Question

Consider an aluminum pan used to cook stew on top of an electric range. The bottom section of the pan is \(L=0.25 \mathrm{~cm}\) thick and has a diameter of \(D=18 \mathrm{~cm}\). The electric heating unit on the range top consumes $900 \mathrm{~W}$ of power during cooking, and 90 percent of the heat generated in the heating element is transferred to the pan. During steady operation, the temperature of the inner surface of the pan is measured to be $108^{\circ} \mathrm{C}$. Assuming temperature-dependent thermal conductivity and onedimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.

Short answer

Answer: The mathematical formulation consists of a differential equation and boundary conditions as follows: Differential equation: $$\frac{d}{dx}\left( k(T) \frac{dT}{dx}\right) = 0$$ Boundary conditions: 1. At \(x = 0\), \(T = 108^{\circ} \mathrm{C}\). 2. At \(x = L\), \(q = \frac{q_{total}}{A}\).

Step-by-step solution

Heat Conduction Equation (Fourier's Law)

We know that the heat conduction equation (Fourier's Law) is: $$q = -k \frac{dT}{dx}$$ where \(q\) is the heat flux, \(k\) is the thermal conductivity, and \(\frac{dT}{dx}\) is the temperature gradient. Since the conductivity is temperature-dependent, we can write \(k\) as a function of \(T\), i.e., \(k(T)\).

The Heat Flux

Now, let's find the heat flux \(q\). We know that 90% of the heat generated by the electric heating unit is transferred to the pan. So, the heat transferred to the pan is: $$q_{total} = 0.9 \times 900 \thinspace \mathrm{W}$$ Considering a one-dimensional heat transfer and that the bottom surface of the pan has an area, \(A = \pi \left(\frac{D}{2}\right)^2\), the heat flux will be: $$q = \frac{q_{total}}{A}$$

Differential Equation

Now, we will write the differential equation representing the heat conduction problem. Using Fourier's Law, we have: $$q = -k(T) \frac{dT}{dx}$$ Since at steady-state, the temperature will be constant, we can rewrite this equation as: $$\frac{d}{dx}\left( k(T) \frac{dT}{dx}\right) = 0$$

Boundary Conditions

To express the boundary conditions, we will consider the temperatures at the inner and outer surfaces of the pan. We know that the inner surface of the pan has a temperature of \(108^{\circ} \mathrm{C}\) during steady operation, and we're assuming one-dimensional heat transfer. So, the boundary conditions for this problem are: 1. At \(x = 0\), \(T = 108^{\circ} \mathrm{C}\). 2. At \(x = L\), \(q = \frac{q_{total}}{A}\). In conclusion, here is the mathematical formulation for this heat conduction problem during steady operation: Differential equation: $$\frac{d}{dx}\left( k(T) \frac{dT}{dx}\right) = 0$$ Boundary conditions: 1. At \(x = 0\), \(T = 108^{\circ} \mathrm{C}\). 2. At \(x = L\), \(q = \frac{q_{total}}{A}\).