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Carnot system

Carnot Engine, Carnot Heat Pump, and Jet Engine: A Mathematical-Physics Perspective

The physics of heat engines provides one of the clearest windows into the structure of the Second Law of Thermodynamics. In this article we compare three systems:

  1. Carnot heat engine — a reversible, ideal cycle establishing the upper bound of heat-to-work conversion.
  2. Carnot heat pump / refrigerator — the reverse of the Carnot cycle, giving the upper bound of heat transfer per unit work.
  3. Jet engine (Brayton cycle) — a real engineering cycle approximated by an irreversible open system.

Our aim is to analyze them through thermodynamic identities, entropy production, and cycle integrals.

1. Carnot Heat Engine

1.1 Fundamental structure

A Carnot engine cyclically converts thermal energy into mechanical work while operating between two reservoirs at constant temperature:

  • Hot reservoir at temperature (T_H)
  • Cold reservoir at temperature (T_C)

The cycle is reversible, so the total entropy generation per cycle is zero:

[ \Delta S_{\text{cycle}} = 0. ]

1.2 The Carnot cycle

The working substance performs four reversible steps:

  1. Isothermal expansion at (T_H) Heat gained (Q_H), entropy change [ \Delta S = \frac{Q_H}{T_H}. ]

  2. Adiabatic expansion (\Delta S = 0), temperature decreases from (T_H) to (T_C).

  3. Isothermal compression at (T_C) Heat released (Q_C): [ -\Delta S = \frac{Q_C}{T_C}. ]

  4. Adiabatic compression (\Delta S = 0), temperature increases from (T_C) back to (T_H).

Reversibility implies entropy balance:

[ \frac{Q_H}{T_H} = \frac{Q_C}{T_C}. \tag{1} ]

1.3 Efficiency derivation

The work per cycle is:

[ W = Q_H - Q_C. ]

Using entropy relation (1):

[ \frac{Q_C}{Q_H} = \frac{T_C}{T_H}. ]

Thus the Carnot efficiency is:

[ \boxed{ \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} } \tag{2} ]

This is the upper bound for any heat engine respecting the second law.

2. Carnot Heat Pump (and Refrigerator)

A Carnot heat pump is simply the reversed Carnot cycle.

  • Work input: (W)
  • Heat transported from cold to hot
  • Reversible, so Eq. (1) still holds.

2.1 Coefficient of Performance (COP)

Heating mode: Heat Pump

Heat delivered to hot reservoir:

[ COP_{\text{HP}} = \frac{Q_H}{W}. ]

Since:

[ W = Q_H - Q_C, ] [ \frac{Q_C}{Q_H} = \frac{T_C}{T_H}, ]

we obtain:

[ \boxed{ COP_{\text{HP}} = \frac{T_H}{T_H - T_C} } \tag{3} ]

Cooling mode: Refrigerator

[ COP_{\text{REF}} = \frac{Q_C}{W} = \frac{T_C}{T_H - T_C}. \tag{4} ]

Relationship to Carnot Engine

From efficiency (2):

[ \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} \quad \Longrightarrow \quad T_H - T_C = T_H,\eta_{\text{Carnot}}. ]

Thus:

[ COP_{\text{HP}} = \frac{1}{\eta_{\text{Carnot}}} \tag{5} ]

This duality illustrates the deep thermodynamic symmetry between heat engines and heat pumps.

3. Jet Engine: Brayton Cycle (Open-System Thermodynamics)

Jet engines operate on a Brayton cycle, approximated by:

  1. Isentropic compression
  2. Constant-pressure heat addition
  3. Isentropic expansion
  4. Constant-pressure heat rejection

However, unlike the Carnot systems:

  • The Brayton cycle is irreversible
  • It is an open cycle (mass enters and leaves)
  • Heat addition is via combustion, not a reservoir

Thus entropy production is positive:

[ \Delta S_{\text{cycle}} > 0 \quad \text{(irreversible)}. ]

3.1 Ideal Brayton cycle efficiency

Assume ideal gas with constant (c_p) and ratio ( \gamma = c_p/c_v ). Let compressor pressure ratio be ( r_p = \frac{p_2}{p_1} ).

For isentropic compression:

[ \frac{T_2}{T_1} = r_p^{\frac{\gamma - 1}{\gamma}}. \tag{6} ]

Similarly, turbine expansion:

[ \frac{T_4}{T_3} = r_p^{-\frac{\gamma - 1}{\gamma}}. \tag{7} ]

Net work is:

[ W_{\text{net}} = c_p \big[ (T_3 - T_4) - (T_2 - T_1) \big]. ]

Heat input:

[ Q_{\text{in}} = c_p (T_3 - T_2). ]

Thus thermal efficiency is:

[ \eta_{\text{Brayton}} = 1 - \frac{T_4 - T_1}{T_3 - T_2}. \tag{8} ]

Using ratios (6)–(7), we obtain the famous closed-form:

[ \boxed{ \eta_{\text{Brayton}} = 1 - r_p^{-\frac{\gamma - 1}{\gamma}} } \tag{9} ]

Unlike the Carnot efficiency, this depends on:

  • compression ratio
  • working fluid properties
  • irreversibilities

and cannot reach Carnot limits because heat is added at varying temperatures.

4. Mathematical Comparison of the Three Cycles

We can now compare their efficiencies rigorously.

4.1 Carnot vs. Brayton

Carnot efficiency between hottest and coldest temperatures of a Brayton engine would be:

[ \eta_{\text{Carnot}} = 1 - \frac{T_1}{T_3}. \tag{10} ]

But Brayton efficiency (9) is always lower, because heat is added over a temperature range. Formally:

[ \eta_{\text{Brayton}} < \eta_{\text{Carnot}}. \tag{11} ]

This inequality follows from the mean-value theorem applied to heat addition at non-uniform temperature.

4.2 Carnot heat pump duality

[ COP_{\text{HP}} = \frac{T_H}{T_H - T_C} = \frac{1}{1 - \eta_{\text{Carnot}}}. \tag{12} ]

Carnot heat pump efficiency diverges as (T_H \to T_C), highlighting that reversible heat transfer at small (\Delta T) is extremely efficient.

4.3 Entropy generation comparison

System Entropy generation Reversible? Cycle type
Carnot engine (0) Yes Closed
Carnot heat pump (0) Yes Closed
Jet engine (Brayton) (>0) No Open

Thus Carnot cycles define bounds, while Brayton cycles describe real engineering behavior.

5. Geometric (T–S) Interpretation

Carnot cycle (rectangular in T–S plane)

  • Isothermal heat transfer at constant (T_H), (T_C)
  • Adiabatic vertical lines

Area enclosed = work.

Brayton cycle (curved in T–S plane)

  • Compression & expansion are nearly vertical (isentropic)
  • Heat addition/removal at varying temperature → tilted trapezoid
  • Smaller area compared to Carnot rectangle operating between same extremes

This geometric difference explains mathematically why:

[ \eta_{\text{Brayton}} < \eta_{\text{Carnot}}. ]

6. Summary Table (Physics Focus)

Feature Carnot Engine Carnot Heat Pump Jet Engine (Brayton)
System Closed, reversible Closed, reversible Open, irreversible
Governing law 2nd law (entropy balance) 2nd law Fluid dynamics + thermodynamics
Efficiency formula (1 - T_C/T_H) (T_H/(T_H-T_C)) (1 - r_p^{-(\gamma-1)/\gamma})
Entropy generation (0) (0) (>0)
Heat addition Isothermal Isothermal Constant pressure, varying (T)
Existence Ideal limit Ideal limit Real engines

Closing Remarks

From a mathematical physics point of view, the Carnot engine and heat pump represent the boundary of what the second law permits, while the Brayton cycle shows what is achievable in real systems with compressible fluid flow, combustion, and irreversibility.

The fundamental difference between a Carnot system and a jet engine lies in entropy:

  • Carnot cycles minimize and exactly balance entropy change through reversible processes.
  • Jet engines inevitably create entropy through shocks, viscous flow, turbulence, and combustion.

Thus, the comparison of these three systems beautifully illustrates how thermodynamic limits, cycle geometry, and entropy production shape real-world physics.