[marc_vol]

Hi there,

I want to validate the working of a machine with a simple calculation. I do not have much experience with fluid dynamics so I decided to ask here

The machine injects a fluid with a needle (around 300 microgram/L of fluid) in a tube with running fluid (around 14000 L/h).

If I prove that the fluid flow is turbulent is it possible to prove that after x meters of straight tube we will have a homogeneous mix?

Which formulas can I use for this kind of problem?

Thanks in advance!

[LuckyTran]

You can consider it as a pseudo 1D mass diffusion problem (for how long it takes for a drop to spread over the cross-section), but with the simple molecular diffusivity replaced by the effective (molecular + turbulent) mass diffusivity. You still need to estimate this eddy viscosity using a crude turbulence model.

One idea I can think of is to follow all the formulas here. Using the correlation for turbulence intensity and the flow velocity you can get the turbulent kinetic energy. Then you can estimate the turbulent dissipation rate (epsilon) since there is also a correlation for the turbulent length scale provided.

Now you can calculate the turbulent viscosity using the k-epsilon model.

Then you can assume a turbulent schmidt number of 0.9 and this gives you the turbulent mass diffusivity which you can now use in your problem.

There should be analytical solutions to the 1D mass transfer problem, but I bet they'll most likely be in terms of Fourier series or eigenfunction expansion methods. This is a fairly canonical problem (1D heat diffusion of a point source). It might be faster if you cannot dig up this analytical solution to just crank it numerically.

Of course, there are cruder and probably more conservative ways to estimate the length you require.