Question 19.8: In the design work for a model single cylinder reciprocating...
In the design work for a model single cylinder reciprocating engine, the work done during the expansion stroke is to be represented as:
W=P \pi D^{2} C\left(1-Q^{n-1}\right) /(4(n-1))where
Q = C/(C + 2R)
and
C = M – R – L – H
The quantities are defined as follows:
W is the work done during the expansion stroke,
P is the cylinder pressure at start of stroke,
D is the piston diameter,
C is the length of the clearance volume,
n is the expansion stroke polytropic exponent,
R is the crank radius,
M is the distance from the crank center to the underside of the cylinder head,
L is the connecting rod length,
H is the distance from gudgeon pin to piston crown,
Q is the ratio of clearance to clearance-plus-swept volumes.
It is recognized that the independent quantities are subject to variability, and it is desired to determine the consequent extreme and probable variability in the work done.
If
P=20 \times 10^{5} \pm 0.5 \times 10^{5} N / m ^{2},n = 1.3 ± 0.05,
D=4 \times 10^{-2} \pm 25 \times 10^{-6} mR=2 \times 10^{-2} \pm 50 \times 10^{-6} m
M=10.5 \times 10^{-2} \pm 50 \times 10^{-6} m,
H=2 \times 10^{-2} \pm 25 \times 10^{-6} m,
L=6 \times 10^{-2} \pm 50 \times 10^{-6} m,
determine the extreme and probable (normal model) limits of the work (Ellis, 1990).
To help with your solution, certain partial derivatives have already been calculated.
\frac{\partial W}{\partial R}=-2599.46.\frac{\partial W}{\partial M}=2888.39.
\frac{\partial W}{\partial H}=-2888.39.
\frac{\partial W}{\partial L}=-2888.39.
Note: It may be useful to recall that if y=a^{x}, then d y / d x=x \log _{e}(a).
