Air standard Cycle

AIM- To Plot the PV Diagram of Otto cycle and derive the thermal efficiency of the otto cycle

OBJECTIVE

• Write a program to plot the curves of PV- diagram of the Otto Cycle.
  • Find the Thermal Efficiency of the Otto Cycle.

GOVERNING EQUATIONS

Otto cycle - An ideal thermodynamic combustion cycle consists of 4 processes as follows: a compression at constant entropy; a constant-volume heat transfer to the system; an expansion at constant entropy; and a constant-volume heat transfer from the system.

Calculation of different parameters of otto cycle

Engine Geometric Parameters:-

Bore – 0.1

Stroke – 0.1

Connecting Rod Length – 0.15

Compression Ratio (CR) – 12

Swept Volume = ( /4)*Bore^2*Stroke

Clearance Volume = SweptVolume/(CR-1)

Given Parameters

P1 = 101325 Pa

T1 = 500 K

T3 = 2300 K

Gamma = 1.4

Parameters for State Point-1

V1 = Swept Volume + Clearance Volume

V2= Clearance Volume

Parameters for State Point-2

P2V2^ = P1V1^gamma

P2 = P1(V1/V2)^gamma

T2 = P2V2T1/P1V1

Parameters for State Point-3

V3 = V2

P3 = P2V2T3/T2V3

Parameters for State Point-4

V4 = V1

P4 = P3(V3/V4)^gamma

Calculation of Thermal Efficiency Of Otto Cycle

Thermal Efficiency = 1-(1/CR^(gamma-1))

% Thermal Efficiency = Thermal Efficiency* 100

Volume Trace – V/Vc = 1 + ½(CR-1)[R +1 – costheta – (R^2 – sintheta^2)^(1/2)

Where,

a = Stroke/2

R = connecting Rod length/a

PYTHON CODE

import math
import matplotlib.pyplot as plt

def Engine_Kinematics(bore,stroke,con_rod,cr,start_crank,end_crank):


	#Inputs
	a = stroke/2
	R = con_rod/a

	# Volume Computation
	V_s = (math.pi/4)*pow(bore,2)*stroke
	V_c = V_s/(cr-1)

	sc = math.radians(start_crank)
	ec = math.radians(end_crank)

	num_values = 50

	

	dtheta = (ec - sc)/(num_values-1)

	V = []
	for i in range(0,num_values):
		theta = sc + 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)

# Inputs
p1 = 101325
t1 = 500
t3 = 2300
gamma = 1.4

#Geometric Parameters
bore = 0.1
stroke = 0.1
con_rod = 0.15
cr = 12

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

v1 = v_s + v_c

#State Point-2

v2 = v_c
#p1v1^gamma = p2v2^gamma
p2 = p1*pow(v1,gamma)/pow(v2,gamma)

# p1v1/t1 = p2v2/t2 = rhs | t2 = p2v2/rhs
rhs = p1*v1/t1 
t2 = p2*v2/rhs

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

#p2v2/t2 = p3v3/t3 = rhs | p3 = rhs*t3/v3

rhs = p2*v2/t2
p3 = rhs*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)
rhs = p3*v3/t3 
t4 = p4*v4/rhs

# plotting the graph

plt.plot([v2,v3],[p2,p3])
plt.plot(V_compression,P_compression)
plt.plot(V_expansion,P_expansion)
plt.plot([v4,v1],[p4,p1])
plt.show()

# Thermal Efficiency

Therm_eff = 1 - (pow(cr,(1 - gamma)))
print(Therm_eff)

EXPLAINATION OF MAIN CODE

• First we put the Given input values i.e; Gamma, T3,P1,T1,The Engine geometric parameters like Bore , Stroke , Connecting rod length , Compression ratio.

• Then we put calculate the Swept Volume and Clearance Volume using the engine geometric parameters. V1 = Swept volume + Clearance volume

• Then The State variables for State -2 are found i.e; V2 = clearance volume.

• P2 and T2 is also calculated for state-2.

• The values of V_compression and P_compression are calculated using the engine kinematics function.

• Then the values for State variables at state point – 3 are calculated.
• V3 and P3 are calculated.

• V_expansion and P_expansion are calculated using the engine kinematics function.

• State variables at state Point-4 are calculated. P4 and V4 are calculated.

• The plots for P_compression and V_compression are written after the P1,V1 plot command.

• The plots for P_expansion and V_expansion are given after the P3,V3 plot command.

Output Plots

Conclusion

Hence , the otto cycle plot is submitted and the thermal efficiency is calculated.