Can not achieve a pseudo-steady-state condition
 

I am trying to simulate 2D conjugate heat transfer over a finned surface. The simulation works fine and shows some results as shown in Figure 1, which is temperature profile.

But, the problem is that the wall temperature keeps rising even after simulating it for 60 seconds of time (transient) as shown Figure 2.

I expect the wall temperature to reach some pseuso-steady state condition (imaginary plot is shown in Figure 3). But it does not reach that stage.

I have made mesh really small (having minimum cell size = 0.00005 m) and time step of 0.00005 s, which are the smaller values than what I see in literature. So, I think mesh or time step is not the problem. Also I use second order schemes. I tried all SIMPLE, PISO and coupled. and all does not show the desired result (Temp still keeps rising)

I have also seen similar problem with other software as well. I am sure I have been doing something fundamental mistake here. What is it?

You need to run for a very long time (e.g. 15 minutes) to achieve steady state. A smaller mesh size and smaller time-step is unrelated to that requirement and makes it even most costly to achieve

15 min is too much.

Also, with some online search I found two possible problems. One is when I do 2D, I need to adjust the heat flux (by reducing the reference depth) and other can be solid time step value. Solid time step should be much larger for this case (I don't know much large though).