Air Standard Cycle Simulator

AIM: Plotting P-V diagram and efficiency calculation of the Otto Cycle.

GOVERNING EQUATIONS:

Otto cycle is a gas power cycle that is used in a spark-ignition internal combustion engine (modern petrol engines). It is an air standard cycle which approximates the processes in petrol or diesel engines.

An Otto cycle consists of four processes:

1. Two isentropic (reversible adiabatic) processes
2. Two isochoric (constant volume) processes

These processes can be easily understood if we understand p-V (Pressure-Volume) and T-s (Temperature-Entropy) diagrams of the Otto cycle.

In the above diagrams,

p → pressure V → Volume T → temperature s → Entropy V_c → Clearance Volume V_s → Stroke Volume

Processes in Otto Cycle:

Process 1-2: Isentropic compression

In this process, the piston moves from the bottom dead centre (BDC) to top dead centre (TDC) position. Air undergoes reversible adiabatic (isentropic) compression. We know that compression is a process in which volume decreases and pressure increases. Hence, in this process, the volume of air decreases from V₁ to V₂ and pressure increases from p₁ to p₂. Temperature increases from T₁ to T₂. As this an isentropic process, entropy remains constant (i.e., s₁=s₂).

Process 2-3: Constant Volume Heat Addition:

Process 2-3 is isochoric (constant volume) heat addition process. Here, piston remains at the top dead centre for a moment. Heat is added at constant volume (V₂ = V₃) from an external heat source. Temperature increases from T₂ to T₃, pressure increases from p₂ to p₃ and entropy increases from s₂ to s₃.

In this process,

Heat Supplied = mC_v(T₃ – T₂)

where,

m → Mass

C_v → Specific heat at constant volume

Process 3-4: Isentropic expansion

In this process, air undergoes isentropic (reversible adiabatic) expansion. The piston is pushed from the top dead centre (TDC) to bottom dead centre (BDC) position. Here, pressure decreases from p₃ to p₄, volume rises from v₃ to v₄, the temperature falls from T₃ to T₄ and entropy remains constant (s₃=s₄).

Process 4-1: Constant Volume Heat Rejection

The piston rests at BDC for a moment and heat is rejected at constant volume (V₄=V₁). In this process, the pressure falls from p₄ to p₁, temperature decreases from T₄ to T₁ and entropy falls from s₄ to s₁.

In process 4-1,

Heat Rejected = mC_v(T₄ – T₁)

Thermal efficiency (air-standard efficiency) of Otto Cycle,

1 to 2 and from 3 to 4 are isentropic, therefore

Volume swept by the piston, V_s = (π*(bore)²*stroke)/4

Clearance volume, V_c = V_s/(cr-1)

Volume swept as a function of theta:

A=(0.5)*(cr-1)
B=R+1-cosd(theta)
C=((R^2)-((sind(theta))^2))^0.5

V=(1+(A*(B-C)))*Vc

GIVEN:

bore=0.8 meters

stroke=0.2 meters

connecting rod length = 0.15 meters

compression ratio (cr) = 10

p1=101325

y=1.4

t1=450

t3=3050

OBJECTIVES OF THE PROJECT: The main objective of the project is to calculate all unknowns of each state by help of governing equations and given values. After that plotting P-V diagram and calculating efficiency of the cycle.

PROGRAM:

# Otto cycle simulator

import math
import numpy as np
import matplotlib.pyplot as plt

#Engine geometrical parameters
bore=0.8
stroke=0.2
con_rod=0.15
cr=10

def engine_kinematics(bore,stroke,con_rod,cr,theta1,theta2):

	v_stroke=(math.pi*pow(bore,2)*stroke)/4
	v_clear=v_stroke/(cr-1)
	a=stroke/2;
	R=con_rod/a

	theta=np.linspace(theta1,theta2,180)
	V=[]

	for t in theta:

		A=(0.5)*(cr-1)
		B=R+1-math.cos(math.radians(t))
		C=pow((R**2)-(math.sin(math.radians(t))**2),0.5)
		V.append((1+(A*(B-C)))*v_clear)

	return V

#known state variables
p1=101325
#gamma
y=1.4
t1=450
t3=3050

#calculation of volumes
v_stroke=(math.pi*pow(bore,2)*stroke)/4
v_clear=v_stroke/(cr-1)

#state 1
v1=v_stroke+v_clear

#state 2
v2=v_clear
p2=pow(v1/v2,y)*p1
t2=pow(v1/v2,y-1)*t1

#state 3
v3=v2
p3=p2*(t3/t2)

#state 4
v4=v1
p4=pow(v3/v4,y)*p3
t4=pow(v3/v4,y-1)*t3

#During Compression stroke
v_comp=engine_kinematics(bore,stroke,con_rod,cr,180,0)
c1=p1*pow(v1,y)
p_comp=[]

for v in v_comp:
	p_comp.append(c1/(pow(v,y)))


#During Expansion stroke
v_exp=engine_kinematics(bore,stroke,con_rod,cr,0,180)
c2=p3*pow(v3,y)
p_exp=[]

for ve in v_exp:
	p_exp.append(c2/(pow(ve,y)))


#Calculation efficiency 
n=1-((t4-t1)/(t3-t2))
print('Efficiency of engine = ', n);




#plotting state point
plt.plot(v1,p1,'*')
plt.plot(v2,p2,'*')
plt.plot(v3,p3,'*')
plt.plot(v4,p4,'*')
plt.plot(v_comp,p_comp, "-b",linewidth=3,label="Compression Stroke")
plt.plot([v2,v3],[p2,p3], "-r",linewidth=3,label="Heat Addition")
plt.plot(v_exp,p_exp, "-g",linewidth=3,label="Expansion Stroke")
plt.plot([v1,v4],[p1,p4], "-y",linewidth=3,label="Heat Rejection")

plt.xlabel('Volume (m^3)')
plt.ylabel('Pressure (pa)')
plt.legend()
plt.title('OTTO CYCLE')
plt.grid()
plt.show()

• 
  1. All known engine geometry parameters and state variable were defined.
• 2. The formula for volume swept and volume clearance was given.
• 3. With the help of Isentropic formulate for process 1-2 and 3-4, all state variable was calculated.
• 4. By this, we got all four-point of the p-v diagram but for tracing isentropic volume we need to define how the volume changes in the engine piston.

ERROR FACED: No Error Faced

OUTPUT:

The program outputs the efficiency of the Otto cycle and plots P-V diagram. In the P-V diagram we can see that for heat addition and heat removal process, the volume is constant and for the isentropic processes, the graph is following an isentropic path.

CONCLUSION:

Using PYTHON programming, the efficiency of the Otto cycle was calculated and the P-V diagram is plotted. This is very helpful in the case where there is any change in engine geometry, we can easily update that in the code and quickly see the changes in the P-V diagram.