Question

(1.6) A rectangular block has thickness B in the x-direction. The side at x = 0 is held at temperature T? while the side at x = B is held at T?. The other four sides are well insulated. Heat is generated in the block at a uniform rate per unit volume of ?. (a) Use the conduction equation to derive an expression for the steady-state temperature profile, T(x). Assume constant thermal conductivity. (b) Use the result of part (a) to calculate the maximum temperature in the block for the following values of the parameters: T? = 100°C k = 0.2 W/m?K B = 1.0 m T? = 0°C ? = 100 W/m³ Ans. (a) $T(x) = T_1 + \left(\frac{T_2 - T_1}{B} + \frac{?B}{2k}\right)x - \frac{?x^2}{2k}$ (b) $T_{max} = 122.5°C$ at x = 0.3 m. (1.7) Repeat Problem 1.6 for the situation in which the side of the block at x = 0 is well insulated. Ans. (a) $T(x) = T_2 + \frac{?}{2k}(B^2 - x^2)$. (b) $T_{max} = 250°C$. (1.8) Repeat Problem 1.6 for the situation in which the side of the block at x = 0 is exposed to an external heat flux, $\dot{q}_o$, of 20 W/m². Note that the boundary condition at x = 0 for this case becomes $\frac{dT}{dx}|_{x=0} = -\frac{\dot{q}_o}{k}$ Ans. (a) $T(x) = T_2 + \frac{\dot{q}_o}{k}(B - x) + \frac{?}{2k}(B^2 - x^2)$. (b) $T_{max} = 350°C$.

          (1.6) A rectangular block has thickness B in the x-direction. The side at x = 0 is held at temperature T? while the side at x = B is held at T?. The other four sides are well insulated. Heat is generated in the block at a uniform rate per unit volume of ?.
(a) Use the conduction equation to derive an expression for the steady-state temperature profile, T(x). Assume constant thermal conductivity.
(b) Use the result of part (a) to calculate the maximum temperature in the block for the following values of the parameters:
T? = 100°C  k = 0.2 W/m?K  B = 1.0 m
T? = 0°C  ? = 100 W/m³
Ans. (a) $T(x) = T_1 + \left(\frac{T_2 - T_1}{B} + \frac{?B}{2k}\right)x - \frac{?x^2}{2k}$ (b) $T_{max} = 122.5°C$ at x = 0.3 m.
(1.7) Repeat Problem 1.6 for the situation in which the side of the block at x = 0 is well insulated.
Ans. (a) $T(x) = T_2 + \frac{?}{2k}(B^2 - x^2)$. (b) $T_{max} = 250°C$.
(1.8) Repeat Problem 1.6 for the situation in which the side of the block at x = 0 is exposed to an external heat flux, $\dot{q}_o$, of 20 W/m².
Note that the boundary condition at x = 0 for this case becomes
$\frac{dT}{dx}|_{x=0} = -\frac{\dot{q}_o}{k}$
Ans. (a) $T(x) = T_2 + \frac{\dot{q}_o}{k}(B - x) + \frac{?}{2k}(B^2 - x^2)$. (b) $T_{max} = 350°C$.

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