Steady vs Unsteady State Analysis of heat conduction through 2D Square Plate

Number of Iterations to Convergence:

The fastest steady state simulation i.e. SOR converged in 106 iterations, whereas the fastest unsteady state simulation took 2593 iterations to converge.

The probable reason behind this can be attributed to the fact that, in steady state, the only loop iterating is the space loop, through the nodal grid points. Hence, the iterative solver is used to evaluate the T matrix at the grid points along the 2D domain.

For the unsteady state solver, in addition to the space loop, there is a time loop, that loops outside the space loop. Hence, the iterative technique is first used to evaluate T at various nodal points, in the first time step. Then the T matrix is re evaluated for all the nodes at the second time step and the process is repeated for 500 time steps. Hence, the number of iterations to achieve convergence is much more in case of transient/unsteady state solutions.

Effect of Time on Solution

The unsteady state solution, as evident from its name is time dependent, unlike steady state solution. In this project, we have used a time step of 0.01 seconds and the number of time steps considered is 500, for the transient conduction case. It is observed that after (500*0.01)=5 seconds, the transient state solution and the steady case solution looks absolutely identical. This means that, within 5 seconds, heat conduction through the 2D square plate achieves steady state.

Transient Heat Conduction through 2D square plate after 5 seconds:

Steady State heat conduction through 2D square plate:

Now, for the transient solution, keeping the time step instact, if the number of time steps is decreased to 50 i.e. the simulation instead of being run till 5 seconds is restricted to (50*0.01)=0.5 seconds, then a completely different solution matrix is observed, as given below. This is because, the thermal condition of the plate is yet to achieve steady state after 0.5 seconds.

ransient Heat Conduction through 2D square plate after 0.5 seconds:

Hence, it can be concluded that unlike in steady state, time of heat flow plays a significant role in transient state problems. It may also be added that, transient state problems can give steady state solutions, if the heat flow is given adequate time to achieve steady state.