Ideal Rankine cycle
Steam-electric power plants produce nearly 90% of all electricity worldwide. Almost all coal, nuclear, geothermal, solar thermal,
and waste incineration plants use a vapor power cycle based on
the ideal Rankine cycle. The figure below provides a schematic of
a Rankine cycle in relation to the various other components (e.g., heat
source, electric generator, cooling tower) involved in a steam-electric
power plant. An ideal Rankine cycle provides a mechanism for understanding strategies for increasing vapor power cycle efficiency.
turbine
heat
source
electric
generator
Ẇout
to cooling
tower
coal, sun,
nuclear, etc.
from cooling
tower
pump
Ẇp
An ideal Rankine cycle consists of the following internally reversible processes.
• Process 1 → 2: Isentropic expansion of the working fluid through
the turbine from saturated vapor at state 1 to the condenser pressure p2 .
• Process 2 → 3: Heat transfer from the working fluid as it flows
at constant pressure p2 = p3 through the condenser, exiting as a
saturated liquid at state 3.
• Process 3 → 4: Isentropic compression in the pump to state 4 in
the compressed liquid region of the phase diagram.
• Process 4 → 1: Heat transfer to the working fluid as it flows at
constant pressure p4 = p1 through the boiler to complete the cycle.
Figure 1: A schematic of a vapor power
plant capable of undergoing a Rankine
cycle.
Assignments
Complete these assignments as a group during class. Show all work
for credit.
T
isobar
Prob. 1: Label streams 1 through 4 in Fig. 1 according to the description of the
ideal Rankine process in the worksheet introduction.
Prob. 2: Recall that a T-s diagram for any Carnot power cycle is depicted
by the plot on the left. Using the description of the ideal Rankine
process in the worksheet introduction, sketch a T-s diagram for an
ideal Rankine cycle on the axes at the right. Label all stream states
as numbered points. 1
T
s
Hint: The above sketch illustrates an
isobar on a T-s diagram. The vapor
dome is shown in both plots.
1
T
TH
Tsat ( p4 = p1 )
TC
Tsat ( p2 = p3 )
s
s
Prob. 3: Recall that for an internally reversible process
Z
Q = TdS
(1)
int. rev.
Using this expression, show how Q̇/ṁ is graphically represented for
the arbitrary irreversible process shown below.2 Is the heat flow in
this example into or out of the system?
T
s
Hint: Note that s = S/m. You may assume the mass flow rate ṁ is constant.
2
Prob. 4: Based on your answer to the previous question and your T-s diagram
in Prob. 2, how much heat is transferred during process 1 → 2?
during process 3 → 4?
Prob. 5: Sketch only process 2 → 3 on the graph below. Indicate the region of
the graph that corresponds to Q̇out /ṁ.
T
Tsat ( p2 = p3 )
s
Prob. 6: Sketch only process 4 → 1 on the graph below. Indicate the region of
the graph corresponding to Q̇in /ṁ. Then sketch an isothermal process
occurring at average temperature T in that would have the same heat
transfer per unit mass of working fluid. (Use a dashed line.) Your
plot is just a sketch and need not be exact.
T
Tsat ( p4 = p1 )
s
Prob. 7: Sketch a hypothetical cycle that includes processes 1 → 2, 2 → 3,
and 3 → 4, but that approximates process 4 → 1 with an isothermal
process at T in .3
T
Tsat ( p4 = p1 )
Hint: Note that processes 3 → 4 and
1 → 2 will be slightly altered since
states 4 and 1 must now lie on the
isotherm at T in . Indicate T in on the
y-axis.
3
Tsat ( p2 = p3 )
s
Prob. 8: You will have sketched a Carnot cycle in the previous problem.4 In
your drawing above, note which temperature corresponds to the cold
reservoir TC and which corresponds to the hot reservoir TH .
Prob. 9: Write an expression for the thermal efficiency η of a cycle as a function of Q̇in /ṁ and Q̇out /ṁ.5
Prob. 10: How are η of the hypothetical cycle with average input temperature
T in and η of the ideal Rankine cycle related?
Hint: A Carnot cycle has two isothermal and two isentropic processes.
4
5
Hint: η = Wnet /Qin .
Prob. 11: For the hypothetical Carnot cycle, list two strategies you would use
to increase its thermal efficiency.6
Prob. 12: It is costly to reduce the temperature of a cold reservoir below atmospheric values, therefore strategies to increase Rankine cycle efficiency generally aim to increase the average temperature at which
heat input occurs T in . One such strategy is to superheat the working fluid in state 1 rather than restricting it to be a saturated vapor.
Sketch an ‘ideal Rankine cycle with superheat’ on the plot below.7
T
Tsat ( p4 = p1 )
Tsat ( p2 = p3 )
s
Prob. 13: Overlay a hypothetical Carnot cycle with average temperature of heat
input T in on the plot you have just drawn. Depending on how much
superheat you drew, T in could even become greater than Tsat ( p4 =
p1 ).
Prob. 14: Based on your hypothetical Carnot cycle for the Rankine cycle with
reheat, did the reheat process increase or decrease the maximum
efficiency of the ideal Rankine cycle? Why?
Hint: Recall that ( Q H /QC )int. rev. =
TH /TC .
6
Hint: Don’t forget that processes
1 → 2 and 3 → 4 are still isentropic.
Peek back at the beginning of the
worksheet to remind yourself how an
isobar looks on a T-s diagram.
7
Prob. 15: Ethical considerations: Briefly discuss two ethical considerations that
arise from the use of superheat to increase Rankine cycle efficiency.
1) How might economic and environmental impacts be affected by
an increase in cycle efficiency? 2) How might economic and environmental impacts be affected by an increase in the temperature of
stream 1?8
Hint: Might cooling water be affected?
Can this impact the environment?
8