2. 2
Table of Contents
NOMENCLATURE.....................................................................................................................................................................3
GENERAL ASSUMPTIONS.....................................................................................................................................................4
SECTION A....................................................................................................................................................................................4
SECTION B....................................................................................................................................................................................6
SECTION C....................................................................................................................................................................................7
SECTION D...................................................................................................................................................................................9
SECTION E.................................................................................................................................................................................10
REFERENCES............................................................................................................................................................................14
3. 3
Nomenclature
๐ถ ๐ Constant pressure specific heat of dry air
๐ถ ๐ฃ Constant volume specific heat of dry air
k
๐ถ ๐
๐ถ ๐ฃ
โ
๐๐๐ Heat into thermodynamic cycle
๐ ๐๐ข๐ก Heat out of thermodynamic cycle
๐๐๐๐ก Net work of cycle
๐๐๐ ๐๐๐ก๐๐๐๐๐ Isentropic work
๐๐๐๐ก๐ข๐๐ Work considering irreversibilities
๐๐กโ Thermal efficiency
๐๐กโ,โ Thermal efficiency of Humphrey Cycle
๐๐กโ,โ,๐
Thermal efficiency of Humphrey Cycle
considering irreversibilities
๐๐กโ,โ,๐๐๐ฅ
Maximum thermal efficiency of Humphrey
Cycle
๐๐กโ,๐ Thermal efficiency of Brayton Cycle
๐๐ Efficiency of compressor
๐๐ก Efficiency of turbine
๐๐ Compressor pressure ratio
๐๐,๐๐๐ฅ Maximum compressor pressure ratio
๐1 Compressor inlet temperature
๐2 Compressor exit/burner inlet temperature
๐2
โฒ Compressor exit/burner inlet temperature when
considering losses in compressor
๐3
Burner exit temperature/ turbine inlet
temperature
๐4 Turbine exit temperature
๐4
โฒ Turbine exit temperature when considering
losses in turbine
๐3
๐3
๐1
โ
๐4
๐4
๐1
โ
4. 4
General Assumptions
Throughout this paper we will neglect any chemical changes that occur during the combustion
process. We will also hold the specific heat of dry air to be constant. These assumptions are
made in order to simplify the process of analyzing these specific thermodynamic cycles.
Section A
The thermal efficiency of a cycle can be defined as the ratio of net work to the heat introduced
into the cycle. The net work can be defined as the difference between heat introduced and
leaving the cycle. This can be seen below:
๐๐กโ =
๐๐๐๐ก
๐๐๐
=
๐๐๐ โ ๐ ๐๐ข๐ก
๐๐๐
(1)
For the Humphrey Cycle work is introduced via a constant volume process and rejected via a
constant pressure process. Using conservation of energy:
๐๐กโ,โ =
๐ถ ๐ฃ( ๐3 โ ๐2) โ ๐ถ ๐( ๐4 โ ๐1)
๐ถ ๐( ๐3 โ ๐2)
(2)
Simplifying:
๐๐กโ,โ = 1 โ
๐๐1( ๐4 โ 1)
๐2 ( ๐3
๐2
โ 1)
(3)
In order to represent this expression in terms of ฯ4 and ฯc we need a relationship between
๐3
๐2
and
ฯ4. We can find this relationship from Reference [1] and by using conservation of energy we
achieve the relationship:
๐4 =
๐3
๐2
1
๐
(4)
Because there are no irreversibilities the compression process is isentropic. From the definition
of isentropic processes:
๐2
๐1
= ๐๐
๐โ1
๐ (5)
Placing (4) and (5) into (3) we obtain:
๐๐กโ,โ = 1 โ
๐๐
โ๐+1
๐ ( ๐4 โ 1)
๐ ๐ โ 1
(6)
5. 5
In order to compare the thermal efficiency of the Humphrey and Brayton Cycle we will need an
expression for the thermal efficiency of the Brayton Cycle. Using Reference [2] and (5) we
achieve:
๐๐กโ,๐ = 1 โ
1
๐๐
๐โ1
๐
(7)
For a comparison we will use ฯc=20 and ฯ3=6. However our expression for the thermal efficiency
of the Humphrey Cycle is in ฯ4 instead of the more relevant temperature ratio ฯ3. If we assume a
reasonable T1=288K we can calculate T4 using (4) and (5), thus allowing the determination of ฯ4.
Using this method, (6), (7), and k=1.4 we obtain:
๐๐กโ,โ = 63.5%
๐๐กโ,๐ = 57.5%
The Humphrey Cycle is more efficient than the Brayton Cycle because it is able to convert the
heat gained from combustion to a pressure rise in the working fluid. This is a clear indicator of
useful mechanical energy. The Brayton cycle converts this heat into molecular motion of the
working fluid. This is an indicator of a gain in internal energy. The Brayton Cycle produces
significantly more entropy than the Humphrey Cycle. The definition of entropy change for an
ideal gas undergoing heating/cooling and expansion/compression reinforces this statement. The
specific heat of dry air at constant volume is significantly less than the specific heat of dry air at
constant pressure, thus making the production of entropy less for the Humphrey Cycle. The
definition of entropy is the measure of a systems thermal energy unavailability. The Humphrey
Cycle is thermodynamically more available than the Brayton Cycle. Furthermore, if one
examines a T-S diagram of the two cycles it can be seen that T4 is always less for the Humphrey
Cycle. This corresponds to the thermodynamic availability of the Humphrey Cycle. A lower T4
represents more energy being extracted from the working fluid, which represents better
efficiency. Below one can find a plot for thermal efficiency:
6. 6
Figure 1 Thermal Efficiency vs Compressor Pressure Ratio for Ideal Humphrey and Brayton Thermodynamic
Cycles with Varying ๐ ๐ Values
It can be seen from Figure 1 that the Humphrey Cycle is always more efficient. It is noted that
Figure 1 was generated by finding ฯ4 using T1=288K, (4), and (5). Furthermore, Figure 1 was
generated by using (6) and (7).
Section B
In order to begin finding an expression for non-dimensional net work output in terms of ฯ4 and ฯc
we will use the expression for net work in a thermodynamic cycle and conservation of energy. It
is noted that this expression for net work applies directly to the Humphrey Cycle. We achieve:
๐ค ๐๐๐ก = ๐ถ ๐ฃ( ๐3 โ ๐2) โ ๐ถ ๐( ๐4 โ ๐1) (8)
Rearranging terms:
๐ค ๐๐๐ก
๐ถ ๐ฃ ๐1
=
๐2
๐1
(
๐3
๐2
โ 1) โ ๐( ๐4 โ 1) (9)
Using (4) and (5):
๐ค ๐๐๐ก
๐ถ ๐ฃ ๐1
= ๐๐
๐โ1
๐ ( ๐4
๐
โ 1) โ ๐( ๐4 โ 1) (10)
By using the same method to find T4 as in Section A we can plot non-dimensional work output in
terms of ฯ4 and ฯc:
7. 7
Figure 2 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle
with Varying ๐ ๐ Values
It is noted Figure 2 was generated using (10).
Section C
In order to find thermal efficiency in terms of ฯ3 and ฯc we will utilize (3). Combining with (4)
and (5) and simplifying we achieve:
๐๐กโ,โ = 1 โ
๐๐๐
โ๐+1
๐ (๐3
๐โ1
๐๐
โ๐+1
๐2
โ 1)
๐3 ๐๐
โ๐+1
๐ โ 1
(11)
To find non-dimensional net work in terms of ฯ3 and ฯc we will utilize (9). Again combining with
(4) and (5) then simplifying we achieve:
๐๐๐๐ก
๐ถ ๐ฃ ๐1
= ๐๐
๐โ1
๐ (๐3 ๐๐
โ๐+1
๐ โ 1) โ ๐ (๐3
๐โ1
๐๐
โ๐+1
๐2
โ 1)(12)
Below one can find plots for both thermal efficiency and non-dimensional network:
8. 8
Figure 3 Thermal Efficiency vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle with
Varying ๐ ๐ Values
Figure 4 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle
with Varying ๐ ๐ Values
It is noted that Figure 3 and Figure 4 were generated using (11) and (12).
From Figure 3 one can see that as ฯc increases thermal efficiency increases as well. This is to be
expected, as it is known that higher temperatures in a thermodynamic cycle will increase thermal
efficiency. This is the same reason why efficiency is greater in the figure for higher ฯ3 values.
When fixing ฯ3 and increasing ฯc thermal efficiency still increases because of the definition of
thermal efficiency in a thermodynamic cycle, however net work decreases. As one may envision
9. 9
from a T-S diagram with a fixed T3 value the area between the heat addition/rejection curves
diminished until it becomes zero. Thus, in an ideal cycle scenario there is a specific thermal
efficiency value where net work will equal zero. When ฯ3 is not fixed T3 may be increased thus
leading to not only increased efficiencies but also increased net work. In reality T3 is a highly
controlled parameter because of structural concerns relating to the turbine.
From Figure 4 it can be seen that there are ฯc values for maximum net work. As previously
discussed as ฯ3 increases so does T3, thus increasing net work. Thus, for higher ฯ3 values the
maximum net work value is increased. In addition as previously discussed net work decreases
with increasing ฯc . As T2 approaches T3 because of ฯc the area inside the heat addition/rejection
curves, in the cycles T-S diagram, shrinks indicating a loss in net work. Finally as T2 nears T3 the
area is reduced to zero, as there is no heat addition. Figure 4 clearly indicates that there is a
maximum ฯc value where net work becomes zero.
Section D
There is no explicit term for optimal ฯc that maximizes thermal efficiency. Like an ideal Brayton
Cycle thermal efficiency increases with ฯc for an ideal Humphrey Cycle. Eventually at very high
ฯcโs T2 approaches T3 meaning no heat is added to the thermodynamic cycle. With no heat added
to the cycle no work is generated. This defeats the purpose of a propulsion system. The ฯc when
zero net work is generated can be described as the maximum ฯc. At this point thermal efficiency
is also at its highest possible value, while propulsion is still being generated. Thus at maximum
ฯc thermal efficiency is also at its maximum.
In order to find a ฯc value for maximum thermal efficiency we will determine an expression for
maximum ฯc. By using the expression for non-dimensional net work, (12), and setting to zero we
achieve:
0 = ๐๐
๐โ1
๐ (๐3 ๐๐
โ๐+1
๐ โ 1) โ ๐ (๐3
๐โ1
๐๐
โ๐+1
๐ โ 1)(13)
By solving for ฯc we achieve:
๐๐,๐๐๐ฅ = ๐3
๐
๐โ1 โ ๐
2
๐โ1 โ ๐
2
๐โ1 ๐3
โ๐+1
๐๐,๐๐๐ฅ
โ๐+1
๐2โ๐ (14)
By solving for this equation numerically one can find a value for maximum ฯc, which equals the
ฯc that maximizes thermal efficiency.
10. 10
In order to determine an expression for the thermal efficiency, which results from maximum ฯc,
we can simply insert the term ฯc,max into expression (11). This results in:
๐๐กโ,โ,๐๐๐ฅ = 1 โ
๐๐๐,๐๐๐ฅ
โ๐+1
๐ (๐3
๐โ1
๐๐,๐๐๐ฅ
โ๐+1
๐2
โ 1)
๐3 ๐๐,๐๐๐ฅ
โ๐+1
๐ โ 1
(15)
One can interpret this point using graphs that include non-dimensional net work vs ฯc and
thermal efficiency vs ฯc. By locating the ฯc when non-dimensional net work becomes zero one
can locate the maximum thermal efficiency value by using the same ฯc.
Section E
In order to find expression for thermal efficiency and non-dimensional net work in terms of ฯ3,
ฯc, ฮทc, ฮทt and k we will begin by using the definition of compressor efficiency:
๐ ๐ =
๐๐๐ ๐๐๐ก๐๐๐๐๐
๐๐๐๐ก๐ข๐๐
(16)
Using conservation of energy and simplifying we achieve:
๐ ๐ =
๐2 โ ๐1
๐2
โฒ
โ ๐1
(17)
Rearranging terms we can also achieve:
๐2
โฒ
๐1
=
๐ ๐ + ( ๐2
๐1
โ 1)
๐๐
(18)
The same steps will be taken for turbine efficiency:
๐๐ก =
๐๐๐๐ก๐ข๐๐
๐๐๐ ๐๐๐ก๐๐๐๐๐
(19)
๐๐ =
๐3 โ ๐4
โฒ
๐3 โ ๐4
(20)
๐4
โฒ
๐1
= ๐3 โ ๐๐ก (๐3 โ
๐1
๐2
๐โ1
๐3
๐โ1
)(21)
By using (2) in terms of a cycle with irreversibilities and simplifying we begin to achieve an
expression for thermal efficiency with irreversibilities:
๐๐กโ,โ = 1 โ
๐ ( ๐4
โฒ
๐1
โ 1)
(๐3 โ ๐2
โฒ
๐1
)
(22)
After inserting (5), (18), (21), and simplifying we can obtain:
11. 11
๐๐กโ,โ,๐ = 1 โ
๐ [๐3 โ ๐๐ก (๐3 โ ๐3
๐โ1
๐๐
โ๐+1
๐2
) โ 1]
๐3 โ ๐๐
โ1 [๐๐ + (๐๐
๐โ1
๐ โ 1)]
(23)
Similarly using (9) in terms of a cycle with irreversibilities and simplifying we begin to achieve
an expression for non-dimensional net work with irreversibilities:
๐ค ๐๐๐ก
๐ถ ๐ฃ ๐1
=
๐2
โฒ
๐1
(
๐3
๐2
โฒ
๐1
โ 1) โ ๐ (
๐4
โฒ
๐1
โ 1) (24)
Again after plugging in (5), (18), (21), and simplifying we can obtain:
๐ค ๐๐๐ก
๐ถ ๐ฃ ๐1 ๐
= ๐ ๐
โ1
[๐๐ + (๐๐
๐โ1
๐ โ 1)] [๐3 (๐ ๐
โ1
(๐๐ + (๐๐
๐โ1
๐ โ 1)))
โ1
โ 1]
โ ๐ [๐3 โ ๐๐ก (๐3 โ ๐3
๐โ1
๐๐
โ๐+1
๐2
) โ 1] (25)
By setting ฮทc and ฮทt in (23) and (25) to 1 and rearranging terms equations (11) and (12) can be
found which are ideal expressions. This is a quick way to verify the validity of the expressions.
Below one can find plots for both thermal efficiency and non-dimensional network:
Figure 5 Thermal Efficiency vs Compressor Pressure Ratio For Ideal and Non-Ideal Humphrey Thermodynamic
Cyles with Varying ๐ ๐ values
12. 12
Figure 6 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal and Non-Ideal Humphrey
Thermodynamic Cycles with Varying ๐ ๐ values
It is noted that Figure 5 and Figure 6 were generated using (23) and (25).
From Figure 5 one can see the effects of adding losses from the compressor and turbine. One
initially can see that the thermal efficiencies for each set of ฯ3โs across increasing ฯcโs for non-
ideal cycles are lower than the ideal cycles. In addition to this when losses are taken into account
thermal efficiencies do not keep climbing. It can be seen that there are maximum thermal
efficiency points for each fixed ฯ3โs at corresponding ฯcโs. Maximum thermal efficiency points
climb with increased ฯ3โs due to higher cycle temperatures, which provide better thermal
efficiency. In addition to this these points occur at higher ฯcโs for higher ฯ3โs because of the
needed T2 to reach necessary T3. After these maximum thermal efficiency points the values
begin to drop. The reductions in efficiencies are caused by the work needed to drive the
compressor. Just as maximum thermal efficiency points occur at lower ฯcโs for lower ฯ3โs, zero
thermal efficiency points occur at earlier ฯcโs for lower ฯ3โs.
From Figure 6 one can see the effects of adding losses from the compressor and turbine in regard
to non-dimensional net work. Initially one can see that the non-dimensional net work values are
significantly lower than the ideal cycles. This implies that maximum non-dimensional values are
also lower than the ideal cycles. Despite all values being significantly lower the behavior of the
cycles with losses greatly resemble the behavior of the ideal cycles. The only discrepancies are
13. 13
the increased slopes in the non-ideal cycles compared to the ideal cycles. As expected adding
losses form the compressor and turbine greatly reduce net work.
14. 14
References
1) Kamiuto, K. "Comparison of Basic Gas Cycles under the Restriction of Constant Heat
Addition." Science Direct. 1 Sept. 2005. Web. 3 May 2015.
<http://www.sciencedirect.com.ezproxy.cul.columbia.edu/science/article/pii/S030626190500085
1#>
2) Farokhi, Saeed. Aircraft Propulsion. Second ed. Chichester: John Wiley & Sons, 2014. Print.