UNIT – V
Power cycles:
Otto cycle, diesel cycle, duel cycle and Brayton cycle on air standard basis -
Thermodynamic analysis, comparison of Otto, diesel, duel cycles. Simple Rankine cycle.
MODULE 1: Gas Power Cycles
Basics of Air Standard Cycles 1
Otto Cycle 1
Diesel Cycle 3
Dual Cycle 5
Brayton Cycle 9
MODULE 2: Vapour Power Cycles
Rankine Cycle 14
Air-Standard Assumptions
In our study of gas power cycles, we assume the working fluid is air, and the air undergoes a
thermodynamic cycle even though the working fluid in the actual power system does not
undergo a cycle.
To simplify the analysis, we approximate the cycles with the following assumptions:
• The air continuously circulates in a closed loop and always behaves as an ideal gas.
• All the processes that make up the cycle are internally reversible.
• The combustion process is replaced by a heat-addition process from an external
source.
• A heat rejection process that restores the working fluid to its initial state replaces the
exhaust process.
• The cold-air-standard assumptions apply when the working fluid is air and has
constant specific heat evaluated at room temperature (25oC or 77oF).
Terminology for Reciprocating Devices
The following is some terminology we need to understand for reciprocating engines—
typically piston-cylinder devices. Let’s look at the following figures for the definitions of
top dead center (TDC), bottom dead center (BDC), stroke, bore, intake valve, exhaust
valve, clearance volume, displacement volume, compression ratio, and mean effective
pressure.
Thermodynamics R-23
The compression ratio r of an engine is the ratio of the maximum volume to the
minimum volume formed in the cylinder.
V max VBDC
r
V min VTDC
The mean effective pressure (MEP) is a fictitious pressure that, if it operated on the
piston during the entire power stroke, would produce the same amount of net work as
that produced during the actual cycle.
Wnet wnet
MEP
Vmax Vmin vmax vmin
Otto Cycle (Constant Volume Cycle):
This ideal heat engine cycle was proposed in 1862 by Bean de Rochas. In 1876 Dr.
Otto designed an engine to operate on this cycle. The Otto engine immediately became
so successful from a commercial stand point, that its name was affixed to the cycle used
by it.
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 2
Thermodynamics R-23
The ideal p - v and T-s diagrams of this cycle are shown in fig. In working out the air-
standard efficiency of the cycle, the following assumptions are made:
(i) The working fluid (working substance) in the engine cylinder is air, and it
behaves as a perfect gas, i.e., it obeys the gas laws and has constant specific
heats.
(ii) The air is compressed adiabatically (without friction) according to law pvγ = C
(iii) The heat is supplied to the air at constant volume by bringing a hot body in
contact with the end of the engine cylinder.
(iv) The air expands in the engine cylinder adiabatically (without friction) during
the expansion stroke.
(v) The heat is rejected from the air at constant volume by bringing a cold body in
contact with the end of the engine cylinder.
Process 1 2 Isentropic compression
Process 2 3 Constant volume heat addition
Process 3 4 Isentropic expansion
Process 4 1 Constant volume heat rejection
Consider one kilogram of air in the engine cylinder at point (1). This air is compressed
adiabatically to point (2), at which condition the hot body is placed in contact with the
end of the cylinder. Heat is now supplied at constant volume, and temperature and
pressure rise; this operation is represented by (2-3). The hot body is then removed and
the air expands adiabatically to point (4). During this process, work is done on the piston.
At point (4), the cold body is placed at the end of the cylinder. Heat is now rejected at
constant volume, resulting in drop of temperature and pressure. This operation is
represented by (4-1). The cold body is then removed after the air is brought to its original
state (condition). The cycle is thus completed. The cycle consists of two constant volume
processes and two reversible adiabatic processes. The heat is supplied during constant
volume process (2-3) and rejected during constant volume process (4-1). There is no
exchange of heat during the two reversible adiabatic processes (1-2) and (3-4).
The performance is often measured in terms of the cycle efficiency.
W
th net
Qin
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 3
Thermodynamics R-23
Thermal Efficiency of the Otto cycle
Wnet Qnet Qin Qout Q
th 1 out
Qin Qin Qin Qin
Now to find Qin and Qout.
Apply first law closed system, V = constant.
Heat supplied during constant volume operation (2-3), Heat rejected during constant
volume operation (4-1) is
qin u3 u2 cv (T3 T2 )
qout u4 u1 cv (T4 T1 )
Qnet , 23 U 23
Qnet , 23 Qin mCv (T3 T2 )
Qnet , 41 U 41
Qnet , 41 Qout mCv (T1 T4 )
Qout mCv (T1 T4 ) mCv (T4 T1 )
The thermal efficiency becomes
Qout
th , Otto 1
Qin
mCv (T4 T1 )
1
mCv (T3 T2 )
(T4 T1 )
th , Otto 1
(T3 T2 )
T1 (T4 / T1 1)
1
T2 (T3 / T2 1)
Recall processes 1-2 and 3-4 are isentropic, so
1 1
T1 v2 v T4
3
T2 v1 v4 T3
Since V3 = V2 and V4 = V1
T2 T
3
T1 T4
or
T4 T
3
T1 T2
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 4
Thermodynamics R-23
The Otto cycle efficiency becomes
T1
th, Otto 1
T2
Since process 1-2 is isentropic,
1 1
T2 v1 1
T1 v2 r
Where the compression ratio is
vmax v1
r
vmin v2
1
th , Otto 1
r k 1
Air-Standard Diesel Cycle (or constant pressure cycle):
The air-standard Diesel cycle is the ideal cycle that approximates the Diesel combustion
engine
Process Description
1-2 isentropic compression
2-3 Constant pressure heat addition
3-4 isentropic expansion
4-1 Constant volume heat rejection
The P-v and T-s diagrams are
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 5
Thermodynamics R-23
Heat supplied Q1 Q23 mc p (T3 T2 )
Heat Rejected = Q2 Q41 mcv (T4 T1 )
Q2 mc (T T ) (T T )
Effieciency 1 1 v 4 1 1 4 1
Q1 mc p (T3 T2 ) (T3 T2 )
The efficiency may be expressed terms of any two of the following.
v1
Compression Ratio rk
v2
v4
Expansion Ratio re
v3
v3
Cut off ratio rc
v2
rk re .rc
Process 3-4
1
T4 v3 1
1
T3 v4 re
1
v re 1
T4 T3 3 T3
v4 rk 1
Process 2-3
T2 p2v2 v2 1
T3 p3v3 v3 re
1
T2 T3
re
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 6
Thermodynamics R-23
Process 1-2
1
T1 v2 1
1
T2 v1 rk
1 T 1
T1 T2 1 3 1
rk rc rk
By subsisting T1,T2 and T4 in the expression of efficiency
rc 1 T3 1
T3
rk 1 rc rk 1
1
1
T3 T3
rc
1 1 rc 1
diesel 1 .
rc 1 rc 1
Dual Cycle (mixed cycle/ limited pressure cycle):
Process 1 2 Isentropic compression
Process 2 2.5 Constant volume heat addition
Process 2.5 3 Constant pressure heat addition
Process 3 4 Isentropic expansion
Process 4 1 Constant volume heat rejection
Thermal Efficiency:
v3 P3
where rc and
v2.5 P2
Note, the Otto cycle (rc=1) and the Diesel cycle (a=1) are special cases:
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 7
Thermodynamics R-23
Otto 1
1
Diesel 1
1 1 rck 1
r k 1 const cV
r k 1 k rc 1
The use of the Dual cycle requires information about either:
i) The fractions of constant volume and constant pressure heat addition (common
assumption is to equally split the heat addition), or
ii) Maximum pressure P3.
Q1 mcv (T3 T2 ) mc p T4 T3
Q 2 mcv (T5 T1 )
Here Q1 -= heat input
Q2 =Out put
Q2 mcv (T5 T1 ) T5 T1
1 1 1
Q1 mcv (T3 T2 ) mc p T4 T3 (T3 T2 ) T4 T3
v1
Compression Ratio rk
v2
v4
Expansion Ratio re
v3
p3
cons tan t volume pressure ratio rp
p2
rk rc .re
rk
re
re
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 8
Thermodynamics R-23
process3 4
v T p T
rc 4 4 3 4
v3 p4T3 T3
T4
T3
rc
Pr ocess 2 3
p2 v2 p3v3
T2 T3
p2 T4
T2 T3
p3 rp rc
process 1 2
1
T1 v2 1
T2 v1 rk 1
T4
T1
rp .rc .rk 1
Process 4-5
T4
T1
rp .rc .rk 1
1
T5 v4 1
1
T4 v5 re
r 1
T5 T4 c 1
rk
SubtitlingT1, T2,T3 and T4values
rc 1 T4
T4 1
rk rp .rc .rk 1 1 rp rc 1
dual 1 1
T4 T4 T rk 1 rp 1 rp rc 1
T4 4
rc rp rc rc
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 9
Thermodynamics R-23
Comparison of cycles:
For the same inlet conditions P1, V1 and the same compression ratio P2/P1:
For the same initial conditions P1, V1 and the same compression ratio:
Otto Dual Diesel
For the same inlet conditions P1, V1 and the same peak pressure P3:
For the same initial conditions P1, V1 and the same peak pressure P3
(Actual design limitation in engines):
Diesel Dual otto
Brayton Cycle (or Joule cycle)
The Brayton cycle is the air-standard ideal cycle approximation for the gas-turbine
engine. This cycle differs from the Otto and Diesel cycles in that the processes
making the cycle occur in open systems or control volumes. Therefore, an open
system, steady-flow analysis is used to determine the heat transfer and work for the
cycle.
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 10
Thermodynamics R-23
We assume the working fluid is air and the specific heats are constant and will
consider the cold-air-standard cycle.
The closed cycle gas-turbine engine
The T-s and P-v diagrams for the
Closed Brayton Cycle
Process Description
1-2 Isentropic compression (in a compressor)
2-3 Constant pressure heat addition
3-4 Isentropic expansion (in a turbine)
4-1 Constant pressure heat rejection
Thermal efficiency of the Brayton cycle
Wnet Q
th, Brayton 1 out
Qin Qin
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 11
Thermodynamics R-23
Now to find Qin and Qout.
Apply the conservation of energy to process 2-3 for P = constant (no work), steady-
flow, and neglect changes in kinetic and potential energies.
E in E out
m 2 h2 Q in m
3 h3
in m
The conservation of mass gives m out
2 m
m 3 m
For constant specific heats, the heat added per unit mass flow is
Q in m
(h3 h2 )
Q in mC
p (T3 T2 )
Q in
qin C p (T3 T2 )
m
The conservation of energy for process 4-1 yields for constant specific heats
m
Q (h4 h1 )
out
mC
Q p (T4 T1 )
out
Q
qout out
C p (T4 T1 )
m
The thermal efficiency becomes
Q q
th , Brayton 1 out 1 out
Qin qin
C p (T4 T1 )
1
C p (T3 T2 )
(T4 T1 )
th , Brayton 1
(T3 T2 )
T1 (T4 / T1 1)
1
T2 (T3 / T2 1)
Recall processes 1-2 and 3-4 are isentropic, so
1
T2 p2
T1 p1
1
T3 p3
T4 p4
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 12
Thermodynamics R-23
Since P3 = P2 and P4 = P1, T2 T3
T1 T4
or
T4 T3
T1 T2
The Brayton cycle efficiency becomes
T1
th, Brayton 1
T2
Since process 1-2 is isentropic,
1
1
T2 p2
rp
T1 p1
T1 1
1
T2
rp
Where the pressure ratio is rp = P2/P1 and
1
th , Brayton 1 ( 1)/
rp
SIMPLE RANKINE CYCLE
Rankine Cycle: The simplest way of overcoming the inherent practical difficulties of the
Carnot cycle without deviating too much from it is to keep the processes 1-2 and 2-3 of the
latter unchanged and to continue the process 3-4 in the condenser until all the vapour has
been converted into liquid water. Water is then pumped into the boiler upto the pressure
corresponding to the state 1 and the cycle is completed. Such a cycle is known as the Rankine
cycle. This theoretical cycle is free of all the practical limitations of the Carnot cycle.
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 13
Thermodynamics R-23
Figure (a) shows the schematic diagram for a simple steam power cycle which works
on the principle of a Rankine cycle.
The Rankine cycle comprises the following processes.
Process 1-2: Constant pressure heat transfer process in the boiler
Process 2-3: Reversible adiabatic expansion process in the steam turbine
Process 3-4: Constant pressure heat transfer process in the condenser and
Process 4-1: Reversible adiabatic compression process in the pump.
Figure (b) represents the T-S diagram of the cycle.
The numbers on the plots correspond to the numbers on the schematic diagram. For
any given pressure, the steam approaching the turbine may be dry saturated (state 2),
wet (state 21) or superheated (state 211), but the fluid approaching the pump is, in each
case, saturated liquid (state 4). Steam expands reversibly and adiabatically in the
turbine from state 2 to state 3 (or 21 to 31 or 211 to 311), the steam leaving the turbine
condenses to water in the condenser reversibly at constant pressure from state 3 (or 31,
or 311) to state 4. Also, the water is heated in the boiler to form steam reversibly at
constant pressure from state 1 to state 2 (or 21 or 211)
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 14
Thermodynamics R-23
Applying SFEE to each of the processes on the basis of unit mass of fluid and
neglecting
changes in KE & PE, the work and heat quantities can be evaluated
For 1kg of fluid, the SFEE for the boiler as the CV, gives,
h4+ Q1 = h1 i.e., Q1 = h1 – h4 --- (1)
SFEE to turbine, h1 = WT + h2 i.e., WT = h1 – h2 --- (2)
SFEE to condenser, h2 = Q2 + h3 i.e., QL = h2 – h3 --- (3)
SFEE to pump, h3 + WP = h4 i.e., WP = h4 – h3 --- (4)
Wnet
The efficiency of Rankine cycle is
Q1
Wnet WT Wp h1 h2 h4 h3
The efficiency of Rankine cycle is
Q1 Q1 h1 h4
Dept. of ME Seshadri Rao Gudlavalleru Engineering College 15
