Simulating Otto Air Standard Cycle

Otto cycle- The air standard Otto cycle analysis was developed to model the thermal efficiency of a spark ignition (SI) engine, which was originally envisioned as an engine in which thermal energy was added at constant volume by combustion of the working fluid. Typical P-V diagram for SI engine are shown in figure 1.

Fig.1 Typical P-V diagram for a SI engine

In order to develop a closed form solution for the indicated thermal efficiency, assumptions must be made to simplify the cycle. These assumptions are:

1. Air is assumed to be the sole working fluid
2. The specific heats of air are assumed to be constant
3. The intake and exhaust stroke are neglected
4. The four-process remaining in the cycle are idealized
   1. The compression process is assumed to be reversible and adiabatic (isentropic)
   2. Neglect the effects of spark timing, the finite duration of combustion, and the relatively large heat losses
   3. Expansion process is assumed to be reversible and adiabatic (isentropic)
   4. The exhaust blowdown process is replaced by an equivalent external heat rejection.

For a piston engine, the volume varies with crack angle, theta, via the relationship:

Equation 1. Volume depending of crank angle

Equation 2 is the air standard Otto cycle model for the indicated thermal efficiency for a SI. It indicates that the compression ratio dominates the indicated thermal efficiency.

Eq. 2 Thermodynamic efficiency of a SI

The first step to produce a P-V diagram is to calculate the volume at a specific cranck angle. the following function calculates the volume given a specific bore, stroke ,connecting lenght, compression ratio, start angle crank, and end angle crank.

function [v] = otto_kinematics (bore, stroke ,connecting_rod, compression_ratio,start_crank, end_crank)

%% Engine Geometry

cranck_radius= stroke/2;                    %crank radius
R= connecting_rod/cranck_radius;

v_swept= (pi/4)*bore^2*stroke;              % volume that piston displace
v_clearance= v_swept/(compression_ratio-1); % volume when piston at TDC

%%

theta= linspace(start_crank,end_crank,100);

term1=  0.5*(compression_ratio-1);
term2=  R+1-cosd(theta);
term3=  (  R*R - sind(theta).^2  ).^0.5;


v= (1+ term1.*(term2-term3)  ).* v_clearance;

end

The following code calls the function to obtain the volume at a specific crank angle. We only use this function when the volume changes in expansion and compression process. We use the angle 180 to 0 when is compression and for expansion 0 to 180. From the assumption, we conclude that p1v1^gamma=p2v2^gamma. From this equation, we can plot the expansion and compression part for the P-V diagram. Also, for the constant volume 2-3 and 4-1 point on the P-V is it just a straight line given the assumptions.

clear all
close all
clc
%% inputs
gamma= 1.4;
T_3= 2300;               %kelvin
%% State variables

P_1= 101325 ;            % atmospheric pressure in pascal
T_1= 500;                % kelvin

%% Engine Geometry
bore= 0.1;
stroke= 0.1;
connecting_rod= 0.15;
compression_ratio= 12;

%% Swept volume

v_swept= (pi/4)*bore^2*stroke;              % volume that piston displace
v_clearance= v_swept/(compression_ratio-1); % volume when piston at TDC

v_1= v_swept+v_clearance;                   % volume when piston at BDC
v_2= v_clearance ;                          % volume when piston at TDC


%% state variables at state 2
%  (p2*v2)^gamma=(p1*v1)^gamma

P_2 = P_1*compression_ratio^gamma;

% p1*v1/t1= p2*v2/t2

T_2= (P_2*v_2*T_1)/(P_1*v_1);
constant_c= P_1*v_1^gamma;
v_comp= otto_kinematics (bore, stroke ,connecting_rod, compression_ratio,180, 0);
P_comp= constant_c./v_comp.^gamma;


%% state variables at state 3

v_3=v_2;

P_3=  P_2*T_3/T_2;


constant_c= P_3*v_3^gamma;
v_exp= otto_kinematics (bore, stroke ,connecting_rod, compression_ratio,0, 180);
P_exp= constant_c./v_exp.^gamma;

%% state variables at state 4
v_4=v_1;

% p3v3^gama=p2v3^gamma

P_4= P_3 *(v_3/v_4)^gamma;

%% plot

figure (1)
hold on
plot ( v_1 ,P_1, '*' , 'color', 'red')
plot (v_comp , P_comp)
plot ( v_2 ,P_2, '*' , 'color', 'red')
plot ( v_3 ,P_3, '*' , 'color', 'red')
plot (v_exp , P_exp)
plot ( v_4 ,P_4, '*' , 'color', 'red')
plot([v_2 v_3], [P_2 P_3],'color', 'red')
plot([v_4 v_1], [P_4 P_1],'color', 'red')
title('P-V Diagram')
xlabel(' Volume ')
ylabel('Pressure ')

%% Thermal Efficiency

N_th= 1   -  (1)/(compression_ratio^(gamma-1))

P-V Diagram