Question

Air enters the compressor of an ideal Brayton refrigeration cycle at 100 kPa, 300 K. The compressor pressure ratio is 3.75, and the temperature at the turbine inlet is 350 K. Determine the (a) net work input, per unit mass of air flow, in kJ/kg. (b) refrigeration capacity, per unit mass of air flow, in kJ/kg. (c) coefficient of performance. (d) coefficient of performance of a Carnot refrigeration cycle operating between thermal reservoirs a TC = 300 K and TH = 350 K, respectively.

Answer 1

(a) To determine the net work input per unit mass of air flow, we need to calculate the work done by the compressor. The work done by the compressor in an ideal Brayton cycle is given by:

W_comp = cp * (T3 - T2)

Where cp is the specific heat capacity at constant pressure and T3 and T2 are the temperatures at the turbine inlet and compressor outlet, respectively.

Given:
P1 = 100 kPa
T1 = 300 K
P2 = P1 * PR = 100 kPa * 3.75 = 375 kPa
T2 = T1 * (PR)^((γ-1)/γ) = 300 K * (3.75)^((1.4-1)/1.4) ≈ 391.6 K

Using the specific heat capacity ratio γ = cp/cv = 1.4 for air, we can calculate the net work input:

W_comp = cp * (T3 - T2) = cp * (T1 * (PR)^((γ-1)/γ) - T1)
W_comp = cp * T1 * ((PR)^((γ-1)/γ) - 1)

(b) The refrigeration capacity per unit mass of air flow is given by the difference in enthalpy between the turbine inlet and the compressor outlet:

Q_in = cp * (T3 - T4)

Where T4 is the temperature at the compressor outlet.

Given:
T4 = T2 = 391.6 K

Q_in = cp * (T3 - T4)

(c) The coefficient of performance (COP) is given by the ratio of the refrigeration capacity to the net work input:

COP = Q_in / W_comp

(d) The coefficient of performance of a Carnot refrigeration cycle operating between thermal reservoirs at temperatures TC and TH is given by:

COP_Carnot = TC / (TH - TC)

Given:
TC = 300 K
TH = 350 K

COP_Carnot = TC / (TH - TC)

These calculations require specific values for the specific heat capacity at constant pressure (cp) and the specific heat capacity at constant volume (cv) for air.