2D Heat Conduction Equation
1.1 Introduction:
Conduction is a fundamental transport phenomenon of Heat. In this study, the 2D steady state and Unsteady state Heat conduction equation was analyzed. For steady state, the primary iterative solvers have been implemented to study the rate of convergence. The iterative solvers are Jacobi, Gauss-Seidel, and SOR (Successive Over-Relaxation). For Transient 2D Heat equation, the stability for both implicit and explicit schemes was studied. Moreover, Finite difference Method is used to discretize the Partial differential equation, and the discretized equation is solved using Matlab.
1.2 Boundary:
The defined boundary would be a square, with the wall temperature around the plate would be 400K on the left, 900K in the bottom, 600K at the top, 800K at the right.
1.3 Steady State Simulation:
In the steady state heat diffusion equation, the primary iterative solvers are used to solve the equation. The general 2D steady-state heat conduction equation is written as `(partial^2 phi)/(partialx^2) + (partial^2 phi)/(partialy^2) = 0`. Assuming, there is no source term. The equation was discretized using Finite difference Method. Since, both Forward differencing and backward differencing are of Order`(ODeltax)`.To reduce the error, the central difference scheme was used, as the order of accuracy of Central difference would be `(ODeltax^2)`.The discretized Heat Conduction equation is, `T_p = 1/k((T_L+T_R)/(Deltax^2) +(T_T+T_B)/(Deltay^2))`. Where, `k = 2((Deltax^2+Deltay^2))/(Deltax^2Deltay^2)`
1.4 Results- Steady state:
The Successive Over-Relaxation method converged faster than the other iterative methods. The Jacobi Method took around 217 iterations to converge; however, the Gauss-Seidel method took about 116 iterations to converge, which is about half of the Jacobi. In a case for Successive Over-Relaxation which took only 30 iterations to converge. Which clearly states that the Successive Over-Relaxation is faster than other iterative methods. On the other hand, the relaxation factor plays a vital role in the rate of convergence on the SOR method. The plots explain the effect of the relaxation factor on the convergence of the solution. When the system is over relaxed, the number of iteration for convergence is less. On the contrary to this, the Under relaxed system has a larger number of iterations. For figure 1.1, the relaxation factor is 1.6, where the number of iterations for convergence is 30. However, in figure 1.3, with the relaxation factor 0.9, the number of iterations was increased to 141.
1.5 Unsteady State Simulation:
When it comes to Unsteady state simulation, Stability is one of the major concerns on the equation. In Unsteady state simulation, the equation has been discretized using the central differencing method, which is Forward in Time and Central in Space (FTCS). The solution was solved for both explicit and the implicit schemes, and moreover the stability of the two schemes was noted. The general unsteady heat conduction equation is `(partialphi)/(partialt) + \alpha((partial^2phi)/(partialx^2)+(partial^2phi)/(partialy^2))`. where, `alpha`is thermal diffusivity. The discretized implicit equation is, `T_P^(n+1) = (T_P^n-k_1T_L^(n+1)-k_1T_R^(n+1)-k_2T_T^(n+1)-k_2T_B^(n+1))/(1-2k_1+2k_2)`. The explicit discretized equation is, `T_p^(n+1) = T_p^n + k_1(T_R^n-2T_P^n+T_L^n)+k_2(T_T^n-2T_P^n+T_B^n)`,Where, `k_1 = alpha(Deltat)/(Deltax^2)` and `k_2 = alpha(Deltat)/(Deltay^2)`
1.6 Results-Unsteady state:
The plots describe the stability of explicit methods. The two plots are calculated for different CFL(Courant–Friedrichs–Lewy) numbers. For Figure 1.4, the CFL number is greater than 0.5. For a stable solution, the CFL number should be less than 0.5, which is `k1+k2 < 0.5`.Hence, at the 60 time step, the solution becomes unstable. In the Figure 1.5, the CFL number is 0.45 which shows a stable solution. This concludes that the Explicit Method is Conditionally stable. However, the implicit Scheme is Unconditionally stable, for various CFL number, the solution will always stable.