# Simplify coupled system of equations in fluid dynamics

I am currently working on the structural mechanical side some model. Since the contains a fluid flow inside, I wanted to model the flow inside my structure.

I assume that my flow is similar to a "simple" pipe flow. Basically this is what I have:

• A list of crosssections with a specified area and the distance between each crosssection
• The mass flow rate through my pipe (which is constant for every crosssection)
• I have given temperatures of the pipe which varries from crosssection to crosssection

What I eventually want to get is:

• the fluid temperatures and the heat transfer coefficients at each crosssection

Since I do not want to solve a coupled system of differential equations (full navier stokes, 1D). I went with the following aproach:

1. assume constant density to begin with
2. compute velocities (continuity eq.)
3. compute pressure loss (bernoulli)
4. compute heat transfer coefficients (mainly based on velocity)
5. incrementally compute the fluid temperature using the heat transfer coefficients using the external temperature

This aproach fails in the central aspect that the changing temperature which results from step 5. is not considered in the first few steps.

My question is:

1. Is this aproach sufficient for a very vague temperature field inside my fluid and how close to the real solution could I expect this solution to be?
2. Is there a simplification I could do to solve the full navier stokes equations on each section and get an accurate result?

I am very happy for any help :)

• The third option would be to somehow use the temperature field in step 5 to recompute the velocities/densities/pressures in step 1-3. I am not sure how the bernoulli equation should be adjusted for this. Aug 17, 2021 at 6:27

I dont know exactly how much things vary. For example, for all I know the pipe areas barely change and there is very high heat transfer, etc. But inferring from what you say is important and my experience with similar problems, here’s what I’d say:

To start off, differing pipe areas are probably biggest source of error (velocity errors are most affected by lack of accounting for pipe cross section)

1. assume constant density to begin with

2. compute velocities (continuity eq.)

It’s easy to get pressure, so:

1. compute pressure loss (bernoulli)

Now exclusion of pressure probably biggest source of error (density and velocity errors are most affected by lack of accounting for pressure):

1. Get new densities with pressure included

2. Compute velocities

3. Compute heat transfer coefficients (mainly based on velocity)

4. incrementally compute the fluid temperature using the heat transfer coefficients using the external temperature

Now exclusion of temperature probably biggest source of error (density and velocity errors are most affected by lack of accounting for temperature):

1. Get new densities with temperature included

2. Compute velocities

3. Compute heat transfer coefficients (mainly based on velocity)

4. incrementally compute the fluid temperature using the heat transfer coefficients using the external temperature

I know doing the heat transfer twice (6,7 and 10,11) is annoying, but unless the temperature differences are small, you must account for density differences due to temperature.

That process would reduce estimation error probably an order of magnitude from what you had. Just a guess of course. But it should be big.

This problem is easy with software packages. Really easy. There are some hvac packages. It has been a couple decades for me so I can’t recommend, but look around. You have a great clear boundary condition of constant temperature.

Good luck.

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– rob
Aug 17, 2021 at 13:08