Scripting and Plotting of a P-V diagram of an Otto Cycle using Python

Objective: Write code that can solve an otto cycle and make plots for it.

Explaination:

As in an ideal Diesel cycle four distinct processes are also followed in the Otto cycle: reversible adiabatic compression, addition of heat at constant volume, reversible adiabatic expansion, and rejection of heat at constant volume as follows:

1. Air enters the cycle at point 1 when compression starts and it ends at point 2. The air is compressed in an isentropic compression through a volume ratio v₁/v₂, known as the compression ratio r_c (the ratio of the volume of the working fluid before the compression process to its volume after).

2. Heat addition starts at point 2 and ends at point 3. A quantity of heat Q_inis added at constant volume, and the working fluid temperature rises.

3. The isentropic expansion process starts at point 3 and it ends at point 4 to the original volume.

4. The heat Qout is rejected at constant volume from point 4 to point 1. Air gets cooled and the working fluid temperature goes down at a constant volume until the cycle is completed.

PROCEDURE:

1. Consider some given data:

p1 = 101325 pa

T1 = 500 k

T3 = 2300 k

gamma = 1.4

Stroke length = 0.1 m

Bore dia. = 0.1 m

connecting rod = 0.15 m

compression ratio (cr = 12)

Crankpin radius = stroke length/2

R = constant = con_rod/crankpin radius

2. Some Formulas used:

Swept vol. (v_s) = π/4 d²L

Clearance vol. (v_c) = swept vol./(cr -1)

V = v_c + v_s

p1v1/t1 = p2v2/t2 | Ideal gas equation

V/Vc= 1 + 1 (rc - 1)[R + 1 - cos θ - (R² – sin² θ)1/2]

This eq. used to describe a relation between crank angle & vol. inside the combustion chamber. By this

eq. we get volume trace and we need this to finish our program.

Coding:

# otto cyle simulator

import math
import matplotlib.pyplot as plt

def engine_kinematics(bore, stroke, con_rod, cr, start_crank, end_crank):

	#geometric expressions
	a = stroke/2
	R = con_rod/a

    # different volume paramters
	V_s= (math.pi/4)*pow(bore,2)*stroke
	V_c = V_s/(cr-1)

	sc = math.radians(start_crank)
	ec = math.radians(end_crank)

	num_values = 30
	dtheta = (ec-sc)/(num_values-1)

	V= []

	for i in range(0,num_values):
		theta = sc + 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) 

		V.append(( 1 + term1*(term2 - term3))*V_c)

	return V
		

p1 = 101325  # in pascal = 1 atm
t1 = 500     # in kelvin 
t3 = 2300
gamma = 1.4

# engine parameters
bore = 0.1 
stroke = 0.1 
con_rod = 0.15
cr = 7
V_s= (math.pi/4)*pow(bore,2)*stroke
V_c = V_s/(cr-1)

v1 = V_s + V_c

v2 = V_c  #state point 2

#isentropic relation 
#p2v2/t2 = p1v1/t1 | rhs = p1v1/t1 | p1v2/t2 = rhs | t2 = p2v2/rhs 

# p2v2^gamma = p1v1^gamma

p2 = p1*pow(v1,gamma)/pow(v2,gamma)

rhs = p1*v1/t1   #ideal gas law
t2 = p2*v2/rhs

V_compression = engine_kinematics(bore,	stroke, con_rod, cr, 180, 0)

constant = p1*pow(v1,gamma)

P_compression = []

for v in V_compression:
 	P_compression.append(constant/pow(v,gamma))


# state point 3 
v3 = v2

#p3v3/t3 = p2v2/t2 | rhs = p2v2/t2 | p3 = rhs*t3/v3

rhs = p2*v2/t2
p3 = rhs*t3/v3

V_expansion = engine_kinematics(bore, stroke, con_rod, cr, 0, 180)

constant = p3*pow(v3,gamma)

P_expansion = []

for v in V_expansion:
 	P_expansion.append(constant/pow(v,gamma))

# state point 4
v4 = v1

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

# p4v4/t4 = rhs

t4 = p4*v4/rhs  

thermal_efficiency = 1-((t4-t1)/(t3-t2))

print(thermal_efficiency)

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.xlabel('Volume')
plt.ylabel('Pressure')
plt.savefig('pv_diagram')
plt.show()

Thermal efficiency of this program: 0.5408434500405659

Note that the thermal efficiency of otto cycle is a function of compression ratio r and the ratio of specific

heats. As gamma is assumed to be constant for any working fluid, the efficiency is increased by

increasing the compression ratio. Further, the effeciency is independent of heat supplied and pressure

ratio. The use of gases with higher gamma values would increase efficiency of otto cycle.

Conclusion:

In depth use of functions, for loop gave good understanding of it.

with the help of this program we can observe the results at any given crank angle.