Hi Sahil Kala,

if your scalar transport quanity is passive, meaning it does not influence the flow, you may perform a calculation of a steady state flow field (time averaged for turbulent flow if "steady-state") and subsequently solve the scalar transport equation using the computed flow field.

Whether or not you have to include the diffusion term depends essentially on the magnitude of inertial scalar transport with the flow in relation to the transport by molecular diffusion or hydrodynamic dispersion. I believe the latter is what you mean by diffusion due to flow, isn't it? A useful quantity to determine wether or not you have to include a diffusion term at all, is the Bodenstein or Peclet number. Mathematically they are the same but in engineering they're used a little differently.

On time scale a convenient measure of inertial transport is the superficial residence time so characteristic length (of the flow path in the main direction of the flow ) divided by the time scale of the diffusion phenomenon. The latter will be characteristic length squared divided by the diffusion coefficient.

N = U/L * L²/D = UL/D (which is the Peclet number).

If this number is large diffusion occurs on a smaller time scale than advection and you should include diffusion. If it is the other way around you can neglect it. In CFD usually numerical schemes exist that account for simultaneous advection and diffusion for 0,1

Dear Andre Rieger,

Thanks a lot for your response. The hint on Peclet number was really very helpful.

Answering to your queries:

After getting a hint on the use of Peclet number, I stumbled upon this paper (https://orbi.ulg.ac.be/bitstream/2268/3305/2/PecletpaperHJ.pdf). Although this paper is great but with so many new terms and definitions I found myself abit lost. I have a few findings and few queries which i am listing down:

Findings:

Queries:

Please forgive me if I have asked any dumb question.

https://orbi.ulg.ac.be/bitstream/2268/3305/2/PecletpaperHJ.pdf

Dear Sahil Kala,

there is no need to be lost and also not to apologize for asking questions. Acutally, asking questions is what science is all about.

Regarding your findings:

1) This is correct. Since dispersion and diffsuion are phenomenologically the same, even though the reasons for there occurence is different, The passive scalar transport equation is an advection-diffusion equation. Actually this equation is nothing more than a statement of mass conservation of the passive scalar quantity.

2) Of course it always exists, but when it comes to solving the equation you should always ask wether or not both modes of transport are equal in magnitude. If diffusion for example is 1000 time slower than advection in a fast flow, it will probably have no effect in the results. If however the flow is very very slow, sso slow in fact, that diffusion is faster, than you do not need to account for advection. This is helpful, because in these cases where one of the transport modes is dominant, the transport equation will reduce to a simple advection or diffusion equation, which are especially numerically easier to solve.

3) is absolutely correct.

Regarding your querries:

1) All the definitions in the paper are the same. Peclet number always is the ratio of diffusion time (L^2/D) over fluid mean residence time (L/U). All the definitions in the paper still stick to this general definition, but define these time scales in different ways.If you need one in particular, you would use eqn. (6) or (7). Wether or not you need to account for the dispersion depends on the geometry you solved the flow for. Can you give me any information on it?

2)You may calculate them in different ways but since these are substance and system related properties you may find them in data collections. Can you give any information on the passive scalar quantity that is diffusing through what flowing medium? Are the particles fixed and the flow is moving around them or are the particles unfixed and move with the flow? Are these particles the scalar quantities you want to "transport"? How large are they and at what density?

Beste regards,

André

It is standard to model the diffusion term of a scalar as: divergence of (diffusion coefficient * gradient of the scalar).

The diffusion term has physical contribution at the molecular level and also a represents a pseudo-diffusion effect at the macro level due to the flow in case of turbulence.

From the numerical aspect, ignoring the diffusion term altogether is something significant because it changes the character of the partial differential equation of the scalar variable to hyperbolic and eliminates the stabilizing dissipation.

Dear Andre Rieger,

Thank you for your response. 

The geometry used to solve the flow is very similar to a wheat/flour kneading machine. 

In reality the solid particles are moving with the flow but for the numerical simulation I took the average properties of the mixture and solved the flow; then on the converged flow i am solving a scalar equation to find the mixing time of the solid particles. 

Yes, these particles are the scalar quantities that I want to transport.

The particles are of different sizes viz. 10 micron, 100 micron and 500 micron. These particles have same true densities 1200 Kg/m3. 

Dear Sahil Kala,

are you sure that the scalars are passive meaning that they do not influence the flow at all. I could imagine that in dependence of the particle concentration or even particle size distribution may alter flow properties suchs as viscosity and therefore does influence the flow. Hence the scalar is not passive but active and you need to solve the transport equations and the flow equations simultaneously. Also in dependence of the overall particle load there might be some sort of paricle-fluid coupling meaning the particles introduce some additional force acting on the fluid such as a drag force.

Do you have any theory on that? Did you do experiments where you investigated the flow with and without particles and compared the flow fields by PIV?

What type of software do you use to model the flow? I wonder wether or not you can use some existing code such as fluid flow and discrete particle coupling already implemented in OpenFOAM. I do not know the exact name of the solver, but there certainly are solvers that can solve flow eequations for fluidized beds. Maybe you can check that and see wether or not you can use these instead of generating your own code. Also withput any experimental data it might be difficult to find wether or not your dispersion coefficients are accurate.

Best regards and good luck with your research,

André