AIR STANDARD CYCLE
AIM
To calculate the thermal efficiency of an OTTO CYCLE and to plot the PV diagram using MATLAB software.
GOVERNING EQUATIONS
`eta`OTTO = 1 - `1⁄〖CR〗^((γ¬1))`
Where,
`eta` Efficiency of otto cycl`
CR Compression ratio
`γ` Adiabatic index`
Otto cycle is a thermodynamic cycle which describes the functioning of spark ignition engine. Otto cycle consists of the following process
Process 0-1 Air is drawn into the cylinder at constant pressure
Process 1-2 Isentropic compression where the piston moves from BDC to TDC
Process 2-3 Constant volume heat addition
Process 3-4 Isentropic expansion also known as power stroke
Process 4-1 Constant volume heat rejection.
Process 1-0 The exhaust gas is released into the atmosphere.
The isentropic process follows the following relation
`PV^γ`= C
where P and V are Pressure and Volume at their respective points.
Let `P_1``V_1 and T_1`be the pressure, volume and temperature at point 1.
`V_1=V_s+V_c`
where `V_s and V_c` are swept volume and clearance volume.
`V_s= π/4 d^2 l` `
`V_c = V_s/(cr-1)`
To calculate the state variables at point 2
`P_(2 )= P_(1 ) CR^γ` Pressure at point 2
`T_(2 = ) (T_2 V_2 P_2)/(P_1 V_1 )` Temperature at point 2
To calculate the state variables at point3
`P_( 3 = ) (P_2 T_3)/T_2`
`V_2=V_3`
To calculate the state variables at point 4
`V_4=V_1`
`P_(4 )= P_(3 ) (V_3/V_4 )^γ`
The following formula is used to determine the change in volume inside the cylinder by taking engine kinematics into consideration
`V/V_c = 1+ 1/(2 ) (CR-1)[R+1-costheta-(R^2 -sin^2 theta )^(1/2) ]`
The following code is used in MATLAB to perform the necessary operation
% To write a program to calculate the efficiency of OTTO cycle
clear all
close all
clc
% state variables
p1 = 101325 % Pressure at point1
t1 = 500 % Temperature at point1
gamma = 1.4 % value for gamma
t3 = 2300 % Temperature at point3
start_crank_angle_compression = 180;
end_crank_angle_compression = 0;
start_crank_angle_expansion = 0;
end_crank_angle_expansion = 180;
theta1 = linspace(start_crank_angle_compression,end_crank_angle_compression,90)
theta2 = linspace(start_crank_angle_expansion,end_crank_angle_expansion,90)
% Engine parameters
b = 0.1 % Assigning value of bore in m
l = 0.1 % Assigning value for stroke in m
cond_rod = 0.15 % Length of connencting rod
cr = 12 % compression ratio
a = l/2 % crank pin radius
R = cond_rod/a
v_s = (pi/4)*b^2*l; % formula to calculate swept volume
v_c = v_s/(cr-1); % formula to calculate clearance volume
% To calculate the change in volume inside the cylinder
term1 = 0.5*(cr-1)
term2 = R+1-cosd(theta1);
term3 = (R^2-(sind(theta1).^2)).^0.5;
v_compression = (1+term1*(term2-term3))*v_c;
term4 = R+1-cosd(theta2);
term5 = (R^2-(sind(theta2).^2)).^0.5;
v_expansion = (1+term1*(term4-term5)).*v_c;
% State variables at point 1
v1 = v_s+v_c; % Volume at point1
v2 = v_c;
v3 = v_c;
c1 = p1*(v1)^gamma;
p_compression = c1./(v_compression.^gamma)
% State variables at point 2
p2 = p1*(cr)^gamma % To calculate pressure at point 2
t2 = (t1*v2*p2)/(p1*v1) % To calculate temperature at point 2
% State variables at point 3
p3 = p2*(t3/t2)
c2 = p3*(v3)^gamma;
p_expansion = c2./(v_expansion.^gamma);
% State variable at point 4
v4 = v1
term_1_1 = ((v3/v4))^gamma
p4 = p3*term_1_1;
% PV Diagram
hold on
xlabel('Volume')
ylabel('Pressure')
title('PV Digaram')
plot(v1,p1,'*','color','r')
plot(v_compression,p_compression,'color','b')
plot([v2 v3],[p2 p3],'color','g')
plot(v2,p2,'*','color','r')
plot(v3,p3,'*','color','r')
plot(v_expansion,p_expansion,'color','b')
plot(v4,p4,'*','color','r')
plot([v4 v1],[p4 p1],'color','g')
% Efficency of OTTO Cycle
term6 = cr^(gamma-1)
efficiency_of_otto_cycle = 1-(1/(term6))RESULT
The thermal efficiency of Otto cycle is determined and P-V diagram is plotted.