Explanation:

Carnot cycle:

• The Carnot cycle consists of 4 processes
  • 1-2 isothermal heat addition
  • 2-3 reversible adiabatic expansion
  • 3-4 isothermal heat rejection
  • 4-1 reversible adiabatic compression
• A cycle is said to be reversible only when each process in a cycle is reversible.
• Carnot cycle consists of 2 isothermal and 2 adiabatic processes. An isothermal process is a very slow process and the adiabatic process is a very fast process and the combination of a slow process and fast process are very difficult.
• So a Carnot cycle is also known as an impractical cycle. It is used only to compare other actual cycles.

Carnot theorem: For different engines operating between the same temperature limit, no engine has an efficiency greater than Carnot cycle efficiency.

Important points regarding the Carnot cycle

• All reversible cycles operating between the same temperature limit will have the same efficiency.
• The efficiency of a reversible cycle depends only on the temperature limit.
• The efficiency of the reversible cycle is independent from the working fluid.

Kelvin-Plank statement: It is impossible for a heat engine to produce net work in a complete cycle if it exchanges heat only with bodies at a single fixed temperature.

For heat engine to operate, the working fluid has to exchange heat with a heat sink as well with the heat source.

If QL = 0, then \(\eta = 1 - \frac{{{Q_L}}}{{{Q_H}}} = 1\)

Therefore, the second law claims that no heat engine can be 100% efficient.

So, W = Q (is impossible)

There will be always some heat loss to the surrounding.

W = QH – QL (is possible)

As (QH – QL) ≥ 0 so Wcycle ≥ 0

Clausius statement of Second law of Thermodynamics:

It is impossible to construct a system which will operate in a cycle, transfers heat from the low-temperature reservoir (or object) to the high-temperature reservoir (or object) without any external effect or work interaction with the surrounding.

• It is impossible to construct a device that operates on a cycle and produces no effect other than the transfer of heat from a lower temperature body to a higher temperature body.
• The Clausius statement is related to refrigerators as well as heat pumps.

Concept:

(COP)Heat Pump = 1 + (COP)Refrigerator

Refrigerator: A refrigerator is a device which works on reverse Carnot cycle and extracts heat from a lower temperature body to keep the temperature of the body lower than the surrounding temperature. And by taking work input it transfers heat to the higher temperature body or surrounding.

Refrigerating Effect R.E.= QL

Work input = QH - QL

For a reversible engine, \(\frac{{{Q_H}}}{{{Q_L}}} = \frac{{{T_H}}}{{{T_L}}}\)

\({\bf{Coefficient}}\;{\bf{of}}\;{\bf{Performance}}\;\left( {{\bf{COP}}} \right) = \frac{{{\bf{Refrigerating}}\;{\bf{Effect}}}}{{{\bf{Work}}\;{\bf{Input}}}} = \frac{{{{\bf{Q}}_{\bf{L}}}}}{{\bf{W}}}\)

\({\bf{CO}}{{\bf{P}}_{{\bf{refrigerator}}}} = \frac{{{{\bf{Q}}_{\bf{L}}}}}{{{{\bf{Q}}_{\bf{H}}} - {{\bf{Q}}_{\bf{L}}}}} = \frac{{{{\bf{T}}_{\bf{L}}}}}{{{{\bf{T}}_{\bf{H}}} - {{\bf{T}}_{\bf{L}}}}}\)

Heat Pump: A heat pump is a device which works on reversed Carnot cycle and transfers heat from a lower temperature body to a higher temperature body. A heat pump maintains a body at a temperature higher than the surrounding temperature or a lower temperature body.

\({\rm{CO}}{{\rm{P}}_{{\rm{heat\;pump}}}} = \frac{{{\rm{Refrigeration\;Effect}}}}{{{\rm{Work\;Input}}}} = \frac{{{{\rm{Q}}_{\rm{H}}}}}{{{{\rm{Q}}_{\rm{H}}} - {{\rm{Q}}_{\rm{L}}}}} = \frac{{{{\rm{T}}_{\rm{H}}}}}{{{{\rm{T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}}}\)

\({\rm{CO}}{{\rm{P}}_{{\rm{heat\;pump}}}} = \frac{{{{\rm{T}}_{\rm{H}}}}}{{{{\rm{T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}}} = \frac{{{{\rm{T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}}}{{{{\rm{T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}}} + \frac{{{{\rm{T}}_{\rm{L}}}}}{{{{\rm{T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}}}\)

\({\rm{CO}}{{\rm{P}}_{{\rm{heat\;pump}}}} = 1 + {\rm{\;CO}}{{\rm{P}}_{{\rm{Refrigerator}}}}\)

The difference between the COP of heat pump and refrigerator

\({\rm{CO}}{{\rm{P}}_{{\rm{heat\;pump}}}}- {\rm{\;CO}}{{\rm{P}}_{{\rm{Refrigerator}}}}=1\)

Concept:

The efficiency of Carnot or any reversible engine is given by:

 \({\eta _{Carnot}} = {\eta _{Rev}} = 1 - \frac{{{T_2}}}{{{T_1}}}\)

where, η = Efficiency of Carnot Engine, T₁ = Source temperature in K and T₂ = Sink Temperature in K.

Calculation:

Given:

η_I = 0.25 when T₁ = Source temperature (K) and T₂ = Sink Temperature (K)

η_II = 0.30 when  T₁ = Source temperature(K) and (T₂)_new = Sink Temperature (K); (T₂)_new = T₂ – 20

\({\eta _I} = 1 - \frac{{{T_2}}}{{{T_1}}}\)

\( \Rightarrow 0.25 = 1 - \frac{{{T_2}}}{{{T_1}}}\)

\( \Rightarrow \frac{{{T_2}}}{{{T_1}}} = 0.75\;\)  

⇒ T₂ = 0.75T₁    …….eqn (i)

Now,

\({\eta _{II}} = 1 - \frac{{{{\left( {{T_2}} \right)}_{new}}}}{{{T_1}}}\)

\( 0.3 = 1 - \frac{{{T_2} - 20}}{{{T_1}}}\)

\(\frac{{{T_2} - 20}}{{{T_1}}} = 0.7\)    ……eq(ii)

Putting values from eq(i) in eq(ii),

\(\frac{{0.75{T_1} - 20}}{{{T_1}}} = 0.7\)

⇒ 0.05T₁ = 20

⇒ T₁ = 400 K

∴ The source temperature is 400 K.

1) Temperature in the efficiency formula is only taken in terms of Kelvin.

2) The difference in degree celsius scale is the same as the Kelvin scale.

Concept:

Heat Pump is a device which delivers the heat from low temperature to a high temperature in a cyclic process.

On the other hand, a refrigerator is a device which extracts heat from low temperature to a high temperature in a cyclic process.

The difference is that in heat pump concern is at high temperature (T₁) and in the refrigerator, the focus is on low temperature (T₂).

Units of refrigeration is commonly expressed as ton of refrigeration.

One ton of refrigeration is defined as the capacity to freeze one ton of water from and at 0°C in 24 hours (or refrigerating effect produced by the melting of 1 toon of ice from and at 0°C in 24 hours).

1 TR = 210 kJ/min = 3.5 kW 

1 kCal = 4.18 kJ

∴ 1 kJ = 0.239 kCal

∴ 210 kJ = 50.19 kCal

∴ 1 TR = 50.19 kCal/min = 3011.52 kCal/hr

CONCEPT:

• Carnot engine: A theoretical thermodynamic cycle proposed by Leonard Carnot. This engine gives the maximum possible efficiency that a heat engine can have during the conversion process of heat into work or vice-versa, working between two reservoirs.
  • So practically and theoretically there can not be any engine with more efficiency than Carnot engine.

The efficiency of the Carnot's Heat engine is given by:

 \(η=1-\frac{T_L}{T_H}\)

where TH is the hot reservoir temperature and TL is the cold reservoir temperature.

Calculation:

Given:

T_H = 900°C = 1173 K, T_L = 300°C = 573 K

\(\eta =1- \frac{573}{1173}=51.15 \)%

• The efficiency of this type of engine does not depend on the nature of the working substance and is only dependent on the temperature of the hot and cold reservoirs.

The efficiency of the Carnot engine is given as follows

\(\eta = \frac{{{T_H} - {T_L}}}{{{T_H}}} = 1 - \frac{{{T_L}}}{{{T_H}}}\)

\(\eta \; = \;1 - \frac{{{T_{sink}}}}{{{T_{source}}}}\)

 \(\therefore {T_{source}} \uparrow and\;\left( {\frac{{{T_{sink}}}}{{{T_{source}}}}} \right) \downarrow\)         ∴ η↑

CONCEPT:

First Law of Thermodynamics: It is a statement of conservation of energy in the thermodynamical process.

• According to it, heat given to a system (ΔQ) is equal to the sum of the increase in its internal energy (ΔU) and the work done (ΔW) by the system against the surroundings.

ΔQ = ΔU + ΔW

Second Law of Thermodynamics:

1. Clausius statement: It is impossible for a self-acting machine to transfer heat from a colder body to a hotter one without the aid of an external agency
2. Kelvin-Planck’s statement: It is impossible to design an engine that extracts heat and fully utilizes it into work without producing any other effect.

EXPLANATION:

The diagram below shows the essential components of a heat engine:

HE = Heat Engine, T_H = Temperature of Source/Hot reservoir and T_L = Temperature of Sink/Cold Reservoir, Q₁ = Heat absorbed from Source and Q₂ = Heat rejected to Sink, W = Work Done

The efficiency(η) of a heat engine is defined as:

\(\eta = \frac{\text{Work done by heat engine}}{\text{Heat absorbed from the source}}\)

Now, W = Q1  - Q2 

\(\therefore \eta = \frac{W}{Q_1} = \frac{Q_1-Q_2}{Q_1} =1-\frac{Q_2}{Q_1}\)

• The efficiency of the heat engine is always less than 1. 
• This is because in reality none of the engines can convert absorbed heat completely into work, as some of the heat will always be dissipated during the process.
• Also stated, according to the second Law no heat engine can have 100% efficiency. So option 2 is correct.

Explanation:

Clausius statement:

• "It is impossible to construct a system which will operate in a cycle, transfers heat from the low-temperature reservoir (or object) to the high-temperature reservoir (or object) without any external effect or work interaction with the surrounding." Hence according to Clausius statement if we aid an external source, then heat can flow from low temperature reservoir to high-temperature reservoir or cold body to a hot body.
• Refrigeration is based on Clausius statement of the second law of thermodynamics.

Kelvin-Planck statement:

• "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." Hence according to kelvin Planck statement, we will require at least two reservoirs to extract work from a given heat source.
• The heat engine works on Kelvin Planck’s statement of the second law of thermodynamics.

Gay Lussac's statement:

"For a given mass at constant volume, the pressure of a fixed amount of gas varies directly with temperature."

Hence, P ∝ T

where P = pressure of the gas, T = absolute temperature 

Joule statement:

"The internal energy of a given quantity of ideal gas depends only on the temperature. It is independent of pressure and volume."

Mathematically: ΔU = C_vΔT

where ΔU = change in internal energy, C_v = specific heat at constant volume, ΔT = change in temperature