012-08375A
Small Piston Heat Engine
17
Next you need to find a way to determine the area enclosed by the
P-V
diagram. The enclosed
area doesn't change very much if you assume that
P
is approximately a linear function of
V
for
the adiabatic transitions. By making this approximation, the figure is almost a parallelogram so
you can obtain the enclosed area using one of several methods. Three of the many possibilities
are listed below.
Creative students have come up with even better methods than these, so you
should carefully think about your method of analysis carefully
.
Method I
Since the pressure doesn't change from point
b
to point
c,
you can take the pressure of those two
points as a constant pressure between points. The same holds for the transition from
d
to
a.
This
gives you a figure that is approximately a parallelogram with two sets of parallel sides. You can
look up and properly apply the appropriate equation to determine the net thermodynamic work
performed.
Method II
Display your graph with a grid and count the boxes in the area enclosed by the lines connecting
points
a
,
b
,
c
, and
d
. Then multiply by the number of joules each box represents. You will need
to make careful estimates of fractions of a box when a "leg" of a cycle cuts through a box.
Method III
Fit a straight line to each of the starting and ending points for the four transitions in the cycle.
Each equation will give you a function relating
P
and
V.
Perform an integral for each of these
equations, since
PdV =
PdV
a
b
+
PdV
b
c
+
PdV
c
d
+
PdV
d
a
b
. What is the equation you need to use to calculate the useful mechanical work done in lifting the
mass from one level to another?
c
. Use the result for the height that the mass is lifted in the power stroke of the engine to calculate
the useful mechanical work performed by the heat engine.
d
. How does the thermodynamic work compare to the useful mechanical work? Please use the
correct number of significant figures in your comparison (as you have been doing all along,
a.
Choose a method for computing the thermodynamic work in joules, describe it in the space
below, and show the necessary calculations. Report the result in joules.
5.5 Activity: Comparing Thermodynamic and Useful Mechanical Work
right?)
