ENGINE KINEMATICS OF AN OTTO CYCLE

ABSTRACT:

The Otto cycle was given by Dr. Nikolaus August Otto. It is a gas power cycle that is used in petrol engine for its working. The entire modern petrol engine works on Otto cycle. It consist of four processes, Two isentropic (reversible adiabatic) processes and two isochoric (constant volume) processes. It has low compression ratio ranges from 7:1 to 10:1. Here we will try to understand this cycle with help of its P-V diagram.

PROCESS: 

The four processes of this cycle is as follows:

Isentropic ( reversible adiabatic) compression.

Constant volume (Isochoric) heat addition.

Isentropic (reversible adiabatic) Expansion.

Constant volume heat rejection.

Criteria will be provided while computing such problems, in our case V1,T1 are given. with this we can derive P1. With Cr(compression ratio) given we can calcultate V2, once we have it we can calculate P2, T2 with V2 and P2.

T3(temp. after combustion) is given usually with the problem. So, either P3 is provided or T3 is provided. V3 = V2 becasue of constant volume heat additon process. Once these two parameters are known P3 can be calculated.

Once having all the inputs, we can compute all the points in a PV diagram.

To construct the points we need to know how to the volume changes from state point 1 to state point 2. This depends on engine kinematics.

So, we start with engine geometric parameters rahter than starting with the inputs. Provided this we can calculate the kinematics of the piston and volume trace, swept volume and clearance volume.

Piston Kinematics explain how the piston moves from BDC to TDC, also the volume change inside the combustion chamber as the function of the crank angle. 

 THE CODE:

% simulation of air standard/otto cycle
clear all
close all
clc

% Inputs at state-1
p1 = 101325;
t1 = 500;
gamma = 1.4;
t3 = 2300;

% Engine geometric parameters
bore = 0.1;
stroke = 0.1;
con_rod = 0.15;
Cr = 12;

% Cr is the ratio of total cylinder volume and clearance_v
% Calculating the swept volume
% swept volume is the difference between total cylinder volume and the clearance volume 
swept_v = (pi/4)*bore^2*stroke;

% Calculating the clearance volume
% clearance_v is the minimum volume inside the cylinder when the piston is approaching towards the TDC after compression stroke
clearance_v = swept_v/(Cr-1);

% Calculating v1
v1 = swept_v + clearance_v;

% Calculating v2
v2 = clearance_v;

% Inputs at state-2

% p2(v2)^gamma = p1(v1)^gamma;
% p2 = p1(v1/v2)^gamma;
% p2 = p1*(Cr)^gamma;
% p2 is the pressure at state point-2
p2 = p1*(Cr)^gamma;

% applying IDEAL GAS LAW
% (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;

% Inputs at state-3
v3 = v2;
% to  facilitate the process let us assume t3 = 2300K;
% the remaining parameter to be defined at state point-3 is pressure and can be determined by ideal gas law;
% 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;

% Inputs at state-4
% volume at state-4 is going to be equal to volume at state-1
v4 = v1;
% computing the pressure at state-4;
% p4v4^gamma = p3v3^gamma;
% p4 = p3(v3/v4)^gamma;
p4 = p3*(v3/v4)^gamma;
t4 = t1;

% Computing otto cycle effeciency
% ration of work done during the cycle to the heat supplied during the otto cycle
% Thermal Effeciency N = 1-((t4-t1)/(t3-t2))
N = 1-((t4-t1)/(t3-t2));


% plotting the otto cycle
figure(1)
hold on
plot(v1,p1,'*','color','r')
plot(V_compression,P_compression,'linewidth',3,'color','yellow')
plot(v2,p2,'*','color','r')
plot([v2,v3],[p2,p3],'linewidth',3,'color','blue')
plot(v3,p3,'*','color','r')
plot(V_expansion,P_expansion,'linewidth',3,'color','green')
plot(v4,p4,'*','color','r')
plot([v4,v1],[p4,p1],'linewidth',3,'color','blue')
legend('v1,p1','compression','v2,p2','heat addition','v3,p3','expansion','v4,p4','heat rejection')
hold off
xlabel('volume')
ylabel('pressure')


function [Volume_trace] = engine_kinematics(bore,stroke,con_rod,Cr,start_crank_angle,end_crank_angle)
 bore = 0.1;
 stroke = 0.1;
 con_rod = 0.15;
 Cr = 12;
 a = stroke/2;
 R = con_rod/a;
 start_crank_angle = 0;
 end_crank_angle = 180;
 swept_v = (pi/4)*bore^2*stroke;
 clearance_v = swept_v/(Cr-1);
 theta = linspace(start_crank_angle,end_crank_angle,180);
 % this eqn explains how the volume inside the combustion chamber changes;
 % V/Vc = 1+1/2*(Cr-1)[R+1 cosd(theta) - (R^2-sind^2(theta))^1/2];
 % breaking up the equation in to three terms we get;
 term1 = 0.5*(Cr-1);
 term2 = R +1 -cosd(theta);
 term3 = (R^2 - sind(theta).^2).^0.5;
 Volume_trace = (1+term1*(term2 - term3)).*clearance_v;
end

OUTPUT:

THE PV DIAGRAM

THE ERRORS:

Results in the command window, workspace and errors thrown during the progress.

CONCLUSION:
The efficiency of the otto cycle seems to be 62.98%.