The Second law of Thermodynamics and Air standard cycles

Cyclic processes:

A cyclic process is a single process or a series of processes arranged so that at the end of the cycle the system is at original state. The properties of the system will be the same at the start and the end of the cycle. 

Thus , for a cyclic process ;

 Q + W = 0 = ΔU

This may also be written as ,

 φ dQ + φ dW = 0

Cyclic processes are used in heat engines to convert heat into work. Some cycles are of special importance to heat engines. 

Air standard cycles:

Cycles that are suited for using air as the working fluid are known as air standard cycles. Three air standard cycles will be dealt with here and the cycles will be described in this section.

Since the purpose of an engine cycle is to convert heat into work , its suitability to do that task is measured in terms of thermal efficiency ; 

η_thermal = -W_net / Q_in 

Here ,

-W_net (= -φ dW) is the net work coming out of the system(really , the working fluid of the engine).

Q_in is the sum of all heat added to the system( Sum of all positive heat interactions between the system and surroundings). 

The thermal efficiency of an air standard cycle is known as its air standard efficiency (η_{air standard}) or called after the name of the cycle , as Otto efficiency (η_Otto) , Diesel efficiency (η_Diesel) etc.

The Otto cycle consists of four processes carried out with a perfect gas (in this case air) ;

1. The air is compressed adiabatically by a ratio r.
2. Heat is added at constant volume until the air reaches a suitable temperature and pressure.
3. The air is expanded adiabatically to its original volume.
4. The air is cooled at constant volume to its original state.     

The net work output (= -φ dW) can be found by finding the work done during the processes 1-2 and 3-4 (As no work is done during the other two processes).

On the other hand , heat is given to the air only during process 2-3 and rejected during process 4-1 . The net work can be found more easily as ;

  -φ dW = φ dQ = Q_2-3 + Q_4-1

The air standard efficiency ;

                                                          η_Otto = ( Q_2-3 + Q_4-1 ) / Q_2-3

This value is depends on the compression ratio , r (=V₁/V₂) only and is given by ;

η_Otto = 1 - (1/r^γ-1)

The Diesel cycle differs from the Otto cycle in that heat is added to the air at constant pressure.

Heat is added until the volume increases from V₂ to V₃ . If the compression ratio (V₁/V₂) is r and the ratio V₃/V₂ (known as the cut-0ff ratio) is α , the standard efficiency can be calculated in a way similar to that before to give ;

η_Diesel = (1 - 1/r^γ-1) [ (α^γ – 1) / γ (α – 1) ]

The carnot cycle is the most important thermodynamic cycle in many ways and comprises a pair of adiabatic processes (1-2 and 3-4) and a pair of isothermal  (2-3 and 4-1) processes. As two processes are adiabatic , heat interaction takes place only during the two isothermal processes.

The following equations can be written for the two adiabatic processes ; 

p₁V₁^γ = p₂V₂^γ

and ;

p₃V₃^γ = p₄V₄^γ

For the two isothermal processes ;

   p₂V₂ = p₃V₃

And ;

 p₄V₄ = p₁V₁

Using these four equations we can show that ;

V₂ / V₁ = V₃ / V₄

The net work output may be obtained by calculating the work done during each of the four processes or as ;

-φ dW = φ dQ = Q_2-3 + Q_4-1

In an isothermal process a-b for a perfect gas , the heat supplied ;

Q_a-b = -W_a-b = mRT_a ln(V_b / V_a)

And ;

 V₂ / V₁ = V₃ / V₄

Using these , it can be shown that the efficiency of the Carnot cycle ;

η_Carnot = (1 – T₂/T₁)

The Second law of Thermodynamics:

The second law of thermodynamics is the most powerful law of the physics and has been stated in different ways , which are equally valid and useful in their own way. Engineers find the following statements very useful.

Kelvin-planck statement of the second law:

It is impossible for a device that operates on a cycle to exchange heat with just a single reservoir and produce a net amount of work.

This means that a perpetual motion machine of the second kind (PMM2) , i.e. a machine that takes heat from a single heat reservoir and coverts all of it into work , is not possible. In other words , the working fluid of the engine needs to exchange heat with a heat source and a heat sink (usually the environment). 
It also means that no heat engine can have a thermal efficiency of 100%. 

Clausius statement of the second law:

It is impossible to construct a device that operates on a cycle to produce no effect other than the transfer of heat from a lower-temperature body to a higher-temperature body.

The second law also tells us that the efficiency of an irreversible engine cannot be more than that of a reversible engine operating between the same two thermal reservoirs.

It also tells us that the efficiencies of all reversible engine , we need to understand the meaning of thermodynamic reversibility(reversibility from here on).

Reversibility:

A process is reversible if it is possible to fully reverse the direction of the process and reserve each of its effects on the system(or systems) concerned and restore the system(or systems) to the initial state.

All fully resisted processes are reversible in themselves. If a process involves motion against solid friction or viscous forces , the flow of electrical current against a resistance , Plastic bending , mixing of fluids at different pressures or at different temperatures or of different chemical composition , non-equilibrium chemical reactions , inelastic collision , electric discharge across a voltage difference , and heat flow between bodies with a temperature difference are among common irreversible processes.

Reversibility requires frictionless motion between sliding surfaces , in-viscid fluid motion , elastic deformation and elastic collision of solids , electric current flow against zero electrical resistance , equilibrium reaction between chemicals , transfer of charges without potential difference , transfer of heat without a difference in temperature. 

While each process in a thermodynamic cycle of a simple thermodynamic system may be reversible , interactions with the surroundings can be irreversible. for example , when heat is exchanged with another system at a different temperature.

If we compare engines exchanging heat with a hot reservoir at temperature T_h and a cold reservoir at T_c  , the maximum temperature during the cycle cannot be more than T_h and the minimum temperature cannot be less than T_c.

The working fluid of a Carnot engine receives heat only at T_h and rejects heat only at T_c.

The working fluid in an Otto or a Diesel cycle receives heat while the temperature increases from a value less than T_h until it reaches T_h , it loses heat while the temperature decreases from a value more than T_c until it reaches T_c.

Thus the heat interactions between the working fluid and the two reservoirs are irreversible. Thus η_Otto and η_Diesel will be less than η_Carnot for cycles operating between the same temperature limits.

An absolute scale of temperature can be defined on the basis of reversible engines using the formula ;

η_rev = (1 – T₂ / T₁)

Measurement of the efficiency of a reversible engine operating between two temperatures , of which one is known (or defined) allows us to determine the other. Thus , the second law allows us to define a temperature scale that is independent of the working fluid. In practice this scale coincides with the perfect gas scale of temperature.

Clausius inequality:

For a system undergoing a cyclic process ;

 φ dQ / T ≤ 0

The equal sign applies for reversible processes only. From this , we arrive at a property called entropy.

Entropy:

Change in entropy( Symbol S , specific entropy s ) of a system undergoing a reversible process 1-2 is given by ; 

S₁ – S₂ =  ₁∫² dQ/T

The difference in entropy between two states of any system can be found by identifying a combination of reversible processes that will take the system from one state to the other. It is not permissible to use the above expression or the forms ;

dS = dQ / T   and   dQ = T dS 

for an irreversible process.(note that same rule applies for dW = -p dV)

Many more useful results come from the Second law. A particularly useful result that follows from Clausius inequality is that the entropy of an isolated system can either remain the same or increase.

It can also be said that if a system with work interactions has no heat interaction its entropy will remain the same or increase.

This rule can be used to test the reversibility of a process in a more general way , if for a process ;

If ,   S₂ – S₁ = ₁∫² dQ / T    ; The process is reversible.

If ,   S₂ – S₁ > ₁∫² dQ / T    ; The process is irreversible.

If ,   S₂ – S₁ < ₁∫² dQ / T    ; The process is not possible.

Vapour power cycles : The Rankine cycle:  

The operation of the Carnot cycle with a gas has practical limitations. Vapour at saturation conditions allows heat interactions without to change of temperature. (This is due to the interdependence of pressure and temperature at saturation.)

Thus a vapour operated Carnot cycle is possible with two constant pressure processes and two adiabatic processes arranged inside the vapour dome. Practical considerations make it necessary to deviate from the Carnot cycle and use instead the Rankine cycle. The Rankine cycle is shown below as P-V and T-S plots. 

                           

                           

The Rankine cycle is executed as a series of flow processes and the work done in the turbine is        (h₂ – h₃) for unit mass flow. The heat supplied is (h₂ – h₁) per unit mass flow. The cycle is analyzed using the steam tables. The expansion 2-3 is adiabatic reversible and S₂ = S₁. The feed-pump work is neglected so that h₁ = h₄ , and the net work is the turbine work.

Rankine efficiency ;

η_Rankine  = (h₂ – h₃) /  (h₂ – h₁)