calculate-thermodynamic-efficiency

Calculate Thermodynamic Efficiency

Analyze thermodynamic efficiency of heat engines, power cycles, and thermal systems by applying first and second law analysis — computing Carnot efficiency bounds, actual cycle efficiency, entropy generation, and exergy destruction to identify where losses occur.

Why This Is Best Practice

Adopted by: Every power plant, refrigeration system, and chemical process is designed and analyzed using thermodynamic cycle analysis. ASHRAE (American Society of Heating, Refrigerating and Air-Conditioning Engineers) standards require thermodynamic efficiency analysis for HVAC system certification. The DOE uses exergy analysis to identify where US industrial energy use is most inefficient (DOE Bandwidth Studies). Impact: Çengel & Boles (2015) demonstrate that the Carnot efficiency sets an inviolable upper bound — any claimed efficiency exceeding the Carnot limit for the temperature ratio is physically impossible and indicates an error. Real power plant efficiencies are 30-50% of the Carnot limit; identifying where the remaining 50-70% is lost requires second law (entropy/exergy) analysis. Bejan (2016) established exergy analysis as the tool that reveals the location, magnitude, and source of irreversibilities — giving engineers a clear optimization target.

Steps

1. Define the system boundary and state the problem

Before any calculation:

• Identify: working fluid, heat source temperature (TH), heat sink temperature (TL)
• State: is this a heat engine (work output) or heat pump/refrigerator (work input)?
• Define: steady-state or transient? Open or closed system?
• List: all energy interactions crossing the system boundary

2. Apply the first law (energy balance)

First Law: energy is conserved

For a cycle (net change in stored energy = 0):

W_net = Q_H − Q_L     (heat engine)
Q_H = W_net + Q_L     (heat pump)

For a process (steady-flow open system):

Q̇ − Ẇ = ṁ[(h₂ − h₁) + ½(V₂² − V₁²) + g(z₂ − z₁)]

Where h = specific enthalpy (from steam tables, ideal gas relations, or refrigerant charts).

3. Calculate Carnot efficiency (theoretical maximum)

The Carnot efficiency is the upper bound for any heat engine operating between TH and TL (temperatures must be in Kelvin):

η_Carnot = 1 − TL/TH

For a refrigerator or heat pump (COP — Coefficient of Performance):

COP_Carnot,R = TL / (TH − TL)    (refrigerator: heat removed per unit work)
COP_Carnot,HP = TH / (TH − TL)   (heat pump: heat delivered per unit work)

Example: steam power plant with boiler at 600°C and condenser at 40°C:

η_Carnot = 1 − (40+273)/(600+273) = 1 − 313/873 = 0.642 = 64.2%

Actual plant efficiency of 35-40% is 54-62% of Carnot maximum — identifying where the remaining loss occurs requires second law analysis.

4. Analyze the actual cycle (Rankine, Brayton, Otto, Diesel)

Rankine cycle (steam power):

η_thermal = W_net / Q_H = (W_turbine − W_pump) / Q_boiler
η_thermal = [(h₁ − h₂) − (h₄ − h₃)] / (h₁ − h₄)

Read h values from steam tables at the relevant T,P states.

Isentropic efficiency of turbine/compressor:

η_turbine = W_actual / W_isentropic = (h₁ − h₂a) / (h₁ − h₂s)
η_compressor = W_isentropic / W_actual = (h₂s − h₁) / (h₂a − h₁)

Typical isentropic efficiencies: turbine 85-92%; compressor 75-85%.

5. Apply the second law — entropy and irreversibility

Entropy balance for a closed system:

S₂ − S₁ = ∫(δQ/T) + S_gen

Where S_gen ≥ 0 (equality for reversible; strict inequality for irreversible).

Entropy generation for a process:

Ṡ_gen = ṁ(s₂ − s₁) − Q̇/T_boundary ≥ 0

Sources of entropy generation (irreversibilities):

• Heat transfer across finite temperature difference
• Fluid friction (pressure drops)
• Mixing of streams at different temperatures/compositions
• Chemical reactions far from equilibrium

6. Conduct exergy analysis to locate losses

Exergy = maximum useful work extractable relative to environment (T₀, P₀):

Exergy of a stream: ex = (h − h₀) − T₀(s − s₀)
Exergy destruction: Ẋ_destroyed = T₀ · Ṡ_gen  (Gouy-Stodola theorem)

Exergy efficiency (second-law efficiency):

η_II = Exergy output / Exergy input = 1 − (Ẋ_destroyed / Ẋ_input)

Exergy analysis reveals which component loses the most work potential — the one with largest Ẋ_destroyed is the priority for engineering improvement.

Common Mistakes

• Using Celsius instead of Kelvin in Carnot formula: η = 1 − TL/TH requires absolute temperatures. Using Celsius gives physically meaningless and incorrect results.
• Confusing thermal efficiency and second-law efficiency: A device with 90% first-law efficiency (very little heat loss) can have 10% second-law efficiency (enormous irreversibilities from mixing or friction).
• Treating reversible efficiency as achievable: Carnot and isentropic processes are ideal references — no real device achieves them. Always compare real vs. Carnot to assess performance.

When NOT to Use

• Non-equilibrium thermodynamics (coupled transport processes, biological systems): equilibrium thermodynamics applies to quasi-static processes; for systems far from equilibrium, use irreversible thermodynamics (Onsager coefficients, entropy production rate density).