Air Standard Cycle (PV diagram visualizer and thermal efficiency)

Aim:

To generate a simulation of the otto cycle the with help of pv plot and to find thermal efficiency.

Theory:

Otto Cycle

Otto cycle basically represents four strokes of the piston inside the cylinder, which are isentropic compression, constant volume heat addition, isentropic expansion, and constant volume heat rejection.

Figure 1- Showing the different strokes performed by a piston inside an IC engine.

PV Diagram

Consider a gas sealed in a container or a cylinder with a piston. When the force is applied to the piston it compresses the gas inside the cylinder the volume of the gas decreases and pressure increases, when the force is released from the piston the gas exerts a force on the piston and pushes it back(shown in figure 2).

When we subject the gas to these thermodynamics processes, the pressure and volume of the gas can change.

A convenient way of visualizing these changes of pressure and volume is by using the Pressure-Volume diagram or simply PV diagram.

The pressure is represented by the y-axis and volume by the x-axis. Each point on a pv diagram represents the different state of a gas.

Figure 2

Figure 3- Graph showing the different states of gas in terms of pressure and volume.

Mathematical Equation:

`V /(vc)= 1+½(rc-1)[R+1-cos(theta)-(R^2-sin(theta)^2)^(½)]`

Where, P- Pressure

V- Total Volume of cylinder

T- Temperature

`vc`- Clearance volume

`vs`- Swept volume

Thermal efficiency

`eta = (1 - 1/(cr^(gamma-1)))100`

cr- Compression ratio

Coding:

1. Creating a function

% engine kinenamatics
function [v] = engine_kinematics(bore, stroke, con_rod, cr, start_crank, end_crank)  
 
 % inputs 
 
 a = stroke/2;
 R = con_rod/a;
 
 theta = linspace(start_crank,end_crank,200);
 v_s = (pi/4)*bore^2*stroke;
 v_c = v_s/(cr-1);
 
 % Formula
 % v/vc = 1 + 0.5(cr-1) * [R+1-cos(theta) - (R^2 - Sin(theta)^2)]^0.5
 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

Coding for pv plot and to find thermal efficiency

% Otto cycle
 
clear all
close all
clc

% Inputs
gamma = 1.4
t3 = 2000

%state variables
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 variables at state point 2
 % p2v2^gamma = p1v1^gamma
 % p2 = p1*(v1/v2)^gamma p2 = p1*cr^gamma
 p2 = p1*cr^gamma;

 % p1v1/t1 = p2v2/t2 t2 = p2*v2*t1/(p1*v1)
 t2 = p2*v2*t1/(p1*v1);

 % calling function
 constant_c = p1*v1^gamma;
 V_compression = engine_kinematics(bore, stroke, con_rod, cr, 180, 360) 
 P_compression = constant_c./V_compression.^gamma
  
 % State variables at state point 3
 v3=v2

 % p3v3/t3 = p2v2/t2  p3 = p2*t3/t2
 p3 = p2*t3/t2
 constant_c = p3*v3^gamma;
 V_expansion = engine_kinematics(bore, stroke, con_rod, cr, 0, 180)
 P_expansion = constant_c./V_expansion.^gamma
 
 % state variables at state piont 4
 v4 = v1
 
 % p3v3^gamma = p4v4^gamma  p4 = p3(v3/v4)^gamma
 p4 = p3*(v3/v4)^gamma
 
 
 % Thermal Efficiency or air standard efficiency of otto cycle
 thermal_eff = (1 - 1/cr^(gamma-1))*100;
 
 figure(1)
 hold on
 plot(v1,p1,'*','color','r')
 plot(V_compression,P_compression)
 plot(v2,p2,'*','color','r')
 plot(v3,p3,'*','color','r')
 plot(V_expansion,P_expansion,'color','g')
 plot(v4,p4,'*','color','r')
 plot([v2 v2],[p2 p3])
 plot([v1 v1],[p1 p4])
 xlabel('Volume')
 ylabel('Pressure')

Output:

Results and Conclusions:

1. PV plot is obtained by using MATLAB coding and the thermal efficiency is calculated, where `eta = 63%`

2. The volume of a cylinder at any given crank angle can be calculated by this program.