[Deeds]

Hello all!

I have been solving a one-dimensional heat conduction equation of quenching phenomena (along the axial direction) on MATLAB. I have discretized the governing equation using the FDM scheme(implicit) and solved using TDMA algorithm. The equation is subject to some initial and boundary conditions and the temperature of the rod is cooled from 573K to 303K (B.C at n=1: 303K and B.C. at n=6000: 573K). The rod domain consists of 6000 nodes (uniform mesh).

In the considered problem, the quench front or the wet front is propagating at a certain quench velocity and each curve(varying colours) shows the temperature profile of the rod as the quench front propagates in the increasing direction of axial length (downwards along the length of rod). Thus, temperature plotted is a function of time and axial length (and heat transfer coefficient, as the governing equation consists of the 'h' and its variation is shown in subplot no. 2). The position of the quench front changes with each time step. I have obtained the following plots on MATLAB (images attached):

1. The initial temperature values for nodes no. 1 to 200 is 303K and rest of them are at 573 K for all the time steps.

2. The initial temperature values for nodes no. 1 to 1200 is 303K and rest of them are at 573 K for all the time steps.

I want to use the plot no. 1 but a great variation is observed during the initial iterations of the code (rest of the plots are okay and validated against experimental data).

I am not able to understand the behaviour of the solution obtained and its dependency on initial values. Though it is well known and obvious that the solution depends on the initial values, how can I decide what nodes should be kept at the temperature of 303K and the 573K? If I keep all the nodes at initial temperature 573K, the results are not satisfying.

I am a beginner and I could not find an appropriate source of finding a solution to such a problem. Please provide me with some possible explanation or let me know about the source/reference where I can read and learn more about the issue. Any kind of guidance is really appreciated.

Thanks!