Engine Kinematics 4.4L V8 gasoline engine (using Otto cycle)

AIM: To obtain a plot for ‘Otto cycle’ using geometric parameters of a 4.4L V8 gasoline engine using MATLAB.

OBJECTIVES:

• To thoroughly understand the working of an Otto cycle and to able to derive meaningful observations from its P-V diagram.
• Write MATLAB code that can solve an Otto cycle and make plots (of P-V diagram) for it.
• To obtain output as the thermal efficiencyof the engine.

THEORY:

Otto cycle is the one of most common thermodynamic cycles that can be found in automobile engines and describes the functioning of a typical spark ignition piston engine. In an ideal Otto cycle, the system executing the cycle undergoes a series of four internally reversible processes: two isentropic (reversible adiabatic) processes alternated with two isochoric processes.

1. Isentropic compression (compression stroke) – The gas (fuel-air mixture) is compressed adiabatically from state 1 to state 2, as the piston moves from bottom dead centre to top dead centre. The surroundings do work on the gas, increasing its internal energy (temperature) and compressing it. On the other hand, the entropy remains unchanged. The changes in volumes and it’s the ratio (V₁ / V₂) is known as the compression ratio.

1. Isochoric compression (ignition phase) – In this phase (between state 2 and state 3) there is a constant-volume (the piston is at rest) heat transfer to the air from an external source while the piston is at rest at top dead centre. This process is intended to represent the ignition of the fuel–air mixture injected into the chamber and the subsequent rapid burning. The pressure rises and the ratio (P₃ / P₂) is known as the “explosion ratio”.

1. Isentropic expansion (power stroke) – The gas expands adiabatically from state 3 to state 4, as the piston moves from top dead centre to bottom dead centre. The gas does work on the surroundings (piston) and loses an amount of internal energy equal to the work that leaves the system. Again, the entropy remains unchanged. The volume ratio (V₄ / V₃) is known as the isentropic expansion ration, but for Otto cycle, it is equal to the compression ratio.

1. Isochoric decompression (exhaust stroke) – In this phase the cycle completes by a constant-volume process in which heat is rejected from the air while the piston is at bottom dead centre. The working gas pressure drops instantaneously from point 4 to point 1. The exhaust valve opens at point 4. The exhaust stroke is directly after this decompression. As the piston moves from bottom dead centre (point 1) to top dead centre (point 0) with the exhaust valve opened, the gaseous mixture is vented to the atmosphere and the process starts anew.

Geometric parameters considered for the code:

Abbreviations used in the code:

1. gamma = Adiabatic index
2. t1, t2, t3 = Temperatures at points 1, 2, and 3 respectively
3. p1, p2, p3, p4 = Pressures at points 1, 2, 3, and 4 respectively
4. v1, v2, v3, v4 = volumes at points 1, 2, 3, and 4 respectively
5. connecting_rod = Length of connecting rod
6. comp_ratio = Compression ratio
7. Vs = Swept volume
8. Vc = Clearance volume
9. v_comp and p_comp = Volume and pressure at end of compression
10. v_exp and p_exp = Volume and pressure at end of expansion
11. theta = Crank angle

Formulae used:

1. Ideal Gas Law for Isentropic process:

`p*V^γ=const`

2. Ideal Gas law for Isochoric process:

`p/T=const`

3. Governing eq of engine kinematics:

`V=Vc*[1+1/2 (C_r-1)*(R+1-cos⁡〖θ-〗 √(R^2-〖sin⁡θ〗^2 ))]`

Where: Vc = Clearance volume

Cr = Compression Ratio

R = Ratio of connecting rod length to half of stroke length

ϴ = Crank angle

4. Thermal Efficiency:

`η_t=1-(t_4-t_3)/(t_2-t_1 ) =1- (V_2/V_1 )^(γ-1)=1-1/(C_r)^(γ-1)`

MATLAB CODES:

1) Function code of Governing Eqaution:

% Governing eq of engine kinematics curve
function[V] = engine_kinematics(bore, stroke, connecting_rod, comp_ratio, start_crank, end_crank)

a = stroke/2;           %crank radius
R = connecting_rod/a;

Vs = (pi/4)*bore^2 * stroke;
Vc = Vs/(comp_ratio-1);

% theta is the crank angle
theta = linspace(0,180,180);

%spliting the governing eq to make it look simple
term1 = 0.5*(comp_ratio-1);
term2 = R + 1 - cosd(theta);
term3 = (R^2 - sind(theta).^2).^0.5;

%putting the eq together
V = (1 + term1.*(term2 - term3)).*Vc;
end

2) Actual code used to solve and plot the Otto cycle:

clear all 
close all
clc
% Otto cycle calculations

%Inputs
gamma = 1.4;    %adiabatic index
t3 = 2300;      %(K)

%State variables
p1 = 101325;                    %(Pa)
t1 = 500;                       %(K)

%Engine geometric parameters
bore = 0.092;                   %(mm)
stroke = 0.0827;                %(mm)
connecting_rod = 0.143;         %(mm)
comp_ratio = 10;

%Calculating Vs and Vc
Vs = (pi/4)*bore^2 * stroke;    %swept volume(m^3)
Vc = Vs/(comp_ratio-1);         %clearance volume(m^3)

v1 = Vs + Vc;
v2 = Vc;

%State variables at point 2
const_c = p1*v1^gamma;
    % p2 = p1*(v1/v2)^gamma,
    % v1/v2 = comp_ratio
p2 = p1*comp_ratio^gamma;
    % p1*v1/t1 = p2*v2/t2
t2 = (p2*v2*t1)/(p1*v1);    

v_comp = engine_kinematics(bore,stroke,connecting_rod,comp_ratio, 180,0);
p_comp = const_c./v_comp.^gamma ;

%State variables at point 3
v3 = v2;                    
     % p3v3/t3 = p2v2/t2
     % p3 = p2v2t3/t2v3
p3 = p2*t3/t2;             

const_c = p3*v3^gamma;

v_exp = engine_kinematics(bore,stroke,connecting_rod,comp_ratio, 0, 180);
p_exp = const_c./v_exp.^gamma  ;

%State variables at point 4
v4 = v1;
      % p3v3^gamma = p4v4^gamma
p4 = p3*(v3/v4)^gamma;      

%Thermal Efficiency
thermal_eff = 1-(1/comp_ratio^(gamma-1))

% Ploting
hold on
grid on
plot(v_comp, p_comp, 'color', 'r')
plot([v2,v3],[p2,p3], 'color', 'g')
plot(v_exp, p_exp, 'color', 'b')
plot([v1,v4],[p1,p4], 'color', 'y')

xlabel('Volume [m^3]')
ylabel('Pressure [Pa]')
legend('Issentropic Compression','Heat addition','Issentroppic Expansion','Heat rejection')

EXPLAINATION OF THE CODE:

1. First of all since we have to use the complicated ‘governing eq’, to keep our code neat and tidy we need to create a separate function(named engine_kinematics) for it.
2. Now to fet started with our main code we frist initialize the constants and input variables such as gamma, p1, t1 and the geometric parameters of the engine.
3. Then we calculate the much needed values of swept and clearance volumes.
4. Thus we can now calculate pressure and volume values from ideal gas equations for the other states.
5. For compression and expansion stroke we need to make use of governing eq to the right curves.
6. Next we calculate the thermal efficiency by using the standard formula.
7. Lastly to plot all the calculated values on a P-V diagram we need to carefully choose the values for compression and expansion stroke (calculated using governing eq.). since the are arrays with each value calculated with respect to crank angle, we get a smoot curve.

MATLAB OUTPUTS:

1) P-V diagram plot of the calculated values.

2) The thermal efficiency of the engine:

CONCLUSION:

We studied the Otto cycle in detail and hence were able to plot a nice curve(P-V diagram) for the given V8 engine’s geometric parameters. Also we found that its thermal efficiency is 60.19%.