**5. Applications to silicon carbide ceramic foam volumetric receiver**

In order to overcome the problems associated with thermal spots and flow instabilities, we would like to study fluid flow and heat transfer characteristics in silicon carbide ceramic foams based on the analytical expressions of pressure and temperature fields within a solar volumetric receiver. The performance of the receiver may be assessed in terms of the receiver efficiency *η* under equal pumping power*PP*. Thus, the effects of the pore diameter *dm* on the receiver efficiency *η* are presented in **Figure 5**, since *dm* is a crucial geometry parameter affecting hydrodynamic and thermal characteristics of foam shown in Eqs. (23), (24) and (27). The pore diameter *dm* is varied whereas the other parameters are fixed as follows:

$$\left\langle \rho\_f \right\rangle\_0^f = \mathbf{1}.\mathbf{2}[\mathbf{kg}/\mathbf{m}^3], \ \left\langle T \right\rangle\_0^f = \mathbf{300}[\mathbf{K}](\langle p \rangle\_0^f = \mathbf{10}^5[\mathbf{Pa}]), \ c\_p = \mathbf{1000}[\mathbf{J}/\mathbf{kg}\mathbf{K}],$$

$$L = \mathbf{0}.\mathbf{03}[m], \ I\_0 = \mathbf{10}^6[\mathbf{W}/\mathbf{m}^2], \ \xi = \mathbf{0}, h\_{conv} = \mathbf{0}[\mathbf{W}/\mathbf{m}^2\mathbf{K}], \ k\_t = \mathbf{150}[\mathbf{W}/\mathbf{m}\mathbf{K}], \ \varepsilon = \mathbf{0}.9.$$

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

**Figure 5.** *Effects of the pore diameter on the receiver efficiency.*

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 design a volumetric receiver, and can be interpreted in what follows.

As indicated in Eq. (46), it can be easily deduced that *G* ∝ ffiffiffiffiffiffi *PP* <sup>p</sup> for low *PP* and *G* ∝ ffiffiffiffiffiffi *PP* <sup>p</sup><sup>3</sup> for high *PP*, which results in that the amount of heat carried by the air, *G Teq* � h i *<sup>T</sup> <sup>f</sup>* 0 � �<sup>∝</sup> ffiffiffiffiffiffi *PP* <sup>p</sup> , increases drastically on increasing the pumping power *PP* from zero. Nevertheless, its rate of increase diminishes for the higher *PP* range, in which *G Teq* � h i *<sup>T</sup> <sup>f</sup>* 0 � �<sup>∝</sup> ffiffiffiffiffiffi *PP* <sup>p</sup><sup>3</sup> . Moreover, it can also be concluded that the sharp rise in the receiver efficiency occurs around the transition from the Darcy to Forchheimer regime, namely,

$$\frac{\mu\_0}{K} \left( \frac{T\_{eq}}{\langle T \rangle\_0^f} \right)^n G\_{tr} \cong bG\_{tr}^{-2} \tag{65}$$

or

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.

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

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 conver-

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

**5. Applications to silicon carbide ceramic foam volumetric receiver**

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

In order to overcome the problems associated with thermal spots and flow instabilities, we would like to study fluid flow and heat transfer characteristics in silicon carbide ceramic foams based on the analytical expressions of pressure and temperature fields within a solar volumetric receiver. The performance of the receiver may be assessed in terms of the receiver efficiency *η* under equal pumping power*PP*. Thus, the effects of the pore diameter *dm* on the receiver efficiency *η* are presented in **Figure 5**, since *dm* is a crucial geometry parameter affecting hydrodynamic and thermal characteristics of foam shown in Eqs. (23), (24) and (27). The pore diameter *dm* is varied whereas the other parameters are fixed as follows:

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

*<sup>L</sup>* <sup>¼</sup> <sup>0</sup>*:*03½ � *<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 , *<sup>ε</sup>* <sup>¼</sup> <sup>0</sup>*:*9*:*

½ �Þ Pa , *cp* ¼ 1000 J½ � *=*kgK ,

. It can be clearly seen

gence criteria, the residuals of all equations are less than 10�<sup>5</sup>

local thermal non-equilibrium model.

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

*ρ f* D E*<sup>f</sup>*

**64**

**Figure 4.**

*Foams - Emerging Technologies*

*analysis.*

$$G\_{tr} \cong \left(\frac{\mu\_0}{bK} \left(\frac{I\_0 \cos \xi}{c\_p \langle T \rangle\_0^f}\right)^n\right)^{\frac{1}{1+n}} \tag{66}$$

since

$$\frac{T\_{eq}}{\langle \langle T \rangle\_0^f \rangle} \cong \frac{c\_p G\_{tr} \langle T \rangle\_0^f + I\_0 \cos \xi}{c\_p G\_{tr} \langle T \rangle\_0^f} \cong \frac{I\_0 \cos \xi}{c\_p G\_{tr} \langle T \rangle\_0^f} \tag{67}$$

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

$$\begin{split} PP &= \frac{\mathbf{G}\_{lr}}{\left(\left<\rho\right>\_{0}^{\circ}\right)^{2}} \left(2b\mathbf{G}\_{lr}^{\circ}\right)^{2} \left(\frac{T\_{eq}}{\left\_{0}^{\circ}}\right)^{2} L \cong \frac{2bL\mathbf{G}\_{lr}^{\circ}}{\left(\left<\rho\right>\_{0}^{\circ}\right)^{2}} \left(\frac{I\_{0}\cos\xi}{c\_{p}\mathbf{G}\_{lr}\left\_{0}^{\circ}}\right)^{2} \\ &= \frac{2bL}{\left(\left<\rho\right>\_{0}^{\circ}\right)^{2}} \left(\frac{I\_{0}\cos\xi}{c\_{p}\left\_{0}^{\circ}}\right)^{2} \left(\frac{\mu\_{0}}{bK} \left(\frac{I\_{0}\cos\xi}{c\_{p}\left\_{0}^{\circ}}\right)^{n}\right)^{\frac{1}{1+n}} \end{split} \tag{68}$$

on the convective, conductive and radiative heat transfer within a ceramic foam receiver, is presented based on the two-energy equation model of porous media. Both the Rosseland approximation and the P1 model are applied to account for the radiative heat transfer through the solar receiver, while the low Mach approximation is exploited to investigate the compressible flow through the receiver. Based on the P1 model, two positive roots were found from the characteristic equations of the fifth-order differential equation, indicating possible occurrence of hydrodynamic and thermal instabilities. However, it has been found that 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. Due to their advantages, such as high thermal conductivity and fluid mixing, silicon carbide ceramic foams are considered as a possible candidate for the receiver, which can overcome the problems associated with thermal spots and flow instabilities. The results show that the pore diameter must be larger than its critical value to achieve high receiver efficiency. As a result, there exists an optimal pore diameter for achieving the maximum receiver efficiency under the equal pumping power. The optimal pore diameter yielding the maximum receiver efficiency may be found around the critical value given by Eq. (71). A simple relation is derived for determining the length

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

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

of the volumetric solar receivers of silicon carbide ceramic foam.

)

)

)

K)

)

)

)

*Aint* interfacial surface area between the fluid and solid (m<sup>2</sup>

*cp* specific heat at constant pressure (J/kg K)

*hv* volumetric heat transfer coefficient (W/m<sup>3</sup>

*n <sup>j</sup>* normal unit vector from the fluid side to solid side () *PP* pumping power per unit frontal area (W/m<sup>2</sup>

*V* representative elementary volume (m<sup>3</sup>

)

The authors declare no conflict of interest.

*A* surface area (m2

*dm* pore diameter (m) *G* mass flux (kg/m<sup>2</sup> s) *h* specific enthalpy (J/kg)

*K* permeability (m2

*L* receiver length (m)

*Pr* Prandtl number () *q* heat flux (W/m<sup>2</sup>

*R* gas constant (J/kg K) *T* temperature (K) *ui* velocity vector (m/s)

*xi* Cartesian coordinates (m) *x* axial coordinate (m)

*b* inertial coefficient (1/m) *c* specific heat (J/kg K)

*I*<sup>0</sup> intensity of radiation (W/m2

*k* thermal conductivity (W/m K)

**Conflict of interest**

**Nomenclature**

**67**

which, for given PP, gives the minimum value of the pore diameter *dmtr*:

$$\frac{d\_{\rm mtr}}{L} = f(\varepsilon) \left( \frac{2}{\left( \langle \rho \rangle\_0^{\boldsymbol{f}} \right)^2 P P} \left( \frac{I\_0 \cos \xi}{c\_p \langle T \rangle\_0^{\boldsymbol{f}}} \right)^3 \right)^{\frac{1+\kappa}{2+\kappa}} \left( \frac{\mu\_0}{\left( \frac{I\_0 \cos \xi}{c\_p \langle T \rangle\_0^{\boldsymbol{f}}} \right) L} \right)^{\frac{1}{2+\kappa}} \tag{69}$$

$$f(\varepsilon) = \left(\frac{(bd\_m)^n}{K/d\_m}\right)^{\frac{1}{1+\varepsilon}} = \left(\frac{(12(1-\varepsilon))^n}{0.00073(1-\varepsilon)^{-0.224}\left(\frac{1.18}{1-\varepsilon^{-(1-\varepsilon)/0.04}}\sqrt{\frac{1-\varepsilon}{3\pi}}\right)^{-1.11}}\right)^{\frac{1}{1+\varepsilon}}\tag{70}$$

For *PP* = 300, 500 and 1000 W/m<sup>2</sup> studied here, Eq. (69) gives *dmtr*= 0.0022, 0.0016 and 0.0010 m, respectively. It is consistent with what is observed in **Figure 5**, since an increase in *dm* (i.e., decrease in*β*) from *dmtr* makes further penetration of the solar radiation possible. This works to keep the solid temperature at the inlet comparatively low such that heat loss to the ambient by radiation is suppressed. As a result, high receiver efficiency can be achieved. However, the increase in *dm* on the other hand results in decreasing the volumetric heat transfer coefficient, as can be seen from Eq. (27). Too large *dm* deteriorates interstitial heat transfer from the solid to air. Thus, as can be seen from the figure, the optimal size of *dm* exits under the equal pumping power constraint.

In order to achieve local thermal equilibrium for the two phases within the receiver, the length of the receiver is assumed to be sufficiently long in the present study. In view of minimizing the required pumping power, however, it is noticeable that shorter length is better, as clearly seen from Eq. (46). Hence, a minimum length required to approach local thermal equilibrium may be chosen to design a receiver, which would guarantee both maximum receiver efficiency and minimum pumping power. Therefore, we may roughly set the optimal receiver length as

$$L = \frac{\mathfrak{Z}}{\mathfrak{N}}\tag{71}$$

such that

$$\frac{\langle T\rangle^f \Big|\_{\mathbf{x}=L} - T\_{eq}}{\langle T\rangle\_0^f - T\_{eq}} = \frac{\langle T\rangle^s \vert\_{\mathbf{x}=L} - T\_{eq}}{\langle T\rangle\_0^s - T\_{eq}} = e^{-3} \cong \mathfrak{F}\mathfrak{G} \tag{72}$$

Eq. (71) together with Eq. (69) provides useful information for designing a volumetric solar receiver of silicon carbide ceramic foam.
