A plane wall of thickness \(L\) is subjected to convection at both surfaces with ambient temperature \(T_{\infty 1}\) and heat transfer coefficient \(h_{1}\) at inner surface, and corresponding \(T_{\infty 2}\) and \(h_{2}\) values at the outer surface. Taking the positive direction of \(x\) to be from the inner surface to the outer surface, the correct expression for the convection boundary condition is (a) \(\left.k \frac{d T(0)}{d x}=h_{1}\left[T(0)-T_{\mathrm{o} 1}\right)\right]\) (b) \(\left.k \frac{d T(L)}{d x}=h_{2}\left[T(L)-T_{\infty 2}\right)\right]\) (c) \(\left.-k \frac{d T(0)}{d x}=h_{1}\left[T_{\infty 1}-T_{\infty 2}\right)\right]\) (d) \(\left.-k \frac{d T(L)}{d x}=h_{2}\left[T_{\infty 1}-T_{\infty 22}\right)\right]\) (e) None of them

The convection boundary conditions are expressions relating the heat transfer through a surface to the temperature difference between the surface and the ambient air. At the inner surface (at \(x=0\)), we have a heat transfer coefficient \(h_1\) and ambient temperature \(T_{\infty 1}\). Similarly, at the outer surface (at \(x=L\)), we have a heat transfer coefficient \(h_2\) and ambient temperature \(T_{\infty 2}\).

We will use Fourier's law to describe heat transfer through the wall. It states that the heat flux (rate of heat transfer per unit area) is proportional to the temperature gradient along the wall: \(q_x=-k\frac{dT}{dx}\) where \(q_x\) is the heat flux in the x-direction, \(k\) is the thermal conductivity, and \(\frac{dT}{dx}\) is the temperature gradient along the wall.

The heat transfer coefficients \(h_1\) and \(h_2\) represent the proportionality of the heat flux on the inner and outer surfaces of the wall, respectively. We will apply these coefficients at the inner surface (x=0) and the outer surface (x=L): 1. At the inner surface (x=0), the heat transfer coefficient \(h_1\) represents the proportionality between the heat flux and the temperature difference as follows: \(q_x = h_1[T(0) - T_{\infty 1}]\) 2. At the outer surface (x=L), the heat transfer coefficient \(h_2\) represents the proportionality between the heat flux and the temperature difference as follows: \(q_x = h_2[T(L) - T_{\infty 2}]\)

Now, let's examine the given expressions to find the correct convection boundary condition expression: (a) \(\left.k \frac{dT(0)}{dx} = h_1\left[T(0)-T_{\infty 1}\right]\right]\) This expression represents the correct convection boundary condition at the inner surface (x=0) since it describes how the heat flux is related to the heat transfer coefficient \(h_1\) and the temperature difference between the inner surface and its ambient temperature. (b) \(\left.k \frac{dT(L)}{dx} = h_2\left[T(L)-T_{\infty 2}\right]\right]\) This expression represents the correct convection boundary condition at the outer surface (x=L) since it describes how the heat flux is related to the heat transfer coefficient \(h_2\) and the temperature difference between the outer surface and its ambient temperature. (c) \(\left.-k \frac{dT(0)}{dx} = h_1\left[T_{\infty 1}-T_{\infty 2}\right]\right]\) This expression is incorrect because it does not represent the temperature difference between the inner surface and its ambient temperature. (d) \(\left.-k \frac{dT(L)}{dx} = h_2\left[T_{\infty 1}-T_{\infty 2}\right]\right]\) This expression is incorrect because it does not represent the temperature difference between the outer surface and its ambient temperature. (e) None of them Since we have already found two expressions (a and b) that represent the correct convection boundary conditions at the inner and outer surfaces, this answer is incorrect. The correct answer is (a) and (b).