PV diagram for the Otto Cycle &The thermal efficiency of the engine using MATLAB

process 1-2 → Isentropic compression.

process 2-3 → Constant volume heat addition.

process 3-4 → Isentropic expansion and

process 4-1 → Constant volume rejection.

figure: PV diagram for the Otto Cycle

Given Data:

function [V] = engine_kinematics(bore, stroke, con_rod, cr, start_crank, end_crank)

a = stroke/2
R = con_rod/a

v_swept = (pi/4)*bore^2*stroke;
v_clearance = v_swept/(cr-1);

theta = linspace(start_crank,end_crank,100);

term1 = 0.5*(cr-1);
term2 = R + 1 - cosd(theta);
term3 = (R^2 - sind(theta).^2).^0.5;

V = (1+term1*(term2 - term3)).*v_clearance

 

end
​

PV Diagram for Otto Cycle

clear all
close all
clc

% input
gamma = 1.4
t3 = 2300

% state variable
p1 = 101325
t1 = 500
%Engine geometric parameters
bore = 0.1;
stroke = 0.1;
con_rod = 0.15;
cr = 8;

% calcuclate the swept volume and the clearance volume
v_swept = (pi/4)*bore^2*stroke;
v_clearance = v_swept/(cr-1);

v1 = v_swept + v_clearance
v2 = v_clearance

% state variable at state point 2
%p2v2^gamma = p1v1^gamma
%p2 = p1*(v1/v2)^gamma = p1*cr^gamma

p2 = p1*cr^gamma;

%p1v1/t1 = p2v2/t2 | t2 = p2*v2*t1/(p1*v1)

t2 = p2*v2*t1/(p1*v1);

constant_c = p1*v1^gamma;

V_compression = engine_kinematics(bore, stroke, con_rod, cr, 180, 0);

P_compression = constant_c./V_compression.^gamma;


% state variable at state point 3
v3 = v2


%p3v3/t3 = p2v2/t2 | p3 = p2*t3/t2

p3 = p2*t3/t2

constant_c = p3*v3^gamma;

V_expansion= engine_kinematics(bore, stroke, con_rod, cr, 0, 180);

P_expansion = constant_c./V_expansion.^gamma;

% state variable at state point 4
v4 = v1

%p4v4^gamma = p3v3^gamma | p4 = p3(v3/v4)^gamma

p4 = p3*(v3/v4)^gamma



figure(1)
hold on 
 
 title('PV Diagram for Otto Cycle'),xlabel('Volume(V)'),ylabel('Pressure(P)')

 plot(V_compression, P_compression,'linewidth',1)

plot([v2,v3],[p2,p3],'linewidth',1)
 
plot(V_expansion, P_expansion,'linewidth',1)
 
plot([v4,v1],[p4,p1],'linewidth',1)

text(v1,p1,'1')
text(v2,p2,'2')
text(v3,p3,'3')
text(v4,p4+0.5*p4,'4')

legend('Isentropic compression','CV Heat addition','Isentropic expansion','CV Heat rejection')
grid on
 
%efficiency

efficieency = (1-(1/cr^(gamma-1)))*100

output

Thermal efficiency

efficiency = (1-(1/cr^(gamma-1)))*100 = 56.4725%

Error facing during executions:

1 unrecognized function c_r because for compression ratio all symbols are cr but only on c_r

2 because I do not define the value of gamma