A Thermodynamic Analysis of a Turbojet Engine

ME 2334 – Course Project

By Jeffrey Kornuta

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Introduction

This paper looks into the thermodynamic analysis of an ideal turbojet engine,

focusing on the relationships between the compression ratio (RC), max temperature

(Tmax), mass specific thrust (MST), and thrust specific fuel consumption (TSFC). Also,

this paper will explore the effects of an aircraft’s Mach number on engine performance

and why supersonic flight differs so much from subsonic flight.

Figure A

The air-standard power cycle for a turbojet engine differs very little from the

well-known Brayton cycle; however, unlike the Brayton cycle, this engine relies on the

rapid acceleration of air, or thrust, to produce the desired power. Thrust is defined as

! = !m

a(1+ ƒ)V

6"V

1[ ] , (1)

where

�

˙ m a is mass flow rate of air, ƒ is the fuel to air ratio, and V is the velocity of the air.

As one sees from Equation 1, thrust relies on the difference in air velocities between the

intake and exhaust of the engine. As a result, a diffuser and nozzle is added to the basic

structure of this Brayton cycle to produce an overall increase in

�

!V1"6

, thus forming a

basic jet propulsion cycle (Figure A).

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Analysis and Results

The Matlab program included in this paper calculates and graphs MST kN ! s kg[ ]

and TSFC kg kN ! s[ ] as a function of compression ratio RC for Tmax values of 1500,

1600, and 1700 K. For an engine traveling at subsonic Mach number 0.85, the MST

increases sharply, hits a maximum value, and then decreases slowly as RC increases. At

the same Mach number, TSFC decreases sharply then proceeds to decrease gradually as

RC increases.

These results make sense; as RC increases, the specific volume of the air will

decrease, causing the overall amount of air per unit volume to increase. As a result, the

ratio of air to fuel will increase until the mixture reaches optimum stoichiometric

conditions for combustion. After the ideal pressure is surpassed, the surplus air acts as a

cooling agent and absorbs the heat generated from the combustion process, thus

decreasing the availability of the gas. Availability is defined as

! = [h " T0s + 1

2V2+ gZ ]" [h

0" T

0s0+ gZ

0] , (2)

where h is specific enthalpy, s is specific entropy, and gZ represents specific potential

energy. After a few simplifications for our process, availability reduces to the following:

! = CP(T " T

0)

! h

! "# $#+ T

0(s0" s) + 1

2V2 , (3)

where Cp is the constant pressure specific heat of the gas. As one may observe, this

cooling of the gas caused by the excess air passing through the combustor will overall

decrease the ∆T of the gas, thus decreasing the energy available for thrust. Similarly, as

Tmax increases, the conditions remain the same with an exception for a higher temperature

gas exiting the combustor. Thus, the same RC will yield a higher MST.

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Likewise, TSFC decreases continually as RC increases because of the overall

increase in air per unit volume, resulting in a decrease of the fuel to air ratio ƒ. Realizing

that TSFC is defined as

!mf

!=

f

! !ma

=f

(1+ f )V6"V

1

, (4)

one sees that a decrease in ƒ will ultimately decrease the TSFC. Similarly, realizing that

the fuel to air ratio is defined as

f =T4T3!1

qf (CpT3) ! T4 T3, (5)

where qf is the specific heat addition, a higher value for the max temperature T4 will yield

a higher ƒ, resulting in greater TSFC.

Clearly, the optimum RC for mass specific thrust is given by a pressure ratio that

produces a maximum MST value. On the other hand, the optimum RC for the thrust

specific fuel consumption is given by a pressure ratio that produces a minimum TSFC

value. What is the overall optimum compression ratio when considering MST and TSFC?

The answer to this question depends on the application of the engine. If one is

considering a commercial airline aircraft, a low TSFC is crucial, so a compression ratio

which causes a low TSFC but produces just enough thrust to fly at cruise speed would be

ideal. However, if the application requires that the engine produce the maximum thrust

possible, the compression ratio needs to be set accordingly.

When considering the same analysis of the turbojet engine at Mach number 2.0,

the results change quite drastically. The max MST value occurs at a lower RC, and the rate

at which MST decreases is much greater than the previous Mach number (Figure D).

When considering the TSFC for the higher Mach number, one may observe that as RC

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increases, TSFC decreases, reaches a minimum, and then begins to increase to divergence

(Figure E).

The reasoning for the changes in the MST as a result of the increased Mach

number begins with the diffuser: as the air experiences a greater ∆V, the overall pressure

and temperature of the air will increase continually as it proceeds through the cycle. This

effect will ultimately reduce the availability ! of the gas as it passes through the

combustor by producing a smaller value for ∆T (Equation 3), thus resulting in a

maximum MST at a lower compression ratio and an overall lower MST. On the other

hand, the logic behind the changes in TSFC as a result of the increased Mach number is a

bit more puzzling. As compression ratio increases, the TSFC decreases as expected;

however, because the MST begins to decrease so rapidly, the TSFC begins to diverge

upward as the MST approaches zero (Equation 4).

Surprisingly, an optimum compression ratio for this Mach number is easier to

distinguish. Although actual results depend on the application, MST and TSFC have clear

maximum and minimum values, thus the mean RC value between the ratio producing the

maximum MST and minimum TSFC can most likely be considered ideal for supersonic

flight. In addition, assuming that the overall optimum RC for the lower Mach number will

usually be lower than the optimum RC for the Mach number 2.0, one concludes that the

ideal compression ratio decreases as Mach number increases. Again, this makes sense

when considering that the speed increase at the diffuser inlet will produce a greater

pressure entering the compressor, thus requiring a lower RC for optimum operation.

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Conclusions

The analysis and discussion of this turbojet engine assumes some ideal conditions.

In reality, the engine will experience losses in performance due to a number of factors.

One of the main factors that would hinder performance is heat loss throughout the

system. In this analysis, the diffuser, compressor, turbine, and nozzle are all assumed to

be adiabatic; however, in a real life scenario, the cycle would experience a loss in heat

from these components, causing the available energy of the gas to gradually decrease.

The second major assumption in this analysis is the elimination of entropy generation

within the diffuser and nozzle. In reality, the air would experience a greater temperature

leaving the diffuser and leaving the nozzle, affecting the engine performance. Finally,

this analysis negates pressure drops throughout the system. In a practical case, the flow of

air would experience pressure drops due to the friction within the pipes, causing an

overall decrease in engine performance.

In retrospect of the results from the analysis, one comes to the conclusion that

subsonic flight differs greatly from supersonic flight. In addition, realizing that the

Brayton-specific part of the analysis (mainly the compressor, combustor, and turbine)

remains constant, one may decide that the differences in the results for the different Mach

numbers lie with the diffuser and the nozzle. As the Mach number increases into

supersonic regions, the diffuser and nozzle must be redesigned in order to handle the high

velocity air. Ideally, supersonic turbojet engines would benefit from a diffuser that

created less of an increase in pressure and a nozzle that had more surface area facing the

rear of the engine, even though one might argue that the complete elimination of the

compressor and turbine (ie, a ramjet) would be best for high velocity situations.

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Figure B

Figure C

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Figure D

Figure E

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Matlab Program

% Jeff Kornuta % ME 2334-01 % Course Project % Define constants Cp = 1.004; k = 1.4; R = .287; qf = 4.5e4; Nc = .85; Nt = .9; M1 = .85; T1 = 216.7; P1 = 18.75; Rc_array = 2:.01:100; max1 = 0; max2 = 0; max3 = 0; % Analyze diffuser V1 = M1*sqrt(1000*k*R*T1); T2 = V1^2/(1000*2*Cp) + T1; P2 = P1*(T2/T1)^(k/(k-1)); % Start loop to vary temperatures for T4 = 1500:100:1700 i = 0; % Start loop to vary compression ratio Rc for Rc = 2:.01:100 i = i + 1; % Analyze compressor T3 = (T2*Rc^((k-1)/k) - T2)/Nc + T2; P3 = Rc*P2; wc = Cp*(T3 - T2); P4 = P3; % Analyze turbine wt = wc; T5 = T4 - wt/Cp; T5s = T4 - (T4 - T5)/Nt; P5 = P4*(T5s/T4)^(k/(k-1)); % Analyze nozzle P6 = P1; T6 = T5*(P6/P5)^((k-1)/k); V6 = sqrt(1000*2*Cp*(T5 - T6));

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% Crunch numbers for different temperatures & % find MST maxes

f = (T4/T3 - 1)/(qf/(Cp*T3) - T4/T3); if T4 == 1500 MST1(i) = (1 + f)*V6 - V1; TSFC1(i) = f/MST1(i); if MST1(i) > max1 max1 = MST1(i); Rc_max1 = Rc; end else if T4 == 1600 MST2(i) = (1 + f)*V6 - V1; TSFC2(i) = f/MST2(i); if MST2(i) > max2 max2 = MST2(i); Rc_max2 = Rc; end else MST3(i) = (1 + f)*V6 - V1; TSFC3(i) = f/MST3(i); if MST3(i) > max3 max3 = MST3(i); Rc_max3 = Rc; end end end end end fprintf(1,'Max MST for M=%.2f and Tmax=1500 = %.2f\n',M1,Rc_max1); fprintf(1,'Max MST for M=%.2f and Tmax=1600 = %.2f\n',M1,Rc_max2); fprintf(1,'Max MST for M=%.2f and Tmax=1700 = %.2f\n',M1,Rc_max3); % Plot the data plot(Rc_array,MST1,Rc_array,MST2,Rc_array,MST3); axis([0 105 500 950]); xlabel('Compression Ratio [P3/P2]'); ylabel('Mass Specific Thrust [kN-s/kg]'); title('Mass Specific Thrust as a Function of Compression Ratio [Mach 0.85]'); % plot(Rc_array,TSFC1,Rc_array,TSFC2,Rc_array,TSFC3); % axis([0 105 1.5e-5 5e-5]); % xlabel('Compression Ratio [P3/P2]'); % ylabel('Thrust Specific Fuel Consumption [kg/(kN-s)]'); % title('TSFC as a Function of Compression Ratio [Mach 0.85]');