**2. Volume averaged governing equations**

As illustrated in **Figure 1**, the structure of silicon carbide ceramic foam volumetric receiver may be treated as homogeneous porous medium. Since the dependence of the Darcian velocity on the transverse direction can only be observed in a small region very close to the walls of the passage, we may neglect the boundary effects (i.e. Brinkman term).

Based on a theoretical derivation of Darcy's law, Neuman [10] pointed out that the application of Darcy's law to compressible fluids is justified as long as Knudsen numbers are sufficiently small to ensure the no-slip conditions at the solid–gas interface. This is usually the case for the volumetric receivers. Thus, allowing the density to vary through the receiver, the following Forchheimer extended Darcy law should hold:

*Turbulent Heat Transfer Analysis of Silicon Carbide Ceramic Foam as a Solar… DOI: http://dx.doi.org/10.5772/intechopen.93255*

**Figure 1.** *Volumetric receiver.*

destruction although the average temperature is comparatively low. These difficulties encountered in the receiver must be overcome to run the power

turbulence mixing on the heat transfer were not considered.

heat transfer within a ceramic foam receiver.

**2. Volume averaged governing equations**

effects (i.e. Brinkman term).

law should hold:

**50**

In consideration of these requirements, ceramic foams have come to draw attention as a possible candidate to replace the conventional extruded monoliths with parallel channels. Many researchers including Becker et al. [4], Fend et al. [5] and Bai [6] focused on porous ceramic foams as a promising absorber material. Recently, Sano et al. [7] carried out a local non-thermal equilibrium analysis to investigate the receiver efficiency under the equal pumping power. For the first time, the complete set of analytical solutions based on the two-energy equation model of porous media was presented, so as to fully account for the combined effects of tortuosity; thermal dispersion and compressibility on the convective, conductive and radiative heat transfer within a ceramic foam receiver. In their analysis, however, the Rosseland approximation was applied to account for the radiative heat transfer through the solar receiver. It is well known that the Rosseland approximation ceases to be valid near boundaries. Although no wall boundaries exist for the case of the one-dimensional analysis of the solar volumetric receiver, the validity of applying the Rosseland approximation near the inlet boundary of the receiver has not been investigated yet. Furthermore, the effects of

In this study, the validity of the Rosseland approximation [7] will be examined by comparing the results based on the Rosseland approximation and the results obtained from solving the irradiation transport equation based on the P1 model. The set of the equations will be reduced to a fifth-order ordinary differential equation for the air temperature. Once the air temperature distribution is determined, the pressure distribution along the flow direction can readily be estimated from the momentum equation with the low Mach approximation. Thus, the receiver efficiency, namely, the ratio of the air enthalpy flux increase to the concentrated solar heat flux, can be compared under the equal pumping power, so as to investigate the optimal operating conditions. Some analytical and numerical investigations [3–8] have been reported elsewhere. However, none of them appeared to elucidate well the combined effects of turbulence, compressibility, radiation, convection and conduction within the volumetric receiver on the developments of air and ceramic temperatures as well as the pressure along the flow direction. This study appears to be the first to provide the complete set of analytical solutions based on the twoenergy equation model of porous media [9], fully accounting for the combined effects of turbulence, tortuosity, thermal dispersion, compressibility and radiative

As illustrated in **Figure 1**, the structure of silicon carbide ceramic foam volumetric receiver may be treated as homogeneous porous medium. Since the dependence of the Darcian velocity on the transverse direction can only be observed in a small region very close to the walls of the passage, we may neglect the boundary

Based on a theoretical derivation of Darcy's law, Neuman [10] pointed out that the application of Darcy's law to compressible fluids is justified as long as Knudsen numbers are sufficiently small to ensure the no-slip conditions at the solid–gas interface. This is usually the case for the volumetric receivers. Thus, allowing the density to vary through the receiver, the following Forchheimer extended Darcy

plant safely.

*Foams - Emerging Technologies*

$$-\frac{\partial \langle \mathbf{p} \rangle^{f}}{\partial \mathbf{x}\_{i}} = \frac{\langle \mu \rangle^{f}}{K} \langle u\_{i} \rangle + b \langle \rho \rangle^{f} \sqrt{\langle u\_{j} \rangle \langle u\_{j} \rangle} \langle u\_{i} \rangle \tag{1}$$

where *K* and *b* are the permeability and the inertial coefficients, respectively. Furthermore, by virtue of the volume averaging procedure [11–13], the microscopic energy equations of the compressible fluid flow phase and the solid phase may be integrated over an elemental control volume *V*, so as to derive the corresponding macroscopic energy equations. Since the porous medium is considered to be homogeneous, the integration of the two distinct energy equations gives:

For the air:

$$\begin{split} & \varepsilon \frac{\partial}{\partial t} \left\langle \rho\_f \left( h\_{\text{stag}} - \frac{\mathbf{p}}{\rho} \right) \right\rangle^f + \varepsilon \frac{\partial}{\partial \mathbf{x}\_j} \langle \rho \rangle^f \langle u\_j \rangle^f \langle h\_{\text{st}} \rangle^f \\ & \qquad = \frac{\partial}{\partial \mathbf{x}\_j} \left( \varepsilon \langle k\_f \rangle^f \frac{\partial \langle T \rangle^f}{\partial \mathbf{x}\_j} + \frac{1}{V} \int\_{A\_{\text{im}}} k\_f T n\_j dA - \varepsilon \langle \rho \rangle^f \left\langle \bar{h}\_{\text{stag}} \bar{u}\_j \right\rangle^f + \varepsilon \langle u\_i \tau\_{\vec{y}} \rangle^f \right) \\ & \qquad + \frac{1}{V} \int\_{A\_{\text{im}}} k\_f \frac{\partial T}{\partial \mathbf{x}\_j} n\_j dA \end{split} \tag{2}$$

For the solid matrix:

$$\rho \left( 1 - \varepsilon \right) \rho\_s \varepsilon\_s \frac{\partial \langle T \rangle^s}{\partial t} = \frac{\partial}{\partial \mathbf{x}\_j} \left( (1 - \varepsilon) k\_s \frac{\partial \langle T \rangle^s}{\partial \mathbf{x}\_j} - \frac{k\_s}{V} \int\_{A\_{\text{int}}} T n\_j dA \cdot \mathbf{q}\_R \right) - \frac{1}{V} \int\_{A\_{\text{int}}} k\_f \frac{\partial T}{\partial \mathbf{x}\_j} n\_j dA \tag{3}$$

where the intrinsic volume average of a certain local variable *ϕ* in the fluid phase and solid matrix phase can be defined as.

$$\langle \phi \rangle^f \equiv \frac{1}{V\_f} \int\_{V\_f} \phi dV, \langle \phi \rangle^m \equiv \frac{1}{V\_m} \int\_{V\_m} \phi dV \tag{4}$$

Note that subscripts *f* and *m* refer to the fluid phase and solid matrix phase, respectively. The decomposition of the local variable *ϕ* can be expressed in terms of its intrinsic average and the spatial deviation from it:

*Foams - Emerging Technologies*

$$
\phi = \langle \phi \rangle^f + \tilde{\phi} \tag{5}
$$

or

*kstag*

The term *ε ρ <sup>f</sup>*

such that

h i *<sup>h</sup> <sup>f</sup>* <sup>¼</sup> *cp*h i *<sup>T</sup> <sup>f</sup>*

*εcp ∂ ∂x <sup>j</sup> ρ f* D E *<sup>f</sup>*

*<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup>*

**53**

*<sup>∂</sup>*h i *<sup>T</sup> ∂xi*

¼ *ε k <sup>f</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.93255*

*hu*~*<sup>i</sup>* D E *<sup>f</sup>*

D E *<sup>f</sup>* ~

� � *<sup>f</sup>* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>ε</sup> ks* � � *<sup>∂</sup>*h i *<sup>T</sup>*

*Turbulent Heat Transfer Analysis of Silicon Carbide Ceramic Foam as a Solar…*

stagnant thermal conductivity from its upper bound *εk <sup>f</sup>* þ ð Þ 1 � *ε ks*

*<sup>ε</sup>* <sup>∗</sup> ð Þ � *<sup>ε</sup> <sup>∂</sup>*h i *<sup>T</sup>*

*∂xi*

Using the effective porosity *<sup>ε</sup>* <sup>∗</sup> and the equation of state h i *<sup>p</sup> <sup>f</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>f</sup>*

*ε* <sup>∗</sup> *k <sup>f</sup>*

*<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup> ∂x <sup>j</sup>*

Note that the assumption of equal temperature gradients, *<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup>*

sion hypothesis [14], the thermal dispersion term is usually expressed as:

¼ *ρ <sup>f</sup>* D E*<sup>f</sup>*

� �

� � *<sup>f</sup> <sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup> ∂x <sup>j</sup>*

þ *qR*

*<sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>*, has been discarded. This practice has been proven to be quite effective in a series of computations (e.g. [8, 9]). According to the gradient diffu-

*cp T*~*u*~ *<sup>j</sup>*

while the interfacial heat transfer between the solid and fluid phases is modeled

where *hv* is the volumetric heat transfer coefficient. The Maxwell approximations may be used for the dynamic viscosity and thermal conductivity of the air:

� � *<sup>f</sup>* ¼ �*kdisjk*

¼ 1 *V* ð *A*int

, the volume average energy equations Eqs. (2) and (3) may be

!

þ *εkdisjk*

*<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup> ∂xk*

� *hv* h i *<sup>T</sup> <sup>s</sup>* � h i *<sup>T</sup> <sup>f</sup>* � � <sup>¼</sup> 0 (14)

*<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup> ∂xk*

*<sup>n</sup> jdA* <sup>¼</sup> *hv* h i *<sup>T</sup> <sup>s</sup>* � h i *<sup>T</sup> <sup>f</sup>* � � (16)

Yang and Nakayama [9] introduced the effective porosity *ε* <sup>∗</sup>

*<sup>ε</sup>* <sup>∗</sup> <sup>¼</sup> *ks* � *kstag ks* � *k <sup>f</sup>*

concisely rewritten for the steady state for air as:

h i *<sup>T</sup> <sup>f</sup>* <sup>¼</sup> *<sup>∂</sup>*

<sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks*

*hu*~ *<sup>j</sup>* D E*<sup>f</sup>*

1 *V* ð *A*int *k f ∂T ∂x <sup>j</sup>*

*∂x <sup>j</sup>*

*u j* � � *<sup>f</sup>*

for the solid matrix phase as: *∂ ∂x <sup>j</sup>*

> *ρ f* D E *<sup>f</sup>* ~

using Newton's cooling law:

*∂xi* þ 1 *V* ð

vector, which serves an additional heat flux resulting from the hydrodynamic mixing of fluid particles passing through pores. On the other hand, the second term on the right-hand side term in Eq. (10) is associated with the surface integral, and it describes the effects of the tortuosity on the macroscopic heat flux, which adjusts the level of the

*A*int

in Eq. (9) describes the thermal dispersion heat flux

<sup>¼</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>ε</sup><sup>k</sup> <sup>f</sup>* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>ε</sup> ks* � *kstag ks* � *k <sup>f</sup>*

*k <sup>f</sup>* � *ks*

� �*TnidA* (10)

� � to a correct one.

*TnidA* (12)

D E *<sup>f</sup>*

� *hv* h i *<sup>T</sup> <sup>f</sup>* � h i *<sup>T</sup> <sup>s</sup>* � �

*<sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi

, which is defined as

(11)

*R T*h i*<sup>f</sup>* and

(13)

(15)

Moreover, *qR* is the radiative heat flux, *A*int is the interfacial surface area between the fluid and solid matrix phases, while *n <sup>j</sup>* is the normal unit vector from the fluid phase to the solid matrix phase.

In order to simplify the foregoing set of the equations, the low Mach approximation is applied due to the relatively low Mach number when the air flows through a porous medium. Thus, the dynamic pressure change is sufficiently small as compared to the absolute pressure prevailing over the system, such that the stagnant enthalpy is approximated by *hstag* ¼ *h* þ *ukuk=*2 ffi *h*. Combining the foregoing two energy equations namely Eqs. (2) and (3), and, then, noting the continuity of temperature and heat flux at the interface, we obtain the one-equation model for the steady state as follows:

$$
\epsilon \frac{\partial}{\partial \mathbf{x}\_j} \left< \rho\_f \right>^f \left< \mathbf{u}\_j \right>^f \left< \mathbf{h} \right>^f = \frac{\partial}{\partial \mathbf{x}\_j} \left( \epsilon \left< \mathbf{k}\_f \right>^f \frac{\partial \langle T \rangle^f}{\partial \mathbf{x}\_j} + (\mathbf{1} - \epsilon) \mathbf{k}\_s \frac{\partial \langle T \rangle^f}{\partial \mathbf{x}\_j} + \frac{1}{V} \int\_{A\_{\text{int}}} \left( \mathbf{k}\_f - \mathbf{k}\_s \right) \mathbf{T} n\_j d\mathbf{A} \right) \tag{6}
$$

$$
$$

$$
\tag{7}
$$

For the time being, let us assume *<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>* (this assumption will be relaxed shortly). Then, the equation reduces to

$$\begin{split} e\frac{\partial}{\partial \mathbf{x}\_{j}} \Big/ \langle \rho\_{f} \rangle^{f} \langle \mathbf{u}\_{j} \rangle^{f} \langle \mathbf{h} \rangle^{f} &= \frac{\partial}{\partial \mathbf{x}\_{j}} \Bigg( \Big( \varepsilon \langle \mathbf{k}\_{f} \rangle^{f} + (\mathbf{1} - \varepsilon) \mathbf{k}\_{i} \Big) \frac{\partial \langle T \rangle}{\partial \mathbf{x}\_{j}} + \frac{1}{V} \int\_{A\_{\mathrm{int}}} \Big( \mathbf{k}\_{f} - \mathbf{k}\_{i} \Big) \mathbf{T} \mathbf{n}\_{j} dA \\ &- e \Big/ \rho\_{f} \Big)^{f} \left\langle \bar{\mathbf{h}} \bar{\boldsymbol{u}}\_{j} \right\rangle^{f} - q\_{R} \Bigg) \tag{7} \tag{7} \tag{7}$$

where

$$
\langle \phi \rangle \equiv \frac{1}{V} \int\_{V} \phi dV \tag{8}
$$

is the Darcian average of the variable *ϕ* such that *u <sup>j</sup>* � � <sup>¼</sup> *<sup>ε</sup> <sup>u</sup> <sup>j</sup>* � � *<sup>f</sup>* is the Darcian velocity vector. From the foregoing equation, that is, Eq. (6), the macroscopic heat flux vector *qi* ¼ *qx*, *qy*, *qz* � � and its corresponding stagnant thermal conductivity *kstag* may be defined as follows:

$$\begin{split} q\_i &= -k\_{\text{stag}} \frac{\partial \langle T \rangle}{\partial \mathbf{x}\_i} + q\_R + e \Big\langle \rho\_f \Big\rangle^f \left\langle \bar{h} \bar{\boldsymbol{u}}\_i \Big\rangle^f \\ &= -\left( e \langle \mathbf{k}\_f \rangle^f + (\mathbf{1} - e) \mathbf{k}\_i \right) \frac{\partial \langle T \rangle}{\partial \mathbf{x}\_i} - \frac{\mathbf{1}}{V} \int\_{A\_{\text{int}}} (\mathbf{k}\_f - \mathbf{k}\_i) T \mathbf{n}\_i d\mathbf{A} + q\_R + e \Big\langle \rho\_f \Big\rangle^f \left\langle \bar{h} \bar{\boldsymbol{u}}\_i \Big\rangle^f \right\rangle \end{split} \tag{9}$$

*Turbulent Heat Transfer Analysis of Silicon Carbide Ceramic Foam as a Solar… DOI: http://dx.doi.org/10.5772/intechopen.93255*

or

*<sup>ϕ</sup>* <sup>¼</sup> h i *<sup>ϕ</sup> <sup>f</sup>* <sup>þ</sup> *<sup>ϕ</sup>*<sup>~</sup> (5)

Moreover, *qR* is the radiative heat flux, *A*int is the interfacial surface area between the fluid and solid matrix phases, while *n <sup>j</sup>* is the normal unit vector from

In order to simplify the foregoing set of the equations, the low Mach approximation is applied due to the relatively low Mach number when the air flows through a porous medium. Thus, the dynamic pressure change is sufficiently small as compared to the absolute pressure prevailing over the system, such that the stagnant enthalpy is approximated by *hstag* ¼ *h* þ *ukuk=*2 ffi *h*. Combining the foregoing two energy equations namely Eqs. (2) and (3), and, then, noting the continuity of temperature and heat flux at the interface, we obtain the one-equation model for

the fluid phase to the solid matrix phase.

h i *<sup>h</sup> <sup>f</sup>* <sup>¼</sup> *<sup>∂</sup> ∂x <sup>j</sup>*

For the time being, let us assume *<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup>*

h i *<sup>h</sup> <sup>f</sup>* <sup>¼</sup> *<sup>∂</sup> ∂x <sup>j</sup>* *ε k <sup>f</sup>*

0

B@

� *ε ρ <sup>f</sup>* D E *<sup>f</sup>* ~

� � *<sup>f</sup> <sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup> ∂x <sup>j</sup>*

> *hu*~ *<sup>j</sup>* D E*<sup>f</sup>*

(this assumption will be relaxed shortly). Then, the equation reduces to

*ε k <sup>f</sup>*

*hu*~ *<sup>j</sup>* D E *<sup>f</sup>*

h i *<sup>ϕ</sup>* � <sup>1</sup> *V* ð *V*

velocity vector. From the foregoing equation, that is, Eq. (6), the macroscopic heat

0

B@

� *ε ρ <sup>f</sup>* D E *<sup>f</sup>* ~

is the Darcian average of the variable *ϕ* such that *u <sup>j</sup>*

D E *<sup>f</sup>* ~

*∂xi*

*hu*~*<sup>i</sup>* D E*<sup>f</sup>*

> � 1 *V* ð

> > *A*int

*k <sup>f</sup>* � *ks*

� �

� � *<sup>f</sup>* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>ε</sup> ks* � � *<sup>∂</sup>*h i *<sup>T</sup>*

þ *qR* þ *ε ρ <sup>f</sup>*

þ ð Þ 1 � *ε ks*

1

CA

*<sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup>*

� *qR*

� � *<sup>f</sup>* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>ε</sup> ks* � � *<sup>∂</sup>*h i *<sup>T</sup>*

� *qR*

1

CA

*<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup> ∂x <sup>j</sup>* þ 1 *V* ð

*∂x <sup>j</sup>* þ 1 *V* ð

*A*int

*<sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>*

*A*int

*ϕdV* (8)

� � <sup>¼</sup> *<sup>ε</sup> <sup>u</sup> <sup>j</sup>*

and its corresponding stagnant thermal conductivity

� �*TnidA* <sup>þ</sup> *qR* <sup>þ</sup> *ε ρ <sup>f</sup>*

*k <sup>f</sup>* � *ks* � �*Tn jdA*

� � *<sup>f</sup>* is the Darcian

D E *<sup>f</sup>* ~

*hu*~*<sup>i</sup>* D E *<sup>f</sup>*

(9)

*k <sup>f</sup>* � *ks* � �*Tn jdA*

(6)

(7)

the steady state as follows:

*Foams - Emerging Technologies*

*u j* � � *<sup>f</sup>*

> *u j* � � *<sup>f</sup>*

*ε ∂ ∂x <sup>j</sup> ρ f* D E*<sup>f</sup>*

> *ε ∂ ∂x <sup>j</sup> ρ f* D E *<sup>f</sup>*

where

flux vector *qi* ¼ *qx*, *qy*, *qz*

*qi* ¼ �*kstag*

**52**

¼ � *ε k <sup>f</sup>*

*kstag* may be defined as follows:

*<sup>∂</sup>*h i *<sup>T</sup> ∂xi*

$$k\_{\rm tag} \frac{\partial \langle T \rangle}{\partial \mathbf{x}\_i} = \left( \varepsilon \langle k\_f \rangle^f + (\mathbf{1} - \varepsilon) k\_\varepsilon \right) \frac{\partial \langle T \rangle}{\partial \mathbf{x}\_i} + \frac{\mathbf{1}}{V} \int\_{A\_{\rm int}} (k\_f - k\_\varepsilon) T n\_i dA \tag{10}$$

The term *ε ρ <sup>f</sup>* D E *<sup>f</sup>* ~ *hu*~*<sup>i</sup>* D E *<sup>f</sup>* in Eq. (9) describes the thermal dispersion heat flux vector, which serves an additional heat flux resulting from the hydrodynamic mixing of fluid particles passing through pores. On the other hand, the second term on the right-hand side term in Eq. (10) is associated with the surface integral, and it describes the effects of the tortuosity on the macroscopic heat flux, which adjusts the level of the stagnant thermal conductivity from its upper bound *εk <sup>f</sup>* þ ð Þ 1 � *ε ks* � � to a correct one. Yang and Nakayama [9] introduced the effective porosity *ε* <sup>∗</sup> , which is defined as

$$\varepsilon^{\*} = \frac{k\_{\rm s} - k\_{\rm stag}}{k\_{\rm s} - k\_{f}} = \varepsilon + \frac{\varepsilon k\_{f} + (1 - \varepsilon)k\_{\rm s} - k\_{\rm stag}}{k\_{\rm s} - k\_{f}} \tag{11}$$

such that

$$(\left(\varepsilon^{\*} - \varepsilon\right)\frac{\partial \langle T\rangle}{\partial \mathbf{x}\_{i}} = \frac{1}{V}\int\_{A\_{\rm int}} T n\_{i} dA \tag{12}$$

Using the effective porosity *<sup>ε</sup>* <sup>∗</sup> and the equation of state h i *<sup>p</sup> <sup>f</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>f</sup>* D E *<sup>f</sup> R T*h i*<sup>f</sup>* and h i *<sup>h</sup> <sup>f</sup>* <sup>¼</sup> *cp*h i *<sup>T</sup> <sup>f</sup>* , the volume average energy equations Eqs. (2) and (3) may be concisely rewritten for the steady state for air as:

$$\mathrm{ac}\_{p}\frac{\partial}{\partial \mathbf{x}\_{j}} \left< \rho\_{f} \right>^{f} \left< \mathbf{u}\_{j} \right>^{f} \left< \mathbf{T} \right>^{f} = \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \mathbf{e}^{\*} \left< \mathbf{k}\_{f} \right>^{f} \frac{\partial \langle \mathbf{T} \rangle^{f}}{\partial \mathbf{x}\_{j}} + \mathrm{c} \mathbf{k}\_{\mathrm{dir}\_{k}} \frac{\partial \langle \mathbf{T} \rangle^{f}}{\partial \mathbf{x}\_{k}} \right) - \mathbf{h}\_{v} \left( \langle \mathbf{T} \rangle^{f} - \langle \mathbf{T} \rangle^{f} \right) \tag{13}$$

for the solid matrix phase as:

$$\frac{\partial}{\partial \mathbf{x}\_j} \left( (\mathbf{1} - \boldsymbol{\varepsilon}^\*) \mathbf{k}\_s \frac{\partial \langle T \rangle^{\boldsymbol{\varepsilon}}}{\partial \mathbf{x}\_j} + q\_R \right) - h\_v \left( \langle T \rangle^{\boldsymbol{\varepsilon}} - \langle T \rangle^{\boldsymbol{\varepsilon}} \right) = \mathbf{0} \tag{14}$$

Note that the assumption of equal temperature gradients, *<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>* ffi *<sup>∂</sup>*h i *<sup>T</sup> <sup>=</sup>∂<sup>x</sup> <sup>j</sup>*, has been discarded. This practice has been proven to be quite effective in a series of computations (e.g. [8, 9]). According to the gradient diffusion hypothesis [14], the thermal dispersion term is usually expressed as:

$$
\left\langle \rho\_f \right\rangle^f \left\langle \tilde{h}\tilde{u}\_j \right\rangle^f = \left\langle \rho\_f \right\rangle^f c\_p \left\langle \tilde{T}\tilde{u}\_j \right\rangle^f = -k\_{d\bar{u}jk} \frac{\partial \langle T \rangle^f}{\partial \mathbf{x}\_k} \tag{15}
$$

while the interfacial heat transfer between the solid and fluid phases is modeled using Newton's cooling law:

$$\frac{1}{V} \int\_{A\_{\rm int}} k\_f \frac{\partial T}{\partial \mathbf{x}\_j} n\_j dA = h\_v \left( \langle T \rangle^s - \langle T \rangle^f \right) \tag{16}$$

where *hv* is the volumetric heat transfer coefficient. The Maxwell approximations may be used for the dynamic viscosity and thermal conductivity of the air:

$$
\mu \left( \langle T \rangle^f \right) = \mu\_0 \left( \frac{\langle T \rangle^f}{\langle T \rangle\_0^f} \right)^n = \mathbf{1.8} \times \mathbf{10}^{-5} \left( \frac{\langle T \rangle^f}{\mathbf{300K}} \right)^{0.7} [\mathbf{Pa} \cdot \mathbf{s}] \tag{17}
$$

*<sup>K</sup>* <sup>¼</sup> <sup>0</sup>*:*00073 1ð Þ � *<sup>ε</sup>* �0*:*<sup>224</sup> <sup>1</sup>*:*<sup>18</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93255*

*Turbulent Heat Transfer Analysis of Silicon Carbide Ceramic Foam as a Solar…*

*hv* <sup>¼</sup> <sup>8</sup>*:*72 1ð Þ � *<sup>ε</sup>* <sup>1</sup>*=*<sup>4</sup> <sup>1</sup> � *<sup>e</sup>*�ð Þ <sup>1</sup>�*<sup>ε</sup> <sup>=</sup>*0*:*<sup>04</sup>

*β* is calculated by the following correlation:

For a given mass flux *G* ¼ *ρ <sup>f</sup>*

and *ρ <sup>f</sup>* D E *<sup>f</sup>*

*x* ¼ 0 (inlet):

such that *ρ <sup>f</sup>*

� *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup> <sup>f</sup>*

**55**

D E*<sup>f</sup>*

� � *<sup>f</sup>* <sup>þ</sup> *<sup>ε</sup>kdisxx* � �*d T*h i*<sup>f</sup>*

<sup>¼</sup> *<sup>I</sup>*<sup>0</sup> cos *<sup>ξ</sup>* � ð Þ <sup>1</sup> � *<sup>ε</sup> <sup>a</sup><sup>σ</sup>* h i *<sup>T</sup> <sup>s</sup>*

¼ *ρ <sup>f</sup>* D E*<sup>f</sup>*

and

using the following expression:

the following correlations:

1 � *e*�ð Þ <sup>1</sup>�*<sup>ε</sup> <sup>=</sup>*0*:*<sup>04</sup>

*<sup>b</sup>* <sup>¼</sup> 12 1ð Þ � *<sup>ε</sup> dm*

*<sup>ε</sup>kdisxx* <sup>¼</sup> <sup>1</sup>*:*2*cpG* ffiffiffiffi

With respect to the stagnant thermal conductivity and the volumetric heat transfer coefficient for foams, Calmidi and Mahajan [15, 16] empirically provided

� �<sup>1</sup>*=*<sup>2</sup> *Gdm*

Kamiuto et al. [21] experimentally affirmed that the Rosseland model is quite effective. Therefore, it can be deduced that the Rosseland model is also applicable for the present case of silicon carbide ceramic foam. Based on the measurements made on cordierite ceramic foams by Kamiuto et al., the mean extinction coefficient

*ε*

D E*<sup>f</sup>*

the equation of state may be solved for the four unknowns, namely, h i *<sup>T</sup> <sup>f</sup>*

h i *<sup>T</sup> <sup>f</sup>* <sup>¼</sup> h i *<sup>T</sup> <sup>f</sup>*

h i *<sup>p</sup> <sup>f</sup>* <sup>¼</sup> h i *<sup>p</sup> <sup>f</sup>*

0*=R T*h i*<sup>f</sup>*

<sup>0</sup> <sup>¼</sup> h i *<sup>p</sup> <sup>f</sup>*

� � � � � *x*¼0

0 � �<sup>4</sup> � h i *<sup>T</sup> <sup>f</sup>*

*dx*

<sup>0</sup> <sup>¼</sup> 105

<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup>

0 � �<sup>4</sup> � � <sup>þ</sup> *hconv* h i *<sup>T</sup> <sup>s</sup>*

16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* ð Þ<sup>3</sup> � �*d T*h i*<sup>s</sup>*

� � � � (31)

� <sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks* <sup>þ</sup>

. The boundary conditions are given as follows:

*K*

*kstag* <sup>¼</sup> *<sup>ε</sup><sup>k</sup> <sup>f</sup>* <sup>þ</sup> <sup>0</sup>*:*19 1ð Þ � *<sup>ε</sup>* <sup>0</sup>*:*<sup>763</sup>*ks* (26)

!<sup>1</sup>*=*<sup>2</sup>

*β* ¼ 8 1ð Þ � *ε =dm* (28)

h i *u* , the foregoing three equations along with

<sup>0</sup> ¼ 300 K½ � (29)

½ � P*a* (30)

*dx*

<sup>0</sup> � h i *<sup>T</sup> <sup>f</sup>* 0

� � � � *x*¼0

*<sup>=</sup>*ð Þ¼ <sup>287</sup> � <sup>300</sup> <sup>1</sup>*:*2 kg*=*m3 ½ �

h i *<sup>μ</sup> <sup>f</sup>*

<sup>p</sup> (25)

*Pr*<sup>0</sup>*:*<sup>37</sup> *<sup>k</sup> <sup>f</sup>*

*dm*<sup>2</sup> (27)

,h i *<sup>T</sup> <sup>s</sup>*

, h i *<sup>p</sup> <sup>f</sup>*

respectively, where *dm* is the pore diameter of foam. The longitudinal dispersion coefficient is roughly about 20 times more than the transverse one. Thus, following Calmidi and Mahajan [16], we may evaluate the longitudinal dispersion coefficient

ffiffiffiffiffiffiffiffiffiffi 1 � *ε* 3*π*

*dm*<sup>2</sup> (23)

(24)

! r �1*:*<sup>11</sup>

and

$$k\_f \left( \langle T \rangle^f \right) = k\_0 \left( \frac{\langle T \rangle^f}{\langle T \rangle\_0^f} \right)^n = 0.025 \left( \frac{\langle T \rangle^f}{300 \text{K}} \right)^{0.7} \left[ \text{W/mK} \right] \tag{18}$$

where the exponent n is 0.7 according to [4]. The specific heat capacity of the air *cp* <sup>¼</sup> 1000 J½ � *<sup>=</sup>*kgK and the Prandtl number *Pr* <sup>¼</sup> <sup>1</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>5</sup> � <sup>1000</sup>*=*0*:*<sup>025</sup> <sup>¼</sup> <sup>0</sup>*:*72 are assumed to be constant.
