Air Standard Cycle

1.AIM:

• Write a program that can solve an OTTO-Cycle and to make a plot of Pressure VS Volume during the Otto-cycle and also to calculate the thermal efficiency of the OTTO-Cycle.

2.GOVERNING EQUATIONS IF ANY:

3.OBJECTIVES OF THE PROJECT:

• To solve the Otto-Cycle.
• Plot the P-V graph of the cycle.
• Calculate the efficiency of the Otto-Cycle.

• The workflow of the project.
• Collect the given data.
• Calculate the swept volume and clearance volume from the given data using formulas 1&2.
• Using V_s & V_c calculate the volume conditions at each stage i.e. V₁, V₂, V₃, V₄ by using formulas 3&4.
• Using formula 5 calculate the pressure at each stage i.e. P₁, P₂, P₃, P₄.
• Using formula 6 calculate the pressure at each stage i.e. T₁, T₂, T₃, T₄.
• Since from stage 2-3 & stage 4-1 volume remains constant, so it gets a straight line on the P-V graph
• As stage 1-2 & stage 3-4 are adiabatic compression & adiabatic expansion its volume changes continuously. And its volume is calculated using form 7.
• Now using formula 8, the pressure is calculated for stage 1-2 i.e. adiabatic compression, and also for stage 3-4 i.e. adiabatic expansion.
• Now the process of stage 1-2 and stage 3-4 can plot on a P-V graph.
• And finally using formula 9 the efficiency of the given Otto-Cycle is calculated.

4.BODY OF THE CONTENT:

• Matlab Programming language is used.
• Main Program is attached below.

• %Write code that can solve an otto cycle and make plots for it.
  %Here are the requirements
      %Your code should create a PV diagram
      %You should output the thermal efficiency of the engine.
  
  %Given Data.
  % P1=0.1MPa, T1=35 Degree Celcius, T3= 1100 Degree Celcius
  % Bore=12cm, Stroke=10cm, Connecting Rod=15cm, Compression Ratio=9, Gamma=1.4
      
  close all
  clear all
  clc
  
  %Given Data/Conditions 
  p1=0.1; %in MegaPascals %Pressure at Stage 1.
  t1=308; %in kelvin %Temperature at Stage 1.
  t3=1373; %in kelvin %Temperature at Stage 3.
  bore = 0.12; %in m 
  stroke = 0.1; %in m
  con_rod = 0.15; %in m
  cr = 9;
  gamma=1.4;
  
  %calculatig v1,v2,v3,v4
  v_swept = (pi/4)*bore^2*stroke; %in m^3 %Swept Volume by piston.
  v_clearance = v_swept/(cr-1); %in m^3 %&Clearance Volume.
  
  v1 = v_swept + v_clearance; %in m^3 %Volume at Stage 1.
  v2 = v_clearance; %in m^3 %Volume at Stage 2.
  v3=v2; %in m^3 %Volume at Stage 3.
  v4=v1; %in m^3 %Volume at Stage 4.
  
  %Calculating State Variables at 2
  %p1v1^gamma=p2v2^gamma
  p2=p1*(v1/v2)^gamma; %in MPa %Pressure at Stage 2.
  
  %p1v1/t1=p2v2/t2
  t2=(p2*v2)/(p1*v1)*t1; %in kelvin %Temperature at Stage 2.
  
  %Supplementary programs: Function(engine_kinematics) is created in the MatLab to calculate the instantaneous Volume.
  %and by calling the function volume change is calculated from stage 1-2.
  V_com=engine_kinematics(bore,stroke,con_rod,cr,180,0); %in m^3 %Volume change from stage 1-2.
  P_com=p1*v1^gamma./V_com.^gamma; %in MPa %Pressure change from stage 1-2.
  
  %Calculating State Variables at 3
  %p3v3/t3=p2v2/t2
  p3 =p2*v2/t2*t3/v3; %in MPa %Pressure at Stage 3.
  
  
  %Calculating State Variables at 4
  %p1v1^gamma=p2v2^gamma
  p4=p3*(v3/v4)^gamma; %in MPa %Pressure at Stage 4.
  
  %p3v3/t3=p4v4/t4
  t4=(p4*v4)/(p3*v3)*t3; %in kelvin %Temperature at Stage 4.
  
  %Supplementary programs: Function(engine_kinematics) is created in the MatLab to calculate the instantaneous Volume.
  %and by calling the function volume change is calculated from stage 3-4.
  V_exp=engine_kinematics(bore,stroke,con_rod,cr,0,180); %in m^3 %Volume change from stage 3-4.
  P_exp=p3*v3^gamma./V_exp.^gamma; %in MPa %Pressure change from stage 3-4.
  
  %efficiency=1-1/(cr)^(gamma-1)
  efficiency=1-1/(cr)^(gamma-1); %Efficiency Of the Air-Standard_Cycle.
  
  %Displaying the Conditions of Pressure, Volume and Temperature at each stage.
  disp('Conditions at State 1')
  disp(['P1=' num2str(p1) ' MPa ; V1=' num2str(v1) ' m^3 ; T1=' num2str(t1) ' k'])
  disp(' ')
  disp('Conditions at State 2')
  disp(['P2=' num2str(p2) ' MPa ; V2=' num2str(v2) ' m^3 ; T2=' num2str(t2) ' k'])
  disp(' ')
  disp('Conditions at State 3')
  disp(['P3=' num2str(p3) ' MPa ; V3=' num2str(v3) ' m^3 ; T3=' num2str(t3) ' k'])
  disp(' ')
  disp('Conditions at State 4')
  disp(['P4=' num2str(p4) ' MPa ; V4=' num2str(v4) ' m^3 ; T4=' num2str(t4) ' k'])
  disp(' ')
  disp(['Efficiency=' num2str(efficiency*100) '%'])
  
  %Ploting the Otto-Cycle.
  figure(1)
  hold on 
  plot(v1,p1,'*','color','r')%Ploting the Conditions of Pressure and Volume stage 1.
  plot(v2,p2,'*','color','r')%Ploting the Conditions of Pressure and Volume stage 2.
  plot(v3,p3,'*','color','r')%Ploting the Conditions of Pressure and Volume stage 3.
  plot(v4,p4,'*','color','r')%Ploting the Conditions of Pressure and Volume stage 4.
  plot(V_com,P_com,'linewidth',3,'color','r') %Ploting the Process change of Pressure and Volume from stage 1-2.
  plot([v2,v3],[p2,p3],'linewidth',3,'color','g') %Ploting the Process change of Pressure and Volume from stage 2-3.
  plot(V_exp,P_exp,'linewidth',3,'color','r') %Ploting the Process change of Pressure and Volume from stage 3-4.
  plot([v4,v1],[p4,p1],'linewidth',3,'color','g') %Ploting the Process change of Pressure and Volume from stage 4-1.
  xlabel('Volume in m^3') %Labeling the X-Axis.
  ylabel('Pressure in MPa') %Labeling the Y-Axis.

• Supplementary Program to calculate Instantaneous Volume

• %Supplementary programs:Function(engine_kinematics) is to calculate the instantaneous Volume.
  
  function [V] = engine_kinematics(bore,stroke,con_rod,cr,crank_start,crank_end)
  
  a=stroke/2; %Crank-Pin Radius.
  R=con_rod/a; 
  
  v_s=(pi/4)*bore^2*stroke; %Swept Volume.
  v_c= v_s/(cr-1); %Clearance Volume.
  
  theta=linspace(crank_start,crank_end,100); %Range and no. of outputs to be calculated.
  
  %Formula to calculate instantaneous Volume during movement of the piston.
  % V/v_c=1+(cr-1)./2.*(1+R-cosd(theta)-(R^2-sind(theta).^2).^0.5)
  V=v_c.*(1+(cr-1)./2.*(1+R-cosd(theta)-(R^2-sind(theta).^2).^0.5));
  
  end 

• Results
• 

5.CONCLUSION:

• Output Results of the Air-Standard-Cycle Program
• 

• Graph Of Otto-Cycle.
•