Unsteady Heat Conduction in a Solid

To compare the Flow Simulation predictions with the analytical solution (Ref. 12), we will solve a one-dimensional problem.

A warm solid rod having the specified initial temperature and the heat-insulated side surface suddenly becomes and stays cold (at a constant temperature of Tw = 300 K) at both ends (see Figure 1). The rod inner temperature evolution is studied. The constant initial temperature distribution along the rod is considered: Tini(x) = 350 K.

The problem is described by the following differential equation:

where ρ, Cp, and k are the solid material density, specific heat, and thermal conductivity, respectively, and t is the time, with the following boundary condition: T = T0 at x = 0 and at x = L.

In the general case, i.e., at an arbitrary initial condition, the problem has the following solution:

where coefficients Cn are determined from the initial conditions (see Ref. 12).

With the uniform initial temperature profile, according to the initial and boundary conditions, the problem has the following solution:

To perform the time-dependent analysis with Flow Simulation, a geometry model representing a solid parallelepiped with dimensions 1 x 0.2 x 0.1 m has been created (see Figure 2).

The evolution of maximum rod temperature, predicted with Flow Simulation and compared with theory, is presented in Figure 3. The Flow Simulation prediction has been performed at result resolution level 5. One can see that it coincide with the theoretical curve.

The temperature profiles along the rod at different time moments, predicted by Flow Simulation, are compared to theory and presented in Figure 4.

One can see that the Flow Simulation predictions are very close to the theoretical profiles. The maximum prediction error not exceeding 2 K occurs at the ends of the rod and is likely caused by calculation error in the theoretical profile due to the truncation of Fourier series.