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 ∂^{2}T/∂X^{2}
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^{-1}k^{-1}