OTTO CYCLE PV DIAGRAM PLOT

OTTO CYCLE PRESSRE-VOLUME DIAGRAM PLOT

INTRODUCTION

Otto cycle is the ideal cycle for spark-ignition reciprocating engines. Named after Nikolaus A. Otto, who built a successful four-stroke engine in 1876 in Germany. Can be executed in two or four strokes. Four stroke: Intake, compression, power and exhaust stroke.

OTTO CYCLE

Otto cycle consists of four processes – Isentropic compression (1-2), Isochoric (constant volume) heat addition (2-3), Isentropic expansion (3-4), Isochoric (constant volume) heat rejection (4-1). All the processes are internally reversible.

During intake stroke - crank rotates from 0 to 180 degree.

During compression stroke - crank rotates from 180 to 360 degree.

During expansion stroke - crank rotates from 360 to 540 degree.

During exhaust stroke - crank rotates from 540 to 720 degree.

On the whole SI engines completes a cycle in 720 degrees.

FORMULA FOR THERMAL EFFICIENCY

CODE FOR P-V PLOT OF OTTO CYCLE

clear all
close all
clc

%Initial parameters @ stage 1
p1=101325;
t1=500;
g=1.4;

%Engine parameters
bore=0.1;
stroke_length=0.1;
c_rod=0.15;
cr=12;

%Volumetric parameters
swept_vol=(pi/4)*(bore)^2*stroke_length; 
clearance_vol=swept_vol/(cr-1);
v1=swept_vol+clearance_vol;
v2=clearance_vol;

%Pressure @ stage 2 (p2)     p1v1^g=p2v2^g    p2=p1*(v1/v2)^g
p2=p1*cr^g;

%Temperature @ stage 2 (t2)     p1v1/t1=p2v2/t2   t2=p2v2t1/p1v1
t2=p2*v2*t1/(p1*v1);

%Volume(v3), Temperature(t3), Pressure(p3) @ stage 3
v3=v2;
t3=2300;
p3=p2/t2*t3;

%Volume(v4),Pressure(p4) @ stage 4
v4=v1;
p4=p3*(v3/v4)^g;

v_cr=otto_cycle_kinematics(bore,stroke_length,c_rod,cr,180,360);
c1=p1*v1^g;
p_cr=c1./v_cr.^g;

v_exp=otto_cycle_kinematics(bore,stroke_length,c_rod,cr,360,540);
c3=p3*v3^g;
p_exp=c3./v_exp.^g;

%calculation of thermal efficiency of otto cycle
thermal_efficiency=1-(1/(cr^(g-1)))


hold on

plot(v_cr, p_cr, 'color','green');
plot([v2 v3],[p2 p3], 'color','red');
plot(v_exp,p_exp, 'color','yellow');
plot([v4 v1],[p4 p1],'color','blue');
xlabel('Volume')
ylabel('Pressure')
legend('Isentropic compression', 'Isochoric heat addition', 'Isentropic expansion','Isochoric heat rejection')
title('Otto cycle PV diagram')














function [V]= otto_cycle_kinematics(bore,stroke_length,c_rod,cr,st_ang,end_ang)


crank_pin_radius=stroke_length/2;
R=c_rod/crank_pin_radius;


swept_vol=(pi/4)*(bore)^2*stroke_length; 
clearance_vol=swept_vol/(cr-1);

theta=linspace(st_ang,end_ang,100);
T1=0.5*(cr-1);
T2=R+1-cosd(theta);
T3=(R^2-(sind(theta)).^2).^0.5;

V=(1+T1*(T2-T3)).*clearance_vol;


end

CODING PROCEDURE

1. Engine parameters

Engine parameters are assumed from real world existence. Bore refers to the diameter of the piston or the inner diameter of the cylinder. Stroke length is the distance from Top Dead Centre (TDC) to Bottom Dead Centre (BDC). Compression ratio is the ratio of volume inside the cylinder before and after compression. c_r refers to connecting rod length.

2. Volumes

Swept volume refers to product of area of the piston and stroke length. In other words, it refers to the volume swept by the piston from BDC to TDC. Clearance volume is the volume in the cylinder when the piston is at TDC.

3. Parameters at each stage

Initial parameters are also assumed.

Parameters at stage 2 is found from ideal gas equation and isentropic process relation between 1 and 2.

Volume at stage 3 is equal to that of state 2. Using ideal gas equation and assuming isochoric process, pressure at stage 3 is found. Temperature is stage 3 is assumed.

From PV plot of Otto cycle, we can see that volumes at stage 1 and 4 are equal. Pressure at stage 4 is found from ideal gas equation and assuming constant volume.

Volume inside the cylinder varies sinusoidally as per the equation

Where R is the ratio of connecting rod length to crank pin radius.

Crank pin radius is half the stroke length

Since processes 1-2 and 3-4 are isentropic and follow a curved path, we make use of this relation to plot the curves. So we are calling the function otto_cycle_kinematics for plotting compression and expansion curves.

Since compression process occurs between 180 to 360 degrees and expansion process occurs between 360 to 540 degrees we give these angles as inputs. Thus volume is computed. Pressure at these point are found by ideal gas law.

Now everything is plotted using plot command.

Thermal efficiency is also found.

GRAPH

MISTAKES MADE AND SOLUTION

1. Missed to give compression ratio (cr) as an input argument when calling the function

Mistake rectified.

RESULT

The thermal efficiency of Otto cycle has been found to be 63%.