Calculate the change of entropy of 1 kg of air expanding polytropically in a cylinder behind a
piston from 7 bar and 600°C to 1.05 bar. The index of expansion is 1.25.
To calculate the change in entropy, we need to use the formula for entropy change in a polytropic process:
ΔS = Cp * ln(T2/T1) - R * ln(P2/P1)
Where:
ΔS = Change in entropy
Cp = Specific heat at constant pressure
T1 = Initial temperature
T2 = Final temperature
P1 = Initial pressure
P2 = Final pressure
R = Specific gas constant
Given:
Cp = 1005 J/kg⋅K (specific heat of air at constant pressure)
T1 = 600°C = 873K
T2 = ?
P1 = 7 bar = 700,000 Pa
P2 = 1.05 bar = 105,000 Pa
n = 1.25 (Index of expansion)
R = 287 J/kg⋅K (specific gas constant for air)
First, let's calculate the final temperature (T2) using the polytropic process equation:
P1 * V1^n = P2 * V2^n
Assuming the mass of air remains constant, we can rewrite this as:
P1 / T1^n = P2 / T2^n
T2^n = T1^n * (P2/P1)
T2 = (T1^n * (P2/P1))^(1/n)
T2 = (873^1.25 * (105,000/700,000))^(1/1.25)
T2 ≈ 452.21 K
Now let's calculate the change in entropy using the given formula:
ΔS = Cp * ln(T2/T1) - R * ln(P2/P1)
= 1005 * ln(452.21/873) - 287 * ln(105,000/700,000)
≈ -129 J/K
Therefore, the change in entropy for 1 kg of air expanding polytropically in a cylinder from 7 bar and 600°C to 1.05 bar, with an index of expansion of 1.25, is approximately -129 J/K.