Question

Consider a steel pan used to boil water on top of an electric range. The bottom section of the pan is L = 0.5 cm thick and has a diameter of D = 20 cm. The electric heating unit on the range top consumes 1250 W of power during cooking, and 85 percent of the heat generated in the heating element is transferred uniformly to the pan. Heat transfer from the top surface of the bottom section to the water is by convection with a heat transfer coefficient of h.

Required:
Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.

Answer 1

Answer:

-Differential equation: d²T/dx² = 0

-The boundary conditions are;

1) Heat flux at bottom;

-KAdT(0)/dx = ηq_e

2) Heat flux at top surface;

-KdT(L)/dx = h(T(L) - T(water))

Explanation:

To solve this question, let's work with the following assumptions that we are given;

- Heat transfer is steady and one dimensional

- Thermal conductivity is constant.

- No heat generation exists in the medium

- The top surface which is at x = L will be subjected to convection while the bottom surface which is at x = 0 will be subjected to uniform heat flux.

Will all those assumptions given, the differential equation can be expressed as; d²T/dx² = 0

Now the boundary conditions are;

1) Heat flux at bottom;

q(at x = 0) is;

-KAdT(0)/dx = ηq_e

2) Heat flux at top surface;

q(at x = L):

-KdT(L)/dx = h(T(L) - T(water))