Question

Need help with part B (the sketching!) thanks

Problem 3: A thin flat plate of length L-120 mm, thickness t=4 mm, width W=30 mm, and thermal conductivity k=20 W/m-K is thermally joined to heat sinks at different temperatures, where T(x = 0) = To-100 C and T(x = L) = T,-35°C. The top surface of the plate is subjected to uniform heat flux q"-20 kW/m2, and the bottom surface of the plate is subjected to uniform convection with a convection coefficient of h=50 W/m2-K and freestream temperature Too 25°C. (a) Derive the differential equation that describes the steady-state temperature T(x) in the plate. You MUST include appropriate boundary conditions with your differential equation. You DO NOT need to solve the differential equation to find T (x). (b) Sketch what T(x) might look like for two cases: 1) specified q", and 2) q"=0. Clearly show the differences between the two cases on your sketch.

Answer 1

Consider a small strip of thin flat plane of length dx at a distance of x from the left end Write the expression for the conductive heat transfer through the small strip dT Q kxtxwx dc Write the expression for the uniform heat flux on this small area Q q'wdx Write the expression for the convection heat transfer through the bottom of the plate Qhw(T-T)dx Write the expression for the heat balance from the plate dT + q*wdx= hw(T-T)dx ktw dT -+ q*dx=h(T-T)dc kt dx Since the heat sinks are present at the right and the left end so The Boundary conditions are ) At x 0, T T 2) At x L, T T

dT Thus, the required differential equation is kt- +q'dx=h(T-T.)d| dx 2) i) The curve will follow a parabolic curve with maximum temperature at the middle of the plate ii) The curve will follow a straight line with negative slope. The maximum temperature will be at the left end and the minimum temperature will at the right end