Question

Find the final temperature of the gas which expands in a polytropic process, with \[n = 1.2\] from \[{T_i}{\text{ }} = 500\,K\] , \[10bar\] to \[0{\text{ }}bar\] .
A. $335.3\,K$
B. $353.3\,K$
C. $332.5\,K$
D. $354.2\,K$

Answer 1

Hint:The term "polytropic" was first used to define any reversible process on any open or closed system of gas or vapour that involves both heat and work transport and maintains a specific combination of attributes constant throughout the operation.

Complete step by step answer:
A polytropic process is a reversible process involving a gas or vapour in a closed or open system that involves both heat and work transmission and maintains a consistent combination of attributes. It is calculated as,
\[PVn = C\]
where \[P\] is pressure, \[V\] is volume, \[n\] is the polytropic index, and \[C\] is a constant.

A polytropic process with a polytropic exponent \[n\] fulfils the following condition:
$P{V^n} = constant = C$
Assuming ideal gas behaviour, volume \[V\] can be replaced by \[V = \dfrac{{nRT}}{P}\].So,
$P{\left( {\dfrac{{nRT}}{P}} \right)^n} = C \\
\Rightarrow {P^{1 - n}}{T^n} = C{\left( {nR} \right)^{ - n}} = constant\;\left( {if{\text{ }}n{\text{ }}is{\text{ }}constant} \right)\; \\ $
As a result, pressure and temperature in the initial state (1) and end state (2) are connected as follows.
$P_i^{1 - n}T_i^n = P_f^{1 - n}T_f^n$
As a result, the final temperature is given by
\[{T_f} = {T_i}\,\left[ {\dfrac{{{P_i}}}{{{P_f}}}} \right]{\,^{\dfrac{{(1 - n)}}{n}}}\]
The initial temperature is now \[T = 500{\text{ }}K\] .
The initial and final pressure pressures are as follows:
${P_i} = {P_{gauge}} + {P_{atm}} \\
\Rightarrow {P_i} = 10\,bar\, + 1\,bar \\
\Rightarrow {P_i} = 11\,bar \\
\Rightarrow {P_i} = 1100\,kPa$
$\Rightarrow {P_f} = {P_{gauge}} + {P_{atm}} \\
\Rightarrow {P_f} = 0\,bar + 1\,bar \\
\Rightarrow {P_f} = 1\,bar \\
\Rightarrow {P_f} = 100\,kPa$
So for \[n = 1.2\]
${T_f} = 500{\left[ {\dfrac{{1100}}{{100}}} \right]^{\dfrac{{\left( {1 - 1.2} \right)}}{{1.2}}}} \\
\therefore {T_f} = 335.3K \\ $
Hence, the correct option is A.

Note: To some extent, gas compressors and gas turbines operate under a polytropic process. The polytropic process implies constant entropy, whereas in actuality, minor entropy changes occur during gas compression or expansion. When calculating how much work is necessary to compress a gas, the most accurate way is usually assuming it is polytropic and then introducing efficiency to account for the fact that it is not truly polytropic and there are variations in entropy.