Chapter 4:Selection of Motor Power Rating

Example 4.1,Page no:49

In [3]:
import math 
from scipy.optimize import fsolve
#Variable declaration
P=30.0 #[Power  KW
theta1=30.0 #Temperature degree C
t1=40.0 # time in  min
theta2=45.0 #Final temperature in degree C

#Calculation
t2=80.0 #min(t2=2*t1)
#theta=theta_f*(1-exp(-t/T))"
def f(T):
    return((1-math.exp(-80.0/T))/(1-math.exp(-40/T))-(theta2/theta1))
T=fsolve(f,1)
theta_f=theta1/(1.0-math.exp(-t1/T)) #degreeC
#Result
print"(a).Thermal time constant: ",round(T[0],2),"minutes"
print"(b).Final temperature rise: ",theta_f,"degrees"

 (a).Thermal time constant:  57.71 minutes
(b).Final temperature rise:  60.0 degrees

Example 4.2,Page no:50

In [7]:
import math 

#Variable declaration
P=30.0 #Power in KW
theta1=20.0 #Temerature degree C
t1=30.0 #Time in min
theta2=30.0 #Temperature in degree C
t2=60.0 #Final time innmin(t2=2*t1)

#Calculation
#Let exp(-t1/T)=x then exp(-t2/T)=x**2
#theta1/theta2=(1-x)/(1-x**2)
#x**2*theta1-x*theta2+theta2-theta1=0
def f(T):
    return(((1-math.exp(-t2/T))/(1-math.exp(-t1/T)))-theta2/theta1)
T=fsolve(f,1)
theta_f=theta1/(1-math.exp(-t1/T)) #degreeC

#Result
print"Thermal time constant  : ",round(T[0],2),"minutes"
print"Final temperature rise  : ",theta_f,"C"

 Thermal time constant  :  43.28 minutes
Final temperature rise  :  40.0 C

Example 4.3,Page no:51

In [11]:
import math 

#Variable declaration
T_ambient=30        #Ambient tmeperature in dgree C
P=30.0 #Power  in KW
theta1=54.0-T_ambient #Temperature durin g full load operation degree C
t1=1.0 #Time in hour
theta2=67.0-T_ambient #Temperature in degree C
t2=2.0 #Time in hour(t2=2*t1)

#Calculation
def f(T):
    return(((1-math.exp(-2/T))/(1-math.exp(-1/T)))-theta2/theta1)
T=fsolve(f,1)
theta_f=theta1/(1-math.exp(-t1/T)) #degreeC
theta_steady=theta_f+30 #degreeC

#Result
print"(a)  Final steady state temperature  : ",round(theta_steady,2),"C"
print"(b)  Heating time constant :",round(T[0],2),"hour"
theta2=theta_f #degree C
t=2.7 #hour
theta=40-30 #degree C
Tdash=-t/math.log(theta/theta2) #hour
print"(c)  Cooling time constant: ",round(Tdash,2),"hour"

 (a)  Final steady state temperature  :  82.36 C
(b)  Heating time constant : 1.63 hour
(c)  Cooling time constant:  1.63 hour

Example 4.4,Page no:52

In [13]:
import math 

#Variable declaration
T=110.0 #Time in min
Tdash=150.0 #Final time in min
t=30.0 #Run time of motor min
tdash=45.0 #Switch off time in min
theta_f=50.0 #final temperature rise in degree C

#Calculation
#theta=theta_f-(theta_f-theta1)*exp(-t/T)
#theta1=theta*exp(-tdash/Tdash) 
theta=(theta_f-theta_f*math.exp(-t/T))/(1-math.exp(-tdash/Tdash)*math.exp(-t/T)) #degreeC

#Result
print"Maximum temperature rise of the motor: ",round(theta,2),"C"

Maximum temperature rise of the motor:  27.37 C

Example 4.5,Page no:52

In [14]:
import math 

#Variable declaration
theta1=20.0 #Initial temperature in degreeC
theta2=28.0 #Final temperature in degreeC
dthetaBYdt1=0.08 #Rate of temp rise in degreeC/min
dthetaBYdt2=0.06 #Rate of temp rise in degreeC/min

#Calculation
#theta=theta_f-(theta_f-theta1)*exp(-t/T)
#dtheta/dt=(theta_f-theta)/T
#dthetaBYdt1/dthetaBYdt2=(theta_f-theta1)/(theta_f-theta2)
theta_f=(theta2*dthetaBYdt1-theta1*dthetaBYdt2)/(dthetaBYdt1-dthetaBYdt2)
T=(theta_f-theta1)/dthetaBYdt1 #min

#Result
print"Final temperature rise,theta_f= : ",theta_f,"C"
print"Heating time constant: ",T,"minutes"

Final temperature rise,theta_f= :  52.0 C
Heating time constant:  400.0 minutes

Example 4.6,Page no:59

In [15]:
import math 

#Variable declaration
cycle1=50.0 #Duty cycle hp
t1=20.0 #Time in sec
cycle2=100.0 #Duty cycle in hp
t2=20.0 #Time in sec
cycle3=150.0 #3rd duty cycle in hp
t3=10.0 #time sec
cycle4=120.0 #4th duty cyclein hp
t4=20.0 #time sec
cycle5=0.0 #th duty cycle in hp
t5=15.0 #sec

#Calculation
hp_rms=math.sqrt((cycle1**2*t1+cycle2**2*t2+cycle3**2*t3+cycle4**2*t4+cycle5**2*t5)/(t1+t2+t3+t4+t5)) #hp

#Result
print"hp(rms) for the motor : ",round(hp_rms,2),"hp"
print"We should choose 100hp motor."

hp(rms) for the motor :  94.74 hp
We should choose 100hp motor.

Example 4.7,Page no:61

In [17]:
import math 

#Variable declaration
t_on=15.0 #Time for full load [min]
t_off=25.0 #time for off load in [min]
T=100.0 #Heating time constant in [min]
Tdash=140.0 #Cooling time constant in [min]
theta_f=55.0 #Continuous full load operation time in [degree C]

#Calculation
#theta=theta_f-(theta_f-theta1)*exp(-t/T)
#theta1=theta*exp(-tdash/Tdash) 
theta_max=theta_f*(1-math.exp(-t_on/T))/(1-math.exp(-(t_off/Tdash+t_on/T))) #degreeC

#Result
print"Maximum temperature rise: ",round(theta_max,2),"C"

Maximum temperature rise:  27.36 C

Example 4.8,Page no:62

In [18]:
import math 


#Variable declaration
Rating=100.0 #Rating in KW
alfa=0.9 #Ratio rating unitless
ts=20.0 #Running time in min
T=100.0 #Heating time constant in min

#Calculation
S=math.sqrt((1.0+alfa)/(1.0-math.exp(-ts/T))) 
ShortTimeRating=S*Rating #KW

#Result
print"Short time rating: ",round(ShortTimeRating,1),"kW"
print"NOTE:Answer is wrong in the textbook."

Short time rating:  323.8 kW
NOTE:Answer is wrong in the textbook.

Example 4.9,Page no:62

In [20]:
import math 

#Variable declaration
T=80.0 #Heating time constant in min
Tdash=110.0 #cooling time constant in min
Rating=50.0 #Full load rating in [KW]
ts=15.0 #Runt time in [min]

#Calculation
S=math.sqrt(1.0/(1.0-math.exp(-ts/T))) 
ShortTimeRating=S*Rating #KW
t_off=20.0 #min
S=math.sqrt((1.0-math.exp(-(ts/T+t_off/Tdash)))/(1.0-math.exp(-(ts/T))))
DutyRating=S*Rating #KW

#Result
print"(a)  Short time rating of motor: ",round(ShortTimeRating,1),"kW"
print"(b)  Intermittent periodic duty rating: ",round(DutyRating,2),"kW"

(a)  Short time rating of motor:  120.9 kW
(b)  Intermittent periodic duty rating:  67.2 kW

Example 4.10,Page no:63

In [22]:
import math 

#Variable declaration
T=90.0 #Heating time constant in [min]
t=25.0 #time in min 
ShortTimeRating=50.0 #Short time rating in KW
Eff=80/100.0 #Efficiency

#Calculation
#Let full load rating is P KW and Losses=Pc
#CuLoss=(P/(P*Eff))**2 & alfa=Pc/CuLoss
alfa=(Eff)**2 #unitless
S=math.sqrt(((1.0+alfa)/(1.0-math.exp(-t/T))-alfa)) 
ContinuousRating_fl=ShortTimeRating/S #KW

#Result
print"Full load continuous rating: ",round(ContinuousRating_fl,1),"kW"

Full load continuous rating:  20.2 kW

Example 4.11,Page no:63

In [23]:
import math 

#Variable declaration
Rating=25.0 #Rating in [KW]
T=90.0 #Heating time constant in min
ts=30.0 #time in min

#Calculation
S=math.sqrt(1.0/(1.0-math.exp(-ts/T))) 
HalfHourRating=S*Rating #KW

#Result
print"Half hour rating of motor: ",round(HalfHourRating,2),"kW"
print"Answer wrong in textbook."

Half hour rating of motor:  46.96 kW
Answer wrong in textbook.

Example 4.12,Page no:63

In [25]:
#Variable declaration:
ts=20        #rating itme in [min]
W=300         #rating in [W]
T=60         #time constant in [minutes]
max_eff_load=0.8      #Maximum effeciency of motor at load

#Calculations
import sympy
P=sympy.Symbol('P')
pc=sympy.Symbol('pc')
Full_load_CL=(P/(0.8*P))**2*pc
alpha=pc/Full_load_CL
S=math.sqrt((1.0+alpha)/(1-math.exp(-ts/64.0))-alpha)
Cont_rating=W/round(S,1)

#Result
print"Continuous rating of motor is :",round(Cont_rating,1),"W"
print"NOTE:There is a slight error in book in calculation of S,approximate value is taken"

Continuous rating of motor is : 130.4 W
NOTE:There is a slight error in book in calculation of S,approximate value is taken
Continuous rating of motor is : 130.4 W
NOTE:There is a slight error in book in calculation of S,approximate value is taken

Example 4.13,Page no:67

In [26]:
import math 

#Variable declaration
P=6.0 #poles
f=50.0 #Frequency in [Hz]
MoI=9.5 #Moment of inertia in [Kg-m**2]
Tr=550.0 #Torque in [N-m]
S=5.0/100.0 #Slip
Tmax=720.0 #MAximum torque in [N-m]
T_LH=1020.0 #Torque requirement in [N-m]
th=12.0 #Time in [sec]
Tmin=220.0 #Light torque req in [N-m]
Snl=3.0/100.0 #No load slip
Ns=120.0*f/P #Speed in rpm

#Calculation
Nnl=Ns-Ns*Snl #rpm
Nrated=Ns-Ns*S #rpm
omega_mo=Nnl*2.0*math.pi/60.0 #rad/s
omega_mr=Nrated*2.0*math.pi/60.0 #rad/s
J=(Tr/(omega_mo-omega_mr))*(th/math.log((T_LH-Tmin)/(T_LH-Tmax))) #Kg-m**2
MoI_flywheel=J-MoI #Kg-m**2

#Result
print"Moment of inertia of flywheel : ",MoI_flywheel,"kg-m^2"
print"NOTE:Answer in the book is wrong."

Moment of inertia of flywheel :  3203.36081501 kg-m^2
NOTE:Answer in the book is wrong.