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Second law of thermodynamics, thermodynamics note,thermo note chapter 5

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- 1. R Gnyawali / P Timilsina Page 1 5 Chapter: 5 Second Law of Thermodynamics 5.1 Necessity of Formulation of Second Law Drawbacks of First Law of Thermodynamics The first law of thermodynamics states that during any cycle that a system undergoes, the cyclic integral of the heat is equal to the cyclic integral of the work. ( ∫ ∫= WQ ) The first law, however, places no restriction on the direction of flow of heat and work. A cycle in which a given amount of heat is transferred from the system and an equal amount of work is done on the system satisfies the first law just as well as a cycle in which the flows of heat and work are reversed. However, we know from the experience that because a proposed cycle doesn’t violate the first law doesn’t ensure that the cycle will actually occur. This leads to the formulation of second law of thermodynamics. Thus, a cycle will occur only if both the first law and second law are satisfied. The second law tells that processes proceed in a certain direction but not in the opposite direction. 1. A hot cup of coffee cools by virtue of heat transfer to the surroundings, but heat will not flow from the cooler surroundings to the hotter cup of coffee. 2. Steam is kept in a chamber and other chamber is vacuum. If suddenly the partition is removed, the steam fills up the vacuum but the reverse doesn’t happen spontaneously. 3. Two blocks at different temperature eventually come to a same temperature but the process doesn’t occur in reverse direction.
- 2. R Gnyawali / P Timilsina Page 2 Thus, the second law needs to describe the direction for processes, and the direction is described in terms of a system property that characterizes the system’s randomness, disorder, or uncertainty. Qualitative differences between heat and work The first law of thermodynamics states that a certain energy balance will hold when a system undergoes a change of state or a process but it doesn’t give any information on whether that change of state or the process is at all feasible or not. Joule’s experiment demonstrate that energy, when supplied to a system in the form of work, can be completely converted into heat but the complete conversion of heat into work in a cycle is not possible. So heat and work are not completely interchangeable forms of energy. Work Internal Energy Heat (100% conversion) Heat can’t be converted completely and continuously into work in a cycle. Some heat has to be rejected without conversion. The conversion of low grade of energy (heat) into high grade of energy (work) in a cycle is impossible.
- 3. R Gnyawali / P Timilsina Page 3 Heat Work (100% conversion impossible) System Q W Q > W 5.2 Entropy and Second Law of Thermodynamics for an Isolated System Entropy The entropy of a system is a thermodynamic property which is a measure of the degree of molecular disorder existing in the system. It describes the randomness or uncertainty of the system. It is a function of a quantity of heat which shows the possibility of conversion of heat into work. Thus, for maximum entropy, there is minimum availability for conversion into work and for minimum entropy there is a maximum availability for conversion into work. Characteristics: 1. It increases when heat is supplied irrespective of the fact whether temperature changes or not. 2. It decreases when heat is removed whether the temperature changes or not. 3. It remains unchanged in all adiabatic reversible processes. 4. The increase in entropy is small when heat is added at a high temperature and is greater when heat addition is made at a lower temperature. Entropy and Second Law of Thermodynamics for an Isolated System The microscopic disorder of a system is prescribed by a system property is called entropy. “The entropy S, an extensive equilibrium property, must always increase or remain constant for an isolated system.” This is expressed mathematically as, 0≥isolateddS ……………………………..eq(1) Or, 0)( ≥− IsolatedInitialFinal SS …………………………………eq(2) Entropy, like our other thermodynamic properties, is defined only at equilibrium states or for quasi-equilibrium processes. Equation (2) shows that the entropy of the final state is never less than that of the initial state for any process which an isolated system undergoes. Entropy is a measure of the molecular disorder of the substance. Larger values of entropy imply larger disorder or uncertainty and lower values imply more microscopically organized states.
- 4. R Gnyawali / P Timilsina Page 4 The term entropy production or entropy generation Sgen is considered in eq (1) to eliminate the inequality sign. 0)( =− isolatedgenSdS δ Or 0)( =−− IsolatedInitialgenFinal SSS Here, genSδ is the entropy generated during a change in system state and is always positive or zero. 5.3 Reversible and Irreversible Process A reversible process for a system is defined as a process that once having taken place can be reversed so that the system and surroundings can be restored to their initial state. An irreversible process is a process which once having taken place can’t be restored to its initial state. The reversible process is an idealization, and all the actual processes are normally irreversible process. 5.3.1 Types of irreversibility 1. Mechanical and Thermal Irreversibility 2. Internal and External Irreversibility Mechanical irreversibility is associated with the fluid friction among the molecules and the mechanical friction between surfaces. Thermal irreversibility is associated with heat transfer with finite temperature difference between the parts of system or between a system and surroundings. Internal irreversibility is associated with fluid friction and temperature variation within the fluid. Combustion and diffusion also cause internal irreversibility. External irreversibility is associated with friction between the atmosphere and rotating members. All these absorb some work developed by the system. External irreversibility occurs outside the boundary of the system. 5.3.2 Factors that makes process irreversible 1. Friction 2. Unrestricted Expansion 3. Heat transfer through a finite temperature difference 4. Mixing of two different fluids 5.3.3 Examples Reversible Processes Irreversible Processes 1. Frictionless adiabatic expansion or compression 2. Friction less isothermal expansion or compression 3. Condensation or boiling of liquid 1. Free Unrestricted Expansion 2. Combustion, diffusion 3. Electric current flow through resistor 4. Mixing of two fluids 5. Heat transfer from high temperature to low temperature 4. Process involving friction
- 5. R Gnyawali / P Timilsina Page 5 5.3.4 For a process to be reversible, it should obey following conditions. 1. The process should not involve friction of any kind. 2. Heat transfer should not take place with finite temperature difference. 3. There should be no mixing. 4. There should be no free or unrestricted expansion. The process must proceed in a series of equilibrium states. It should move at an infinitely slow pace. 5.4 Heat Engine and Refrigeration Cycles Processes that return to their initial state are called cyclic processes. Energy Reservoirs A thermal energy reservoir (TER) is defined as a large body of infinite heat capacity, which is capable of absorbing or rejecting an unlimited quantity of heat without suffering appreciable changes in its thermodynamic properties. For example: River, Sea, Atmosphere etc. Heat Engine A heat engine cycle is a thermodynamic cycle in which there is a net heat transfer to the system and a net work transfer from the system. The device which executes a heat engine cycle is called a heat engine. This device takes heat as input and converts into work as output. The cycles have a common feature – they operate between two limiting temperatures. The high temperature results from the combustion process in the steam generator or within the cylinder. The low temperature results from the cooling process. The characteristics of these two temperature cycles are shown from a general viewpoint as a high temperature heat transfer reservoir or source (hot reservoir) at TH and a low temperature heat transfer reservoir or sink (cold reservoir) at TL. The cycle operating between these two temperatures is arbitrary. The general power cycle or engine is shown schematically in figure below. The first law for an arbitrary cycle states; ∫∫ = WQ ……………….eq(1) which is valid for an arbitrary collection of processes and for both reversible and irreversible cycles. For the heat engine cycle shown in above figure;
- 6. R Gnyawali / P Timilsina Page 6 LH QQW −= ………………..eq(2) Second law for all heat engine cycles is: 0)( ≤∫ T Qδ ……………………..eq(3) For reversible heat transfer with the two heat transfer reservoirs, the equation(3) becomes as; 0≤− L L H H T Q T Q or H L H L Q Q T T ≤ ……………………..eq(4) The efficiency of a cycle η is defined as; InputHeat OutputWorkOutputDesired ⋅ ⋅⋅ = )( η A power cycle or engine as shown in above figure has a work output W and an input from the high-temperature reservoir QH. Thus, this efficiency is; HQ W =η ………………eq(5) From equation (2) , the eq(5) becomes as; H L H LH Q Q Q QQ −= − = 1η …………..eq(6) The ratio of heat in eq(6) can be eliminated with eq(4) as; H L T T −≤1η …………………..eq(7) Where, equality applies to reversible cycle and inequality applies to irreversible cycle. Therefore, it can be written as; H L vIrrev T T −=< 1Reηη ……………………….eq(8) This shows that the maximum efficiency of a heat engine operating between two heat transfer reservoirs occurs for a reversible cycle. Refrigerator and Heat Pump A refrigerator is a device operating in a cycle which maintains a body, at a temperature lower than the temperature of the surroundings. The performance for a refrigerator is termed as the coefficient of performance (COP) and is given as LH LL R QQ Q W Q InputWork EffectcoolingEffectDesired COP − === . ).(. ………………….eq(1) First law; HL QQW −=− ………………….eq(2) Second law; for cyclic process ∫ ≤ 0 T Q 0≤− H H L L T Q T Q or
- 7. R Gnyawali / P Timilsina Page 7 L H L H T T Q Q ≥ ………………….eq(3) From eq(1) and eq(3), 1 1 1 1 − ≤ − = − = L H L HLH L R T T Q QQQ Q COP ………………….eq(4) revRIrrevR COPCOP ,, ≤ The equality applies to the reversible refrigerator and the inequality to the irreversible refrigerator. A Heat pump is a device operating in a cycle which maintains a body, at a temperature higher than the temperature of the surroundings. Coefficient of Performance of Heat Pump LH HH HP QQ Q W Q InputWork EffectHeatingEffectDesired COP − === . ).(. ………………….eq(5) 5.5 Carnot Cycle Carnot cycle is a heat engine cycle which operates between the given high-temperature and low-temperature reservoirs such that every process is reversible. If every process is reversible, the cycle is also reversible and if the cycle is reversed, the heat engine cycle becomes refrigerator or heat pump. This cycle was proposed by a French engineer Sadi Carnot (1796-1832). The working substance for Carnot cycle is air, an ideal gas. Assumptions for Carnot cycle 1. No friction is involved. 2. The transfer of heat doesn’t affect the temperature of source or sink. 3. Working medium air is perfect gas. 4. Compression and expansion are reversible. 5. Perfectly reversible adiabatic conditions are achieved.
- 8. R Gnyawali / P Timilsina Page 8 Process 1-2 It is a reversible isothermal process in which heat is transferred from the high- temperature reservoir at constant temperature (T1). The source at a high temperature is brought in contact with the bottom of the cylinder so that air expands isothermally from V1 to V2. )ln()ln()ln( 1 1 2 1 1 2 1121 crmRT V V mRT V V VPQ ===− Process 2-3 It is a reversible adiabatic process in which the temperature of the fluid decreases from T1 to T2. The source is removed and the bottom of the cylinder is insulated so that the air is allowed to expand adiabatically so that the volume increases from V2 to V3. 032 =−Q Process 3-4 It is a reversible isothermal process in which heat is rejected to the low temperature reservoir at T2. The bottom of the cylinder is brought in contact with the sink at temperature T2 so that air is compressed isothermally from V3 to V4. )ln()ln()ln( 3 3 4 3 3 4 3343 crmRT V V mRT V V VPQ ===− Process 4-1 It is a reversible adiabatic process in which temperature of working fluid is increased from T1 to T2 by compressing it reversibly and adiabatically. The sink is removed and the bottom of the cylinder is insulated so that the air is allowed to compress adiabatically so that the volume decreases from V4 to V1.
- 9. R Gnyawali / P Timilsina Page 9 014 =−Q Now, )ln()ln( 31 4321 cc rmRTrmRT QQ utNetHeatInpputNetWorkOut −= −= = −− And, the efficiency of a Carnot cycle is: H L c c T T T T T TT rmRT TTrmR HeatInput Workoutput −= −= − = − = = 1 1 )ln( ))(ln( 1 3 1 31 1 31 η It is impossible to construct an engine operating between only two heat reservoirs which will have a higher efficiency that a reversible engine operating between the same two reservoirs. In other words, for the same source and sink, a reversible engine has the higher efficiency. This is also known as Carnot’s Theorem. All reversible engines (Carnot Engines) operating between the same two reservoirs have the same efficiency. 5.6 Classical Statements of Second Law of Thermodynamics 5.6.1 Kelvin-Planck Statement It is impossible to construct a device that will operate in a cycle and produce no effect other than the extraction of heat from a single reservoir and the performance of equivalent amount of work.
- 10. R Gnyawali / P Timilsina Page 10 5.6.2 Clausius Statement It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a body at low temperature to a body at high temperature without work input. 5.6.3 Equivalence of Kelvin-Planck and Clausius Statement It can be proved that these statements are equivalent by proving the violation of one statement leads to the violation of other statement. 1. Violation of Kelvin-Planck statement leads to the violation of Clausius Statement: Let us assume a cyclic Heat Pump (HP) extracting heat Q2 from a low temperature reservoir at T2 and discharging heat to high temperature reservoir at T1 with the expenditure of work (W). Let’s assume a cyclic Heat Engine (HE) extracting heat Q1 from a high temperature reservoir and produces net work (W = Q1) without rejecting any heat thus violating Kelvin-Planck statement. These two constitute a device extracting heat Q2 from low temperature reservoir and rejecting to high temperature reservoir without any work input. Thus Clausius statement is violated. So, a violation of Kelvin-Planck statement leads to the violation of Clausius statement. 2. Violation of Clausius statement leads to the violation of Kelvin-Planck Statement: Let us consider a Heat Pump (HP) that requires no work and thus violates the Clausius statement. Let an amount of heat Q2 be transferred from the low temperature reservoir to the HP, and let same amount of heat Q2 be transferred to the high-temperature reservoir.
- 11. R Gnyawali / P Timilsina Page 11 Let us consider a Heat Engine (HE) where an amount of heat Q1 be transferred from the high-temperature reservoir and let the engine reject the amount of heat Q2 and does net work (W = Q1-Q2). Because there is no net heat transfer to the low-temperature reservoir, the low-temperature reservoir along with the HE and HP can be considered together as a device that operates in a cycle and produces no effect other than the exchange of heat with a single reservoir and performance of equal amount of work. Thus Kelvin-Planck Statement is also violated. So a violation of Clausius Statement leads to the violation of Kelvin-Planck statement.

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