PV Diagram of Air Standard Cycle using MATLAB.

AIM

1. To visualize PV diagram of air standard cycle.
2. Calculate the thermal efficiency of the engine.

Coding to visualize PV diagram and thermal efficiency of the engine is done by using MATLAB.

% PV diagram & Thermal efficiency of Air Standard Cycle.

%Inputs.
bore=.1; %Diameter of engine's cylinder (m).
stroke=.1; %Length from BDC to TDC (m).
con_rod=.15; %Length of connecting rod (m)
cr=12; %compression ratio
gamma=1.4;
p1=101325;
t1=500;
t3=2300;

v_swept=(pi/4)*bore^2*stroke;
v_clearance=v_swept/(cr-1);

v1=v_swept+v_clearance;
v2=v_clearance;

%State variables at poinnt 2(Isentropic comptession).

%p1*v1^gamma=p2*v2^gaama
p2=p1*cr^gamma;
%Ideal gas law, p1v1/t1=p2v2/t2
t2=p2*v2*t1/(p1*v1);

constant_c=p1*v1^gamma;
V_compression=engine_kinematics(bore,stroke,con_rod,cr,180,0);
%P_compression*V_compression^gamma=constant_c
P_compression=constant_c./(V_compression.^gamma);

%State variables at poinnt 3(contant volume heat addition)
v3=v2;
%p2v2/t2=p3v3/t3
p3=p2*t3/t2;

constant_c=p3*v3^gamma;
V_expansion=engine_kinematics(bore,stroke,con_rod,cr,0,180);
%P_expansion*V_expansion^gamma=constant_c
P_expansion=constant_c./(V_expansion.^gamma);

%State variables at poinnt 4(Isentropic expansion).
%p4v4^gamma=p3v3^gamma
v4=v1;
p4=p3*v3^gamma/(v4^gamma);
%p4v4/t4=p3v3/t3
t4=p4*v4*t3/(p3*v3);

 Thermal_efficiency_in_percentage =  (1-(t1/t2))*100 

figure(1)
hold on

plot(V_compression,P_compression,'linewidth',2,'color','r')
plot([v2 v3],[p2 p3],'linewidth',2,'color','b')
plot(V_expansion,P_expansion,'linewidth',2,'color','g')
plot([v4 v1],[p4 p1],'linewidth',2,'color','b')
xlabel('Volume (V)')
ylabel('Pressure (P)')

grid off

Coding of function engine_kinematics is shown below -

%Engine kinematics.
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*bore^2*stroke/4;
v_c = v_s/(cr-1);

%V /v_c = 1+0.5*(cr-1)[R+1-cosd(theta)-(R^2-sind(theta)^2]^0.5.

theta = linspace(0,720,500);
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

• p1,p2,p3,p4 = Presure at state point 1,2,3 and 4 respectively.
• v1,v2,v3,v4 = Volume at state point 1,2,3 and 4 respectively.
• t1,t2,t3,t4= Temprature at state point 1,2,3 and 4 respectively.
• v2=v3 (Constant volume heat addition)
• v4=v1 (Constant volume heat rejection)

Procedure

Step 1: All the values of required inputs like Bore, Stroke, Connecting rod, Compression ratio, p1,t1,t3 were taken into consideration.

Step 2: v1 and v2 were computed by calculating the swept volume(V_swept ) and clearance volume(V_clearance). Where v1 implies Volume at Top Dead Center(TDC) and v2 implies Volume at Bottom Dead Center(BDC).

Step 3: All the state variable at point 2, point3 and point 4 were calculated.

V_compression and V_expansion were calculated by using calling of the function engine_kinematics.

Step 4: Thermal efficiency of the cycle was computed by the formula (1-(t1/t2))*100.

Step 5: PV diagram of the cycle is shown by using plot command.

Error Faced:

Resolved:

The error was resolved by using the element-wise operator between constant_c and / as it is an array.

Results:

1) PV diagram is attached above.

2) Thermal efficiency of the engine carried out to be 62.98 %.