# Chapter 11 : Turbomachines:Elementary Analysis¶

## Example 11.1 Page No : 426¶

In [26]:
import math
from sympy.functions.elementary.trigonometric import acot

#initialisation of variables
rt= 1.3 	#ft
rr= 0.6 	#ft
Q= 75.   	#ft**3 flow rate
rm= 0.95
w1= 40.  	#rev/sec
bim= 153. 	#degrees blade inlet angle
bom= 147. 	#degrees blade outlet angle
w= 62.4 	#lb/ft**3
g= 32.2 	#ft/sec**2

#CALCULATIONS
A= round(math.pi*(rt**2-rr**2),2)
Va= round(Q/A,2)
Vbm= rm*w1
#a= -1/math.degrees(math.atan(-Vbm/Va))
a = math.degrees(acot(-Vbm/Va))
im= a-bim
vwm= Vbm+Va*1/math.tan(math.radians(bom))
dvwm= rm*vwm
C= w*Q*dvwm/g
Cw= C*w1
dp= Cw/Q

#RESULTS
print  ' Incidence = %.1f degrees'%(im)
print  ' Oulet velocity = %.2f ft/sec'%(vwm)
print  ' Change of whirl at the mean radius = %.2f ft**2/sec'%(dvwm)

print  ' Torque = %.f lbf/ft'%(C)
print  ' Rate of working = %.f ft lbf/sec'%(Cw)
print  ' Workdone by the rotor = %.f lbf/ft**2'%(dp)

# note : answer in book is wrong. please check manually.
Incidence = -178.3 degrees
Oulet velocity = 10.37 ft/sec
Change of whirl at the mean radius = 9.86 ft**2/sec
Torque = 1432 lbf/ft
Rate of working = 57300 ft lbf/sec
Workdone by the rotor = 764 lbf/ft**2

## Example 11.2 Page No : 428¶

In [3]:
import math

#initialisation of variables
vbm= 38. 	#ft/sec
va= 17.94 	#ft/sec
a= 147.5 	#degrees
vwm= 10.37 	#ft/sec
C= 1430. 	#lbf/ft
P= 763. 	#lbf/ft**2

#CALCULATIONS
vwm1= vbm+va*1/math.tan(math.radians(a))
p= (vwm-vwm1)/vwm
C1= C*(1-p)
P1= P*(1-p)

#RESULTS
print  ' Oulet Velocity = %.2f ft/sec'%(vwm1)
print  ' Torque = %.f lbf/ft'%(round(C1,-1))
print  ' Workdone by the rotor = %.f lbf/ft**2'%(P1)
Oulet Velocity = 9.84 ft/sec
Torque = 1360 lbf/ft
Workdone by the rotor = 724 lbf/ft**2

## Example 11.3 Page No : 430¶

In [27]:
import math
from sympy.functions.elementary.trigonometric import acot

#initialisation of variables
a= 154 	    #degrees
vbm= 38 	#ft/sec
bom= 147 	#degrees outlet angle
vwm= -7.78 	#ft/sec outlet whirl velocity
w= 62.4 	#lbf/ft**3
g= 32.2 	#ft/sec**2
vb= 38    	#ft/sec velocity
A= 4.18 	#ft**2 flow area
e= 0.95

#CALCULATIONS
vat= (vwm-vb)*math.tan(math.radians(bom))
Q= vat*A
#a1= 1/math.tan(math.radians(-vbm/vat))
a1 = math.degrees(acot(-vbm/vat))
imt= a1-a
C= w*Q*vwm*e/g

#RESULTS
print  ' Flow rate = %.1f ft**3'%(Q)
print  ' Incidence angle= %.f degrees'%(imt)
print  ' Torque= %.f lbf ft'%(C)
#Incorrect value for a1 in textbook. Hence the difference in answers
Flow rate = 124.3 ft**3
Incidence angle= -192 degrees
Torque= -1780 lbf ft

## Example 11.4 Page No : 435¶

In [28]:
import math
from numpy import *

#initialisation of variables
rt= 0.5 	#ft radius
rr= 0.16 	#ft root radius
dv1= 88.3 	#ft/sec
b= 150.  	#degrees
r= array([0.16, 0.3, 0.5])
vw= array([2.5, 5, 7.5])
vb= array([46.6, 88.3, 132.5])
vrb= array([44.16, 88.3, 132.5])
v1= array([-1.154, -0.385])

#CALCULATIONS
A= math.pi*(rt**2-rr**2)
Va= -dv1*math.tan(math.radians(b))
Q= Va*A
ari = degrees((arctan(Va/(vw - vb)))) + 180
ari = array([ari[0],ari[2]])
#a= tan(radians(v1))+180
b = degrees(math.tan(0.577))
i = ari - 150

#RESULTS
print  ' Velocity = %.2f ft/sec'%(Va)
print  ' Flow rate = %.1f ft**3'%(Q)

print (v1)
print (ari)
print (i)

# rounding off error. and for 'i' answer is wrong. please check.
Velocity = 50.98 ft/sec
Flow rate = 35.9 ft**3
[-1.154 -0.385]
[ 130.86126801  157.81238868]
[-19.13873199   7.81238868]

## Example 11.5 Page No : 436¶

In [6]:
import math
from numpy import *

#initialisation of variables
rt= 0.5 	#ft
rr= 0.16 	#ft
dv1= 88.3 	#ft/sec
b= 150.  	#degrees
a= 5.    	#degrees mean radius
v1= array([-0.933 ,-0.311])
i= array([1.0, 5.0 ,6.7])

#CALCULATIONS
b1= b+a
A= math.pi*(rt**2-rr**2)
Va= -dv1*math.radians(math.tan(b1))
Q= Va*A
a1= degrees(tan(v1))+180

#RESULTS
print  ' Velocity = %.2f ft/sec'%(Va)
print  ' Flow rate = %.1f ft**3/sec'%(Q)

print (v1)
print (a1)
print (i)
#Incorrect calculations in textbook
Velocity = -2.76 ft/sec
Flow rate = -1.9 ft**3/sec
[-0.933 -0.311]
[ 102.69071396  161.58339075]
[ 1.   5.   6.7]

## Example 11.6 Page No : 439¶

In [29]:
import math

#initialisation of variables
r= 1.    	#in
b= 0.75 	#in rotor inlet width
w= 180.  	#rev/sec
B= 120. 	#degrees blade inlet angle
Bo= 150. 	#degrees blade outlet angle
ro= 3.   	#ft
bo= 0.5 	#ft
Vbo= 180. 	#ft/sec
w1= 62.4 	#lbf/ft**3 density
g= 32.2 	#ft/sec**2

#CALCULATIONS
Q= -2*math.pi*(r/12)**2*(b/12)*w*math.tan(math.radians(B))
Vfo= Q/(2*math.pi*(ro/12)*(bo/12))
Vwo= Vbo*(ro/12)+Vfo*1/math.tan(math.radians(Bo))
C= w1*Q*Vwo*(ro/12)/g
dp= w1*Vwo*w*(ro/12)/g
ari= degrees(math.atan((-Q*0.8/(2*math.pi*(r/12)**2*(b/12)*w))))+180
i1= ari-B

#RESULTS
print  ' Flow rate = %.2f ft**3/sec'%(Q)
print  ' radial velocity= %.2f ft/sec'%(Vfo)
print  ' outlet whirl velocity= %.2f ft/sec'%(Vwo)
print  ' Torque= %.2f lbf ft'%(C)
print  ' Stagnant pressure = %.f lbf/ft**2'%(dp)
print  ' Incidence angle = %.1f degrees'%(i1)
Flow rate = 0.85 ft**3/sec
radial velocity= 12.99 ft/sec
outlet whirl velocity= 22.50 ft/sec
Torque= 9.27 lbf ft
Stagnant pressure = 1962 lbf/ft**2
Incidence angle = 5.8 degrees

## Example 11.7 Page No : 447¶

In [30]:
import math

#initialisation of variables
r= 1.4
Mai= 0.5    	#ft/sec mach number
T= 582.     	#R temperature
psi= 3040.  	#lbf/in**2 pressure
R= 53.3 	    #ft lbf/lbm gas
g= 32.2      	#ft/sec**2
Vwi= 300.    	#ft/sec velocity
m= 35.       	#lb/sec
rm= 0.7 	    #ft radius
rp= 4.25
w= 1200.     	#rev/sec
cp= 0.24
J= 778.      	#lb

#CALCULATIONS
tr= 1+0.5*(r-1)*Mai**2
Ti= round(T/tr)
pr= tr**(r/(r-1))
pi= psi/pr
ai= pi/(R*Ti)
Vi= Mai*(r*R*g*Ti)**0.5
Vai= math.sqrt(Vi**2-Vwi**2)
h= m/(2*math.pi*ai*rm*Vai)
pr1= rp**(1./12)
Vwo= Vwi+(pr1**((r-1)/r)-1)*(cp*J*g*T/(rm*w))
BO= 1/math.tan(math.radians((Vwo-w*rm)/Vai))

#RESULTS
print  ' Absolute air velocity = %.f ft/sec'%(Vi)
print  ' air velocity = %.f ft/sec'%(Vai)
print  ' Blade height = %.3f ft'%(h)
print  ' velocity = %.f ft/sec'%(Vwo)
print  ' outlet balde angle = %.1f degrees'%(BO) #incorrect answer in the textbook
Absolute air velocity = 577 ft/sec
air velocity = 493 ft/sec
Blade height = 0.186 ft
velocity = 446 ft/sec
outlet balde angle = -71.7 degrees