Efficiency of Stationary Fixed Matrix Regenerators
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Most Stirling engines use the simplest kind of regenerator: an array of tubes that let the flow pass both ways. There are no valves or any other device that prevents the flow from going through the regenerator.
These regenerators increase and decrease the temperature depending on whether the flow goes either one direction or the other. This mission is accomplished forcing a temperature gradient in the regenerator itself. For instance, if the cold flow goes from the cold side to the hot side its temperature will increase. This gradient is ideally kept constant and any variation can be considered a loss.
There is a severe lack of literature about Stirling Engines. We can find a lot of blueprints and ideas on how to actually build an engine but very few authors have faced the whole problem from a theoretical point of view. This is the cause why very few designs have become successful. There is a very poor understanding of the thermal processes that make a design more suitable than the rest. Every single idea has been patented before hesitating if it was good or bad.
The simplest piece of the whole engine is the fixed matrix regenerator. I have not found a single textbook or paper where the efficiency of the regenerator is defined. All the phenomena that one could learn in a Fluid Mechanics course is coupled in a regenerator: heat transfer, pressure losses, compressibility, turbulence... That may be the reason why there has not been (or I have not found) any successful attempt to give a single parameter description of a regenerator. In this post I want to suggest my own.
The quantitative description of any forced convection phenomenon is done with its Nusselt number. We can define a Nusselt number for the regenerator NuD, where D is a characteristic length of its width. The Nusselt number depends on the Reynolds number, the Prandtl number and its geometry. Only the latter is difficult to measure. Expanding the meaning of the Nusselt number and multiplying it for the area of a slice of the regenerator we reach:
Tw-T = (m' cp)/(NuDk f)·dT/dx
Where f is a form factor (2π for a cylinder), m' is the mass flow, cp is the heat capacity and k is the thermal conductivity. Tw-T is the difference between the wall surface temperature and the bulk temperature. We have seen that, with the given hypothesis:
- The temperature gradient is constant
- The part of the regenerator where the heat transfer processes are not stationary can be neglected.
is constant.
We can average the whole equation and tadaaaaaa.
Tw-T = (m' cp)/(NuDk f)·Δ T/L
Where Δ T is the difference of the wall surface temperatures in the inlet and the outlet of the regenerator and L is the length of the regenerator. Then, the heat transfer from the regenerator body to the flow is
Δh r = m' cp Δ T(1-(m'cp)/(Nu k f L))
Δh r = m' cp Δ T(ηL)
Where the proposed regenerator efficiency ηL=1 is the impossible case where the flow is always at the wall temperature.
Comentarios
Most Stirling engines use the simplest kind of regenerator: an array of tubes that let the flow pass both ways. There are no valves or any other device that prevents the flow from going through the regenerator.
These regenerators increase and decrease the temperature depending on whether the flow goes either one direction or the other. This mission is accomplished forcing a temperature gradient in the regenerator itself. For instance, if the cold flow goes from the cold side to the hot side its temperature will increase. This gradient is ideally kept constant and any variation can be considered a loss.
There is a severe lack of literature about Stirling Engines. We can find a lot of blueprints and ideas on how to actually build an engine but very few authors have faced the whole problem from a theoretical point of view. This is the cause why very few designs have become successful. There is a very poor understanding of the thermal processes that make a design more suitable than the rest. Every single idea has been patented before hesitating if it was good or bad.
The simplest piece of the whole engine is the fixed matrix regenerator. I have not found a single textbook or paper where the efficiency of the regenerator is defined. All the phenomena that one could learn in a Fluid Mechanics course is coupled in a regenerator: heat transfer, pressure losses, compressibility, turbulence... That may be the reason why there has not been (or I have not found) any successful attempt to give a single parameter description of a regenerator. In this post I want to suggest my own.
The quantitative description of any forced convection phenomenon is done with its Nusselt number. We can define a Nusselt number for the regenerator NuD, where D is a characteristic length of its width. The Nusselt number depends on the Reynolds number, the Prandtl number and its geometry. Only the latter is difficult to measure. Expanding the meaning of the Nusselt number and multiplying it for the area of a slice of the regenerator we reach:
Tw-T = (m' cp)/(NuDk f)·dT/dx
Where f is a form factor (2π for a cylinder), m' is the mass flow, cp is the heat capacity and k is the thermal conductivity. Tw-T is the difference between the wall surface temperature and the bulk temperature. We have seen that, with the given hypothesis:
- The temperature gradient is constant
- The part of the regenerator where the heat transfer processes are not stationary can be neglected.
We can average the whole equation and tadaaaaaa.
Tw-T = (m' cp)/(NuDk f)·Δ T/L
Where Δ T is the difference of the wall surface temperatures in the inlet and the outlet of the regenerator and L is the length of the regenerator. Then, the heat transfer from the regenerator body to the flow is
Δh r = m' cp Δ T(1-(m'cp)/(Nu k f L))
Δh r = m' cp Δ T(ηL)
Where the proposed regenerator efficiency ηL=1 is the impossible case where the flow is always at the wall temperature.