A
to B : volume constant
B
to C : adiabatic
C
to D : volume constant
D
to A :adiabatic
\[{{V}_{C}}=\,{{V}_{D}}=\,2{{V}_{A}}=\,2{{V}_{B}}\]
(a)
In which part of the cycle heat is supplied to the engine from outside?
(b)
It which part of the cycle beat is being given to die surrounding by the
engine?
(c)
What is the work done by the engine in one cycle? Write your answer in term of\[{{P}_{A}},\text{
}{{P}_{B}},{{V}_{A}}\].
(d)
What is die efficiency of the engine?
[[\[\gamma
={\scriptstyle{}^{5}/{}_{3}}\] for the gas], (\[{{C}_{\upsilon }}=\frac{3}{2}R\]
for one
mole)]
Answer:
(a) Heat is
supplied to increase the pressure at constant volume, so heat is supplied to
die engine from outside while going from A to B.
(b)
Heat is given to the surrounding while going from C to D.
(c)
\[{{W}_{AB}}=\,\int\limits_{A}^{B}{P\,dV}\,=\,0\] \[(\because
\,{{V}_{A}}=\,{{V}_{B}})\]
\[{{W}_{BC}}=\,\frac{1}{\gamma
-1}\,[{{P}_{B}}\,{{V}_{B}}-{{P}_{C}}{{V}_{C}}]\]
This
is work done during adiabatic expansion.
or \[{{W}_{BC}}\,=\frac{1}{1-\gamma
}\,[{{P}_{C}}{{V}_{C}}-\,{{P}_{B}}{{V}_{B}}]\]
\[=\frac{1}{1-\gamma
}\,[{{P}_{C}}\,\times 2{{V}_{B}}-{{P}_{B}}{{V}_{B}}]\]
\[=\,\frac{{{V}_{B}}}{1-\gamma
}\,[2{{P}_{C}}-\,{{P}_{B}}]\]
Now
\[{{P}_{B}}V_{B}^{\gamma }\,=\,{{P}_{C}}\,V_{C}^{\gamma }\]\[\therefore
\]\[\,{{P}_{C}}\,={{\left( \frac{{{V}_{B}}}{{{V}_{C}}} \right)}^{\gamma }}\]
\[=\,{{P}_{B}}{{\left(
\frac{1}{2} \right)}^{\gamma }}\]
\[={{2}^{-\gamma
}}\,{{P}_{B}}\]
\[\therefore
\] \[{{W}_{BC}}=\,\frac{{{P}_{B}}{{V}_{B}}}{1-\gamma }\,[{{2}^{-\gamma
+1}}-1]\] ?..
(i)
\[{{W}_{CD}}=\,\int\limits_{C}^{D}{PdV}\,=\,0\] \[(\because
\,\,{{V}_{C}}=\,{{V}_{D}})\]
\[{{W}_{DA}}=\,\frac{1}{\gamma
-1}\,[{{P}_{D}}\,{{V}_{D}}\,-\,{{P}_{A}}{{V}_{A}}]\,=\,\frac{1}{\gamma -1}\]
\[\left(
{{P}_{D}}\times \text{ }2{{V}_{A}}\text{ }{{P}_{A}}{{V}_{A}} \right)\]
\[=\,\frac{{{V}_{A}}}{\gamma
-1}\,[2{{P}_{D}}-{{P}_{A}}]\]
Also
\[{{P}_{D}}V_{D}^{\gamma }\,=\,{{P}_{A}}V_{A}^{\gamma }\] or \[{{P}_{D}}=\,{{\left(
\frac{{{V}_{A}}}{{{V}_{D}}} \right)}^{\gamma }}\,{{P}_{A}}\,\]
\[={{2}^{-\gamma
}}{{P}_{A}}\]
\[\therefore
\] \[{{W}_{DA}}=\,\frac{{{P}_{A}}{{V}_{A}}}{\gamma -1}\,[{{2}^{-\gamma
+1}}-1]\] ?..
(ii)
(d)
Head supplied (during the process A to B).
\[Q=\,{{C}_{\upsilon
}}\,({{T}_{B}}-\,{{T}_{A}})\,=\,\frac{3}{2}\,\,R\,({{T}_{B}}-\,{{T}_{A}})\]
But
\[{{P}_{A}}{{V}_{A}}=R{{T}_{A}}\] and \[{{P}_{B}}{{V}_{B}}=\text{
}R{{T}_{B}}\]
\[\therefore
\] \[Q=\frac{3}{2}\,\,({{P}_{B}}{{V}_{B}}-\,{{P}_{A}}{{V}_{A}})\,=\frac{3}{2}\,\,({{P}_{B}}-\,{{P}_{A}})\,{{V}_{A}}\]
\[(\because
\,{{V}_{A}}\,={{V}_{B}})\]
\[=1.5\text{
(}{{P}_{B}}{{P}_{A}})\text{ }{{V}_{A}}\]
Now
\[\eta \,=\frac{W}{Q}\,=\,\frac{0.55}{1.5}\,=\,37%\]
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