. Yang, K. Vafai, Transient aspects of heat flux bifurcation in porous media: an exact solution, J. Heat Transfer 133 (5) (2011) 052602. [5] G.R. Imani, M. Maerefat, K. Hooman, Estimation of heat flux bifurcation at the heated boundary of a porous medium using a pore-scale numerical simulation, Int. J. Therm. Sci. 54 (2012) 109–118. [6] G.R. Imani, M. Maerefat, K. Hooman, Pore-scale numerical experiment on the effect of the pertinent parameters on heat flux splitting at the boundary of a porous medium, Transp. Porous Media 98 (2013) 631–649. [7] X. Ouyang, K. Vafai, P. Jiang, Analysis of thermally developing flow in porous media under local thermal non-equilibrium conditions, Int. J. Heat Mass Transfer 67 (2013) 768–775. [8] Y. Mahmoudi, N. Karimi, K. Mazaheri, Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition: effects of different thermal boundary conditions at the porous–fluid interface, Int. J. Heat Mass Transfer 70 (2014) 875–891. [9] Y. Mahmoudi, N. Karimi, Numerical investigation of heat transfer enhancement in a pipe partially filled with a porous material under local thermal non-equilibrium condition, Int. J. Heat Mass Transfer 68 (2014) 161– 173. [10] Y. Mahmoudi, Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local
M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
[11]
[12]
[13] [14]
[15]
[16]
[17]
[18] [19]
[20]
[21]
[22]
[23]
[24]
thermal non-equilibrium condition, Int. J. Heat Mass Transfer 85 (2015) 524– 542. A. Amiri, K. Vafai, T.M. Kuzay, Effect of boundary conditions on non-Darcian heat transfer through porous media and experimental comparisons, Numer. Heat Transfer J. Part A 27 (1995) 651–664. A.V. Kuznetsov, Thermal nonequilibrium, non-Darcian forced convection in a channel filled with a fluid saturated porous medium – a perturbation solution, Appl. Sci. Res. 57 (1997) 119–131. K. Vafai, S.J. Kim, Forced convection in a channel filled with a porous medium: an exact solution, ASME J. Heat Transfer 111 (1989) 1103–1106. K. Vafai, S.J. Kim, Discussion of the paper by A. Hadim, ‘‘forced convection in a porous channel with localized heat sources”, ASME J. Heat Transfer 117 (1995) 1097–1098. D.A. Nield, S.L.M. Junqueira, J.L. Lage, Forced convection in a fluid saturated porous medium channel with isothermal or isoflux boundaries, J. Fluid Mech. 322 (1996) 201–214. D.A. Nield, A.V. Kuznetsov, Local thermal nonequilibrium effects in forced convection in a porous medium channel: a conjugate problem, Int. J. Heat Mass Transfer 42 (1999) 3245–3252. D.Y. Lee, K. Vafai, Analytical characterization and conceptual assessment of solid and fluid temperature differentials in porous media, Int. J. Heat Mass Transfer 31 (1999) 423–435. B. Alazmi, K. Vafai, Analysis of variants within the porous media transport models, ASME J. Heat Transfer 122 (2000) 303–326. B. Alazmi, K. Vafai, Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions, Int. J. Heat Mass Transfer 45 (2002) 3071–3087. D.A. Nield, A.V. Kuznetsov, M. Xiong, Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects, Int. J. Heat Mass Transfer 46 (2003) 643–651. A.V. Kuznetsov, M. Xiong, D.A. Nield, Thermally developing forced convection in a porous medium: circular duct with walls at constant temperature, with longitudinal conduction and viscous dissipation effects, Transp. Porous Media 53 (2003) 331–345. D.A. Nield, A.V. Kuznetsov, Thermally developing forced convection in a channel occupied by a porous medium saturated by a non-Newtonian fluid, Int. J. Heat Mass Transfer 48 (2005) 1214–1218. P.X. Jiang, X.C. Lu, Numerical simulation and theoretical analysis of thermal boundary characteristics of convection heat transfer in porous media, Int. J. Heat Fluid Flow 28 (2007) 1144–1156. M. Dehghan, M.S. Valipour, S. Saedodin, Perturbation analysis of the local thermal non-equilibrium condition in a fluid saturated porous medium bounded by an iso-thermal channel, Transp. Porous Media 102 (2) (2014) 139–152.
823
[25] M. Dehghan, M.T. Jamal-Abad, S. Rashidi, Analytical interpretation of the local thermal non-equilibrium condition of porous media imbedded in tube heat exchangers, Energy Convers. Manage. 85 (2014) 264–271. [26] M. Dehghan, Y. Rahmani, D.D. Ganji, S. Saedodin, M.S. Valipour, S. Rashidi, Convection–radiation heat transfer in solar heat exchangers filled with a porous medium: homotopy perturbation method versus numerical analysis, Renewable Energy 74 (2015) 448–455. [27] M. Dehghan, Y. Mahmoudi, M.S. Valipour, S. Saedodin, Combined conduction– convection–radiation heat transfer of slip flow inside a micro-channel filled with a porous material, Transp. Porous Media 108 (2015) 413–436, http://dx. doi.org/10.1007/s11242-015-0483-z. [28] D.A. Nield, A.V. Kuznetsov, Forced convection in a parallel-plate channel occupied by a nanofluid or a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 70 (2014) 430–433. [29] M. Dehghan, M.S. Valipour, S. Saedodin, Temperature-dependent conductivity in forced convection of heat exchangers filled with porous media: a perturbation solution, Energy Convers. Manage. 91 (2015) 259–266. [30] D.A. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 2006. [31] M. Mirzaei, M. Dehghan, Investigation of flow and heat transfer of nanofluid in microchannel with variable property approach, Heat Mass Transfer 49 (2013) 1803–1811. [32] M. Dehghan, M. Daneshipour, M.S. Valipour, R. Rafee, S. Saedodin, Enhancing heat transfer in microchannel heat sinks using converging flow passages, Energy Convers. Manage. 92 (2015) 244–250. [33] M. Dehghan, M.S. Valipour, S. Saedodin, Analytical study of heat flux splitting in micro-channels filled with porous media, Transp. Porous Media 109 (2015) 571–587. [34] M. Dehghan, Effects of heat generations on the thermal response of channels partially filled with non-Darcian porous materials, Transp. Porous Media (2015), http://dx.doi.org/10.1007/s11242-015-0567-9. [35] M. Dehghan, M.S. Valipour, S. Saedodin, Y. Mahmoudi, Thermally developing flow inside a porous-filled channel in the presence of internal heat generation under local thermal non-equilibrium condition: A perturbation analysis, Appl. Therm. Eng. (2016). in preparation. [36] M. Dehghan, M.S. Valipour, S. Saedodin, Conjugate heat transfer inside microchannels filled with porous media: an exact solution, J. Thermophys. Heat Transfer (2016). in preparation. [37] M. Dehghan, M.S. Valipour, S. Saedodin, Microchannels enhanced by porous materials: heat transfer enhancement or pressure drop increment, Energy Convers. Manage. (2016). in preparation. [38] A. Haji-Sheikh, D.A. Nield, K. Hooman, Heat transfer in the thermal entrance region for flow through rectangular porous passages, Int. J. Heat Mass Transfer 49 (2006) 3004–3015.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
On the thermally developing forced convection through a porous material under the local thermal non-equilibrium condition: An analytical study Maziar Dehghan a,⇑, M.S. Valipour a, Amir Keshmiri b,c, S. Saedodin a, Nima Shokri d,⇑ a
Faculty of Mechanical Engineering, Semnan University, Semnan, Iran School of Engineering, Manchester Metropolitan University, Manchester, UK School of Mechanical Aerospace and Civil Engineering, , University of Manchester, Manchester, UK d School of Chemical Engineering and Analytical Science, University of Manchester, Manchester, UK b c
a r t i c l e
i n f o
Article history: Received 1 June 2015 Accepted 30 August 2015 Available online 1 October 2015 Keywords: Thermally developing forced convection Analytical study Iso-flux thermal boundary condition modeling Porous medium Local thermal non-equilibrium
a b s t r a c t The aim of the present analytical study is to investigate a thermally developing forced convective heat transfer inside a channel filled with a porous medium whose walls are imposed to a constant heat flux (i.e. the iso-flux thermal boundary condition). The Darcy’s law of motion and the two-energy equation (i.e. local thermal non-equilibrium, LTNE) model are considered. A perturbation analysis is conducted to avoid using any model for the iso-flux thermal boundary condition. Thermally developing forced convection inside a porous-filled channel has previously not been analyzed without implementing a heuristic model for the iso-flux thermal boundary condition. The temperature difference between the fluid and solid phases (called the LTNE intensity) is analytically obtained. Results concerning the LTNE intensity is compared with the available models of the iso-flux thermal boundary condition. In addition, the Nusselt number of a thermally developing convection through a porous material is obtained for the LTNE condition. Also, an expression representing the developing length, i.e. the dimensionless axial location at which the flow becomes fully developed, for a porous-filled channel is proposed for the first time. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Local thermal non-equilibrium (LTNE) may be observed in various engineering applications including microchannel heat sinks, heat pipes, fluidized beds, dryers, and catalytic reactors, thus, motivating numerous researchers to study the LTNE condition with an aim of accurately predicting the thermal behavior of porous media [1–10]. Amiri et al. [11] numerically focused on the inertial as well as viscous effects of the equation of motion of a fluid saturated porous medium. For the first time, they discussed on the iso-flux thermal boundary condition modeling. Kuznetsov [12] used the Brinkman–Forchheimer-extended Darcy equation to analytically study the LTNE condition for the first time. Kuznetsov [12] discussed the thermal non-equilibrium effects in a channel filled with a fluid saturated porous medium based on a perturbation analysis. Kuznetsov used the results of the previous studies (Vafai and Kim [13,14], and Nield et al. [15]) for the flow ⇑ Corresponding authors. E-mail addresses: [email protected], [email protected] (M. Dehghan), [email protected] (N. Shokri). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.091 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
field and a two-equation energy model [16,17] for the temperature field and proposed a simplified relation for measuring the LTNE. Using the Darcy’s law of motion, Nield and Kuznetsov [16] analytically investigated the conjugate heat transfer problem of a saturated porous channel. Lee and Vafai [17] analytically characterized the forced convective flow through a channel filled with a porous material and obtained the dimensionless temperature for the fluid and solid phases as well as the Nusselt number. Alazmi and Vafai [18] investigated four important categories in modeling the fluid flow and heat transfer through porous media, namely constant and variable porosity, thermal dispersion, and local thermal non-equilibrium phenomena. They numerical experiments showed that the above-mentioned variants have stronger influences on the flow field than the heat transfer. Alazmi and Vafai [19] numerically studied two primary assumptions for the iso-flux thermal boundary condition with seven different sub-models. They found the Nusselt number obtained by each model. Nield et al. [20] applied a modified Graetz methodology to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial
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Nomenclature asf specific surface area (m1) b a constant value Bi Biot number cp specific heat at constant pressure (J kg1 K1) d0, dn constant values di, i = 1–4 constant values pore diameter (m) dp f an intermediate function g an intermediate function H half of the channel gap (m) hsf fluid–solid heat transfer coefficient (W m2 K1) k conductivity ratio (kf,eff /ks,eff) kf conductivity of fluid phase (W m1 K1) kf,eff effective conductivity of fluid phase (W m1 K1) effective conductivity of the medium (kf,eff + ks,eff) km (W m1 K1) ks conductivity of solid phase (W m1 K1) effective conductivity of solid phase (W m1 K1) ks,eff MA Model A MB Model B MC Model C n counter Nu Nusselt number O order of magnitude Pe Peclet number Pr Prandtl number q00 w heat flux at the wall (W m2)
conduction and viscous dissipation included. Their analysis based on the LTNE assumption led to expressions for the local Nusselt number. Kuznetsov et al. [21] followed a similar methodology to the one used in Nield et al. [20] to investigate the thermally developing forced convection inside a fluid saturated pipe with an isothermal wall. Later, Nield and Kuznetsov [22] applied the classical Graetz methodology to study the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at constant temperature, for the case of a non-Newtonian fluid of power-law type. Nield and Kuznetsov [22] obtained the Nusselt number at LTNE condition. Based on a pore-scale numerical simulation, Jiang and Lu [23] explored effects of the thermal contact resistance between a porous medium and an impermeable wall. They showed that the first approach of the iso-flux thermal boundary condition models (i.e. Model A of Ouyang et al. [7]) gives the closest results to the experiments when the thermal contact resistance is negligible and the wall has a finite thickness. Yang and Vafai [4] analytically investigated the effects of internal heat generations within the fluid and solid phases on the thermal behavior of a channel filled with a porous material. Later on, Yang and Vafai [5] analytically solved the problem of transient forced convection inside a saturated porous medium incorporating the LTNE model. Ouyang et al. [7] derived analytical solutions for thermally developing flows in porous media under LTNE condition for the constant wall heat flux boundary condition. They used three different models to obtain the local Nusselt number and to predict the dimensionless thermal entry length. Recently based on a perturbation analysis, Dehghan et al. [24,25] investigated the LTNE situation in a fluid saturated channel or pipe filled with a saturated porous material. They proposed a dimensionless number representing the LTNE intensity in terms of pertinent parameters. Following the work of Dehghan et al. [26] who analyzed the heat transfer rate in a solar heat exchanger filled with porous media for the case of combined conduction– convection–radiation heat transfer, Dehghan et al. [27]
Re T Ti u x⁄, y⁄ x, y
Reynolds number temperature (K) inlet temperature (K) Darcian velocity (m s1) dimensional coordinates (m) dimensionless coordinates
Greek letters
DNE
e
h
q rn
/
LTNE intensity (hs hf) small parameter (1/hsfasf, W1 m3 K) dimensionless temperature fluid density (kg m3) parameter in the obtained dimensionless temperature profile porosity of the medium
Subscripts dev f fd i m n s w
developing fluid phase fully developed component inlet bulk-mean value counter solid phase wall
investigated effects of the slip-flow regime on the combined convective–radiative heat transfer rate. Dehghan et al. [27] showed that the heat transfer rate is mainly controlled by the temperature jump phenomenon. Nield and Kuznetsov [28] analytically obtained the Nusselt number for forced convection heat transfer of nanofluids flow through a clear or porous-filled channel at LTE condition. Dehghan et al. [29] solved the non-linear problem of forced convection inside a porous-filled channel with a temperaturedependent conductivity using the perturbation technique. They found that the Nusselt number increases with a variable conductivity approach. In the present study, thermally developing forced convection in a fluid saturated porous medium bounded by two infinite parallelplates held at a constant heat flux is analytically investigated allowing the temperature of fluid and solid phases to be different (LTNE). It is assumed that the Darcy’s law governs the fluid motion in the porous medium. The energy equation has been solved by the method of Fourier series. The temperature difference between the solid and fluid phases, called the ‘LTNE intensity’, is found without the need for implementing any model for the iso-flux thermal boundary condition. In addition, the Nusselt number and thermally developing length are also obtained. To the best of authors’ knowledge, no other study has investigated the thermally developing forced convection heat transfer in a porous-filled channel without implementing a model for the iso-flux thermal boundary condition. 2. Mathematical modeling The schematic diagram of the problem considered in this study is shown in Fig. 1. The following assumptions are invoked in the formulation of the model: The flow in the porous medium is incompressible and uniform along the cross-sectional area (Darcy’s law).
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ks;eff @ 2 T f þ Oðe2 Þ hsf asf @y2
ðT s T f Þ ¼
ð8Þ
To define the temperature difference (or the intensity of LTNE condition), the value of
@2 T f @y2
should be defined. From Eqs. (1), (2)
and (7) it can be shown that:
qcp u
@T f @2T f ¼ km 2 þ OðeÞ @x @y
ð9Þ
where km is the effective thermal conductivity of the medium and equals to ks;eff þ kf ;eff . For the sake of brevity, the subscript ‘‘f” will be dropped hereafter. Now, the solution reduces to solve a fluid saturated porous medium under the LTE condition (Eq. (9)). The boundary conditions for the LTE model under the situation of a constant heat flux imposed at the walls are:
km
Fig. 1. Schematic diagram of the present problem.
The thermally developing forced convection is desired. The porous medium is isotropic and homogenous. Walls are impermeable with infinite dimension perpendicular to the plane of view to ensure the two-dimensionality of the problem. The temperature of the solid and fluid phases could be different (i.e. LTNE condition).
@T ¼0 @y
qcp u
0 ¼ ks;eff
@2T s hsf asf ðT s T f Þ @y2
ð1Þ
ð2Þ
ks;eff ¼ ð1 /Þks
ð3Þ
kf ;eff ¼ /kf
ð4Þ
where / is the porosity of the medium, and hsf and asf are the fluid– solid heat transfer coefficient and specific surface area (surface per unit volume) respectively, and are defined as [30]:
"
hsf ¼
ð5Þ
1 kf qudp 2 þ 1:1Pr 3 dp l
0:6 # ð6Þ
where dp is particle diameter and Pr is the Prandtl number. In Eqs. (1) and (2) it is assumed that the axial heat conduction term is negligible [7]. 3. Analysis and solution
Ts ¼ Tf þ O
e¼
1 hsf asf
ð11Þ
x¼
ðT T 0 Þks;eff q00w H x k f
¼
qcp H u 2
x =H ; Pe
ð12Þ
y¼
y ; H
k¼
kf ;eff ks;eff
ð7Þ
where e is a positive parameter with a small value, i.e. OðeÞ < 1. It is chosen for the perturbation analysis. Combining Eqs. (2) and (7) yields:
ð13Þ
where h is the dimensionless temperature, x is the dimensionless axial coordinate, Pe is the Peclet number, y is the dimensionless vertical coordinate, and k is the effective conductivity ratio. Combining Eqs. (7), (9), (12) and (13) results in:
ð70 Þ
hs ¼ hf þ OðBi Þ k
@h @2h 1 ¼ ðk þ 1Þ 2 þ OðBi Þ @x @y
ð14Þ
and the dimensionless boundary conditions for the LTE model are:
@h 1 ¼ y¼0 @y kþ1 @h ¼0 y¼1 @y
ð15Þ
x¼0
h¼0
ð16Þ
Combining Eqs. (8) and (14) yields:
ðT s T f Þ ¼
k q00w @h þ Oðe2 Þ ðk þ 1Þ hsf asf H @x
ð17Þ
Eq. (17) reveals that the temperature difference between the . Now in fluid and solid phases is proportional to the value of @h @x order to find the dimensionless temperature distribution (h), one can solve Eq. (14) with the boundary conditions (15) and (16) through:
hðx; yÞ ¼ gðx; yÞ þ f ðyÞ
Using the scale analysis and noting that the product of hsfasf is a large value, one can show from Eq. (2) that [2,12,24,25]:
x ¼ 0
1
where subscripts ‘s’ and ‘f’ denote solid and fluid phases, respectively, T is the temperature, cp is the specific heat of the fluid phase, u is the Darcy velocity and, ks,eff and kf,eff are the effective thermal conductivities of the solid and fluid phases given by [30]:
asf ¼ 6ð1 /Þ=dp
ð10Þ
where Ti is the inlet temperature of the fluid phase. To solve Eq. (9), the following dimensionless variables and parameters are defined:
2
@T f @ Tf ¼ kf ;eff 2 þ hsf asf ðT s T f Þ @x @y
y ¼ 0 y ¼ 1
T ¼ Ti
h¼ According to the above assumptions, the steady-state energy conservation equations of the solid and fluid phases are given as [7,24,30]:
@T ¼ q00w @y
ð18Þ
Combining Eqs. (14) and (18) results in:
kg x ¼ ðk þ 1Þg yy þ ðk þ 1Þf
00
ð19Þ
with the following updated boundary conditions:
1 y¼0 kþ1 0 gy þ f ¼ 0 y¼1 0
gy þ f ¼
ð20Þ
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gð0; yÞ þ f ðyÞ ¼ 0
x¼0
ð21Þ
Now, the solution procedure is reduced to the solution of two individual differential equations. Since the boundary conditions in the transverse direction are of Neumann type (the first order of differentiation), an additional step is required to satisfy the boundary conditions:
ðhs hf Þ ¼ DNE ¼
00
f ¼
1 ; kþ1
8 > > <
1 Biðk þ 1Þ > > :fully
0
f ð0Þ ¼
1 0 ; f ð1Þ ¼ 0 kþ1
DNE ¼ ðhs hf Þ ¼ ð23Þ
where b is an unknown constant. Subsequently, combining Eqs. (19) and (22) results in the second differential equation expressed as:
kg x ¼ ðk þ 1Þg yy þ 1
ð24Þ
with the following boundary conditions:
y¼0 y¼1
ð25Þ ð26Þ
Based on the boundary conditions (25), the solution of Eq. (24) is expected to have the following form:
gðx; yÞ ¼ c0 ðxÞ þ
1 X
½cn ðxÞ cosðnpyÞ
ð27Þ
n¼1
Substituting Eq. (27) into Eq. (24) and using the orthogonality relations of the trigonometric functions results in:
c0 ðxÞ ¼
x þ d0 k
cn ðxÞ ¼ dn expðrn xÞ;
ð28Þ n ¼ 1; 2; . . .
ð29Þ
where d0 and dn are constants and rn ¼ pÞ .The inlet condition, Eq. (26), should be used to obtain the constants (d0 and dn). One can derive the following solution by combining Eqs. (23) and (26)–(29): ðkþ1Þ ðn k
gðx; yÞ ¼
2
x 1 þ b k 3ðk þ 1Þ " # 1 X 2 expð r xÞ cosðn p yÞ þ n 2 n¼1 ðk þ 1ÞðnpÞ
ð30Þ
Combining Eqs. (18), (23) and (30) yields the dimensionless temperature distribution for the LTE model under the condition of constant heat flux imposed at the walls:
hðx; yÞ ¼
ðT s T f Þks;eff q00w H
ð34Þ
DNE consists of two components (see Eq. (33)): developing and fully developed components. The fully developed component of the LTNE intensity (DNEfd) is:
DNEfd ¼ ðhs hf Þfd ¼
1 2 þ OðBi Þ Biðk þ 1Þ
ð35Þ
where the Biot number, Bi ¼ hsf asf H2 =ks;eff . Eq. (35) shows that DNEfd is the same as what was proposed by Kuznetsov [12] and Dehghan et al. [25] for thermally fully developed flow under the Darcy’s law and under the constant heat flux thermal boundary condition. Also, one can show that OðeÞ OðBi Þ. In summary, a fundamental analytical relation, Eq. (33), representing the intensity of LTNE condition (DNE) has been obtained. Eq. (33) is a simple and general relation that can be used for the estimation of the LTNE intensity (DNE) as well as validation of numerical simulations. Further parametric studies on DNE and comparison with available models are presented in the next section. 1
x¼0
gð0; yÞ ¼ f ðyÞ
ð33Þ
The LTNE intensity, DNE, represents:
ð22Þ
y 2y þb 2ðk þ 1Þ
gy ¼ 0
And the dimensionless temperature difference between the fluid and solid phases (called LTNE intensity) is:
dev eloping component
2
gy ¼ 0
ð32Þ
9 > > = X1 2 1 þ ½2 expð r xÞ cosðn p yÞ þ OðBi Þ n |{z} n¼1 > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl } > ; dev eloped component
The solution can be written as:
f ðyÞ ¼
( ) 1 X q00w ðT s T f Þ ¼ ½2 expðrn xÞ cosðnpyÞ þ Oðe2 Þ 1þ hsf asf Hðk þ 1Þ n¼1
1 x 3ðy 1Þ2 1 X 2 þ expðrn xÞ cosðnpyÞ k 6ðk þ 1Þ krn n¼1
ð31Þ
The temperature difference between the fluid and solid phases can be found by combining Eqs. (17) and (31):
4. Results and discussions 4.1. The LTNE intensity at the wall Fig. 2 shows the LTNE intensity (hs hf) versus dimensionless axial coordinate for Bi = 10 and different conductivity ratios (k) at the channel wall (y = 0). The figure compares the present dimensionless temperature difference between the fluid and solid phases obtained analytically with those obtained by Ouyang et al. [7] at the channel wall. Ouyang et al. [7] used three different models (i. e. models A, B, and C) to investigate the thermally developing heat transfer in a fluid saturated porous medium imbedded in a channel. These models were used to model the iso-flux thermal boundary condition at the walls [7]; Model A:
hs ¼ hf ;
k
@hf @hs ¼1 @y @y
y¼0
ð36Þ
Model B:
k
@hf @hs ¼ ¼1 @y @y
y¼0
ð37Þ
M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
0.6
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0.4
(θs-θf )
0.2 0 -0.2 -0.4 -0.6 -0.8 0.001
0.01
x
0.1
1
(a) 0.7
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0.6 0.5
(θs-θf )
0.4 0.3 0.2 0.1 0 -0.1 -0.2 0.001
0.01
x
0.1
1
(b) 0.45 0.4 0.35
(θs-θf )
0.3 0.25 Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0.2
0.15 0.1
Meanwhile, the fully developed component of Model A is shown in the figures. In the following figures ‘‘MA”, ‘‘MB”, and ‘‘MC” represent ‘‘Model A”, ‘‘Model B”, and ‘‘Model C” of Ref. [7], respectively. Fig. 2 shows that the LTNE intensity (hs hf) obtained by the present study decreases in the streamwise direction and with increasing the conductivity ratio (k), which is in agreement with the findings of Dehghan et al. [24,25]. The conductivity ratio (k) increases with increasing the porosity (/). At higher porosities, the occupying solid phase has less thermal inertia and would follow the fluid phase more easily and vice versa [24,25,31,32], which lead to lower LTNE intensities. In addition, by increasing the conductivity ratio (k), the ability of heat transfer in the fluid phase compared to the solid phase increases and results in a lower degree of LTNE intensity. Based on the assumptions made for Model A, the absolute value of the LTNE intensity is zero for this model (i.e. Model A) at y = 0. Comparing Fig. 2(b) with Fig. 2(c) reveals that the LTNE intensity obtained by Model B increases with increased conductivity ratio (k). Model B returns negative LTNE intensity for k < 1 but becomes positive for k P 1. Meanwhile, Model C gives negative values for k 6 1. It should be noted that a negative value for the LTNE intensity (hs hf) on the wall (y = 0) is unphysical as there is no internal heat generation in the fluid phase. Since the heat is transferred from the wall to the porous medium and is cooled via entering the fluid phase with a cold temperature at the inlet (Ti), the heat is transmitted through the solid phase to the fluid phase. Consequently, based on the second law of thermodynamics the fluid phase cannot have a higher temperature than the solid phase. Fig. 3 shows the effects of the Biot number on the LTNE intensity for k = 1. It is seen that the LTNE intensity decreases by increasing the Biot number (Bi). A higher Biot number represents an enhanced internal convective heat transfer between the solid and fluid phases and consequently, a lower temperature difference between the two phases (i.e. lower LTNE intensity). For example, the LTNE intensity corresponding to Bi = 10 is about one order of magnitude larger than that corresponding to Bi = 100. Moreover, the LTNE intensity at the wall (y = 0) decreases along the streamwise direction since the fluid phase moves in contact with the solid phase to reach a fully developed LTNE intensity. Another finding to emerge implicitly from Fig. 3 is that the portion of the wall which is in direct contact with the fluid phase has a lower dimensionless temperature (equal to zero) due to the ‘no-jump condition’ which says that the fluid should have the same temperature as the adjacent wall temperature [27,31,32]. On the other hand, the other portion of the wall, which is in direct contact with the solid phase,
0.05 0 0.001
0.01
x
0.1
1
(c) Fig. 2. LTNE intensity versus dimensionless axial direction (Bi = 10 and y = 0); (a) k = 0.1, (b) k = 1, (c) k = 10.
Model C:
k
@hf ¼ /; @y
@hs ¼1/ @y
y¼0
ð38Þ
Since Model C depends on the porosity of the medium and this dependency is not the focus of the present study, a constant value for / = equal to 0.9 was used in plotting the results of Model C.
819
Fig. 3. Effects of the Biot number on the LTNE intensity (k = 1 and y = 0).
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M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
has a dimensionless temperature equal to the value of the LTNE intensity. This matter indicates the need for introducing a model for the iso-flux thermal boundary condition. This has led to, for example the work of Ouyang et al. [7], where three different models for the iso-flux thermal boundary condition have been used which have resulted in different LTNE intensities.
0.2
(θs-θf )
0.15 0.1
0.05 Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
0 -0.05 0.001
0.01
0.1
x
1
(a) 0.1
(θs-θf )
0.05
0
-0.05 0.001
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC 0.01
x
0.1
1
(b) 0.05 0.04
In the previous subsection, the LTNE intensity at the channel wall (y = 0) was discussed for different pertinent parameters. In this subsection, the LTNE intensity at the channel centerline (y = 1) is discussed. The LTNE intensity at the centerline for Bi = 10 is shown in Fig. 4. Similar to Figs. 2 and 3, the LTNE intensity obtained by the present analysis decreases by increasing the conductivity ratio (k). However, in contrast to the results reported earlier at the wall, it is now seen that the LTNE intensity at the channel centerline increases in the streamwise direction (x). Moreover, it is worth noting that in Fig. 4 the present analysis shows that the LTNE intensity is zero at the entrance for different values of conductivity ratio, while the models of Ouyang et al. [7] (i.e. models A, B, and C) return non-zero LTNE intensity at the entrance. Another point to highlight about the results in Fig. 4 is that further downstream, values of the LTNE intensity of the present analysis approach to those of models A and C. Based on the observations presented in Figs. 2 and 4 (particularly in relation to the sign and magnitude of the LTNE), it is concluded that the present analytical results are in agreement with the physics of the problem and the ‘scale analysis’ prediction [11,24,25]. The reason for the discrepancies between the present analysis and the models of Ouyang et al. [7] is that no heuristic model was used to describe the iso-flux thermal boundary condition at the wall (i.e. a constant heat flux imposed at the walls). This matter shows the importance of the modeling of the iso- flux thermal boundary condition. In Fig. 5, the effects of the Biot number on the LTNE intensity are investigated at the centerline for k = 1. It shows that the temperature difference between the solid and fluid phases (LTNE intensity) is negligible in the entrance region due to a high convective heat transfer coefficient of the fluid phase in this region. The LTNE intensity increases along the channel length since the convective heat transfer coefficient of the fluid phase decreases in the streamwise direction. In addition, Fig. 5 indicates that the LTNE intensity decreases with increased Biot number due to a lower internal convective thermal resistance between the two phases. Finally, similar to the argument of Ref. [17], when there is no heat generation within the solid and fluid phases and the heat is imposed from the wall, the dimensionless temperature of the solid phase should be greater than the one of the fluid phase. This occurs since the solid phase acts as an extended surface propagating the heat flux throughout the fluid volume. Consequently, the heat flux must flow through the solid phase in order to be convected via the
Present Analysis Ouyang et al. (2013), MA Ouyang et al. (2013), MB Ouyang et al. (2013), MC
(θs-θf )
0.03
4.2. The LTNE intensity at the centerline
0.02 0.01 0
-0.01 0.001
0.01
x
0.1
1
(c) Fig. 4. LTNE intensity versus dimensionless axial direction (Bi = 10 and y = 1); (a) k = 0.1, (b) k = 1, (c) k = 10.
Fig. 5. Effects of the Biot number on the LTNE intensity (k = 1 and y = 1).
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M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
fluid phase. This matter imposes a limit on the validity of the models in which the dimensionless temperature of the solid phase must be greater than that of the fluid phase resulting in positive LTNE intensities (hs hf > 0). Therefore, the magnitude of the LTNE intensity plays an important role in determining the validity of the models used for the thermal boundary condition. It should be noted that the above-mentioned discussion is valid when there is no internal heat generation/absorption within the porous material. Effects of heat generations/absorptions within the solid and fluid phases of a porous material on the thermal response have been studied by Refs. [33–37] in details. 4.3. The Nusselt number The Nusselt number for a thermally developing flow based on the LTNE approach is defined as [30]:
Nu ¼
q00w 4H km ðT w T f ;m Þ
ð39Þ
where hf ;m ðxÞ is the bulk-mean temperature of the fluid phase given by:
Z hf ;m ðxÞ ¼
1
hf ðx; yÞdy
ð40Þ
0
Based on the present solution, the local Nusselt number is obtained as follows:
NuðxÞ ¼ 1
3
P1
4
1 rx þ n¼0 ðnpÞ e Biðkþ1Þ 2
2
þ OðBi Þ
ð41Þ
2
The terms in the denominator of Eq. (41) represents the following three phenomena involved in a thermally developing force convection through a porous material: (1) the fully developed component (i.e. 1/3) of the Nusselt number of a fluid convected through a Darcian porous material based on the LTE approach, (2) the developing component of the Nusselt number, and (3) the LTNE intensity of the convective heat transfer. It should be noted that such a simple expression for the local Nusselt number has previously not been proposed, which does not use any model for the iso-flux thermal boundary condition. Fig. 6 shows the developing Nusselt number along the dimensionless flow direction (x = x⁄/(H.Pe), where x⁄ is the dimensional axial-direction) for LTE condition (Bi ! 1). As one would expect, the Nusselt number starts off with a high value in the entrance region because of the inlet condition and thin thermal boundary layer thickness. As the fluid moves downstream along the channel,
the Nusselt number decreases until reaches a fully developed value. The fully developed value of Nusselt number for a channel under the iso-flux thermal boundary condition is equal to 12 [30], which is indicated in Fig. 6. Comparison has also been made with the developing Nusselt number obtained analytically by Haji-Sheikh et al. [38], where good agreement can be found with the present analysis. In addition, Fig. 6 shows the effects of the effective conductivity ratio (k) on the developing Nusselt number, where it can be seen that the Nusselt number increases with increasing the conductivity ratio (k). In a porous medium with a higher conductivity ratio, which translates to a medium having a fluid phase with a relatively higher effective conductivity than that of the solid phase, the fluid phase plays the key-role in heat transferring. In other words, the ‘convective’ heat transfer, which is governed by the fluid phase, is enhanced by increasing k. The dimensionless thermally developing length (xdev.) is defined as a dimensionless axial location where:
DNu ¼
NuðxÞ Nufd ¼ 0:01 Nufd
ð42Þ
Combining Eqs. (41) and (42) yields the location at which the flow becomes thermally fully developed. Fig. 7 shows the relationship between thermally developing length, xdev., against the effective conductivity ratio, k. Similar to the observations in Fig. 6, by reducing k, the flow reaches its fully developed condition at a smaller dimensionless axial location. In a porous medium with a lower effective conductivity ratio, the convective heat transfer plays a relatively weaker role in comparison to the conductive heat transfer in the solid phase. It should be noted that in the present study the axial conduction is ignored and the thermal development phenomenon is only associated with the convective heat transfer. The foregoing discussion on the dependency of developing length (xdev.) on k explains why the value of the Nusselt number in the developing region is a function of k, while in the fully developed region it is independent of k as seen in Fig. 6 for LTE situation. Following a curve fitting based on the exponential nature of the present analytical results, the following expression is proposed for the first time to calculate the dimensionless thermally developing length. This expression reveals the validity region of numerous convection heat transfer studies, both analytical and numerical, with the simplified fully-developed assumption.
xdev : ¼ d1 ed2 k þ d3 ed4 k d1 ¼ 0:3608; d3 ¼ 0:3526;
100
d4 ¼ 0:8357
Nu
1
xdev.
10 k=10
0.1
Analytic results Fitted curve
k=1
xdev.= d1*exp(d2*k)+d3*exp(d4*k) d1 = 0.3608, d2 = 0.001339, d3 = -0.3526, d4 = -0.8357 R-square = 0.9978
k=0.1 Haji-Sheikh et al. (2006)
1 0.001
ð43Þ
d2 ¼ 0:001339
Fully developed Nu=12
0.01
x
0.1
1
Fig. 6. Thermally developing Nusselt number for the LTE condition (Bi ! 1).
0.01
0.1
1
k
10
100
Fig. 7. The thermal entrance length (xdev.) versus the effective conductivity ratio (k).
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M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
100
Nu
5. Conclusion
10
The thermally developing forced convection of a Newtonian fluid through a Darcian porous material has been analytically studied based on the local thermal non-equilibrium (LTNE) approach. A constant heat flux is prescribed at the wall boundaries of the channel studied here (i.e. the iso-flux thermal boundary condition). A perturbation analysis has been performed to avoid using any model for the iso-flux thermal boundary condition. The main findings to emerge from the present study are as follows:
Present analysis Yang and Vafai (2010) Ouyang et al. (2013) Model A Ouyang et al. (2013) Model B Ouyang et al. (2013) Model C
1 0.001
0.01
x
0.1
1
Fig. 8. Validation of the local Nusselt number for a LTNE condition (Bi = 10, k = 1).
Nu
100
10
k=10 k=1
1 0.001
k=0.1
0.01
x
0.1
1
The LTNE intensity obtained by the existing models of the isoflux thermal boundary condition showed neither a unique behavior nor a similar trend for different pertinent parameters. While, the LTNE intensity obtained in the present analytical study is in agreement with the physical interpretations. It was shown that the LTNE intensity (hs hf) decreases by increasing the conductivity ratio (k) and the Biot number (Bi). The present analysis revealed that the LTNE intensity increases in the streamwise direction (x) at the centerline (y = 1) and decreases at the channel walls (y = 0). Also, it was shown that the LTNE intensity of the centerline is zero at the entrance (x 0). The validity limit of the iso-flux thermal boundary condition models in the absence of internal heat generation was discussed and based on the physics of the problem it was found that the results of any model would not be valid if the obtained LTNE intensity was negative. The Nusselt number of a thermally developing flow through a Darcian porous material was obtained based on the LTNE approach. The obtained developing Nusselt number has a simple form without the need of using any heuristic model. Finally, an expression representing the thermal entrance region, i.e. a dimensionless axial location where the flow becomes thermally developed, was proposed based on the present analytical solution. This expression reveals the validity region of the simplified fully developed heat transfer studies. Conflict of interest
Fig. 9. Thermally developing Nusselt number for a LTNE condition (Bi = 10).
There is no conflict of interest. To further study the thermally developing Nusselt number for a LTNE situation (Bi = 10), Figs. 8 and 9 are plotted. Fig. 8 depicts the present analysis local Nusselt number for k = 1 and Bi = 10 in comparison with those of Refs. [3,7]. A good agreement is seen between the results. Fig. 9 shows the distributions of the Nusselt number in the streamwise direction for three different effective conductivity ratios. As expected, it is seen that the Nusselt number increases with increasing the effective conductivity ratio. Also, it exhibits that the Nusselt number decreases along the flow direction, similar to the LTE situation. However, each curve, corresponding to a different effective conductivity ratio, reaches a slightly different fully developed value equal to 9.43, 10.43, and 11.68 for k = 0.1, 1, and 10 when Bi = 10, respectively. For LTNE situations, fluid and solid phases of the porous material have their independent roles in heat transferring due to the imperfect inter-phase heat exchange 1
between the solid and fluid phases of the porous material. k 1
and Bi represent the importance of the solid-phase conduction and inter-phase thermal resistances, respectively. So that, the Nusselt number is a function of both k and Bi for LTNE situations. Furthermore, the foregoing statement reveals why the Nusselt number of a thermally developing flow, as well as the fully developed one, for the LTNE condition (Bi = 10) is lower than that for the LTE condition (Bi ! 1).
References [1] P.X. Jiang, R.N. Xu, W. Gong, Particle-to-fluid heat transfer coefficients in miniporous media, Chem. Eng. Sci. 61 (2006) 7213–7222. [2] B. Deng, Y. Qiu, C.N. Kim, An improved porous medium model for microchannel heat sinks, Appl. Therm. Eng. 30 (2010) 2512–2517. [3] K. Yang, K. Vafai, Analysis of temperature gradient bifurcation in porous media – an exact solution, Int. J. Heat Mass Transfer 53 (2010) 4316–4325. [4] K. Yang, K. Vafai, Transient aspects of heat flux bifurcation in porous media: an exact solution, J. Heat Transfer 133 (5) (2011) 052602. [5] G.R. Imani, M. Maerefat, K. Hooman, Estimation of heat flux bifurcation at the heated boundary of a porous medium using a pore-scale numerical simulation, Int. J. Therm. Sci. 54 (2012) 109–118. [6] G.R. Imani, M. Maerefat, K. Hooman, Pore-scale numerical experiment on the effect of the pertinent parameters on heat flux splitting at the boundary of a porous medium, Transp. Porous Media 98 (2013) 631–649. [7] X. Ouyang, K. Vafai, P. Jiang, Analysis of thermally developing flow in porous media under local thermal non-equilibrium conditions, Int. J. Heat Mass Transfer 67 (2013) 768–775. [8] Y. Mahmoudi, N. Karimi, K. Mazaheri, Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition: effects of different thermal boundary conditions at the porous–fluid interface, Int. J. Heat Mass Transfer 70 (2014) 875–891. [9] Y. Mahmoudi, N. Karimi, Numerical investigation of heat transfer enhancement in a pipe partially filled with a porous material under local thermal non-equilibrium condition, Int. J. Heat Mass Transfer 68 (2014) 161– 173. [10] Y. Mahmoudi, Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local
M. Dehghan et al. / International Journal of Heat and Mass Transfer 92 (2016) 815–823
[11]
[12]
[13] [14]
[15]
[16]
[17]
[18] [19]
[20]
[21]
[22]
[23]
[24]
thermal non-equilibrium condition, Int. J. Heat Mass Transfer 85 (2015) 524– 542. A. Amiri, K. Vafai, T.M. Kuzay, Effect of boundary conditions on non-Darcian heat transfer through porous media and experimental comparisons, Numer. Heat Transfer J. Part A 27 (1995) 651–664. A.V. Kuznetsov, Thermal nonequilibrium, non-Darcian forced convection in a channel filled with a fluid saturated porous medium – a perturbation solution, Appl. Sci. Res. 57 (1997) 119–131. K. Vafai, S.J. Kim, Forced convection in a channel filled with a porous medium: an exact solution, ASME J. Heat Transfer 111 (1989) 1103–1106. K. Vafai, S.J. Kim, Discussion of the paper by A. Hadim, ‘‘forced convection in a porous channel with localized heat sources”, ASME J. Heat Transfer 117 (1995) 1097–1098. D.A. Nield, S.L.M. Junqueira, J.L. Lage, Forced convection in a fluid saturated porous medium channel with isothermal or isoflux boundaries, J. Fluid Mech. 322 (1996) 201–214. D.A. Nield, A.V. Kuznetsov, Local thermal nonequilibrium effects in forced convection in a porous medium channel: a conjugate problem, Int. J. Heat Mass Transfer 42 (1999) 3245–3252. D.Y. Lee, K. Vafai, Analytical characterization and conceptual assessment of solid and fluid temperature differentials in porous media, Int. J. Heat Mass Transfer 31 (1999) 423–435. B. Alazmi, K. Vafai, Analysis of variants within the porous media transport models, ASME J. Heat Transfer 122 (2000) 303–326. B. Alazmi, K. Vafai, Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions, Int. J. Heat Mass Transfer 45 (2002) 3071–3087. D.A. Nield, A.V. Kuznetsov, M. Xiong, Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects, Int. J. Heat Mass Transfer 46 (2003) 643–651. A.V. Kuznetsov, M. Xiong, D.A. Nield, Thermally developing forced convection in a porous medium: circular duct with walls at constant temperature, with longitudinal conduction and viscous dissipation effects, Transp. Porous Media 53 (2003) 331–345. D.A. Nield, A.V. Kuznetsov, Thermally developing forced convection in a channel occupied by a porous medium saturated by a non-Newtonian fluid, Int. J. Heat Mass Transfer 48 (2005) 1214–1218. P.X. Jiang, X.C. Lu, Numerical simulation and theoretical analysis of thermal boundary characteristics of convection heat transfer in porous media, Int. J. Heat Fluid Flow 28 (2007) 1144–1156. M. Dehghan, M.S. Valipour, S. Saedodin, Perturbation analysis of the local thermal non-equilibrium condition in a fluid saturated porous medium bounded by an iso-thermal channel, Transp. Porous Media 102 (2) (2014) 139–152.
823
[25] M. Dehghan, M.T. Jamal-Abad, S. Rashidi, Analytical interpretation of the local thermal non-equilibrium condition of porous media imbedded in tube heat exchangers, Energy Convers. Manage. 85 (2014) 264–271. [26] M. Dehghan, Y. Rahmani, D.D. Ganji, S. Saedodin, M.S. Valipour, S. Rashidi, Convection–radiation heat transfer in solar heat exchangers filled with a porous medium: homotopy perturbation method versus numerical analysis, Renewable Energy 74 (2015) 448–455. [27] M. Dehghan, Y. Mahmoudi, M.S. Valipour, S. Saedodin, Combined conduction– convection–radiation heat transfer of slip flow inside a micro-channel filled with a porous material, Transp. Porous Media 108 (2015) 413–436, http://dx. doi.org/10.1007/s11242-015-0483-z. [28] D.A. Nield, A.V. Kuznetsov, Forced convection in a parallel-plate channel occupied by a nanofluid or a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 70 (2014) 430–433. [29] M. Dehghan, M.S. Valipour, S. Saedodin, Temperature-dependent conductivity in forced convection of heat exchangers filled with porous media: a perturbation solution, Energy Convers. Manage. 91 (2015) 259–266. [30] D.A. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 2006. [31] M. Mirzaei, M. Dehghan, Investigation of flow and heat transfer of nanofluid in microchannel with variable property approach, Heat Mass Transfer 49 (2013) 1803–1811. [32] M. Dehghan, M. Daneshipour, M.S. Valipour, R. Rafee, S. Saedodin, Enhancing heat transfer in microchannel heat sinks using converging flow passages, Energy Convers. Manage. 92 (2015) 244–250. [33] M. Dehghan, M.S. Valipour, S. Saedodin, Analytical study of heat flux splitting in micro-channels filled with porous media, Transp. Porous Media 109 (2015) 571–587. [34] M. Dehghan, Effects of heat generations on the thermal response of channels partially filled with non-Darcian porous materials, Transp. Porous Media (2015), http://dx.doi.org/10.1007/s11242-015-0567-9. [35] M. Dehghan, M.S. Valipour, S. Saedodin, Y. Mahmoudi, Thermally developing flow inside a porous-filled channel in the presence of internal heat generation under local thermal non-equilibrium condition: A perturbation analysis, Appl. Therm. Eng. (2016). in preparation. [36] M. Dehghan, M.S. Valipour, S. Saedodin, Conjugate heat transfer inside microchannels filled with porous media: an exact solution, J. Thermophys. Heat Transfer (2016). in preparation. [37] M. Dehghan, M.S. Valipour, S. Saedodin, Microchannels enhanced by porous materials: heat transfer enhancement or pressure drop increment, Energy Convers. Manage. (2016). in preparation. [38] A. Haji-Sheikh, D.A. Nield, K. Hooman, Heat transfer in the thermal entrance region for flow through rectangular porous passages, Int. J. Heat Mass Transfer 49 (2006) 3004–3015.