OTTO CYCLE

AIM:

To understand and to make an ‘OTTO CYCLE’ graph and analyze it.

OBJECTIVES:

• To make a P-V graph of Otto cycle
• To find the thermal efficiency of the process

INTRODUCTION:

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 involved in the cycle are as follows:

• 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.

Formulas Used:

• For adiabatic processes

`PV^γ =`constant

where P = pressure of the ideal gas

V = volume of the ideal gas

`gamma` = `C_p/C_v` where `C_p` = specific heat at constant pressure

`C_v` = specific heat at constant volume

• `(PV)/T` = constant

This is the gas equation for all the ideal gases undergoing any process.

• `V/V_c =``1+1/2(CR-1)[R+1-cos(theta)-(sqrt (R^2-(sin(theta))^2))]`

where V = volume during compression or expansion

`V_c` = clearance volume

`theta` = crank angle

CR = compression ratio

R = `(CR)/a` ,where a = crank pin radius

• `eta = 1 - 1/(CR^(gamma-1))`

where `eta` = thermal efficiency

PROCEDURE OF COMPLETION:

• Here in this case, components of a 5.7L V8 Engine had been taken i.e. the bore length, the stroke, the connecting rod length, the compression ratio and etc.
• Then all the calculations were made as per the formulae above and calculated.
• A function named ‘engine_kinematics’ was used to calculate the volume during compression and expansion.
• Then, the plotting of the graph alongwith the thermal efficiency was calculated.

The following MATLAB code would execute the graph alongwith the thermal efficiency:

%% OTTO CYCLE
% Program to find P-V characteristics of Otto cycle and
% find the thermal efficiency of the cycle
clear all;
close all;
clc;


%% Initialization of parameters
bore    = 0.1016;    % (in m)
stroke  = 0.0884;     % (in m)
CR      = 10;       % compression ratio
c_rod   = 0.148;   % connecting rod length (in m)
k  = 1.4;           % gamma


%% Parameters at state 1
p1 = 100000;        % (in pascals)
t1 = 300;           % (in kelvin)


%% Parameters at state 3
t3 = 2000;          % (in kelvin)


%% Calculation of swept volume and clearance volume
v_swept     = (pi/4)*bore^2*stroke;
v_clearance = v_swept/(CR-1);
v1          = v_swept + v_clearance;            % volume at state 1
v2          = v_clearance;                      % volume at state 2

%% Calculation of pressure,volume and temperature at different states
% State 2:
% p1*v1^gamma = p2*v2^gamma
p2 = p1*(v1/v2)^k;

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

% During compression
constant = p1*v1^k;
V_compression = engine_kinematics(bore,stroke,c_rod,CR,180,0);
P_compression = constant./(V_compression'.^k);

% State 3:
% (p2*v2)/t2 = (p3*v3)/t3
v3 = v2;
p3 = (p2*t3)/t2;

% During expansion
constant = p3*v3^k;
V_expansion = engine_kinematics(bore,stroke,c_rod,CR,0,180);
P_expansion = constant./(V_expansion'.^k);

% State 4:
% p3*v3^gamma = p4*v4^gamma
v4 = v1;
p4 = p3*(v3/v4)^k;


%% Calculating Thermal Efficiency
eff = 1 - (1/(CR^(k-1)));


%% plotting
hold on
plot(V_compression,P_compression,'k','linew',2)
plot([v2 v3],[p2 p3],'k','linew',2)
plot(V_expansion,P_expansion,'k','linew',2)
plot([v4 v1],[p4 p1],'k','linew',2)
title('OTTO CYCLE','fontsize',25)
xlabel('Volume(in m^3)','fontsize',15)
ylabel('Pressure(in Pa)','fontsize',15)
axis square, grid on


%% Labelling the figure
text(4*10^(-4),5.5*10^6,['Thermal efficiency = ' num2str(eff)],'fontsize',12)
text(v1,p1,'State-1','fontsize',12)
text(v2,p2,'State-2','fontsize',12)
text(v3,p3,'State-3','fontsize',12)
text(v4,p4,'State-4','fontsize',12)
text(V_compression(150),P_compression(150),'Compression','fontsize',12)
text(V_expansion(30),P_expansion(30),'Expansion','fontsize',12)

The function named 'engine_kinematics' is as follows:

function [V] = engine_kinematics(bore,stroke,c_rod,CR,start_crank,end_crank)
    
%  The function gives the output of the compression and expansion volume of the
%  otto engine. This helps in providing with the exact P-V characteristics
%  that we learn from the theoretical basis.
%  
%  INPUTS : 
%           1. Bore
%           2. Stroke
%           3. Connecting rod
%           4. Compression ratio
%           5. Crank angle at start
%           6. Crank angle at end
%
%  OUTPUT :
%           1. V_compression at isotropic compression
%              V_expansion at isotropic expansion


%% initialization of parameters and calculation
a = stroke/2;               % crank pin radius
R = c_rod/a;

theta = linspace(start_crank,end_crank,180);

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

term1 = CR-1;
term2 = R+1-cosd(theta);
term3 = sqrt(R^2-sind(theta).^2);

V = (1 + 0.5*(term1).*(term2-term3)).*v_clearance;

end

OBSERVATIONS:

The Otto cycle alongwith its thermal efficiency is as follows:

The arrow direction shows the direction of the cycle.

CONCLUSION:

Otto cycle is one of the fundamental thermodynamic processes that occur in any internal combustion engine. Plotting the cycle in MATLAB is an essential feature of the software which allows us to visualize our data with more precision and helps us to tweak it accordingly.