Week 2 Air standard Cycle

AIM: To write a Python program to plot the PV diagram of an Otto cycle and determine the thermal efficiency of the cycle.

THEORY AND GOVERNING EQUATIONS:

Otto cycle consists of two adiabatic processes and two isochoric processes.

1-2: Adiabatic Compression Process

`p1v1^gamma = p2v2^gamma`

`(p1v1)/(t1) = (p2v2)/(t2)`

2-3: Isochoric Heat Addition Process

`v2 = v3`

`(p2v2)/(t2) = (p3v3)/(t3)`

3-4: Adiabatic Expansion Process

`p3v3^gamma = p4v4^gamma`

`(p3v3)/(t3) = (p4v4)/(t4)`

4-1: Isochoric Heat Rejection Process

`v4 = v1`

Given,

1. Engine Geometric Parameters:-

bore diameter, b = 0.1

stroke length, s = 0.2

connecting rod length, c = 0.15

Crank radius, a = stroke length / 2

2. Other parameters

adiabatic constant, `gamma = 1.4`

compression ratio, cr = 8

Equations:-

Swept Volume, `v_(swept) = pi/4 *b^2 * s`

Clearance Volume, `v_(cl) = v_(swept) / (cr - 1)`

The equation for calculating volume during the adiabatic processes:

`V = (1 + 1/2(cr - 1)[ R + 1 - cos(theta) - (R^2 - sin^2(theta))^(1/2)])*v_(cl)`

Where,

`theta` is the crank angle.

cr is the compression ratio.

`R = c / a` c is the connecting rod length, a is the crank radius.

All 'v' denotes volume in `m^3`

All 'p' denotes pressure in kPa

All 't' denotes temperature in K

OBJECTIVES:

1. To plot the PV diagram of an Otto cycle.

2. To determine the thermal efficiency of the cycle.

PYTHON CODE:

"""
Otto Cycle Simulator
"""

import math
import matplotlib.pyplot as plt

# Function to calculate volume during compression and expansion process

def engine_kinematics(bore, stroke, con_rod, cr, start_crank, end_crank):
	# Engine Geometric parameters
	a = stroke / 2
	R = con_rod / a

	# Volume Computation
	v_s = (math.pi/4) * pow(bore,2) * stroke
	v_c = v_s / (cr - 1)

	# Converting theta values to radian
	sc = math.radians(start_crank)
	ec = math.radians(end_crank)

	num_values = 50
	dtheta = (ec - sc)/(num_values - 1)

	# Calculating Volume
	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(pow(R,2) - pow(math.sin(theta),2),0.5)

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

	return V

# Main Program
# inputs
p1 = 101325
t1 = 500
gamma = 1.4
t3 = 2300

# Geometric Parameters
bore = 0.1
stroke = 0.1
con_rod = 0.15
cr = 8

# Volume Computation
v_s = (math.pi/4) * pow(bore,2) * stroke
v_c = v_s / (cr - 1)

v1 = v_s + v_c

# State point 2
v2 = v_c

# p2*(v2^gamma) = p1*(v1^gamma)  | p2 = p1*(v1/v2)^gamma = p1*cr^gamma
p2 = p1 * pow(cr,gamma)

# p1*v1/t1 = p2*v2/t2 | t2 =t1*(p2*v2/p1v1)
t2 = p2 * v2 * t1 / (p1 * v1)

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

# p3*v3/t3 = p2*v2/t2 | p3 =t3*(p2/t2)
p3 =t3 * (p2 / t2)

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

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

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

# 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])

# Axis Label
plt.xlabel('Volume (m^3)')
plt.ylabel('Pressure (Pa)')

# Title of the Plot
plt.title('PV diagram of an Otto Cycle')
plt.grid()

plt.text(v1,p1,'1')
plt.text(v2,p2,'2')
plt.text(v3,p3,'3')
plt.text(v4,p4,'4')

plt.show()

# Efficiency

eff = 1 - 1/pow(cr,(gamma - 1))
print('The efficiency of the cycle is ' + str(eff * 100) + ' %')

GRAPH:

OUTPUT:

CONCLUSION:

The efficiency of the cycle is found to be 56.47 %.