OTTO CYCLE (P-V diagram and calculation of thermal efficiency )

clear all
close all
clc

%input
gamma=1.4
t3=1926


% state variables at state point 1
p1=100000
t1=293

% engine geometric parameter

bore=.1;
stroke=.1;
cr=9;
co_rod=.15;

%calculating the swept volume and clearance volume
v_swept=pi/4*bore^2*stroke;
v_clearance=v_swept/(cr-1);

v1=v_swept+ v_clearance;
v2=v_clearance;

% state variable at state point 2
%p2v2^gamma= p1v1^gamma
%p2=p1*(v1/v2)^gamma =p1*cr^gamma

p2=p1*(cr)^gamma;

%p1v1/t1=p2v2/t2  t2=p2*v2*t1/(p1v1)
t2=p2*v2*t1/(p1*v1)

constant_c=p1*v1^gamma;
V_compression=engine_kinematics(bore,stroke,cr,co_rod,180,0);
P_compression=constant_c./V_compression.^gamma;


% state variable at state point 3
v3=v2

%p3v3/t3=p2v2/t2 p3=p2*t3/t2
p3=p2*t3/t2

constant_c=p3*v3^gamma;
V_expansion=engine_kinematics(bore,stroke,cr,co_rod,0,180);
P_expansion=constant_c./V_expansion.^gamma;


% state variable at state point 4
v4=v1

%p3v3^gamma= p4v4^gamma  p4=p3*(v3/v4)^gamma 
p4=p3*(v3/v4)^gamma

hold on 
plot (v1,p1,'*','Color','b')
plot (v2,p2,'*','Color','b')
plot (v3,p3,'*','Color','b')
plot (v4,p4,'*','Color','b')
plot(V_compression,P_compression,'Color','r')
plot ([v2,v3],[p2,p3],'Color','g')
plot(V_expansion,P_expansion,'Color','r')
plot ([v4,v1],[p4,p1],'Color','g')

% Thermal efficiency of ottocycle
efficieny_ottoengine=(1-1/(cr^(gamma-1)))*100

To write the program to create p-v diagram and thermal efficiency of otto cycle:

STEP-I Clear all (to clear all the stored data to a variable, close all (to close all Matlab fig. window),clc (to clear the command window) commands are used.

A suitable program name is given

STEP-II Inputs are given

gamma=1.4(adiabatic index)

t3=1926(in k temperature at point 3)

state variables are inputted

p1=100000 (in N/m2)

t1=293 (in k temperature at point 1)

engine geometric parameters are inputted

bore=.1

stroke=.1

cr=9; (compression ratio)

co_rod=.15 (length of connecting rod)

STEP-III To calculate the swept volume and clearance volume following formulas are inputted

v_swept=pi/4*bore^2*stroke (swept volume)

v_clearance=v_swept/(cr-1) (clearance volume)

v1=v_swept+ v_clearance (volume at state 1)

v2=v_clearance (volume at state 2)

STEP-IV state variables at state point 2 are inputted

p2v2^gamma= p1v1^gamma

p2=p1*(v1/v2)^gamma =p1*cr^gamma

p2=p1*(cr)^gamma;

p1v1/t1=p2v2/t2 t2=p2*v2*t1/(p1v1)

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

constant_c=p1*v1^gamma

STEP-V calling the function engine _kinematics for compression

function [v]= engine_kinematics(bore,stroke,cr,co_rod,start_crank,end_crank)

a=stroke/2;
R=co_rod/a;

v_swept=(pi/4)*bore^2*stroke
v_clearance=v_swept/(cr-1)

theta =linspace(start_crank,end_crank,100)

term1= .5*(cr-1);
term2=R+1-cosd(theta);
term3=(R^2 -sind(theta).^2).^0.5;

v=(1+term1*(term2-term3)).*v_clearance

end

where engine_kinematics is a function

a=stroke/2 (crankpin radius)

R=co_rod/a (ratio of stroke to crank pin radius)

start_crank =start_crank angle

end_crank= end_crank angle

to create value of theta linspace command is used(upto 100 )

to find the output v:

v=(1+term1*(term2-term3)).*v_clearance

where,

term1= .5*(cr-1)

term2=R+1-cosd(theta)

term3=(R^2 -sind(theta).^2).^0.5

d:degree

*:multiplication

V/Vc=1 + ½(rc-1)[R+1-cosθ (R2 -sin2 θ)1/2]

V_compression=engine_kinematics(bore,stroke,cr,co_rod,180,0);

start_ crank 180 degrees and end _crank angle 0 degrees are inputted.

P_compression=constant_c./V_compression.^gamma

STEP -VI state variable at state point 3 are inputted

v3=v2

p3v3/t3=p2v2/t2 p3=p2*t3/t2

p3=p2*t3/t2

constant_c=p3*v3^gamma

calling the function engine _kinematics for expansion

V_expansion=engine_kinematics(bore,stroke,cr,co_rod,0,180);

P_expansion=constant_c./V_expansion.^gamma;

start_ crank 0 degrees and end _crank angle 180 degrees are inputted.

STEP -VII state variables at state point 4 are inputted

v4=v1

p3v3^gamma= p4v4^gamma p4=p3*(v3/v4)^gamma

p4=p3*(v3/v4)^gamma

STEP-VIII To plot the p-v diagram

hold on command is inputted

Plot command is inputted for to find the state points and to plot the diagram

for :p1,v1;p2,v2;p3,v3;p4,v4

plot (v1,p1,'*','Color','b')

plot (v2,p2,'*','Color','b')

plot (v3,p3,'*','Color','b')

plot (v4,p4,'*','Color','b')

plot(V_compression,P_compression,'Color','r')

plot ([v2,v3],[p2,p3],'Color','g')

plot(V_expansion,P_expansion,'Color','r')

plot ([v4,v1],[p4,p1],'Color','g')

STEP-IX To find the thermal efficiency of the engine:

The following formula is inputted

efficieny_ottoengine=(1-1/(cr^(gamma-1)))*100

efficieny_ottoengine=58.4756%

[xlabel,ylabel and arrow in the plot are inserted through insert option in figure]

ERROR 1:while putting relation between p4,p3,v4 and v3

Array indices must be positive integers or logical values.

Error in c5 (line 57)
p4=p3(v3/v4)^gamma

Error1 resolved : by putting * between p3 and (v3/v4)

ERROR 2: thermal efficiency calculation

Error using /
Arguments must be numeric, char, or logical.

Error in c5 (line 70)
efficieny_ottoengine=(1-1/{cr^(gamma-1)})*100

Error 2 resolved : by making the follwing correction