Air standard Cycle using MATLAB

Objective: - Write MATLAB code that can solve or check the engine kinematics of an engine operating on the OTTO cycle and make plots for it. Create a PV diagram and also output the thermal efficiency of the engine.

Theory: - An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines.

Processes: -

The processes are described by:

• Process 0–1 a mass of air is drawn into piston/cylinder arrangement at constant pressure.
• Process 1–2 is an adiabatic (isentropic) compression of the charge as the piston moves from bottom dead center (BDC) to top dead center (TDC).
• Process 2–3 is a constant-volume heat transfer to the working gas from an external source while the piston is at top dead center. This process is intended to represent the ignition of the fuel-air mixture and the subsequent rapid burning.
• Process 3–4 is an adiabatic (isentropic) expansion (power stroke).
• Process 4–1 completes the cycle by a constant-volume process in which heat is rejected from the air while the piston is at bottom dead center.
• Process 1–0 the mass of air is released to the atmosphere in a constant pressure process.

Thermal efficiency (air-standard efficiency) of Otto Cycle: - Basic terms used in derivation of air-standard efficiency of Otto cycle:

Compression Ratio:

Compression ratio (r) is the ratio of total cylinder volume to the clearance volume.

Now that we know the basic terms, let us derive expressions for T₂ and T₃. These expressions will be useful for us to derive the expression for air-standard efficiency of Otto cycle. For finding T₂, we take process 1-2 and for finding T₃, we take process 3-4.

Process 1-2:

This process is an isentropic (reversible adiabatic) process. For this process, the relation between T and V is as follows:

Process 3-4:

This is also an isentropic process. The relation between T and V in this process is similar to the relation between T and V in process 1-2:

Here,

Air-standard efficiency of Otto cycle:

It is defined as the ratio between work done during Otto cycle to the heat supplied during Otto cycle.

Air-Standard Efficiency (thermal efficiency) of Otto cycle,

thermal eff. calculates by using different values of temperatures.

nth = Thermal efficiency and r = compression ratio.

The Geometric parameters of Engine: -

• bore = 0.1
• stroke = 0.1
• con_rod = 0.15 #connecting rod
• cr = 12 #compression ratio
• a = stroke/2 #crank pin radius
• R = con_rod/a
• gamma = 1.4

Inputs: -

State point 1: -

• p1 = 101325 #unit - pascal
• t1 = 500 #unit – kelvin
• v1 = v_swept + v_clearance (v_s is swept volume and v_c is clearance volume)

State point 3: -

• t3 = 2300 #unit – kelvin

The following equation shows the relationship between crank angle and volume inside the crank chamber: -

Formula for thermal efficiency: -

Coding: -

First we have to define a funtion named as engine kinematics:

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

a = stroke/2;
R = con_rod/a;

v_s = (pi/4)*bore^2*stroke;
v_c = v_s/(cr-1);

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;

V = (1 + term1*(term2 - term3)).*v_c;

end

Main code:

clear all
close all
clc

% Inputs
gamma = 1.4;
t3 = 2300;

% State Variables
p1 = 101325;
t1 = 500;

% Engine gepmetric parameters 
bore = 0.1;
stroke = 0.1;
con_rod = 0.15;
cr = 12;

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

v1 = v_swept + v_clearance;
v2 = v_clearance;

% State varaible 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)/(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;


% State varaible 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, con_rod, cr, 0, 180);
P_expansion = constant_c./V_expansion.^gamma;

% State varaible at state point 4
v4 = v1;

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

% p4v4/t4 = p3v3/t3 | t4 = (p4*v4*t3)/(p3*v3)
t4 = (p4*v4*t3)/(p3*v3);

figure(1)
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([v1 v4],[p1 p4],'color','b')
xlabel('Volume')
ylabel('Pressure')
title('OTTO CYCLE')

% Thermal efficiency
term4 = t4-t1;
term5 = t3-t2;
Thermal_efficiency = (1-(term4/term5))*100

Explanation:-

• First we have to define inputs like bore, stroke, connecting rod, compression ratio, gamma, and temp and pressure at state point 1 and temp at state point 3.
• Then by using the formul like p1v1^gamma = p2v2^gamma and ideal gas law p1v1/t1 = p2v2/v3. By using this equation we can find the other value of pressure, temp and volume at different state points.
• By this point we can plot the pv diagram, but the diagram is not accurate, all the lines are straight. For getting the accurate P-V diagram we have to use engine kinematics.
• We have to define a fuction called engine kinematics by this we can change the angle and find he actual volume in engine.
• In main code we call that fuction called engine kinematics, and find the the different vaule of volume at different angles for compression and expansion.
• By this we get accurate vaule of pressure, temp and volume at differnet state point from 1 to 4.
• By this point we get the accurate P-V diagram as shown belown in the results.

Result: -

P-V Diagram: -

Thermal efficiency is 62.9893

Conclusion: -The above Matlabcode was written to plot a P-V Diagram of the Air standard cycle or Otto Cycle and also calculates thermal efficiency of Otto cycle.

Thermal efficiency is 62.9893