Week 2 Air standard Cycle

AIM: To write a Python code that plots an otto cycle and calculates the thermal efficiency.

BASICS OF THERMODYNAMIC PROCESSES AND THEIR EQUATIONS:

• Isentropic Process: An adiabatic and reversible process is termed as isentropic process. The equation describing an Isentropic process is `p*v^gamma=constant` ---- Eq. 1
• Ideal gas relation is `(p*v)/T=constant` ---- Eq. 2
• Efficiency of otto cycle: `eta_(th)= 1-1/(r^gamma-a)`

INPUTS REQUIRED:

• Engine geometry
• Bore
• Stroke Length
• Length of connecting rod
• Initial Pressure and Temperature
• Temperature immediately after combustion
• Compression Ratio
• Value of Gamma (ratio of specific heats)

OTTO CYCLE: It is an idealized thermodynamic cycle that describes the functioning of a typical Spark Ignition (SI) engine.

REQUIRED FORMULAE:

1. Swept volume (`v_s`)= `pi/4 * d^2 * l`
2. Clearance volume (`v_c`)= `v_s/(r-1`
3. Variation of volume inside the cylinder with crank angle
   `v/v_s= 1+1/2 (r-1)[R+1-cos theta-(R^2-sin^2theta)^(1/2)]`

The Otto cycle comprises of four processes:

• Isentropic compression (1-2)
• Constant volume heat addition (2-3)
• Isentropic expansion (3-4)
• Constant volume heat rejection (4-1)

Fig. 1 Ideal otto cycle

The thermodynamic changes occurring throughout the cycle are:

1. Isentropic compression process (1-2)
   The charge is compressed in an adiabatic and reversible process (isentropic).
   The state properties undergo change according to the equation 1.
   Hence pressure at state to is calculated as `p_2= p_1* (v_2/v_1)^gamma` The temperature is calculated using the equation `T_2= T_1*(p_2/p_1).(v_2/v_1)`
2. Constant volume heat addition process (2-3)
   Compressed charge is heated at constant volume. Temperature at the state point 3 is known to us. Hence the equation 1 is made used to determine the pressure at state point 3 as `T_3= T_2*(p_3/p_2)`.
3. Isentropic expansion process (3-4)
   The heated charge is allowed to expand isentropically and the properties state point 4 are found using the equations 1 and 2 by substituting respective values.
4. Constant volume heat rejection process (4-1)
   The charge is cooled at constant volume which results in a pressure drop.

Using the above equations we determine the properties at all the state points.

CODE:

import math
import matplotlib.pyplot as plt

#Function to incorporate engine kinematics and calculate the pressure corresponding to the volume
def engine_kinematics(bore,stroke,con_rod,cr,start_crank,end_crank):
		
	#Calculating volume
	V_s=math.pi*(1/4)*pow(bore,2)*stroke
	V_c=V_s/(cr-1)

	start_angle=math.radians(start_crank)
	end_angle=math.radians(end_crank)    

	#Connecting rod geometry
	a=stroke/2
	R=con_rod/a

	num_values=100 #Number of divisions 
	
	dtheta=(end_angle-start_angle)/(num_values-1)
	V=[]

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

#Kmown primitives 
p1=101325
t1=500
gamma=1.4
t3=2300

#Engine geometry
bore=0.1
stroke=0.1
con_rod=0.15
cr=8

#Calculating the volume
v_s=(math.pi/4)*pow(bore,2)*stroke
v_c=v_s/(cr-1)
v1=v_c+v_s


#state point 2
v2=v_c

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



t2=(p2*v2)/(p1*v1/t1)
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

p3=(p2*v2/t2)*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
p4=p3*pow(v3,gamma)/pow(v4,gamma)
t4=(p4*v4/(p3*v3))*t3

#Plotting the results
plt.plot([v2,v3],[p2,p3],color='blue')
plt.plot(V_compression,P_compression,color='blue')
plt.plot(V_expansion,P_expansion,color='blue')
plt.plot([v4,v1],[p4,p1])
plt.xlabel('Volume (m^3)')
plt.ylabel('Pressure (Pa)')
plt.title('P-V diagram of otto cycle')

#Thermal efficiency
Thermal_efficiency=(1-(1/(pow(cr,gamma-1))))*100
print('Thermal efficiency of this cycle is:',Thermal_efficiency,'%')

EXPLAINATION OF CODE:

• All known variables (Pressure(s), Temperature(s), adiabatic index, engine geometry) have to be defined and assigned
• The swept and clearance volumes have to be calculated
• The properties at all state points have to be calculated using respective formulae
• The corrected volume has to be calculated based on the formulae decsribed. This has been written as a function named engine_kinematics. This function is called an an array of volumes for different crank angles at 1 degree. This volume vector is used to calculate the corresponding pressure.
• The plot commands have to be used in order to visualize the graph
• The equation to calculate the thermal efficiency has to be calculated and the same has to be displayed.

OUTPUT:

1. Plot of the otto cycle

RESULTS:

1. The otto cycle was plotted
2. The thermal efficiency of the otto cycle was calculated to be 56.472471835193794 %

NOMENCLATURE:

p- pressure
v- volume
T- Temperature
`eta_(th)`- Thermal efficiency

r- Compression Ratio
R- Ratio of length of connecting rod to crank pin radius
`theta`- Crank angle