Ideal Rankine Cycle

(a) Schematic representation of an ideal Rankine

cycle (b) T-s diagram of an ideal Rankine cycle
Application of the First law of thermodynamics
to the control volume (pump, steam generator,
turbine and condenser), gives
Work done on pump, per kg of water, W
P
= h
2
-h
1
Energy added in steam generator, q
1
= h
3
-h
2
Work delivered by turbine, W
T
= h
3
-h
4
Energy rejected in the condenser, q
2
= h
4
-h
1
The thermal efficiency of the Rankine cycle is
given by,
2 3
1 2 4 3
2 3
1 4 2 3
1
2 1
) ( ) ( ) ( ) (
h h
h h h h
h h
h h h h
q
q q

= Net work done
----------------------
Energy absorbed
Practical Rankine cycle
Pump and Turbine do not operate isentropically
in practice.
The practical Rankine cycle is shown as 1-2-3-
4-1.
In the actual turbine, the work delivered is less
than the isentropic turbine. Similarly, the work
consumed by an actual pump is greater than the
work consumed by an isentropic pump.
That is,
h
3
-h
4
< h
3
-h
4
h
2
-h
1
> h
2
-h
1
Thermal efficiency of a practical Rankine cycle,
'
2 3
1
'
2
'
4 3
) ( ) (
h h
h h h h

The performance of an actual turbine or pump is
usually expressed in terms of isentropic
efficiency.
Isentropic efficiency of turbine (
T
) is defined as
the ratio of Work delivered by actual turbine to
Work delivered by an isentropic turbine.
4 3
'
4 3
h h
h h
T

Isentropic efficiency of pump (
P
) is defined as
the ratio of Work required by isentropic pump
to Work required by actual pump
1
'
2
1 2
h h
h h
P

Methods to increase the efficiency of the
Rankine cycle
Basic idea: Increase the average temperature at
which heat is transferred to the working fluid in
the boiler, or decrease the average temperature
at which heat is rejected from the working fluid
in the condenser.
1. Lowering the condenser Pressure:-
Lowering the operating pressure of the
condenser lowers the temperature at which heat
is rejected. The overall effect of lowering the
condenser pressure is an increase in the thermal
efficiency of the cycle.
2. Superheating the steam to high temperatures:-
The average temperature at which heat is added
to the steam can be increased without increasing
the boiler pressure by superheating the steam to
high temperatures.
Superheating the steam to higher temperatures
has another very desirable effect: It decreases
the moisture content of the steam at the turbine
exit.
3. Increasing the Boiler pressure:-
Increasing the operating pressure of the boiler,
automatically raises the temperature at which
boiling takes place.
This raises the average temperature at which
heat is added to the steam and thus raises the
thermal efficiency of the cycle..
Reheat Rankine Cycle
(a) schematic representation of a reheat Rankine
cycle (b) T-s diagram of a reheat Rankine cycle
The energy added ( per unit mass of steam ) in
the steam generator is given by,
) ( ) (
4 5 2 3 1
h h h h q +
The energy rejected in the condenser,
1 6 2
h h q
The thermal efficiency,
) ( ) (
) ( ) ( ) (
4 5 2 3
1 6 4 5 2 3
h h h h
h h h h h h
+
+

Regenerative Cycle
(a) schematic diagram (b) T-s diagram
Consider the feed water heater as the control
volume and apply the first law of
thermodynamics to obtain,
3 8 2
m m m +

and
3 3 8 8 2 2
h m h m h m +
or
8 8 2 8 3 3 3
) ( h m h m m h m +
or
2
3
8
8
3
8
3
1 h
m
m
h
m
m
h

,
_

,
_

Let,
3
8
m
m

=Y= the fraction of steam extracted

from the turbine for preheating
( )
2 8 3
1 h Y h Y h

+

Energy added in the boiler per unit mass of the

working fluid,
4 7 1
h h q
Energy rejected in the condenser,
( ) ) ( ' 1
1 9 2
h h Y q
Thermal efficiency,
( )
) (
) ( ' 1 ) (
4 7
1 9 4 7
h h
h h Y h h

The work output of the turbines =
( ) ) ( ' 1 ) (
9 8 8 7
h h Y h h
Work spent on the pumps =
( ) ) ( ' 1 ) (
1 2 3 4
h h Y h h
Air standard Otto Cycle
Air standard Otto cycle on (a) P-v diagram (b)
T-s diagram
Processes: -
0-1: a fresh mixture of fuel-air is drawn into the
cylinder at constant pressure
1-2: isentropic compression
2-3: energy addition at constant volume
3-4: isentropic expansion
4-1: combustion products leave the cylinder
1-0: the piston pushes out the remaining
combustion products at constant pressure
Since the net work done in processes 0-1 and 1-0
is zero, for thermodynamic analysis, we consider
the 1-2-3-4 only.
The thermal efficiency of the cycle is given by
1
2 1
1
Q
Q Q
Q
W
net


where Q
1
and Q
2
denote the energy absorbed and
rejected as heat respectively.
For a constant volume process Q=U. If m is
the mass of the air which is undergoing the
cyclic process,
T mC U
v

Energy is absorbed during the process 2-3
Energy is rejected during the process 4-1
Hence,
( )
2 3
1 4
1 4 1 4 2
2 3 2 3 1
1
) (
T T
T T
T T mC U U Q
T T mC U U Q
v
v

For an ideal gas undergoing an isentropic

process (process 1-2 and 3-4),
1
Tv
= constant
Hence,
1
1
2
2
1

,
_

v
v
T
T
and
1
4
3
3
4

,
_

v
v
T
T
But v
1
=v
4
and v
2
=v
3
. Hence we get,
3
4
2
1
T
T
T
T

or
3
2
4
1
T
T
T
T

3
2
4
1
1 1
T
T
T
T

or
3
2 3
4
1 4
T
T T
T
T T

2
1
3
4
2 3
1 4
T
T
T
T
T T
T T

Hence,
1
0
1
1
2
2
1
1
1 1 1

,
_

,
_

r v
v
T
T
Where the compression ratio r
0
is defined as
2
1
0
v
v
r
Sometimes it is convenient to express the
performance of an engine in terms of Mean
effective Pressure, P
m
, defined as the ratio of
Net work done to Displacement volume
2 1
v v
W
P
m

) (
2 1
v v P W
m

Thermal efficiency of the ideal Otto cycle as a
function of compression ratio (=1.4)
The thermal efficiency of the Otto cycle
increases with the specific heat ratio, of the
working fluid.
Air standard Diesel cycle
Diesel cycle on (a) P-v diagram (b) T-s diagram
Processes: -
0-1: fresh air is drawn into the cylinder
1-2: isentropic compression
2-3: constant pressure energy addition
3-4: isentropic expansion
4-1: combustion products leave the cylinder
1-0: remaining combustion products are
exhausted at constant pressure
Defining cutoff ratio, r
c
as,
2
3
v
v
r
c

For a constant pressure process (2-3),
Q=H.
Hence, the energy addition during process 2-3,
) ( ) (
2 3 2 3 2 3 1
T T mC h h m H H Q
p

where m is the mass of gas undergoing the
cyclic change.
The energy rejection during the process 4-1,
) ( ) (
1 4 1 4 1 4 2
T T mC u u m U U Q
v

The thermal efficiency, is given by

,
_

,
_

1
1
1
) (
1
) (
) ( ) (
2
3
2
1
4
1
2 3
1 4
2 3
1 4 2 3
1
2 1
T
T
T
T
T
T
T T
T T
T T mC
T T mC T T mC
Q
Q Q
p
v p

Since the process 1-2 is isentropic,

1
0
1
1
2
2
1
1

,
_

,
_

r v
v
T
T
Since the process 4-1 is a constant volume
process,

,
_

,
_

1
2
3
4
1
3
3
4
1
4
1
4
P
P
P
P
P
P
P
P
P
P
T
T
since P
2
=P
3
The processes 1-2 and 3-4 are isentropic. Hence,

,
_

4
3
3
4
v
v
P
P
and

,
_

2
1
1
2
v
v
P
P
Hence we get,

c
r
v
v
v
v
v
v
T
T

,
_

,
_

,
_

2
3
2
1
4
3
1
4
For the constant pressure process,
c
r
v
v
T
T

2
3
2
3
Hence the efficiency becomes,
1
]
1

1
1 1
1
1
0 c
c
r
r
r

The mean effective pressure of an air standard

diesel cycle is given by,
( ) [ ]
) 1 )( 1 (
1 ( ) 1
0
0 0
1

r
r r r r
P P
c c
m

Thermal efficiency of the ideal diesel cycle as a
function of compression and cutoff ratios
(=1.4)
Air standard Dual cycle
Dual cycle on (a) P-v diagram (b) T-s diagram
Energy addition is in two stages: Part of energy
is added at constant volume and part of the
energy is added at constant pressure
Energy added, q
1
) ( ) (
3 4 2 3 1
T T C T T C q
p v
+
Energy rejected, q
2
) (
1 5 2
T T C q
v

Thermal efficiency,
) ( ) (
) (
1
) ( ) (
) (
1
1
3 4 2 3
1 5
3 4 2 3
1 5
1
2
T T T T
T T
T T C T T C
T T C
q
q
p v
v
+

The efficiency can be expressed also in terms of,

Compression ratio, r
0
= V
1
/V
2
Cut-off ratio, r
c
= V
4
/V
3
Constant volume pressure ratio, r
vp
= P
3
/P
2