**1. Introduction**

A solar volumetric receiver is required to have the resistance to temperature as high as 1000 degree Celsius, high porosity for sufficiently large extinction volume such that the concentrated solar radiation penetrates through the receiver, high cell density to achieve large specific surface area and sufficiently high effective thermal conductivity to avoid possible thermal spots. Extruded monoliths with parallel channels (i.e. honeycomb structure) are being used in some solar power plants in Europe, including the solar power tower plant of 1.5 MW built in 2009, in Julich in Germany [1, 2]. However, in such conventional receivers, both thermal spots [3] and flow instabilities [4] have been often reported. In the monolith receiver, locally high solar flux leads to a low mass flow with high temperature, whereas locally low solar flux leads to a high mass flow with low temperature. This causes the absorber material to exceed the design temperature locally, which then leads to its

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

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 turbulence mixing on the heat transfer were not considered.

� *<sup>∂</sup>*h i *<sup>p</sup> <sup>f</sup> ∂xi*

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

For the air:

¼ *∂ ∂x <sup>j</sup>*

> þ 1 *V* ð

ð Þ 1 � *ε ρscs*

**51**

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup> <sup>f</sup> hstag* � *<sup>p</sup>*

0

B@

*A*int *k f ∂T ∂x <sup>j</sup>*

For the solid matrix:

*<sup>∂</sup>*h i *<sup>T</sup> <sup>s</sup> <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup> ∂x <sup>j</sup>*

and solid matrix phase can be defined as.

� � � � *<sup>f</sup>*

*ε k <sup>f</sup>*

*ρ*

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

*ε ∂*

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

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

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

geneous, the integration of the two distinct energy equations gives:

*A*int

ð Þ 1 � *ε ks*

0

B@

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

its intrinsic average and the spatial deviation from it:

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

h i*<sup>ρ</sup> <sup>f</sup> <sup>u</sup> <sup>j</sup>* � � *<sup>f</sup>*

<sup>þ</sup> *<sup>ε</sup> <sup>∂</sup> ∂x <sup>j</sup>*

*<sup>K</sup>* h i *ui* <sup>þ</sup> *<sup>b</sup>*h i*<sup>ρ</sup> <sup>f</sup>*

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 homo-

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

*<sup>k</sup> fTn jdA* � *ε ρ*h i*<sup>f</sup>* <sup>~</sup>

� *ks V* ð

where the intrinsic volume average of a certain local variable *ϕ* in the fluid phase

*<sup>ϕ</sup>dV*,h i *<sup>ϕ</sup> <sup>m</sup>* � <sup>1</sup>

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

*A*int

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u j* � � *u <sup>j</sup>*

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

*n jdA* (2)

*Tn jdA*‐*qR*

*Vm* ð *Vm* 1 CA � <sup>1</sup> *V* ð

<sup>q</sup> � �h i *ui* (1)

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

*A*int *k f ∂T ∂x <sup>j</sup>*

*ϕdV* (4)

1

CA

*n jdA*

(3)

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 heat transfer within a ceramic foam receiver.
