Project

MATLAB CODE TO SOLVE OTTO CYCLE AND MAKE PLOTS FOR IT

RRushikesh Patil1 Nov 2021

CODE TO SOLVE OTTO CYCLE AND MAKE PLOTS FOR IT

Problem taken: In an I C Engine working with the Otto cycle, the cylinder diameter is 250 mm and a stroke is 375 mm. If the clearance volume is 0.00263 𝑚^3, and the initial pressure and temperature are 1 bar and 50℃, calculate the thermal efficiency of engine and plot the PV graph. The maximum cycle pressure is limited to 25 bar and the length of connecting rod is 450mm. (Take gamma=1.4)

Reference: From GTU Winter 2014 Engineering Thermodynamics question paper.

CODE:

% PROBLEM:
% In an I C Engine working with the Otto cycle, the cylinder diameter is 250 mm and a stroke is 375 mm.
% If the clearance volume is 0.00263 𝑚^3, and the initial pressure and temperature are 1 bar and 50℃, 
% calculate the thermal efficiency of engine and plot the PV graph. 
% The maximum cycle pressure is limited to 25 bar and the length of
% connecting rod is 450mm. (Take gamma=1.4)


% Code to solve otto cycle, create plots and to calculate the thermal efficiency of the engine 
close all
clear all
clc

% Given Data:
bore= 0.25; % in meters.
stroke= 0.375; % in meters.
v_clearance= 0.0026; % in cubic meters.
p1=100; % K*N/m^2.
t1=50+273; % Kelvin.
p3=2500; % K*N/m^2.
con_rod=0.45; % in meters.
gamma=1.4;
% Calculating Swept volume.
disp('Swept volume in cubic meters')
v_swept=(pi/4)*bore^2*stroke % in cubic meters

% Calculating Compression Ratio.
disp('Compression Ratio')
c_r=(v_swept+v_clearance)/v_clearance

% State variables at point-1
disp('State variables at point-1')
p1=100   % K*N/m^2.
v1=v_swept+v_clearance    % in cubic meters.
t1=50+273   % Kelvin.

% process 1-2 is Isentropic Compression
constant_c1= p1*v1^gamma;
v_compression=engine_kinematics(stroke,con_rod,c_r,v_clearance,180,0);
p_compression = constant_c1./v_compression.^gamma;

%State variables at point-2
disp('State variables at point-2')
v2=v_clearance;     % in cubic meters.

% since PV^gamma=Constant.
p2=p1*((v1/v2)^gamma)   % in K*N/m^2.
v2 
% Ideal Gas law---> PV/T = Constant
t2=(p2*v2*t1)/(p1*v1)   % Kelvin.

%State variables at point-3
disp('State variables at point-3')
p3   % in K*N/m^2. (In Given Data)
v3=v2;   % in cubic meters. ( Since process 2-3 is constant volume heat addition)
% Ideal Gas law---> PV/T = Constant
t3=(p3*v3*t2)/(p2*v2)  % Kelvin.

% process 3-4 is Isentropic Expansion
constant_c2= p3*v3^gamma;
v_expansion=engine_kinematics(stroke,con_rod,c_r,v_clearance,0,180);
p_expansion = constant_c2./v_expansion.^gamma;

%State variables at point-4
disp('State variables at point-4')
v4=v1;   % in cubic meters. ( Since process 4-1 is constant volume heat subtraction)

% since PV^gamma=Constant.
p4=p3*((v3/v4)^gamma)   % in K*N/m^2.
v4
% t/v^(gamma-1) is constant.
t4=((v3/v4)^(gamma-1))*t3  % Kelvin.


% Plotting P-V graph
hold on
xlabel('VOLUME');
ylabel('PRESSURE');
plot(v1,p1,'*','color','k')                      % point-1
plot(v_compression,p_compression,'Color','b')    % curve joining points 1 and 2
plot(v2,p2,'*','color','k')                      % point-2
plot(v3,p3,'*','color','k')                      % point-3
plot([v2,v3],[p2,p3],'color','r')                % line joining points 2 and 3
plot(v4,p4,'*','color','k')                      % point-4
plot(v_expansion,p_expansion,'Color','g')        % curve joining points 1 and 2
plot([v4,v1],[p4,p1],'color','r')                % line joining points 4 and 1

% Calculating Thermal Efficiency of engine.
Thermal_Efficiency=(1-(c_r^(1-gamma)))*100; % in percentage.
fprintf('Thermal_Efficiency of the engine is %d percentage.n',Thermal_Efficiency)

FUNCTION: (File name=engine_kinematics.m) (The function file and the main code file should be in the same directory)

function V=engine_kinematics(stroke,con_rod,c_r,v_clearance,start_crank,end_crank)

R=con_rod/(stroke/2);
theta=linspace(start_crank,end_crank,100);

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

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

CODE EXPLAINATION:

Line-1 to 6: ---> The problem which is solved under comment section.
Line-10: close all ----> closes all the Plots.
Line-11: clear all ----> clears all the variables in the workspace.
Line-12: clc ----> clears the command window.
Line-14 to 22: ---> The date given in problem.
Line-23 to 25: ---> Calulation of swept volume using formula : v_swept=(pi/4)*bore^2*stroke in cubic meters.
Line-27 to 29: ---> Calculating Compression Ratio using formula : Compression Ratio=(v_swept+v_clearance)/v_clearance.
Line-31 to 35: ---> State variables at point-1 are calculated. Since the value of v1=Swept volume+Clearance volume in cubic meters.
Line-37 to 40: ---> Calculation of compression volume and compression pressure using P*V^Gamma=CONSTANT . In line-39, use of function is made.

FUNCTION CODE:

Line-1: ---> A function of name engine_kinematics is made using function command with output variable V and input variables (stroke,con_rod,c_r,v_clearance,start_crank,end_crank).
Line-2: ---> Calculation of R using formula R=con_rod/(stroke/2).
Line-4: ---> Values of crank angle is stored in an array with name theta using linspace command.
Line-6 to 8: ---> Splitting the formula V=(1+0.5*(c_r-1)*(R+1-cosd(theta)-(R^2-sind(theta).^2).^0.5)).*v_clearance into three terms. The usuage of Element wise operator is made in the formula because the values of theta is an array.
Line-10: ---> The value of V is calculated using the formula and the value is returned to the output variable V in the function command.

Line-42 to 50: ---> State variables at point-2 are calculated using relations P*V^Gamma=CONSTANT and Ideal Gas law PV/T = Constant, since the value of v2=Clearance volume process1-2 is Iscentropic compression process.
Line-52 to 57: ---> State variables at point-3 are calculated using relations P*V^Gamma=CONSTANT and Ideal Gas law PV/T = Constant, since the value of v3=v2 because the process 2-3 is Constant volume heat addition.
Line-59 to 62: ---> Calculation of expansion volume and expansion pressure using P*V^Gamma=CONSTANT .
Line-64 to 71: ---> State variables at point-4 are calculated using relations P*V^Gamma=CONSTANT and t/v^(gamma-1)= constant.
Line-76: ---> hold on command is used to add a second line plot without deleting the existing line plot.
Line-77: ---> Labels X-axis of plot naming as VOLUME.
Line-78: ---> Labels Y-axis of plot naming as PRESSURE.
Line-79,81,82,84: ---> Plots points (v1,p1), (v2,p2), (v3,p3), (v4,p4) with marker * of colour black.
Line-80: ---> Creates a curve of colour blue joining points 1 and 2 using compression volume and compression pressure values of colour blue.
Line-83: ---> Creates a line of colour red joining points 2 and 3 using points (v2,p2), (v3,p3).
Line-85: ---> Creates a curve of colour green joining points 3 and 4 using expansion volume and expansion pressure values.
Line-85: ---> Creates a line of colour red joining points 4 and 1 using points (v3,p3), (v4,p4).
Line-89: ---> Formula for calculation of Thermal efficiency of engine. Thermal_Efficiency=(1-(c_r^(1-gamma)))*100 %. and storing the thermal efficiency value in the variable Thermal_Efficiency.
Line-90: ---> fprintf is used to show the string 'Thermal_Efficiency of the engine is %d percentage.' along with the value of thermal efficiency in the output.