Porous media - heat transfer - set up of mean values

Wowbagger · May 10, 2024, 08:33

Hi Foamers,

I am working on a case with heat transfer in porous media.
This post is related to this older one but with a new main topic.
The old one:
See also: https://www-cfd--online-com.translat..._x_tr_pto=wapp

Regarding the setup of this - or any other porous media case with heat transfer - I got some thoughts which I am not sure about the correctness.

I set up my case which is a silo with rocks.
For better understanding, lets assume it is filled with spheres.

As usual there are a lot of values to set for the simulation.
In this case I am wondering about the definition of length in "interRegionHeatTransferCoeffs" the header file says:
"Model for inter-region heat exchange. Requires specification of a model for the heat transfer coefficient (htc) and the area per unit volume (Av)."

The time behavior is only related to the area per volume if all other coefficients stays the same.

For my case I could calculate the surface area for a given volume of a packed bed.
But thinking about this led me to a more general approach and I am not sure, if this is the right approach for such a porous approach which deal with mean values.

There are two calculations for the transfered heat:
Solid:

$\dot{Q} = m \cdot c_{p} \cdot \frac{\Delta T}{t}$

Gas:

$\dot{Q} = \alpha \cdot A \cdot \Delta T$

Having a look to those two formulas I could use them to have a look to the dimensions.
For the solid side there is mass which is proportional to L³.
For the gas side there is the area which is proportional to L².

For the better understanding: Lets assume the smallest object of heat transfer in the packed bed which is one single sphere.
For an individual cell (with a given volume) I could say that:

$\rho \cdot c_{p} \cdot \frac{\Delta T}{t} = \alpha \cdot \frac{L^{2}}{L^{3}} \cdot \Delta T$

For a sphere diameter of 0.01 this would lead to a value of 100 m²/m³ (instead of ~1579m²/m³ for a given porosity of 0.38).
Which is more a value of 1/100 m, which is an value for the heat boundary layer given by the geometry of the packed bed.

There are some constants missing due to the simple look at the dimensions. Not having in mind pi(), etc.

But is that main assumption valid?
Has someone experience in such a case setup? Maybe some formulations for packed beds?

I am happy for every thought on this.

Best regards
Michael

Wowbagger · May 13, 2024, 14:58

I found my concern in this wikipage:

https://en.wikipedia.org/wiki/Lumped...stive_circuits

Best regards Michael

May 10, 2024, 08:33	Porous media - heat transfer - set up of mean values	#1
Wowbagger Member Michael S. Join Date: May 2020 Location: Hamburg, Germany Posts: 53 Rep Power: 6	Hi Foamers, I am working on a case with heat transfer in porous media. This post is related to this older one but with a new main topic. The old one: See also: https://www-cfd--online-com.translat..._x_tr_pto=wapp Regarding the setup of this - or any other porous media case with heat transfer - I got some thoughts which I am not sure about the correctness. I set up my case which is a silo with rocks. For better understanding, lets assume it is filled with spheres. As usual there are a lot of values to set for the simulation. In this case I am wondering about the definition of length in "interRegionHeatTransferCoeffs" the header file says: "Model for inter-region heat exchange. Requires specification of a model for the heat transfer coefficient (htc) and the area per unit volume (Av)." The time behavior is only related to the area per volume if all other coefficients stays the same. For my case I could calculate the surface area for a given volume of a packed bed. But thinking about this led me to a more general approach and I am not sure, if this is the right approach for such a porous approach which deal with mean values. There are two calculations for the transfered heat: Solid: $\dot{Q} = m \cdot c_{p} \cdot \frac{\Delta T}{t}$ Gas: $\dot{Q} = \alpha \cdot A \cdot \Delta T$ Having a look to those two formulas I could use them to have a look to the dimensions. For the solid side there is mass which is proportional to L³. For the gas side there is the area which is proportional to L². For the better understanding: Lets assume the smallest object of heat transfer in the packed bed which is one single sphere. For an individual cell (with a given volume) I could say that: $\rho \cdot c_{p} \cdot \frac{\Delta T}{t} = \alpha \cdot \frac{L^{2}}{L^{3}} \cdot \Delta T$ For a sphere diameter of 0.01 this would lead to a value of 100 m²/m³ (instead of ~1579m²/m³ for a given porosity of 0.38). Which is more a value of 1/100 m, which is an value for the heat boundary layer given by the geometry of the packed bed. There are some constants missing due to the simple look at the dimensions. Not having in mind pi(), etc. But is that main assumption valid? Has someone experience in such a case setup? Maybe some formulations for packed beds? I am happy for every thought on this. Best regards Michael Last edited by Wowbagger; May 10, 2024 at 08:35. Reason: Kindness ;)

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May 13, 2024, 14:58		#2
Wowbagger Member Michael S. Join Date: May 2020 Location: Hamburg, Germany Posts: 53 Rep Power: 6	I found my concern in this wikipage: https://en.wikipedia.org/wiki/Lumped...stive_circuits Best regards Michael