Question

An ideal Stirling cycle filled with air uses a 75 F energy reservoir as a sink. The engine is designed to so that the maximum air volume is 0.5 ft3 , the minimum air volume is 0.05 ft3 , and the minimum pressure is 15 psi. It is to be operated such that the engine produces 2 Btu of net work when 5 Btu of heat are transferred externally to the to the engine. Determine the temperature of the energy source, the amount of air contained inside the engine, and the maximum air pressure during the cycle.

Answer 1

Answer:

Explanation:

For the given stirling circle with air as the working fluid, temperature, pressure, and volume i^th stage are $T_t, P_i$ and $V_i$ respectively

Obtain the properties of air at room temperature from molar mass, gas constant and critical point properties table

Gas constant R = 0.3704 psia ft³ / ibm.R

Express the thermal efficiency of an ideal stirling engine

$n_{th}=\frac{W_{net}}{Q_{in}} =1-\frac{T_L}{T_H}$

Here, network done by the stirling engine is $W_{net}$

heat input to the engine is $Q_{in}$

Temperature of the heat source is $T_H$

Temperature of the sink is $T_L$

substitute 535 for  $T_L$ , 2Btu for $W_{net}$ , 5Btu for $Q_{in}$  to find $T_H$

$\frac{2Btu}{5Btu}=1-\frac{535R}{T_H} \\\\T_H=\frac{535}{0.6}\\\\=891.7R$

Hence , the temperature of the source of energy reservoir is 891.7R

Express the idea gas equation for air at state 3 of the cycle

$m=\frac{P_3V_3}{RT_3}$

substitute

$15psia \ \ for \ \ p_3 \\\\0.5ft^3 \ \ for \ \ V_3 \\\\ 0.3704 psia.ft^/ibm.R \ \ for \ \ R \\\\535R \ \ for \ \ R$

$m=\frac{(15)(0.5)}{(0.3785)(535)} \\\\=0.3785Ibm$

Hence ,the amount of air conditioned in the engine is 0.3785 Ibm

Express the ideal gas equation for air at state 1 of the cycle

$P_1=\frac{mRT_1}{V_1}$

substitute 0.03785 for m,

$0.3704 psia ft^3/Ibm.R for R,\\\\ 891.7R for T_1 \ \ and \ \ 0.06ft^3 for V_1$

$P_1=\frac{(0.03785)(0.3704)(891.7)}{(0.05)} \\\\=250 \texttt{psia}$

Hence, the maximum air pressure during the cycle is 250 psia