A Carnot engine whose sink is at 300 K has an efficiency of 40%. By how much should the temperature of source be increased so as to increase its efficiency by 50% of original efficiency?
- 275 K
- 175 K
- 250 K
- 225 K
The Correct Option is C
Solution and Explanation
Temperature of sink TL = 300 K
Original efficiency η = 40% = 0.4
Let the Initial temperature of the source be TH
Using η = 1−THTL
∴ 0.4 = 1−TH x 300 ⟹ TH = 500 K
Now the efficiency of the engine is increased by 50% of original efficiency,
∴ New efficiency η′=40% +20% =60%
∴ η′ = 1−TH'TL
OR 0.6 = 1−TH′ x 300
OR TH′ x 300 = 0.4 ⟹TH′ = 750 K
Increase in source temperature ΔTH = TH′ − TH = 750 − 500 = 250 K
so, the correct option is (C): 250 K
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Concepts Used:
Carnot Engine
The Carnot engine is a theoretical heat engine that operates on the principle of a reversible thermodynamic cycle. It was developed by French engineer Nicolas Léonard Sadi Carnot in the early 19th century and is considered one of the most efficient heat engines that can be constructed.
The Carnot engine consists of four stages: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. In the first stage, the working fluid in the engine is heated isothermally by a heat source, which provides the energy needed to expand the fluid and do work. In the second stage, the fluid expands adiabatically, meaning that there is no heat transfer between the fluid and its surroundings. In the third stage, the fluid is cooled isothermally by a heat sink, which removes energy from the fluid and allows it to contract and do work. In the fourth stage, the fluid is compressed adiabatically, returning it to its original state and completing the cycle.
Read More: Carnot’s Theorem
The Carnot engine is the most efficient heat engine that can be constructed, as it operates at the maximum possible efficiency for a given temperature difference between the heat source and sink. The efficiency of the Carnot engine is given by the ratio of the temperature difference between the heat source and sink to the absolute temperature of the heat source, or 1 minus the ratio of the absolute temperature of the heat sink to the absolute temperature of the heat source.
Although the Carnot engine is a theoretical construct, it has important practical applications in the design and optimization of real-world heat engines, such as internal combustion engines and steam turbines.