Air standard cycle

AIM: To 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.

DESCRIPTION: The Otto cycle is the ideal cycle for the spark-ignition reciprocating engines, and it consists of four internally reversible processes:

1. isentropic compression,
2. constant-volume heat addition,
3. isentropic expansion, and
4. constant-volume heat rejection.

Figure: Otto cycle (P-v diagram)

GOVERNING EQUATIONS:-

The ratio of the maximum volume formed in the cylinder to the minimum

(clearance) volume is called the compression ratio r of the engine:

r = `v_max/v_min =(displacement \ volume)/(cIearance\ volume)`

To find the efficiency of the cycle T₃ and T₄ are needed.

Process 1-2:

This process is an isentropic process. Therefore

`T_2/T_1 =(V_1/V_2 )^(γ-1) = r^(γ-1)`

Process 3-4:

This is also an isentropic process. Therefore

`T_3/T_4 =(V_3/V_4 )^(γ-1) = r^(γ-1)`

Air-standard efficiency of the Otto cycle:

The thermal efficiency of the ideal Otto cycle under the cold air standard w_net

assumptions are:

`eta_(th=w/q =1-(T_4-T_1)/(T_3-T_2 )`

On simplification

`eta_(th= 1-1/r^(γ-1)`

The relationship between crank angle and volume inside the crank chamber is: -

`V/V_c =1+0.5*(r-1)*[R+1-cos⁡〖θ-(R^2-〖sin⁡θ〗^2 )^0.5 〗]`

OBJECTIVES OF THE PROJECT

• To write a 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.

Figure 2: Flow of the project

Code

First of all, a separate function is made for engine kinematics

function [V] = engine_kinematics_otto(bore, stroke, connecting_rod, compression_ratio, start_crank, end_crank)

%defining the parameters of engine

a = stroke/2;

R = connecting_rod/a;

v_s = (pi/4)*bore^2*stroke;

v_c = v_s/(compression_ratio-1);

theta = linspace(start_crank,end_crank,100);

% here x,y,z are the three variables of equations

x = 0.5*(compression_ratio-1);

y = R + 1 -cosd(theta);

z = (R^2 - sind(theta).^2).^0.5;%elementwise

V = (1 + x*(y - z)).*v_c;

end

main code

clear all

close all

clc

% Input values

gamma = 1.4;

t3 = 2500;

% State Variables

p1 = 101325;

t1 = 300;

% Engine design parameters

bore = 0.15;

stroke = 0.15;

connecting_rod = 0.25;

compression_ratio = 14;

% Calculating the swept volume and the clearance volume

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

v_clearance = v_swept/(compression_ratio-1);

v1 = v_swept + v_clearance;

v2 = v_clearance;

% State variable at point 2

% p2v2^gamma = p1v1^gamma

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

p2 = p1*compression_ratio^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_otto(bore, stroke, connecting_rod, compression_ratio, 180, 0);

P_compression = constant_c./V_compression.^gamma;

% State variable at 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_otto(bore, stroke, connecting_rod, compression_ratio, 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;

% 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','g')

plot(v2,p2,'*','color','r')

plot(v3,p3,'*','color','r')

plot(V_expansion, P_expansion,'color','g')

plot(v4,p4,'*','color','r')

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

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

xlabel('Volume')

ylabel('Pressure')

title('OTTO CYCLE')

% Thermal efficiency

term1 = t4-t1;

term2 = t3-t2;

Thermal_efficiency = (1-(term1/term2))*100

Explanation of code

1. A function is written for the following equation to find V:
2. In the main code the Input values are given (gamma, t₃, engine parameters, etc)
3. Equations for swept volume and clearance volume are written.
4. p_2,t_2, and t₃ are calculated using the isentropic condition.
5. Plot command is used for plotting the P-V diagram.
6. Thermal efficiency is calculated using the following equation:

Conclusions

1. Thermal efficiency is calculated in an effective manner by using MATLAB 18.0.

Figure 3: output in the command window

2. P-V diagram is plotted using the plot command

Figure 4: P-V diagram of the Otto cycle using MATLAB