Air standard Cycle For The Subaru Ej225 Engine

PROJECT OBJECTIVE

This project is intended to model the otto-cycle or the air standard cycle for the ej255 Subaru engine.

The project aims to achieve the following.

• Give a basic thermodynamics understanding of the otto-cycle
• Give a basic understanding of engine parameters and the engine kinematic parameters
• Use Matlab to create plots for volume vs crank angle
• Create a plot for volume vs pressure for an engine cylinder
• Calculate the efficiency of the ej255 engine
• Creating a plot of efficiency vs intake temperature
• Creating a plot of efficiency vs intake compression ratio

REQUIRED THERMODYNAMIC PRINCIPLES FOR AN OTTO-CYCLE

State: State of a system is the physical condition that is defined by the system properties.

Process: Phenomena that occur due to a state change.

Adiabatic-process: Is a process in which no heat is lost or gained in a system. The adiabatic process gas equation is.

`pv^gamma=K`

`(p_1*v_1^gamma)=(p_2*v_2^gamma)`

`gamma=c_p/c_v`

Where

• P=presure
• v=volume
• gamma=specific heat ratio of the gas.
• c_p=Specific heat at a constant pressure
• c_v=Specific heat at constant volume

Ideal-gas: An ideal gas is a gas that satisfies the equation below.

`(p_1*v_1)/(t_1)=(p_2*v_2)/(t_2)`

Where

• p1 is pressure 1
• v1 is volume 1
• t1 is temparature 1
• p2 is pressure 2
• v2 is volume 2
• t2 is temparature 2

ENGINE GEOMETRIC PARAMETERS

fig 1.

B=bore: is the diameter of the cylinder

TDC= Top dead center. It is the maximum top position in which the piston can move. The crank angle is 0.

BDC= Bottom Dead center. It is the maximum bottom position in which the piston can move. The crank angle is 180.

l=connecting rod length. named con_rod in the Matlab program

s=stroke. This is the distance between the TDC and BDC.

a=crank radius it is equal to.

`a=1/2s`

theta=is the crank angle

Understanding the engine cylinder geometry and thermodynamics is necessary in order to create a mathematical model of the otto cycle.

OTTO CYCLE

An Otto cycle is a simplified thermodynamic analysis of the internal combustion engine. It is used to conduct elementary thermodynamics analysis of the internal combustion engine.

A detailed study of the ic with spark ignition would take into account many features these include.

• The friction of moving parts
• Internal combustion occurring within the cylinder
• Temperature gradient
• Heat transfer between the gases, cylinder, and coolant. For water-cooled engine
• Exhaust and intake valves opening and closing

The following are assumptions made while analyzing the otto cycle.

• A fixed amount of air is modeled as an ideal gas in the working fluid
• The combustion process is replaced by heat transfer from an external source
• There is no exhaust and intake process in an actual engine. The cycle is completed by a constant volume heat transfer process taking place while the piston is at the bottom dead center position.
• All process is internally reversible in the otto cycle.

The Otto cycle has four processes and four states

fig2

• Process 1-2: Is an adiabatic compression of air as piston moves from BDC to TDC
• Process 2-3: Is a constant volume heat transfer from an external source.While the piston is at TDC. This represents ignition of the fuel-air mixture
• Process 3-4: Is an adiabatic expansion. This represents the power stroke of the engine
• process 4-1: its a constant volume heat rejection process

SUBARU Ej 255 ENGINE PARAMETERS

For calculating the efficiency and programming in Matlab the PV diagram. The Subaru e255 engine will be used. The following parameters can be found on the Subaru website or forester ej255 manual. The Subaru ej255 is used on most Subaru cars. Its a four-cylinder boxer engine.

• con_rod=0.1305 m
• bore=0.0995 m
• cr=8.4
• stroke=79 m
• displacement=2457 cc

con_rod length, engine bore, and stroke, the values are in m.

cr is the compression ratio and its equal to

`c_r=(v_c+v_s)/v_c`

Where v_s=v_d which is the displaced volume as the piston moves from TDC to BDC.

v_c is the clearance volume which is the volume that is left when the piston is at TDC

displacement is the value of v_d or v_s multiplied by the number of cylinders in this case four.

The following values are the inputs. These are found through measurement or can be given to solve the ott-cycle.

• Temperature after combustion=2500 K =t3
• Pressure before compression =10000 =p1
• temperature before compression =300 K=t1
• gamma=1.4

All units are in Si units

Calculating Values Required for Pressure and Values for the states in PV Diagram

The following values are required for the PV diagram

State 1:Piston at BDC

p1=10000 is given

t1=300 k is given

v1=v_dispaced+v_clerance : V1 can be calculated if v_displaced and v_clearance is known

`v_1=v_c+v_d`

v_displaced is the displaced volume when the piston moves from TDC to BDC

v_dispaced=pi*r*r*h. Where:

• pi=3.142
• r=bore radius
• h=stroke

v_clearance=v_displaced/(cr-1) Where:

`v_c=v_d/(c_r-1)`

• v_clearance is the volume that remains after the piston is at TDC
• cr=compresion ratio

hence values of v1 p1 and t1 have been calculated

State 2 piston at TDC

p2=(p1*v1^gamma)/(v2^gamma) where:

`p_2=(p_1*v_1^gamma)/(v_2^gamma)`

P2 is the pressure at state 2. The formulae above is an expression of the ideal gas for an adiabatic process Which is.

`p1*v1^gamma=p2*v2^gamma`

calculating t2

`t_2=(p_2*v_2*t_1)/(p_1*v_1)`

where t2 is the temperature at state 2: The formulae above is an expression of the ideal gas formulae which is:

`(p1*v1)/(t1)=(p2*v2)/(t2)`.

State 3 piston at TDC

`v3=v2=v_c` : This is because the piston is TDC

`p3=(p2*t3)/(t2)` The formula is an expression of the ideal gas formula

t3= temperature after combustion. And is given.

State 4 piston at BDC

v4=v1 since the piston is at BDC

`p_4=(p_3*v_3^gamma)/(v_4^gamma)`

t4=(p4*v4*t3)/(p3*v3)

CYLINDER VOLUME AS A FUNCTION OF CRANK ANGLE

Since there are adiabatic compression and expression in process 1-2 and process 3-4.

P_compression and v_compression ,p_expansion and v_expansion need to be computed . Since the values vary with change in crank angle. The following formulae is used. The derivation of the formulae can be found at https://www.engr.colostate.edu/~allan/thermo/page2/page2.html.

vd=displaced volume

R=conection-rod/(stroke/2)

r=compression ratio

v_compression=is calculated from engine kinematics function.=V

p_compresssion=k./v_compression.^gamma: where

`p_1*v_1^gamma=K`

`pcomp=(k.)/(vcomp.^gamma)`

v_expansion=is calculated from engine kinematics function.

p_expansion=k./v_expansion.^gamma: where k=p3*v3^gamma

`p*v^gamma=K`

Plot showing how the volume changes due to change in crank angle from 0 to 180

fig 3

An increase in theta increases volume. But the relationship is not linear.

EFFICIENCY OF THE ENGINE

The efficiency of the ej255 engine can be calculated from the following two formulae.

formulae 1:

where u3,u2,u4,u1 are t1 t2 t3 and t4 respectively.

formulae 2:

where r= compression ratio

k=is a constant=gamma

The derivation of the above formulae can be found in the book referenced in the conclusion.

The efficiency was calculated by a Matlab program show in the code section and was found to be

57.3137 %

From the first formula, the parameter which can be easily controlled is t1 which is the intake air temperature. A Matlab function is created to plot how change in the intake temperature changes efficiency. The plot is given below.

fig 4

From the above plot intake air temperature, t1 is directly proportional to an increase in efficiency linearly. This is true if the number of oxygen molecules is not taken into account. The number of oxygen molecules decreases with an increase in air temperature. Hence reducing engine efficiency.

From the second formula, the parameter which can be easily controlled is r which is the compression ratio. A Matlab function is created to plot how a change in the compression ratio affects efficiency. The plot is given below.

fig 5

From the plot above an increase in the compression ratio leads to an increase in engine efficiency. Their relationship is not linear.

In the petrol engine, a high compression ratio can lead to engine knocking. This is a phenomenon caused by the self-ignition of the fuel-air mixture, this can lead to engine destruction.

Matlab Code

Main program to calculate and plot PV diagram and efficiency of the ej255 engine

% Otto cycle Matlab program for the Subaru ej255 engine
% Process 1-2: Is an adiabatic compression of air as piston moves from BDC to TDC
% Process 2-3: Is a constant volume heat transfer from an external source.While the piston is at TDC. This represents ignition of the fuel-air mixture
% Process 3-4: This is an adiabatic isentropic expansion. This represents the power stroke of the engine
% process 4-1: its a constant volume heat rejection process

% Ej255 parameters
pi=3.1416
con_rod=0.1305 
bore=0.0995 
cr=8.4
stroke=79 
displacement=2457 

% % con_rod, bore is the radius of bore and stroke are in m. Displacement is in cc. cr_compression ratio is
% % a ratio hence no units
% The following input value is given or can be computed by measurement

p1=10000 
t1=350
gamma=1.4

% calculation for state 1 where p1 v1 and t1 are required.The piston is at BDC

% Calculating displaced volume
v_d=pi*bore*bore*stroke
% V_c=clearance volume when piston is TDC
v_c=v_d/(cr-1)

v1=v_d+v_c
p1=10000


% calculation for state 2 where p2 v2 and t2 are required piston is at TDC
v2=v_c
p2=(p1*v1^gamma)/(v2^gamma)
t2=(p2*v2*t1)/(p1*v1)

% calculation for state 3 where p3 v3 and t3 are required piston is at TDC
v3=v_c
t3=2300
p3=(p2*t3)/(t2)


% t3 is temperature after combustion and it is given in Kelvin

% calculation for state 4 where p4 v4 and t4 are required piston is at  BDC
v4=v1
p4=(p3*v3^gamma)/(v4^gamma)
t4=(p4*v4*t3)/(p3*v3)

% Calculating p compression=p_comp and v compression=v_comp
v_comp=engine_kinematics(bore,stroke,con_rod,cr,180,0,180)
k=p1*v1^gamma
p_comp=k./v_comp.^gamma

% Calculating p Expansion=p_exp and v expansion=v_exp
v_exp=engine_kinematics(bore,stroke,con_rod,cr,0,180,180)
k=p3*v3^gamma
p_exp=k./v_exp.^gamma

% Plotting 
figure(1)
hold on
plot(v1,p1,'*','color','r')
plot(v_comp,p_comp)
plot(v2,p2,'*','color','r')
plot([v2 v3],[p2 p3],'color','g')
plot(v3,p3,'*','color','r')
plot(v_exp,p_exp)
plot([v4 v1],[p4 p1],'color','b')
plot(v4,p4,'*','color','r')
title('PV plot of Ej255 for an otto-cyccle')
xlabel({'volume in m^3',('x-axis')})
ylabel({'Pressure in pa',('y-axis')})

% engine efficiency
% Calculating eficiency1 based on temperature
eficiency1=temp_eficiency(t1,t2,t3,t4)

% Calculating eficiency12 based on compression ratio
eficiency2=cr_eficency(gamma,cr)

% Plotting to show how change in temparature changes efficiency of an engine
start_temp=23+273
end_temp=800
temp_val=20

% Calling the function to plot temp vs eficiecny
plot_temp_eff(start_temp,end_temp,temp_val,t2,t3,t4)

% Plotting to show how the change in compression ratio changes the efficiency of an engine
r_start=1.5
r_end=30
r_val=30

% Calling the function to plot temp vs eficiecny
cr_eff_plot(gamma,r_start,r_end,r_val)

Function for Matlab to calculate volume as crank angle changes.

function [V]=engine_kinematics(bore, stroke, con_rod, cr, start_crank, end_crank,values)
% where values are a number of values required
% The formulae to calculate volume as a function of crank angle is divided into three terms

a=stroke/2
R=con_rod/a

v_s=pi*bore*bore*stroke
v_c=v_s/(cr-1)


theta=linspace(start_crank,end_crank,values)

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

Matlab function for calculating engine efficiency using temperature

function E=temp_eficiency(t1,t2,t3,t4)

% FUNCTION FOR Calculating engine efficiency based on temperature

    E=(1-((t4-t1)/(t3-t2)))*100

end

function for calculating engine efficiency using the compression ratio

function E2=cr_eficency(k,r)

% fUNCTION FOR Calculating engine efficiency based on Compression ratio
    E2=(1-(1/(r^(k-1))))*100
end

Function for plotting efficiency based on how temperature changes

function [E]=plot_temp_eff(t_start,t_end,t_val,t2,t3,t4)

% Function for plotting efficiency based on how temp changes
t1_val=linspace(t_start,t_end,t_val)

E=(1-((t4-t1_val)/(t3-t2)))*100


figure(2)
plot([t1_val],[E])
grid on
grid minor
title('Temperature vs Eficiency plot')
xlabel({'Temparature in Kelvins',('x-axis')})
ylabel({'Eficiency as a percentage',('y-axis')})


end

function for plotting efficiency based on how compression ratio changes

function [E2]=cr_eff_plot(k,r_start,r_end,r_val)

% fUNCTION FOR Calculating engine efficiency based on Compression ratio_
    cr=linspace(r_start,r_end,r_val)
    E2=(1-(1./(cr.^(k-1))))*100
    
figure(3)
plot([cr],[E2])
grid on
grid minor
title('Compresion Ratio vs Eficiency plot')
xlabel({'Compresion ratio',('x-axis')})
ylabel({'Eficiency as a percentage',('y-axis')})

end

Matlab plots

PV plot for ej255 engine for an otto-cycle

fig 6

the red curve represents the process 1-2

the green line represents the process 2-3

the cyan curve represents the process 3-4

the blue line represents the process 4-1

Conclusion

The efficiency of the Matlab program gives a value of 57.31 %. Which is not true since most engine efficiencies range from 30 to 40. This efficiency is the thermodynamic efficiency since it doesn't take into account factors such as friction and other factors when the engine is running.

The change in efficiency vs temperature shows that an increase in intake temperature causes an increase in efficiency and their relationship is linear. This is not true since the increase in intake air temperature would cause a decrease in oxygen molecules hence causing a low efficiency.

All formulae and some facts are from the book: Fundamentals of Engineering Thermodynamics. ref below

Figures 2 to 6 are generated using the above Matlab code.

Figure 1 can be found website given below.

https://www.engr.colostate.edu/~allan/thermo/page2/page2.html

Ref:

Michael J. Moran
The Ohio State University

Howard N. Shapiro
Iowa State University of Science and Technology

Book: Fundamentals of Engineering Thermodynamics