**Conflict of interest**

*PP* <sup>¼</sup> *Gtr*

*Foams - Emerging Technologies*

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

<sup>¼</sup> <sup>2</sup>*bL* h i*<sup>ρ</sup> <sup>f</sup>* 0 � �<sup>2</sup>

*dmtr*

*K=dm*<sup>2</sup>

! <sup>1</sup>

*<sup>f</sup>*ð Þ¼ *<sup>ε</sup>* ð Þ *bdm <sup>n</sup>*

such that

**6. Conclusions**

**66**

*<sup>L</sup>* <sup>¼</sup> *<sup>f</sup>*ð Þ*<sup>ε</sup>* <sup>2</sup>

2þ*n*

0

B@

0

B@

of *dm* exits under the equal pumping power constraint.

h i *<sup>T</sup> <sup>f</sup>* � � � *x*¼*L*

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

� *Teq*

<sup>0</sup> � *Teq*

volumetric solar receiver of silicon carbide ceramic foam.

<sup>2</sup> � � *Teq*

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

!<sup>2</sup>

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

*PP*

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

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

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

For *PP* = 300, 500 and 1000 W/m<sup>2</sup> studied here, Eq. (69) gives *dmtr*= 0.0022,

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

*<sup>L</sup>* <sup>¼</sup> <sup>3</sup>

*<sup>x</sup>*¼*<sup>L</sup>* � *Teq*

¼ *e*

<sup>0</sup> � *Teq*

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

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

Eq. (71) together with Eq. (69) provides useful information for designing a

For the first time, the complete set of analytical solutions, which fully considers the combined effects of turbulence, tortuosity, thermal dispersion, compressibility

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

> *μ*0 *bK*

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

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

!<sup>3</sup>

!<sup>2</sup>

*L* ffi

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

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

2*bLGtr* 3

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

!<sup>2</sup>

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

*L*

ffiffiffiffiffiffi 1�*ε* 3*π*

*γλ* (71)

�<sup>3</sup> ffi 5% (72)

� � q �1*:*<sup>11</sup>

1

1 2þ*n*

1

1 2þ*n*

CA

CCA

(68)

(69)

(70)

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

1

1þ*n* 2þ*n* 0

BB@

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

CA

1þ*n*

The authors declare no conflict of interest.
