[Dharma Vedula]

I wrote SIMPLER code in Python to solve steady state 3D momentum and energy balance equations in a rectangular channel of H=0.2, W=0.1, L=45mm. I am getting pressure drop and velocity profile correctly. I am not getting temperature profile correctly. The inlet temperature is 21C. Wall temperature is 33C. Upto half length (22.5mm) water (fluid) temperature is increasing, which is correct. But subsequently fluid temperature is decreasing. But it must go on increasing with distance. I am sending below the temperature profile y versus temperature at different x. The y positions are (0.0, 0.03, 0.067, 0.1, 0.13, 0.167, 0.2). Vertical row shows temperatures at different y. Horizontal row shows temperatures at different x (downstream).
0.0000: 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000
0.0333: 26.333 28.059 29.011 29.491 29.660 29.621 29.429 29.093 28.604 27.954 27.152 26.231 25.238 24.234 23.291 22.419 22.419
0.0667: 22.333 29.203 28.285 26.769 25.523 24.510 23.496 22.261 20.658 18.638 16.238 13.577 10.820 8.153 5.722 5.430 5.430
0.1000: 21.000 22.655 24.114 25.072 25.652 25.921 25.909 25.611 25.010 24.098 22.894 21.449 19.842 18.174 16.550 15.126 15.126
0.1333: 22.333 25.408 26.225 26.518 26.695 26.774 26.695 26.394 25.830 24.990 23.897 22.604 21.193 19.758 18.390 17.711 17.711
0.1667: 26.333 27.969 28.693 29.152 29.449 29.602 29.612 29.470 29.169 28.711 28.113 27.411 26.649 25.882 25.161 24.551 24.551
0.2000: 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000 33.000

[LuckyTran]

The SIMPLE family solves only the p-v coupling problem. The energy equation is solved separately. So the fact that you calculate the wrong temperature field, isn't really anything to do with SIMPLER (unless the updated velocity field completely screws with your energy equation). That is, SIMPLER does not solve momentum AND energy, it solves only momentum & continuity.

[Dharma Vedula]

What you said is perfectly true. I got the solution of the momentum (and continuity) equation correctly. I am getting the pressure drop and velocity profiles correctly. But I have a heat transfer application, for which I have to solve the 3-D steady state energy balance equation. This equation contains the u,v and w velocity components. I am using the values of u, v, w obtained from SIMPLE in the solution of energy equation.
I find from the temperature results that heating downstream is not occuring. Heating from wall to bulk is also not happening as effectively as it must be. I don't find any mistake in discretisation. There can be error in implementation of boundary conditions, particularly the exit boundary condition. I am using dT/dx=0 as the boundary condition at the exit.

[Dharma Vedula]

Is there any forum that discusses on the numerical solution of 3-D energy balance equation, in which I can post my problem?
For my problem SIMPLER algorithm is mandatory. I have to add energy equation to it and solve the combined sytem in a conjugate manner.

[LuckyTran]

I am skeptical of discretization. But if you are so sure...

Then just specify a fixed temperature as the boundary condition for the energy equation.

[Dharma Vedula]

I am getting the results fine now. I obtained results for laminar flow and heat transfer in a rectangular microchannel. My pressure drop, friction and heat transfer coefficients are matching/tallying very well with the experimental data of two different authors, their papers published in International journal of heat and mass transfer, which is a very standard journal.
Thank you very much for your help. I used the exit boundary condition for u-velocity as given in Versteeg and Malalasekra, which implements conservation of mass flow at inlet and exit. It gave good result.
I used my own condition for the exit boundary condition for temperature since I couldn't find the exit boundary condition for temperature in any text book or journal paper.