Answer:
1) The power developed by the engine is 14705.7739 kW
2) The thermal efficiency is approximately 61.5%
Explanation:
The given parameters are;
P₁ = 95 kPa
T₁ = 22°C
V₁ = 3.17 liters
The cutoff ratio = 2.5
Displacement volume = 3 liters
The number of times the cycle is executed per minute = 1000 times per minute
We have;
The displacement volume = V₁ - V₂ = 3 l
V₁ = 3.17 l
V₂ = 3 - 3.17 = 0.17 l
Compression ratio = V₁/V₂ = 3.17/0.17 ≈ 18.65
P₂/P₁ = P₂/(95 kPa) = (V₁/V₂)^(k) = 18.65^1.4
P₂ = (95×18.65^(1.4)) ≈ 5710.5 kPa
T₂/T₁ = (V₁/V₂)^(k - 1)
T₂/(295 K)= (18.65)^(1.4 - 1)
T₂ = 295 * (18.65)^(1.4 - 1) = 950.81 K
The cutoff ratio = V₃/V₂ = 2.5
T₃ = T₂ × V₃/V₂ = 2.5 * 950.81 K = 2377.025 K
=
×(T₃ - T₂) = 1.006 × (2377.025 - 950.81) = 1,434.77 kJ/kg
T₄ = T₃ × (V₃/V₄)^(k-1) =
Therefore,
![T_4 = T_3 \times \left (\dfrac{r_c}{r} \right )^{k - 1} = 2377.025 \times \left( \dfrac{2.5}{18.65} \right )^{1.4 - 1} \approx 1064 \ K](https://tex.z-dn.net/?f=T_4%20%3D%20%20T_3%20%5Ctimes%20%5Cleft%20%28%5Cdfrac%7Br_c%7D%7Br%7D%20%5Cright%20%29%5E%7Bk%20-%201%7D%20%3D%202377.025%20%5Ctimes%20%5Cleft%28%20%5Cdfrac%7B2.5%7D%7B18.65%7D%20%5Cright%20%29%5E%7B1.4%20-%201%7D%20%5Capprox%201064%20%5C%20K)
T₄ ≈ 1064 K
= ![-C_v \times (T_4 - T_1)](https://tex.z-dn.net/?f=-C_v%20%5Ctimes%20%28T_4%20-%20T_1%29)
![C_v = C_p/k = 1.006/1.4 \approx 0.7186 \ kJ/kg](https://tex.z-dn.net/?f=C_v%20%3D%20C_p%2Fk%20%3D%201.006%2F1.4%20%5Capprox%200.7186%20%5C%20kJ%2Fkg)
∴
= 0.7186×(1064 - 295) = 552.6034 kJ/kg
1) The net work =
-
= 1,434.77 kJ/kg - 552.6034 kJ/kg ≈ 882.17 kJ/kg
The number of cycle per minute = 1000 rpm
The number of cycle per minute = 1000 rpm/60 = 16.67 cycles per second
The power developed by the engine = The number of cycles per second × The net work of the engine
Therefore;
The power developed by the engine = 16.67 cycles/second × 882.17 kJ/kg
The power developed by the engine = 14705.7739 kW
2) Efficiency,
, is given as follows;
![\eta _{th} = \dfrac{Q_{in}-Q_{out}}{Q_{in}} \times 100 = 1 - \dfrac{Q_{out}}{Q_{in}} \times 100= 1 - \dfrac{552.6034}{1434.77}\times 100 \approx 61.5\%](https://tex.z-dn.net/?f=%5Ceta%20_%7Bth%7D%20%3D%20%5Cdfrac%7BQ_%7Bin%7D-Q_%7Bout%7D%7D%7BQ_%7Bin%7D%7D%20%5Ctimes%20100%20%3D%201%20-%20%5Cdfrac%7BQ_%7Bout%7D%7D%7BQ_%7Bin%7D%7D%20%5Ctimes%20100%3D%201%20-%20%5Cdfrac%7B552.6034%7D%7B1434.77%7D%5Ctimes%20100%20%5Capprox%2061.5%5C%25)
Therefore, the thermal efficiency ≈ 61.5%.