HVAC Engineering World

CARNOT ENGINE /THEOREM
A Carnot engine is an ideal heat engine that operates on a reversible thermodynamic cycle between two heat reservoirs at different absolute temperatures and converts heat into work with maximum possible efficiency.

➡️Carnot Principle / Carnot Theorem
Statement of Carnot Theorem:
No heat engine operating between two given temperatures can be more efficient than a reversible (Carnot) engine operating between the same temperatures.

All reversible heat engines operating between the same two temperatures have the same efficiency, irrespective of the working substance.

Implication:
The efficiency of a heat engine depends only on the temperatures of the heat reservoirs, not on the nature of the engine.

STRUCTURE:
Carnot engine consists of:
• A cylinder filled with an ideal gas.
• A frictionless piston.
• A hot reservoir at temperature Th.
• A cold reservoir at temperature Tc.
• All processes occur reversibly and without friction.

➡️Working of Carnot Engine (Carnot Cycle)
The Carnot cycle consists of four reversible processes:

Process 1: Isothermal Expansion at Temperature Th:

-The gas is kept in contact with the hot reservoir at temperature Th.
-Heat Qh is absorbed by the gas.
-Temperature remains constant.
-The gas expands and does work.

For an ideal gas:
Qh = W1
Qh = n R Th ln(V2 / V1)

where:
n = number of moles
R = gas constant
V1, V2 = initial and final volumes

Process 2:
(Adiabatic Expansion)

The cylinder is thermally insulated.
No heat exchange takes place.
Q = 0
The gas expands and its temperature drops from Th to Tc.

For adiabatic process:
T V^(γ − 1) = constant
or
Th V2^(γ − 1) = Tc V3^(γ − 1)

where:
γ = Cp / Cv

Process 3:
(Isothermal Compression at Temperature Tc):

-The gas is placed in contact with the cold reservoir.

-Heat Qc is rejected to the cold reservoir.
-Temperature remains constant.
-Gas is compressed.

For an ideal gas:
Qc = W3
Qc = n R Tc ln(V3 / V4)
(Note: Qc is negative since heat is rejected)

PROCESS 4:
(Adiabatic Compression);
-The gas is insulated again.
-No heat exchange occurs.
Q = 0

The gas is compressed and temperature rises from Tc back to Th.

Adiabatic relation:
Tc V4^(γ − 1) = Th V1^(γ − 1)
The system returns to its initial state.

➡️ Efficiency of Carnot Engine
Efficiency (η) is defined as:
η = Work output / Heat absorbed
η = W / Qh

Net work done:
W = Qh − Qc
So,
η = (Qh − Qc) / Qh
η = 1 − (Qc / Qh)
From thermodynamics of reversible engines:
Qc / Qh = Tc / Th

Therefore,
η = 1 − (Tc / Th)

where
Th = absolute temperature of hot reservoir (in Kelvin)
Tc = absolute temperature of cold reservoir (in Kelvin)

In general,
•Efficiency depends only on Th and Tc.
•Higher Th or lower Tc increases efficiency.
•Efficiency can never be 100% unless Tc = 0 K (impossible).
•No real engine can exceed Carnot efficiency.

APPLICATIONS OF CARNOT ENGINE:
Although a Carnot engine cannot be built in practice, it is used for:

-Determining maximum possible efficiency of real engines.
-Comparing performances of heat engines.
-Understanding the second law of thermodynamics.
-Designing power plants and thermal systems.
Basis of refrigerators and heat pumps (reversed Carnot cycle).

EXAMPLES
Q1.
Calculate the efficiency of an operating at:
Th = 600 K, Tc = 300 K.

Solution ⭐
Formula:
η = 1 − (Tc / Th)
Substitute values:
η = 1 − (300 / 600)
η = 1 − 0.5
η = 0.5
Efficiency = 50%

Q2.
A Carnot engine operating between two heat reservoirs at temperatures: Th = 500 K and
Tc = 300 K. If the heat absorbed, Qh is 2000J, calculate the work done.

Solution ⭐
Heat absorbed Qh = 2000 J
Step 1: Efficiency
η = 1 − (300 / 500)
η = 1 − 0.6
η = 0.4
Step 2: Work done
W = η × Qh
W = 0.4 × 2000
W = 800 J

Q3.
The heat absorbed, Qh by an engine whose efficiency is 25% is 4000J. Calculate the heat rejected.

Solution ⭐
Given:
Efficiency = 25% = 0.25
Heat absorbed Qh = 4000 J
Work done:
W = η × Qh
W = 0.25 × 4000
W = 1000 J

Heat rejected:
Qc = Qh − W
Qc = 4000 − 1000
Qc = 3000 J

1.54K views10:57