Air standard Cycle - Otto Cycle

AIM

To solve an Otto cycle problem and make plots for it.

OBJECTIVES

• To create a Pressure-Velocity (PV) diagram of the Otto cycle.
• To find out the the thermal efficiency of the engine.

THEORY

An Otto cycle is an ideal thermodynamic cycle which teaches us about the functioning of a typical spark ignition piston engine by having constant volume heat addition. It is the thermodynamic cycle most commonly found in automobile engines.

The Otto cycle is formed by replacing both isothermal processes of Carnot cycle by two isochoric processes i.e., constant volume processes.

Terminologies required for determining the volume and the measurements of the engine cylinder are as follows,

• Bore`(d)` : It is the inside diameter of engine cylinder. It is called Bore as it is made through boring process.
• Stroke`(l)` : It is the linear distance through which the piston travels between the extreme upper and lower positions of the piston. The Stroke is equal to two times the crank radius.
• Top Dead Centre (TDC): When the piston is at the topmost position of the cylinder during its travel, that position is called the Top Dead Centre. At this position the piston velocity is zero and the piston reverses its direction of motion to travel downwards. It is the dead centre when the piston is farthest from the crankshaft.
• Bottom Dead Centre (BDC): When the piston is at the bottom-most position of the cylinder during its travel, the position is called the Bottom Dead Centre. At this position the piston velocity is zero and the piston reverses its direction of motion to travel upwards. It is the dead centre when the piston is nearest to the crankshaft.
• Swept Volume : It is the volume swept through by the piston in moving between the TDC and the BDC. The formula for calculating the swept volume is as follows,

`V_s=π/4 d^2 l`

• Clearance Volume : When the piston is at the TDC position, the volume contained in the cylinder above the top of the piston is called the Clearance Volume. The piston cannot occupy any part of this volume and always keep this volume clear.

`V_c=V_s/(cr-1)` Where,`cr=`Compression Ratio

• Cylinder Volume : It is the sum of the swept volume and the clearance volume.

`V=V_s+V_c`

• Compression Ratio : It is the ratio of total cylinder volume to clearance volume.

`cr= V/V_c =(V_s+V_c)/V_c`

Figure: Otto cycle

The Otto cycle consists:

1.Process 0–1

An amount of air is sucked into piston-cylinder arrangement under constant pressure condition i.e., at atmospheric pressure.

2.Process 1–2

An isentropic (reversible adiabatic) compression of the working gas takes place as the piston moves from crank end (bottom dead centre, BDC) to top dead centre (TDC).

State Variables in this process will be calculated as follows,

`P_1`will be atmospheric pressure i.e., 101325 Pa

`V_1=V_s+V_c`

`T_1`as given

`V_2=V_c`

As this is an isentropic process,

`P_1 V_1^γ=P_2 V_2^γ`

or, `P_2=P_1 (V_1⁄(V_2))^γ`

or, `P_2=P_1 (cr)^γ`

As per ideal gas law,

`(P_1 V_1)/T_1 =(P_2 V_2)/T_2`

or, `T_2=(P_2 V_2 T_1)/(P_1 V_1 )`

3.Process 2–3

In this process heat is added at constant volume (isochoric process) as the piston is at rest at Top Dead Centre.

As the volume remains constant in this process,

`V_2=V_3`

As per ideal gas law,

`(P_2 V_2)/T_2 =(P_3 V_3)/T_3`

or, `P_3=(P_2 T_3)/T_2`

`T_3` as given

Heat added to the cycle during this isochoric process 2 -3 is,

`Q_A=mc_v (T_3-T_2)`

Where,

`m` = mass of the working gas

`c_v` = specific heat of the working gas in constant heat

4.Process 3-4

An isentropic (reversible adiabatic) expansion of the working gas takes place as the piston is pushed towards the crank end (bottom dead centre, BDC) by the application of high pressure. In this process, work is done by the system on the piston.

As this is an isentropic process,

`P_3 V_3^γ=P_4 V_4^γ`

or, `P_4=P_3 (V_3/V_4 )^γ`

5.Process 4-1

In this process heat is rejected at constant volume (isochoric process) as the piston is at rest at Bottom Dead Centre.

As the volume remains constant in this process,

`V_4=V_1`

Heat rejected by the cycle during this isochoric process 4 -1 is,

`Q_R=mc_v (T_4-T_1)`

6.Process 1–0

The gaseous mixture is released to the surrounding at constant pressure and the whole process starts again.

• Calculation of Thermal Efficiency

Efficiency of the Otto cycle is calculated as,

`η=1-(1/(cr^(γ-1) ))`

• Engine kinematics

The cylinder volume, which is dependent on the crank angle, is given by,

`V/V_c =1+1/2 (cr-1)[R+1-cos⁡θ- (R^2-sin^2⁡θ)^(1/2) ]`

The above formula is to be used for calculation of volume at compression and expansion process.

Software used:- MATLAB R2014a

PROGRAM

I.Program to define the function for Engine Kinematics

%Program for Function definition

function [V] = engine_kinematics(bore,stroke,con_rod,cr, start_crank,end_crank)
%Defining the input arguments
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);     

%Defining the output argument cylinder volume 'V' 
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

II.Program to solve an Otto cycle and create a PV diagram of the Otto cycle and find out the thermal efficiency of the engine

%Program to solve an otto cycle and create a PV diagram of the otto cycle
%and find out the thermal efficiency of the engine

clear all 
close all
clc

%inputs                            %Assigning the values of Specific Heat Ratio and temperature at point 3
gamma = 1.4;
t3 = 2300;

%Engine geometric properties       %Assigning the values of Engine geometric properties
bore = 0.1;
stroke = 0.1;
con_rod = 0.15;
cr = 12;

%Calculating the swept volume and clearance volume
v_swept = (pi/4)*bore^2*stroke;    
v_clearance = v_swept/(cr-1);

%State variables at state point 1-

p1 =101325;                       %Assigning the values of pressure and temperature at state point 1        
t1 = 500;
v1 = v_swept+v_clearance;         %Calculating the value of volume at state point 1

%State variables at state point 2-

%p2*v2^gamma = p1*v1^gamma => p2 = p1*(v1/v2)^gamma => p2 = p1*cr^gamma
p2 = p1*cr^gamma;                 %Calculating the value of pressure and volume at state point 2
v2 = v_clearance;
%p1*v1/t1 = p2*v2/t2  => t2 = p2*v2*t1/(p1*v1)
t2 = p2*v2*t1/(p1*v1);            %Calculating the value of temperature at state point 2

%p2*v2^gamma = p1*v1^gamma = constant_c
constant_c = p1*v1^gamma;   
%Calling the function 'engine_kinematics' to get the P-V variation in process 1-2
V_compression = engine_kinematics(bore,stroke,con_rod,cr,180,0);
P_compression = constant_c./V_compression.^gamma;

%State variables at state point 3-

v3 = v2;                          %Calculating the value of volume at state point 3
%p3*v3/t3 = p2*v2/t2 => p3 = p2*t3/t2
p3 = p2*t3/t2;                    %Calculating the value of pressure at state point 3


%p3*v3^gamma = p4*v4^gamma = constant_c
constant_c = p3*v3^gamma;  
%Calling the function 'engine_kinematics' to get the P-V variation in process 3-4
V_expansion = engine_kinematics(bore,stroke,con_rod,cr,0,180);
P_expansion = constant_c./V_expansion.^gamma;

 
%State variables at state point 4-

v4 = v1;                          %Calculating the value of volume at state point 4
%p3*v3^gamma = p4*v4^gamma => p4 = p3*(v3/v4)^gamma 
p4 = p3*(v3/v4)^gamma ;           %Calculating the value of pressure at state point 4


%Plotting                         %To create a P-V diagram of the otto cycle

figure(1)                         %To create a blank figure and put the plot in it
hold on                           %To put all the plots in the same figure
plot(v1,p1,'*','color','r')                           %To get the state point 1 of P-V diagram
plot(V_compression,P_compression,'color','b')         %To get the curve of prceess 1-2 of P-V diagram
plot(v2,p2,'*','color','r')                           %To get the state point 2 of P-V diagram
plot(v3,p3,'*','color','r')                           %To get the state point 3 of P-V diagram
plot([v2 v3],[p2 p3],'color','g')                     %To get the curve of prceess 2-3 of P-V diagram
plot(V_expansion,P_expansion,'color','r')             %To get the curve of prceess 3-4 of P-V diagram
plot(v4,p4,'*','color','r')                           %To get the state point 4 of P-V diagram
plot([v4 v1],[p4 p1],'color','k')                     %To get the curve of prceess 4-1 of P-V diagram
text(v1+.00001,p1,'1');                               %To label the state points from 1 to 4
text(v2+.00001,p2,'2');
text(v3+.00001,p3,'3');
text(v4+.00001,p4,'4');
xlabel('Volume');                                    %To label the x-axis as 'Volume'
ylabel('Pressure');                                  %To label the y-axis as 'Pressure'
title('P-V Diagram for Otto Cycle');                 %To give title for the plot



%Finding out the thermal efficiency

thermal_efficiency = 1-(1/cr^(gamma-1)) 

RESULTS

• Plot

• Thermal efficiency of the engine

thermal_efficiency =

    0.6299

CONCLUSION

1. A MATLAB program is constructed for solving an Otto cycle problem and creating a graphical representation of Pressure-Volume variation of the Otto cycle. The related terminologies are defined. Processes involved in the Otto cycle are elaborated.
2. Another MATLAB program is constructed to define the engine kinematics involved in the compression and expansion processes.
3. Finally Thermal efficiency of the engine is calculated using MATLAB.