The first mathematical theories to describe
regenerator operation were published in the late 1920s, more than
100 years after its invention by Robert Stirling. Significantly,
these and subsequent theories of regenerator operation are based
on assumptions which are neither relevant nor applicable to Stirling
engine regenerators. More recently (1997) Allan Organ has published
a book "**The
Regenerator and the Stirling Engine**" which represents
a significant step towards "bridging the gap" between
Hausen's celebrated regenerator analysis, widely used in the analysis
of gas turbine engines, and the unique conditions that apply to
Stirling engines.

By definition a regenerator is a cyclic device. On the first part of the cycle the hot gas flows through the regenerator from the heater to the cooler, and in so doing transfers heat to the regenerator matrix. This is referred to as a "single blow". Subsequently during the second part of the cycle the cold gas flows in the reverse direction, absorbing the heat that was previously stored in the matrix. Thus at steady state the net heat transfer per cycle between the working gas and the regenerator matrix is zero.

The regenerator quality is usually defined on an enthalpy basis in terms of a regenerator effectiveness ε as follows:

However this definition is not amenable to
usage in Stirling engines. We propose an equivalent definition
in the context of the **Ideal
Adiabatic model**, which represents the limiting maximum
performance measure, as follows:

The regenerator effectiveness ε thus
varies from 1 for an ideal regenerator (as defined in the Ideal
Adiabatic model) to 0 for no regenerative action. Consider for
example the cyclic energy-theta diagram which we obtained in the
**Ideal Adiabatic analysis**
of the Ross D-90 engine:

The thermal efficiency of the Ideal Adiabatic
cycle (suffix "i") is given in terms of the energy values
accumulated at the end of the cycle by:

ηi = Wi / Qhi = (Qhi + Qki) / Qhi

Note from the diagram that Qki is a negative quantity, thus ηi = 0.627 for the D-90 engine as shown. Notice also the significant amount of heat tranferred during a single blow of the regenerator given by Qri^. Thus for the D-90 engine as shown the ratio Qri^/Qhi = 5.66.

Now for a system having a non-ideal regenerator, during the single blow when the working gas flows from the cooler to the heater, on exit from the regenerator it will have a temperature somewhat lower than that of the heater. This will result in more heat being supplied externally over the cycle by the heater in increasing the temperature of the gas to that of heater and can be written quantitatively as follows:

Qh = Qhi + Qri^(1 - ε)

Similarly, when the working gas flows from the heater to the cooler, then an extra cooling load will be burdened on the cooler, as follows:

Qk = Qki - Qri^(1 - ε)

The thermal efficiency of the non-ideal engine (without the suffix "i") is given by:

η = W / Qh = (Qh + Qk) / Qh

Substituting for Qh, Qk, and ηi from the above equations we obtain:

The following diagram shows a plot of the above equation for the specific case of the D-90 engine and shows the effect of regerator effectiveness ε on thermal efficiency η.

Notice that as ε varies from 1 for an ideal regenerator (Ideal Adiabatic cycle) to 0 for no regenerative action, the thermal efficiency η drops from more than 60% to less than 10%. Furthermore, differentiating the efficiency equation η with respect to ε, and substituting ε = 1 we obtain:

Thus we see that for highly effective regenerators (close to ε = 1) a 1% reduction in regenerator effectiveness results in a more than 5% reduction in thermal efficiency η. Furthermore we see that if one has a regenerator that has an effectiveness of 0.8, the thermal efficiency has dropped by half to around 30%. This not only means a significantly less efficient machine, but one that has to have a significantly larger cooler. Obviously we need to have a means of determining the actual regenerator effectiveness in any specific machine.

We now consider the regenerator effectiveness in terms of the temperature profile of the 'hot' and 'cold' gas streams with respect to the regenerator matrix. We assume an equal difference in temperature ΔT on the hot and the cold sides, and linear temperature profiles, leading to the definition of regenerator effectiveness ε in terms of temperatures, as shown.

Combining the two equations in the figure, we obtain:

Now from energy balance considerations of the hot stream, the change in enthalpy of the hot stream is equal to the heat transfer from the hot stream to the matrix, and subsequently from the matrix to the cold stream, thus:

where(watts) is the heat transfer power, h is the overall heat transfer coefficient (hot stream / matrix / cold stream), Awg refers to the wall/gas, or "wetted" area of the heat exchanger surface, cp the specific heat capacity at constant pressure, and(kg / s) the mass flow rate through the regenerator. Substituting in the effectiveness equation we obtain:

We now introduce the concept of Number of Transfer
Units (NTU) which is a well known measure of heat exchanger effectiveness,
and is defined in the section on **Scaling
Parameters**.

Thus:

Notice that the NTU value is a function of
the type of heat exchanger as well as its physical size. It includes
the actual wetted area Awg, as well as the actual mass flow through
the regenerator(kg/s). In heat exchanger analysis
it is more usual to evaluate local heat exchanger parameters in
terms of fluid property values which are independent of size.
Thus we define a Stanton number (refer to the section on **Scaling Parameters**) as follows:

NST = h / (ρ u cp)

where ρ is the fluid density, and u is the fluid velocity, thus:

where A is the free flow area through the matrix. Tables and graphs of empirical values of Stanton number vs Reynolds number are available from heat exchanger texts for various heat exchanger types. We use two types of matrices in our regenerator analysis - woven mesh and coiled annular foil. The NTU value can then be obtained in terms of the Stanton number as follows:

NTU = NST (Awg / A) / 2

The factor 2 in this equation is unusual, and stems from the fact that the Stanton number is usually defined for the transfer of heat from the gas stream to the matrix alone, whereas the NTU usage in this section is for overall transfer of heat from the hot stream to the regenerator matrix, and subsequently to the cold stream.