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1. Introduction, Assumptions, and Definitions ## 1. Introduction, Assumptions, and DefinitionsThe 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 V
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 dm The following assumptions are basis of the subsequent analysis : - The engine turns at constant speed. Therefore, time and crank angle
Θ are proportional to each other.
- 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.
- Leakage of gas to the outside of the engine, including crank case, is
assumed to be zero.
- Both, the compression and expansion space, volumes
V
_{c}and V_{e}, 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. - The heat transfer conditions for the kooler space are sufficiently
good to keep the gas inside the kooler space, volume V
_{k}, at constant uniform temperature T_{c}at all times. The same holds true for the heater space, volume V_{h}and uniform temperature T_{h}. - The heat transfer conditions are sufficient to keep the temperature
distribution inside the regenerator, volume V
_{r}, linear, varying from T_{c}where the regenerator is connected to the kooler to T_{h}at the heater side. - 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, c _{v}, and constant pressure, c_{p}, are connected to the specific gas constant :(1.2) c _{p}- c_{v}= RWe furthermore assume that the working gas has temperature independent values for the specific heat capacities, c _{v}and c_{p}, and their ratio κ is known :(1.3) κ = c _{p}/c_{v}(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 = c _{v}( T - T_{0}) [kJ/kg]And then by definition (h=u+p*v) the specific enthalpy becomes : (1.4) h = c _{p}T - c_{v}T_{0}[kJ/kg]The temperature T _{0}is inserted to provide the best fit between the real behavior of u and constant c_{v}relationship. For monatomic gases like He, Ne, Ar etc. Eq. (1.4) holds true with T_{0}=0 over a temperature range of a few thousand degrees. We will later see that T_{0}itself will drop out of the important equations to follow.
## 2. Derivation of relevant differential equations## 2.1 Conservation of mass and energyIn 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)
dm 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(m The term h (2.1.3)
dm 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 dQ
## 2.2 Compression SpaceThe ideal gas law states : (2.2.1)
p V As the crank angle varies, so do pressure p, mass m (2.2.2)
R T In order to facilitate sustitution of dm (2.2.3)
d(m We substitute now u
m Note that the temperature T (2.2.4)
m The newly introduced temperature
T
dT (2.2.6)
dm which we later substitute into the total mass balance, Eq. (2.1.1),
to eliminate from there the quantity dm ## 2.3 Expansion SpaceThe 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 V (2.3.4)
m
(2.3.6)
dm ## 2.4 Kooler SpaceThe ideal gas law states : (2.4.1)
p V Because volume and temperature are constant for this space, differentiation of Eq. (2.2.1) gives : (2.4.2)
dm which is all what we need for substitution
into the total mass balance, Eq. (2.1.1)
to eliminate dm
dQ Here the term (dm
(2.4.3)
dQ ## 2.5 Heater SpaceThe 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)
dm (2.5.2)
dQ in which the temperature T ## 2.6 RegeneratorThe 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 T The effective temperature of the regenerator is : (2.6.1)
T 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, V (2.6.2)
dm 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)
dQ ## 2.7 Putting it all togetherWe 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 (
V
V
V
dV
dV 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 T
In order for that to succeed we also have to know the temperatures
in the expansion and compression space, T We can do that by simultaneously integrating : (2.2.6)
dm and (2.3.6)
dm and calculate the temperatures in the compression and heater space using the ideal gas law for either space : (2.2.4)
m (2.3.4)
m ## 3. Sample Calculations and Optimization## 3.1 Some Observation on an Example Calculation |

Last revised: 12/14/14