Estimating the temperature distribution in aluminium bar

Question

The non-steady-state temperature distribution in a thin metal bar is described by the following partial differential equation:

ρ C_p ∂T/∂T =K ∂²T/∂X²

where ρ is the material density, C_p its heat capacity, and k its thermal conductivity.

A 6 cm long aluminum bar initiatly has a temperature distribution as follows:

T(x,0)=O°C for O cm ≤x ≤ 6cm

The sides of the bar are insulated and the two ends are subject to the following boundary conditions:

T(x,t)=20°C           at x = 0 cm for t> 0

q_x= 540 kW m^-2      at x = 6cmfor t>0

Use a numerical technique to estimate the temperature distribution in the bar at the following times: 0.5 sec, 1 sec, and 1.5 sec. You must use a simple explicit technique with a spatial discretization Δx = 1 cm and a time increment Δt = 0.25 sec. Check that the method will be convergent and stable for these spatial and temporal increments.

Using a sketch indicate how you think the temperature distribution will change with time. What is the final steady state temperature distribution in the aluminium bar?

For aluminium: k = 0.27 kW m^-1 K^-l

ρ= 3000 kg m^-3

C_p = 0.9 kJ kg^-1k^-1