Air-Standard Cycle - Otto Cycle

Air-Standard Cycle – Otto Cycle

Aim:

Development of code for air standard otto cycle to extract the pressure vs volume diagram and to calculate the thermal efficiency of otto cycle.

Objective:

• To study the working of otto cycle.
• To develop the algorithm for calculation of state variable in otto cycle.
• To plot the state variable in the graph called pv-diagram.
• To calculate the thermal efficiency of cycle.

Description & Governing equations:

An engine is a machine which generate power through the combustion air-fuel mixture by providing heat at utmost pressure and temperature. During this combustion process the piston which is main part of engine goes back and forth to repeat the cycle. This cycle of intake, compression, combustion and expansion are governed by the specific air standard cycles. This air standard cycle provides the information about the process how they are happing inside the cylinder. The type engine decides which kind of cycle it is running in the process. As if it is petrol engine then it performs Otto cycle and if it is diesel engine then it performs diesel engine. Both the process is different but involve under the same piston-cylinder of engine.

Otto cycle is called constant volume cycle as the heat addition process happens at constant volume and heat rejection also. The process involves isentropic compression and expansion of air inside the cylinder. The heat addition is done by the spark plug in case of petrol engine and in case of diesel engine it is done by high pressured fuel spray following by auto-ignition phenomenon. The PV diagram tells us the pressure and volume condition at which the heat added and rejected during the process. There is T-S diagram which is temperature and entropy diagram for same PV diagram which tell us about the temperature changes with respect to constant entropy process. The actual PV diagram is totally different than that of air-standard cycle because of neglection of friction, mass factors of components and inertia of air flow. The main most important aspect of air-standard cycle and actual cycle is that in air-standard the working medium is ideal gas.

From the above PV diagram, we can see that the process is in cyclic nature. The process always starts from the heat rejection which means from point 4-1 and during this process the new volume of medium is taken inside the cylinder in case of actual cycle, but in above case the process starts from the point 0 which is consider as intake point and at point 1 the piston is supposed to be at BDC (Bottom Dead Center). In process 1-2 the isentropic compression process is done where fluid is compressed whose pressure and temperature is get increased. After the peak pressure and temperature are reached during the process 2-3 is heat addition process is done which is given by:

`Qs=mCv (T3-T2)`

After adding this heat at peak pressure condition the air get ignited and expand spontaneously which result in generation of high temperature and piston traveled from top dead center to bottom dead center. This process eventually generates the power which in turn gives the torque to crankshaft resulting in repeating of cycle. After the combustion isentropic expansion process is carried out from 3-4. The constant heat rejection process from 4-1 is done when exhaust valve is open which release the by-product of combustion which is given as:

`Qr=mCv (T4-T1)`

From these two equations we can find the efficiency of otto cycle as:

`ηotto=1-(T4-T1)/(T3-T2)`

During isentropic process the relation between temperature and volume can be written as:

`(T2)/(T1)= ((V1)/(V2))^((γ-1))`

Similarly, for process 3-4

`(T3)/(T4)= ((V4)/(V3))^((γ-1))`

To calculate each and every state point in PV diagram we need to know some basic terminology factor of engine such as bore, stroke, connecting rod length and compression ratio. Similarly, we need to calculate the clearance volume and swept volume which is given by:

`Swept Volume= (π/4)*(B)^2*Stroke`

And clearance volume is given as:

`C_(Volume)=(Swept Volume)/((Compression ratio-1) )`

The relation from which pressure is calculate as:

`(P2V2)^γ=(P1V1)^γ`

Explanation of Code:

1. The code starts from the basic inputs for calculation of state points which are bore, stroke, connecting rod length and compression ratio. The code is divided in two different parts one in which we calculate the state point and in other we calculate the volume swept by the piston thought the whole cycle.
2. We assume some atmospheric condition for pressure and temperature which are very much required for the calculation of other parameters. From these parameters we calculate the pressure at state point 2 and after that using ideal gas law equation, we calculate the temperature at point 2. As volume at point 1 is addition of swept volume and clearance volume which is further calculated from the bore, stroke and compression ratio. Similarly, volume at point 2 is nothing but clearance volume.
3. After getting the pressure, temperature and volume at point 1 & 2 we calculate the further state point by assuming the temperature at point 3 which is around 2300 K. We also know that as process 2-3 is constant volume heat addition the volume at point 2 is equals to volume at point. So, again by using ideal gas law we can calculate the pressure at point 3 which should be higher of any other pressure point in the otto cycle.
4. Considering another isentropic process to calculate the state point 4 where the volume at point 4 is equal to volume at point 1 due constant volume heat rejection process. The rest of parameter can be calculated by ideal gas law.
5. After calculation all the point, the engine kinematic need to be taken into consideration to calculate the actual volume swept by the piston. This is important because, if we plot all the state point it will show a linear line connection between each of the pressure and volume data which is not suitable.
6. To calculate the swept volume by the piston we are going to call the function called ‘engine_kinematics’ which have one output variable and 6 input parameters which are defined earlier. The relation by which the swept volume is calculated is given at end of 9th statement:
7. In which R is the ratio of connecting rod length to the crack radius. The above equation will give the swept volume from piston but still we need to calculate the swept volume for compression stroke and expansion stroke so that we can plot in the PV diagram.
8. For the compression stroke we can calculate the volume from the changing the crank angle from 180 to 0. Similarly, we know that PV^γ is constant from which we can calculate the pressure variation for compression stroke. Similar terminology can be used for expansion stroke in which crank angle changes from 0 to 180 degree.
9. After calculation of all this equation we put the plot accordingly. The code for whole process is given as below.

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

clear all
close all
clc

% Inputs for Air-standard Cycle
gamma = 1.4;
p1 = 101325;    %State variable at point 1 (Pressure)
t1 = 500;       %State variable at point 1 (Temperature)
t3 = 2300;      %Temperature at ignition point

%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*cr^gamma;

% According to ideal gas law- p1v1/t1 = p2v2/t2

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

% Piston kinematic equation function for compression stroke
const_c = p1*v1^gamma;
V_compression = piston_kinematics(bore,stroke,con_rod,cr,180,0);

% As we know P_compression*V_compression^gamma = const_c

P_compression = const_c./(V_compression.^gamma);

%State variable at point 3
v3 = v2;

% According to ideal gas law - p3v3/t3 = p2v2/t2
p3 = p2*t3/t2;

% Piston kinematic equation function for expansion stroke

const_c = p3*v3^gamma;
V_expansion = piston_kinematics(bore,stroke,con_rod,cr,0,180);

% As we know P_expansion*V_expansion^gamma = const_c

P_expansion = const_c./(V_expansion.^gamma);

% State variable at state point 4
v4 = v1;

% From idal gas law expression we can say that p3*v3^gamma = p4*v4^gamma  

p4 = p3*(v3/v4)^gamma; 

t4 = t3/((v4/v3)^(gamma-1));

% Thermal efficiency of the engine is given by

term1 = cr^(gamma-1);
n_otto = (1-(1/term1))*100;

% Work Output for otto cycle
rp = p3/p2;

W = (p1*v1/gamma-1)*(rp-1)*((cr^(gamma-1))-1);

figure(1)
hold on
plot(v1,p1,'*','color','r')
plot(V_compression, P_compression,'linewidth',1.5,'color','b')
plot(v2,p2,'*','color','r')
plot(v3,p3,'*','color','r')
plot(V_expansion,P_expansion,'linewidth',1.5,'color','b')
plot(v4,p4,'*','color','r')
axis([0 0.0009 0 5900000])
line([v2,v3],[p2,p3],'color','r')
line([v4,v1],[p4,p1],'color','r')
xlabel('Volume')
ylabel('Pressure in Pascals')
title('PV Diagram for Otto Cycle')

The function for Engine Kinematics is given as:

function [V] = piston_kinematics(bore,stroke,cond_rod,cr,start_crank,end_crank)

a = stroke/2;                   % Crank-pin radius
R = cond_rod/a;                 

v_s = pi/4*bore^2*stroke;       % Swept volume by piston
v_c = v_s/(cr-1);               % Clearance volume in combustion chamber
theta = linspace(start_crank,end_crank,720);

% the original equation which determined the volume moved by piston is
% given by = V/v_c =
% (1+0.5*(cr-1)*(R+1-cosd(theta)-(R^2-sind(theta)^2)^0.5))

V = v_c.*(1+0.5*(cr-1)*(R+1-cosd(theta)-(R^2-sind(theta).^2).^0.5));

end

Inference from Output:

From the plot it can be seen that the pressure point and volume points are in marker and the curves are in blue color for second graph. The straight line between the points are shown by red line. If we compare the first graph just by joining the line by each and every point then area under the curve for first graph will not give better performance or otherwise, we can say that the amount of volume which is taken inside is not enough. From the second graph it can be seen that the area under the curve is very less compared to first one.

Conclusion: -

In this program we have calculated the each and every state point in PV diagram also we have plotted the same. The maximum pressure in the engine is around 55.2 Mpa and thermal efficiency of engine is around 66.2%. The thermal efficiency of otto cycle is function of compression ratio and ratio of specific heat. As specific heat is assumed to be a constant for any working fluid, the efficiency is increased by increasing the compression ratio.