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# Time to heat a packed bed

## Time to heat a packed bed

(OP)
Hello,

I'm trying to determine the center line temperature of a fixed bed reactor over time during heat up. The reactor has a 6" diameter and will be heated externally by electric heaters to 700C. Two heaters are currently being considered, one providing a heat flux up to 4 W/in^2 and another up to 10 W/in^2. The bed is filled with silica sand (k=0.2 W/mK, Cp=830 J/kgK, density=1400 kg/m^3, dp=500 microns). During heat up there will be no gas flowing through the reactor, thus I expect nearly all the heat transfer to be via conduction and radiation. Therefore, I would also assume that the heaters can maintain a constant wall temperature, since the conduction into the sand is likely the limiting factor (very large Biot number). I would like to plot the centerline temperature of the reactor versus time for the different heat flux cases. Then I would also like to see how the centerline temperature versus time changes with a small purge gas flow through the sand. Does anyone have any good examples of how to calculate this?

Any help is much appreciated!

Thank you!

### RE: Time to heat a packed bed

Sounds like a fun one, though not an easy one.

Conduction will certainly be a significant factor, but it will be limited by the contact between particules - do you have an idea of the void fraction of the material? Essentially, there is an additional resistance to heat transfer that is a function of the contact area between particles, usually correlated empirically for experimental data. The thermal conductivity you are using may already incorporate this factor, I'm not certain.

If you truly care about a quiescent system, remember that natural convection cellular flows may also play a role. I have a feeling that radiation is insignificant. Order of magnitude calculations may let you exclude different phenomena to simplify your model (parallel heat xfer means implicit functions!)

I have encountered a few papers for packed beds and radial heat transfer, though I believe they only covered a bed under flowing conditions. Unfortunately, I don't have those references handy but I remember the radial heat transfer coefficient being on the order of 100-400 W/m2-K range for our laminar system with fairly large catalyst particles.

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