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The Thermodynamics of Rocket Engine

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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;
Exhaust_velocity

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;

Exhaust_velocity_02

Equations from the book

Rocket and Spacecraft Propulsion Principles Practice and New Developments by Martin J.L. Turner – 3rd Edition

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