**3. One-dimensional analysis for volumetric receiver**

In this section, we perform one-dimensional analysis to obtain analytic solutions for convective-radiative heat transfer in volume receiver. Prior to that, the radiative heat flux *qR* needs to be determined in advance. In the literature, there are two models, namely, the Rosseland approximation and the P1 model.

## **3.1 Analysis based on the Rosseland approximation**

In the Rosseland approximation, the radiative heat flux is given by

$$q\_R = -\frac{16\sigma}{3\beta} (\langle T \rangle^s)^3 \frac{\partial \langle T \rangle^s}{\partial \mathbf{x}\_j} \tag{19}$$

where *<sup>σ</sup>* <sup>¼</sup> <sup>5</sup>*:*<sup>67</sup> � <sup>10</sup>�<sup>8</sup> <sup>W</sup>*=*m2K4 � � is the Stephan-Boltzmann constant while *<sup>β</sup>* is the mean extinction coefficient.

As schematically shown in **Figure 1**, the air is flowing through a passage of length *L* at the rate of the mass flux *G* ¼ *ρ <sup>f</sup>* D E *<sup>f</sup>* h i *u* . Under the low Mach number approximation, namely, *ρ <sup>f</sup>* D E *<sup>f</sup>* <sup>∝</sup>1*=*h i *<sup>T</sup> <sup>f</sup>* , the macroscopic governing equations Eqs. (1), (13) and (14) can be simplified to be a one-dimensional set of equations as follows:

$$-\frac{d\langle\mathbf{p}\rangle^{f}}{d\mathbf{x}} = \frac{\langle\mu\rangle^{f}}{K}\frac{\mathbf{G}}{\left\langle\rho\_{f}\right\rangle^{f}} + b\frac{\mathbf{G}^{2}}{\left\langle\rho\_{f}\right\rangle^{f}} = \frac{R}{\langle\mathbf{p}\rangle^{f}}\left(\frac{\langle\mu\rangle^{f}}{K}\mathbf{G} + b\mathbf{G}^{2}\right) \langle\mathbf{T}\rangle^{f} \tag{20}$$

$$\epsilon c\_p \mathbf{G} \frac{d \langle T \rangle^f}{d \mathbf{x}} = \frac{d}{d \mathbf{x}} \left( \epsilon^\* \left< k\_f \right>^f + \epsilon k\_{\rm disx} \right) \frac{d \langle T \rangle^f}{d \mathbf{x}} - h\_v \left( \langle T \rangle^f - \langle T \rangle^s \right) \tag{21}$$

$$\frac{d}{d\mathbf{x}}\left(\left(\mathbf{1}-\boldsymbol{\varepsilon}^{\*}\right)\mathbb{I}\_{\mathbf{t}}+\frac{\mathbf{1}6\sigma}{3\boldsymbol{\beta}}\left(\left<\boldsymbol{T}\right>^{\boldsymbol{s}}\right)^{3}\right)\frac{d\left<\boldsymbol{T}\right>^{\boldsymbol{s}}}{d\mathbf{x}}-h\_{\boldsymbol{\nu}}\left(\left<\boldsymbol{T}\right>^{\boldsymbol{s}}-\left<\boldsymbol{T}\right>^{\boldsymbol{f}}\right)=\mathbf{0}\tag{22}$$

According to Calmidi and Mahajan [15, 16], Dukhan [17], Kuwahara et al. [18] and Yang et al. [19, 20], the permeability and inertial coefficient of foams are given by

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

$$K = 0.00073(1 - \varepsilon)^{-0.224} \left( \frac{1.18}{1 - e^{-(1 - \varepsilon)/0.04}} \sqrt{\frac{1 - \varepsilon}{3\pi}} \right)^{-1.11} d\_m^{-2} \tag{23}$$

and

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

*Foams - Emerging Technologies*

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

assumed to be constant.

the mean extinction coefficient.

approximation, namely, *ρ <sup>f</sup>*

*dx* <sup>¼</sup> h i *<sup>μ</sup> <sup>f</sup> K*

*dx* <sup>¼</sup> *<sup>d</sup>*

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

� *d p*h i*<sup>f</sup>*

*cpG d T*h i*<sup>f</sup>*

*d*

follows:

given by

**54**

length *L* at the rate of the mass flux *G* ¼ *ρ <sup>f</sup>*

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

*G*

D E *<sup>f</sup>* <sup>þ</sup> *<sup>b</sup> <sup>G</sup>*<sup>2</sup>

*ρ f*

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

and

¼ *μ*<sup>0</sup>

¼ *k*<sup>0</sup>

**3. One-dimensional analysis for volumetric receiver**

models, namely, the Rosseland approximation and the P1 model.

In the Rosseland approximation, the radiative heat flux is given by

*qR* ¼ � <sup>16</sup>*<sup>σ</sup>*

<sup>∝</sup>1*=*h i *<sup>T</sup> <sup>f</sup>*

*ρ f*

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

**3.1 Analysis based on the Rosseland approximation**

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

!*<sup>n</sup>*

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

!*<sup>n</sup>*

<sup>¼</sup> <sup>1</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>5</sup> h i *<sup>T</sup> <sup>f</sup>*

<sup>¼</sup> <sup>0</sup>*:*<sup>025</sup> h i *<sup>T</sup> <sup>f</sup>*

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

In this section, we perform one-dimensional analysis to obtain analytic solutions for convective-radiative heat transfer in volume receiver. Prior to that, the radiative heat flux *qR* needs to be determined in advance. In the literature, there are two

<sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* ð Þ<sup>3</sup> *<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup>*

where *<sup>σ</sup>* <sup>¼</sup> <sup>5</sup>*:*<sup>67</sup> � <sup>10</sup>�<sup>8</sup> <sup>W</sup>*=*m2K4 � � is the Stephan-Boltzmann constant while *<sup>β</sup>* is

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

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

According to Calmidi and Mahajan [15, 16], Dukhan [17], Kuwahara et al. [18]

and Yang et al. [19, 20], the permeability and inertial coefficient of foams are

h i *<sup>μ</sup> <sup>f</sup>*

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

*<sup>K</sup> <sup>G</sup>* <sup>þ</sup> *bG*<sup>2</sup> !

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

As schematically shown in **Figure 1**, the air is flowing through a passage of

Eqs. (1), (13) and (14) can be simplified to be a one-dimensional set of equations as

D E *<sup>f</sup>* <sup>¼</sup> *<sup>R</sup>*

*∂x <sup>j</sup>*

h i *u* . Under the low Mach number

h i *<sup>T</sup> <sup>f</sup>* (20)

¼ 0 (22)

(21)

, the macroscopic governing equations

300*K* !0*:*<sup>7</sup>

300*K* !0*:*<sup>7</sup>

½ � Pa � s (17)

½ � W*=*mK (18)

(19)

$$b = \frac{12(1 - \varepsilon)}{d\_m} \tag{24}$$

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 using the following expression:

$$
\varepsilon k\_{\text{diš}\_{\text{xx}}} = \mathbf{1}.2c\_p G \sqrt{K} \tag{25}
$$

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

$$k\_{\text{stag}} = \varepsilon k\_f + 0.19(1 - \varepsilon)^{0.763}k\_s \tag{26}$$

$$h\_v = 8.72(1 - \varepsilon)^{1/4} \left(\frac{1 - e^{-(1 - \varepsilon)/0.04}}{\varepsilon}\right)^{1/2} \left(\frac{\text{G}d\_m}{\langle\mu\rangle^f}\right)^{1/2} Pr^{0.37} \frac{k\_f}{d\_m^{-2}} \tag{27}$$

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 *β* is calculated by the following correlation:

$$
\beta = 8(1 - \varepsilon) / d\_m \tag{28}
$$

For a given mass flux *G* ¼ *ρ <sup>f</sup>* D E*<sup>f</sup>* h i *u* , the foregoing three equations along with 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>s</sup>* , h i *<sup>p</sup> <sup>f</sup>* and *ρ <sup>f</sup>* D E *<sup>f</sup>* . The boundary conditions are given as follows: *x* ¼ 0 (inlet):

$$
\langle T \rangle^f = \langle T \rangle\_0^f = \mathbf{300[K]} \tag{29}
$$

$$
\langle p \rangle^f = \langle p \rangle\_0^f = \mathbf{10^5} [\mathbf{Pa}] \tag{30}
$$

$$\begin{aligned} \text{such that } \left\langle \rho\_f \right\rangle^f &= \left\langle \rho\_f \right\rangle\_0^f = \left\langle p \right\rangle\_0^f / \mathbb{R} \langle T \rangle\_0^f = \mathbf{10}^\S / (2\mathbf{87} \times \mathbf{300}) = \mathbf{1.2} [\mathbf{kg}/\mathbf{m}^3] \\\\ -\left( \varepsilon^\* \left\langle k\_f \right\rangle^f + \varepsilon k\_{\text{dirax}} \right) \frac{d \langle T \rangle^f}{d \mathbf{x}} &\quad -\left( (\mathbf{1} - \varepsilon^\*) k\_i + \frac{\mathbf{16} \sigma}{\mathbf{3} \beta} (\langle T \rangle^\varepsilon)^3 \right) \frac{d \langle T \rangle^\varepsilon}{d \mathbf{x}} \end{aligned}$$

$$\begin{aligned} &-\left(\varepsilon^\* \left^\circ + \varepsilon k\_{\rm diffxxx}\right) \frac{u\left<1\,^\circ\right>}{d\mathfrak{x}}\bigg|\_{\mathfrak{x}=0} - \left(\left<1\,^\circ - \varepsilon^\*\right>k\_\circ + \frac{1\mathfrak{x}\sigma}{3\beta} \left<\langle T\rangle^\circ\right>^\circ\right) \frac{u\left<1\,^\circ\right>}{d\mathfrak{x}}\bigg|\_{\mathfrak{x}=0} \\ &= I\_0 \cos\xi - (1-\varepsilon)\left(\mathfrak{a}\sigma\left(\left<\langle T\rangle^\circ\_0\right>^4 - \left<\langle T\rangle^\circ\_0\right>^4\right) + h\_{\rm conv}\left(\langle T\rangle^\circ\_0 - \langle T\rangle^\circ\_0\right)\right) \end{aligned} \tag{31}$$

where *I*<sup>0</sup> is the intensity of radiation and *ξ* is the incidence angle. Moreover, *a* ffi 0*:*9 is the emissivity of the front surface of the receiver, while *hconv* is the convective heat transfer coefficient at the frontal surface. The properties of the air depend on the temperature, which makes the integrations of the foregoing governing equations formidable. In order to obtain analytic expressions for the unknown variables, we may approximate these properties by their representative values evaluated at the average air temperature over the receiver as given by

$$\overline{\langle T\rangle^f} = \frac{1}{L} \int\_0^L \langle T\rangle^f d\mathbf{x} \tag{32}$$

This ordinary differential equation, with the boundary conditions in Eqs. (29),

0 � �*<sup>e</sup>*

0 � �*e*

*kstag* <sup>þ</sup> *<sup>ε</sup>kdisxx* <sup>þ</sup> <sup>16</sup>*<sup>σ</sup>* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> � �*<sup>λ</sup>*

<sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> � �

<sup>0</sup> and temperature at the thermal

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

h i *<sup>T</sup> <sup>s</sup>* <sup>¼</sup> *Teq* � *Teq* � h i *<sup>T</sup> <sup>s</sup>*

where *γ* is the positive real root, which can uniquely be determined from the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> � �*hv*

<sup>∞</sup>, are given by

<sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks* <sup>þ</sup> <sup>16</sup>*<sup>σ</sup>*

*Gcp*

1 *γλ<sup>L</sup>* h i *<sup>T</sup> <sup>f</sup>*

1 *γλ<sup>L</sup>* h i *<sup>T</sup> <sup>s</sup>*

h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>* <sup>þ</sup> *<sup>ε</sup>kdisxx* � �*γλ*

<sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> � �*γλ*

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

� � � �

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>1</sup>

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>1</sup>

*γλL* � �*Teq* (42)

*γλL* � �*Teq* (43)

*Gcp* <sup>þ</sup> *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>*

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

respectively. Usually, the receiver length *L* is sufficiently long to reach the local thermal equilibrium. Thus, the average air and solid temperatures are evaluated

As one of the most important performance parameters, the receiver efficiency is

*kstag* <sup>þ</sup> *<sup>ε</sup>kdisxx* <sup>þ</sup> <sup>16</sup>*<sup>σ</sup>*

<sup>þ</sup> *<sup>ε</sup>kdisxx* � � <sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks* <sup>þ</sup> <sup>16</sup>*<sup>σ</sup>*

<sup>∞</sup> <sup>¼</sup> h i *<sup>T</sup> <sup>s</sup>*

0

� �

*<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>*

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>1</sup> � *<sup>e</sup>*�*γλ<sup>L</sup>*

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>1</sup> � *<sup>e</sup>*�*γλ<sup>L</sup>*

*γλL* � �*Teq* ffi

*γλL* � �*Teq* ffi

*<sup>γ</sup>*<sup>2</sup> � *<sup>γ</sup>* � *Gcp*

<sup>0</sup>*=dx* <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

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

�*γλ<sup>x</sup>* (36)

�*γλ<sup>x</sup>* (37)

<sup>0</sup>*=dx*<sup>2</sup> <sup>¼</sup> 0,

¼ 0 (38)

(39)

(40)

(41)

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

(30) and (31) and the auxiliary asymptotic condition *d T*h i*<sup>f</sup>*

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

yields

and

*<sup>γ</sup>*<sup>3</sup> <sup>þ</sup>

where

following cubic equation:

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>* <sup>þ</sup> *<sup>ε</sup>kdisxx* � �*<sup>λ</sup>*

*λ* ¼

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

and

from

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

<sup>0</sup> þ

h i *<sup>T</sup> <sup>f</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>e</sup>*�*γλ<sup>L</sup>*

h i *<sup>T</sup> <sup>s</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>e</sup>*�*γλ<sup>L</sup>*

defined by

**57**

*γλ<sup>L</sup>* h i *<sup>T</sup> <sup>f</sup>*

*γλ<sup>L</sup>* h i *<sup>T</sup> <sup>s</sup>*

vuuuuut

equilibrium, namely, *Teq* <sup>¼</sup> h i *<sup>T</sup> <sup>f</sup>*

*Gcp*

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

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

<sup>0</sup> <sup>¼</sup> *Teq* <sup>þ</sup> *Teq* � h i *<sup>T</sup> <sup>f</sup>*

The solid phase temperature at the inlet h i *<sup>T</sup> <sup>s</sup>*

Likewise we shall define the solid phase average temperature as follows:

$$\overline{\langle T\rangle^{s}} = \frac{1}{L} \int\_{0}^{L} \langle T\rangle^{s} d\mathbf{x} \tag{33}$$

The two energy equations, that is, Eqs. (21) and (22) may be added together and integrated using the boundary conditions in Eqs. (29) and (31) to give

$$\begin{split} c\_{p}G\Big(\langle T\rangle^{f} - \langle T\rangle\_{0}^{f}\Big) &= \left(e^{\*}k\_{0}\overline{\left(\frac{\langle T\rangle^{f}}{\langle T\rangle\_{0}^{f}}\right)^{n}} + ek\_{\text{dlex}}\right)\frac{d\langle T\rangle^{f}}{d\mathbf{x}} \\ &+ \left(\left(1-e^{\*}\right)k\_{\text{s}} + \frac{16\sigma}{3\beta}\left(\overline{\langle T\rangle^{s}}\right)^{3}\right)\frac{d\langle T\rangle^{f}}{d\mathbf{x}} + I\_{0}\cos\xi \\ &- \left(\mathbf{1} - e\right)\left(a\sigma\left(\left(\langle T\rangle\_{0}^{\prime}\right)^{4} - \left(\langle T\rangle\_{0}^{\prime}\right)^{4}\right) + h\_{\text{cov}}\left(\langle T\rangle\_{0}^{s} - \langle T\rangle\_{0}^{f}\right)\right) \end{split} \tag{34}$$

This equation is substituted into Eq. (21) to eliminate h i *<sup>T</sup> <sup>s</sup>* in favor of h i *<sup>T</sup> <sup>f</sup>* . The resulting ordinary differential equation for h i *<sup>T</sup> <sup>f</sup>* runs as

$$\begin{split} \frac{d^2(T)^f}{dx^3} &= \frac{G\_p}{\epsilon^\* k\_0 \left(\frac{\langle T\rangle^f}{\langle T\rangle\_o^f}\right)^n} + \epsilon k\_{\rm dmax} \frac{d^2 \langle T\rangle^f}{dx^2} \\ &+ h\_p \frac{k\_{\rm dag} + \epsilon k\_{\rm dmax} + \frac{16\sigma}{3\beta} \left(\overline{\langle T\rangle^f}\right)^3}{\left(\epsilon^\* k\_0 \left(\frac{\langle T\rangle^f}{\langle T\rangle\_o^f}\right)^n + \epsilon k\_{\rm dmax}\right) \left((1 - \epsilon^\*)k\_s + \frac{16\sigma}{3\beta} \left(\overline{\langle T\rangle^f}\right)^3\right)^3} \frac{d \langle T\rangle^f}{dx} \\ &- h\_p \frac{G\_p}{\left(\epsilon^\* k\_0 \left(\frac{\langle T\rangle^f}{\langle T\rangle\_o^f}\right)^n + \epsilon k\_{\rm dmax}\right) \left((1 - \epsilon^\*)k\_s + \frac{16\sigma}{3\beta} \left(\overline{\langle T\rangle^f}\right)^3\right)} \langle T\rangle^f \\ &+ h\_p \frac{I\_0 \cos\xi - (1 - \epsilon) \left(a\sigma \left(\left(\langle T\rangle^f\_o\right)^4 - \left(\langle T\rangle^f\_o\right)^4\right) + h\_{\rm cov} \left(\langle T\rangle^f\_o - \langle T\rangle^f\_o\right)\right)}{\left(\epsilon^\* k\_0 \left(\frac{\langle T\rangle^f}{\langle T\rangle^f\_o}\right)^n + \epsilon k\_{\rm dmax}\right) \left((1 - \epsilon^\*)k\_s + \frac{16\sigma}{3\beta} \left(\overline{\langle T\rangle^f}\right)^3\right)} \end{split} \tag{35}$$

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

This ordinary differential equation, with the boundary conditions in Eqs. (29), (30) and (31) and the auxiliary asymptotic condition *d T*h i*<sup>f</sup>* <sup>0</sup>*=dx* <sup>¼</sup> *<sup>d</sup>*<sup>2</sup> h i *<sup>T</sup> <sup>f</sup>* <sup>0</sup>*=dx*<sup>2</sup> <sup>¼</sup> 0, yields

$$
\langle T \rangle^f = T\_{eq} - \left( T\_{eq} - \langle T \rangle\_0^f \right) e^{-\gamma \mathbf{k} \mathbf{x}} \tag{36}
$$

and

where *I*<sup>0</sup> is the intensity of radiation and *ξ* is the incidence angle. Moreover, *a* ffi 0*:*9 is the emissivity of the front surface of the receiver, while *hconv* is the convective heat transfer coefficient at the frontal surface. The properties of the air

depend on the temperature, which makes the integrations of the foregoing governing equations formidable. In order to obtain analytic expressions for the unknown variables, we may approximate these properties by their representative values evaluated at the average air temperature over the receiver as given by

> h i *<sup>T</sup> <sup>f</sup>* <sup>¼</sup> <sup>1</sup> *L* ð*L* 0 h i *<sup>T</sup> <sup>f</sup>*

h i *<sup>T</sup> <sup>s</sup>* <sup>¼</sup> <sup>1</sup> *L* ð*L* 0 h i *<sup>T</sup> <sup>s</sup>*

integrated using the boundary conditions in Eqs. (29) and (31) to give

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

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

resulting ordinary differential equation for h i *<sup>T</sup> <sup>f</sup>* runs as

� �

� �

*<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>*

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

þ *εkdisxx*

� ð Þ <sup>1</sup> � *<sup>ε</sup> <sup>a</sup><sup>σ</sup>* h i *<sup>T</sup> <sup>s</sup>*

!*<sup>n</sup>*

!

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

*d*3 h i *T dx*<sup>3</sup>

*f*

þ*hv*

�*hv*

þ*hv*

**56**

� �

*Foams - Emerging Technologies*

0

<sup>¼</sup> *Gcp <sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

> *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

> *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

<sup>¼</sup> *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup>

Likewise we shall define the solid phase average temperature as follows:

The two energy equations, that is, Eqs. (21) and (22) may be added together and

þ *εkdisxx*

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

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

This equation is substituted into Eq. (21) to eliminate h i *<sup>T</sup> <sup>s</sup>* in favor of h i *<sup>T</sup> <sup>f</sup>*

*f*

*Gcp*

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

þ *εkdisxx*

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

1 � *ε* <sup>∗</sup> ð Þ*ks* þ

1 � *ε* <sup>∗</sup> ð Þ*ks* þ

� �<sup>4</sup> � �

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

16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> � � h i *<sup>T</sup> <sup>f</sup>*

� � � �

<sup>þ</sup> *<sup>h</sup>*cov h i *<sup>T</sup> <sup>s</sup>*

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

0

1 � *ε* <sup>∗</sup> ð Þ*ks* þ

*d*2 h i *T dx*<sup>2</sup>

*kstag* þ *εkdisxx* þ

þ *εkdisxx*

þ *εkdisxx*

� �

� �<sup>4</sup> � �

*d T*h i*<sup>f</sup> dx*

0

� � � �

*dx* (32)

*dx* (33)

*dx* <sup>þ</sup> *<sup>I</sup>*<sup>0</sup> cos *<sup>ξ</sup>*

<sup>þ</sup> *<sup>h</sup>*cov h i *<sup>T</sup> <sup>s</sup>*

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

*d T*h i *dx*

*f*

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

(35)

(34)

. The

$$
\langle T \rangle^{s} = T\_{eq} - \left( T\_{eq} - \langle T \rangle\_{0}^{s} \right) e^{-\gamma \mathbf{k} \mathbf{x}} \tag{37}
$$

where *γ* is the positive real root, which can uniquely be determined from the following cubic equation:

$$\gamma^3 + \frac{Gc\_p}{\left(\varepsilon^\* k\_0 \left(\frac{\overline{\langle T\rangle'}}{\langle T\rangle\_0^f}\right)^n + \varepsilon k\_{\rm difxx}\right) \lambda} \gamma^2 - \gamma - \frac{Gc\_p}{\left(k\_{\rm tag} + \varepsilon k\_{\rm dissx} + \frac{16\sigma}{3\beta} \left(\overline{\langle T\rangle'}\right)^3\right) \lambda} = 0 \tag{38}$$

where

$$\lambda = \sqrt{\frac{\left(k\_{\text{tag}} + ek\_{\text{difxxx}} + \frac{16\sigma}{3\theta} \left(\overline{\langle T\rangle}^{\circ}\right)^{3}\right)h\_{v}}{\sqrt{\left(e^{\*}k\_{0}\overline{\left(\frac{\langle T\rangle}{\langle T\rangle\_{0}^{\circ}}\right)^{n}} + ek\_{\text{difxxx}}\right)\left(\left(1-e^{\*}\right)k\_{\text{s}} + \frac{16\sigma}{3\theta} \left(\overline{\langle T\rangle^{\circ}}^{\circ}\right)^{3}\right)}}\tag{39}$$

The solid phase temperature at the inlet h i *<sup>T</sup> <sup>s</sup>* <sup>0</sup> and temperature at the thermal equilibrium, namely, *Teq* <sup>¼</sup> h i *<sup>T</sup> <sup>f</sup>* <sup>∞</sup> <sup>¼</sup> h i *<sup>T</sup> <sup>s</sup>* <sup>∞</sup>, are given by

$$\langle T \rangle\_0^\varepsilon = T\_{eq} + \left( T\_{eq} - \langle T \rangle\_0^f \right) \frac{\mathbf{G} \mathbf{c}\_p + \left( \varepsilon^\* k\_0 \left( \frac{\overline{\langle T \rangle^f}}{\langle T \rangle\_0^f} \right)^n + \varepsilon k\_{d\text{fixx}} \right) \chi \lambda}{\left( (\mathbf{1} - \varepsilon^\*) k\_s + \frac{16\sigma}{3\theta} \left( \overline{\langle T \rangle^s} \right)^3 \right) \chi \lambda} \tag{40}$$

and

$$T\_{eq} = \langle T \rangle\_0^f + \frac{I\_0 \cos \xi - (1 - \varepsilon) \left( a \sigma \left( \left( \langle T \rangle\_0^s \right)^4 - \left( \langle T \rangle\_0^f \right)^4 \right) + h\_{\rm cov} \left( \langle T \rangle\_0^s - \langle T \rangle\_0^f \right) \right)}{G c\_p} \tag{41}$$

respectively. Usually, the receiver length *L* is sufficiently long to reach the local thermal equilibrium. Thus, the average air and solid temperatures are evaluated from

$$\overline{\langle T\rangle^f} = \frac{\mathbf{1} - e^{-\gamma \mathbf{L}}}{\gamma \lambda L} \langle T\rangle\_0^f + \left(\mathbf{1} - \frac{\mathbf{1} - e^{-\gamma \lambda L}}{\gamma \lambda L}\right) T\_{eq} \cong \frac{\mathbf{1}}{\gamma \lambda L} \langle T\rangle\_0^f + \left(\mathbf{1} - \frac{\mathbf{1}}{\gamma \lambda L}\right) T\_{eq} \tag{42}$$

$$\overline{\langle T\rangle^{s}} = \frac{\mathbf{1} - e^{-\gamma \lambda L}}{\gamma \lambda L} \langle T\rangle\_{0}^{s} + \left(\mathbf{1} - \frac{\mathbf{1} - e^{-\gamma \lambda L}}{\gamma \lambda L}\right) T\_{eq} \cong \frac{\mathbf{1}}{\gamma \lambda L} \langle T\rangle\_{0}^{s} + \left(\mathbf{1} - \frac{\mathbf{1}}{\gamma \lambda L}\right) T\_{eq} \tag{43}$$

As one of the most important performance parameters, the receiver efficiency is defined by

*Foams - Emerging Technologies*

$$\eta = \frac{I\_0 \cos \xi - (1 - \varepsilon) \left( a \sigma \left( \left( \langle T \rangle\_0^s \right)^4 - \left( \langle T \rangle\_0^f \right)^4 \right) + h\_{\text{cov}} \left( \langle T \rangle\_0^s - \langle T \rangle\_0^f \right) \right)}{I\_0 \cos \xi} \tag{44}$$

foam is optically thick, the radiant energy emitted from other locations in the domain is quickly absorbed such that the radiative heat flux is given by

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

*qR* ¼ � <sup>1</sup> 3*β ∂G ∂x <sup>j</sup>*

where the diffuse integrated intensity *Gr* satisfies the irradiation transport

<sup>þ</sup> *<sup>κ</sup>* <sup>4</sup>*<sup>σ</sup>* h i *<sup>T</sup> <sup>s</sup>* ð Þ<sup>4</sup> � *Gr*

Moreover, the effects of turbulence mixing on the heat transfer are also consid-

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

� � <sup>¼</sup> <sup>0</sup> (48)

*cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT* ! *<sup>∂</sup>*h i *<sup>T</sup> <sup>f</sup>*

*∂xk*

h i *<sup>T</sup> <sup>f</sup>* (50)

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

(49)

(51)

þ *ε kdisjk* þ

*<sup>δ</sup>jk* !

equation based on the P1 model as follows:

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

where *κ* is the absorption coefficient.

*u j* � � *<sup>f</sup>*

� *d p*h i*<sup>f</sup>*

*cpG d T*h i*<sup>f</sup>*

**59**

*dx* <sup>¼</sup> h i *<sup>μ</sup> <sup>f</sup> K*

*dx* <sup>¼</sup> *<sup>d</sup>*

*d*

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

*∂ ∂x <sup>j</sup>*

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

one-dimensional set of the equations as follows:

*G*

D E*<sup>f</sup>* <sup>þ</sup> *<sup>b</sup> <sup>G</sup>*<sup>2</sup>

*ρ f*

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

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

stays nearly constant within the receiver.

structure, the eddy viscosity is given by

*d dx*

1 3*β dGr*

1 3*β*

*∂Gr ∂x <sup>j</sup>* � �

ered. Therefore, the energy equation for the air will be written as

*ε* <sup>∗</sup> *k <sup>f</sup>*

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

where turbulent Prandtl number *σ<sup>T</sup>* ¼ 0*:*9 is assumed to be constant. Under the low Mach number approximation, namely, we may reduce the macroscopic governing equations namely Eqs. (1), (49), (14) and (48) to a

*ρ f*

� � *<sup>f</sup>* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup>

*d T*h i*<sup>s</sup> dx* þ

D E *<sup>f</sup>* <sup>¼</sup> *<sup>R</sup>*

! ! *d T*h i*<sup>f</sup>*

1 3*β dGr*

*dx* � � <sup>þ</sup> *<sup>κ</sup>* <sup>4</sup>*<sup>σ</sup>* h i *<sup>T</sup> <sup>s</sup>* ð Þ<sup>4</sup> � *Gr*

The turbulence kinetic energy is dropped from the momentum equation since it

Nakayama and Kuwahara [22] established the macroscopic two-equation turbulence model, which does not require any detailed morphological information for the structure. The model, for given permeability and Forchheimer coefficient, can be used for analyzing most complex turbulent flow situations in homogeneous porous media. For the case of fully developed turbulent flow in an isotropic porous

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

*cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT*

*dx* � � � *hv* h i *<sup>T</sup> <sup>s</sup>* � h i *<sup>T</sup> <sup>f</sup>* � � <sup>¼</sup> 0 (52)

h i *<sup>μ</sup> <sup>f</sup>*

*<sup>K</sup> <sup>G</sup>* <sup>þ</sup> *bG*<sup>2</sup> !

� � <sup>¼</sup> <sup>0</sup> (53)

*∂x <sup>j</sup>*

(47)

Having established the temperature development, the momentum equation, that is, Eq. (20) along with the equation of state can easily be solved to find out the pressure distribution along the receiver as

$$\langle p \rangle^f = \sqrt{\left( \langle p \rangle\_0^f \right)^2 - 2R \left( \frac{\mu\_0}{K} \left( \frac{\langle T \rangle^f}{\langle T \rangle\_0^f} \right)^n G + bG^2 \right) \left( \frac{1 - e^{-\gamma kx}}{\gamma \lambda} \langle T \rangle\_0^f + \left( x - \frac{1 - e^{-\gamma kx}}{\gamma \lambda} \right) T\_{eq} \right)} \tag{45}$$

Under the low Mach approximation, the required pumping power per unit frontal area may be evaluated from

$$\begin{split} PP &= G\left[\frac{1}{\phi} \frac{d\langle p\rangle^f}{\langle \rho\_f\rangle^f} = G\left[\frac{1}{\phi} \frac{1}{\langle \rho\_f\rangle^f} \frac{d\langle p\rangle^f}{dx} dx \right. \\ &= \left(\frac{R}{\langle p\rangle\_0^f}\right)^2 G\left(\frac{\mu\_0}{K} \left(\frac{\langle T\rangle^f}{\langle T\rangle\_0^f}\right)^n G + bG^2\right) \int\_0^L \left(\langle T\rangle^f\right)^2 dx \\ &= \left(\frac{R}{\langle p\rangle\_0^f}\right)^2 G\left(\frac{\mu\_0}{K} \left(\frac{\langle T\rangle^f}{\langle T\rangle\_0^f}\right)^n G + bG^2\right) L\left(T\_{eq}^{-2} - \frac{2\langle 1 - e^{-\gamma L}\rangle}{\gamma kL} T\_{eq} \left(T\_{eq} - \langle T\rangle\_0^f\right) \\ &\quad + \frac{1 - e^{-2\gamma iL}}{2\gamma iL} \left(T\_{eq} - \langle T\rangle\_0^f\right)^2 \end{split}$$

$$\cong \frac{G}{\left(\langle p\rangle\_0^f\right)^2} \left(\frac{\mu\_0}{K} \left(\frac{\langle T\rangle^f}{\langle T\rangle\_0^f}\right)^2 G + bG^2\right) \frac{1}{2\gamma i} \left((2\gamma iL - 3) \left(\frac{T\_{eq}}{\langle T\rangle\_0^f}\right)^2 + 2\left(\frac{T\_{eq}}{\langle T\rangle\_0^f}\right) + 1\right) \tag{46}$$

Note that the dynamic pressure change is sufficiently small as compared to the absolute pressure such that *ρ <sup>f</sup>* D E *<sup>f</sup>* <sup>∝</sup>1*=*h i *<sup>T</sup> <sup>f</sup>* .

### **3.2 Analysis based on the P1 model**

Since the Rosseland approximation used in the previous analysis ceases to be valid near boundaries, the validity of applying the Rosseland approximation near the inlet boundary of the receiver should be investigated. In order to examine the validity of the Rosseland approximation, the results based on the Rosseland approximation will be compared with the results obtained from solving the irradiation transport equation based on the P1 model. Since the silicon carbide ceramic

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

foam is optically thick, the radiant energy emitted from other locations in the domain is quickly absorbed such that the radiative heat flux is given by

$$q\_R = -\frac{1}{3\beta} \frac{\partial G}{\partial \mathbf{x}\_j} \tag{47}$$

where the diffuse integrated intensity *Gr* satisfies the irradiation transport equation based on the P1 model as follows:

$$\frac{\partial}{\partial \mathbf{x}\_j} \left( \frac{\mathbf{1}}{\Re \beta} \frac{\partial \mathbf{G}\_r}{\partial \mathbf{x}\_j} \right) + \kappa \left( 4\sigma (\langle T \rangle^s)^4 - \mathbf{G}\_r \right) = \mathbf{0} \tag{48}$$

where *κ* is the absorption coefficient.

*η* ¼

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

*PP* ¼ *G*

ð *L*

0

!<sup>2</sup>

!<sup>2</sup>

<sup>1</sup> � *<sup>e</sup>*�2*γλ<sup>L</sup>* 2*γλL*

*G* h i*<sup>ρ</sup> <sup>f</sup>* 0 � �<sup>2</sup>

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

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

þ

ffi

**58**

h i *<sup>p</sup> <sup>f</sup>* 0 � �<sup>2</sup>

*Foams - Emerging Technologies*

vuut

*<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>*

pressure distribution along the receiver as

� <sup>2</sup>*<sup>R</sup> <sup>μ</sup>*<sup>0</sup> *K*

> ð *L*

� <sup>1</sup> *ρ f* D E *<sup>f</sup>*

! ð

0

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

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

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

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

!*<sup>n</sup>*

� �<sup>2</sup>

!*<sup>n</sup>*

0

!

�

*<sup>G</sup>* <sup>þ</sup> *bG*<sup>2</sup>

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

!

!*<sup>n</sup>*

frontal area may be evaluated from

*<sup>G</sup> <sup>μ</sup>*<sup>0</sup> *K*

*<sup>G</sup> <sup>μ</sup>*<sup>0</sup> *K*

> *μ*0 *K*

absolute pressure such that *ρ <sup>f</sup>*

**3.2 Analysis based on the P1 model**

� *d p*h i*<sup>f</sup> ρ f* D E *<sup>f</sup>* <sup>¼</sup> *<sup>G</sup>*

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

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

!*<sup>n</sup>*

!

� �<sup>4</sup> � �

is, Eq. (20) along with the equation of state can easily be solved to find out the

*<sup>G</sup>* <sup>þ</sup> *bG*<sup>2</sup>

Under the low Mach approximation, the required pumping power per unit

*L*

h i *<sup>T</sup> <sup>f</sup>* � �<sup>2</sup>

*dx*

<sup>2</sup> � 2 1 � *<sup>e</sup>*�*γλ<sup>L</sup>* � � *γλL*

� � �

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

!<sup>2</sup>

þ 2

0

1

0 @

Note that the dynamic pressure change is sufficiently small as compared to the

.

Since the Rosseland approximation used in the previous analysis ceases to be valid near boundaries, the validity of applying the Rosseland approximation near the inlet boundary of the receiver should be investigated. In order to examine the validity of the Rosseland approximation, the results based on the Rosseland

approximation will be compared with the results obtained from solving the irradiation transport equation based on the P1 model. Since the silicon carbide ceramic

<sup>∝</sup>1*=*h i *<sup>T</sup> <sup>f</sup>*

*L Teq*

<sup>2</sup>*γλ* ð Þ <sup>2</sup>*γλ<sup>L</sup>* � <sup>3</sup>

*d p*h i*<sup>f</sup> dx dx*

*<sup>G</sup>* <sup>þ</sup> *bG*<sup>2</sup>

*<sup>G</sup>* <sup>þ</sup> *bG*<sup>2</sup>

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � *e*�*γλ<sup>x</sup> γλ* h i *<sup>T</sup> <sup>f</sup>*

Having established the temperature development, the momentum equation, that

� � � �

<sup>þ</sup> *<sup>h</sup>*cov h i *<sup>T</sup> <sup>s</sup>*

*<sup>I</sup>*<sup>0</sup> cos *<sup>ξ</sup>* (44)

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

<sup>0</sup> <sup>þ</sup> *<sup>x</sup>* � <sup>1</sup> � *<sup>e</sup>*�*γλ<sup>x</sup>*

� �

*γλ* � �

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

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

!

0

þ 1

(46)

1 A

*Teq*

(45)

Moreover, the effects of turbulence mixing on the heat transfer are also considered. Therefore, the energy equation for the air will be written as

$$\begin{split} \left( \text{ec}\_{p} \frac{\partial}{\partial \mathbf{x}\_{j}} \Big/ \rho\_{f} \right)^{f} \langle \mu\_{j} \rangle^{f} \langle T \rangle^{f} &= \frac{\partial}{\partial \mathbf{x}\_{j}} \Big/ \left( \varepsilon^{\*} \left< \mathbf{k}\_{f} \right>^{f} \frac{\partial \langle T \rangle^{f}}{\partial \mathbf{x}\_{j}} + \varepsilon \left( \mathbf{k}\_{\text{dir}\_{jk}} + \frac{\mathbf{c}\_{P\_{f}} \langle \mu\_{t} \rangle^{f}}{\sigma\_{T}} \right) \frac{\partial \langle T \rangle^{f}}{\partial \mathbf{x}\_{k}} \delta\_{jk} \right) \\ & \qquad \qquad - \mathbf{h}\_{v} \Big( \langle T \rangle^{f} - \langle T \rangle^{f} \Big) \end{split} \tag{49}$$

where turbulent Prandtl number *σ<sup>T</sup>* ¼ 0*:*9 is assumed to be constant.

Under the low Mach number approximation, namely, we may reduce the macroscopic governing equations namely Eqs. (1), (49), (14) and (48) to a one-dimensional set of the equations as follows:

$$-\frac{d\langle p\rangle^f}{d\mathbf{x}} = \frac{\langle \mu\rangle^f}{K} \frac{\mathbf{G}}{\left\langle \rho\_f \right\rangle^f} + b\frac{\mathbf{G}^2}{\left\langle \rho\_f \right\rangle^f} = \frac{R}{\langle p\rangle^f} \left(\frac{\langle \mu\rangle^f}{K} \mathbf{G} + b\mathbf{G}^2\right) \langle T\rangle^f \tag{50}$$

$$\varepsilon\_{p}G\frac{d\langle T\rangle^{f}}{d\mathbf{x}} = \frac{d}{d\mathbf{x}}\left(\varepsilon^{\*}\left\langle\mathbf{k}\_{f}\right\rangle^{f} + \varepsilon\left(\mathbf{k}\_{\mathrm{div}\_{\mathbf{x}}} + \frac{\varepsilon\_{p\_{f}}\langle\mu\_{t}\rangle^{f}}{\sigma\_{T}}\right)\right)\frac{d\langle T\rangle^{f}}{d\mathbf{x}} - h\_{v}\left(\langle T\rangle^{f} - \langle T\rangle^{s}\right) \tag{51}$$

$$\frac{d}{d\mathbf{x}}\left((1-\varepsilon^\*)k\_\circ \frac{d\langle T\rangle^s}{d\mathbf{x}} + \frac{\mathbf{1}}{3\beta}\frac{dG\_r}{d\mathbf{x}}\right) - h\_v\left(\langle T\rangle^s - \langle T\rangle^f\right) = \mathbf{0} \tag{52}$$

$$\frac{d}{d\mathbf{x}}\left(\frac{1}{3\beta}\frac{dG\_r}{d\mathbf{x}}\right) + \kappa \left(4\sigma (\langle T\rangle^s)^4 - G\_r\right) = \mathbf{0} \tag{53}$$

The turbulence kinetic energy is dropped from the momentum equation since it stays nearly constant within the receiver.

Nakayama and Kuwahara [22] established the macroscopic two-equation turbulence model, which does not require any detailed morphological information for the structure. The model, for given permeability and Forchheimer coefficient, can be used for analyzing most complex turbulent flow situations in homogeneous porous media. For the case of fully developed turbulent flow in an isotropic porous structure, the eddy viscosity is given by

$$\langle \mu\_{\mathfrak{t}} \rangle^{f} = \mathfrak{Z}bbK \tag{54}$$

*d*5 h i *T dx*<sup>5</sup> *f*

<sup>¼</sup> *Gcp <sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

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

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

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

þ 3*βκ*

�3*βκhv*

�3*βκhv*

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

*ε* <sup>∗</sup> *k*<sup>0</sup>

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

*kstag* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

*kstag* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

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

!*<sup>n</sup>*

*<sup>γ</sup>*<sup>5</sup> <sup>þ</sup>

3*βκ hv*

�

�

þ

**61**

where

þ

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

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

BBBBB@

0

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

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

1 � *ε* <sup>∗</sup> ð Þ*ks*

*kstag* <sup>þ</sup> *<sup>ε</sup> kdisjk* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

! !

3*βκhvGcp*

! !

*Gcp*

! !

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

!

*σT*

!

3*βκGcp*

*σT*

!

! !

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

þ *hv* 3*βκ*

þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup>

> þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup>

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

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

*σT* !

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

þ *hv* 3*βκ*

þ *hv* <sup>3</sup>*βκ* � � <sup>3</sup>*βκGcp*

> *σT* !

> > <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

*σT*

ð Þ <sup>1</sup> � ð Þ <sup>1</sup> � *<sup>a</sup>* ð Þ <sup>1</sup> � *<sup>ε</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>*

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

which can be determined from the following characteristic equation:

*λ γ*4

!

*σT*

3*βκGcp hv<sup>λ</sup> <sup>γ</sup>*<sup>2</sup> <sup>þ</sup>

> *λ*3 ¼ 0

*σT*

*σT*

*kstag* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

*d*4 h i *T dx*<sup>4</sup>

þ

*f*

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

*<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>*

þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup>

*σT*

*hv*

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

*d T*h i*<sup>f</sup> dx*

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

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

� � � �

1 � *ε* <sup>∗</sup> ð Þ*ks*

þ *hv*

<sup>3</sup>*βκ* � � <sup>þ</sup> <sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks*

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

*σT*

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

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

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

þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup>

> 3*βκ λ*2 *γ*

� �

! !

1 � *ε* <sup>∗</sup> ð Þ*ks*

1 � *ε* <sup>∗</sup> ð Þ*ks*

! !

This ordinary differential equation, with the boundary conditions in Eqs. (29), (56) and (57) and the zero derivative conditions far downstream (*x* ! ∞: Note *L* is sufficiently large), yields Eqs. (36) and (37). Note that *γ* is the positive real root

*σT* !

*σT*

1

CCCCCA *d*3 h i *T dx*<sup>3</sup>

1 � *ε* <sup>∗</sup> ð Þ*ks*

*f*

*d*2 h i *<sup>T</sup> <sup>f</sup> dx*<sup>2</sup>

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

(60)

*γ*3

(61)

Note that

$$\left(\mathbb{A}\_{\text{dis}\_{\text{xx}}} / \left(\mathbb{c}\_{\mathcal{P}\_f} \langle \mu\_\mathbf{t} \rangle^f / \sigma\_T \right) = \mathbf{0}. \mathsf{G} \sigma\_T / \epsilon b \sqrt{K} > 1 \right)$$

such that the dispersion thermal conductivity usually overwhelms the eddy thermal conductivity.

For absorption coefficient *κ*, the measurements made on cordierite ceramic foams by Kamiuto et al. [22] give the following correlation:

$$\kappa = 4a(\mathbf{1} - \varepsilon)/d\_m \tag{55}$$

The boundary conditions of h i *<sup>T</sup> <sup>f</sup>* and h i *<sup>p</sup> <sup>f</sup>* are the same as Eqs. (29) and (30). The other boundary conditions are given as follows:

$$q\_{R\_x} = -\frac{\mathbf{1}}{3\beta} \frac{dG\_r}{d\mathbf{x}} = -\frac{G\_r}{2} \tag{56}$$

and

$$\begin{aligned} &-\left(\varepsilon^\* \left< k\_f \right>^f + \varepsilon \left( k\_{d\text{times}} + \frac{c\_{p\_f} \left< \mu\_t \right>^f}{\sigma\_T} \right) \right) \frac{d \langle T \rangle^f}{d\mathbf{x}} \Bigg|\_{\mathbf{x}=0} - (1 - \varepsilon^\*) k\_s \frac{d \langle T \rangle^s}{d\mathbf{x}} \Bigg|\_{\mathbf{x}=0} - \frac{G\_{\parallel}|\_{\mathbf{x}=0}}{2} \\ &= (1 - (1 - a)(1 - \varepsilon)) I\_0 \cos \xi \end{aligned} \tag{57}$$
 
$$ - (1 - \varepsilon) \left( a \sigma \left( \left< \langle T \rangle^s\_0 \right>^4 - \left< \langle T \rangle^f\_0 \right>^4 \right) + h\_{conv} \left( \langle T \rangle^s\_0 - \langle T \rangle^f\_0 \right) \right) \tag{58}$$

Furthermore, the streamwise gradients of the dependent variables h i *<sup>T</sup> <sup>f</sup>* , h i *<sup>T</sup> <sup>s</sup>* and *Gr* are set to zero sufficiently far downstream at *x* = *L*.

The two energy equations, namely, Eqs. (51) and (52) may be added together and integrated using the boundary conditions in Eqs. (29) and (57) to give

$$\begin{split} &c\_{p}G\Big(\langle T\rangle^{f} - \langle T\rangle\_{0}^{f}\Big) \\ &= \left(\varepsilon^{\*}k\_{0}\left(\frac{\langle T\rangle^{f}}{\langle T\rangle\_{0}^{f}}\right)^{u} + \varepsilon\Big(k\_{\mathrm{dir}\_{\mathrm{ax}}} + \frac{c\_{p\_{f}}\langle\mu\_{t}\rangle^{f}}{\sigma\_{T}}\right)\right)\frac{d\langle T\rangle^{f}}{d\mathbf{x}} + (\mathbbm{1} - \varepsilon^{\*})k\_{\mathrm{r}}\frac{\partial\langle T\rangle^{\mathrm{r}}}{\partial\mathbf{x}\_{j}} + \frac{\mathbbm{1}}{3\beta}\frac{dG\_{r}}{d\mathbf{x}} \\ &+ (\mathbbm{1} - (\mathbbm{1} - a)(\mathbbm{1} - \varepsilon))I\_{0}\cos\xi - (\mathbbm{1} - \varepsilon)\Big(a\sigma\Big(\big(\langle T\rangle^{\mathrm{r}}\_{0}\big)^{4} - \big(\langle T\rangle^{f}\_{0}\big)^{4}\big) + h\_{\mathrm{conv}}\Big(\langle T\rangle^{\mathrm{r}}\_{0} - \langle T\rangle^{f}\_{0}\big)\Big) . \end{split} \tag{58}$$

Eqs. (52) and (53) are combined to give

$$\mathcal{G}\_r = 4\sigma (\langle T\rangle^s)^4 + \frac{h\_v}{\kappa} \left( \langle T\rangle^s - \langle T\rangle^f \right) - \frac{(\mathbf{1} - \varepsilon^\*)k\_t}{\kappa} \frac{d^2 \langle T\rangle^s}{d\kappa^2} \tag{59}$$

This equation, Eq. (59), and Eq. (51) are substituted into Eq. (58) to eliminate *Gr* and h i *<sup>T</sup> <sup>s</sup>* in favor of h i *<sup>T</sup> <sup>f</sup>* . The resulting ordinary differential equation for h i *<sup>T</sup> <sup>f</sup>* runs as

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

$$\begin{split} \frac{d^2 \langle \nabla f^{\prime\prime} \rangle^{\prime}}{ds^2} &= \frac{G\_F}{e^\* k\_0 \left(\frac{\langle \nabla f^{\prime} \rangle}{\langle \nabla f \rangle} \right)^n + e \left(k\_{\mathrm{duk}\_{\mathrm{ox}}} + \frac{c\_{\mathrm{f}} \langle \mu\_{\mathrm{f}} \rangle^{\prime}}{\sigma T} \right)}{\left(3\hbar \kappa - \frac{16\pi}{3\ell} \left(\frac{\langle \nabla f \rangle}{\langle \nabla f \rangle} \right)^3 + \frac{k\_{\mathrm{f}}}{3\hbar \kappa} + \frac{c\_{\mathrm{f}}}{\left(k\_{\mathrm{duk}\_{\mathrm{ox}}} + \frac{c\_{\mathrm{f}} \langle \mu\_{\mathrm{f}} \rangle^{\prime}}{\sigma T} \right) \right) \frac{d^3 \langle \nabla f^{\prime} \rangle^{\prime}}{dx^3}} \\ &+ \left( 3\hbar \kappa - \frac{16\pi}{3\ell} \left(\frac{\langle \nabla f \rangle}{\langle \nabla f \rangle^3} \right)^3 + \frac{k\_{\mathrm{f}}}{3\hbar \kappa} \right) \frac{16\pi}{\left(\kappa^\* k\_0 \left(\frac{\langle \nabla f \rangle}{\langle \nabla f \rangle} \right)^3 + \kappa \left(k\_{\mathrm{duk}\_{\mathrm{ox}}} + \frac{c\_{\mathrm{f}} \langle \mu\_{\mathrm{f}} \rangle^{\prime}}{\sigma T} \right) \right) (1 - \epsilon^\*) k\_{\mathrm{e}}} \frac{d^2 \langle \nabla f^{\prime} \rangle^{\prime}}{dx^2} \\ &- \left( 3\hbar \kappa - \frac{1}{3\ell} \left(\frac{k\_{\mathrm{duk}\_{\mathrm{red}}} + \frac{c\_{\mathrm{f}}}{\mathrm{(\$$

This ordinary differential equation, with the boundary conditions in Eqs. (29), (56) and (57) and the zero derivative conditions far downstream (*x* ! ∞: Note *L* is sufficiently large), yields Eqs. (36) and (37). Note that *γ* is the positive real root which can be determined from the following characteristic equation:

*<sup>γ</sup>*<sup>5</sup> <sup>þ</sup> *Gcp <sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 � �*<sup>n</sup>* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT* ! ! *λ γ*4 � 3*βκ hv ε* <sup>∗</sup> *k*<sup>0</sup> h i *<sup>T</sup> <sup>f</sup>* h i *<sup>T</sup> <sup>f</sup>* 0 !*<sup>n</sup>* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT* ! ! <sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks* <sup>þ</sup> 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> þ *hv* <sup>3</sup>*βκ* � � <sup>þ</sup> <sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks kstag* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT* ! þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> *γ*3 � <sup>1</sup> � *<sup>ε</sup>* <sup>∗</sup> ð Þ*ks* <sup>þ</sup> 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> þ *hv* 3*βκ kstag* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT* ! þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> 3*βκGcp hv<sup>λ</sup> <sup>γ</sup>*<sup>2</sup> <sup>þ</sup> 3*βκ λ*2 *γ* þ 3*βκGcp kstag* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT* ! þ 16*σ* <sup>3</sup>*<sup>β</sup>* h i *<sup>T</sup> <sup>s</sup>* � �<sup>3</sup> !*λ*3 ¼ 0 (61)

where

*<sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>* <sup>¼</sup> <sup>2</sup>*GbK* (54)

*K* <sup>p</sup> <sup>&</sup>gt;<sup>1</sup>

*κ* ¼ 4*a*ð Þ 1 � *ε =dm* (55)

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

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

> *<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup> ∂x <sup>j</sup>* þ 1 3*β dGr dx*

> > *d*2 h i *<sup>T</sup> <sup>s</sup>*

<sup>þ</sup> *hconv* h i *<sup>T</sup> <sup>s</sup>*

0

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

. The resulting ordinary differential equation for h i *<sup>T</sup> <sup>f</sup>*

� � � �

and h i *<sup>p</sup> <sup>f</sup>* are the same as Eqs. (29) and (30).

<sup>2</sup> (56)

*d T*h i*<sup>s</sup> dx*

� � � � *x*¼0

� *Gr*<sup>j</sup> *x*¼0 2

(57)

and

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

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

*dx*<sup>2</sup> (59)

(58)

<sup>¼</sup> <sup>0</sup>*:*6*σT=ε<sup>b</sup>* ffiffiffiffi

Note that

*Foams - Emerging Technologies*

thermal conductivity.

and

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

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

<sup>¼</sup> *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</sup>

*Gr* and h i *<sup>T</sup> <sup>s</sup>*

runs as

**60**

� �

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

!*<sup>n</sup>*

0

The boundary conditions of h i *<sup>T</sup> <sup>f</sup>*

� � *<sup>f</sup>* <sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup>

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

! !

*kdisxx <sup>=</sup> cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

foams by Kamiuto et al. [22] give the following correlation:

*qRx* ¼ � <sup>1</sup>

*cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup> σT*

� �<sup>4</sup> � �

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

*Gr* are set to zero sufficiently far downstream at *x* = *L*.

<sup>þ</sup> *<sup>ε</sup> kdisxx* <sup>þ</sup> *cp <sup>f</sup> <sup>μ</sup><sup>t</sup>* h i*<sup>f</sup>*

! !

þð Þ <sup>1</sup> � ð Þ <sup>1</sup> � *<sup>a</sup>* ð Þ <sup>1</sup> � *<sup>ε</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>*

Eqs. (52) and (53) are combined to give

*Gr* <sup>¼</sup> <sup>4</sup>*<sup>σ</sup>* h i *<sup>T</sup> <sup>s</sup>* ð Þ<sup>4</sup> <sup>þ</sup>

in favor of h i *<sup>T</sup> <sup>f</sup>*

*σT*

*hv*

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

This equation, Eq. (59), and Eq. (51) are substituted into Eq. (58) to eliminate

3*β dGr dx* ¼ � *Gr*

*d T*h i*<sup>f</sup> dx*

� � � �

The two energy equations, namely, Eqs. (51) and (52) may be added together

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

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

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

� �<sup>4</sup> � �

0

Furthermore, the streamwise gradients of the dependent variables h i *<sup>T</sup> <sup>f</sup>*

and integrated using the boundary conditions in Eqs. (29) and (57) to give

� � � � � *x*¼0

<sup>þ</sup> *hconv* h i *<sup>T</sup> <sup>s</sup>*

The other boundary conditions are given as follows:

*=σ<sup>T</sup>* � �

such that the dispersion thermal conductivity usually overwhelms the eddy

For absorption coefficient *κ*, the measurements made on cordierite ceramic

### *Foams - Emerging Technologies*

$$\lambda = \sqrt{\frac{\left(k\_{\text{stag}} + \varepsilon \left(k\_{\text{dis}\_{\text{xx}}} + \frac{c\_{p\_f} \langle \mu\_t \rangle^f}{\sigma\_T} \right) + \frac{16\sigma}{3\theta} \left(\overline{\langle T\rangle^f} \right)^3 \right) h\_v}{\sqrt{\left(\varepsilon^\* \, k\_0 \left(\frac{\langle T\rangle^f}{\langle T\rangle\_0^f} \right)^n + \varepsilon \left(k\_{\text{dis}\_{\text{xx}}} + \frac{c\_{p\_f} \langle \mu\_t \rangle^f}{\sigma\_T} \right) \right) (1 - \varepsilon^\* \,) k\_t}} \tag{62}$$

The solid phase temperature at the inlet h i *<sup>T</sup> <sup>s</sup>* <sup>0</sup> and temperature at the thermal equilibrium, namely,*Teq* <sup>¼</sup> h i *<sup>T</sup> <sup>f</sup>* <sup>∞</sup> <sup>¼</sup> h i *<sup>T</sup> <sup>s</sup>* <sup>∞</sup>, are determined from the following implicit equations:

$$\begin{split} \text{Gc}\_{p}\left(T\_{eq} - \langle T\rangle\_{0}^{f}\right) &= -\left(\epsilon^{\*}k\_{0} + \epsilon\left(k\_{\text{dirax}} + \frac{c\_{p}\left\langle\mu\_{t}\right\rangle^{f}}{\sigma\_{T}}\right)\right)\text{y}\lambda\left(T\_{eq} - \langle T\rangle\_{0}^{f}\right) \\ &- (\mathbf{1} - \epsilon^{\*})k\_{i}\mathbf{y}\lambda\left(T\_{eq} - \langle T\rangle\_{0}^{\prime}\right) \\ &- \frac{1}{2}\left(4\sigma\left(\langle T\rangle\_{0}^{\prime}\right)^{4} + \frac{h\_{\text{v}}}{\kappa}\left(\langle T\rangle\_{0}^{\prime} - \langle T\rangle\_{0}^{f}\right) + \frac{(\mathbf{1} - \epsilon^{\*})k\_{\text{v}}}{\kappa}\left(\mathbf{y}\lambda\right)^{2}\left(T\_{eq} - \langle T\rangle\_{0}^{\prime}\right)\right) \end{split} \tag{\text{fGd}}$$

$$\begin{split} T\_{\eta} &= \langle T \rangle\_{0}^{f} \\ &+ \frac{(1 - (1 - a)(1 - \epsilon))I\_{0}\cos\xi - (1 - \epsilon)\left(a\sigma\left(\left(\langle T \rangle\_{0}^{s}\right)^{4} - \left(\langle T \rangle\_{0}^{f}\right)^{4}\right) + h\_{\text{conv}}\left(\langle T \rangle\_{0}^{s} - \langle T \rangle\_{0}^{f}\right)\right)}{\text{Gr}\_{P}} \end{split} \tag{64}$$

characteristic Eq. (61) under a possible range of the silicon carbide parameters yields two positive roots *γ<sup>h</sup>* and *γl*, which are fairly close to each other. The corresponding temperature variations of both phases however depend strongly on its value, which results in a non-unique value of equilibrium temperature. Since flow instability is inferred by an unexpected nature of the quadratic pressure difference with respect to equilibrium temperature, the existence of two positive roots may be responsible for possible hydrodynamic and thermal instabilities reported previously. A further investigation based on an unsteady procedure is definitely needed to explore possible causes of these instabilities, closely related to the radia-

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

The third-order characteristic Eq. (38) based on the Rosseland approximation, on the other hand, yields only one positive root *γ*1. The corresponding fluid and

*Comparison of the temperature developments with the Rosseland approximation and P1 model.*

tive heat transfer mode.

*Residual of the fifth-order characteristic equation.*

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

**Figure 2.**

**Figure 3.**

**63**

where the boundary condition in Eq. (56) is utilized. Usually, the receiver length *L* is sufficiently long to reach the local thermal equilibrium. Thus, the average air and solid temperatures are evaluated from Eqs. (42) and (43).
