Engine Kinematics - Air Standard Otto Cycle

Problem Statement:

In an Air Standard Otto Cycle, the compression ratio is 7 and compression begins at 35°c and 0.1Mpa. The maximum temperature of cycle is 1100°c. Bore and Stroke of engine cylinder are 69mm and 76.8mm. Length of connecting rod is 100mm. Consider Cv = 0.718 KJ/Kg k. Develope a Python Program for:

1. Thermal Efficiency of cycle

2. PV diagram.

Python Program:

# Program for Air Standard Otto cycle Efficiency Calculation #

import math
import matplotlib.pyplot as plt

# Input
bore = 0.069
stroke = 0.0768
l = 0.1 # Length of connecting rod
a = stroke/2 # Crank Radius
cr = 7 # Compression ratio
p1 = 100000 # Presure at which compression process begins
t1 = 308 # Temperature at which compression begins
t3 = 1373 # Temperature after combustion process ends
R = l/a # Any constant teken under consideration
gamma = 1.4
Cv = 0.718 # Specific heat at constant volume 

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

# Defining function for calculating volume of cylinder at any crank angle
def engine_kinematics(bore, stroke, l, cr, start_crank, end_crank):

	# As crank angle may vary from 0 - 90 degrees, considering 50 intervals
	n_theta = 50

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

	# Volume of cylinder at any crank angle
	V = []

	for i in range(0, n_theta):
		d_theta = (sc-ec)/(n_theta - 1)
		theta = sc + (i * d_theta)

		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

# Calculating parameters at each state point

# Process 1-2 = Isentropic Compression
v1 = v_s + v_c
v2 = v_c

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

t2 = pow((v1/v2),(gamma-1)) * t1

	# Calling function to calculate variation of V & P with varying crank angle from 180 - 0 deg.
v_compression = engine_kinematics(bore, stroke, l, cr, 180, 0)

constant = p1 * pow(v1, gamma)

p_compression = []

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


# Process 2-3 = Constant Volume heat addition
v3 = v2

p3 = (t3/t2) * p2

	# Calling function to calculate variation of V & P with varying crank angle from 0 - 180 deg.
v_expansion = engine_kinematics(bore, stroke, l, cr, 0, 180)

constant = p3 * pow(v3, gamma)

p_expansion = []

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

# Process 3-4 = Isentropic Expansion

v4 = v1 # as Process 4-1 = Constant Volume Heat Rejection

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

t4 = pow((v3/v4), (gamma-1)) * t3 

# Calcuation of thermal efficiency of otto cycle

q_in = Cv * (t3 - t2) # Heat Input 
q_out = Cv * (t4 - t1) # Heat Output

w_net = q_in - q_out # Net work done

efficiency_otto = (w_net / q_in)*100

print(\"The Efficiency of Otto Cycle is \" + \"%.2f\" % efficiency_otto + \"%\")
plt.xlabel(\'Volume (m^3)\')
plt.ylabel(\'Pressure (Pa)\')
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.savefig(\'Air_Standard_Cycle_PV_Plot.png\')
plt.show()

Program Explanation:

1. Input:

bore = 0.069m

stroke = 0.0768m

Length of connecting rod = l = 0.1m

Crank Radius = a = stroke/2 = 0.0384m

Compression Ratio = cr = 7

Pressure before compression = p1 = 0.1MPa

Temperature before compression = t1 = 35°c = 308K

Max. temperature after combustion = t3 = 1100°c = 1373K

R = Length of connecting rod / crank radius = 2.60

gamma = 1.4

Specific heat at constant volume = 0.718 KJ/Kg k

2. As per given dimensions of cylinder, swept and clearance volume can be calculated.

Swept Volume = v_s = (pi/4)*(bore^2)*(stroke)

Total Volume = swept volume + clearance volume

therefore,

Clearance Volume = v_c = v_s / (cr - 1)

3. As we need to calculate volume of cylinder at any crank angle, we define a function which

can be used in place in program to calculate volume at that poit and corresponding

pressure.

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

under above function we calculate V at any crank angle using following formula.

As Θ in above equation varies from 0° to 180°, 10 equal intervals are considered to

calculate V at those intervals. As required Θ in above quation are in radians, Θ in degree

is converted to radian using math.radians().

To reduce complexity of equation, it is devided into 3 terms

term1 = 0.5 * (rc - 1)

term2 = R + 1 - cos(Θ)

term3 = ((r^2) - (sin^2(Θ))^0.5

by considering volume at any crank angle as \"V\"

V = []

By using For loop, value of is appended by using term

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

for storing value of V in array, \"return V\" is used after FOR LOOP..

4.

Above is P-V plot for Air Standard Otto Cycle.

Process 1 - 2: Isentropic Compression

using above equation p2 & t2 are calculated. At this point by calling function defined

earlier, v_compression and p_compression are calculated.

v_compression = engine_kinematics(bore, stroke, l, cr, 180, 0)

as while compression crank angle varies from 180° to 0°, start_crank is replaced by 180

and end_crank is replaced by 0.

By using For Loop, for each value of v_compression, p_compression is calculated.

Process 2 - 3: Constant Volume Heat Addition (Combustion)

For Constant Volume process;

p3 is calculated using above equation, as all other values are known.

Process 3 - 4: Isentropic Expansion

v4 = v1 since process 4-1 is constant volume process.

Similar to process 1-2, pressure and volume at after expansion process i.e. p4 & t4 are

calculated using same relations between p,v & t.

same as that of calculation of v_compression and p_compression, v_expansion and

p_expansion are calculated, similarly by calling function. Only change being crank angle.

Crank angle while expansion varies from 0° to 180°.

v_expansion = engine_kinematics(bore, stroke, l, cr, 0, 180)

5. After calculating values of pressure and volume at all state point, efficiency can be

calculated by using following equations;

Heat Added = q_in = Cv * (t3 - t2) KJ/Kg

Heat Rejected = q_out = Cv * (t4 - t1) KJ/Kg

Net work done = w_net = q_in - q_out

Thermal Efficeiency = efficiency_otto = w_net / q_in

The result is printed as output of program.

6. Using all calculated values of pressure and volume at each state point, P-V plot is plotted.

Plot is plotted for curves ([v2,v3],[p2,p3]) = Isochoric Heat addition

([v4,v1],[p4,p1]) = Isochoric Heat Rejection

([v_compression, p_compression]) = Isentropic Compression

([v_expansion, p_expansion]) = Isentropic Expansion

Obtainted P-V plot is saved by using savfig() and shown using show() under matplotlib

module.

Output:

Efficiency of Otto cycle for given condition is 54.08%.

Above is P-V plot for Air Standard Otto cycle.

1. Green Curve shows process 1-2 i.e. Isentropic Compression, where pressure

increases and volume decreases.

2. Blue Vertical line shows process 2-3 i.e. Constant Volume or Isochoric

Combustion.

3. Red curve indicates process 3-4 i.e. Isentropic Expansion, where pressure seems

to be dropping with increase in volume.

4. Yellow line indicates process 4-1 i.e. Constant Volume or Isochoric Heat

rejection through exhaust.