Week 2 Air standard Cycle

Otto Cycle

Aim- To study otto cycle

Objective-

• To study otto cycle
• To write program in python to get P-V diagram of otto cycle
• To calculate thermal efficiency of otto cycle

Theory-

Otto cycle was developed by Nicolas Otto in 1861 for spark ignition engines used petrol/gasoline as a fuel.

Otto cycle consists of total five processes

1. 0-1 Suction Stroke-Air fuel mixture taken into engine cylinder
2. 1-2 Compression Stroke- Air and fuel mixture compressed adiabatically
3. 2-3 Ignition Stroke- Constant Volume Heat addition i.e burning of fuel by ignition
4. 3-4 Expansion Stroke-Air fuel mixture will expand
5. 4-0 Exhaust Stroke- Burnt out gases will be removed from engine cylinder

• 

Fig1. Otto Cycle

Problem Statement-

4S SI engine have following parameters

Bore=0.15m

Stroke=0.15m

Connecting rod Length =0.25m

Compression ratio=6

Atmospheric Pressure =101325Pa

Atmospheric Temperature=300K

Maximum Temperature=1400

Adiabatic Index=1.4

For getting plots pressure vs. volume, we can program it by using python

Calculation-

Bore = d =0.15m

Stroke= s= 0.15m

Connecting rod Length = l = 0.25m

Compression ratio = cr = 6

Atmospheric Pressure = p1 = 101325Pa

Atmospheric Temperature = t1 = 300K

Maximum Temperature = t3 = 1400K

Adiabatic Index = γ =1.4

1. Calculation of stroke volume

vs= π/4 d² s

2. Calculation of compressed volume

vc=vs/(rc-1)

3. Calculation of suction volume

v1=vc+vs

4. Calculation of v2

As per relation

v2=vc

5. Calculation of p2

Using adiabatic equation

p1v1^γ = p2v2^γ

p2 = (p1v1^γ)/ v2^γ

6. Calculation of t2

Using ideal gas equation

p1v1/t1 = p2v2/t2

t2 = (p2v2t1)/(p1v1)

7. Calculation of v3

As per constant volume heat addition process

v3=v2

8. Calculation of p3

Using ideal gas equation

P2v2/t2 = p3v3/t3

p3=(p2v2t3)/(t2v3)

9. As per constant volume heat rejection process

v4=v1

10. Calculation of p4

p3v3^γ = p4v4^γ

p4 = (p3v3^γ)/ v4^γ

11. Calculation of t4

Using ideal gas equation

p3v3/t3=p4v4/t4

t4=(p4v4t3)/(p3v3)

12. For getting volume at any given crank angle

V/Vc=1+0.5(rc-1)[R+1-cosϴ-(R²-sin²ϴ)^0.5

13. Cal0culation of thermal efficiency

Ƞ=1/(1-r_c^γ-1)

#importing required libraries
import math
import matplotlib.pyplot as plt
# taking all engine inputs
bore=0.15
stroke=0.15
con_rod=0.25
cr=6
p1=101325
t1=300
t3=1400
gamma=1.4
#calculation of all engine volumes,pressures,temperatures
vs=math.pi/4*pow(bore,2)*stroke
vc=vs/(cr-1)
v1=vc+vs
v2=vc
v3=v2
v4=v1
p2=p1*pow(v1,gamma)/pow(v2,gamma)
t2=(p2*v2*t1)/(p1*v1)
p3=(p2*v2*t3)/(t2*v3)
p4=p3*pow(v3,gamma)/pow(v4,gamma)
t4=(p4*v4*t3)/(p3*v3)
#defining function to calculate and store volume in array
def engine_kinematiks(bore,stroke,cr,start_angle,end_angle):
	V=[]
	n=500
	a=stroke/2
	R=con_rod/a
	sc=math.radians(start_angle)
	ec=math.radians(end_angle)
	for i in range(0,n):
		theta=sc+i*(ec-sc)/(n-1)
		vs=math.pi/4*pow(bore,2)*stroke
		vc=vs/(cr-1)
		v1=vs+vc
		tr1=cr-1
		tr2=R+1-math.cos(theta)
		tr3=pow(R,2)-pow(math.sin(theta),2)
		tr5=1+0.5*(tr1)*(tr2-pow(tr3,0.5))
		V.append(vc*tr5)
	return V
# defining function to calculate thermal efficiency of otto cycle
def otto_cycle_efficiency(compression_ratio):
	cr=compression_ratio
	tr7=pow(cr,(gamma-1))
	tr8=1/tr7
	efficiency=1-tr8
	return efficiency
E=otto_cycle_efficiency(cr)
print(E)   
# calling fuction to get arrays of volume and pressure during compression
v_compression=engine_kinematiks(bore,stroke,cr,180,0)
constant1=p1*pow(v1,gamma)
p_compression=[]
for v5 in v_compression:
	p_compression.append(constant1/pow(v5,gamma))
# calling fuction to get arrays of volume and pressure during expansion
v_expansion=engine_kinematiks(bore,stroke,cr,0,180)
constant2=p3*pow(v3,gamma)
p_expansion=[]
for v6 in v_expansion:
	p_expansion.append(constant2/pow(v6,gamma))
# plotting p-v diagram
plt.plot([v2,v3],[p2,p3])
plt.plot([v4,v1],[p4,p1])
plt.plot(v_compression,p_compression)
plt.plot(v_expansion,p_expansion)
plt.show()

Explanation-

• Imported math module for calculation
• Imported matplotlib model for plotting PV diagram
• Defined engine parameters i.e. lengths, diameter, connecting rod length etc.
• Calculated all volumes, pressures, temperatures
• Defined function to return volume array at any given crank angle
• Defined function to calculate thermal efficiency
• Called function for getting compression volumes and compression at different crank angles
• Called function for getting expansion volumes and expansion at different crank angles
• Plotted PV diagram using plt.

Result-

• Thermal efficiency of otto cycle is 0.511
• P-V diagram

Conclusion-

Successfully calculated thermal efficiency of otto cycle and plotted P-V diagram

References-

[1]https://www.google.com/search?q=otto+cycle&authuser=2&tbm=isch&source=iu&ictx=1&fir=tZyeb2LbiCbADM%252CjcBxSZwdTm4v-M%252C_%253BDrn6EOBZ2wzO_M%252Cc2-G-L-1qY_irM%252C_%253BFDTFxa_xYpodmM%252CpXSulY5jUZDVEM%252C_%253BkWoQH78UkQOuYM%252C_gdC5SXH8L5omM%252C_%253BFy7fjbppdZQGXM%252CUI2eBRNQw2wftM%252C_&vet=1&usg=AI4_-kTiEvxc8QNGvv8A_SfE3ADJa3GdgQ&sa=X&ved=2ahUKEwiqiZ6mnbXzAhXCV3wKHZQBDVkQ_B16BAg9EAE#imgrc=tZyeb2LbiCbADM