Week 2 Air standard Cycle

AIM: To write MATLAB 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: The Otto cycle is the ideal cycle for the spark-ignition reciprocating engines, and it consists of four internally reversible processes:

1. isentropic compression,
2. constant-volume heat addition,
3. isentropic expansion, and
4. constant-volume heat rejection.

Figure: Otto cycle (P-v diagram)

GOVERNING EQUATIONS:-

The ratio of the maximum volume formed in the cylinder to the minimum

(clearance) volume is called the compression ratio r of the engine:

r = vmaxvmin=displacement volumecIearance volumevmaxvmin=displacement volume clearance volume

To find the efficiency of the cycle T₃ and T₄ are needed.

Process 1-2:

This process is an isentropic process. Therefore

T2T1=(V1V2)γ−1=rγ−1T2T1=(V1V2)γ-1=rγ-1

Process 3-4:

This is also an isentropic process. Therefore

T3T4=(V3V4)γ−1=rγ−1T3T4=(V3V4)γ-1=rγ-1

Air-standard efficiency of the Otto cycle:

The thermal efficiency of the ideal Otto cycle under the cold air standard w_net

assumptions are:

ηth=wq=1−T4−T1T3−T2ηth=wq=1-T4-T1T3-T2

On simplification

ηth=1−1rγ−1ηth=1-1rγ-1

The relationship between crank angle and volume inside the crank chamber is: -

VVc=1+0.5⋅(r−1)⋅[R+1−cos〖θ−(R2−〖sinθ〗2)0.5〗]VVc=1+0.5⋅(r-1)⋅[R+1-cos⁡〖θ-(R2-〖sin⁡θ〗2)0.5〗]

OBJECTIVES OF THE PROJECT

• To write a 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.

Figure 2: Flow of the project

Program:

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)

Output:

Explanation of code

1. A function is written for the following equation to find V:
2. In the main code the Input values are given (gamma, t₃, engine parameters, etc)
3. Equations for swept volume and clearance volume are written.
4. p_2,t_2, and t₃ are calculated using the isentropic condition.
5. Plot command is used for plotting the P-V diagram.

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.