PV diagram of Otto Cycle using Python

Objective: To write a program to plot the PV diagram of an otto cycle.

Program:

#otto cycle
import math
import matplotlib.pyplot as plt 
import numpy as np

P1 = 101325                		#presure at stage 1
T1 = 500                   		#temperature at stage 1
T3 = 2500                  		#temperature after combustion (stage 3)
G = 1.4                    		#heat capacity ratio (gamma)
D = 0.1					   		#bore diameter
L = 0.2					   		#stroke length
C = 0.15						#connecting rod length
CR = 12                    		#compression ratio

Vs = (math.pi*D*D*L)/4       	#swept volume
Vc = Vs/(CR-1)             		#clearance volume

def crank_angle(start_angle,end_angle):       #function for volume points on isentropic curve for given crank angles
	Sa = math.radians(start_angle)
	Ea = math.radians(end_angle)
	theta = np.linspace (Sa,Ea,50)
	V = []
	
	for i in theta:                           
		X1 = 0.5*(CR-1)	 
		X2 = (2*C/L)+ 1 - math.cos(i)
		X3 = pow((2*C)/L,2)-pow(math.sin(i),2)
		X4 = pow(X3,0.5)
		Z = Vc*(1+X1*(X2-X4))
		V.append(Z)

	return V

V1 = Vs+Vc                 		#stage 1 

V2 = Vc                    		#stage 2
P2 = (P1*pow(V1,G))/pow(V2,G)
k = (P1*V1)/T1
T2 = (P2*V2)/k

V3 = V2							#stage 3
P3 = (k*T3)/V3

V4 = V1							#stage 4
P4 = (P3*pow(V3,G))/pow(V4,G)
T4 = (P4*V4)/k


V_comp = crank_angle(180,0)    #isentropic compression curve points
P_comp = []
for v in V_comp:    
	x = (P1*pow(V1,G))/pow(v,G)
	P_comp.append(x)

V_exp = crank_angle(0,180)    #isentropic exoansion curve points
P_exp = []
for v in V_exp:    
	y = (P3*pow(V3,G))/pow(v,G)
	P_exp.append(y)
	
plt.plot([V2,V3],[P2,P3])
plt.plot([V4,V1],[P4,P1])
plt.plot(V_comp,P_comp)
plt.plot(V_exp,P_exp)
plt.show()

N = (1 - (T4-T1)/(T3-T2))*100			# efficiency of otto cycle      
print(\'The efficiency is \'+str(N))

Procedure:

• math, matplotlib.pyplot and numpy modules were imported for using mathematical functions, plotting and using linspace command respectively.
• The inputs given for the program are as follows:
  • Pressure at stage 1 (P1) = 1 bar
  • Temperature at stage 1 (T1) = 500K
  • Temperature after combustion (T3) = 2500K
  • Heat capacity ratio (`gamma`) = 1.4
  • Bore diameter (D) = 0.1 m
  • Stroke length (L) = 0.2 m
  • Connecting rod length (C) = 0.15 m
  • Compression ratio (CR) = 12
• Swept and clearnce volumes were calculated by the following formula:
  • `Vs = (pi*D^2*L)/4`
  • `Vc = V_s/(CR-1)`
• The values of volume, pressure and temperaure were calculated at the 4 stages of the otto cycle by using the ideal gas equation `(P.V )/T =`constant.
• For plotting the points on the isentropic expansion and compression curves a function was created with the start and end crank angles as the input. The range between the crank angles was split in 50 parts for a smooth curve. This angle was used as the input in the formula `V /V_c = 1+0.5(CR-1)[R+1-costheta-(R^2-sin^2theta)]^(1/2)` where `R= (2*C)/L`, to find the value of volume. These values were stored in a volume array.
• To calculate the volume values the function was called with input angles 180 to 0 (BDC to TDC) for the compression process and 0 to 180 for the expansion process. A loop was created to input the values from volume array and form the corresponding pressure array using the formula `P*V^gamma =`constant.
• The calculated values of 4 stages of the otto cycle and the isentropic curves were plotted.
• Efficiency was calculated using the formula `eta = (1 - (T4-T1)/(T3-T2))*100`

Results:

• The P-V diagram of the otto cycle with the given inputs was created as shown in the figure below.
• The efficiency value for the given inputs was found to be 62.99%.