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• Second law of Thermodynamics

# Second law of Thermodynamics

## Introduction

One commonly considered problem is of a simple steam power plant and by using the SFEE (i.e. the first law) and the properties of steam it is possible to calculate the work done and heat transfers for individual components.  However, we are not yet able to understand ways of improving the steam engine efficiencies.  It is with the help of the second law of TD that we should be able to answer the important questions we want to know:
i) What is the maximum achievable efficiency of a steam engine
ii) How can we reduce thermal pollution by decreasing the amount of cooling water.
The second law is one of the most important - if not the most important of all physical laws.  Like all laws it is based on experimental observation.  In fact it is the result of the question "how efficient can one make a steam engine".  So perhaps it is best if we start by considering engines and define them with the precision that Thermodynamic requires. The sort of steam engines we shall discuss are devices in boxes with no fluid entering or leaving but with just Heat and work crossing the boundaries.

## Heat Engines

Definition of heat engine:
"A heat engine (or cyclic Heat Power Plant CHPP) is a continuously operating Thermodynamic system at the boundary of which there are heat and work transfers".
Notes:
a) Continuously operating" means that the state of the system exhibits only periodic (cyclic) changes.
b) The heat engine is a Thermodynamic system and so no matter crosses the boundary e.g. simple steam power plant and closed-cycle gas turbine plant. However, a diesel engine (or IC engines in general) is not a heat engine (CHPP) because matter crosses its boundaries. Jet engine also is not a CHPP because matter, air, fuel exhaust crosses the boundary of system (see hand-out).
c) The definition says nothing about the direction of heat and work transfers so that a domestic refrigerator is also a heat engine,The refrigerator consists of four components, a condenser, a vapour compressor, an evaporator and a throttle.  If we compare this with the steam, or gas turbine plants we see that they are similar but with the role of components reversed.  The boiler (an evaporator) is a condenser, the turbine a compressor, the condenser an evaporator and the pump has been replaced by an expander, throttle valve.
d) We may visualise the direct and reverse heat engine operation diagrammatically.

The direct engine produces a net supply of work (per unit time or for unit mass flow of working substance) from a supply of heat and rejects heat at a lower temperature than it was supplied.
The Reversed Heat Engine requires work to be supplied to it.  The effect of work supply is to cause a heat transfer from the cold reservoir to the Hot reservoir. A Heat Reservoir is a source of thermal energy such that transfer heat from it or to it does not cause a change in temperature e.g. the ocean is a good example of a heat reservoir.

Performance of heat engines:
In order to assess and compare the performance of different heat engines we need define an efficiency or a performa w efficient.  For direct engines, such as steam or gas turbine plants, what we require is the work output and we pay by the fuel to be used in the boiler, or the heater so we define efficiency of a direct heat engine by:
Where  W  and QH are measured per unit time or per unit mass flow rate of the working substance.
From first law:
so for a cyclic process W = QH - QC

The term above is often called the thermal efficiency

For reversed engines, the "output" is QC in the case of "refrigerators" and QH in the case of Heat pumps and the input is W:

Of course both the first and second ratio can be greater than 1 and so usually the term "coefficient of performance" is used.

A typical COP for a domestic refrigerator is about 3-4 i.e. a 250 watt electrical supply will cause a 1 to 1.25 kW heat flow from evaporator (Food Cabinet).

## Improving efficiency of direct heat engines

So maximum efficiency is attained when Qc = 0 and the engin elooks as below:

This idea becomes even more impressive when we use this engine together with a reversed engine in the following arrangement;
Arrange for the reversed engine to transfer to the hot body as much heat as is used by the direct engine, so that the temperature stays at 1000oC.  Now:
Wr=Q-Qc, while Wd = QH so that there is a net work output i.e. No Fuel is Needed

This is known as The Perpetual Motion Machine Of The Second Kind i.e. the most environmentally friendly and cheapest way of producing power!  Of course, we all know that such a machine is impossible to make.  The question is why?  Because this machine does not violate the First Law.  The answer lies in the Second law of Thermodynamics.  There are two statements of the Second Law of Thermodynamics, one due to Kelvin and Planck and the other due to Clausius.

Kelvin and Planck Statement
"It is impossible to construct a device that will operate in a cycle and produce no other effect than the raising of a weight and the exchange of heat with a single reservoir"
or
"No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work".
That is impossible to make a heat engine which operates in a cycle that receives a given amount of heat from a high temperature body and does an equal amount of work.  Thus it is necessary that heat is transferred from the working fluid at a lower temperature to a lower-temperature reservoir.  This work can be done by the transfer of heat only if there are two temperature levels involved, and heat is transferred from the high temperature body to the heat engine and also from the heat engine to the low temperature body.  This implies that it is impossible to build a heat engine that has a thermal efficiency of 100% i.e_._ Complete Conversion of heat into work is impossible, while work can be completely converted into heat*.*  This is the directional implication of the Second Law.
This definition of course deals with the direct heat engine.  The Clausius definition however, concerns the reversed Heat Engine or refrigerator.
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 cooler body to a hotter body".
i.e. impossible to construct a refrigerator that operates without an input of work. Therefore, COP of refrigerators is always less than infinity".
Observations:
i) The two statements of the Second Law are both negative statements. Of course impossible to "prove" a negative statement.  However, the Second Law is based on experimental evidence.  Relevant experiments have directly or indirectly verified the Second Law and no experiment has ever been conducted that contradicts the Second Law.
ii) The two statements of the Second Law are equivalent i.e. it can be shown that the truth of each statement implies the truth of the other.  We shall show this later.
iii) The Second Law can also be stated as the impossibility of the perpetual motion machine of the second kind.

## Reversibility

As a result of second law we know that the complete conversion of heat to work is impossible  and so an efficiency of 100% cannot be achieved for a heat engine, or then we could have made PMM2. Then, the question is,  "what is the maximum efficiency of a CHPP?"
The answer to this was provided by Sadi Carnot (1800-1831) who stated the maximum achievable efficiency of a CHPP is that which can be achieved by a reversible CHPP.
What do we mean by reversible? Say you have a gas turbine which gives up x Joules of work when one kg of air expands from a high to a low pressure. Now if the turbine is driven backwards ie. As a compressor then the turbine is reversible if it needs x Joules of work to increase the pressure from P1 to P2.
"A heat engine which engages in heat transfer with two systems of fixed, but different temperatures, is reversible if its efficiency when operating directly is equal to the reciprocal of its coefficient of performance when operating as a heat pump."

Reversible if:
If:
Now we can state the theorem;
Theorem: The maximum efficiency of a cyclic heat engine operating between two fixed temperature levels is attained when the cycle is reversible.
Proof: (by a thought experiment)

(1) Assume that an irreversible cyclic engine can be found which is more efficient than the reversible one. For the same Q1:
W1>WR; Q2I > Q2R
(2) Use the irreversible engine to drive the reversible one backwards. The combination produces net work (WI-WR)>0
(3) We can then ignore the hot reservoir. Thus we end up with a CHPP producing (WI-WR) work and exchanging heat with a single reservoir at temperature T2, ie PMM2 which is impossible.

Hence:
Now we know that in order to obtain the maximum efficiency we have to have a reversible CHPP, ie a cycle device which undergoes reversible processes, we have to see what sort of processes are reversible or are close to being reversible. The ideal TD is known as reversible process.

Definition:
"A process is reversible if after the process has been completed, means can be found to restore the system and all elements of its surroundings to their respective initial states."
The second process which restores the system into the original state is known as an effacing process. Any process which is not reversible is irreversible.
The way to decide whether a process is reversible or irreversible is to apply the only natural law at our disposal which concerns directions, ie the 2nd law of TD.
To show a process is irreversible assume that it is reversible and has a effacer and then use the effacer in a cyclic process to produce a PMM2.

Example 1
Mechanical Friction: disc sander

The effacer is:

We do not need to explain how the effacer works for if process is reversible then effacer must exist.
Of course the effacer violates the second law, ie heat transfer from a single reservoir è Mechanical Friction is an irreversible process.

Example 2
Heat transfer across a finite temperature difference

Process
Effacer

The effacer obviously violates the Claussius definition of the second law and so heat transfer across a finite temperature difference is an irreversible process. Alternatively:

Obviously a PMM2.

Example 3
Un-resisted Expansion
Process: Puncture Partition and let gas expand into evacuated space

State of system moves from A to B. Process path cannot be drawn because the process is not quasi-static.
Q=0, W=0,DU=0 ie. No effect on the surroundings.
Assume there exist an effacing process EBA which restores state from B to A
i.e. volume decreases
pressure increases
U is unchanged
Now let us consider the following device:

So the gas goes through a turbine produce work and then goes to evacuated section. Then we apply heat Q1 to restore internal energy of gas to original value. Now we can use EBA to get back gas behind partition and repeat the cycle.
Thus we produce work W by exchanging heat with a single reservoir so we have PMM2.
Thus unresisted expansion is an irreversible process. Other examples of irreversible processes are:
(4) Inelastic (or plastic) deformation.
(5) Flow of electric current through a resistor
(6) Magnetisation of material exhibiting hystersis
(7) Mixing of two different liquids
In general all irreversible processes have the following features.
1. The conditions for TD equilibrium are not satisfied
2. Dissipative effects, such as viscosity, friction, inelasticity, electric resistance, and magnetic hystersis are present.
Thus if a process is performed "quasi-statically" and no dissipative effects are present, then the system passes through states of TD equilibrium, which may be traversed just as well in one direction as in the other. So we can conclude that a process isreversibleif:
1. It is performed quasi-statically
2. It is not accompanied by dissipative effects.
Obviously, a reversible process is purely an ideal abstraction like weightless springs and frictionless pulling inMechanics. In fact all natural spontaneous process are irreversible. But we can by continuous refinement visualise the limiting case of a reversible process.

First Law of Thermodynamics
Combined first and second laws