The net heat received in the cycle is 3.15 kJ.
Given: Initial pressure (P1) = 7 bar ,Initial temperature (T1) = 206°C = 479, Initial volume (V1) = 0.03 m³ ,Final volume (V2) = 0.09 m³ ,Polytropic index (n) = 1.5 ,Gas constant (R) = 0.287 kJ/kgK ,Specific heat at constant volume (Cv) = 0.713 kJ/kgK
Step 1: Constant pressure process (1-2)
In a constant pressure process, the pressure remains constant throughout the process. Therefore, P1 = P2 = 7 bar.
Using the ideal gas law, we can find the initial moles of air:
n = PV/RT = (7 bar * 0.03 m³)/(0.287 kJ/kgK * 479 K) = 0.0149 kmol
Step 2: Polytropic process (2-3)
In a polytropic process, the relationship between pressure and volume is given by:
PV^n = constant
Substituting the given values, we get:
7 bar * (0.09 m³)^1.5 = constant
Solving for the constant, we get:
constant = 0.5226 bar m³^(1.5)
Using this constant, we can find the pressure at point 3 (P3):
P3 = 0.5226 bar m³^(1.5) / (0.03 m³)^1.5 = 4.67 bar
Step 3: Constant temperature process (3-4)
In a constant temperature process, the temperature remains constant throughout the process. Therefore, T3 = T4.
Using the ideal gas law, we can find the volume at point 4 (V4):
V4 = nRT/T3 = 0.0149 kmol * 0.287 kJ/kgK * T3 / T3 = 0.0149 m³
Step 4: Calculating heat received and heat rejected
Heat received (Q12) in the constant pressure process (1-2):
Q12 = nCv(T2 - T1) = 0.0149 kmol * 0.713 kJ/kgK * (479 K - 206 K) = 3.15 kJ
Heat rejected (Q34) in the constant temperature process (3-4):
Q34 = nCp(T4 - T3) = 0.0149 kmol * 1.005 kJ/kgK * (479 K - 479 K) = 0 kJ
Therefore, the net heat received in the cycle is 3.15 kJ.