Air standard Cycle

Aim:

To plot a P-V diagram of an Otto cycle and find the thermal efficiency by using Matlab Tool.

Formulae:

• Swept Volume (`Vswept``: `pi/4*d^2*L`
• clearance volume (`Vc`): `(Vs)/(gamma-1)`
• compression Ratio (`r`)= `(V2)/(V1)`
• Thermal Efficiency of an engine (`eta``o`): `1-1/(r)^(gamma-1)`
• Adiabatic process:`pv^gamma=c`
• where

`d`=Bore diameter (m)

`L`= Stroke Length (m)

Governing equation used to plot the P-V diagram is

`V/(Vc)=1+1/2(r-1)[R+1-cosd (theta)-((R^2-sin^2 (theta))^(1/2)]`

Body of the content:

Theoretical plot of P-V and T-S diagram of Otto cycle is

where

• P = Pressure
• V= volume
• T=Temperature
• P=Pressure
• S=entropy

Note: Assumed required values to generate P-V diagram and to find the Thermal efficiency.

• P1=101325 Pa
• T1=500k
• T3=2300k
• r=12
• `gamma`=1.4
• Bore diameter (d)=0.1m
• Stroke Length (L)=0.1m
• Length of Connecting rod (R)=0.15m

Matlab code:

• Program used to plot a P-V diagram of an Otto cycle and find the thermal efficiency by using Matlab Tool is

%function code
function [V]=engine_kinematics(bore,stroke,con_rod,cr,start_crank,end_crank);
a=stroke/2
R=con_rod/a
V_swept=(pi/4)*bore^2*stroke;
  V_c=V_swept/(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

%otto cycle

clear all 
close all
clc 
%inputs

  gamma=1.4
  %pressure at state point 1 (Pa)
  P1=101325
  %temperature at state point 1 (k)
  T1=500
  %temperature at state point 3 (k)
  T3=2300
  %engine geometric parameters (meters)
  bore=0.1;
  stroke=0.1;
  con_rod=0.15;
  %compression ration 
  cr=12;
  %thermal efficiency of engine
  neta=(1-1/(cr)^(gamma-1))*100
  
  %calculating the swept volume and the clearence volume
  V_swept=(pi/4)*bore^2*stroke;
  V_clearance=V_swept/(cr-1);
  V1=V_swept+V_clearance;
  V2=V_clearance;
  
  %state variables at state point 2
  %P2*V2^gamma=P1*V1^gamma
  %P2=P1*(V1/V2)^gamma=P1*cr^gamma
  P2=P1*cr^gamma;
  %P1*V1/T1=P2*V2/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 variables at state point 3
  V3=V2;
  %P3*V3/T3=P2*V2/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 variables at state point 4
  V4=V1;
  %P3*V3^gamma =P4*V4^gamma | P4=P3*(V3/V4)^gamma
  P4=P3*(V3/V4)^gamma;
  
  hold on
  plot(V_compression,P_compression,'linewidth',3)
  plot([V2 V3],[P2 P3],'linewidth',3)
  plot(V_expansion,P_expansion,'linewidth',3)
  plot([V4 V1],[P4 P1],'linewidth',3)
  legend('Compression','Addition','Expansion','Heat_Rejection')
  xlabel('volume')
  ylabel('pressure')
  grid on
 
  

Procedure:

• of otto cycle and also to calculate the Thermal Efficiency of Engine
• Initially I have defined the finction code by using the governing equation.
• In the main code initially I have defined the inputs required.
• Next, I have calculated the swept valume and clearence volume.
• Next, I have calculated the thermal efficiency of engine.
• Next, I have defind the state variables at each state points using Adiabatic relation. Through that we got vaules of Temperature, Pressure and volume at all points.
• After calculating all the required values, I have plotted the graph of pressure and volume.
• Finally I have saved & run the code.

Output:

The above graph shows the P-V diagram of Otto cycle created by Matlab Tool

• Thermal Efficiency of engine is `eta`=62.98%`

Conclusion:

Hence the Thermal efficiency is solved and P-V diagram of otto cycle is plotted.