Question

Air within a piston-cylinder assembly executes a Carnot heat pump cycle, . For the cycle, TH 5 600 K and TC 5 300 K. The energy rejected by heat transfer at 600 K has a magnitude of 250 kJ per kg of air. The pressure at the start of the isothermal expansion is 325 kPa. Assuming the ideal gas model for the air, determine (a) the magnitude of the net work input, in kJ per kg of air, and (b) the pressure at the end of the isothermal expansion, in kPa.

Answer 1

Answer:

a) w = 125 kJ/kg

b) P = 76.23 kPa

Explanation:

Given details:

T_H = 600 k

T_c = 300 k

Rjected energy Q 250 kg

Pressure is 325 kPa

a) coefficient of performance is given as

$COP = \frac{Q}{W} = \frac{T_H}{T_H - T_C}$

$\frac{Q}{W} = \frac{600}{600 - 300}$

$\frac{Q}{W} =  = 2$

SOLVING FOR W

$W = \frac{250}{2} = 125 kJ/kg$

b) from figure

$Q12 - W12 = \Delta U$

$\Delta U = 0$  due to same temp at point 1 and 2

Q12 = W12

$W12 =  mRT_C ln\frac{V_2}{V_1}$

and we know that

P1V1 =P2V2 so

$W12 =  mRT_C ln\frac{P_1}{P_2}$

$125 = 0.287 \times 300 ln\frac{P_1}{P_2}$

$ln\frac{P_1}{P_2} = 1.45$

$\frac{P_1}{P_2} = e^{1.45}$

$P_2 = \frac{325}{e^{1.45}}$

$P_2 = 76.23 kPa$

Answer 2

A) The magnitude of the net work input, in kJ per kg of air is; W = 125 kJ/kg

B) The pressure at the end of the isothermal expansion is; P₂  = 76.1 kPa

We are given;

T_H = 600 K

T_C = 300 K

Q = 250 kJ/kg

P₁ = 325 kPa

A) The magnitude of the net work input can be gotten from the formula for Coefficient of performance;

COP = Q/W = T_H/(T_H - T_C)

Thus;

250/W =600/(600 - 300)

250/W = 2

W = 250/2

W = 125 kJ/kg

B) From first law of thermodynamics, we know that;

Q - W = ΔU

However, we know that ΔU = 0 because the system is isothermal.

Thus;

Q - W = 0

Q = W = 125 kJ/kg

Formula for W is also;

W = mRT_c(In (P₁/P₂))

where;

R is gas constant of air = 0.287 kJ/kg.K

P₁ is pressure at the start

P₂ is pressure at the end

Thus;

125 = 0.287 × 300 × In(325/P₂)

In(325/P₂) = 125/(0.287 × 300)

In(325/P₂) = 1.4518

325/P₂ = e^(1.4518)

325/P₂ = 4.2708

P₂  = 325/4.2708

P₂  = 76.1 kPa

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