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

$W12 = mRT_C ln\frac{P_1}{P_2}$

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

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

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

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

$P_2 = 76.23 kPa$