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Suppose I have a flow of hot air around a cold and unevenly shaped object with holes and tunnels (think about it as a bed packed with some objects). I would like to know the Reynolds number of this flow and its convective heat transfer coefficient. The definition of the Reynolds number contains a "characteristic length" that is somehow mysterious to me and that I do not have at hand. And I am a bit reluctant to use the formula for the packed bed Reynolds number. Is it possible to measure it? How would I design an experiment for this?

I would like to avoid temperature readings of the object. In this article I found a relation between the Reynolds number of a packed bed and the heat transfer coefficient, so measuring only the Reynolds number would be a good start, even though I do not have a real packed bed. Could this be done by simple pressure drop readings?

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From what you describe there are two different length scales associated with this problem. The one ascociated with flow through the porous medium (the 'bed packed with some objects'), and the second ascociated with the flow around the 'bed'. I will assume you are talking about the flow through the porous 'bed'.

Normally the characteristic dimension or length scale for internal flows is taken to be the hydraulic diameter. This is defined to be four times the cross-sectional area (of the fluid), divided by the wetted perimeter. However, for such things as 'pebble beds' etc. the Reynolds number is defined differently.

For flow of fluid through a bed of approximately spherical particles of diameter D in contact, if the voidage (fraction of the bed not filled with particles) is ε and the superficial velocity V (that is, the fluid velocity through the bed as if the spheres/objects were not present) then a Reynolds number can be defined as:

$$Re = \frac{\rho V D}{\mu(1 - \epsilon)}$$

Laminar conditions apply up to Re = 10, fully turbulent from 2000 (Wikipedia). There are more advanced formulas for this, and they work in a variety of regimes; from not-so-packed beds, to very packed-beds, also with a variaty of pebble/object shapes.

Many experiments have been done on convective and radiative heat transfer in pebble bed nuclear reactors and other such heat exchangers. I am sure you should be able to find some journal papers on this stuff along with the standard correlations you need for your particular flow.

For the convective heat transfer coefficient for this flow however, you should be using the Nusselt Number which is a measure of the ratio of convective to conductive heat transfer a solid-fluid boundary.

I hope this helps.

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