Otto Cycle thermal efficiency and P-V plot

1. AIM: To plot the P-V diagram and find out the thermal efficiency of an Otto cycle.
2. Introduction: The Otto cycle is a set of processes used by spark-ignition internal combustion engines (2-stroke or 4-stroke cycles). It consists mainly of 5 processes:

• process1: a mass of air is drawn into the piston-cylinder arrangement at constant pressure.
• process2: an adiabatic (isentropic) compression of the charge as the piston moves from the bottom dead centre (BDC) to top dead centre (TDC).
• process3: a constant-volume heat transfer to the working gas from an external source while the piston is at top dead centre. This process is intended to represent the ignition of the fuel-air mixture and the subsequent rapid burning.
• process4: an adiabatic (isentropic) expansion (power stroke).
• process5: heat rejection process.

In this project, we only plot 4 processes (2,3,4,5) on the P-V diagram.

Thermal efficiency:

ɳ = 1-CR^1-γ (γ=gamma)

Description: Now to plot the P-V diagram we need to find out pressure and volume at every point (stage) of process.

For start, we have to assume engine geometric parameter like-

1. Bore=0.1m
2. Stroke=0.1m
3. Connecting rod length=0.15m
4. Compression ratio=12
5. Pressure at stage 1(p1) = 101.325
6. Temperature at stage 1(T1) = 500K
7. Temperature at stage 3(T3) = 2300K

Now we have to calculate volume at stage 1(V1), for this, we have to calculate clearance and swept volume.

V_swept =( pi/4)*bore^2*stroke

V_clearance=V_swept/(cr-1)

V1(volume at stage 1)=V_swept+V_clearance

V2(volume at stage 2)=V_clearance

Now from this information we find out pressure and volume each and every point

In this we use mainly two formula

1. P*V = mRT
2. P*V^gamma=Constant

Values of pressure and volume at other stage-

1. p2=p1*cr^ gamma
2. t2 = t1*(p2*v2/p1*v1)
3. v3=v2;
4. p3 = p2*(v2*t3/v3*t2);
5. v4=v1;
6. p4 = p3*(v3/v4)^gamma;

We need to run the matlab code for get the plot of the P-V diagram.

Main Code-

clear all
close all
clc

%Inputs
gamma=1.4;
t3=2300;

%state variable at state point 1
p1 = 101.325;
t1=500;

%engine geometric parameter
bore=0.1;
stroke=0.1;
con_rod=0.15;
cr=12;

% calculating 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 variable at state pointr 2
%p2v2^gamma=p1v1^gamma
%p2=p1*(v1/v2)^gamma  | p2=p1*cr^gamma
p2=p1*cr^gamma;

%p1v1/t1=p2v2/t2 | t2 = t1*(p2v2/p1v1)
 t2 = t1*(p2*v2/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 variable at state point 3
 v3=v2;
 
 %p2v2/t2= p3v3/t3  | p3 = p2*(v2t3/v3t2)
  p3 = p2*(v2*t3/v3*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 the variables at state point 4
  v4=v1;
  
  %p3v3^gamma=p4v4^gamma  | p4 = p3*(v3/v4)^gamma
  p4 = p3*(v3/v4)^gamma;
 % thermal efficiency
 thermal_efficiency=1-1/cr^(gamma-1)
  
  figure(1)
  hold on
  plot(v1,p1,'*','color','r')
  plot(V_compression,P_compression)
  plot(v2,p2,'*','color','r')
  plot([v2 v3],[p2 p3],'color','b')
  plot(v3,p3,'*','color','r')
   plot(V_expansion,P_expansion)
  plot(v4,p4,'*','color','r')
  plot([v4 v1],[p4 p1],'color','b')
  xlabel('volume')
  ylabel('pressure(in pa)')
  grid on

In the above code, we use a function engine_kinematics for this function other code is required.

supplement code-

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

CONCLUSION:

1. We get a perfect plot of P-V.

2. We get thermal efficiency of the otto cycle is 62.99%.

3. We can get the P-V plot and thermal efficiency of any otto cycle just varying the initial values.