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The Thermodynamics of Rocket Engine
Contents
Exhaust velocity
clc; clear all;hold on; grid on; Tc = 3200; % [K] Temperature of the gas in the combustion chamber M = 16e-3; % [kg/mol] Molecular weight R = 8.3144621; % Universal gas constant pc = 5E6; % [Pa] Combustion chamber pressure % Pressure ratio (combustion chamber pressure divided by pressure at exit plane) pr = 0:0.001:1; % Exhaust velocity gamma = 1.3; % cp/cv u_e = (2*gamma/(gamma - 1)*R*Tc/M*(1 - (pr).^(1 - 1/gamma))).^0.5; scatter(pr,u_e,'.b'); gamma = 1.2; % cp/cv u_e = (2*gamma/(gamma - 1)*R*Tc/M*(1 - (pr).^(1 - 1/gamma))).^0.5; scatter(pr,u_e,'.r'); gamma = 1.1; % cp/cv u_e = (2*gamma/(gamma - 1)*R*Tc/M*(1 - (pr).^(1 - 1/gamma))).^0.5; scatter(pr,u_e,'.k'); set(gca,'XDir','Reverse') xlabel('Pressure ratio p/p_c '); ylabel('Exhaust velocity [m/s]'); legend('\gamma = 1.3','\gamma = 1.2','\gamma = 1.1'); title('Gas velocity as a function of pressure ratio, Tc = 3200 K, M = 16 g/mol '); text(0.3,400,'smallsats.org','Color',[0 0 1], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); hold off;
Mass flow rate per unit cross-sectional area of the nozzle
figure(2); hold on;grid on; gamma = 1.3; % cp/cv Mfr = pc*(2*gamma/(gamma - 1)*M/(R*Tc)*pr.^(2/gamma).*... (1 - (pr).^(1 - 1/gamma))).^0.5; scatter(pr,Mfr,'.b'); gamma = 1.2; % cp/cv Mfr = pc*(2*gamma/(gamma - 1)*M/(R*Tc)*pr.^(2/gamma).*... (1 - (pr).^(1 - 1/gamma))).^0.5; scatter(pr,Mfr,'.r'); gamma = 1.1; % cp/cv Mfr = pc*(2*gamma/(gamma - 1)*M/(R*Tc)*pr.^(2/gamma).*... (1 - (pr).^(1 - 1/gamma))).^0.5; scatter(pr,Mfr,'.k'); set(gca,'XDir','Reverse') xlabel('Pressure ratio p/p_c '); ylabel('Mass flow density [kg/m^2]'); legend('\gamma = 1.3','\gamma = 1.2','\gamma = 1.1'); title('Mass flow density variation through the nozzle, Tc = 3200 K, Pc = 5 Mpa,M = 16 g/mol '); text(0.3,200,'smallsats.org','Color',[0 0 1], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); hold off;
Equations from the book
Rocket and Spacecraft Propulsion Principles Practice and New Developments by Martin J.L. Turner – 3rd Edition
Tsiolkovsky’s Rocket equation
The Tsiolkovsky rocket equation or ideal rocket equation, describes the motion of rocket vehicles. Rocket vehicle accelerates by expelling part of its mass(propellant) with high speed and move due to the conservation of momentum. The Tsiolkovsky’s equation relates the change of speed of the rocket(in absence of other external forces) with the effective exhaust velocity and the mass ratio(ratio of initial and final mass of a rocket).
clc; clear all; hold on; grid on; % M0 - initial mass of a Rocket % m - current mass of a Rocket R = 1:0.1:25; % R = M0/m; mass ratio v_exit = [1000,1500,2000,3000,4000,5000]; for i = 1:6 V(:,i) = v_exit(i)*log(R); scatter(R,V(:,i),'.'); end legend('v_e = 1 km/s','v_e = 1.5 km/s','v_e = 2 km/s','v_e = 3 km/s',... 'v_e = 4 km/s','v_e = 5 km/s'); set(gca,'XLim',[1 25]); xlabel('Mass ratio (M0/m)'); ylabel('Rocket velocity [m/s]') title('Tsiolkovskys rocket equation') text(17,1000,'smallsats.org','Color',[0 0 0], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); hold off;
Equations from the book
Rocket and Spacecraft Propulsion Principles Practice and New Developments by Martin J.L. Turner – 3rd Edition
Small Satellites 