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Thermodynamics

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Published in: Physics
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Sidhant S / Bathinda

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  1. Chapter 4 THE SECOND LAW OF THERMODYNAMICS 4.1 Limitations of First Law of Thermodynamics If a well insulated tank of fluid is stirred by a rotating paddle wheel, the energy of the fluid increases. If the stirrer is stopped, however the energy of the fluid will not decrease and cause the stirrer to rotate in the opposite direction. The possibility of this process proceeding in the opposite direction is not excluded by the first law of Thermodynamics. Hence first law of thermodynamics does not allow us to predict whether a proposed conceived energy conversion is possible or not. In all the internal combustion engines fuel and air mixture is supplied at room temperature. This mixture undergoes combustion inside the engine and gives out work. Exhaust gases coming out of the engine are always at higher temperature, indicating that some heat is taken away into atmosphere. Hence, in all the IC engines only a part of the heat is converted into work. From our experience we know that if any attempt is made to convert all the heat into work, our effort will go in vain. This limitation in the extent of energy conversion has also not been addressed in first law of thermodynamics. 4.2 The Second law of Thermodynamics Kelvin Planck's statement It is impossible to construct a device that, operating continuously, will produce no effect other than transfer of heat from a single thermal reservoir and performance of an equal amount of work. The term thermal reservoir refers to a very large system in stable equilibrium, to which or from which, any amount of heat can be transferred at constant temperature. A thermal reservoir supplying heat continuously at constant temperature is known as source. (Example : Sun) A thermal reservoir receiving heat continuously at constant temperature is known as sink. (Examples : River, Sea) From Kelvin-Planck statement it is clear that for any system to operate in a cycle and to give out work continuously it should interact with a minimum of two reservoirs at different temperatures. The system will receive heat from the high temperature reservoir and reject heat to the low temperature reservoir. Such devices are known as heat engines. Performance (or) Efficiency of a heat engine can be expressed as the ratio of desired output to the required input. In a heat engine the desired output is net work output and the required input is total heat input
  2. Source Desired Effect 00 Qin Required Effect Heat Engine Qout Sink Figure 4.1 Heat Engine From first law of thermodynamics net wnet Clausius statement : Unaided by an external agency heat can not be transferred from a body at lower temperature to a body at higher temperature. Devices that are used to transfer heat from a body at lower temperature to a body at higher temperature are known as refrigerators (or) heat pumps. If the high temperature side is atmosphere it is a refrigerator. If the low temperature side is atmosphere it is known as a heat pump. The performance index here is called coefficient of performance (COP). In refrigerator (and heat pumps) the performance is the ratio of two independent parameters and hence the possibility of getting the value more than unity is always there. But the term efficiency is restricted to a maximum of unity. Hence the term efficiency is not used here. Desired Effect COP Re quired Effect COP 43
  3. Taking work as external agency, for refrigerators (Figure 4.2) From first law COP Required Effect Desired Effect (4.3) Sink [Atmosphere] Refrige rator Source [conditioned Space] Figure 4.2 Refrigerator 44
  4. Desired Effect Required Effect Sink [Conditioned Space] 000 QI Heat Pump Source [Atmosphere] Figure 4.3 Heat Pump Similarly for a heat pumps (Figure 4.3) Desired Effect COP Re quired Effect COP Since, COP 45
  5. 4.3 Equivalence of Kelvin-Planck and Clausius Statements The Clausius and Kelvin-Planck statements of the second law are entirely equivalent. This equivalence can be demonstrated by showing that the violation of either statement can result in violation of the other one. Referring to Figure 4.4(a) the device marked Clausius violator is pumping QI amount of heat from a low temperature reservoir at Tl to a high temperature reservoir at T without the aid of any external agency. This is an impossible arrangement. If such an arrangement is possible it would also violate Kelvin-Planck statement. Let a heat engine operating between the same reservoirs at T 2 and Tl take in Q2 as heat input at T . It converts a part of this heat into work and rejects heat Q to the sink at T l. Since the Clausius violator is rejecting the same quantity Q at T 2, it can be supplied directly into the heat engine so that the reservoir at T 2 can be eliminated. This combination as shown in Figure 4.4 (b) is producing continuous work with a single reservoir at T . Hence it violates the Kelvin-Planck statement. Reservoir Reservoir w Heat Clausius Engine violator Reservoir at Tl (a) 46
  6. Q2 Clausius violator Q2 w Heat Engine Reservoir at Tl (b) Figure 4.4 Illustration of the equivalence of Clausius and Kelvin-Planck's statement Referring to Figure 4.5 a Kelvin-planck violator is converting all heat (OH taken from the reservoir at TH into work. If such an impossible heat engine is assumed to exist it will violate the Clausius statement. Consider a refrigerator pumping QL heat from the low temperature reservoir at T to the reservoir at higher temperature T H. Combined with the Kelvin-Planck violator, the arrangement is pumping QL heat from TL to T H , without any external agency. Hence it violate the Clausius statement. 4.4 Reversible Process A process is said to be reversible if it can be reversed without leaving any trace on the surroundings. For example, let a system be taken from state 1 to state 2 with a work transfer of +5 kJ and heat transfer of —10 kJ. If the process is reversible, while taking the system from state 2 to state 1, the work transfer must be 5 kJ and heat transfer must be +10 kJ. So that, both the system and surroundings are returned to their initial states at the end of the process 2 to 1. 4.5 Irreversibility and Causes of Irreversibility The factors that make a process irreversible are known as irreversibilities. Various forms of irreversibilities are listed below. 47
  7. a) Friction b) Heat transfer: through finite temperature difference Friction occurs at the interface of two bodies moving relative to each other. It is the main cause of irreversibility in many processes. Energy spent in overcoming friction is dissipated in the form of heat which can never be restored. Once heat is transferred from a body at higher temperature to a body at lower temperature, it can never be reversed without the aid of an external agency. c) Unresisted expansion : Consider a vessel with two chambers as given in the arrangement as shown in Fig. 4.6. If the members separating the gas from vacuum is removed, gas will expand and occupy the entire space. Since the expansion has no influence on the surroundings, there is no work output in this process. But to restore the initial arrangement, a definite work input is required. d) Mixing of two gases : Consider a vessel with two chambers, one with 02 and the other with e) Throttling N . When the member separating O & N 2 is removed, uniform mixing is taking place without any work output. But such a process can not be reversed without any work input. It is a totally irreversible process. Gas or vapour will expand through a restricted passage with its pressure decreasing rapidly without any work output. Such an expansion can not be reversed. 4.6 Externally and internally reversible processes As mentioned earlier if no irreversibilities occur outside the system boundaries during the process, it is known as externally reversible. If no irreversibilities occur within the boundary of the system during a process, it is known as internally reversible process. For such a process, the path of the reverse process will follow exactly that of the forward process in any property diagram. To be totally reversible or simply reversible both external and internal reversibilities must be ensured. 48
  8. 4.7 The Carnot Cycle In 1824, Nicholas Sadi Carnot proposed a classical ideal cycle consisting of four processes. All processes are individually reversible and hence the cycle as a whole is a reversible cycle. The processes that make up the Carnot cycle are : Process 1-2 The working substance is taken in a piston cylinder arrangement as given in Figure 4.8(a). Heat is added reversibly and isothermally from a high temperature reservoir at T H. Since the process is to be reversible, the temperature TH of the reservoir should be equal to or infinitesimally greater than that of the working substance. Vol. 4.0 Press. Vol. Temp. isothermal process 2 Figure 4.8(a) Process 2-3 2.0 300 200 Press. adiöatic process 2 3 Figure 4.8(b) The working substance is allowed to expand reversibly and adiabatically until its temperature falls down to Tv The process is represented by Figure 4.8(b) Process 3-4 Heat is rejected by the working substance to a low temperature reservoir kept TL or at temperature infinitesimally smaller than T Vol. vol. 3.0 Press. Tetpp, isothermal process 3 > 4 5.0 300 200 Press. adiabatic process 4 49
  9. Process 4-1 The working substance is then compressed reversibly and adiabatically until its temperature becomes TH and the cycle continues. The cycle has been represented in a p-V diagram in Figure 4.9. The included area represents the net work done in the cycle. From first law of thermodynamics net workdone is equal to net heat transfer in the cycle. Since QH is the heat added to system and QL is the heat rejected by the system, the neat heat transfer is QH QL Isothermal heat addition 1 Adiabatic compression Isothermal heat rejection Adiabatic expa sion Efficiency of Carnot Engine = Where 4 net 3 3 Since the process is isothermal U U 3 50
  10. Similarly QH P3 V 3 In mRT In mRTH In Process 2-3 is reversible adiabatic 3 Process 4-1 is also reversible adiabatic 4 194 From the above two expressions Substituting the above condition we get Carnot in mRTL In mRT In It shows that efficiency of carnot engine is purely a function of TH and T . Since the carnot cycle being completely reversible, if carried out in reverse direction, the magnitudes of all energy transfers remain the same but their sign change. This reversed carnot 51
  11. cycle can be applied for a refrigerator or a heat pump. Figure 4.10 shows the p-V diagram of a reversed carnot cycle. 52

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