Fisica

SCHMIDT THEORY FOR STIRLING ENGINES
Tentative version on January 20, 1997
KOICHI HIRATA
e-mail: khirata@gem.bekkoame.ne.jp

1. INTRODUCTION
The Schmidt theory is one of the isothermal calculation methods for Stirling engines. It is the most simple method and very useful during Stirling engine development. 
This theory is based on the isothermal expansion and compression of an ideal gas.2. ASSUMPTION OF SCHMIDT THEORY
The performance of the engine can be calculated a P-V diagram. The volume in the engine is easily calculated by using the internal geometry. When the volume, mass of the working gas and the temperature is decided, the pressure is calculated using an ideal gas method as shown in equation (1).
(1)
The engine pressure can be calculated under following assumptions:(a) There is no pressure loss in the heat-exchangers and there are no internal pressure differences.
(b) The expansion process and the compression process changes isothermal.
(c) Conditions of the working gas is changed as an ideal gas.
(d) There is a perfect regeneration.
(e) The expansion dead space maintains the expansion gas temperature - TE, the compression dead space maintains thecompression gas temperature - TC during the cycle.
(f) The regenerator gas temperature is an average of the expansion gas temperature - TE and the compression gas temperature - TC.
(g) The expansion space - VE and the compression space - VC changes according a sine curves.
Table 1 shows symbols used the Schmidt Theory.
Table 1 Symbols

3. ALPHA-TYPE STIRLING ENGINE
Figure 1 shows the calculationmodel of Alpha-type Stirling engine.

Fig. 1 Alpha-type Stirling Engine
The volumes of the expansion- and compression cylinder at a given crank angle are determined at first. The momental volumes is described with a crank angle - x. This crank angle is defined as x=0 when the expansion piston is located the most top position (top dead point). 
The momental expansion volume - VE is describedin equation (2) with a swept volume of the expansion piston - VSE, an expansion dead volume - VDE under the condition of assumption (g).
 (2)
The momental compression volume - VC is found in equation (3) with a swept volume of the compression piston - VSC, a compression dead volume - VDC and a phase angle - dx.
 (3)
The total momental volume is calculated in equation (4).
 (4)
By theassumptions (a), (b) and (c), the total mass in the engine - m is calculated using the engine pressure - P, each temperature - T , each volume - V and the gas constant - R.
 (5)
The temperature ratio - t, a swept volume ratio - v and other dead volume ratios are found using the following equations.
 (6)
 (7)
 (8)
 (9)
 (10)
The regenerator temperature - TR is calculated in equation (11), by usingthe assumption (f).
 (11)
When equation (5) is changed using equation (6)-(10), the total gas mass - m is described in the next equation.
 (12)
Equation (12) is changed in equation (13), using equation (2) and (3).
 (13)
Now;
 (14)
 (15)
 (16)
The engine pressure - P is defined as a next equation using equation (13).
 (17)
The mean pressure - Pmean can be calculated as follows:
 (18)c is defined in the next equation.
 (19)
As a result, the engine pressure - P, based the mean engine pressure - Pmean is calculated in equation (20).
 (20)
On the other hand, in the case of equation (17), when cos(x-a)=-1, the engine pressure - P becomes the minimum pressure - Pmin, the next equation is introduced.
 (21)
Therefore, the engine pressure - P, based the minimum pressure -Pmin is described in equation (22).
 (22)
Similarly, when cos(x-a)=1, the engine pressure - P becomes the maximum pressure - Pmax. The following equation is introduced.
 (23)
The P-V diagram of Alpha-type Stirling engine can be made with above equations.

4. BETA-TYPE STIRLING ENGINE
Similarly, the equations for Beta-type Stirling engine are declared. Figure 2 shows a calculation model of a...