**4. Validations of the Rosseland approximation**

Smirnova et al. [23] numerically studied the compressible fluid flow and heat transfer within the solar receiver with silicon carbide monolithic honeycombs. In their paper, the following input data were collected to obtain the analytic solutions based on the present local thermal non-equilibrium model:

$$\begin{aligned} \left\langle \rho\_f \right\rangle\_0^f &= \mathbf{1.2[kg/m^3]}, \ \langle T \rangle\_0^f = \mathbf{300[K]} \langle \langle p \rangle\_0^f = \mathbf{10^5[Pa]} \rangle, \ c\_p = \mathbf{1000[J/kgK]},\\ \mathbf{G} &= \mathbf{1.2[kg/m^2s]}, \ \mathbf{L} = \mathbf{0.05[m]}, \ \mathbf{l}\_0 = \mathbf{10^6[W/m^2]}, \ \xi = \mathbf{0}, \ \mathbf{h}\_{conv} = \mathbf{0[W/m^2K]},\\ \mathbf{k}\_t &= \mathbf{150[W/mK]} \mathbf{k}\_{dii} = \mathbf{0[W/mK]}, \ \mathbf{h}\_v = \mathbf{8.8} \times \mathbf{10^4[W/m^3K]}, \ \varepsilon = \mathbf{0.5}, \ \beta = \mathbf{50[1/m]}. \end{aligned}$$

However, it should be noticed that the porosity of the silicon carbide monolithic honeycombs is not available in Smirnova et al. [23], its value was estimated to be *ε* ¼ 0*:*5 from the figure provided by Agrafiotis et al. [24]. The mean extinction coefficient *β* for silicon carbide monolithic honeycombs is not available in their paper. Finally, the value was estimated to be 50[1/m] by correlating the present results against theirs. It should also be noted that the convective heat transfer coefficient was set to zero since radiation predominates over convection in the receiver front.

As for possible instabilities, the fifth-order characteristic Eq. (61) based on the P1 model should be examined carefully. **Figure 2** shows the residual of the fifthorder characteristic equation *f <sup>R</sup>*ð Þ*γ* . The figure clearly shows that the fifth-order

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

**Figure 2.** *Residual of the fifth-order characteristic equation.*

*λ* ¼

*Foams - Emerging Technologies*

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

0

implicit equations:

� �

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

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

þ

*ρ f* D E*<sup>f</sup>*

receiver front.

**62**

<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*2 kg*=*m<sup>3</sup> ½ �, h i *<sup>T</sup> <sup>f</sup>*

vuuuuut

� 1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

*σT*

0 � �

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

� �

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

*Gcp*

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

Smirnova et al. [23] numerically studied the compressible fluid flow and heat transfer within the solar receiver with silicon carbide monolithic honeycombs. In their paper, the following input data were collected to obtain the analytic solutions

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

However, it should be noticed that the porosity of the silicon carbide monolithic honeycombs is not available in Smirnova et al. [23], its value was estimated to be *ε* ¼ 0*:*5 from the figure provided by Agrafiotis et al. [24]. The mean extinction coefficient *β* for silicon carbide monolithic honeycombs is not available in their paper. Finally, the value was estimated to be 50[1/m] by correlating the present results against theirs. It should also be noted that the convective heat transfer coefficient was set to zero since radiation predominates over convection in the

As for possible instabilities, the fifth-order characteristic Eq. (61) based on the P1 model should be examined carefully. **Figure 2** shows the residual of the fifthorder characteristic equation *f <sup>R</sup>*ð Þ*γ* . The figure clearly shows that the fifth-order

� �<sup>4</sup> � �

<sup>þ</sup> <sup>16</sup>*<sup>σ</sup>*

*σT*

*hv*

(62)

0

(63)

(64)

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

1 � *ε* <sup>∗</sup> ð Þ*ks*

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

0

*<sup>κ</sup>* ð Þ *γλ* <sup>2</sup> *Teq* � h i *<sup>T</sup> <sup>s</sup>*

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

<sup>∞</sup>, are determined from the following

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

þ

0

� � � �

� �

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

� � � �

½ �Þ Pa , *cp* ¼ 1000 J½ � *=*kgK ,

*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>*

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

! !

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

¼ � *<sup>ε</sup>* <sup>∗</sup> *<sup>k</sup>*<sup>0</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*ksγλ Teq* � h i *<sup>T</sup> <sup>s</sup>*

and solid temperatures are evaluated from Eqs. (42) and (43).

**4. Validations of the Rosseland approximation**

based on the present local thermal non-equilibrium model:

<sup>0</sup> <sup>¼</sup> 300 K½ �ðh i *<sup>p</sup> <sup>f</sup>*

*<sup>G</sup>* <sup>¼</sup> <sup>1</sup>*:*2 kg*=*m2 ½ �<sup>s</sup> , *<sup>L</sup>* <sup>¼</sup> <sup>0</sup>*:*05½ � *<sup>m</sup>* , *<sup>I</sup>*<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>6</sup> <sup>W</sup>*=*m<sup>2</sup> ½ �, *<sup>ξ</sup>* <sup>¼</sup> 0, *hconv* <sup>¼</sup> 0 W*=*m<sup>2</sup> ½ � <sup>K</sup> , *ks* <sup>¼</sup> 150 W½ � *<sup>=</sup>*mK *kdis* <sup>¼</sup> 0 W½ � *<sup>=</sup>*mK , *hv* <sup>¼</sup> <sup>8</sup>*:*<sup>8</sup> � <sup>10</sup><sup>4</sup> <sup>W</sup>*=*m<sup>3</sup> ½ � <sup>K</sup> , *<sup>ε</sup>* <sup>¼</sup> <sup>0</sup>*:*5, *<sup>β</sup>* <sup>¼</sup> 50 1½ � *<sup>=</sup>*<sup>m</sup> *:*

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

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>*

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

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 radiative heat transfer mode.

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

**Figure 3.** *Comparison of the temperature developments with the Rosseland approximation and P1 model.*

### **Figure 4.**

*Axial developments of the fluid and solid phase temperatures: comparison of the present analysis and FEM analysis.*

solid temperature variations generated under the Rosseland approximation are compared with those based on the P1 model with the larger root *γh*. **Figure 3** shows that both sets of the temperature developments agree fairly well with each other. Thus, the Rosseland approximation for this case, despite its failure near the inlet boundary, is fairly accurate and may well be used for quick estimations and further analysis.

All other parameters are evaluated using Eqs. (17), (18) and from Eq. (23) to (28). As shown in **Figure 5**, it is interesting to note that *η* suddenly increases at some critical value of *dm* for a given value of *PP*, which means that the pore diameter *dm* must be larger than this critical value to achieve high *η*. This finding is useful to

*PP* <sup>p</sup><sup>3</sup> for high *PP*, which results in that the amount of heat carried by the air,

from zero. Nevertheless, its rate of increase diminishes for the higher *PP* range, in

*PP* <sup>p</sup> , increases drastically on increasing the pumping power *PP*

*Gtr* ffi *bGtr*

*I*<sup>0</sup> cos *ξ cp*h i *<sup>T</sup> <sup>f</sup>* 0

! !*<sup>n</sup>* <sup>1</sup>

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

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

Thus, Eq. (46) may be written for the case in which the sharp rise in *η* takes

*PP* <sup>p</sup><sup>3</sup> . Moreover, it can also be concluded that the sharp rise

1þ*n*

*I*<sup>0</sup> cos *ξ cpGtr*h i *<sup>T</sup> <sup>f</sup>* 0

ffi

*PP* <sup>p</sup> for low *PP* and

<sup>2</sup> (65)

(66)

(67)

design a volumetric receiver, and can be interpreted in what follows. As indicated in Eq. (46), it can be easily deduced that *G* ∝ ffiffiffiffiffiffi

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

in the receiver efficiency occurs around the transition from the Darcy to

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

!*<sup>n</sup>*

*μ*0 *K*

*Gtr* ffi *<sup>μ</sup>*<sup>0</sup> *bK*

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

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

*G* ∝ ffiffiffiffiffiffi

**Figure 5.**

or

since

place as follows:

**65**

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

0 � �<sup>∝</sup> ffiffiffiffiffiffi

Forchheimer regime, namely,

0 � �<sup>∝</sup> ffiffiffiffiffiffi

*Effects of the pore diameter on the receiver efficiency.*

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

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

In **Figure 4**, the present analytic solutions are compared against the large-scale FEM numerical calculations based on COMSOL, reported by Smirnova et al. [23]. It should be mentioned that the direct numerical integrations of Eqs. (20)–(22) were also carried out using the finite volume method code, SAINTS [12]. As the convergence criteria, the residuals of all equations are less than 10�<sup>5</sup> . It can be clearly seen that the air temperature increases as receiving heat from the monolithic receiver. Eventually, these two phases reach local thermal equilibrium near the exit. Both sets of solutions agree very well with each other, indicating the validity of the present local thermal non-equilibrium model.
