Spherical Bed Regenerators

I recently had some email communication with Tom Gentry who runs the Stirling Engine Society USA (sesusa) list. He has been considering the use of spherical beds as an inexpensive and easily constructed regenerator for a Stirling engine. These consist of small randomly stacked spherical pebbles. In his literature research he did find a set of flow resistance and heat transfer equations from the KTA (Kerntechnischer Ausschusses) - a German Atomic Energy Commission - which addresses flow through beds of hot 'pebbles' (radioactive and thermally hot). Two of the papers that they make available in .pdf format are particularly useful in this context:

3102.2 - Heat Transfer in Spherical Fuel Elements
3102.3 - Loss of Pressure through Friction in Pebble Bed Cores

Tom Gentry was wondering about the previous use of randomly stacked spheres in Stirling engine regenerators. I do recall that spherical beds of lead shot were once popular for cryogenic cooler regenerators, however could find no literature on the subject. It is interesting that the heat transfer and friction data are presented by the KTA, since Ivo Kolin (The Evolution of the Heat Engine 1972 - reprinted by Moriya Press, 1998) has previously presented a convincing argument for using a Stirling engine to convert nuclear energy to mechanical energy. One could conceive of a bed of radioactive and thermally hot pebbles inside the heater section of a Stirling engine, in series with a bed of steel pebbles to act as the regenerator, the working fluid being helium. In this context the high reliability offered by the free-piston alpha configuration recently proposed by William Beale of Sunpower, Inc could be used to advantage.

In the heat transfer or flow friction analysis of a heat exchanger or regenerator, (such as Similarity approach to design developed by Dr Allan Organ) one of the important parameters needed is an equivalent length, such as the Hydraulic Radius (refer: Scaling Parameters), however the KTA papers did not present any method of determining this parameter.

In the following analysis we determine the Hydraulic Radius rh of the spherical bed, which is defined as the void volume V divided by the wetted area Awg. We determine these two values as follows:

Volume of each spherical ball having diameter d is

For n balls the total matrix metal volume is

The surface area of each spherical ball is

For n balls the total wetted area is

Thus the ratio

The matrix porosity is defined as the void volume V divided by the total volume (V + Vm), thus:

Substituting (1) into (2) and simplifying, we obtain

We can now evaluate the Hydraulic Radius rh, which we recall is defined as the void volume V divided by the wetted area Awg

Tom Gentry uses a porosity range between = 0.37 to 0.43. Thus if we use a mean value of 0.4 then the Hydraulic Radius rh = d / 9 (recall that d is the mean diameter of each spherical ball).

The porosity can however be easily determined experimentally by comparing the weight of a spherical bed in a known volume to a solid having the same volume.

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