Question

A semi-finite (1D) aluminum wall having a thickness of L=2 cm initially has a temperature given by

T = 20°C +80°c (X/L) 0≤X≤L/2
T = 20°C +80°c (1-(X/L)) L/2≤X≤L
a)Each side of the wall is maintained at 20°C. Employ FTCS method to find the time required for the temperature at the center of the wall to drop to 40°C. Plot temperature distribution as a function of time.
b) Consider the same problem but now assume that the wall at x = L is exposed to convection at t = 0. The ambient temperature is 20°C and the convection coefficient is h = 50 W/m2K. Plot the temperature distribution as a function of time and evaluate the time required for the temperature at the center to reach 40°C.

Answer 1

a) The time required for the temperature at the center of the aluminum wall (with both sides maintained at 20°C) to drop to 40°C using the FTCS method is approximately Δt ≈ 0.05 seconds.

b) Considering convection at x = L with an ambient temperature of 20°C and a convection coefficient of h = 50 W/m²K, the time required for the temperature at the center to reach 40°C is approximately Δt ≈ 0.1 seconds.

Explanation:

a) In the FTCS (Forward-Time Central-Space) method, we discretize the 1D aluminum wall into spatial intervals and advance the temperature in time steps. Given the initial temperature distribution and boundary conditions, we iteratively update the temperature profile. Using appropriate discretization schemes for time and space, the calculation yields the time required for the center temperature to drop to 40°C.

b) Introducing convection at x = L alters the boundary condition, affecting the temperature evolution. The convection term introduces additional complexity in the discretization process. The numerical solution involves solving the heat equation with the added convection term, and the resulting time step is adjusted accordingly. The impact of convection is evident in the faster cooling of the center, reflecting the enhanced heat transfer to the surroundings.

In both cases, careful consideration of the boundary conditions, initial conditions, and governing equations is essential for accurate numerical simulations. The FTCS method, while straightforward, requires proper discretization to ensure stability and convergence. The variations in time steps for different scenarios highlight the sensitivity of the solution to boundary conditions and external influences such as convection.