Week 2 Air standard Cycle

Objective - To write a program to plot a P-V diagram of the Otto Cycle and To calculate Thermal Efficiency of Otto Cycle using Python.

Air Standard Cycle -

The air-standard Otto cycle is the idealized cycle for spark-ignition internal combustion engines.

The Otto cycle 1 → 2 → 3 → 4 consists of the following four processes:

Process 1-2: Isentropic compression of air;

Process 2-3: Heat supply at constant volume;

Process 3-4: Isentropic expansion of air;

Process 4-1: Heat rejection at constant volume.

Python Code:

"""

otto cycle simulator

"""

import math

import matplotlib.pyplot as plt

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

"""

Engine Kinematics

"""

# Geometric parameters

a = stroke/2 # crank pin radius

R = con_rod/a

# Volume parameters

v_s = math.pi*(1/4)*pow(bore,2)*stroke # Swept Volume

v_c = v_s / (cr-1) # Clearance volume

sc = math.radians(start_crank) # Starting crank angle

ec = math.radians(end_crank) # Ending crank angle

num_values = 50

dtheta = (ec-sc)/(num_values-1) # Crank angle spacing

V = [] # empty array

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

# Inputs

p1 = 101325 # Pressure at State-1

t1 = 500 # Temperature at State-1

gamma = 1.4

t3 = 2300

# Geometric parameters

bore = 0.1 # Engine bore size

stroke = 0.1 # Stroke length of engine

con_rod = 0.15 # Connecting rod length

cr = 12 # compression ratio

# Calculating the swept volume and the clearance volume

v_s = (math.pi/4)*pow(bore,2)*stroke # Volume of cylinder at start of compression or end of expansion

v_c = v_s/(cr-1) # Volume of cylinder at end of compression and start of expansion

v1 = v_c + v_s

# State point 2

v2 = v_c # Volume after isentropic compression

# p2v2^gamma = p1v1^gamma

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

# p2v2/t2 = p1v1/t1 | rhs = p1v1/t1 | p2v2/t2 = rhs | t2 = p2v2/rhs

rhs = p1*v1/t1

t2 = p2*v2/rhs

# Position of crank pin angle when expansion start from 180 to 0

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

# position of crank pin angle when expansion start from 0 to 180

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 calculation

x = (1-gamma)

Thermal_efficiency = 1 - (pow(cr,x)) # Thermal efficiency of Otto cycle

Otto_Cycle_efficiency = Thermal_efficiency * 100 # Efficiency in percentage

print('Thermal_efficiency_of_Otto_cycle')

print(Otto_Cycle_efficiency)

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 - PV diagram')

plt.show()

Code Explanation:

• The library module ‘math’ and ‘matplotlib.pyplot’ are imported using import command.
• The ‘Engine kinematics’ function created using def function statement with proper input variables
• The parameters such as crank pin radius, R value, swept volume and clearance volume calculated using formula
• Inside the function the volume formula is defined and values of volume calculated for 50 items and saved in empty array
• Each State variables pressure and volume are calculated. geometric parameters are defined, based on that swept volume and clearance volume calculated
• Function is called out to fine volume, pressure during expansion and compression cycle
• The thermal efficiency of otto cycle calculated using the formula and resulted are printed in display window
• Then, using all state variables (pressure, volume) the PV diagram is plotted

Output:

# PV diagram of Otto cycle

# Thermal Efficiency of Otto cycle

Conclusion:

• The P-V diagram for all four processes has been plotted with proper markers of all 4 states and labels of all 4 process
• Thermal efficiency of the air standard cycle is calculated as 62.98%