Air standard Cycle

Aim:

Solving an Otto Cycle and deriving a PV plot for it.

Governing Equations:

• The Ideal gas equation: PV=mR'T
• The ratio of total volume to clearence volume, which basically used to derive the inside engine kinematics: `V/V_c=1+0.5(cr-1)[R+1-costheta-(R^2-sin^2theta)^0.5`

Where:

P is Pressure in Pascals

V is volume in `m^3`

m is mass in kg

R' is universal gas constant in `J/((mol) K`

T is temperature in Kelvin 'K'

`theta` is in degrees

cr is compression ratio

Vc is clearence volume

Vs is swept volume

R is crank length in m

Objective:

To plot a Otto-cycle on a Pressure-Volume graph considering the engine kinematics inside a cylinder and also deriving the equation for efficiency of the given cycle.

Creating the function for engine kinematics.

The given equation is:

`V/V_c=1+0.5(cr-1)[R+1-costheta-(R^2-sin^2theta)^0.5`

%% Function for engine Kinematics

function [V]= engine_kinematics(bore,stroke,con_rod,cr,start_crank,end_crank)% function is created with varaible V by passing the inputs as 'bore','stroke','connecting rod length','compression ratio','start crank angle','last crank angle'

a=stroke/2; % is crank pin radius
R=con_rod/a; % crank length

Vs= pi/4 *bore^2*stroke; % Swept volume
Vc= Vs/(cr-1); %clearence volume

theta=linspace(start_crank,end_crank,100);

%Breaking the formula into several terms
term1=0.5*(cr-1);
term2=(R+1-cosd(theta));
term3=(R^2-sind(theta).^2).^(0.5);

V= (1+term1*(term2-term3)).*Vc;
plot(theta,V,'linewidth',3)
end

Creating the rest of the plot with given below codes:

close all
clear all
clc


%% Otto Cycle plot

% Inputs pistion cylinder arrengment
bore= 0.1; % cylinder bore radius in meter
stroke=0.1; %piston strole length in meters
con_rod= 0.15; %connecting rod length in meters
cr= 12; % commpresion ratio
gaama=1.4; %specific heat ratio

%operating parameters
P1= 101325; %Pressure at point1 in pascals
T1=500; %temperature at point1 in kelvin
T3=2300; %temperature at point3

%calculating the swept volume and clearence volume
Vs= pi/4 *bore^2*stroke %Swept volume
Vc= Vs/(cr-1) %clearence volume
V1= Vs+Vc %total cylinder volume
V2= Vc

%calculation for state2, from pion1 to 2 is reversible adiabatic condition
P2= P1*(V1^gaama/V2^gaama) %calculating P2 by adiabatic gas condition equation 
T2=T1*(V1/V2)^(gaama-1) %the reversible adiabatic property equation
C_constant= P1*V1^gaama;
V_compression= engine_kinematics(bore,stroke,con_rod,cr,0,180); %calling the function with alloted inputs
P_compression= C_constant./V_compression.^gaama; %calculating Pressure with same input function

%calculation for point3
V3=V2;
P3=P2*(T3/T2)

%calculation for state4, from pion3 to 4 is reversible adiabatic condition
V4=V1
P4=P3*(V3^gaama/V4^gaama) %calculating P2 by adiabatic gas condition equation
C_constant= P3*V3^gaama;
V_expansion= engine_kinematics(bore,stroke,con_rod,cr,180,0); %calling the function with alloted inputs
P_expansion= C_constant./V_expansion.^gaama; %calculating Pressure with same input function


%Plotting
plot(V1,P1,'*','color','r') %plotting P1 againt V1 marked as * in red
hold on
plot(V2,P2,'*','color','r') %plotting P2 againt V2 marked as * in red
plot(V_compression,P_compression)
plot(V3,P3,'*','color','r') %plotting P3 againt V3 marked as * in red
plot(V_expansion,P_expansion)
plot(V4,P4,'*','color','r') %plotting P4 againt V4 marked as * in red

plot([V2 V3],[P2 P3]) %connecting P2V2 WITH P3V3
plot([V1 V4],[P1 P4]) %connecting P2V2 WITH P3V3
title('Otto Cycle') %Giving the title to plot
xlabel('Volume') %x axis label
ylabel('Pressure') %y axis label

The PV plot foe Otto cycle:

Deriving the efficiency of Otto Cycle:

Otto cycle is also called as constant volume cycle, i.e heat is added at constant volume in this cycle.

To prove:

`eta=1/(cr)^(gamma-1)`

Proof:

`cr= V_1/V_2`

`eta`= work obtained/heat added

`eta=(mC_v(T_3-T_2) - mC_v(T_4-T_1))/(mC_v(T_3-T_2)`

or it can be further reduced to:

`eta=1-(T_4-T_1)/(T_3-T_2)`

and it though the property of reversible adiabatic process:

`T_3/T_4=(V_1/V_2)^(gamma-1) and V_1/V_2 = cr`

Therefore,

`eta=(T_4-T_1)/(T_4(cr)^(gamma-1)-T_1(cr)^(gamma-1)`

hence

`eta=1/(cr)^(gamma-1)`