Answer:
Explanation:To solve this problem, we'll use the given information and the ideal gas law to find the answers step by step. The ideal gas law is not directly given, but we can re-arrange the equation \(PV^{1.25} = \text{Constant}\) to \(P = \text{Constant} \times V^{-1.25}\), which is essentially a modified version of the ideal gas law \(PV = mRT\).
Given:
- Rate of compression (\(\dot{m}\)) = 9 kg/min
- Initial pressure (\(P_1\)) = 3.45 bar
- Initial temperature (\(T_1\)) = 38°C = 38 + 273.15 K
- Final pressure (\(P_2\)) = 69 bar
- Gas constant (\(R_0\)) = 8.314 kJ/kgK
- Compression law: \(PV^{1.25} = \text{Constant}\)
Let's solve the three parts step by step:
(i) *Required Cylinder Volume (V):
We can use the ideal gas law to relate the initial and final states:
\[\frac{P_1V_1^{1.25}}{T_1} = \frac{P_2V_2^{1.25}}{T_2}\]
Since the process is isentropic (no heat exchange), \(T_1 = T_2\).
Also, \(V_1 = \frac{\dot{m}}{\rho_1}\), where \(\rho_1\) is the initial density.
Rearranging and solving for \(V_2\):
\[V_2 = V_1 \left(\frac{P_1}{P_2}\right)^{0.8}\]
(ii) *Temperature after Compression:
Since the process is isentropic, \(T_2 = T_1 = 38 + 273.15\) K.
(iii) *Work Done per Second:
The work done per unit mass is given by:
\[w = c_v (T_2 - T_1)\]
where \(c_v\) is the specific heat at constant volume.
The work done per second (\(W\)) can be calculated as:
\[W = \dot{m} \times w\]
Substitute \(w\) and \(\dot{m}\) to get the value of \(W\).
Remember to convert pressures to absolute (in Pa) and temperatures to Kelvin for consistent units.