A Thermodynamic Analysis of a Turbojet Engine ME 2334 Course Project By Jeffrey Kornuta
Kornuta 2 Introduction This paper looks into the thermodynamic analysis of an ideal turbojet engine, focusing on the relationships between the compression ratio (R C ), max temperature (T max ), 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!v 1"6, thus forming a basic jet propulsion cycle (Figure A).
Kornuta 3 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 R C for T max 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 R C increases. At the same Mach number, TSFC decreases sharply then proceeds to decrease gradually as R C increases. These results make sense; as R C 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 " T 0 s + 1 2 V 2 + gz] " [h 0 " T 0 s 0 + 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:! = C P (T T 0 )!# "# $ + T (s " s) + 1 V 2 0 0 2, (3)! h where C p 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 T max increases, the conditions remain the same with an exception for a higher temperature gas exiting the combustor. Thus, the same R C will yield a higher MST.
Kornuta 4 Likewise, TSFC decreases continually as R C 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!m f! = f f =, (4)!!m a (1 + f )V 6 " V 1 one sees that a decrease in ƒ will ultimately decrease the TSFC. Similarly, realizing that the fuel to air ratio is defined as f = T 4 T 3! 1 q f (C p T 3 )! T 4 T 3, (5) where q f is the specific heat addition, a higher value for the max temperature T 4 will yield a higher ƒ, resulting in greater TSFC. Clearly, the optimum R C for mass specific thrust is given by a pressure ratio that produces a maximum MST value. On the other hand, the optimum R C 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 deps 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 R C, 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 R C
Kornuta 5 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 dep on the application, MST and TSFC have clear maximum and minimum values, thus the mean R C 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 R C for the lower Mach number will usually be lower than the optimum R C 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 R C for optimum operation.
Kornuta 6 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.
Kornuta 7 Figure B Figure C
Kornuta 8 Figure D Figure E
Kornuta 9 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));
Kornuta 10 % 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; 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; else MST3(i) = (1 + f)*v6 - V1; TSFC3(i) = f/mst3(i); if MST3(i) > max3 max3 = MST3(i); Rc_max3 = Rc; 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]');