Improving an Air-Standard Power Cycle
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Improving an Air-Standard Power Cycle
Our Purpose
We will examine several improvements that can be made to the simple air-standard cycle which
increase its efficiency. We look at the addition of a regenerative heat exchanger, and then stages
of reheating and intercooling.
The basic Brayton cycle
In air-standard cycles, such as the Brayton cycle shown below in Figure 1, power is produced by
a turbine expanding high temperature, high pressure air whose pressure has been achieved using
a compressor. Unlike in liquid-vapor compression cycles such as a Rankine Cycle, the work
needed to compress the working fluid in an air standard cycle is a large fraction of the work
extracted by the turbine (often over 50%). The basic cycle's thermal efficiency ηth is determined
by the pressure ratio; the modified cycles' efficiencies depend upon the temperatures as well.
Figure 1 shows the layout of a basic Brayton cycle.
Figure 1: A typical Brayton Cycle
We will start by constructing a typical Brayton cycle which we will use to judge the effects of
our proposed changes. A Brayton cycle is the air-standard analog of the Rankine cycle. As such,
its turbines and compressors are isentropic and adiabatic and its heaters, coolers, and heat
exchangers are isobaric.
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Improving an Air-Standard Power Cycle
The example Brayton cycle (bray1.dsn) to which we will compare has the following parameters:
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turbine inlet pressure of 1 MPa
turbine inlet temperature of 1000°C
turbine exhaust pressure of 100 kPa
turbine exhaust temperature of 15°C
and we note that its efficiencies with these operating parameters are
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Carnot efficiency: 77.37%
thermal efficiency: 48.21%
We will compare the efficiencies of our modified Brayton cycles to these to see if the
modifications have improved the cycle.
Improving air standard cycle efficiencies
Improving cycle efficiency almost always involves making a cycle more like a Carnot cycle
operating between the same high and low temperature limits. The Carnot cycle is maximally
efficient because it receives all of its heat addition at the same temperature, which is the highest
temperature in the cycle and rejects all of its heat at the lowest temperature in the cycle.
Figure 2 shows the T-s diagram for our example Brayton Cycle. We notice two things about this
cycle. First, during the cooling process, we are throwing away quit a bit of heat. The second is
that not all of the heat addition and rejection is at (or even near) the maximum and minimum
cycle temperatures. We will look at each of these issues in turn.
Figure 2: Brayton Cycle T-s Diagram
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Improving an Air-Standard Power Cycle
Most cycles don't have all of their heat addition or rejection at one temperature. So, when we
look to improve a cycle's efficiency, we often consider the mean temperature of heat addition, Ta
and the mean temperature of heat rejection, Tr. These reflect what the temperature would have
been if the same amount of heat had been added (or rejected) all at one temperature. They allow
us to treat improving cycle efficiencies as we would for a Carnot cycle: by raising Ta or lowering
Tr. For reversable heat transfer, the average temperature of heat addition is
Ta = Qin / ∆S
and the average temperature of heat rejection is
Tr = Qout / ∆S
Using these mean temperatures, we can find the cycle's thermal efficiency by applying the
relation for Carnot efficiency to the mean cycle temperatures:
ηth = (Ta - Tr) / Ta
Which will give same answer we get applying the usual ηth = Wnet / Qhi relation.
For more efficient cycles, we would like to add heat at a higher temperature and reject it at a
lower temperature. Obviously, if we could add all of our heat at the maximum cycle temperature
and reject all of it at the minimum cycle temperature, our cycle's efficiency would equal the
Carnot efficiency.
For instance, Table 1, shows these parameters in the simple Brayton cycle we have just
considered.
Maximum Cycle Temperature
1000°C
Mean Temperature of Heat Addition
592.7°C
Minimum Cycle Temperature
15°C
Mean Temperature of Heat Rejection
175.3°C
Carnot Efficiency
77.37%
Thermal Efficiency, ηth
48.21%
Table 1: Efficiency Parameters for simple Brayton Cycle
We are adding heat at a much lower temperature (over 400°C lower) than the maximum cycle
temperature. At the other end, we are removing heat from the system at 160°C above the
minimum cycle temperature. We will now examine some modifications to the basic cycle and
see if we cannot bring these numbers closer to the ideal.
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Improving an Air-Standard Power Cycle
Improving Cycle Efficiency with a Regenerative Heat
Exchanger
Could we use some of the heat rejected between states S2 and S3 to do some of the heating
required between stages S4 and S1? The temperatures in the cycle are as follows:
T1 = 1000°C
T2 = 386°C
T3 = 15°C
T4 = 283°C
This seems to indicate that we can use the heat thrown out between S2 and S3 that is above
283°C to do some heating between S4 and S1. To take advantage of this potential heat, we add a
heat exchanger to our cycle. Figure 3 shows the addition of such a heat exchanger (bray1x.dsn).
Figure 3: Brayton Cycle with Heat Exchanger
The heat exchanger shown in Figure 3 is assumed to be an ideal, counter-current one. For our
purposes, this not only means that it is isobaric, but also that the cool side outlet can be heated up
all the way to the temperature of the hot side inlet. So S8 = S5 and S6 = S7. Note that these are
the maximum temperature differences we can achieve. With a less-than-ideal heat exchanger, we
would not get quite such performance. (That is, S8 < S5 and S6 > S7.)
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Improving an Air-Standard Power Cycle
Figure 4: Brayton Cycle with Heat Exchanger T-s diagram
Simple
Brayton
This Cycle
Maximum Cycle Temperature
1000°C 1000°C
Mean Temperature of Heat Addition 659.7°C 592.7°C
Minimum Cycle Temperature
15°C
15°C
Mean Temperature of Heat Rejection 134.5°C 175.3°C
Carnot Efficiency
77.37% 77.37%
Thermal Efficiency, ηth
56.30% 48.21%
Table 2: Brayton Cycle with Heat Exchanger
Efficiency Parameters
Examining the heating and cooling for the modified cycle in Figure 4, we can see that all of the
heat transfer between S5 and S6 that was formerly thrown out in the cooler is now used to heat
the fluid from state S7 and S8. Because of this, we are now only adding new heat to the cycle
between S8 and S1, raising the mean temperature of heat addition. Similarly, we are now only
removing heat from the cycle between S6 and S3, lowering the mean temperature of heat
rejection. The result is that the cycle efficiency has risen substantially.
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Improving an Air-Standard Power Cycle
Improving Cycle Efficiency with a Reheat Stage
To bring the average temperature at which we add heat to the cycle closer to the peak
temperature, we can add a reheat stage to the cycle. Figure 5 shows the addition of such a stage
to the standard Brayton cycle (bray1r.dsn). The intermediate pressure for the reheat stage is 300
kPa, giving each turbine roughly equal pressure ratios.
Figure 5: Brayton Cycle with Reheat Stage
And, looking at Table 3 and the T-s diagram for this cycle shown in Figure 6, we can see that the
average temperature of heat addition has increased. The original Brayton cycle is overlaid in
light green in Figure 6 for comparison.
Figure 6: Brayton Cycle with Reheat Stage T-s diagram
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Improving an Air-Standard Power Cycle
Simple
Brayton
This Cycle
Maximum Cycle Temperature
1000°C
Mean Temperature of Heat Addition
654.7°C 592.7°C
Minimum Cycle Temperature
15°C
1000°C
15°C
Mean Temperature of Heat Rejection
274.7°C 175.3°C
Carnot Efficiency
77.37%
Thermal Efficiency, ηth
40.96% 48.21%
77.37%
Table 3: Brayton Cycle with Reheat Efficiency Parameters
However, the cycle efficiency of the modified cycle has actually decreased by over 7%. Why is
this? Looking at Figure 6, we can see not only that the mean temperature of heat addition has
increased, but also that the mean temperature of heat rejection has increased. So, while the
former increases efficiency, the latter decreases it. In this case, the net effect is a decrease in
cycle efficiency (and a large decrease at that!).
Improving Cycle Efficiency with Compressor Intercooling
In a similar fashion, we might look at Figure 2, an decide that we might improve the cycle by
lowering the average temperature of heat rejection for the cycle.
To bring the average temperature at which we remove heat to the cycle closer to the lowest
temperature, we can add an intercooling stage to the cycle. Figure 7 shows the addition of such a
stage to a Brayton cycle (bray1i.dsn). Analogous to the reheat stage, the intermediate pressure
for the intercooler is 300 kPa, giving each compressor roughly equal pressure ratios.
Figure 7: Brayton Cycle with Intercooling
And, looking at the T-s diagram for this cycle shown in Figure 8, we can see that the average
temperature of heat rejection has decreased. In the original Brayton cycle (overlaid in light green
in Figure 8), the mean temperature of heat rejection was 175.3°C and the cycle shown in Figure
8 has a mean temperature of heat rejection of 145.0°C.
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Improving an Air-Standard Power Cycle
Figure 8: Brayton Cycle with Intercooling T-s diagram
Simple
Brayton
This Cycle
Maximum Cycle Temperature
1000°C
1000°C
Mean Temperature of Heat Addition
485.9°C
592.7°C
15°C
15°C
Mean Temperature of Heat Rejection
145.0°C
175.3°C
Carnot Efficiency
77.37%
77.37%
Thermal Efficiency, ηth
44.90%
48.21%
Minimum Cycle Temperature
Table 4: Brayton Cycle with Intercooling Efficiency Parameters
However, as we saw with the addition of a reheat stage, the cycle efficiency of the modified
cycle has once again decreased. Again, though we have met our objective of decreasing the
mean temperature of heat rejection, it is also clear that the mean temperature of heat addition has
decreased as well, having a negative impact on cycle efficiency.
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Improving an Air-Standard Power Cycle
Combining Regeneration with Compressor Intercooling
and Turbine Reheat
In the last three sections, we considered the effect of additional components on the efficiency of
the Brayton cycle. We added a regenerative heat exchanger, a turbine reheat stage, and a
compressor intercooling stage. However, we saw that only the regenerative heat exchanger had a
positive effect on cycle efficiency.
The reheat and intercooling stages have their places, though. When we added a heat exchanger to
the Brayton cycle, we took advantage of waste heat being thrown away in the cooler and used it
to provide heat that would be used in the turbine. However, we were limited by the temperature
difference between the air leaving the turbine and the air leaving the compressor. For our
example, that temperature difference was about 100°C. The larger that temperature difference,
the more heat we can grab from the cooling process and inject into the air before it enters the
heater and the greater portion of our added heat is near the maximum cycle temperature.
Can we make that temperature difference greater? When we look at the T-s diagrams in Figure 6
and Figure 8, we see that the difference in temperature between the air leaving the compressor
and the air leaving the turbine in either cycle is larger than it was for the unmodified Brayton
cycle in both cases. Below, Figure 9 shows the T-s diagram for a cycle with both intercooling
and reheating (bray1ri.dsn). For reference, the unmodified Brayton cycle's T-s diagram is shown
in light green. Also shown is the temperature difference for the unmodified Brayton cycle ∆T1
and the temperature difference of the cycle with both intercooling and reheat stages ∆T2. We can
see that the latter is much greater. In fact, whereas ∆T1 is about 100°C, ∆T2 is over 500°C!
Figure 9: Brayton Cycle with both Reheat and Intercooling T-s diagram
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Improving an Air-Standard Power Cycle
Simple
Brayton
This Cycle
Maximum Cycle Temperature
1000°C
1000°C
Mean Temperature of Heat Addition
559.6°C
592.7°C
15°C
15°C
Mean Temperature of Heat Rejection
230.4°C
175.3°C
Carnot Efficiency
77.37%
77.37%
Thermal Efficiency, ηth
39.52%
48.21%
Minimum Cycle Temperature
Table 5: Brayton Cycle with Reheat and Intercooling Efficiency Parameters
We note that the thermal efficiency of this cycle is still worse than that for the basic Brayton
cycle (in fact, it's worse than for the cycles with only reheat or intercooling as well). However,
we can take advantage of the larger temperature difference with a heat exchanger. This new
cycle (bray1rix.dsn) is shown below in Figure 10.
Figure 10: Brayton Cycle with Reheat, Intercooling, and Regererative Heat Exchange
The T-s diagram and efficiency parameters for this new cycle is shown below. We note that the
cycle's thermal efficiency is now 20% better than that of the basic Brayton cycle and within 10%
of the Carnot efficiency, which is very impressive.
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Improving an Air-Standard Power Cycle
Figure 11: Brayton Cycle with Regeneration,Reheat, and Intercooling T-s diagram
Simple
Brayton
This Cycle
Maximum Cycle Temperature
1000°C
1000°C
Mean Temperature of Heat Addition
811.3°C
592.7°C
15°C
15°C
Mean Temperature of Heat Rejection
68.2°C
175.3°C
Carnot Efficiency
77.37%
77.37%
Thermal Efficiency, ηth
68.51%
48.21%
Minimum Cycle Temperature
Table 6: Brayton Cycle with Regeneration, Reheat, and Intercooling Efficiency Parameters
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Improving an Air-Standard Power Cycle
Additional Stages of Regeneration, Reheat, and
Intercooling
The last section investigated the addition of a single stage of regenerative heat exchange,
reheating, and intercooling to the basic Brayton cycle and revealed a substantial improvement in
the cycle's thermal effciency. It seems natural to ask how cycle efficiency would be improved
with even more stages. Figure 12 below shows the layout of such a cycle with 5 turbines and 5
compressors (bray5rix.dsn). The pressures are chosen to give each turbine and compressor
roughly the same pressure ratio as the others.
Figure 12: Brayton Cycle with Regeneration, Reheat, and Intercooling
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Improving an Air-Standard Power Cycle
Figure 13: Brayton Cycle with Multiple Stages of Regeneration, Reheat, and Intercooling
T-s diagram
Additionally, we can see from Figure 13, that the cycle is approaching the ideal Carnot cycle,
with all of the heat addition near the maximum cycle temperature and all of the heat rejection
near the minimum cycle temperature. For comparison, the T-s diagram of the Carnot cycle is
overlaid in light green.
Number of Turbines
Simple
Brayton
1
Maximum Cycle Temperature
Mean Temperature of Heat
Addition
3
4
5
1000°C
592.7°C 659.7°C 811.8°C 870.1°C 900.9°C 920.0°C
Minimum Cycle Temperature
Mean Temperature of Heat
Rejection
2
15°C
275.3°C 134.5°C
Carnot Efficiency
68.0°C
49.0°C
40.2°C
34.8°C
73.3%
74.2%
77.37%
Thermal Efficiency, ηth
48.2%
56.3%
68.6%
71.8%
Table 7: Brayton Cycle with Multiple Regeneration,
Reheat, and Intercooling Stages Efficiency Parameters
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Improving an Air-Standard Power Cycle
Choosing Intermediate Pressures
In each Brayton cycle with more than one turbine, we have chosen the pressures so that the
pressure ratio (PR) for each turbine is roughly the same as for all the rest of them. The same is
true for the compressors. Why do we take this approach?
There are two reasons. First, in keeping with our overall goal of improving cycle efficiency, we
want the mean temperature of heat addition to be as high as possible and the mean temperature
of heat rejection to be as low as possible. It turns out (and this can be shown with a little math)
that we achieve this by keeping the PRs of the turbines even.
The other reason is a purely pragmatic design consideration. If we are going to order turbines,
larger ones tend to be more expensive than smaller ones (both in initial cost and in overhead for
physical space and maintenance). The same will tend to be true of compressors.
Figure 14 below shows the effect of varying the turbine outlet pressure on the thermal efficiency
of the cycle for a two stage plant. We note that the efficiency peaks when the outlet pressure of
the first turbine is about 316 kPa, which gives both turbines PRs of 3.16.
Figure 14: Cycle efficiency vs. turbine outlet pressure
The overall PR for the cycle is 1000 kPa / 100 kPa = 10. For each turbine to have the same PR,
each PRturbine must be (PRoverall)1/(#turbines).
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Improving an Air-Standard Power Cycle
CyclePad Design Files
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CyclePad design of the basic Brayton cycle bray1.dsn.
CyclePad design of the Brayton cycle with a regenerative heat exchanger bray1x.dsn.
CyclePad design of the Brayton cycle with a reheat stage bray1r.dsn.
CyclePad design of the Brayton cycle with an intercooling stage bray1i.dsn.
CyclePad design of the Brayton cycle with a regenerative heat exchanger, intercooling,
and reheat bray1rix.dsn.
CyclePad design of the 5 turbine Brayton cycle with a regenerative heat exchanger,
intercooling, and reheat bray5rix.dsn.
Sources
Haywood, R.W. 1980. Analysis of Engineering Cycles. Pergamon Press. ISBN: 0-08025440-3
Van Wylen, Sonntag, Borgnakke. 1994. Fundamentals of Classical Thermodynamics,
4th Ed. John Wiley and Sons. ISBN: 0-471-59395-8
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