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Effects of Heat Generations on the Thermal Response of Channels Partially Filled with Non-Darcian Porous Materials

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Abstract

The forced convective thermal response of channels partially filled with porous materials is analytically studied. It is assumed that internal heat generations exist within both the fluid and solid phases. Effects of internal heat generations within the both phases on the thermal response of the channel resulting in the heat flux bifurcation phenomenon are discussed for the first time. No study previously analyzed the heat flux bifurcation in a non-Darcian porous medium with heat-generating porous materials. To obtain the most general thermo-hydraulic behavior, the Darcy–Brinkman equation of motion and the two-energy model (local thermal non-equilibrium) along with two practical thermal boundary conditions (models A and B) are used. Consequently, two possible thermal responses are obtained for each phase. Results show that insertion of a porous material inside a heat-convecting fluid leads to have a more uniform temperature distribution which translates to a lower thermal resistance and an enhanced heat transfer. Furthermore, it is seen that increasing the porous material thickness increases the ratio of heat transferred by the porous medium compared to that convected by the clear fluid flow. In addition, the internal heat generations drastically change the temperature distribution. Finally, it is shown that the internal heat generations can inverse the heat flux direction at the porous–fluid interface (the heat flux bifurcation phenomenon). Criteria for the heat flux bifurcation for a partially porous-filled channel are presented under the Darcy’s law of motion.

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Abbreviations

\(a_\mathrm{sf}\) :

Specific area per unit volume of porous material \(\left( \hbox {m}^{-1}\right) \)

A :

Constant defined by Eq. (18)

B :

Constant defined by Eq. (19)

Bi :

Biot number, \(a_\mathrm{sf} h_\mathrm{sf} h_{0}^{2}/k_\mathrm{s,eff}\)

C :

Constant defined by Eq. (21)

\(C_\mathrm{p}\) :

Specific heat of the fluid at constant pressure, \(\left( \hbox {J\,kg}^{-1}\mathrm{K}^{-1}\right) \)

Da :

Darcy number, \(K/{h}_{0}^{2}\)

\(D_\mathrm{h}\) :

Hydraulic diameter of the channel \(\left( 4{h}_{0}\right) \)

\(h_\mathrm{sf}\) :

Interphase fluid to solid heat transfer coefficient \(\left( \hbox {W\,m}^{-2}\mathrm{K}^{-1}\right) \)

\({h}_{0}\) :

Height of the channel (m)

\({h}_{\mathrm{p}}\) :

Porous substrate thickness (m)

K :

Permeability of the porous medium \(\left( \hbox {m}^{2}\right) \)

k :

The ratio of solid effective thermal conductivity to that of the fluid, \((1-\varepsilon )k_\mathrm{s}/( \varepsilon k_\mathrm{f})\)

\(k_\mathrm{f}\) :

Thermal conductivity of the fluid \((\hbox {W\,m}^{-1}\mathrm{K}^{-1})\)

\(k_\mathrm{f,\mathrm{eff}}\) :

Effective thermal conductivity of the fluid, \(\varepsilon k_\mathrm{f}\)

\(k_\mathrm{s}\) :

Thermal conductivity of the solid \((\hbox {W\,m}^{-1}\mathrm{K}^{-1})\)

\(k_\mathrm{s,eff}\) :

Effective thermal conductivity of the solid, \((1-\varepsilon )k_\mathrm{s}\)

Nu :

Nusselt number

\(O_{1}\) :

Constant defined by Eq. (45b)

\(O_{2}\) :

Constant defined by Eq. (45c)

p :

Pressure (Pa)

q :

Heat flux per area \((\hbox {W\,m}^{-2})\)

Re :

Reynolds number, \({Re}=\rho \overline{{u}}{h}_{0} /\mu \)

S :

Ratio of the porous medium thickness to the channel height, \({h}_{\mathrm{p}}/{h}_{0}\)

\(S_\mathrm{f}\) :

Internal heat generation within the fluid phase \((\hbox {W\,m}^{-3})\)

\(S_\mathrm{s}\) :

Internal heat generation within the solid phase \((\hbox {W\,m}^{-3})\)

T :

Temperature \((\mathrm{K})\)

\(T_\mathrm{m}\) :

Average temperature \(\mathrm{(K)}\)

u :

Longitudinal velocity \((\hbox {m}/\mathrm{s})\)

\(\overline{{u}}\) :

Average velocity \((\hbox {m}/\mathrm{s})\)

\(u_{r}\) :

Characteristic velocity, \(-\frac{h_{0}^{2}}{\mu }\frac{\partial p}{\partial x}\)

U :

Dimensionless velocity, \(u/{u_{r}}\)

\(\overline{{U}}\) :

Normalized velocity

x :

longitudinal coordinate (m)

y :

Transverse coordinate (m)

Y :

Dimensionless y coordinate, \(y/{h}_{0}\)

Z :

Constant, \(\sqrt{1/Da}\)

\(\beta \) :

Dimensionless internal heat generation defined by Eq. (26)

\(\gamma \) :

Ratio of wall heat flux to the heat flux at the interface, \(q_\mathrm{w}/q_{\mathrm{interface}}\)

G :

Constant defined by Eq. (48d)

\(\varepsilon \) :

Porosity of the porous medium

\(\varTheta \) :

Dimensionless temperature

\(\mu \) :

Viscosity \((\hbox {kg\,m}^{-1}\mathrm{s}^{1})\)

\(\rho \) :

Density \((\hbox {kg}/\hbox {m}^{3})\)

\(\xi \) :

Constant used in Eq. (37)

\(\phi _{i};i=1-3\) :

Constants defined by Eqs. (48a)–(48c)

eff:

Effective property

f:

Fluid

f1:

Fluid in the clear region

f2:

Fluid in the porous medium

in:

Inlet

m:

Mean

s:

Solid

w:

Wall

interface:

The interface between the porous medium and the clear fluid region

-:

Mean value

’,”,”’,””:

First, second, third, and forth derivatives with respect to Y

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Acknowledgments

The author would like to express his sincere thanks to Dr. Yasser Mahmoudi (University of Cambridge) for his valuable help during preparation of the present study. In addition, the author would like to thank the support by the Young Researchers and Elite Club of Islamic Azad University (Semnan Branch).

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  1. Young Researchers and Elite Club, Semnan Branch, Islamic Azad University, Semnan, Iran

    Maziar Dehghan

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  1. Maziar Dehghan
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Dehghan, M. Effects of Heat Generations on the Thermal Response of Channels Partially Filled with Non-Darcian Porous Materials. Transp Porous Med 110, 461–482 (2015). https://doi.org/10.1007/s11242-015-0567-9

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