UNIT 3UNIT IIILimitations of the First Law Thermal Reservoir, Heat Engine, Heat pump, Parameters of performance, Second Law of Thermodynamics, Kelvin-Planck and Clausius Statements and their Equivalence / Corollaries, PMM of Second kind, Carnots principle, Carnot cycle and its specialties, Thermodynamic scale of Temperature,Clausius Inequality, Entropy, Principle of Entropy Increase Energy Equation, Availability and Irreversibility Thermodynamic Potentials, Gibbs and Helmholtz Functions, Maxwell Relations Elementary Treatment of Third Law of ThermodynamicsLimitations of first law The first law of thermodynamics has its own limitations in actual practice. Some situations are given below.1. According to the first law of thermodynamics, heat and work are mutually convertible .complete conversion is not possible in real practice.2. According to the first law of thermodynamics, there is no restriction on the direction of flow of work and heat, which is not true.3. According to the first law of thermodynamics, in energy cyclic process work and heat are exchangeable completely but from experience it is not.4. In natural way heat is not completely converted to work, but reverse is not automatically true.5. Heat flows from hot to cold region, but reverse is not automatically true.6. high pressure gas expands to low pressure but reverse is not atomatically true.7. from the above cases some external source of energy is required for reverse processto occur which again violates the 1st law of thermodynamics.8.Joules experiment amply demonstrate that energy, when supplied to a system in the form of work, can be completely converted into heat(work transfer -+ internal energy increase -+ heat transfer). 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. When work is converted into heat, we always have

but when heat is converted into work in a complete closed cycle process

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 coordinates. The changes that do take place in the large body as heat enters or leaves are so very slow and so very minute that all processes within it are quasi-static. The thermal energy reservoir TERH from which heat QI is transferred to the system operating in a heat engine cycle is called the source. The thermal energy reservoir TERL to which heat Q2 is rejected from the system during a cycle is the sink. A typical source is a constant temperature furnace where fuel is continuously burnt, and a typical sink is a river or sea or the atmosphere itself.

A mechanical energy reservoir (MER) is a large body enclosed by an adiabatic impermeable wall capable of storing work as potential energy (such as a raised weight or wound spring) or kinetic energy (such as a rotating flywheel). All processes of interest within an MER are essentially quasi-static. An MER receives and delivers mechanical energy quasi-statically,

Cyclic heat engine with source and sinkKelvin planck statement of Second law and PMM2 : IT is impossible for a heat engine to produce net work in a complete cycle if it exchanges heal only with bodies at a single fixed temperature.

If Q2 =0, the heat engine will produce net work in a complete cycle by exchanging heat with only one reservoir, thus violating the Kelvin-Planck statement fig 1 Such a heat engine is called a perpetual motion machine of the second kind, abbreviated to PMM2. A PMM2 is impossible .A heat engine has, therefore, to exchange heat with two thermal energy reservoirs at two different temperatures to produce net work in a complete cycle fig 2 .So long as there is a difference in temperature, motive power(i.e. work) can be produced. If the bodies with which the heat engine exchange heat are of finite heat capacities, work will be produced by the heat engine till the temperatures of the two bodies are equalized.

FIG 1 A PMM2

If the second law were not true, it would be possible to drive a ship across theocean by extracting heat from the ocean or to run a power plant by extracting heat from the surrounding air. Neither of these impossibilities violates the first law of thermodynamics. Both the ocean and the surrounding air contain an enormous store of internal energy, which, in principle, may be extracted in the form of a flow of beat. There is nothing in the first law to preclude the possibility of converting this heat completely into work. The second law is, therefore, a separate law of nature, and not a deduction of the first law ".The first law denies the possibility of creating or destroying energy; the second denies the possibility of utilizing energy in a particular way. The continual operation of a machine that creates its own energy and thus violates the first law is called the PMMI. The operation of a machine that utilizes the internal energy of only one TER, thus violating the second law, is called the PMM2.Clausius Statement of the second law Heat always flows from a body at a higher temperature to a body at a lower temperature. The reverse process never occurs spontaneously. Clausius' statement of the second law gives: It is impossible to construct a device which, operating in a cycle, will produce no effect other than tile transfer of heat from a cooler to a hotter body.Heat canot flow of itself from a body at a lower temperature to a body at a higher temperature. Some work must be expended to achieve this.

REFRIGERATOR AND HEAT PUMP: A refrigerator is a device which, operating in a cycle, maintains a body at a temperature lower than the temperature of the surroundings. Let the body A FIG 1 be maintained at t2, which is lower than the ambient temperature t1Even though A is insulated, there will always be heat leakage Q2 into the body from the surroundings by virtue of the temperature difference. In order to maintain, body A at the constant temperature t2, heat has to be removed from the body at the same rate at which heat is leaking into the body. This heat (Q2) is absorbed !by a working fluid, called the refrigerant, which evaporates in the evaporator EI at a temperature lower than t2 absorbing the latent heat of vaporization from the.body A which is cooled or refrigerated (Process 4-1). The vapour is first compressed in the cornpressor C, driven by a motor which absorbs work Wc (process 1-2), and is then condensed in the condenser C2, rejecting the latent heat of condensation Q1 at a temperature higher than that of the atmosphere (at t1) for heat transfer to take place (Process 2-3). The condensate then expands adiabatically through an expander (an engine or turbine) producing work (We),when the temperature drops to a value lower than t2 such that heat Q2 flows from the body A to make the refrigerant evaporate (process 3-4). Such a cyclic device of flow through E1-C1-C2-E2 is called a refrigerator. In a refrigerator cycle,attention is concentrated on the body A. Q2 and W are of primary interest. Just like efficiency in a heat engine cycle, there is a performance parameter in a refrigerator cycle, called the coefficient of performance, abbreviated to COP, which is defined as

1

Fig1 :A cyclic refrigeration plant

A heat pump is a device which, operating in a cycle, maintains a body, say B(Fig.2), at a temperature higher than the temperature of the surroundings. By virtue of the temperature difference, there will be heat leakage Q1 from the body to the surroundings. The body will be maintained at the constant temperature t1, if heat is discharged into the body at the same rate at which heat leaks out of the body. The heat is extracted from the low temperature reservoir, which is nothing but the atmosphere, and discharged into the high temperature body 0, with the expenditure of work W in a cyclic device called a heat pump. The working fluid operates in a cycle flowing through the evaporator E1',compressor C1, condenser C2 and expander E2, similar to a refrigerator, but the attention is here focussed onthe high temperature body B. Here Q1 and W are of primary interest, and the COP is defined as

Fig 2 A Cyclic heat pump . 2

..3

From the above equations 1 & 3 4T he COP of a heat pump is greater than the COP of a refrigerator by unity..5Q1 is always greater than W.EXAMPLE:For an electrical resistance heater, if W is the electrical energy consumption, then the heat transferred to the space at steady state is W only, i.e., QI = W.A 1 kW electric heater can give I kW of heat at steady state and nothing more.In other words, 1 kW of work (high grade energy) dissipates to give 1 kW of heat (low grade energy), which is thermodynamically inefficient.However, if this electrical energy W is used to drive the compressor of a heat pump, the heat supplied QI will always be more than W, or Q1 > W. Thus, a heat pump provides a thermodynamic advantage over direct heating. For heat to flow from a cooler to a hotter body, W cannot be zero, and hence,the COP (both for refrigerator and heat pump) canot be infinity. Therefore, W> 0, and COP T2 fig 1For reservoir A, It is negative because heat Q flows out of the reservoir. For reservoir B, It is positive because heat flows into the reservoir.

Fig 1The rod connecting the reservoirs suffers no entropy change. because, once in the steady state, its coordinates do not change.Therefore, for the isolated system comprising the reservoirs and the rod, andsince entropy is an additive property

since is positive, and the process is irreversible and possible.If is zero, and the process is reversible. Ifnegative and the process is impossible.ENTROPY TRANSFER MECHANISMSENTROPY TRANEFER IN THE FORM OF HEAT TRANSFER:

Since when heat is added to a systemis positive, and the entropy of the system increases. When heat is removed fromthe system, dQ is negative, and the entropy of the system decreases.Heat transferred to the system of fixed mass increases the internal energy ofthe system, as a result of which the molecules (of a gas) move with higherkinetic energy and collide more frequently, and so the disorder in the systemincreases. Heat is thus regarded as disorganised or disordered energy transferwhich increases molecular state. If heat Q flows reversiblyfrom the system to the surroundings at (To) fig ,the entropy increase of the surroundings is

The entropy of the system is reduced byThe temperature of the boundary where heat transfer occurs is the constanttemperature To. It may be said that the system has lost entropy to thesurroundings. Alternatively, one may state that the surroundings have gainedentropy from the system. Therefore, there is entropy transfer from the systemto the surroundings along with heat flow. In other words, since the heat inflowincreases the molecular disorder, there is flow of disorder along with heat. Thesign of entropy transfer is the same as the sign of heat transfer: positive, if intothe system, and negative, if out of the system.

On the other hand, there is no entropy transfer associated with work. InFig. 2, the system delivers work to a flywheel, where energy is stored in afully recoverable form. The flywheel molecules are simply put into rotationaround the axis in a perfectly organised manner, and there is no dissipation andhence no entropy increase of the flywheel. The same can be said about worktransfer in the compression of a spring or in the raising of a weight by a certainheight. There is thus no entropy transfer along with work. Ifwork is dissipatedadiabatically into internal energy increase of the system ,there is an entropy increase in the system, but there is as such no entropy transfer to it.Work is thus entropy-free, and no entropy is transferred with work. Energyis transferred with both heat and work, whereas entropy is transferred onlywith heat. The first law of thermodynamics makes no distinction between heattransfer and work. It considers them as equals. The distinction between heattransfer and work is brought about by the second law: an energy interactionwhich is accompanied by entropy transfer is heat transfer. and an energyinteraction which is not accompanied by entropy transfer is work. Thus. onlyenergy is exchanged during work interaction, whereas both energy and entropyare exchanged during heat transfer.(b)Entropy transfer through Mass Flow Mass contains entropy as well as energy, and the entropy and energy of a system are proportional to the mass. When the mass of a system is doubled, so are the entropy and energy of the system. Both entropy and energy are carried into or out of a system by streams of matter, and the rates of entropy and energy transport into or out of a system are proportional to the mass flow rate. Closed systems do not involve any mass flow and thus any entropy transport. When an amount of mass m enters or leaves a system, an entropy of amount ms, s being the specific entropy, accompanies it. Therefore, the entropy of a system increases by ms when the mass of amount m enters it, and decreases by by same amount when it leaves it at by same state.

Entropy Change in an Irreversible ProcessFor any process undergone by a system,1Consider the cycles as shown in fig 1

where A and B are reversible processes and C is an irreversible process. For thereversible cycle consisting of A and B2For the irreversible cycle consisting of A and C, by the inequality of Clausius,.3From eqs 2&3 ,.4Since the path B is reversible,..5Since entropy is a property,entropy changes for the paths B and C would bethe same. Therefore,6From eqs 4 & 6

Thus, for any irreversible process,

Whereas for a reversible process

Therefore, for the general case, we can write

The equality sign holds good for a reversible process and the inequality signfor an irreversible process.AVAILABLE ENREGY , AVAILABILITY & IRREVERSIBILITY:Available energy is also called exergy . Unavailable energy is also called anergy The sources of energy can be divided into two groups, viz. high grade energy and low grade energy.High grade energy Low grade energy

The conversion of high grade energy to shaft work is exemptedfrom the limitations of the second lawConversion of low grade energy is subject to limitations of 2nd law

Ex:Mechanical work, electrical work, water power, wind power, kinetic energy of a jet, tidal powerEx:Heat or thermal energy , Heat derived from nuclear fission or fusion, Heat derived from combustion of fossil fuels

The bulk of the high grade energy in the form of mechanical work or electricalenergy is obtained from sources of low grade energy, such as fuels, through themedium of the cyclic heat engine. The complete conversion of low grade energy,heat, into high grade energy, shaft-work, is impossible by virtue of the second law of thermodynamics. That part of the low grade energy which is available for conversion is referred to as available energy, while the part which, according to the second law, must be rejected, is known as unavailable energy.The originator of availability concept is Josiah Willard Gibbs. He indicated that environment plays an important part in evaluating the available energy.Available Energy referred to a CycleThe maximum work output obtainable from a certain heat input in a cyclic heatengine (Fig. .1) is called the available energy (A.E.), or the available' part of theenergy supplied. The minimum energy that has to be rejected to the sink by thesecond law is called the unavailable energy (U.E), or the unavailable part of theenergy supplied.Therefore,

For the given T1 and T2,

For a given T1, will increase with the decrease of T2 . The lowest practicable temperature of heat rejection is the temperature of the surroundings, To

Let us consider a finite process x-y, in which heat is supplied reversibly to aheat engine (fig 2). Taking an elementary cycle, if is the heat received by the engine reversibly at T1,then

Fig 1 fig 2 For the heat engine receiving heat for the whole process x-y, and rejecting heatat To

1

The unavailable energy is thus the product of the lowest temperature of heatrejection, and the change of entropy of the system during the process of supplying heat (fig 3)

Fig 3 Reversible Work by an Open System Exchanging Heat only with the Surroundings Let us consider an open system exchanging energy only with the surroundings atconstant temperature To and at constant pressure Po (Fig). A mass dm1enters the system at state 1, a mass dm2 leaves the system at state 2, an amount of heat is absorbed by the system, an amount of work is delivered by the system, and the energy of the system (control volume) changes by an amount , applying 1st law we have

1For the maximum work, the process must be entirely reversible. There is atemperature difference between the control volume and the surroundings. Tomake the heat transfer process reversible, let us assume a reversible heat engine E operating between the two. Again, the temperature of the fluid in the controlvolume may be different at different points. It is assumed that heat transfer occurs at points of the control surface where the temperature is T. Thus in aninfinitesimal reversible process an amount of heat is absorbed by the engineE from the surroundings at temperature ,an amount of heat is rejected bythe engine reversibly to the system where the temperature is T, and an amount of work is done by the engine. For a reversible engine,.2The work is always positive and is independent of the direction of heatflow. When heat will flow from the surroundings to the system, is positive and hence in eqn 2 positive and hence heat will flow from the system surroundings, is negative, and hence wouldbe positive.Now, since the process is reversible, the entropy change of the system will beequal to the net entropy transfer, and Therefore, 3

.45Substituting eq 1 for in eq.5.6On substituting the value of from eq.3

.7Eqn 7 is the general expression for the maximum work of an open system which exchanges heat only with the surroundings at THERMODYNAMIC POTENTIAlS The thermodynamical state of a system can be described in terms of the basic independent coordinates p,v,T & s. These four coordinates are insufficient to obtain complete knowledge of the system we use certain energy terms which are easily measurable, known as thermodynamics potentials. Thermodynamic energy potentials are energy functions that are mathematically formed by combining basic thermodynamic coordinates p,v,T& s in different ways .We adopt H,F,G for the thermodynamic functions giving extensive values and h.f.g. for their specific values (per unit mass). The four important thermodynamic potentials,u specific internal energyf.specific Helmholtz function gspecific gibbs free energyhspecific enthalpy.Internal energy: Form 1st law and 2nd law of thermodynamics dQ=du+ dW..1If we allow only pdv work then dQ=du+pdv.2dQ=Tds (if it is reversible from 2nd )3form equ 2 and 3 du+pdv= Tds du=Tds- pdv4Or Tds=du+pdv.5Equations 2&3 contains path function dQ and in exact differentials Equation 4 all are exact differentials (point function representing the thermodynamic state)From eqn 4 du=Tds-pdv..6 du is increment of energy in system )Enthalpy: (specific enthalpy h)H = U +PV WE know h = u + pv.. 7To express h is terms of p,v,T and s We eliminate u from eqn 7 can be written as in the following differential form dh = du+pdv+vdpfrom 5 du = T ds pdv dh = tds- pdv +pdv+vdpdh= Tds+vdp .. 8Tds =dh vdp Helmholt Free energy (f): Consider a real isothermal process at constant volume which is always irreversible .The entropy increses in an irreversible process never decreases.Therefore ds dQ/T Tds dQ d(Ts) dQ ( since T is constant) d(Ts) du + pdv d(Ts) du pdv.9 Since isothermal process is taking place at constant volume , Therefore d(Ts-u) 0 d(u-Ts) 0 10u-ts is thus a physical quantity representing energy which never increses during an Isothermal iso volumic process.This u-Ts is called Helmholtz free energy or Helmholtz free energy function f this f= u-Ts df =du-(Tds+sdT) df= -pdv-sdt df=-pdv-sdtGibbs free energy:Considering a real isothermal process at constant pressure (isobaric) which is always irreversible , from eqn 9d(Ts) du d(pv)d (Ts-u-pv) 0 d(u+pv-Ts) 0 this physical quantity representing energy which never increases in a isothermal process isobaric process is called Gibbs free energy (g) g=u+pv-Ts dg= du+pdv +vdp-tds-sdtdg=tds-pdv+pdv+vdp-tds-sdt dg = vdp-sdtGibbs free energy (g) :Let us now consider a system which is in both pressure and temperatureequilibrium with the surroundings before and after the process. When the volume of the system increases some work is done by the system against the surroundings (pdVwork), and this is not available for doing useful work. The availability of the system, as defined by neglecting the Ke and Pe changes can be expressed in the form the maximum work obtainable during a change of state is the decrease inavailability of the system,as given

If the initial and final equilibrium states of the system are at the same pressureand temperature of the surroundings, say Then, 11 The Gibbs function G is defined as 12Then for two equilibrium states at the same pressure P and temperature T

..13From eqs 12 &13..14...15

The decrease in the Gibbs function of a system sets an upper limit to the workthat can be performed, exclusive of pdV work, in any process between twoequilibrium states at the same temperature and pressure, provided the systemexchanges heat only with the environment which is at the same temperature andpressure as the end states of the system If theprocess is irreversible, the usefulwork is less than the maximum.MAXWELLS EQUATION :These equations are differential relations among the basic thermodynamic coordinates from thermodynamics potentials