Humphrey Cycle Analysis

Frederick Avyasa Smith

MECE E4305: Mechanics and Thermodynamics of Propulsion

Prof. Dr. P. Akbari

May 4th, 2015

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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

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NomenclatureC p Constant pressure specific heat of dry airC v Constant volume specific heat of dry air

kCp

C v

Q¿ Heat into thermodynamic cycleQout Heat out of thermodynamic cycleW net Net work of cycle

W isentropic Isentropic workW actual Work considering irreversibilities

ηth Thermal efficiencyηth ,h Thermal efficiency of Humphrey Cycle

ηth ,h , iThermal efficiency of Humphrey Cycle

considering irreversibilities

ηth ,h , maxMaximum thermal efficiency of Humphrey

Cycleηth ,b Thermal efficiency of Brayton Cycleηc Efficiency of compressorηt Efficiency of turbineπc Compressor pressure ratio

πc ,max Maximum compressor pressure ratioT 1 Compressor inlet temperatureT 2 Compressor exit/burner inlet temperature

T 2' Compressor exit/burner inlet temperature when

considering losses in compressor

T 3Burner exit temperature/ turbine inlet

temperatureT 4 Turbine exit temperature

T 4' Turbine exit temperature when considering

losses in turbine

τ3

T3

T1

τ 4

T 4

T 1

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General AssumptionsThroughout 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 AThe 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=W net

Q¿=

Q¿−Qout

Q¿(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 )−Cp (T 4−T 1 )

CV (T3−T2 )(2)

Simplifying:

ηth ,h=1−k T1 (τ 4−1 )

T 2(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:

τ 4=T3

T 2

1k (4)

Because there are no irreversibilities the compression process is isentropic. From the definition

of isentropic processes:

T2

T1

=πc

k−1k (5)

Placing (4) and (5) into (3) we obtain:

ηth ,h=1−k π

−k+1k (τ 4−1 )τ k−1

(6)

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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

π c

k−1k

(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:

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Figure 1 Thermal Efficiency vs Compressor Pressure Ratio for Ideal Humphrey and Brayton Thermodynamic Cycles with Varying τ3 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 BIn 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:

wnet=C v (T 3−T 2 )−C p (T 4−T 1) (8)

Rearranging terms:

wnet

C v T 1

=T 2

T 1(T3

T 2

−1)−k ( τ4−1 )(9)

Using (4) and (5):

wnet

C v T 1

=πc

k−1k (τ 4

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:

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Figure 2 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle with Varying τ3 Values

It is noted Figure 2 was generated using (10).

Section CIn 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−k π c

−k+1k ( τ3

k−1

πc

−k +1

k2

−1)τ3 πc

−k +1k −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:

W net

C v T 1

=πc

k−1k (τ3 π c

−k+1k −1)−k (τ3

k−1

πc

−k+1

k2

−1)(12)

Below one can find plots for both thermal efficiency and non-dimensional network:

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Figure 3 Thermal Efficiency vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle with Varying τ3 Values

Figure 4 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal Humphrey Thermodynamic Cycle with Varying τ3 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 τ 3 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

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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 DThere 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=π c

k−1k ( τ3 πc

−k +1k −1)−k (τ3

k−1

πc

−k +1k −1)(13)

By solving for πc we achieve:

πc ,max=τ3

kk−1−k

2k−1−k

2k−1 τ3

−k+1 π c, max

−k+1k2−k (14)

By solving for this equation numerically one can find a value for maximum πc, which equals the

πc that maximizes thermal efficiency.

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In order to determine an expression for the thermal efficiency, which results from maximum π c,

we can simply insert the term πc,max into expression (11). This results in:

ηth ,h , max=1−k π c, max

−k+1k (τ3

k−1

πc ,max

−k+1

k2

−1)τ3 π c, max

−k+1k −1

(15)

One can interpret this point using graphs that include non-dimensional net work vs π c 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 EIn 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:

ηc=W isentropic

W actual

(16)

Using conservation of energy and simplifying we achieve:

ηc=T 2−T 1

T 2'−T1

(17)

Rearranging terms we can also achieve:

T2'

T 1

=

ηc+(T2

T1

−1)ηc

(18)

The same steps will be taken for turbine efficiency:

ηt=W actual

W isentropic

(19)

ηc=T 3−T 4

'

T 3−T 4

(20)

T 4'

T 1

=τ3−ηt (τ3−T 1

T 2

k−1

τ3k−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:

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ηth ,h=1−

k (T 4'

T1

−1)(τ3−

T2'

T 1)(22)

After inserting (5), (18), (21), and simplifying we can obtain:

ηth ,h , i=1−k [τ3−ηt (τ3−τ3

k−1

πc

−k+1

k2 )−1]τ3−ηc

−1[ηc+(π c

k−1k −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:

wnet

C v T 1

=T 2

'

T 1 ( τ3

T 2'

T 1

−1)−k (T 4'

T1

−1)(24)

Again after plugging in (5), (18), (21), and simplifying we can obtain:

wnet

C v T 1 i

=ηc−1[ηc+(π c

k−1k −1)][τ3(ηc

−1(ηc+(πc

k−1k −1)))

−1

−1]−k [τ3−ηt (τ3−τ3k−1

π c

−k+1k2 )−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:

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Figure 5 Thermal Efficiency vs Compressor Pressure Ratio For Ideal and Non-Ideal Humphrey Thermodynamic Cyles with Varying τ3 values

Figure 6 Non-Dimensional Net Work vs Compressor Pressure Ratio for Ideal and Non-Ideal Humphrey Thermodynamic Cycles with Varying τ3 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

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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

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.

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References1) 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/

S0306261905000851#>

2) Farokhi, Saeed. Aircraft Propulsion. Second ed. Chichester: John Wiley & Sons, 2014. Print.

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