Week 2 Air standard Cycle

AIM:

To plot a PV diagram and calculatw the thermal efficiency of an Otto cycle using the concept of Engine Kinematics with given data.

Theory :

In an ideal otto cycle, the mechanism works as follows where we calculate the Compression volume using the formula:

`V/(V_c)=1+1/2(cr-1)[R+1-costheta-sqrt(R^2-sintheta^2)]`

where

V=Volume Of the Cylinder

V_c= Displacement Volume of one Cylinder

cr= Compression Ratio

R=Connecting rod length/Crank Pin Radius.

The processes involved in an ideal otto cycle are:

1. Intake stroke, gasoline vapor and air drawn into engine (5 to 1 ).
2. Compression stroke, P , T increase ( 1 to 2 ).
3. Combustion (spark), short time, essentially constant volume ( 2 to 3). Model: heat absorbed from a series of reservoirs at temperatures T2 to T3 .
4. Power stroke: expansion (3 to 4).
5. Valve exhaust: valve opens, gas escapes.
6. Rejection of heat to series of reservoirs at temperatures T4 to T1 .(4 to 5)
7. Exhaust stroke, piston pushes remaining combustion products out of chamber ( 1 to 5 ).

Given Data:

The geometry I have assumed is of Buggati Veyron engine which has the following dimensions:

Bore=0.086

Stroke=0.086

Connnecting Rod=0.023

Compression Ratio=9

And I have assumed

P1=101325Pa

t1=288K

t3=2179K

Specific Heat Ratio(gamma)=1.4

Python Code:

import math
import matplotlib.pyplot as plt


# Function for kinematics of engine

'''The parameters are the bore length,stroke length, connecting rod length,
the compression ratio and the start and the end crank angle'''
def engine_kinematics(bore, stroke, con_rod, cr, start_crank, end_crank):

	a = stroke/2                    #crank pin radius
	R = con_rod/a                  

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

	# Conversion to Radians using math library fuction
	startc = math.radians(start_crank) 
	endc = math.radians(end_crank)

	n = 75 # Number of divisions
	dt = (endc-startc)/(n-1) # Spacing between each division
	V = [] # Defining empty array for volume

	for i in range(0,n): #Using loop to append to the volume array
		theta = startc + i*dt
		V.append((1 + 0.5*(cr-1)*(R+1-math.cos(theta)-(pow((pow(R,2)-pow(math.sin(theta),2)),0.5))))*Vc)
		
	return V

    
    

# Inputs of the geometric parameters
    #Taking the engine parameters of Buggati Veyron
bore = 0.086
stroke = 0.086
con_rod = 0.23
p1 = 101325 #Initial Pressure
t1 = 288 #Initial Temperature
t3 = 2179
cr = 9 #Compression ratio 
gamma = 1.4

# Calculating Swept volume and Clearance volume
Vs = (math.pi/4)*pow(bore,2)*stroke
Vc = Vs/(cr-1)
v1 = Vs + Vc
v2 = Vc

# Process 1-2: Isentropic compression

# p1*v1^gamma = p2*v2^gamma; t2/t1 = (v1/v2)^(gamma-1) = cr^(gamma-1)
t2 = t1*pow(cr,(gamma-1)) #Calculating T2
p2 = p1*pow(cr,gamma)     #Calculating p2

# Calling the function to determine the Compression Volume
V_comp = engine_kinematics(bore, stroke, con_rod, cr, 180, 0) 
constant_c = p1*pow(v1,gamma)
P_comp = []


for j in V_comp: # Finding the Compression Pressure
	P_comp.append(constant_c/pow(j,gamma))

# Process 2-3: Constant volume heat addition
v3 = v2
p3 = (p2*t3)/t2

# Process 3-4: Isentropic expansion

v4 = v1
p4 = p3*pow((v3/v4),gamma)

# Calling the function for Expansion Volume
V_exp = engine_kinematics(bore, stroke, con_rod, cr, 0, 180)
constant_c = p3*pow(v3,gamma)
P_exp = []

for j in V_exp: #Determining the Expansion Pressure
	P_exp.append(constant_c/pow(j,gamma))

#Calculating the Thermal efficiency
eff = 1 - (1/pow(cr,(gamma-1)))
eff = str(eff*100)

print("Thermal Efficiency of the Otto Cycle = " + eff + " %")

# Plotting the PV Diagram
plt.figure()
plt.plot(V_comp,P_comp, label='Isentropic compression')
plt.plot([v2,v3],[p2,p3], label='Heat addition')
plt.plot(V_exp,P_exp, label='Isentropic expansion')
plt.plot([v4,v1],[p4,p1], label='Heat Rejection')
plt.xlabel('Volume')
plt.ylabel('Pressure')
plt.title('P-V Diagram of Otto Cycle')
plt.legend()
plt.show()

 

Output :

The PV Plot of an Otto Cycle:

Result:

The Thermal Efficiency of the Otto Cycle is =58.47%