Stirling Cycle vs. Carnot Cycle: Formulas, Diagrams and Efficiency

Both the Stirling cycle and the Carnot cycle are idealized thermodynamic cycles that achieve high efficiency by operating between two temperature reservoirs. However, they differ in their practical implementation and working principles.

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1. Carnot Cycle (Theoretical Maximum Efficiency)

Processes (Ideal Gas Assumption)

1. Isothermal Expansion (Heat Addition, T_H)

   • Gas expands while absorbing heat (${�}_{�}$) at constant temperature ${�}_{�}$.

   • Work done: ${�}_{1-2}={�}_{�}=��{�}_{�}\ln \left(\frac{{�}_2}{{�}_1}\right)$

2. Adiabatic Expansion (No Heat Transfer)

   • Gas expands further, cooling to ${�}_{�}$.

3. Isothermal Compression (Heat Rejection, T_L)

   • Gas is compressed while releasing heat (${�}_{�}$) at constant temperature ${�}_{�}$.

   • Work input: ${�}_{3-4}={�}_{�}=��{�}_{�}\ln \left(\frac{{�}_3}{{�}_4}\right)$

4. Adiabatic Compression (No Heat Transfer)

   • Gas is compressed back to the initial state, heating up to ${�}_{�}$.

Carnot Cycle Efficiency (Maximum Possible)

$${�}_{\text{Carnot}}=1-\frac{{�}_{�}}{{�}_{�}}$$

• Depends only on the temperature limits (${�}_{�}$ = hot reservoir, ${�}_{�}$ = cold reservoir).

• No real engine can exceed Carnot efficiency (2nd Law of Thermodynamics).

Carnot Cycle Diagram (P-V & T-S)

• P-V Diagram: Two isotherms + two adiabats.

• T-S Diagram: A rectangle (constant ${�}_{�}$ and ${�}_{�}$ heat addition/rejection).

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2. Stirling Cycle (Practical High-Efficiency Engine)

Processes (Ideal Regenerative Case)

1. Isothermal Expansion (Heat Addition, T_H)

   • Gas expands while absorbing heat (${�}_{�}$) at ${�}_{�}$.

   • Work done: ${�}_{1-2}={�}_{�}=��{�}_{�}\ln \left(\frac{{�}_2}{{�}_1}\right)$

2. Constant-Volume Heat Removal (Regeneration)

   • Gas passes through a regenerator, transferring heat to a thermal store.

3. Isothermal Compression (Heat Rejection, T_L)

   • Gas is compressed while releasing heat (${�}_{�}$) at ${�}_{�}$.

   • Work input: ${�}_{3-4}={�}_{�}=��{�}_{�}\ln \left(\frac{{�}_4}{{�}_3}\right)$

4. Constant-Volume Heat Addition (Regeneration)

   • Gas reabsorbs stored heat from the regenerator.

Stirling Cycle Efficiency (Ideal Case)

$${�}_{\text{Stirling}}=1-\frac{{�}_{�}}{{�}_{�}}(\text{Same as Carnot})$$

• Only possible with perfect regeneration (all heat from step 2 is reused in step 4).

• Real Stirling engines have lower efficiency due to imperfect regeneration, friction, and heat losses.

Stirling Cycle Diagram (P-V & T-S)

• P-V Diagram: Two isotherms + two isochores (constant volume).

• T-S Diagram: Similar to Carnot but with regeneration (heat exchange at constant volume).

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3. Key Differences Between Stirling & Carnot Cycles

Feature	Carnot Cycle	Stirling Cycle
Processes	2 Isotherms + 2 Adiabats	2 Isotherms + 2 Isochores
Efficiency	$�=1-\frac{{�}_{�}}{{�}_{�}}$ (Maximum possible)	Same as Carnot (if regenerator is perfect)
Practicality	Impossible to build (infinitely slow processes)	Buildable (used in real engines)
Regeneration	Not used	Critical for efficiency (recovers heat)
Applications	Theoretical benchmark	Solar power, submarines, cryocoolers

4. Why Stirling Engines Don’t Achieve Carnot Efficiency in Reality

• Imperfect regeneration (some heat is lost).

• Mechanical losses (friction, leakage).

• Finite heat transfer rates (real processes are not perfectly isothermal).

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Conclusion

• Carnot cycle is the theoretical limit for heat engine efficiency.

• Stirling cycle can match Carnot efficiency in an ideal case with perfect regeneration.

• Real Stirling engines are less efficient but still among the most efficient practical heat engines.

Both cycles are important in thermodynamics:

• Carnot sets the upper bound.

• Stirling provides a practical approach to high-efficiency energy conversion.