Air Refrigeration Cycles: Reversed Brayton and Aircraft Air-Cycle Systems

A calculation-ready guide to air refrigeration: the ideal reversed Brayton (Bell–Coleman) cycle, real component effects, aircraft configurations (simple, bootstrap, regenerative), heat-exchanger modeling, COP equations, sizing for load, and worked examples. Math is rendered via MathJax.

Symbols and assumptions

Thermophysical

Ideal-gas air: \(R = 287\,\text{J/(kg\cdot K)}\), \(k = \gamma = c_p/c_v\) (use \(k \approx 1.4\) if constant)
Heat capacities: \(c_p \approx 1.004\,\text{kJ/(kg\cdot K)}\) (moderate variation with \(T\))
Pressure ratio: \(r_p = \dfrac{p_2}{p_1}\)

States and processes

State 1: Low-pressure air entering compressor (from cooled space)
State 2: Compressor discharge
State 3: After cooler/heat exchanger (rejecting heat to ambient/ram air)
State 4: Turbine discharge supplying refrigeration; \(4 \rightarrow 1\) absorbs heat from space

Efficiencies

Compressor isentropic: \[ \eta_c = \frac{T_{2s} - T_1}{T_2 - T_1} \]
Turbine isentropic: \[ \eta_t = \frac{T_3 - T_4}{T_3 - T_{4s}} \]
Heat-exchanger effectiveness: \[ \varepsilon = \frac{T_{\text{hot,in}} - T_{\text{hot,out}}}{T_{\text{hot,in}} - T_{\text{cold,in}}} \]

Ideal reversed Brayton (Bell–Coleman) cycle

Isentropic relations (ideal gas, constant k)

Compression: \[ \frac{T_{2s}}{T_1} = r_p^{\tfrac{k-1}{k}}, \qquad \frac{p_2}{p_1} = r_p \]
Expansion: \[ \frac{T_{4s}}{T_3} = \left(\frac{p_1}{p_2}\right)^{\tfrac{k-1}{k}} = r_p^{-\tfrac{k-1}{k}} \]

Heat and work per unit mass

Refrigeration effect: \[ q_L = c_p\,(T_1 - T_4) \]
Compressor work (ideal): \[ w_{c,ideal} = c_p\,(T_{2s} - T_1) \]
Turbine work (ideal): \[ w_{t,ideal} = c_p\,(T_3 - T_{4s}) \]
Net work input: \[ w_{net} = w_c - w_t \]
Coefficient of performance: \[ \text{COP}_R = \frac{q_L}{w_{net}} = \frac{c_p\,(T_1 - T_4)}{c_p\,(T_2 - T_1) - c_p\,(T_3 - T_4)} \]

Real effects and component efficiencies

Compressor outlet temperature: \[ T_2 = T_1 + \frac{T_{2s} - T_1}{\eta_c}, \qquad T_{2s} = T_1\,r_p^{\tfrac{k-1}{k}} \]
Turbine outlet temperature: \[ T_4 = T_3 - \eta_t\,(T_3 - T_{4s}), \qquad T_{4s} = T_3\,r_p^{-\tfrac{k-1}{k}} \]
Works and COP: \[ w_c = c_p\,(T_2 - T_1), \quad w_t = c_p\,(T_3 - T_4), \quad \text{COP}_R = \frac{c_p\,(T_1 - T_4)}{w_c - w_t} \]

Merit parameters

Temperature lift: \(\Delta T_{lift} = T_{space,in} - T_{amb,in}\)
Approach temps: HX approaches on hot and cold sides govern feasibility and dictate \(\varepsilon\) targets.
Optimal \(r_p\) trend: COP peaks at moderate \(r_p\); too low gives weak \(T_4\) reduction, too high inflates \(w_c\).

Worked example: simple air-cycle with real components

Given

Ambient/ram at inlet: \(T_{amb} = 300\,\text{K}\), \(p_{amb} = 1\,\text{bar}\) (neglect ram rise)
Targets: Cabin air supplied at \(T_4 \approx 260\,\text{K}\); space return \(T_1 = 285\,\text{K}\)
Cycle: Single compressor–cooler–turbine (simple)
Parameters: \(k=1.4\), \(c_p = 1.004\,\text{kJ/(kg\cdot K)}\), \(\eta_c=0.82\), \(\eta_t=0.86\), \(\varepsilon_c=0.75\)
Pressure ratio: \(r_p = 3.0\); neglect HX pressure loss

Step 1: Compressor outlet temperature

Isentropic rise: \[ T_{2s} = T_1\,r_p^{\frac{k-1}{k}} = 285 \cdot 3^{0.2857} \approx 285 \cdot 1.369 \approx 390.2\,\text{K} \] Actual: \[ T_2 = T_1 + \frac{T_{2s} - T_1}{\eta_c} = 285 + \frac{105.2}{0.82} \approx 413.3\,\text{K} \]

Step 2: Cooler outlet temperature

\[ T_3 = T_2 - \varepsilon_c\,(T_2 - T_{amb}) = 413.3 - 0.75\,(413.3 - 300) \approx 328.3\,\text{K} \]

Step 3: Turbine outlet temperature

Isentropic drop: \[ T_{4s} = T_3\,r_p^{-\frac{k-1}{k}} = 328.3 \cdot 3^{-0.2857} \approx 328.3 \cdot 0.731 \approx 239.9\,\text{K} \] Actual: \[ T_4 = T_3 - \eta_t\,(T_3 - T_{4s}) = 328.3 - 0.86\,(88.4) \approx 252.4\,\text{K} \] Target \(260\,\text{K}\) is met with margin.

Step 4: Performance and sizing

Specific refrigeration: \[ q_L = c_p\,(T_1 - T_4) \approx 1.004\,(285 - 252.4) \approx 32.7\,\text{kJ/kg} \] Specific works: \[ w_c = c_p\,(T_2 - T_1) \approx 1.004\,(128.3) \approx 128.8\,\text{kJ/kg},\quad w_t = c_p\,(T_3 - T_4) \approx 1.004\,(75.9) \approx 76.2\,\text{kJ/kg} \] Net: \[ w_{net} \approx 52.6\,\text{kJ/kg},\quad \text{COP}_R = \frac{32.7}{52.6} \approx 0.62 \] For \(\dot{Q}_L = 20\,\text{kW}\), \[ \dot{m} = \frac{20}{32.7} \approx 0.612\,\text{kg/s},\quad \dot{W}_{sha} \approx \dot{m}\,w_{net} \approx 0.612 \cdot 52.6 \approx 32.2\,\text{kW} \]

Improving \(\varepsilon_c\) or adding a second cooler (bootstrap) can lower \(T_3\) and boost COP notably; raising \(r_p\) further would reduce \(T_4\) but increase \(w_c\), often reducing COP beyond an optimum.

Design checks and quick formulas

Quick relations

Isentropic temperature ratios: \[ \tau_c = \frac{T_{2s}}{T_1} = r_p^{\frac{k-1}{k}},\quad \tau_t = \frac{T_{4s}}{T_3} = r_p^{-\frac{k-1}{k}} \]
Non-ideal outlet temps: \[ T_2 = T_1 + \frac{(\tau_c - 1)T_1}{\eta_c},\quad T_4 = T_3 - \eta_t(1 - \tau_t)T_3 \]
Cooler outlet: \[ T_3 = T_2 - \varepsilon_c (T_2 - T_{amb}) \]
Mass flow for load: \[ \dot{m} = \frac{\dot{Q}_L}{c_p\,(T_1 - T_4)} \]