Answer:
(a) The amount of heat transferred to the air,
is 215.5077 kJ/kg
(b) The net work output,
, is 308.07 kJ/kg
(c) The thermal efficiency is 58.8%
(d) The Mean Effective Pressure, MEP, is 393.209 kPa
Explanation:
(a) The assumptions made are;
= 1.005 kJ/(kg·K),
= 0.718 kJ/(kg·K), R = 0.287 kJ/(kg·K),
Process 1 to 2 is isentropic compression, therefore;
![T_{2}= T_{1}\left (\dfrac{v_{1}}{v_{2}} \right )^{k-1} = 300.15\times 9.2^{0.4} = 729.21 \, K](https://tex.z-dn.net/?f=T_%7B2%7D%3D%20T_%7B1%7D%5Cleft%20%28%5Cdfrac%7Bv_%7B1%7D%7D%7Bv_%7B2%7D%7D%20%20%5Cright%20%29%5E%7Bk-1%7D%20%3D%20300.15%5Ctimes%209.2%5E%7B0.4%7D%20%3D%20729.21%20%5C%2C%20K)
From;
![\dfrac{p_{1}\times v_{1}}{T_{1}} = \dfrac{p_{2}\times v_{2}}{T_{2} }](https://tex.z-dn.net/?f=%5Cdfrac%7Bp_%7B1%7D%5Ctimes%20v_%7B1%7D%7D%7BT_%7B1%7D%7D%20%3D%20%5Cdfrac%7Bp_%7B2%7D%5Ctimes%20v_%7B2%7D%7D%7BT_%7B2%7D%20%7D)
We have;
![p_{2} = \dfrac{p_{1}\times v_{1}\times T_{2}}{T_{1} \times v_{2}} = \dfrac{98\times 9.2\times 729.21}{300.15 } = 2190.43 \, kPa](https://tex.z-dn.net/?f=p_%7B2%7D%20%3D%20%5Cdfrac%7Bp_%7B1%7D%5Ctimes%20v_%7B1%7D%5Ctimes%20T_%7B2%7D%7D%7BT_%7B1%7D%20%5Ctimes%20v_%7B2%7D%7D%20%3D%20%5Cdfrac%7B98%5Ctimes%209.2%5Ctimes%20729.21%7D%7B300.15%20%7D%20%3D%202190.43%20%5C%2C%20kPa)
Process 2 to 3 is reversible constant volume heating, therefore;
![\dfrac{p_3}{T_3} =\dfrac{p_2}{T_2}](https://tex.z-dn.net/?f=%5Cdfrac%7Bp_3%7D%7BT_3%7D%20%3D%5Cdfrac%7Bp_2%7D%7BT_2%7D)
p₃ = 2 × p₂ = 2 × 2190.43 = 4380.86 kPa
![T_3 = \dfrac{p_3 \times T_2}{p_2} =\dfrac{4380.86 \times 729.21}{2190.43} = 1458.42 \, K](https://tex.z-dn.net/?f=T_3%20%3D%20%5Cdfrac%7Bp_3%20%5Ctimes%20T_2%7D%7Bp_2%7D%20%3D%5Cdfrac%7B4380.86%20%20%5Ctimes%20729.21%7D%7B2190.43%7D%20%3D%201458.42%20%5C%2C%20K)
Process 3 to 4 is isentropic expansion, therefore;
![T_{3}= T_{4}\left (\dfrac{v_{4}}{v_{3}} \right )^{k-1}](https://tex.z-dn.net/?f=T_%7B3%7D%3D%20T_%7B4%7D%5Cleft%20%28%5Cdfrac%7Bv_%7B4%7D%7D%7Bv_%7B3%7D%7D%20%20%5Cright%20%29%5E%7Bk-1%7D)
![1458.42= T_{4} \times \left (9.2 \right )^{0.4}](https://tex.z-dn.net/?f=1458.42%3D%20T_%7B4%7D%20%5Ctimes%20%5Cleft%20%289.2%20%5Cright%20%29%5E%7B0.4%7D)
![T_4 = \dfrac{1458.42}{(9.2)^{0.4}} = 600.3 \, K](https://tex.z-dn.net/?f=T_4%20%3D%20%5Cdfrac%7B1458.42%7D%7B%289.2%29%5E%7B0.4%7D%7D%20%20%3D%20600.3%20%5C%2C%20K)
![q_{out} = m \times c_v \times (T_4 - T_1) = 0.718 \times (600.3 - 300.15) = 215.5077 \, kJ/kg](https://tex.z-dn.net/?f=q_%7Bout%7D%20%3D%20m%20%5Ctimes%20c_v%20%5Ctimes%20%28T_4%20-%20T_1%29%20%3D%200.718%20%20%5Ctimes%20%28600.3%20-%20300.15%29%20%3D%20215.5077%20%5C%2C%20kJ%2Fkg)
The amount of heat transferred to the air,
= 215.5077 kJ/kg
(b) The net work output,
, is found as follows;
![W_{net} = q_{in} - q_{out}](https://tex.z-dn.net/?f=W_%7Bnet%7D%20%3D%20q_%7Bin%7D%20-%20q_%7Bout%7D)
![q_{in} = m \times c_v \times (T_3 - T_2) = 0.718 \times (1458.42 - 729.21) = 523.574 \, kJ/kg](https://tex.z-dn.net/?f=q_%7Bin%7D%20%3D%20m%20%5Ctimes%20c_v%20%5Ctimes%20%28T_3%20-%20T_2%29%20%3D%200.718%20%20%5Ctimes%20%281458.42%20-%20729.21%29%20%3D%20523.574%20%5C%2C%20kJ%2Fkg)
![\therefore W_{net} = 523.574 - 215.5077 = 308.07 \, kJ/kg](https://tex.z-dn.net/?f=%5Ctherefore%20W_%7Bnet%7D%20%3D%20523.574%20-%20215.5077%20%3D%20308.07%20%5C%2C%20kJ%2Fkg)
(c) The thermal efficiency is given by the relation;
![\eta_{th} = \dfrac{W_{net}}{q_{in}} \times 100= \dfrac{308.07}{523.574} \times 100= 58.8\%](https://tex.z-dn.net/?f=%5Ceta_%7Bth%7D%20%3D%20%5Cdfrac%7BW_%7Bnet%7D%7D%7Bq_%7Bin%7D%7D%20%5Ctimes%20100%3D%20%20%5Cdfrac%7B308.07%7D%7B523.574%7D%20%5Ctimes%20100%3D%2058.8%5C%25)
(d) From the general gas equation, we have;
![V_{1} = \dfrac{m\times R\times T_{1}}{p_{1}} = \dfrac{1\times 0.287\times 300.15}{98} =0.897\, m^{3}/kg](https://tex.z-dn.net/?f=V_%7B1%7D%20%3D%20%5Cdfrac%7Bm%5Ctimes%20R%5Ctimes%20T_%7B1%7D%7D%7Bp_%7B1%7D%7D%20%3D%20%5Cdfrac%7B1%5Ctimes%200.287%5Ctimes%20300.15%7D%7B98%7D%20%3D0.897%5C%2C%20m%5E%7B3%7D%2Fkg)
The Mean Effective Pressure, MEP, is given as follows;
![MEP =\dfrac{W_{net}}{V_1 - V_2} = \dfrac{W_{net}}{V_1 \times (1- 1/r)}= \dfrac{308.07}{0.897\times (1- 1/9.2)} = 393.209 \, kPa](https://tex.z-dn.net/?f=MEP%20%3D%5Cdfrac%7BW_%7Bnet%7D%7D%7BV_1%20-%20V_2%7D%20%3D%20%5Cdfrac%7BW_%7Bnet%7D%7D%7BV_1%20%5Ctimes%20%281-%201%2Fr%29%7D%3D%20%5Cdfrac%7B308.07%7D%7B0.897%5Ctimes%20%281-%201%2F9.2%29%7D%20%3D%20393.209%20%5C%2C%20kPa)
The Mean Effective Pressure, MEP = 393.209 kPa.