Otto cycle

Introduction

Air standard cycles are the reference cycles which give an idea on performance of internal combustion engines by combination of various process conditions like Isentropic, Isothermal, Adiabatic etc. Carnot cycle defines the maximum temperature that could be attained while operating between source and sink. Similarly, air standard cycle helps assess the maximum possible efficiency that could be obtained while adhering to the process.

Otto cycle forms the basis of spark ignited gasoline engines. No SI gasoline engine can have a efficiency greater than otto cycle efficiency. It has two isochores and two adiabatics.

Assumptions

1) Air is the operating fluid

2) It follows ideal gas state

3) pV^(gamma) is constant

Octave code

Assumed geometries, assumed thermal states are defined and maintained constant throughout the code, volumes are calculated at each state by finding swept volume which is the volume that is swept by piston, clearance volume, volume of cylinder that is not swept by piston, finally by utilising the definition of isochores.

Code then calculates pressure and temperature at each of 1-2-3-4 state points by utilising information derived and two equations, pV^(gamma) = constant and pV/T = constant. A function is written to calculate the nature of curves during compression and expansion as it is not a straight line.

Then, the points are plotted and efficiencies are calculated.

%This program is used to calculate the state values at 4 corners of otto cycle
%Otto cycle is an Air standard cycle forming the basis for operation of gasoline engines
%Otto cycle is characterised by the constant volume process
%1-End of suction 2-End of compression 3-Start of expansion 4-end of expansion
%Following formulas are extensively used in the code below
%pV = RT Ideal gas equation
%pV^(gamma) = constant

close all
clear all
clc

% Assumed Geometries
bore = 0.150;
stroke = 0.180;
conn_rod = 0.37;
cr = 12.95;

%Assumed thermal states
p1 = 300000; %Pressure at the intake
t1 = 300;    %Temperature at the intake
t3 = 2300;   %Temperature at the start of expansion stroke
gamma = 1.3; % cp/cv ratio

%volume calculations
v_swept = (pi()/4)*bore^2*stroke; %cylinder volume as seen by piston
v_clear = v_swept/(cr-1); %volume between cylinder head and piston at TDC
v1 = v_swept+v_clear; %Total cylinder volume
v2 = v_clear; %Volume at the end of compression
v3 = v2;  %Isochoric process
v4 = v1;  %Isochoric process

%state calculations at the end of compression
p2 = p1*cr^gamma;  %pV^(gamma) = constant, cr=v1/v2
t2 = (p2*v2*t1)/(p1*v1); %pV = RT or pV/T = constant

%To plot a curve joining state 1-2
V_compression = volumetrace(bore,stroke,conn_rod,cr,180,0,180);
P_compression = p1.*(v1./V_compression).^gamma;

%state calculation at the start of expansion
p3 = p2*t3/t2; %pV = RT or pV/T = constant

%To plot a curve joining state 3-4
V_expansion = volumetrace(bore,stroke,conn_rod,cr,0,180,180);
P_expansion = p3.*(v3./V_expansion).^gamma;

%state calculation at the end of expansion
p4 = p3*(v3/v4)^gamma; %pV^(gamma) = constant, cr=v1/v2
t4 = p1*t1/p4; %pV = RT or pV/T = constant

%plotting the state points on the plot window
figure(1)
hold on
plot(v1,p1,'*','color','r')
%plot(V_compression,P_compression)
plot(v2,p2,'*','color','r')
plot(v3,p3,'*','color','r')
%plot(V_expansion,P_expansion)
plot(v4,p4,'*','color','r')

plot([v1 V_compression v2 v3 V_expansion v4 v1],[p1 P_compression p2 p3 P_expansion p4 p1])
xlabel('volume(m3)')
ylabel('pressure(pa)')
grid on
title('PV diagram of otto cycle')

%To calculate and print efficiency values
%carnot efficiency is the ideal engine efficiency and no engine can have efficiency
%greater than carnot efficiency
%Carnot efficiency = 1-t1/t3
%Otto cycle efficiency = 1-1/(Compression ratio)^(gamma-1)

carnot_efficiency = (1-((273.15+t1)/(273.15+t3)))*100;
Otto_cycle_efficiency = (1-(1/(cr^(gamma-1))))*100;

printf('Maximum thermal efficiency between temperature source %d C and sink %d C is %f\n',t3,t1,carnot_efficiency)
printf('Maximum otto cycle thermal efficiency for compression ratio %f is %f\n',cr,Otto_cycle_efficiency)

function [V] = volumetrace(bore,stroke,conn_rod,cr,start_crank,end_crank,number_div)

  a = stroke/2;
  R = conn_rod/a;

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

  theta = linspace(start_crank,end_crank,number_div);
  term1 = 0.5*(cr-1);
  term2 = R+1-cosd(theta);
  term3 = (R^2-sind(theta).^2).^0.5;

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

Errors

This is an erraneous plot as we cannot connect states 1-2 and 3-4 directly as shown, volume doesnt change linearly as assumed.

Results and plots

Report and Discussion on Results

1) Carnot efficiency is the maximum possible efficiency and Carnot efficiency>Otto cycle efficiency>real SI engine.

2) Volume trace calculation during compression and expansion is vital for correctly plotting the pV diagram

3) Otto cycle efficiency is heavily dependent on compression ratio