Answer to Question #273278 in Mechanics | Relativity for LOUIE

Question

An ideal Stirling engine with air as the working fluid operates between specified pressure limits. The heat added to the cycle and the network produced by the cycle is to be determined.

Assumptions: Air is an ideal gas with constant specific heats. Properties: The properties of air at room temperature are R = 0.287 kPa·m3 /kg.K, cp = 1.005 kJ/kg·K, cv = 0.718 kJ/kg·K, and k = 1.4 

Answer 1

Properties:

The properties of air at room temperature are 

$R = 0.287 \large\frac{kPa·m^3}{kg·K}$

$c_p = 1.005 \large\frac{KJ}{kg·K}$

$c_v = 0.718 \large\frac{kJ}{kg·K}$

$k = 1.4$

Analysis: Applying the ideal gas equation to the isothermal process -4 gives

$P_4=P_3\frac{V_3}{V_4}=(50kPa)(12)=600kPa$

Since process 4-1 is one of constant volume

$T_1=T_4(\frac{P_1}{P_4})=(298K)(\frac{3600kPa}{600kPa})=1788K$

Adapting the first and work integral to the heat addition process gives

$q_{in}=W_{1-2}=RT_1ln\frac{V_2}{V_1}=(0.287\frac{KJ}{kg*K})(1788K)ln(12)=1275\frac{KJ}{kg}$

Similarly

$q_{out}=W_{3-4}=RT_3ln\frac{V_4}{V_3}=(0.287\frac{KJ}{kg*K})(298K)ln(\frac{1}{12})=212.5\frac{KJ}{kg}$

The net work is then 

$W_{net}=q_{in}-q_{out}=1275-212.5=1063\frac{KJ}{kg}$