Creating an Otto Cycle P-V Diagram and Calculating its Efficiency using MATLAB.

Writing down the required Function Code:

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

% Crank Pin Radius
a = stroke/2;
% Ratio of Connecting Rod and Crank Pin Radius.
R = con_rod/a;
% Swept Volume Calculation
v_s = (pi/4)*bore^2*stroke;
% Clearance Volume Calculation
v_c = v_s/(cr-1);
% Creating an array for diffrent theta values
theta = linspace(start_crank,end_crank,100)
% Combustion Volume Formula Calculation
term1 = 0.5*(cr-1);
term2 = R +1 - cosd(theta);
term3 = (R^2 - sind(theta).^2).^0.5;

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

end

Code Explanation:

• So in the above code, I have created a function named "challenge_engine_kinematics" which has the ability to take input values and assign value to the output array.
• Then in inputs, I wrote down the formula for Crankpin radius and Ratio of Connecting Rod and Crankpin Radius.
• Then in the next steps, I wrote the formula for the calculation of the Swept Volume and Clearance Volume.
• For the different values of theta, I created an array using linspace command.
• In the final step, I wrote down the formula for the calculation of the combustion volume, while writing this down I split the main formula into 3 small parts and named each part as term1, term2, term3 and finally combined all the terms into a single main formula.

Writing down the required code for PV Diagram and Thermal Efficiency:

clear all
close all 
clc

% Inputs
gama = 1.4;
t3 = 2300;

% State Variable values for Stage 1
p1 = 101325;
t1 = 500;

% Engine Geometric Parameters     
bore = 0.1;
stroke = 0.1;
con_rod = 0.15;
cr = 12;

% Formula for calculation of Swept Volume and Clearnace Volume
v_swept = (pi/4)*bore^2*stroke;
v_clearance = v_swept/(cr-1);

% Volume at satge 1 and 2
v1 = v_swept + v_clearance;
v2 = v_clearance;

% State Variable values for Stage 2
% p2v2^gama = p1v1^gama | p2 = p1(v1/v2)^gama = p1*cr^gama
p2 = p1*cr^gama;

% Temperature at point 2 
% p1v1/t1 = p2v2/t2 | t2 = p2*v2*t1(p1*v1)

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

% For Constant C
constant_c = p1*v1^gama;

% For compresion Stroke Volume and Pressure Calculation
V_compression = challenge_engine_kinematics(bore, stroke, con_rod, cr,180,0);
P_compression = constant_c./V_compression.^gama;

% State Variable values for Stage 3
v3 = v2;

% Pressure at point 3
% p3v3/t3 = p2v2/t2 | p3 = p2*t3/t2
p3 = p2*t3/t2;

% For Constant C
constant_c = p3*v3^gama;

% For compresion Stroke Volume and Pressure Calculation
V_expansion = challenge_engine_kinematics(bore, stroke, con_rod, cr,0,180);
P_expansion = constant_c./V_expansion.^gama;

% State Variable values for Stage 4
v4 = v1;

% p3v3^gama = p4v4^gama | p4 = p3(v3/v4)^gama
p4 = p3*(v3/v4)^gama;

% Formula for Thermal Efficiency
thermal_efficiency = (1-(1/cr^(gama - 1)))*100;

% Plotting of P-V Diagram
hold on
plot(V_compression, P_compression)
plot([v2,v3],[p2,p3])
plot(V_expansion, P_expansion)
plot([v4,v1],[p4,p1])
title ('Otto Cycle P-V Diagram')
xlabel('Volume')
ylabel('Pressue')
legend('Isentropic Compression','Constant Volume Heat Addition','Isentropic Expansion','Constant Volume Heat Rejection')

Code Explanation:

• Started the code with clear all, close all, and clc.
• Then I wrote the input values of "gama" and "t3".
• Now I wrote down values for State Variables at Stage 1 "p1" and "t1".
• After this, I included the Engine Geometric Parameters, which included the values of the "bore", "stroke", "connecting rod" and "compression ratio".
• After this, I wrote the formula for the calculation of the "Swept Volume" and "Clearance Volume".
• Then I wrote down the formula for volume at stages 1 and 2; "v1" and "v2".
• Now again values for State Variables but at Stage 2 wrote the formula for "p2".
• Then the formula for the temperature at stage 2 "t2".
• Now formula for the "constant_c", now as we got the formula for the "constant_c" we can now easily calculate the "Compression Stroke Volume" and "Compression Stroke Pressure" so wrote down those formulas also.
• Now again values for State Variables but at Stage 3 now "v3".
• The formula for "Pressure" at point 3 and "Constant_c". As we got the "Constant_C" we can now easily calculate the "Expansion Stroke Volume" and "Expansion Stroke Pressure" so wrote down those formulas also.
• For the final State Variable stage, 4 wrote formulas of "v4" and "p4".
• Now for the "Calculation of the Thermal Efficiency" wrote the formula for the same.
• Towards the final part i.e. the plotting of the P-V Diagram started with the "hold on" command followed by the plotting of the points (V_compression, V_expansion,v1,v2,v3,v4 and P_compression, P_expansion,p1,p2,p3,p4) also give the title and x-axis and y-axis names. I even included the legend command for a little more detail adding purpose.

Results of the Code:

Output for Thermal Efficiency of Engine:

The plot of the P-V Diagram:

The Errors I faced while Programming:

I faced the above two errors actually, it was a small typo on my function file "challenge_engine_kinematics" I made a typo in writing the formula for Ratio of Connecting Rod and Crank Pin Radius the formula I wrote was "R = con_rod/2;" which was wrong as the correct formula was "R = con_rod/a;"