Air Standard Cycle

• Programe for ploting the (PV) diagram.

clear all
close all 
clc


%input
gamma=1.4
t3=2300


%state variable
p1=101325
t1=500

%engine geometric parameters
bore=0.1;
stroke=0.1;
con_rod=0.15;
cr=12;

 %calculating the swept volume and the clearance volume
 v_swept=(pi/4)*bore^2*stroke;
 v_clearance=v_swept/(cr-1);
 
 v1=v_swept+v_clearance
 v2=v_clearance
 
 %state variable at state 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)
 t2=p2*v2*t1/(p1*v1);
 
 
 constant_c=p1*v1^gamma;
 v_compression=piston_otto_cycle(bore,stroke,con_rod,cr,180,0);
  p_compression=constant_c./v_compression.^gamma;
  
  
 % state varibale for state point 3
 v3=v2
 %|p3=p2*t3/t2
 p3=p2*t3/t2
 
 constant_c=p3*v3^gamma;
 v_expansion=piston_otto_cycle(bore,stroke,con_rod,cr,0,180)
  p_expansion=constant_c./v_expansion.^gamma;
 
 %state variable at state point 4 
 v4=v1
 %|p4=p3*(v3/v4)^gamma
 p4=p3*(v3/v4)^gamma
 
 figure(1)
 hold on
 plot(v1,p1,'*','color','r')
 plot(v_compression,p_compression,'color','r')
 plot(v2,p2,'*','color','r')
 plot(v3,p3,'*','color','r')
 plot(v_expansion,p_expansion,'color','r')
 plot(v4,p4,'*','color','r')
 plot([v2 v3],[p2 p3],'color','b')
 plot([v4 v1],[p4,p1],'color','b')
 xlabel('volume')
 ylabel('pressure')

• EXPLANATION
• Starting the programe with (clear all,close all,clc)for clearing the screen.
• We have given the input and the starting variable such as (gamma,t3,p1,v1) as shown in the programe.
• Put the engine geometry parameter (bore,stroke,con_rod,cr).
• In the 1st stage we have v1 and v2 as the swept volume ,and the clearance volume.
• In the state 2 we have t2 which is showed as .

t2=p2×v2×t1÷(p1×v1)

• We are now providing the function for proper ploting of (PV) diagram.

 function [v]=piston_otto_cycle(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 

• In this we have defined the (V) for the given equation whis is subdivided in the different terms as (term1 ,term2,term3).
• Put this in the equation of (v) as shown in the programe.
• Now we call this (V) function in our otto cycle programe as a compression in stage 2,and same for the expansion process.
• Same as for the stage variable 3 and 4 as.

p4=p3×〖v3÷v4〗^γ

• Now plot the values of each state of (1,2,3,4).
• We get our desired output as a (PV) diagrame.

• ERRORS

• As shown in the error we have used the undefined line as used the upper case (V) so we used the (v) as this was defined previously.

• As shown the stroke is not defined so we used command window and used v_compression=piston_otto_cycle(0.1,0.10,0,12,180,0) and defined the values of all (bore,stroke,con_rod,cr).

• Programe for getting output of thermal efficiency.

clear all
close all
clc

%thermal efficiency of an otto cycle

cr=linspace(0,10,100);
gamma=1.5;

%efficiency
etta=1-1./(cr).^(gamma-1);

%plot
plot(cr,etta,'linewidth',1,'color','r');
grid on
xlabel('compression ratio')
ylabel('thermal efficiency')

• For calculating the thermal efficiency we used the following programe.
• Result