Air Standard Cycle

AIM:

To plot the PV diagram for ‘OTTO CYCLE’ and find it’s thermal efficiency.

GOVERNING EQUATION:

1)For Swept Volume(V_s )

V_s = π/4×bore²×stroke

2)For Clearance Volume(V_c)

V_{c = `( Vs ) /(cr-1)`}

3)For Volume Trace(V)

`V = (1+1/2(cr−1)[R+1−cosθ−(R^2−sin^2θ)^0.5])Vc`

4)For thermal efficiency(η)

`η =1 - (1/(cr)^(gamma-1))`

Where,

cr = compression ratio

Crank pin radius(a) = stroke/2

R = length of connecting rod / crank pin radius(a)

OBJECTIVE:

The main objective of the program is to plot the PV diagram for the OTTO CYCLE and find out it's thermal efficiency using MATLAB.

The above figure shows the PV diagram of otto cycle.

WORKFLOW OF THE PROJECT:

1. Initially we assume the values for state variables like(pi,t1,t3).
2. Then we assign the values for the input parameters of the engine like(bore,stroke,connecting rod length,compression ratio).
3. Then we find the values of different state variables like(v1,v2,p2 etc).
4. At last we plot the graph and find thermal efficiency of the otto cycle.

ASSUMPTIONS MADE:

State variables value:

p1 = 101325 pascal

t1 = 500 Kelvin

t3 = 2000 Kelvin

Engine Parameters:

bore = 0.1m

stroke = 0.1m

connecting rod length = 0.15m

compression ratio = 10

gamma = 1.4(for air)

PROGRAMMING LANGUAGE USED: MATLAB

MAIN PROGRAM:

%A program to plot the PV diagram of otto cycle 
%And find its thermal efficiency
clear all 
close all
clc

%inputs
gamma=1.4; %The value for air

%state variables assumed
%initial condition
p1=101325; 
t1=500;
%after combustion
t3=2000; 

%engine parameters
bore = 0.1;
stroke = 0.1;
cr = 10; %compression ratio assumed
con_rod = 0.15;

%calculatin 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 vairiables at point 2
%Process 1-2 is isentropic compression 
% p1v1^gamma = p2v2^gamma
p2 = p1*(v1/v2)^gamma;

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

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

%State vairiables at point 3
%process 2-3 is constant volume heat addition
v3=v2;

%p2/t2 = p3/t3
p3 = (p2*t3)/t2;

constant_e = p3*v3^gamma;
V_expansion = engine_parameters(bore,stroke,con_rod,cr,0,180);
P_expansion = constant_e./V_expansion.^gamma;

%State vairiables at point 4
%Process 3-4 is isentropic compression 
v4 = v1;

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

%plotting of the pv diagram
hold on
plot(v1 , p1,'*','color','r');
plot(V_compression, P_compression,'color','b');
plot(v2 , p2,'*','color','r');
plot(v3 , p3,'*','color','r');
plot(V_expansion, P_expansion, 'color', 'b');
plot(v4 , p4,'*','color','r');
plot([ v2 v3 ],[ p2 p3 ], 'color', 'b' );
plot([ v4 v1 ],[ p4 p1 ], 'color',  'b');
xlabel('VOLUME');
ylabel('PRESSURE');
title('PV DIAGRAM FOR OTTO CYCLE');
grid on

%THermal efficiency
t_efficiency = (1-(1/cr^(gamma-1)))*100;
fprintf('Thermal efficiency of otto cycle is %f percent',t_efficiency)

FUNCTION PROGRAM:

function [V] = engine_parameters(bore,stroke,con_rod,cr,start_crank,end_crank)

%crank pin radius(a)
a = stroke/2;
R = con_rod/a;

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

%Crank angle over which graph is plotted
theta = linspace(start_crank,end_crank,100);
term1 = 0.5*(cr-1);
term2 = R+ 1-cosd(theta);
term3 = (R^2-sind(theta).^2).^0.5;

%calculating the volume trace
V = (1 + term1*(term2-term3)).*v_c;
end

CODE EXPLAINATION:

1. Initially the value of state variables and engine parameters are defined.
2. After that other state variables which are unknown are found by using the respective formula.
3. Then we write the function program and define all the variables.One thing to be remember here is that the file name and function name should be same.
4. Then we call the function program in the main program for the process 1-2 and 3-4.
5. At last we plot the PV diagram and find the thermal efficiency of the cycle.

OUTPUT:

PV DIAGRAM:

THERMAL EFFICIENCY:

CONCLUSION:

Figure 1 shows the PV diagram of the otto cycle and figure 2 shows the thermal efficiency of the otto cycle.The

thermal efficiency obtained for otto cycle is 60.189283 percent.