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Power Cycle for a Piston-Cylinder Device

A power cycle for a piston-cylinder device is described by the following four processes:

1→2 Isothermal compression from T1 = 300K, P1 = 100 kPa to P2 = 600 kPa.
2→3 Constant pressure heat addition until the temperature is T3 = 800K.
3→4 Isentropic expansion until the volume at state 4 equals the volume at state 1.
4→1 Constant volume heat rejection until the temperature is 300K.

Assume the working fluid is an ideal gas with constant specific heats and has properties as follows: CV = 0.600 kJ/(kg-K), CP = 0.900 kJ/(kg-K), R = 0.300 kJ/(kg-K), k = 1.500.

(a) Sketch the P-v diagram for this cycle
(b) Sketch the T-s diagram for this cycle
(c) Determine the heat rejected during process 1→2, in kJ/kg.
(d) Determine the heat added during process 2→3, in kJ/kg.
(e) Determine the total cycle expansion work done by the gas, in kJ/kg.

Could you please fill in this table, this will help organize all of the processes.
State P (kPa) T (K) V (m3/kg)
1 100 300
2 600 300
3 600 800
4

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SOLUTION

For an ideal gas following equations are applicable :

i) PV = nRT where n = quantity of gas in moles, P = Pressure, V = Volume, T = Absolute temperature, R = Universal gas constant = 8.31 J/mole/K ..........(1)

Or PV = m(R/M)T where m = mass of gas, P = Pressure, V = Volume, T = Absolute temperature, M = Molecular mass of the gas, R/M = Gas constant for unit mass of the given gas (different for different gases) ............(2)

ii) For adiabatic expansion or compression, PVk = Constant where k = Cp/Cv ......(3)

iii) From the first law of thermodynamics we have : dQ = dU + dW where dQ = Quantity of heat given to the system, dU = Change in the internal energy of the system(gas), dW = External work done by the gas. .............(4)

iv) Work done by the gas at constant pressure : dW = PdV .........(5)

v) Work done by the gas at constant volume = 0

vi) Heat transferred to/from the gas at constant volume = mcvdT where cv= specific heat of the gas at constant volume ......(6)

vii) Heat transferred to/from the gas at constant pressure = mcpdT where cp= specific heat of the gas at constant pressure ......(7)

viii) Cp - Cv = R where Cp = Molar specific heat of the gas at constant pressure (i.e. specific heat for one mole), Cv = Molar specific heat of the gas at constant volume ......(8)

Cp = Mcp and Cv = Mcv

Substituting in (8) we get : Mcp - Mcv = R or cp - cv = R/M .......(9)

(a) Sketch the P-v diagram for this cycle

1→2 Isothermal compression from T1 = 300K, P1 = 100 kPa to P2 = 600 kPa.

Assuming 1 kg of gas, m = 1 kg.

From (9) R/M = cp - cv = (0.9 - 0.6) kJ/(kg-K) = 0.3 kJ/(kg-K) = 0.3x103 J/(kg-K)

Substituting ...

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