Week 2 Air standard Cycle

• AIM:

               The main aim of the challenge is to code a program to plot P-V diagram of Air Standard cycle using python.

• OBJECTIVE:

                Otto cycle or constant volume cycle,on this cycle petrol engine ,gas and many types of engine work.It is a standard or comparision cycle to the internal combustion engine.The p-v and T-s diagram are shown in the figure.

                   

• THERMAL EFFICIENCY OF AN OTTO CYCLE:

              Let us consider 1kg of air(working substance)

                                        Heat supplied at constant volume = c_v*(t3-t2)                                     c_v = specific heat at constant volume in j/kg-k                 

                                        Heat rejected at constant pressure = c_v*(t4-t1) 

                                             efficiency = workdone/heat supplied 

                                                           = 1-(t4-t1)/(t3-t2) ............................. (1)

                                                          compression ratio c_r = v1/v2

                 from 1 - 2

                              As it is isentropic compression   p1*v1^gamma = p2*v2^gamma                              gamma = c_p/c_v

                                                          t2/t1 = (v1/v2)^(gamma-1) ........................(2)

                 from 3-4

                             As it is isentropic expansion      p3*v3^gamma = p4*v4^gamma                             

                                                            t3/t4 = (v4/v3)^(gamma-1)........................(3)

                    substitute the equation (2) and (3) in equation (1)

                                       EFFICIENCY = 1-1/(C_R)^(GAMMA-1)

                         This expression is known as air standard or thermal efficiency of an otto cycle.

• CODING:

#Importing the required modules
import math
import matplotlib.pyplot as plt

#Defining the function
def engine_kinematics(bore,stroke,con_rod,cr,start_crank,end_crank):

    #Geometric parameters
    a = stroke/2
    R=con_rod/a

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

    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 = 0.1  #pressure at state1
t1 = 500  #Temperature at state1
gamma = 1.4
t3=2300   #Temperature at state 3

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

#Volume calculation
V_s = (math.pi/4)*pow(bore,2)*stroke  #swept volume
V_c = V_s/(cr-1)                      #clearence volume
V1 = V_c+V_s

#At state point2
V2=V_c

#P2V2^gamma = P1V1^gamma
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))

#At stste point3
V3 = V2
#P3V3/t3 = P2V2/t2 |rhs = P2V2/t2 | P2 = 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))

#At state point4
V4 = V1

#P4V4^gamma = P3V3^gamma
P4 = P3*pow(V3,gamma)/pow(V4,gamma)
t4 = P4*V4/rhs

#Thermal Efficiency of engine
Thermal_eff = 1-(pow(cr,(1-gamma)))
print(Thermal_eff)

#Plotting
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('Otto Cycle')
plt.show()

• EXPLANATION:

1. Import math and plot module because the code has to plot p-v diagram of otto cycle.
2. Enter the initial conditions `P_1`,`t_1`and `t_3` and `gamma=1.4`.
3. Enter all the geometrical parameters which consists of bore,stroke,length of connecting rod and compression ratio and calculate swept volume and clearence volume using the formula.
4. State variables are calculated using Adiabatic relation and ideal gas equation.
5. Volume change during compression and expansion process is determined by the relation.
6. Plot pressure vs volume using plot command as shown.
7. Calculate the thermal efficiency using the relation.

• PLOT:

    

• OUTPUT:

• CONCLUSION:

                            Thermal efficiency of an otto cycle is found to be 62.98% and p-v diagram of otto cycle has been plotted.