The Schmidt Analysis
In the previous section we derived
the basic set of equations which describe the Ideal Isothermal
model, as shown in the following table.
Gustav Schmidt of the German Polytechnic Institute of Prague Published
an analysis in 1871 in which he obtained closed form solutions
of these equations for the special case of sinusoidal volume variations
of the working spaces with respect to the cycle angle q.
Consider the following diagram showing the volume variations of
the compression and expansion spaces (Vc and Ve) over a single
cycle. Notice the phase advance angle a
of the expansion space volume variation with respect to the compression
space volume variation:
The sinusoidal volume variations
of the compression and expansion spaces are respectively as follows:
Vc = Vclc + Vswc (1 + cos q)
/ 2
Ve = Vcle + Vswe (1 + cos(q
+ a)) / 2
where Vcl and Vsw represent respectively clearence and swept
volumes, and q is the cycle angle.
Substituting for Vc and Ve in the pressure equation above and
simplifying we obtain
In order to simplify the pressure equation we now consider
a trigonometric substitution of b and
c as defined by the following right-angled triangle
Substituting for b and c in the
pressure equation above and simplifying
where
f = q
+ b
b = c / s
The maximum and minimum values of pressure can now be evaluated
for the extreme values of cos f
The average pressure over the cycle is given by
From tables of integrals, this reduces to
This equation is the most convenient way of relating the total
mass of working gas in the cycle to the more conveniently specified
mean operating pressure.
The net work done by the engine is the sum of the work done
by the compression and expansion spaces. Over a complete cycle
W = Wc + We
The volume derivatives are obtained by differentiating Vc and
Ve above
Substituting these and the pressure equation into the equations
for Wc and We
The solution of these integrals requires the judicious use
of tables of integrals and is done in the book by Urieli &
Berchowitz, "Stirling Cycle Machine Analysis", Adam
Hilger 1984. The book itself is out of print, however the relevant
appendix in this book that deals with the Schmidt analysis has
been placed on the web by Global
Cooling, and can be downloaded
in Acrobat pdf format. Finally we obtain
Schmidt
Analysis - Equation Summary
back to Isothermal
Analysis