*3.2. Code Validation*

*3.2. Code Validation*  Due to the lack of experimental data for conjugate heat transfer of nanofluid in an annulus filled with porous medium, we have compared our results with the numerical results of Abhishek Kumar Singh and Tanmay Basak et al. [37] for a square cavity filled with base fluid (*ϕ =* 0). The comparisons are presented in Table 4, when *ε* = 0.4, and the deviations are 0%, 0%, and 0.84% with *Ra* = 103, 104, and 105, respectively. When *ε* = 0.9, the deviations are Due to the lack of experimental data for conjugate heat transfer of nanofluid in an annulus filled with porous medium, we have compared our results with the numerical results of Abhishek Kumar Singh and Tanmay Basak et al. [37] for a square cavity filled with base fluid (*φ =* 0). The comparisons are presented in Table 4, when *ε* = 0.4, and the deviations are 0%, 0%, and 0.84% with *Ra* = 10<sup>3</sup> , 10<sup>4</sup> , and 10<sup>5</sup> , respectively. When *ε* = 0.9, the deviations are 0.5%, 1.2%, and 1.3% with *Ra* = 10<sup>3</sup> , 10<sup>4</sup> , and 10<sup>5</sup> , respectively. It is clear that the current results are in good agreement with the earlier work, and the maximum deviation is 1.3%. In addition, it can be seen that the deviation is increased with the increase of Rayleigh number and porosity. The validation work has enhanced the confidence in the numerical solution of the current study.

**Table 4.** Comparison of the average Nusselt number with those of Singh et al. [37] for different Rayleigh number when *ϕ* = 0, *ε* = 0.4, 0.9, *Da* = 10−<sup>2</sup> , and *Pr* = 1.

