THERMODYNAMICS  THEORY



Exergy Balance for Control Volume



The exergy balance for control volumes differ from those for closed
systems in that they involve one more mechanism of exergy transfer: exergy
transfer
by mass transfer. Hence, the exergy balance for control volume can be
developed by adding the net exergy transfer by mass on the right side
of the relation
of exergy balance for closed system.
Recall, the
exergy balance for closed system is:
Net exergy transfer by mass is:
Therefore, the exergy balance for a control volume is:
where
i = inlet
e = exit
1 = initial state
2 = final state
ψ =
flow exergy
In rate form, the exergy balance for control volume is:






Exergy Balance for SteadyFlow Systems



Most devices encountered in practice are steadyflow
devices, such as
turbines and
nozzles.
Their mass, energy, entropy and volume remain constants during a
steadyflow
process. Thus, its exergy does not change with time.
dX_{CV}/dt
= 0
Then the rate form of exergy balance for a control volume undergoing
a steadyflow process becomes
If it is an oneinletoneexit steadyflow device, the relation above
further reduces to
where the change in
flow exergy is given by






Secondlaw Efficiency



Thermal efficiency and coefficient of performance for devices defined in
previous sections are a measure of their performance. They are based
on the
first law of thermodynamics and referred to as the firstlaw efficiencies.
The following gives the definition of the secondlaw efficiency.
The secondlaw efficiency is defined as the ratio of the actual thermal
efficiency to the maximum possible (reversible) thermal efficiency under
the same condition (for
heat engine). It can be expressed as the ratio
of the useful work output and the maximum possible work output (for workproducing
device, such as turbine), or the ratio of the minimum work input to the
actual useful work input (for workconsuming device, such as compressor).
For refrigerators or heat Pumps,
it is defined in terms of the coefficient of performance as the ratio of the actual
COP to the COP at reversible process. For devices that involves no work, such as
mixing chambers,
the secondlaw efficiency is defined as the ratio of the exergy recovered to the exergy
supplied. They are summarized in the
below table.




devices

secondlaw
efficiency

Heat Engine 
ε = η_{th}/η_{th,rev} 
Workproducing Device 
ε = W_{u}/W_{rev} 
Workconsuming Device 
ε = W_{rev}/W_{u} 
Refrigerators/Heat Pumps 
ε = COP/COP_{rev} 
Mixing chambers 
ε = Exergy recovered/exergy supplied 




hs Diagram of Adiabatic Turbine


The secondlaw efficiencies for steadyflow devices like turbines,
compressors, heat exchangers, and mixing chambers can be determined
from
their definitions. For a workproducing device
such as turbine, the secondlaw efficiency is defined as
where W_{u} is the actual useful work and W_{rev} is
the reversible work.
W_{u} can be obtained from the energy balance of the turbine. Usually
the kinetic and potential energies associated with a fluid flowing through
a turbine is neglectable compared with the enthalpy change of the fluid.
In this case, the energy balance of the turbine is reduced to
The reversible work can be determined by setting the exergy destruction
to zero in the exergy balance relation. For oneinletoneexit
steadyflow devices, such as turbines, it is,
If the turbine is a adiabatic turbine, the reversible work equals the
difference of the flow exergy at the inlet and the exit.
Therefore, the secondlaw efficiency of an adiabatic turbine can be
determined as




hs Diagram of Adiabatic Compressor 

The secondlaw efficiency of a compressor can be determined using
the similar method. It is




Heat Exchanger with two Unmixed Fluid Stream


Since no work is involved in heat exchangers and mixing chambers,
their secondlaw efficiencies are defined as the ratio of exergy recovered
to
exergy supplied. A heat exchanger with two unmixed fluid streams is
shown on the left. Hot stream enters at state 1 and leaves ad
state 2,
cold stream
enters
at state
3
and leaves at state 4. During the heat exchange process, exergy supplied
equals
the exergy lost in the hot stream and exergy recovered equals the exergy
gained in the cold stream. Thus, the secondlaw efficiency of a heat exchanger
is




Adiabatic
Mixing Chamber 

The secondlaw efficiency of a mixing chamber can be obtained using
the similar method.
The secondlaw efficiencies of these steadyflow devices are summarized
in the following table. 








