Week 2 Air standard Cycle

AIM

To write a program in python to solve the otto cycle and plot the graph.

OBJECTIVE

1. To solve different state variables in the otto cycle and plot p-v diagram.
2. To calculate thermal efficiency for the given parameters in the otto cycle.

PROCESS IN OTTO CYCLE

• Process 0–1 A large amount of air is drawn into the piston/cylinder assembly at constant pressure.
• Process 1–2 is the adiabatic (isentropic) compression of the charge as the piston moves from bottom dead center (BDC) to top dead center (TDC).
• Process 2–3 is a constant amount of heat transfer from an external source to the working gas while the piston is at top dead center. This process is intended to represent the ignition and subsequent rapid combustion of a fuel-air mixture.
• Processes 3–4 are adiabatic (isentropic) expansion (power stroke). The
• process 4-1 completes the cycle with a constant volume process removing heat from the air while the piston is at bottom dead center.
• Process 1–0 Air parcels are released to the atmosphere in a constant pressure process.

The Otto cycle consists of isentropic compression, heat input at constant volume, isentropic expansion, and heat loss at constant volume. Technically, the 4-stroke Otto cycle has two additional processes. One is the process of removing waste heat and combustion products at constant pressure (isobaric) and the other is the process of taking in cool, oxygen-rich air at constant pressure. However, these are often omitted in simplified analyses. Both of these processes are critical to the functioning of a real engine, but the details of heat transfer and combustion chemistry are involved, whereas a simplified analysis of the thermodynamic cycle suggests that all waste heat can be transferred to a single engine It is more convenient to assume that volume change.

FORMULA

The formula to find volume is given below.

The formula to find thermal efficiency is given below

Here the term k is denoted as gamma in the program

r is denoted as cr in the program

PROCEDURE

• Since the otto cycle is an air standard cycle we will take the gamma value as 1.4.
• At the initial condition, we will consider the T1 and P1 values as 500K and 101.325KPa respectively.
• The geometrical parameters are stoke, diameter, connecting rod length, crank start and end angle are defined to the function in the program.
• From the above parameters, the v_c and v_s are calculated accordingly.
• The other states are calculated using adiabatic and ideal gas equations.
• Similarly, the thermal efficiency is calculated using the above formulas.
• For plotting p-v diagram plt.plot command is used from matplotlib.pyplot module.
• For the compression stage, the starting and ending crank angles are 180 and 0.
• For the expansion stage, the starting and ending crank angles are 0 and 180.
• Let us consider the value to be iterated is 50 accordingly v null set is created to store the value using append command.

PROGRAM

import math
import matplotlib.pyplot as plt 
def engine_kinematics(bore,stroke,con_rod,cr,start_angle,end_angle):
	#engine parameters
	a = stroke/2
	R = con_rod/a
	v_s = (math.pi/4) * pow(bore,2) * stroke
	v_c = v_s/(cr -1)
	sc = math.radians(start_angle)
	ec = math.radians(end_angle)
	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

gamma = 1.4

t3 = 2300

#Geomentric parameter
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

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

#p2v2/t2 = p1v1/t1 | rhs = 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

#p3v3/t3 = p2v2/t2 | rhs = p2v2/t2 | 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

#p4v4^gama = p3v3^gamma
p4 = p3*pow(v3,gamma)/pow(v4,gamma)

#p4v4/t4 = rhs
t4 = p4*v4/rhs

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.xlabel('volume')
plt.ylabel('pressure')
plt.title('pressure vs volume')


plt.show()

#thermal efficiency
thermal_efficiency = 1-(1/(pow(cr,(gamma-1))))
print ('Thermal efficiency is : ',thermal_efficiency)