Week 2 Air standard Cycle

AIM:- Write Python code that can solve or check the engine kinematics of an engine operating on the OTTO cycle and make plots for it. Create a PV diagram and also output the thermal efficiency of the engine.

DESCRIPTION:- An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines.

The Otto cycle is a description of what happens to a mass of gas as it is subjected to changes of pressure, temperature, volume, addition of heat, and removal of heat. The mass of gas that is subjected to those changes is called the system. The system, in this case, is defined to be the fluid (gas) within the cylinder. By describing the changes that take place within the system, it will also describe in inverse, the system's effect on the environment. In the case of the Otto cycle, the effect will be to produce enough net work from the system so as to propel an automobile and its occupants in the environment.

PROCESS:-

• Process 0–1 a mass of air is drawn into piston/cylinder arrangement at constant pressure.
• Process 1–2 is an adiabatic (isentropic) compression of the charge as the piston moves from bottom dead centre (BDC) to top dead centre (TDC).
• Process 2–3 is a constant-volume heat transfer to the working gas from an external source while the piston is at top dead centre. This process is intended to represent the ignition of the fuel-air mixture and the subsequent rapid burning.
• Process 3–4 is an adiabatic (isentropic) expansion (power stroke).
• Process 4–1 completes the cycle by a constant-volume process in which heat is rejected from the air while the piston is at bottom dead centre.
• Process 1–0 the mass of air is released to the atmosphere in a constant pressure process

GOVERNING EQUATIONS:-

Compression ratio (r) is the ratio of total cylinder volume to the clearance volume.

`r = V_1/V_2`

We are going to find efficiency of system by using T_3,T_4.

Process 1-2:

This process is an isentropic (reversible adiabatic) process. For this process, the relation between T and V is as follows:

`T_2/T_1 =(V_1/V_2 )^(γ-1) = r^(γ-1)…………..1`

Process 3-4:

This is also an isentropic process. The relation between T and V in this process is similar to the relation between T and V in process 1-2:

Here,

`T_3/T_4 =(V_3/V_4 )^(γ-1) = r^(γ-1)……………2`

Air-standard efficiency of Otto cycle:

It is defined as the ratio between work done during Otto cycle to the heat supplied during Otto cycle.

`n_th= 1-1/r^(γ-1) ,,,,,,by Eq 1,2`

The following equation shows the relationship between crank angle and volume inside the crank chamber: -

`V/V_c =1+0.5*(r_c-1)*[R+1-cos⁡(θ-(R^2-(sin⁡θ)^2 )^0.5 )]`

CODE:-

import math
import matplotlib.pyplot as plt

def engine_kinematics(bore,stroke,con_rod,cr,start_angle,end_angle):
	"""
	Engine Kinematics
	"""
	# Engine Parameters
	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_angle)
	ec = math.radians(end_angle)

	num_values = 50
	dtheta = (sc-ec)/(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

# engine parameters
bore = 0.1
stroke = 0.1
con_rod = 0.15
cr = 8

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

# Stage2
v2 = v_c

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

# Stage3
v3 = v2
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))

# Stage4
v4 = v1

p4 = p3*pow(v3,gamma)/pow(v4,gamma)

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.show()

# Thermal efficiency
n = 1 - (pow(cr,(1-gamma)))
print(n)

CONCLUSION :- In the process 1–2 the piston does work on the gas and in process 3–4 the gas does work on the piston during those isentropic compression and expansion processes, respectively. Processes 2–3 and 4–1 are isochoric processes; heat is transferred into the system from 2—3 and out of the system from 4—1 but no work is done on the system or extracted from the system during those processes. No work is done during an isochoric (constant volume) process because addition or removal of work from a system requires the movement of the boundaries of the system; hence, as the cylinder volume does not change, no shaft work is added to or removed from the system.