Project

Air standard Cycle(Otto cycle) using PYTHON

SSaransh Dimri29 Nov 2020

Objective: -

Develop a code that can solve an otto cycle and make plots for it.

Principle: -

An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark-ignition engine. It is the thermodynamic cycle most commonly found in automobile engines. The Otto Cycle is an air-standard cycle that approximates the processes in petrol or diesel engines. It is based on constant volume heat addition (combustion) and heat rejection processes, and Isentropic compression and expansion

There are three air-standard cycles:

constant volume ‘combustion’ (Otto)
constant pressure ‘combustion’ (Diesel)
dual ‘combustion’ – this is a combination of constant volume and constant pressure combustion and results in a slightly more realistic cycle.

Solution Procedure: -

The basic Otto cycle has four processes:

•Isentropic compression
•Constant volume heat addition
•Isentropic expansion
•Constant volume heat rejection

Processes: -

Process 0 - 1 Intake Stroke
Process 1 - 2 Compression Stroke
Process 2 - 3 Ignition Phase
Process 3 - 4 Expansion Stroke
Process 4 - 1 Idealized Heat Rejection
Process 1 - 0 Exhaust Stroke

Governing Equations: -

Change of Volume Inside the combustion chamber of an engine -

`(V)/(Vc)=1+0.5(CR-1)[R+1-cosθ-(R^2-sin^2 θ)^0.5]`

Isentropic Process

`PV^r=constant`

Isenchoric Process

`P/T = constant`

Compression Ratio

`CR = (Vs + Vc )/(Vc)`

Thermal Efficiency

`η=1-1/(RC)^(γ-1)`

CODE for the Function used: -

def engine_kinematics(bore , stroke, con_rod , cr ,start_crank , end_crank):
	
	# a is crank pin radius
	a=stroke/2

	#R is the ratio for the connecting rod to the crank pin radius
	R = con_rod/a

    #swepth volume and clearance volume
	v_swept =math.pi*(1/4)*pow(bore,2)*stroke
	v_c= v_swept/(cr-1)


	st_cr=math.radians(start_crank)
	en_cr = math.radians(end_crank)

	num_val= 50

	dtheta = (en_cr - st_cr)/(num_val - 1)
	V=[]

	for i in range (0,num_val):
		theta = st_cr + i*dtheta

		term1 = 0.5*(cr-1)
		term2 = R + 1-math.cos(theta)
		term3 = pow(R,2) - pow(math.sin(theta),2)
		term3 = pow(term3,0.5)
        #Equation
		V.append((1+term1*(term2 -term3))*v_c)
		
	return V

Explanation of the above code: -

def engine_kinematics(bore, stroke, con_rod, cr,start_crank, end_crank) - function is used to make the code easy to implement, it is a group of statements that together perform a task
# - comment makes a program user friendly
a=stroke/2 - a is crankpin radius
R = con_rod/a - R is the ratio for the connecting rod to the crank pin radius
#swepth volume and clearance volume - comments
v_swept =math.pi*(1/4)*pow(bore,2)*stroke - calculation of the swept volume(m^3)
v_c= v_swept/(cr-1) - calculation of the clearance volume (m^3)
st_cr=math.radians(start_crank) - initializing value in radians to crank start angle
en_cr = math.radians(end_crank) - initializing value in radians to crank end angle
num_val= 50 - initializing the num val
dtheta = (en_cr - st_cr)/(num_val - 1)
V=[] - assigning a volume array
for i in range (0,num_val): - execution of the statements in the for loop till the value
theta = st_cr + i*dtheta - initializing the value of the theta
term1 = 0.5*(cr-1) - term 1 according to the equation
term2 = R + 1-math.cos(theta) - term 2 according to the equation
term3 = pow(R,2) - pow(math.sin(theta),2) - term 3 according to the equation
term3 = pow(term3,0.5) - term 3 according to the equation
#Equation - comments
V.append((1+term1*(term2 -term3))*v_c) - append the calculated value in the V array
return V - return the value of the V array as output when the function is called

Code of the Otto cycle Program including the above function: -

#Otto cycle simulator
import math
import matplotlib.pyplot as plt

def engine_kinematics(bore , stroke, con_rod , cr ,start_crank , end_crank):
	
	# a is crank pin radius
	a=stroke/2

	#R is the ratio for the connecting rod to the crank pin radius
	R = con_rod/a

    #swepth volume and clearance volume
	v_swept =math.pi*(1/4)*pow(bore,2)*stroke
	v_c= v_swept/(cr-1)


	st_cr=math.radians(start_crank)
	en_cr = math.radians(end_crank)

	num_val= 50

	dtheta = (en_cr - st_cr)/(num_val - 1)
	V=[]

	for i in range (0,num_val):
		theta = st_cr + i*dtheta

		term1 = 0.5*(cr-1)
		term2 = R + 1-math.cos(theta)
		term3 = pow(R,2) - pow(math.sin(theta),2)
		term3 = pow(term3,0.5)
        #Equation
		V.append((1+term1*(term2 -term3))*v_c)
		
	return V

#inputs

gamma =1.4
t3 =2300
#State variable 1
p1 = 101325
t1 = 500

#Engine geometric parameters
bore= 0.1
stroke = 0.1
con_rod = 0.15 #lenght of the connecting rod
cr= 12

#calculating the swept volume and clearence volume

v_swept =(math.pi/4)*pow(bore,2)*stroke
v_clearance= v_swept/(cr-1)
v1= v_clearance+v_swept

#state point 2
v2 = v_clearance

#v1/v2 is the compression ratio

'''p2v2^gamma = p1v1^gamma 
p2 = p1*(v1/v2)^gamma = p1*cr^gamma'''

p2= p1*pow(cr,gamma)


#p1v1/t1 = p2v2/t2 | t2 =p2*v2*t1/(p1*v1) 
#temperature at state point 2 is t2
t2 =p2*v2*t1/(p1*v1)
v_compression = engine_kinematics(bore,stroke,con_rod,cr,180,0)
constant_c = p1*pow(v1,gamma)

p_compression = []

for v in v_compression:
	p_compression.append(constant_c/pow(v,gamma))


#state point 3 determing pressure
v3=v2

#p3v3/t3=p2v2/t2 | p3 =p2*t3/t2
p3 = p2*t3/t2

v_expansion = engine_kinematics(bore,stroke,con_rod,cr,0,180)
constant_exp = p3*pow(v3,gamma)

p_expansion = []

for v in v_expansion:
	p_expansion.append(constant_exp/pow(v,gamma))

# state variables at state point 4
v4=v1

#p3v3^gamma = p4v4^gamma | p4 = p3(v3/v4)^gamma
k=v3/v4
p4 = p3*pow(k,gamma)

#p4v4/t4 = p3v3/t3

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

#Thermal Efficiency
x=1-gamma
thermal_effi = 1-pow(cr,x)
print('Thermal efficiency = ',thermal_effi)

#plotting

plt.plot([v2,v3],[p2,p3]) 
plt.plot(v_compression, p_compression)
plt.plot(v_expansion, p_expansion)
plt.plot([v4,v1],[p4,p1]) 
plt.xlabel('Volume')
plt.ylabel('Pressure')
plt.title('Air standard cycle(otto cycle)')
plt.show()

Explanation of the above code: -

#Otto cycle simulator - comments
import math - importing the math
import matplotlib.pyplot as plt - importing matplot library
#inputs -comments
gamma =1.4 - initializing gamma value as 1.4
t3 =2300 - initializing to as 2300 kelvin

#State variable 1 - comments

p1 = 101325 - 101325 initializing p1 as 101325 pascal
t1 = 500 - initializing t1 as 500 kelvin

#Engine geometric parameters - comments

bore= 0.1 - initializing with 0.1 meter
stroke = 0.1 - initializing with 0.1 meter
con_rod = 0.15 #lenght of the connecting rod - initializing connecting rod length as 0.15 meter
cr= 12 - initializing compression ration as 12

#calculating the swept volume and clearence volume - comments

v_swept =(math.pi/4)*pow(bore,2)*stroke - calculating the swepth volume
v_clearance= v_swept/(cr-1) - calculating the clearance volume
v1= v_clearance+v_swept - calculating v1 volume

#state point 2

v2 = v_clearance -calculating v2 volume

#v1/v2 is the compression ratio - comments

'''p2v2^gamma = p1v1^gamma
p2 = p1*(v1/v2)^gamma = p1*cr^gamma''' - comments

p2= p1*pow(cr,gamma) - calculating p2 pressure

#p1v1/t1 = p2v2/t2 | t2 =p2*v2*t1/(p1*v1) - comments for making the program user friendly
#temperature at state point 2 is t2 - comments

t2 =p2*v2*t1/(p1*v1) - calculating temperature t2
v_compression = engine_kinematics(bore,stroke,con_rod,cr,180,0) - calling of the function to calculate compression volume
constant_c = p1*pow(v1,gamma) - calculating constant c
p_compression = [] - assigning the pressure compression array
for v in v_compression: - execution of the statements in the for loop till the value
p_compression.append(constant_c/pow(v,gamma)) - append the calculated value in the compression pressure array

#state point 3 determing pressure - comments

v3=v2 initializing volume v3 to volume v2

#p3v3/t3=p2v2/t2 | p3 =p2*t3/t2 - comments

p3 = p2*t3/t2 - calculating p3 pressure
v_expansion = engine_kinematics(bore,stroke,con_rod,cr,0,180) - calling of the function to calculate expansion volume
constant_exp = p3*pow(v3,gamma) -calculating constant
p_expansion = [] - assigning the pressure expansion array
for v in v_expansion: - for executing a given statements
p_expansion.append(constant_exp/pow(v,gamma)) - append the calculated value in the expansion pressure array

# state variables at state point 4

v4=v1 initializing volume v4 to volume v1

#p3v3^gamma = p4v4^gamma | p4 = p3(v3/v4)^gamma

k=v3/v4 - ratio of v3 to v4 used to calculate p4 pressure
p4 = p3*pow(k,gamma) - calculating p4 pressure

#p4v4/t4 = p3v3/t3

t4 = (p4*v4*t3)/(p3*v3) - t4 temperature calculation

#Thermal Efficiency

x=1-gamma - initializing x as 1-gamma to use in calculating thermal efficiency
thermal_effi = 1-pow(cr,x) - calculating thermal efficiency
print('Thermal efficiency = ',thermal_effi) - printing thermal efficiency

#plotting

plt.plot([v2,v3],[p2,p3]) - plotting the volume and pressure v2 and v3 and p2 and p3
plt.plot(v_compression, p_compression) - plotting the compression volume and compression pressure
plt.plot(v_expansion, p_expansion) - plotting the expansion volume and expansion pressure
plt.plot([v4,v1],[p4,p1]) - plotting the volume and pressure v1 and v4 and p1 and p4
plt.xlabel('Volume') - labeling the x axis
plt.ylabel('Pressure') - labeling the y axis
plt.title('Air standard cycle(otto cycle)') - labeling the title of the plot
plt.show() - showing the plot in the output screen

Output: -

Thermal efficiency in the command window

Thermal efficiency = 0.6298 or 62.98%

The output of the otto cycle plot: -

Conclusion and Discussion: -

Thermal Efficiency was calculated to be 0.6299 or 62.99%
The graph was plotted between Pressure and volume for an otto cycle

Air standard Cycle(Otto cycle) using PYTHON

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Ship work like thisin your own course.

Talk to a career counsellor

Ship work like this
in your own course.