|br| *Process 2 to 3* : reversible adiabatic expansion |br| *Process 3 to 4* : reversible isothermal compression |br| *Process 4 to 1* : reversible adiabatic compression This Carnot cycle can be applied in reverse and applied to a heat pump. The P-v diagram of a reversed Carnot cycle is given below. .. figure:: reversed-carnot.png :scale: 50 % :alt: P-v diagram of reverse carnot cycle While Carnot cycle is no doubt interesting, what is even more are the two principles which have been proposed by Carnot. In some sense, these two carnot principles are just corollaries of the second law. They will be true as long as second law is true. They provide a new insight to our understanding of the second law. .. important:: It is impossible to construct an engine operating between only two thermal reservoirs which will have a higher thermal efficiency than a reversible engine operating between the same two reservoirs. .. figure:: carnot-first-principle.png :scale: 100 % :alt: Illustration of Carnot First Principle *Proof*: To the contrary, let us assume that there exists an irreversible engine (:math:`X`) which is more efficient than a reversible engine (:math:`R`) when operating between same thermal reservoirs. Let both of them draw a heat of :math:`Q` from the hot reservoir. The work produced by irreversible engine (:math:`W_X`) would be greater than the work produced by the reversible engine (:math:`W_R`). The heat rejected by the irreversible engine would be :math:`Q_{irrev}=Q-W_X` while the heat rejected by the reversible engine would be :math:`Q_{rev}=Q-W_R`. Now if the reversible engine is made to operate in reverse like a pump and is powered by the irreversible engine a net work output would be obtained. However the hot reservoir would now become redundant since the heat sourced by the engine would be same as the heat sunk by the pump. The heat could be directly routed from the pump to the engine. We can now look upon this combination as a device that takes heat input from a single reservoir and delivers net work (:math:`W_X-W_R`). As this combination device is in violation of the second law of thermodynamics we can say that our initial assumption about the irreversible machine was wrong. .. important:: All reversible engines operating between the same two reservoirs have the same efficiency. *Proof*: To the contrary, let us assume that there exists a reversible engine (:math:`R1`) which is more efficient than a reversible engine (:math:`R2`) when operating between same thermal reservoirs. We can here onwards follow the same steps as we did in the previous proof. The resulting combination machine again in this case would violate the second law. So this assumption must be wrong again. .. index:: absolute temperature scale, kelvin temperature scale The Thermodynamic Temperature Scale (Kelvin Scale) -------------------------------------------------- The second carnot principle states that the efficiency of all reversible engines operating between same two thermal reservoirs would be the same. It is noteworthy that this efficiency is independent of the working fluid or any details of the way the cycle is implemented. Now the only parameter that characterises a thermal reservoir is its temperature. Therefore thermal efficiency of a reversible cycle should be a function of temperature of the hot and cold thermal reservoirs. Previously in :eq:`thermal_efficiency_heat_engine` an expression of this efficiency has already been defined. This expression can also be stated as .. math:: \eta_{thermal} = 1 - \frac{Q_L}{Q_H} So we can also conclude that the ratio :math:`\frac{Q_H}{Q_L}` is a function of the reservoir temperatures. Expressed mathematically: .. math:: :label: heat-temp-ratio \frac{Q_H}{Q_L} = f(T_H, T_L) Let us now try to evaluate what this function :math:`f` can look like. To help us do this, consider the arrangement of reversible heat engines as shown in the figure below. .. figure:: thermodynamic-temperature-scale.png :scale: 100 % :alt: Thermodynamic Temperature Scale Illustration The engines A and B on the left operate in a heat series fashion. The heat rejected by A is received as heat input by engine B. Since A and B are reversible their combination is also a reversible engine. Then as per carnot's second principle, the thermal efficiency of the combination of engines and engine A must be same. The combined engine (A+B) and engine (C) receive the same amount of heat from the hot reservoir :math:`Q_1` and reject the same amount of heat to the low temperature reservoir (:math:`Q_3`). The work done :math:`W_A + W_B = W_C`. Applying equation :eq:`heat-temp-ratio` to all three engines we get: .. math:: \frac{Q_1}{Q_2} = f(T_1, T_2), \;\; \frac{Q_2}{Q_3} = f(T_2, T_3), \;\;and \;\; \frac{Q_1}{Q_3} = f(T_1, T_3) The following identity : .. math:: \frac{Q_1}{Q_3} = \frac{Q_1}{Q_2}\frac{Q_2}{Q_3} can be written as: .. math:: f(T_1, T_3) = f(T_1, T_2)f(T_2, T_3) The above equation can only be satisfied if the function :math:`f` is of the form: .. math:: f(T_H, T_L) = \frac{\phi(T_H)}{\phi(T_L)} Various options of :math:`\phi` can satisfy the above form of equation. Kelvin proposed to take :math:`\phi(T)=T`. This new temperature scale is called the absolute temperature scale or the Kelvin scale. .. important:: The Absolute or Kelvin Temperature Scale On the Kelvin scale, the temperature ratio of two reservoirs is equal to the ratio of heat transfer between a reversible heat engine and the reservoirs. .. math:: :label: absolute-temp-scale \frac{T_H}{T_L} = (\frac{Q_H}{Q_L})_{reversible} In order to fix the magnitude of a unit of Kelvin, the International Conference Weights and Measures assigned a value of 273.16 Kelvin to the triple point of water. The magnitudes of temperature units on the Kelvin and Celsius scales are identical (1 K = 1°C). Efficiencies of Carnot Devices ------------------------------ .. index:: carnot efficiency - heat engine **Heat Engine** From :eq:`thermal_efficiency_heat_engine`, we can write: .. math:: \eta_{thermal, reversible} = 1 - (\frac{Q_L}{Q_H})_{reversible} Using the definition of Kelvin scale, :math:`(\frac{Q_L}{Q_H})_{reversible} = \frac{T_L}{T_H}`, and therefore by substitution we get: .. important:: Efficiency of Carnot Heat Engine .. math:: \eta_{thermal, reversible} = 1 - \frac{T_L}{T_H} |br| .. index:: carnot efficiency - refrigerator **Refrigerator** From :eq:`COP_refrigerator`, we can write: .. math:: COP_{refrigerator, reversible} = (\frac{Q_L}{Q_H-Q_L})_{reversible} = (\frac{1}{\frac{Q_H}{Q_L} - 1})_{reversible} Using the definition of Kelvin scale, :math:`(\frac{Q_H}{Q_L})_{reversible} = \frac{T_H}{T_L}`, and therefore by substitution we get: .. important:: Efficiency of Carnot Refrigertor .. math:: COP_{refrigerator, reversible} = \frac{1}{\frac{T_H}{T_L}-1} |br| .. index:: carnot efficiency - heat pump **Heat Pump** From :eq:`COP_heat_pump`, we can write: .. math:: COP_{heatpump, reversible} = (\frac{Q_H}{Q_H-Q_L})_{reversible} = (\frac{1}{1-\frac{Q_L}{Q_H}})_{reversible} Using the definition of Kelvin scale, :math:`(\frac{Q_L}{Q_H})_{reversible} = \frac{T_L}{T_H}`, and therefore by substitution we get: .. important:: Efficiency of Carnot Heat Pump .. math:: COP_{heatpump, reversible} = \frac{1}{1- \frac{T_L}{T_H}}