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
𝐶 𝑝 Constant pressure specific heat of dry air
𝐶 𝑣 Constant volume specific heat of dry air
k
𝐶 𝑝
𝐶 𝑣
⁄
𝑄𝑖𝑛 Heat into thermodynamic cycle
𝑄 𝑜𝑢𝑡 Heat out of thermodynamic cycle
𝑊𝑛𝑒𝑡 Net work of cycle
𝑊𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 Isentropic work
𝑊𝑎𝑐𝑡𝑢𝑎𝑙 Work considering irreversibilities
𝜂𝑡ℎ Thermal efficiency
𝜂𝑡ℎ,ℎ Thermal efficiency of Humphrey Cycle
𝜂𝑡ℎ,ℎ,𝑖
Thermal efficiency of Humphrey Cycle
considering irreversibilities
𝜂𝑡ℎ,ℎ,𝑚𝑎𝑥
Maximum thermal efficiency of Humphrey
Cycle
𝜂𝑡ℎ,𝑏 Thermal efficiency of Brayton Cycle
𝜂𝑐 Efficiency of compressor
𝜂𝑡 Efficiency of turbine
𝜋𝑐 Compressor pressure ratio
𝜋𝑐,𝑚𝑎𝑥 Maximum compressor pressure ratio
𝑇1 Compressor inlet temperature
𝑇2 Compressor exit/burner inlet temperature
𝑇2
′ Compressor exit/burner inlet temperature when
considering losses in compressor
𝑇3
Burner exit temperature/ turbine inlet
temperature
𝑇4 Turbine exit temperature
𝑇4
′ Turbine exit temperature when considering
losses in turbine
𝜏3
𝑇3
𝑇1
⁄
𝜏4
𝑇4
𝑇1
⁄
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:
𝜂𝑡ℎ =
𝑊𝑛𝑒𝑡
𝑄𝑖𝑛
=
𝑄𝑖𝑛 − 𝑄 𝑜𝑢𝑡
𝑄𝑖𝑛
(1)
For the Humphrey Cycle work is introduced via a constant volume process and rejected via a
constant pressure process. Using conservation of energy:
𝜂𝑡ℎ,ℎ =
𝐶 𝑣( 𝑇3 − 𝑇2) − 𝐶 𝑝( 𝑇4 − 𝑇1)
𝐶 𝑉( 𝑇3 − 𝑇2)
(2)
Simplifying:
𝜂𝑡ℎ,ℎ = 1 −
𝑘𝑇1( 𝜏4 − 1)
𝑇2 ( 𝑇3
𝑇2
− 1)
(3)
In order to represent this expression in terms of τ4 and πc we need a relationship between
𝑇3
𝑇2
and
τ4. We can find this relationship from Reference [1] and by using conservation of energy we
achieve the relationship:
𝜏4 =
𝑇3
𝑇2
1
𝑘
(4)
Because there are no irreversibilities the compression process is isentropic. From the definition
of isentropic processes:
𝑇2
𝑇1
= 𝜋𝑐
𝑘−1
𝑘 (5)
Placing (4) and (5) into (3) we obtain:
𝜂𝑡ℎ,ℎ = 1 −
𝑘𝜋
−𝑘+1
𝑘 ( 𝜏4 − 1)
𝜏 𝑘 − 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:
𝜂𝑡ℎ,𝑏 = 1 −
1
𝜋𝑐
𝑘−1
𝑘
(7)
For a comparison we will use πc=20 and τ3=6. However our expression for the thermal efficiency
of the Humphrey Cycle is in τ4 instead of the more relevant temperature ratio τ3. If we assume a
reasonable T1=288K we can calculate T4 using (4) and (5), thus allowing the determination of τ4.
Using this method, (6), (7), and k=1.4 we obtain:
𝜂𝑡ℎ,ℎ = 63.5%
𝜂𝑡ℎ,𝑏 = 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 𝝉 𝟑 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 τ4 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 τ4 and πc
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:
𝑤 𝑛𝑒𝑡 = 𝐶 𝑣( 𝑇3 − 𝑇2) − 𝐶 𝑝( 𝑇4 − 𝑇1) (8)
Rearranging terms:
𝑤 𝑛𝑒𝑡
𝐶 𝑣 𝑇1
=
𝑇2
𝑇1
(
𝑇3
𝑇2
− 1) − 𝑘( 𝜏4 − 1) (9)
Using (4) and (5):
𝑤 𝑛𝑒𝑡
𝐶 𝑣 𝑇1
= 𝜋𝑐
𝑘−1
𝑘 ( 𝜏4
𝑘
− 1) − 𝑘( 𝜏4 − 1) (10)
By using the same method to find T4 as in Section A we can plot non-dimensional work output in
terms of τ4 and πc:
7. 7
Figure 2 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle
with Varying 𝝉 𝟑 Values
It is noted Figure 2 was generated using (10).
Section C
In order to find thermal efficiency in terms of τ3 and πc we will utilize (3). Combining with (4)
and (5) and simplifying we achieve:
𝜂𝑡ℎ,ℎ = 1 −
𝑘𝜋𝑐
−𝑘+1
𝑘 (𝜏3
𝑘−1
𝜋𝑐
−𝑘+1
𝑘2
− 1)
𝜏3 𝜋𝑐
−𝑘+1
𝑘 − 1
(11)
To find non-dimensional net work in terms of τ3 and πc we will utilize (9). Again combining with
(4) and (5) then simplifying we achieve:
𝑊𝑛𝑒𝑡
𝐶 𝑣 𝑇1
= 𝜋𝑐
𝑘−1
𝑘 (𝜏3 𝜋𝑐
−𝑘+1
𝑘 − 1) − 𝑘 (𝜏3
𝑘−1
𝜋𝑐
−𝑘+1
𝑘2
− 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 𝝉 𝟑 Values
Figure 4 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle
with Varying 𝝉 𝟑 Values
It is noted that Figure 3 and Figure 4 were generated using (11) and (12).
From Figure 3 one can see that as πc 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 τ3 values.
When fixing τ3 and increasing πc 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 τ3 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 πc values for maximum net work. As previously
discussed as τ3 increases so does T3, thus increasing net work. Thus, for higher τ3 values the
maximum net work value is increased. In addition as previously discussed net work decreases
with increasing πc . As T2 approaches T3 because of πc 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 πc value where net work becomes zero.
Section D
There is no explicit term for optimal πc that maximizes thermal efficiency. Like an ideal Brayton
Cycle thermal efficiency increases with πc for an ideal Humphrey Cycle. Eventually at very high
πc’s 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 πc when
zero net work is generated can be described as the maximum πc. At this point thermal efficiency
is also at its highest possible value, while propulsion is still being generated. Thus at maximum
πc thermal efficiency is also at its maximum.
In order to find a πc value for maximum thermal efficiency we will determine an expression for
maximum πc. By using the expression for non-dimensional net work, (12), and setting to zero we
achieve:
0 = 𝜋𝑐
𝑘−1
𝑘 (𝜏3 𝜋𝑐
−𝑘+1
𝑘 − 1) − 𝑘 (𝜏3
𝑘−1
𝜋𝑐
−𝑘+1
𝑘 − 1)(13)
By solving for πc we achieve:
𝜋𝑐,𝑚𝑎𝑥 = 𝜏3
𝑘
𝑘−1 − 𝑘
2
𝑘−1 − 𝑘
2
𝑘−1 𝜏3
−𝑘+1
𝜋𝑐,𝑚𝑎𝑥
−𝑘+1
𝑘2−𝑘 (14)
By solving for this equation numerically one can find a value for maximum πc, which equals the
πc that maximizes thermal efficiency.
10. 10
In order to determine an expression for the thermal efficiency, which results from maximum πc,
we can simply insert the term πc,max into expression (11). This results in:
𝜂𝑡ℎ,ℎ,𝑚𝑎𝑥 = 1 −
𝑘𝜋𝑐,𝑚𝑎𝑥
−𝑘+1
𝑘 (𝜏3
𝑘−1
𝜋𝑐,𝑚𝑎𝑥
−𝑘+1
𝑘2
− 1)
𝜏3 𝜋𝑐,𝑚𝑎𝑥
−𝑘+1
𝑘 − 1
(15)
One can interpret this point using graphs that include non-dimensional net work vs πc and
thermal efficiency vs πc. By locating the πc when non-dimensional net work becomes zero one
can locate the maximum thermal efficiency value by using the same πc.
Section E
In order to find expression for thermal efficiency and non-dimensional net work in terms of τ3,
πc, ηc, ηt and k we will begin by using the definition of compressor efficiency:
𝜂 𝑐 =
𝑊𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐
𝑊𝑎𝑐𝑡𝑢𝑎𝑙
(16)
Using conservation of energy and simplifying we achieve:
𝜂 𝑐 =
𝑇2 − 𝑇1
𝑇2
′
− 𝑇1
(17)
Rearranging terms we can also achieve:
𝑇2
′
𝑇1
=
𝜂 𝑐 + ( 𝑇2
𝑇1
− 1)
𝜂𝑐
(18)
The same steps will be taken for turbine efficiency:
𝜂𝑡 =
𝑊𝑎𝑐𝑡𝑢𝑎𝑙
𝑊𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐
(19)
𝜂𝑐 =
𝑇3 − 𝑇4
′
𝑇3 − 𝑇4
(20)
𝑇4
′
𝑇1
= 𝜏3 − 𝜂𝑡 (𝜏3 −
𝑇1
𝑇2
𝑘−1
𝜏3
𝑘−1
)(21)
By using (2) in terms of a cycle with irreversibilities and simplifying we begin to achieve an
expression for thermal efficiency with irreversibilities:
𝜂𝑡ℎ,ℎ = 1 −
𝑘 ( 𝑇4
′
𝑇1
− 1)
(𝜏3 − 𝑇2
′
𝑇1
)
(22)
After inserting (5), (18), (21), and simplifying we can obtain:
11. 11
𝜂𝑡ℎ,ℎ,𝑖 = 1 −
𝑘 [𝜏3 − 𝜂𝑡 (𝜏3 − 𝜏3
𝑘−1
𝜋𝑐
−𝑘+1
𝑘2
) − 1]
𝜏3 − 𝜂𝑐
−1 [𝜂𝑐 + (𝜋𝑐
𝑘−1
𝑘 − 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:
𝑤 𝑛𝑒𝑡
𝐶 𝑣 𝑇1
=
𝑇2
′
𝑇1
(
𝜏3
𝑇2
′
𝑇1
− 1) − 𝑘 (
𝑇4
′
𝑇1
− 1) (24)
Again after plugging in (5), (18), (21), and simplifying we can obtain:
𝑤 𝑛𝑒𝑡
𝐶 𝑣 𝑇1 𝑖
= 𝜂 𝑐
−1
[𝜂𝑐 + (𝜋𝑐
𝑘−1
𝑘 − 1)] [𝜏3 (𝜂 𝑐
−1
(𝜂𝑐 + (𝜋𝑐
𝑘−1
𝑘 − 1)))
−1
− 1]
− 𝑘 [𝜏3 − 𝜂𝑡 (𝜏3 − 𝜏3
𝑘−1
𝜋𝑐
−𝑘+1
𝑘2
) − 1] (25)
By setting ηc and ηt 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 𝝉 𝟑 values
12. 12
Figure 6 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal and Non-Ideal Humphrey
Thermodynamic Cycles with Varying 𝝉 𝟑 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 τ3’s across increasing πc’s 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 τ3’s at corresponding πc’s. Maximum thermal efficiency points
climb with increased τ3’s due to higher cycle temperatures, which provide better thermal
efficiency. In addition to this these points occur at higher πc’s for higher τ3’s 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 πc’s for lower τ3’s, zero
thermal efficiency points occur at earlier πc’s for lower τ3’s.
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.
