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### 1. Introduction, Assumptions, and Definitions

The Stirling engines to be simulated consist of five subspaces connected in linear fashion as shown in the Figure below. The cooler, regenerator, and heater space have constant volumes Vk, Vr, and Vh, respectively. Both, the compression and the expansion space , are each divided into a fixed clearance volume, Vclc and Vcle respectively, and a time-dependent volume which varies between 0 and their respective maxima, Vswc and Vswe. The precise variation of Vswc and Vswe during a complete cycle depends on the mechanical drives employed. Typical arrangements of the five subspace are shown below. For the alpha-type everything can be stacked into a single cyclinder. In the beta-type both the power piston and displacer are housed in a single cyclinder while in the gamma-type they occupy separate cyclinders.

Most commonly used arrangements

In principle, the analysis brought in this paper is not restricted as to the precise mechanism employed to change the volumes of the compression and expansion space although the programs built upon it offer only a few options.

In this paper we use very much the same notation as employed by Urieli. In particular we use the subscripts c, k, r, h, and e to refer to the compression, kooler, regenerator, heater, and expansion space, respectively. As it is common practice in engineering thermodynamics, we use m [kg] for mass, p [N/m²] for pressure, T [°K] for temperature, V [m³] for volume, u [kJ/kg] for specific internal energy, h [kJ/kg] for enthalpy, Q [J] for heat, and W [J] for work.

We frequently use the letter "d" in front of a quantity to denote a (infinitesimal) small change of the quantity as a result of a small change of the crank angle. i.e dmc is the change in mass of the compression space as the crank angle changes by the amount dΘ. The letter "d" is not used for any other purposes.

The following assumptions are basis of the subsequent analysis :

1. The engine turns at constant speed. Therefore, time and crank angle Θ are proportional to each other.

2. Any pressure drops due to flow resistances and pressure differentials needed to accelerate the working gas are neglected. Hence, the pressure, p , has at a given instant the same value everywhere inside the engine and varies only with time. Correspondingly, kinetic energies of the working gas are neglected in the law of conservation of energy.

3. Leakage of gas to the outside of the engine, including crank case, is assumed to be zero.

4. Both, the compression and expansion space, volumes Vc and Ve, are adiabatic. This means that no heat gets exchanged in these two spaces between the gas and the surrounding walls nor with the piston/displacer surfaces. The temperature in these spaces is uniform but varies during a cycle due to changes in pressure, volume and gas coming from/leaving to the adjacent kooler and heater space, respectively.

5. The heat transfer conditions for the kooler space are sufficiently good to keep the gas inside the kooler space, volume Vk, at constant uniform temperature Tc at all times. The same holds true for the heater space, volume Vh and uniform temperature Th.

6. The heat transfer conditions are sufficient to keep the temperature distribution inside the regenerator, volume Vr , linear, varying from Tc where the regenerator is connected to the kooler to Th at the heater side.

7. An ideal gas is used as working fluid and the ideal gas law connecting pressure p, volume V, mass m, and temperature T to each other via the specific gas constant R :

(1.1)     p V = m R T

Also, the specific heat capacities and constant volume, cv, and constant pressure, cp, are connected to the specific gas constant :

(1.2)     cp - cv = R

We furthermore assume that the working gas has temperature independent values for the specific heat capacities, cv and cp, and their ratio κ is known :

(1.3)     κ = cp/cv    (i.e 1.4 for air and hydrogen)

As a result the specific internal energy, u , of an ideal gas at temperature T is :

(1.4)     u = cv ( T - T0 ) [kJ/kg]

And then by definition (h=u+p*v) the specific enthalpy becomes :

(1.4)     h = cp T - cv T0 [kJ/kg]

The temperature T0 is inserted to provide the best fit between the real behavior of u and constant cv relationship. For monatomic gases like He, Ne, Ar etc. Eq. (1.4) holds true with T0=0 over a temperature range of a few thousand degrees. We will later see that T0 itself will drop out of the important equations to follow.

### 2. Derivation of relevant differential equations

#### 2.1 Conservation of mass and energy

In addition to the above restrictions and assumptions the laws of conservation of mass and energy form the basis of the development of the relevant differential equations.

For mass we have :

(2.1.1)     dmc + dmk + dmr + dmh + dme = 0

which states that the gain of mass in one subspace (dm>0) has to be compensated by losses of mass in other subspaces ( dm<0).

Conservation of energy for a single subspace with arbitrary index "s" which is connected to two adjacent subspaces "s1" and "s2" :

(2.1.2)     d(ms us ) = dQs - p dVs + hs1 dms1 + hs2 dms2

The term hs1 dms1 denotes the energy brought into the subspace from its adjacent subspace "s1" with the gas. The enthalpy hs1 corresponds to the enthalpy of the case inside subspace "s1" if dms1>0 (mass coming from "s1") and corresponds to enthalpy hs for dms1<0 (mass leaving "s"). Because of conservation of mass :

(2.1.3)     dms = dms1 + dms2

Special forms of Eq. (2.1.2) arise for the different subspaces. For the compression and expansion space we have by assumption zero heat transfer, hence dQs = 0 , and only a single connection with an adjacent space. For the kooler , regenerator, and heater we have constant volume. Hence, dVs = 0 for these subspaces.

In the following sections we will investigate the individual subspaces. One objective is to express for each subspace the change in mass in terms of the overall pressure change "dp" and volume change dV (where applicable) and substitute these equations into Eq. (2.1.1) to obtain a single differential equation for the pressure p driven by the volume changes in the compression and expansion space. Equally important is the development of equation for the heat fluxes in each of the subspaces and mass flow rates. Finally, in Section 2.7, we pull everything together and summarize the resulting equations.

#### 2.2 Compression Space

The ideal gas law states :

(2.2.1)     p Vc = mc R Tc

As the crank angle varies, so do pressure p, mass mc and temperature Tc, and volume Vc. Differentiation gives :

(2.2.2)     R Tc dmc = Vc dp + p dVc - mc R dTc

In order to facilitate sustitution of dmc in Eq.(2.1.1) we need to eliminate dTc from this equation. We do that using conservation of energy. Because this space is adiabatic and has only one adjacent subspace ( the kooler) the general energy equation, Eq. (2.1.2) simplifies somewhat.

(2.2.3)     d(mc uc ) = - p dVc + hck dmc

We substitute now uc = cv (Tc - T0) and hck = cp Tck - cv T0 in accordance with Eq. (1.4) and (1.5) :

mc cv dTc + cv ( Tc - T0 ) dmc = - p dVc + (cp Tck - cv T0) dmc

Note that the temperature T0 cancels out and we can rearrange :

(2.2.4)     mc cv dTc = - p dVc + (cp Tck - cv Tc) dmc

The newly introduced temperature Tck reflects the different energy content of the gas being exchanged between compression and kooler space :

 (2.2.5) Tck = Tk if dmc > 0 = Tc if dmc < 0

dTc of Eq. (2.2.4) can now be substituted into Eq. (2.2.2) and after some algebra :

(2.2.6)     dmc = ( κ p dVc + Vc dp ) / ( κ R Tck )

which we later substitute into the total mass balance, Eq. (2.1.1), to eliminate from there the quantity dmc in favor of dp and the known change in compression space volume dVc.

#### 2.3 Expansion Space

The expansion space is treated in exactly the same fashion as the compression space. The relevant equation to be used later can be simply gained from those of compression space by subscript substitution :

(2.3.1)     p Ve = me R Te

(2.3.4)     me cv dTe = - p dVe + (cp Teh - cv Te) dme

 (2.3.5) Teh = Th if dme > 0 = Te if dme < 0

(2.3.6)     dme = ( κ p dVe + Ve dp ) / ( κ R Teh )

#### 2.4 Kooler Space

The ideal gas law states :

(2.4.1)     p Vk = mk R Tk

Because volume and temperature are constant for this space, differentiation of Eq. (2.2.1) gives :

(2.4.2)     dmk = (Vk/(R Tk)) dp

which is all what we need for substitution into the total mass balance, Eq. (2.1.1) to eliminate dmk in favor of dp.
The heat flow rate for the kooler space is determined from the energy balance, Eq. (2.1.2). Again we take into account that volume and temperature are constant and express internal energy and enthalpies in terms of specific heat capacities. We also note that the temperature of the gas exchanged between kooler and regenerator is always Tk.

dQk = cv ( Tk - T0 ) dmk + (cp Tck - cv T0 ) dmc - (cp Tk - cv T0 ) ( dmk + dmc )

Here the term (dmk + dmc) reflects the mass loss of kooler and compression space combined to the regenerator taking with it energy. Again the reference temperature T0 cancels out and terms can be regrouped :

(2.4.3)     dQk = cp ( Tck - Tk ) dmc - R Tk dmk

#### 2.5 Heater Space

The analysis of this subspace is completely identical to that of the kooler space. The relevant equation can be taken over, only subscripts need to be adjusted.

(2.5.1)     dmh = (Vh/(R Th)) dp

(2.5.2)     dQh = cp ( Teh - Th ) dme - R Th dmh

in which the temperature Teh is the same as that defined by Eq. (2.3.5).

#### 2.6 Regenerator

The analysis for the regenerator is identical for the ideal adiabatic simulation and the Schmidt analysis. For details we refer the reader to that paper (Heat transfer inside the regenerator). The only adjustment needed is that for the Schmidt analysis the temperature in the adjacent kooler space was referred to as Tc while it is now Tk. We therefore list only the important equations.

The effective temperature of the regenerator is :

(2.6.1)     Tr = ( Th - Tk ) / ln(Th/Tk)

It is constant in time and can be used to calculate the mass inside the regenerator using the ideal gas law and taking into account a linear temperature distribution. With the volume, Vr, being constant, a change in mass is simply related to a change in pressure :

(2.6.2)     dmr = Vr / ( R Tr ) dp

For the energy balance we have to take into account the heat exchange with the walls, the change in internal energy (due to change in mass) and the energies transported by the gas flowing between the regenerator and its adjacent subspaces (kooler and heater).

(2.6.3)     dQr = cv Tr dmr + cp Tk ( dmk + dmc ) + cp Th ( dmh + dme )

#### 2.7 Putting it all together

We substitute now the Equations (2.2.6), (2.3.6), (2.4.2), (2.5.1), and (2.6.2) into mass balance Eq. (2.1.1) and solve for dp. The results is :

(2.7.1)       dp  =  -p   (

 dVc Tck
+
 dVe Teh
) / (
 Vc κ Tck
+
 Ve κ Teh
+
 Vk Tk
+
 Vr Tr
+
 Vh Th
)

Vc and Ve and their derivatives are determined by the drive geometry. For example for a sinusoidal drive ( see Ideal Isothermal Analysis for definitions of the terms ) :

Vc = Vclc + 0.5 Vswc( 1 + cos(Θ) )

Ve = Vcle + 0.5 Vswe( 1 + cos(Θ + δ) )

dVc = -0.5*Vswcsin(Θ) dΘ

dVe = -0.5*Vswesin(Θ+δ) dΘ

Here δ is the swept volume phase lag (VPL).

When integrating Eq. (2.7.1) as function of crank angle care must be taken of the temperatures Tck and Teh as they change values as the crank angle Θ advances :

 (2.2.5) Tck = Tk if dmc > 0 = Tc if dmc < 0

 (2.3.5) Teh = Th if dme > 0 = Te if dme < 0

In order for that to succeed we also have to know the temperatures in the expansion and compression space, Te and Tc respectively. Additionally, the mass flow rates dme and dmc need to be known.

We can do that by simultaneously integrating :

(2.2.6)     dmc = ( κ p dVc + Vc dp ) / ( κ R Tck )

and

(2.3.6)     dme = ( κ p dVe + Ve dp ) / ( κ R Teh )

and calculate the temperatures in the compression and heater space using the ideal gas law for either space :

(2.2.4)     mc cv dTc = - p dVc + (cp Tck - cv Tc) dmc

(2.3.4)     me cv dTe = - p dVe + (cp Teh - cv Te) dme

### 3. Sample Calculations and Optimization

#### 3.1 Some Observation on an Example Calculation

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