Unit – 4

Second Law of ThermodynamicsSecond Law of Thermodynamics

Limitations of First law of thermodynamics

According to first law of thermodynamics heat and work are mutually convertible during any cycle of a closed system. This law does not specify the conditions which conversions is possible. It also does not give any information regarding the direction of heat and work.

The following systems fallow the first law of thermodynamics. However some limitation will be observed in all such cases.

Example : 1

Fig-1: Example showing the transfer of heat from high temperature system to low temperature system

Fig-1 shows two systems, one at higher temperature and the other at a lower temperature under going a process in which heat transferred from the high temperature system to low temperature system. This process can takes place but it is impossible to complete a cycle by transferring heat from low temperature system to high temperature system, by heat transfer only.

Example-2

Fig-2: A closed system that undergoes a cycle involving heat & work

Fig-2 shows a closed system (gas) and the surroundings. Let the gas constitute the system and as per the discussion of first law, let this system undergoes a cycle in which work is first done on the

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system by the paddle wheel as the weight is lowered. Then the cycle is completed by transferring het to the surroundings. However, these processes cannot be reversed, i.e. if we transfer heat to the gas (as shown by the dotted arrow), the temperature of the gas will increase, but the paddle wheel will not turn and raise the weight.

Example-3

Consider the running automobile vehicle is stopped by applying brakes. The breaks get hot and the kinetic energy lost by the vehicle is gained by the breaks whose temperature increases. The first law of thermodynamics would be satisfied if the break were to cool off and give back its internal energy to vehicle causing the vehicle to resume its motion.

Example -4

A hot cup of coffee cools by virtue of heat transfer to the cooler surroundings but once it is cooled, it can never be heated by addition of heat from the cooler surrounding.

Thermal Reservoir

A thermal reservoir is a body to which and from which heat can be transferred indefinitely without any appreciable change in its temperature. Thus in general it may be considered as a system in which any amount of energy may be dumped or extracted ou6t and there shall be no change in its temperature.

Eg. Atmosphere, large river, sea etc.

Thermal reservoir can be of two types depending upon nature of heat interaction (i.e. heat rejection or heat absorption) from it. Thermal reservoir which rejects heat from it is called ‘source’. While the heat reservoir which absorbs heat is called ‘sink’.

Devices Converting Heat into work:

Direct Heat engine

Heat engine is a device used for converting heat into work and it defined as ‘a device operating in a cycle between high temperature source and low temperature sink and producing work. Heat engine receives heat from the source, transforms some portion of heat into work and rejects the balance heat to sink. All process occurring in heat engine constitute cycle.

Fig-3: Heat engine

A schematic diagram of the direct heat engine is shown in the fig-3, where Q1 is the heat received by the heat engine from the high temperature reservoir (Source at a temperature T1), W is the network

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done by the engine and Q2 is the amount of heat transferred to the low temperature reservoir (Sink at a temperature T2).

The performance of a direct heat engine is assessed quantitatively by using a parameter called ‘Thermal Efficiency’ (η) of the engine, and it is defined as the ratio of network output from the engine to the heat absorbed by the engine from the source. Symbolically this can be written as.

Where Net work (W) = (Heat supplied to Heat-engine) – (Heat rejected by Heat-engine)

W = Q1 – Q2

or, in general

Examples of Heat engine include steam engines, steam and gas turbines, spark-ignition and diesel engines, and the "external combustion" engine. Such engines can provide motive power for transportation, to operate machinery, or to produce electricity.

Steam or Gas Turbine Power Plant

Fig-4: Closed cycle Steam turbine power plant

The Steam turbine plant in fig-4 shows that heat is added to the high pressure water (working fluid) from 1–2 in a Boiler (Source), the high pressure steam is expanded in a turbine from 2-3 and produces positive work. After expansion the low pressure and low temperature steam goes to the Condenser (Sink) where it condenses by giving up the heat to a cooling agent from 3-4. The condensate coming out of the condenser will be at low pressure. The pressure of this condensate is increased by means of a Compressor from 4-1, and the high pressure water is sent back to the Boiler.

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The work required for compression is quite small as compared to positive available in turbine and is supplied by the turbine itself.

Heat Engine model for it shall be as follows,

Fig-5: Heat engine representation of steam turbine plant

The work output from the plant Wnet is

Wnet = (Work produced by Turbine) – (Work supplied to Compressor) = WT – WC

Also Wnet = (Heat supplied to Heat-engine) – (Heat rejected by Heat-engine)

Wnet = Qadded – Qrejected

WT – WC = Qadded – Qrejected

The Efficiency of the Heat-Engine is

=

=

Devices Converting work into heat:

a. Heat Pump

Heat pump refers to a device used for extracting heat from a low temperature surroundings and sending it to high temperature body, while operating in a cycle. In other words heat pump maintains a body or system at temperature higher than surroundings, while operating in cycle. Working Cycle for a heat pump is given below (fig-6)

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Fig-6: Heat Pump Working Cycle

Heat pump works by exploiting the physical properties of an evaporating and condensing fluid known as a refrigerant.

The working fluid, in its gaseous state (1), is pressurized and circulated through the system by a compressor. On the discharge side of the compressor (2), the now hot and highly pressurized gas is cooled in a heat exchanger, called a Condenser, until it condenses into a high pressure, moderate temperature liquid (3) by rejecting heat (Qrejected). The condensed refrigerant then passes through a pressure-lowering device like an expansion valve. This device then passes the low pressure liquid refrigerant (4) to another heat exchanger, the evaporator where the refrigerant evaporates into a gas via heat absorption (Qadded). The refrigerant then returns to the compressor and the cycle is repeated.

The Block Diagram of a Heat Pump is shown in fig-7 below

Fig-7: Heat pump

As the heat pump transfers heat from low temperature from low temperature to high temperature, which is non spontaneous process, so external work is required for realizing such heat transfer. Heat pump shown picks up heat Q2 at temperature T2 and rejects Q1 for maintaining high temperature body at temperature T1. For causing this heat transfer heat pump is supplied with work W as shown.

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Since the heat pump is not work producing device and also its objective is to maintain a body at high temperature, so its performance is measured through a parameter called ‘Coefficient of Performance’ (C.O.P). The Coefficient of performance is defined by ratio of desired effect and net work done for getting the desired effect.

For Heat pump:

Net work = W

Desired Effect = Heat transferred Q1 to high temperature body at temperature, T1.

Also W = Q1 – Q2

So

b. Refrigerator

Refrigerator is device similar to heat pump but with reverse objective. It maintains a body at temperature lower than that of surroundings while operating in a cycle. Working Cycle for a refrigerator is given below (fig-8)

Fig-8: Refrigerator Working Cycle

Refrigerator also works by exploiting the physical properties of an evaporating and condensing fluid known as a refrigerant. (Freon or Ammonia)

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The working fluid (refrigerant), in its gaseous state, is pressurized and circulated through the system by a compressor. As the refrigerant is compressed, it increases in temperature and pressure (1-2). After the compressor the refrigerant passes through the heat exchanger, called a Condenser, until it condenses into a high pressure, moderate temperature liquid by rejecting heat (Qrejected) to a surrounding environment. The condensed refrigerant then passes through a pressure-lowering device like an expansion valve, which reduces the temperature and pressure of the refrigerant (3-4). This device then passes the low pressure liquid refrigerant to another heat exchanger called Evaporator where the refrigerant evaporates into a gas by heat absorption (Qadded). The refrigerant then returns to the compressor and the cycle is repeated (4-1).

The Block Diagram of a Refrigerator is shown fig-9 below

Fig-9: Refrigerator

Refrigerator also performs a non spontaneous process of extracting heat from low temperature body for maintaining it cool, therefore external work W is to be done for realizing it.

The block diagram (fig-9) shows that refrigerator extracts heat Q2 for maintaining body at low temperature T2 at expense of work W and rejects heat Q1 to high temperature surroundings.

Performance of refrigerator is also measured by Coefficient of Performance(C.O.P), which could be defined as:

For Heat pump:

Net work = W

Desired Effect = Heat absorbed Q2 from low temperature body at temperature, T2.

Here W = Q1 – Q2

Note: Heat pump and Refrigerator are also called as Reversed Heat Engines

C.O.P values of heat pump and refrigerator can be related as: (COP)HP = (COP)refrigerator + 1

Statements of Second Law of Thermodynamics:

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There are two classical statements of second law of thermodynamics

1) Kelvin – Planck statement

2) Clausius statement

Kelvin – Planck statement

‘It is impossible to construct a heat engine operating on a cycle whose sole effect is transfer of heat energy from a single heat reservoir and its conversion into equal amount of work’.

Possible

Fig-10: Illustration of Kelvin-Plank Statement

This statement states that it is impossible to construct a heat engine which operates in a cycle and receives a given amount of heat from a high temperature body and does an equal amount of work. The only alternative is that some heat must be transferred from working fluid at a lower temperature to a low temperature body as shown in fig-4.5

This statement also implies that it is impossible to build a heat engine having a thermal efficiency of 100 percent.

Clausius statement

‘It is impossible to construct a device operating on a cycle whose sole effect is transfer of heat from a low temperature body to high temperature body.’

Not Possible Possible

Fig-11: Illustration of Clausius statementThis statement is related to the devices converting work into heat i.e. heat pump or refrigerator, says that it is impossible to construct a heat pump or refrigerator that operates without an input of work.

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Not Possible

Hence, external work is necessary to drive a heat pump or refrigerator which in effect transfers heat from a cooler body to a hotter body or extracts from cooler body to a hotter body.

This statement also implies that the coefficient of performance is always than infinity.

Equivalence of the Two Statements:

The Kelvin-Plank and the Clausius statements are equivalent in their consequences, and either statement can be used as the expression of the second law of thermodynamics. Any device that violates the Kelvin-Plank statement also violates the Clausius statement, and vice versa. This can be demonstrated as follows.

Proof of violation of the Kelvin-Plank statement results in violation of the Clausius statement

Fig-12 System based on Violation of Kelvin-Plank Statement

Consider a heat engine producing net work W by extracting heat with only one reservoir at temperature T1, thus based on violation of Kelvin-Plank Statement. Let us also have a perfect heat pump operating between two reservoirs at temperatures T1 and T2 as shown in fig-12 above. Let the work requirement of heat pump may be met from the work available from heat engine.

From the Fig-12,

The Work output from the Heat engine: W = Q1

The Work input to the Heat pump: W = Q2 – Q3

Since all the work output from the Heat engine is supplied to the Heat Pump

Therefore for Heat Pump Q1 = Q2 – Q3

i.e. the combination of heat engine and heat pump shall thus result in a equivalent system working as heat pump transferring heat from low temperature T2 to high temperature T1 without expense of any external work. This heat pump is based on the violation of Clausius statement and therefore not possible.

Proof of violation of the Clausius statement results in violation of the Kelvin-Plank statement

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≡

Fig-13 System based on Violation of Clausius Statement

Consider a heat pump which operating in cycle transfers heat from low temperature reservoir to high temperature reservoir without expense of any work, thus based on violation of Clausius statement. Heat pump transfers heat Q1 to high temperature reservoir by extracting heat Q2 from low temperature reservoir. Mathematically, as no work is done on pump, so Q2 = Q3

Let us also have a perfect heat engine operating between two reservoirs at temperatures T1 and T2 as shown in fig-13 above.

From the Fig-13,

The Work output from the Heat engine: W = Q1 – Q2

The Work input to the Heat pump: W = Q2 – Q3

= Q2 – Q2 = 0Let us now devise for heat rejected from heat pump be given directly to heat engine. i.e. the amount of heat rejected by heat pump Q3(or Q2) is supplied to heat engine. Arrangement is shown by dotted lines

In such a case, the work done by heat engine will be

W = Q1

Because Q1 is equal to Q3 (i.e heat rejected by heat pump) , and Q3 again equal to Q2 (i.e. heat supplied to heat pump). Therefore there is no transfer to the low temperature reservoir, and hence the combination of heat engine and heat pump thus result in a equivalent system working as heat engine transfers heat T1 from the high temperature reservoir and produces equivalent amount of work.

Perpetual Motion Machine of First Kind (PMM-1)

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≡

It is an imaginary device which produces a continuous supply of work without absorbing any energy from the surrounding or from the system. Such a machine in effect creates energy from nothing and violates the first law of thermodynamics. (fig-14)

As per the law of conservation of energy, no engine can produce mechanical work continuously without some other form of energy disappearing simultaneously.

Perpetual Motion Machine of Second Kind (PMM-2)

Without violating the first law, a machine can be imagined which would continuously absorbs heat from single thermal reservoir and would convert this heat into work. The efficiency of such machine would be 100 percent. This machine is called the perpetual motion machine of second kind (PMM2)

Fig-16 shows the PMM2. a machine of this kind will be evidently violates the second law of thermodynamics.

Reversible and Irreversible process

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When the system undergoes change from its initial state to the final state, the system is said to have undergone a process. During thermodynamic process the one or more of the properties of the system like temperature, pressure, volume, enthalpy or heat, entropy etc changes. The second law of thermodynamics enables to classify all the processes under two main categories: reversible or ideal process and irreversible or natural process.

Reversible Process

A process that, once having taken place, can be reversed and in doing so leaves no change in either the system or the surroundings is called reversible process. In this process every state along some path is an equilibrium state. This process is possible if the net of heat and net work exchange between the system and the surrounding is zero for the combined process.

Fig-17: Reversible Process

A reversible process is shown in the fig-17; let us suppose that the system has undergone change from state A to state B. If the system can be restored from state B to state A, and there is no change in the universe, then the process is said to be reversible process. The reversible process can be reversed completely and there is no trace left to show that the system had undergone thermodynamic change.

The phenomenon of a system undergoing reversible change is also called as reversibility. In actual practice the reversible process never occurs, thus it is an ideal or hypothetical process. All thermodynamic process are attempted to reach close to the reversible process in order to give best performance

The following are some of the examples of reversible process

a) Gradual extension of a spring b) Frictionless motion of solids

c) Slow frictionless adiabatic expansion of gas d) Slow frictionless adiabatic expansion of gas

Irreversible Process

A process for which a system cannot be restored to its initial state and in doing so it influences either the system or surroundings is called irreversible process. In this process the system passes through a series of non-equilibrium states

An irreversible process is shown in the fig-18; let us suppose that the system has undergone change from state A to state B, in which the process will not retrace the reverse path to restore the original state.

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Fig-18: Irreversible process

The irreversible process is also called as natural process because all the processes occurring in the nature are irreversible processes. The phenomenon of system undergoing irreversible process is called as irreversibility.

Some of the examples of irreversible process are:

a) Combustion process b) Mixing two fluids

c) Free or unrestricted expansion d) Heat transfer through finite temperature difference

e) Process involving friction f) Plastic Deformation

Factors that make a process Irreversible:

The factors that cause a process to be irreversible are called irreversibilities. They include friction, unrestrained expansion, mixing of two fluids, heat transfer across a finite temperature difference, electric resistance, inelastic deformation of solids, and chemical reactions.

Some of the frequently encountered irreversibilities are explained below.

1. Friction

Fig-19: Process involving Friction

Friction is the familiar form of irreversibility associated with the bodies in motion. When two bodies in contact are forced to move relative to each other, for example consider the piston cylinder arrangement shown in fig-19, a friction force that opposes the motion develops at the interface of cylinder and the piston, some work is needed to overcome this friction force. The energy supplied as work is eventually converted to heat during the process and is transferred to the bodies in contact. Even when the direction of motion of the piston is reversed, more of the work is converted to heat while overcoming the friction force that also opposes the reverse motion of piston.

Friction is also encountered between a fluid and solid and even between the layers of a fluid moving at different velocities.

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2. Unrestrained Expansion or Free Expansion

Fig-20: Unrestrained Expansion

An unstrained expansion is shown in the fig-20, in which a gas is separated from vacuum by a membrane. When the membrane is ruptured, the gas fills the entire tank (fig-20a). The only way to restore the system to its original state is to compress it to its initial volume, while transferring heat from the gas to the surrounding until it reaches the initial temperature (fig-20b). Since the surroundings are not restored to their initial state, unrestrained expansion is an irreversible process.

3. Heat Transfer through a finite temperature difference

Fig-21: Heat transfer over a finite temperature difference

Heat transfer occurs only when there exist temperature difference between bodies undergoing heat transfer. Consider the transfer of heat from a high temperature body to a low temperature body as fig-21(a). The only way of reversing this process, that is, transferring heat from low temperature body to high temperature body is by means of heat pump as shown in fig-21(b), which requires some external work. Hence, the process is an irreversible one.

4. Mixing of two Substances

Let us consider the mixing of any two gases (say oxygen and nitrogen) as shown in fig-22 to form a mixture by rupturing the membrane.

Fig-22: Illustration of mixing of two substances

Now, if this mixture has to be separated back to oxygen and nitrogen, an air separation plant has to be used, which requires some amount of work and heat transfer. Hence, we conclude that the mixing of two substances is an irreversible process since the surroundings are not restored to their initial state.

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5. Electrical resistance

Fig-23: Electrical resistance

The flow electric current (I) through a wire represents work transfer, because the current can drive motor which can raise a weight. Because of electrical resistance dissipation of electrical work into internal energy or heat takes place as shown in fig-23. The reverse transformation from heat or internal energy to electrical work is not possible.

6. Inelastic Solid Deformation

Deformation of solids, which are inelastic type is also irreversible and thus causes irreversibility in the process. If the deformation occurs within elastic limit then it does not lead to irreversibility as it is of reversible type.

Difference between Reversible and Irreversible process

Reversible Process Irreversible Process

Reversible Process can not be realized in Practice

All practical processes occurring are irreversible processes

The Process can be carried out in a reverse direction following the same path

Process, when carried out in reverse direction follows the path different from that in forward direction

A reversible process leaves no traces of occurrences of process upon the system and surroundings after its reversal.

The evidences of process having occurred are evident even after reversal of irreversible process

Such processes can occur in either directions without violating second law of thermodynamics

Occurrence of irreversible processes in either direction is not possible, as in one direction it shall be accomplished with violation of second law of thermodynamics.

A system undergoing reversible process has maximum efficiency. So the systems with reversible processes are considered as reference systems. (for comparison)

Systems having irreversible processes do not have maximum efficiency as it is accompanied by the wastage of energy.

Reversible process occurs at infinitesimal rate i.e. quasi-static process.

Irreversible processes occur at finite rate.

System remains throughout in thermodynamic equilibrium during occurrence of such process

System does not remain in thermodynamic equilibrium during occurrence of irreversible processes.

Examples:Frictionless motion, Controlled expansion and compression, Elastic deformations, Electric current with no resistance, Electrolysis, Polarization and magnetization process etc.

Examples:Viscous fluid flow, inelastic deformation and hysteresis effect, free expansion, Electric circuit with resistance, Mixing of dissimilar gases, Throttling process etc.

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Reversible Heat Engine

[A theoretical heat engine developed by Sadi Carnot]

A heat engine operating in an ideal, reversible cycle (also called as Carnot cycle) between two reservoirs without any losses is called reversible heat engine. Eg: Carnot Heat Engine

A reversible heat engine has got significantly higher thermal efficiencies than existing heat engines operating within the same temperature ranges.

This reversible heat engine can made to operate efficiently as either a forward heat engine or as a reverse heat engine at any given selected moment; thus, permitting the present heat engine to be utilized as an air conditioner or heat pump, or, alternatively, if a heat source is provided to the engine, as a forward heat engine producing a work output.

Carnot Cycle

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Fig-24: Carnot Cycle

Carnot cycle is a reversible cycle in which the working substance passes through a sequence of processes and it is brought back to its initial state. The cycle is named after Sadi Carnot, it is being discussed below.

Consider a substance contained in an insulated cylinder except its cylinder head with a frictionless piston. Referring to Fig- 24 above the substance is subjected to following sequences of processes.

(a) Heat Q1 is transferred during isothermal process 1-2 to the system at temperature T1 from a heat reservoir which is at a temperature infinitesimally higher than that of the system.

(b) Next the adiabatic cover is placed on the cylinder head and during the process 2-3 the system is allowed to expand adiabatically so that final temperature becomes T2.

(c) The adiabatic cover is removed and the diathermic cover placed on a cylinder head by which cylinder is made in contact sink which is at a temperature lower than that T2, Heat Q2 is transferred isothermally during the process 3-4 to the sink from the system.

(d) Finally the diathermic cover is replaced by adiabatic cover on the cylinder head, and the system undergoes compression process 4-1. The temperature of the system increases from T2 to T1 during the process.

Thermal efficiency of Carnot engine: the thermal efficiency of a heat engine is defined as the ratio of work output, W to the heat input, Q1. Therefore,

Efficiency,

Since the heat transfer during the adiabatic process 2-3 and 4-1 are zero (Adiabatic process), the net work done from first law of thermodynamics, we can determine as fallows:

Q = W = Q1-2 + Q2-3 + Q3-4 + Q4-1

i.e. W = Q1 + 0 – Q2 + 0

Efficiency, ------------- (1)

While describing the cycle no specific characteristics were imposed regarding the working substance. However, let assume that the working substance is an ideal gas. We can apply first law of thermodynamics to various as fallows:

Q = W + ΔU ( Where, ΔU = mcvΔT)

For process (1-2): Isothermal Heat addition

Q1 = W1-2

= m R T1 loge (since ΔU = 0, being an isothermal process)

For process (2-3): Adiabatic expansion

Q2-3 = 0

W2-3 = – (ΔU)2-3 = (U2 – U3) = m cv (T1–T2)

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For process (3-4): Isothermal Heat rejection

ΔU = 0 (isothermal process)

Q2 = W3-4 = m R T2 loge

= – m R T1 loge

For process (4-1): Reversible Adiabatic Compression

Q4-1 = 0 W4-1 = – (ΔU)4-1 = (U4 – U1)

= – m cv (T1–T2)

Net work done during the cycle

W = W1-2 + W2-3 + W3-4 + W4-1

= m R T1 loge + m cv (T1–T2) – m R T1 loge – m cv (T1–T2)

W = m R T1 loge – m R T1 loge

Carnot effieciency,

=

= ----------------- (2)

Considering the adiabatic process (2-3) and (4-1)

and or ----------------(3)

From the equation (2) and (3)

----------------- (4)

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Combining the equation (4) and (1)

or, ---------------(5)

It is obvious from the equation that heat transferred is proportional to the temperature of the heat engine

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