Thinking about Regenerators                                                                                                              

Imagine a regenerator that has room for gas of volume V, in and around the wirey solid  material that it contains. Let’s say the regenerator (and the gas it initially contains) is at a temperature half way between the hot and cold temperatures of the engine. Now, very slowly, push in cold gas from one end. The gas is coming in so slowly that it comes to temperature equilibrium with the surrounding solid phase as it moves along. Push the cold gas in, such that the entire volume V is displaced by new gas. This takes less than volume V of cold gas because the cold gas expands as it warms up. The leading edge of the cold gas warms up to the solid phase temperature quite quickly because it is always encountering new warm solid phase. After the gas is at solid phase temperatures, it no longer cools the regenerator solid phase. The net effect is that the cold temperature front lags behind the new gas front, by quite a bit. In fact, because the heat capacity of a volume of the solid phase is about a hundred times that of the same volume of the gas phase, very little of the regenerator material is cooled down. A temperature curve along the cold gas filled regenerator at equilibrium might look like the solid blue line in the graphic.

In an ideal world, if you played the above scenario in reverse, then all the air coming back out of the regenerator would be cold and would contract to its original size and the regenerator would revert to a midpoint temperature.

If hot air is slowly pumped in from the hot side then the equilibrium temperature curve would look like the solid red line. A larger volume of hot gas needs to be pumped because it contracts inside the regenerator. The volume of cold or hot gas that undergoes expansion or contraction during a cycle is represented by the light red and blue shaded areas on the graph. It is important to note, that during each single hot or cold part of the cycle, that both the hot and cold working volumes are undergoing contraction or expansion. As the hot cycle part of the gas exits the regenerator it expands and, at the same time, the cold cycle part of the gas enters the regenerator and also expands. So the total volume of gas expanding or contracting during each half cycle can be represented by adding the blue and red areas on the graph. The white areas of the graph are volumes where work is not being done.

A longer, thus narrower regenerator (V is being held constant), causes higher gas velocities and causes more resistance to gas flow. Higher gas velocities could cause the gas to move though the regenerator without a chance to come to equilibrium with the surrounding solid phase, or perhaps faster moving air removes (and adds?) heat faster from the solid phase. I think ether of these effects would  widen the ideal sharp temperature gradient shoulder on the graph. 

Also, it is possible to saturate the first part of the regenerator solid phase to the temperature of the incoming gas. The more gas that enters the regenerator per square unit of it’s face (long and narrow regenerator), the more likely that part of the regenerator will go to saturation. None of the gas in the part of the regenerator at saturation undergoes contraction (or expansion) and robs pressure from the power stroke.

A regenerator with a wide face and with very shallow depth would have an extremely sharp temperature gradient from one side to the other. Heat does not like to stay where you put it; it diffuses away. Regenerators can be built with stacked wire screens that run perpendicular to the gas flow, to try and keep heat in the same lateral plane. But if the regenerator is too shallow the temperature gradient could be muddied and heat exchange inefficient. 

I conclude that the amount of regenerator solid phase is not critically important as long as it is in large excess of the gas’s heat capacity. 
A target volume  for the gas phase part of the regenerator is simply the cold piston volume adjusted to the mid- temperature in degrees Kelvin or 
V_Rgas = V_cold x K_mid / K_cold. 
The volume (cM³) of the solid phase part of the regenerator is its weight divided by its density, or for steel  
V_Rsolid = Grams of solid phase ÷ 7.8 G/CM³. 
The total regenerator volume is the sum of the solid and the gas volumes. 
The optimum shape for a regenerator is probably slightly wider across the face than along the axis of gas flow. I’m hoping the shape is not critical over a wide range of values.