Question

Two kg of air within a piston cylinder assembly execute a carnot power cycle with max and min temp of 750K and 300K, respectively. The heat transfer to the air during the isothermal expansion is 60KJ. At the end of the isothermal expansion the volume is 0.4m³. Assuming the ideal gas model for air, determine

a)the thermal efficiency

b)the pressure and volume at the beginning of the isothermal expansion in kpa and m³ respectively.

c) the work and heat transfer for each of the four processes in KJ.

***This question is from "Fundamentals of engineering thermodynamics 8th edition" (chapter 5, 83P)

Answer 1

(a) Thermal efficiency = (Tmax - Tmin) / Tmax = (750 K - 300 K) / 750 K = 0.6 = 60.0 %

(b):

(b):

Given Q1 = 60.0 KJ

V2 = 0.40 m3

For isothermal process, dU = 0. Hence dQ = dW

Hence for state-1 to 2: Q1 = W1 = 60.0 KJ

mass os air = 2 Kg = 2x10³ g

Molecular mass of air = 28.97 g/mol

Hence moles of air, n = 2x10³ g /  28.97 g/mol = 69.04 mol

Now for isothermal process, W1 = 60.0 KJ = 60.0 x 10³ J = nRT(max) x ln(V2/V1)

=>  ln(V2/V1) = 60.0 x 10³ J / 69.04 mol x 8.314JK^-1mol^-1 x 750K = 0.1394

=> 0.40 m3 / V1 = exp(0.1394)

=> V1 = 0.348 m3 (answer)

Hence volume at the beginning of isothermal expansion = 0.348 m3

Now P2 = nRT(max) / V2 = 69.04 mol x 8.314JK^-1mol^-1 x 750K / 0.40 m3 = 1076.2 KPa

For isothermal expansion, P1V1 = P2V2

=> P1 = P2V2 / V1 = 1076.2 KPa x 0.40 m3 / 0.348 m3 = 1237 KPa (answer)

Hence pressure at the beginning of isothermal expansion =1237 KPa

(c): For 1 --- >2 (isothermal expansion)

Work done(W1) = Heat transfer(Q1) = 60.0 KJ (answer)

For 2--->3 (adiabetic expansion):

Heat transfer(Q2) = 0

work done, W2 = m x (u2 - u3) = 2 kg x (551.99 - 214.07) KJ/Kg = 676 KJ

For 3--->4: (isothermal compression)

Q3 = W3 = - 60.0 KJ x (300 K / 750K) = - 24.0 KJ

For 4--->1: (adiabetic compression)

Q4 = 0

W4 = - 676 KJ