OTTO CYCLE, p-v diagram and Efficiency of the cycle

% Program to plot p-v diagram for Otto Cycle
  clear
  close all
  clc

% Inputs
    gamma = 1.4;
    t3 = 2300;

% State variables at State point 1
    p1 = 101325;    % Presure at State 1(1ATM)
    t1 = 500;       % Temperature at state 1

% Engine geometric parameters
    bore = 0.1;     % Bore diameter
    stroke = 0.1;   % Length of stroke
    con_rod = 0.15; % Length of connecting rod
    cr = 12;        % Copmression ratio (cr=v1/v2)

% Calculating the swept volume and clearance volume
   % Swept volume is volume swept by the cylinder
        v_swept = (pi/4)*bore^2*stroke;
   % Clearance volume is volume available in cylinder when piston is at TDC
        v_clearance = v_swept/(cr-1);
   % Volume at state 1
        v1 = v_swept + v_clearance;
   % Volume at TDC
        v2 = v_clearance;

% State variables at State point 2
   % Pressure at state point 2
   % By the relation p1v1^gamma = p2v2^gamma
   % ie. p2 = p1(v1/v2)^gamma = p1*cr^gamma
        p2 = p1*cr^gamma;
   % To compute t2 we use idle ga law ie, pv/t is a constant
   % ie,p1v1/t1 = p2v2/t2, t2 = p2*v2*t1/(p1*v1)
        t2 = p2*v2*t1/(p1*v1);
   % Along Isentropic process, pv^gamma = constant
        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
% State point 3 represents the end of point of constant vol. combustion 
        v3 = v2;
   % Assume t3 = 2300K at the end of combustion process
   % p3 can be calculate by Idle gas law
   % p3v3/t3 = p2v2/t2, ie 
        p3 = p2*t3/t2;
   % Along Isentropic process, pv^gamma = constant
        constant_c  = p3*v3^gamma;
        % Function 'engine_kinematics'
        V_expansion = engine_kinematics(bore,stroke,con_rod,cr,0,180);
        P_expansion = constant_c./V_expansion.^gamma;
        
% State variables at State point 4
  % Volume at state point 4 is equal to state point 1
        v4 = v1;
  % Pressure at state point 4
  % ie. p3v3^gamma = p4v4^gamma, ie p4 = p3*(v3/v4)^gamma
        p4 = p3*(v3/v4)^gamma;

% Now we got all State points and we can plot t he graph
        figure(1)   % Open new figure window
        hold on
% Plot the p-v Diagram with all the four State points values
        grid on
        plot(V_compression,P_compression);  % Reversible + Adiabatic compression
        plot([v2 v3],[p2 p3]);              % Constant volume heat addition
        plot(V_expansion,P_expansion);      % Reversible + Adiabatic expansion
        plot([v4 v1],[p4 p1]);              % Constant volume heat rejection
        title('P-V DIAGRAM FOR OTTO CYCLE');% Title for the Graph
        xlabel('Volume (v)');               % Labelling x-axis
        ylabel('Pressure (p)');             % Labelling y-axis

% Efficiency of Otto Cycle, Efficiency, E = 1-(1/(cr)^(gamma-1))
        Efficieny_E = 1-(1/(cr)^(gamma-1))

% Conclusion
% The efficiency will increase if we increase the value of compression ratio
% But indiscriminately increasing the compression ratio leads to increase in temperature
% Increase in Temperature to high extent will lead to Automatic Combustion
% Therefore, Efficiency of cycle is limited by Compression ratio.

function [V] = engine_kinematics(bore,stroke,con_rod,cr,start_crank,end_crank)

    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(start_crank,end_crank,100);    % Crank angle

% Equation to calculate volume trace,V
% V = v_c(1 + (cr-1)(R+1-cosd(theta)-(R^2-(sind(theta)^2))^0.5)
% Splitting the equation into three called 'X','Y' and 'Z'
    X = 0.5*(cr-1);
    Y = R+1-cosd(theta);
    Z = (R^2-sind(theta).^2).^0.5;

% Now putting all three terms into the equation
    V = (1+X*(Y-Z)).*v_c;
end

PROJECT – 1

OTTO CYCLE

1. AIM

To Plot p-v diagram for Otto Cycle and find out the efficiency of the cycle.

2. GOVERNING EQUATIONS
   • Formula to determine the volumes inside an engine combustion chamber is going to

change

`V=V_(c )×(1+1/2 (cr-1)(R+1-cos⁡theta )-(R^2-(sin)^2 theta)^(1/2) )`

where,

V_c = Clearance volume,

cr = Compression ratio,

R = Length of connecting rod/Crank pin radius,

Crank pin radius = Stroke length/2,

= Crank angle

• Formula to determine the efficiency of the cycle

`Efficiency,E =1-1/(cr^(γ-1) )`

where,

cr = Compression ratio

3. OBJECTIVE
   • Process 1-2, Reversible + Adiabatic Compression
   • Process 2-3, Constant volume heat addition
   • Process 2-3, Constant volume heat addition
   • Process 3-4, Reversible + Adiabatic Expansion
   • The values for bore diameter, stroke length, connecting rod length and compression ratio is being provided.
   • Swept volume is volume swept by the cylinder
   • Clearance volume is volume available in the cylinder when the piston is at TDC

4. BODY OF THE CONTENT

• Programming language used is HTML/XML
• There are four stages of Otto Cycle
• Each state is individually noted and the parameters are assigned
• After computing all the parameters, plot the p-v diagram
• Then finding the efficiency of the cycle
• 
• 
• 5. PROGRAMMING CODE

• % Program to plot p-v diagram for Otto Cycle
  clear
  close all
  clc

  % Inputs
  gamma = 1.4;
  t3 = 2300;

  % State variables at State point 1
  p1 = 101325; % Presure at State 1(1ATM)
  t1 = 500; % Temperature at state 1

  % Engine geometric parameters
  bore = 0.1; % Bore diameter
  stroke = 0.1; % Length of stroke
  con_rod = 0.15; % Length of connecting rod
  cr = 12; % Copmression ratio (cr=v1/v2)

  % Calculating the swept volume and clearance volume
  % Swept volume is volume swept by the cylinder
  v_swept = (pi/4)*bore^2*stroke;
  % Clearance volume is volume available in cylinder when piston is at TDC
  v_clearance = v_swept/(cr-1);
  % Volume at state 1
  v1 = v_swept + v_clearance;
  % Volume at TDC
  v2 = v_clearance;

  % State variables at State point 2
  % Pressure at state point 2
  % By the relation p1v1^gamma = p2v2^gamma
  % ie. p2 = p1(v1/v2)^gamma = p1*cr^gamma
  p2 = p1*cr^gamma;
  % To compute t2 we use idle ga law ie, pv/t is a constant
  % ie,p1v1/t1 = p2v2/t2, t2 = p2*v2*t1/(p1*v1)
  t2 = p2*v2*t1/(p1*v1);
  % Along Isentropic process, pv^gamma = constant
  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
  % State point 3 represents the end of point of constant vol. combustion
  v3 = v2;
  % Assume t3 = 2300K at the end of combustion process
  % p3 can be calculate by Idle gas law
  % p3v3/t3 = p2v2/t2, ie
  p3 = p2*t3/t2;
  % Along Isentropic process, pv^gamma = constant
  constant_c = p3*v3^gamma;
  % Function 'engine_kinematics'
  V_expansion = engine_kinematics(bore,stroke,con_rod,cr,0,180);
  P_expansion = constant_c./V_expansion.^gamma;
  
  % State variables at State point 4
  % Volume at state point 4 is equal to state point 1
  v4 = v1;
  % Pressure at state point 4
  % ie. p3v3^gamma = p4v4^gamma, ie p4 = p3*(v3/v4)^gamma
  p4 = p3*(v3/v4)^gamma;

  % Now we got all State points and we can plot t he graph
  figure(1) % Open new figure window
  % Plot the p-v Diagram with all the four State points values
  grid on
  plot(V_compression,P_compression); % Reversible + Adiabatic compression
  plot([v2 v3],[p2 p3]); % Constant volume heat addition
  plot(V_expansion,P_expansion); % Reversible + Adiabatic expansion
  plot([v4 v1],[p4 p1]); % Constant volume heat rejection
  title('P-V DIAGRAM FOR OTTO CYCLE');% Title for the Graph
  xlabel('Volume (v)'); % Labelling x-axis
  ylabel('Pressure (p)'); % Labelling y-axis

  % Efficiency of Otto Cycle, Efficiency, E = 1-(1/(cr)^(gamma-1))
  Efficieny_E = 1-(1/(cr)^(gamma-1))

  % Conclusion
  % The efficiency will increase if we increase the value of compression ratio
  % But indiscriminately increasing the compression ratio leads to increase in temperature
  % Increase in Temperature to high extent will lead to Automatic Combustion
  % Therefore, Efficiency of cycle is limited by Compression ratio.

• function [V] = engine_kinematics(bore,stroke,con_rod,cr,start_crank,end_crank)

  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(start_crank,end_crank,100); % Crank angle

  % Equation to calculate volume trace,V
  % V = v_c(1 + (cr-1)(R+1-cosd(theta)-(R^2-(sind(theta)^2))^0.5)
  % Splitting the equation into three called 'X','Y' and 'Z'
  X = 0.5*(cr-1);
  Y = R+1-cosd(theta);
  Z = (R^2-sind(theta).^2).^0.5;

  % Now putting all three terms into the equation
  V = (1+X*(Y-Z)).*v_c;
  end

  6. CONCLUSION

• The efficiency, E will increase if we increase the value of compression ratio.
• But indiscriminately increase the compression the compression ratio leads to increase in temperature
• Increase in temperature to high extent will lead to Automatic Combustion
• Therefore efficiency, E is limited by Compression ratio

7. PROBLEMS FACED

• Unable to upload the Programming code in INSERT/EDIT CODE SAMPLE.