Which of the following non-dimensional terms is an estimate of Nusselt number?

The correct option is: Which of the following non-dimensional terms is an estimate of Nusselt number? (which represents the non-dimensional temperature gradient at the surface).

To understand why the Nusselt number is represented by the non-dimensional temperature gradient at the wall, we examine the heat transfer at the boundary of a solid wall and a moving fluid.

By Newton's law of cooling, the convective heat flux ^q from the wall to the fluid is given by:
$$q=h(T_w-T_{\infty })$$ where ^h is the convective heat transfer coefficient, ^T_w is the wall temperature, and ^T_∞ is the free-stream fluid temperature.

At the solid-fluid interface (at ^{y = 0}), the fluid is stationary due to the no-slip condition. Therefore, heat is transferred from the wall into the fluid solely by conduction:
$$q=-k_f{\left(\frac{\partial T}{\partial y}\right)}_{y=0}$$ where ^k_f is the thermal conductivity of the fluid and ^y is the coordinate normal to the wall.

Equating the conduction heat flux to the convection heat flux at the boundary:
$$h(T_w-T_{\infty })=-k_f{\left(\frac{\partial T}{\partial y}\right)}_{y=0}$$

To non-dimensionalize this relation, we introduce the dimensionless temperature ^T* and the dimensionless distance ^y* defined as:
$$T^{*}=\frac{T-T_w}{T_{\infty }-T_w}$$ and
$$y^{*}=\frac{y}{L}$$ where ^L is the characteristic length.

Rearranging the variables gives:
$$T-T_w=T^{*}(T_{\infty }-T_w)$$ and
$$dy=Ld{y}^{*}$$

Substituting these dimensionless definitions back into the boundary equation:
$$h(T_w-T_{\infty })=-k_f\frac{T_{\infty }-T_w}{L}{\left(\frac{\partial T^{*}}{\partial y^{*}}\right)}_{y^{*}=0}$$

Simplifying the negative signs and temperature differences yields:
$$\frac{hL}{k_f}={\left(\frac{\partial T^{*}}{\partial y^{*}}\right)}_{y^{*}=0}$$

Since the left-hand side is the definition of the Nusselt number (Nu):
$$Nu={\left(\frac{\partial T^{*}}{\partial y^{*}}\right)}_{y^{*}=0}$$

Thus, the Nusselt number represents the non-dimensional temperature gradient at the solid boundary.