Air standard Cycle (Otto Cycle) theoretical and actual PV plot and thermal efficiency output.

1. AIM:

Air standard Cycle (Otto Cycle) theoretical and actual PV plot and thermal efficiency output.

Theory:

An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines.

The processes are described by:

Process 0–1 a mass of air is drawn into piston/cylinder arrangement at constant pressure.

Process 1–2 is an adiabatic (isentropic) compression of the charge as the piston moves from bottom dead center (BDC) to top dead center (TDC).

Process 2–3 is a constant-volume heat transfer to the working gas from an external source while the piston is at top dead center. This process is intended to represent the ignition of the fuel-air mixture and the subsequent rapid burning.

Process 3–4 is an adiabatic (isentropic) expansion (power stroke).

Process 4–1 completes the cycle by a constant-volume process in which heat is rejected from the air while the piston is at bottom dead center.

Process 1–0 the mass of air is released to the atmosphere in a constant pressure process.

2. GOVERNING EQUATIONS IF ANY:

Consider ‘m’ be the fixed mass of air undergoing the cycle of operation as described above;

Heat supplied;

Heat rejected;

Efficiency;

…….(1)

Process 1-2;

Process 3-1;

Or

Thus

Therefore from equation one;

We have

Where is called compression ratio and given by;

............Note

3. OBJECTIVES OF THE PROJECT:

To find plot the theoretical and actual PV plot of an Air Standard Cycle (Otto Cycle).

To fine the thermal efficiency of an otto cycle.

% Air Standard Cycle using MatLab

%Inputs
gamma = 1.4;
t3 = 2300;

%state variables
p1 = 101325;
t1 = 500;
% Engine Geometry Parameters
bore = 0.1;
stroke = 0.1;
con_rod = 0.15;
cr = 12;

% Calculating the swept volume and the clearance volume 

v_swept = (p1/4)*bore^2*stroke;
v_clearance = v_swept/(cr-1);
v1 = v_swept + v_clearance;
v2 = v_clearance;

% Stroke variables at point 2
% p2v2^gamma = p1v1^gamma
% p2 = p1*(v1/v2)^gamma = p1*cr^gamma

p2 = p1*cr^gamma;

% p1v1/t1= p2v2/t2 | t2 = p2*v2*t1/(p1*v1)
% compression process from 1-2; v_c and p_c are vol and press during compression

t2 = p2*v2*t1/(p1*v1);
constant_c = p1*v1^gamma;

v_c = engine_Kinematics1(bore,stroke,con_rod,cr,180,0);

p_c = constant_c./v_c.^gamma;

% State variable at point 3
% const vol head addition from 2-3, v_ha and p_ha are vol & press

v3 =v2;
constant_c = p2*v2^gamma;
v_ha = engine_Kinematics1(bore,stroke,con_rod,cr,0,0);
p_ha = constant_c./v_ha.^gamma;


% p3v3/t3 = p2v2/t2 | p3 = p2*t3/t2
% expansion process from 3-4, v_e and p_e are vol and press

p3 = p2*t3/t2;
constant_c = p3*v3^gamma;
v_e = engine_Kinematics1(bore,stroke,con_rod,cr,0,180);
p_e = constant_c./v_e.^gamma;

% state variable at state point 4
% const vol heat loss from 4-1, v_hl and p_hl are vol and press

v4 = v1;

% p3v3^gamma = p4v4^gamma | p4 = p3(v3/v4)^gamma
p4 = p3*(v3/v4)^gamma;
constant_c = p4*v4^gamma;
v_hl = engine_Kinematics1(bore,stroke,con_rod,cr,180,180);
p_hl = constant_c./v_hl.^gamma;

% Thermal efficency of an otto cycle
% Efficency = 1 - (v2/v1)^gamma-1 = 1 - 1/cr^gamma-1

x = cr^0.4;
eff_otto = (1 -1/x)*100;

% figure for actual otto cycle volume and pressure change
figure(1)

plot([v_c v_ha v_e v_hl v_c],[p_c p_ha p_e p_hl p_c],'color', 'r')
title('PV Plot for  actual varaition in V and P')
xlabel('Volume')
ylabel('Pressure')

% figure of ideal/therotical otto cycle
figure (2)
plot([v1 v2 v3 v4 v1],[p1 p2 p3 p4 p1],'color','b')
title('PV Plot for  therotical varaition in V and P')
xlabel('Volume')
ylabel('Pressure')

%Engine_Kinematics1

function [v] = engine_Kinematics1(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

4.BODY OF THE CONTENT:

Start the program writing by creating new script;

Always use the comments in the program for the validation of specific inputs and thus the commands have been use through the program;

*Inputs;

Started with the inputs for the base condition and the some state variables and also provided the engine kinematics parameters as show below.

*Calculating the engine cylinder swept volume and clearance volume as mentioned the questions.

This is very important lines in the program as all the next parameters are dependent on these line and please see the equation and set the variables as per the formulas as shown below;

*Use the equation to set the variables for determine all the remaining state condition parameters for the otto cycle.

This is to get the values for all the state variables of Otto cycle for, it is very formal as the equation are; use the variables to define the formula and so that we can record it in workspace for further use as shown below.

But before setting these condition for the ease of working and the clarity of program, create the second script to create the function, this function is to sum up the big part of program which is very repetitive and thus it very recommended for such function create other script and save it with the function name so that we can embed that function in our program line.

So I have created a function called “engine_Kinematics1”. This function is to plot the movement of piston inside the cylinder for set parameters in main script.

Please see the below screenshot for the function code and the use of it with its name in the main script;

*Code ending with thermal efficiency variable and formula;

Set the variables for the thermal efficiency and use the formula mentioned in the equations;

*Now graph plotting and create 2 figure for actual and theoretical cycle

Plot two figures one for theoretical PV plot and one for actual PV plot; the important things while mentioning the points for the plot, start with compression values for volume and pressure and with the same values in the coordinate line. Please see the below screen shot;

I have also gave the inputs for labels, title and the color of the lines.

5. CONCLUSION:

I can conclude that I have successfully wrote a script to calculate all the state variables for Otto cycle and also able to calculate the thermal efficiency of otto cycle.

Workspace output for state variables;

Efficiency;

Plots;

The difference is very much clear for the actual and theoretical PV plot as the work done is very different is to plots and also the losses were not considered in the theoretical plot as shown below;