Chapter 9 - Hydraulic Turbines

Ex 9.1 Page 406

In [1]:
from __future__ import division

#input data
H=91.5#Head of the pelton wheel at inlet in m
Q=0.04#Discharge of the pelton wheel in m**3/s
N=720#Rotating speed of the wheel in rpm
Cv=0.98#Velocity coefficient of the nozzle 
n0=0.8#Efficiency of the wheel
UC1=0.46#Ratio of bucket speed to jet speed
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
P=dw*g*H*Q*n0*10**-3#Power developed in kw
C1=Cv*(2*g*H)**(1/2)#Jet speed in m/s
U=UC1*C1#Wheel speed in m/s
w=(2*3.1415*N)/60#Angular velocity of the wheel in rad/s
D=(2*U)/w#Diameter of the wheel in m
A=Q/C1#Jet area in m**2
d=((4*A)/3.1415)**(1/2)#Jet diameter in m
Dd=D/d#Wheel to jet diameter ratio at centre line of the buckets
Nsp=((1/(g*H))**(5/4))*(((P*10**3)/dw)**(1/2))*(N/60)*2*3.1415#Dimensionless power specific speed in rad

#output
print '(a)Wheel-to-jet diameter ratio at the centre line of the buckets is %3.1f \n(b)\n    The jet speed of the wheel is %3.2f m/s\n    Wheel speed is %3.1f m/s\n(c)Dimensionless power specific speed is %3.3f rad'%(Dd,C1,U,Nsp)

(a)Wheel-to-jet diameter ratio at the centre line of the buckets is 14.5 
(b)
    The jet speed of the wheel is 41.52 m/s
    Wheel speed is 19.1 m/s
(c)Dimensionless power specific speed is 0.082 rad

Ex 9.2 Page 407

In [2]:
#input data
H=500#Head over which pelton wheel works in m
P=13000#Power which pelton wheel produces in kW
N=430#Speed of operation of pelton wheel in rpm
n0=0.85#Efficiency of the wheel 
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3
Cv=0.98#Veloity coefficient
UC=0.46#Speed ratio

#calculations
Q=(P*10**3)/(dw*g*H*n0)#Discharge of the turbine in m**3/s
C=Cv*(2*g*H)**(1/2)#Jet speed in m/s
U=UC*C#Wheel speed in m/s
D=(U*60)/(3.1415*N)#Wheel diameter in m
d=((Q/C)*(4/3.1415))**(1/2)#Diameter of the nozzle in m

#output
print '(a)Discharge of the turbine is %3.2f m**3/s\n(b)Diameter of the wheel is %3.2f m\n(c)Diameter of the nozzle is %3.3f m'%(Q,D,d)

(a)Discharge of the turbine is 3.12 m**3/s
(b)Diameter of the wheel is 1.98 m
(c)Diameter of the nozzle is 0.202 m

Ex 9.3 Page 409

In [3]:
from math import pi, cos
#input data
D=0.8#Mean diameter of the bucket in m
N=1000#Running speed of the wheel in rpm
H=400#Net head on the pelton wheel in m
Q=0.150#Discharge through the nozzle in m**3/s
g=9.81#Acceleration due to gravity in m/s**2
UC1=0.46#Ratio of bucket speed to jet speed
dw=1000#Density of water in kg/m**3
a=15#Side clearance angle in degree

#calculations
m=dw*Q#Mass flow rate through the nozzle in kg/s
U=(3.1415*D*N)/60#Wheel speed in m/s
C1=U/UC1#Jet speed in m/s
P=(1/2)*m*C1**2*(10**-3)#Power available at the nozzle in kW
W1=C1-U#Relative inlet fluid velocity in m/s
W2=W1#Relative exit fluid velocity in m/s assuming no loss of relative velocity
Wx2=W2*cos(a*pi/180)#Exit whirl velocity component in m/s
Cx2=Wx2-U#Absolute exit whirl velocity in m/s
Cx1=C1#Absolute inlet whirl velocity in m/s
Wm=U*(Cx1+Cx2)#Work done per unit mass flow rate in W/(kg/s)
nH=(Wm/g)/((C1**2/2)/g)#Hydrualic effciency 

#output
print '(a)Power available at the nozzle is %3.3f kW\n(b)Hydraulic efficiency is %.1f %%'%(P,nH*100)

(a)Power available at the nozzle is 621.867 kW
(b)Hydraulic efficiency is 97.7 %

Ex 9.4 Page 409

In [4]:
#input data
n=2#Number of jets 
SP=20000*0.736#Shaft power of the wheel in kW
D=0.15#Diameter of each jet in m
H=500#Net head on the turbine in m
Cv=1.0#Velocity coefficient
g=9.81#Acceleration due to gravity in m/s**2
d=1000#Density of water in kg/m**3

#calculations
C1=Cv*(2*g*H)**(1/2)#Velocity of each jet in m/s
A=(3.1415/4)*D**2#Area of each jet in m**2
Qj=A*C1#Discharge of each jet in m**3/s
Q=2*Qj#Total discharge in m**3/s
P=d*g*Q*H*10**-3#Power at turbine inlet in kW
no=SP/P#Overall efficiency

#output
print 'The overall efficiency of the turbine is %0.1f %%'%(no*100)

The overall efficiency of the turbine is 85.7 %

Ex 9.5 Page 410

In [5]:
#input data
a=170#Jet deflection angle in degree
K=1-0.12#Percentage of effective relative velocity after considering friction
UC1=0.47#Ratio of bucket speed to jet speed
GH=600#Gross head on the wheel in m
P=1250#Actual power developed by the wheel in kW
Hl=48#Head loss in nozzle due to pipe friction in m
D=0.9#Bucket circle diameter of the wheel in m
ATnH=0.9#The ratio between actual and calculated hydraulic efficiency
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3
Cv=0.98#Velocity coefficient

#calculations
H=GH-Hl#Net head after loses at entry to nozzle in m
C1=Cv*(2*g*H)**(1/2)#Jet speed in m/s
U=UC1*C1#Wheel bucket speed in m/s
N=(U*60)/(3.1415*D)#Wheel rotational speed in rpm
Wm=U*((C1-U)*(1-(K*cos(a*pi/180))))#Work done per unit mass flow rate in W/(kg/s)
Tnh=Wm/(C1**2/2)#Theoretical hydraulic efficiency 
Anh=ATnH*Tnh#Actual hydrualic effficiency
m2=(P*10**3)/(Anh*(1/2)*C1**2)#Mass flow rate for both the nozzles in kg/s
m=m2/2#Mass flow rate of each nozzle in kg/s
d=((4*m)/(dw*C1*3.1415))**(1/2)#Nozzle diameter in m

#output
print '(a)theoretical hydraulic efficiency is %3.2f \n(b)Wheel rotational speed is %3.f rpm\n(c)diameter of the nozzle is %0.1f mm'%(Tnh,N,d*1000)

(a)theoretical hydraulic efficiency is 0.93 
(b)Wheel rotational speed is 1017 rpm
(c)diameter of the nozzle is 42.3 mm

Ex 9.6 Page 413

In [6]:
#input data
H=60#Head on the pelton wheel in m
N=200#Speed of the pelton wheel in rpm
P=100#Power developed by the pelton wheel in kW
Cv=0.98#Velocity coefficient
UC1=0.45#Speed ratio 
n0=0.85#Overall efficiency of the wheel
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
C1=Cv*(2*g*H)**(1/2)#Velocity of the jet in m/s
U=UC1*(2*g*H)**(1/2)#Velocity of the buckets in m/s
D=(60*U)/(3.1415*N)#Diameter of the wheel in m
Q=(P*10**3)/(dw*g*H*n0)#Discharge of the wheel in m**3/s
d=((4*Q)/(3.1415*C1))**(1/2)#Diameter of the jet in m
Z=15+(D/(2*d))+1#Number of buckets rounding off to nearest decimal as the final answer has a decimal value less than 0.5
w=5*d#Width of the buckets in m
de=1.2*d#Depth of the buckets in m

#output
print '(a)Diameter of the wheel is %3.2f m\n(b)Diameter of the jet is %3.3f m\n(c)Number of buckets is %3.f\n(d)Size of the buckets is \n    width of the bucket is %3.3f m\n    Depth of the bucket is %3.3f m'%(D,d,Z,w,de)

(a)Diameter of the wheel is 1.47 m
(b)Diameter of the jet is 0.087 m
(c)Number of buckets is  24
(d)Size of the buckets is 
    width of the bucket is 0.435 m
    Depth of the bucket is 0.104 m

Ex 9.7 Page 414

In [7]:
#input data
N=300#Running speed of the wheel in rpm
H=150#OPerating head of the wheel in m
dD=1/12#Ratio of nozzle diameter to wheel diameter
Cv=0.98#Velocity coefficient
UC1=0.46#Speed ratio
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3
n0=0.84#Overall efficiency

#calculations
C1=Cv*(2*g*H)**(1/2)#Velocity of jet in m/s
U=UC1*(2*g*H)**(1/2)#Velocity of the wheel in m/s
D=(60*U)/(3.14*N)#Diameter of the wheel in m
d=D*dD#Diameter of the jet in m
Q=(3.1415/4)*(d**2)*C1#Quantity of water required in m**3/s
Pa=dw*g*Q*H#Power available at the nozzle in kW
P=n0*Pa*10**-3#Power developed in kW
#output
print '(a)Diameter of the wheel is %3.2f m\n(b)Diameter of the jet is %3.3f m\n(c)Quantity of water required is %3.3f m**3/s\n(d)Power developed is %3.1f kW'%(D,d,Q,P)
# Answer in the textbook is wrong.

(a)Diameter of the wheel is 1.59 m
(b)Diameter of the jet is 0.132 m
(c)Quantity of water required is 0.733 m**3/s
(d)Power developed is 905.5 kW

Ex 9.8 Page 415

In [8]:
from math import tan, pi
#input data
N=1260#Rotational speed of the francis turbine in rpm
H=124#The net head in m
Q=0.5#Volume flow rate of the turbine in m**3/s
r1=0.6#Radius of the runner in m
b1=0.03#Height of the runner vanes at inlet in m
b11=72#Angle of inlet guide vanes in radial direction in degree
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3
Cx2=0#Absolute exit whirl velocity in m/s as flow is radial at outlet

#calculations
m=dw*Q#Mass flow rate in kg/s
T1=-m*r1#Torque by the turbine in Nm in terms of Cx1
A=2*3.1415*r1*b1#Area at inlet in m**2
Cr1=Q/A#Inlet flow velocity in m/s
Cx1=Cr1*tan(b11*pi/180)#Absolute inlet whirl velocity in m/s
T=-T1*Cx1#Torque by water on the runner in Nm
w=(2*3.1415*N)/60#Angular velocity of the turbine in rad/s
W=T*w*10**-3#Power exerted in kW
nH=W*10**3/(dw*g*Q*H)#Hydraulic efficiency 

#output
print '(a)Torque by water on the runner is -%3.f Nm\n(b)Power exerted is %3i kW\n(c)Hydraulic efficiency is %0.1f %%'%(T,W,nH*100)

(a)Torque by water on the runner is -4082 Nm
(b)Power exerted is 538 kW
(c)Hydraulic efficiency is 88.6 %

Ex 9.9 Page 416

In [9]:
from math import atan, degrees
#input data
n0=0.74#Overall efficiency
H=5.5#Net head across the turbine in m
P=125#Required Power output in kW
N=230#Speed of the runner in rpm
nH=(1-0.18)#Hydraulic efficiency
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3
U1=0.97*(2*g*H)**(1/2)#Runner tangential velocity in m/s
Cr1=0.4*(2*g*H)**(1/2)#Flow velocity in m/s

#calculations
Cx1=(nH*g*H)/U1#Absolute inlet whirl velocity in m/s    as flow is radial at outlet Cx2=0  in m/s
a11=degrees(atan(Cr1/Cx1))#Inlet guide vane angle in degree
b11=180+degrees(atan(Cr1/(Cx1-U1)))#Angle of inlet guide vanes in radial direction in degree
D1=(U1*60)/(3.1415*N)#Runner inlet diameter in m
Q=(P*10**3)/(n0*dw*g*H)#Flow rate in m**3/s
b1=Q/(3.1415*D1*Cr1)#Height of runner in m

#output
print '(a)Inlet guide vane angle is %3.1f degree\n(b)Angle of inlet guide vanes in radial direction is %3.1f degree\n(c)Runner inlet diameter is %3.3f m\n(d)Height of runner is %3.3f m'%(a11,b11,D1,b1)

(a)Inlet guide vane angle is 43.4 degree
(b)Angle of inlet guide vanes in radial direction is 143.8 degree
(c)Runner inlet diameter is 0.837 m
(d)Height of runner is 0.287 m

Ex 9.10 Page 418

In [10]:
#input data
D=1.4#Diameter of the turbine in m
N=430#Speed of the turbine in rpm
Cr1=9.5#Flow velocity without shock at runner in m/s
C2=7#Absolute velocity at the exit without whirl in /s
dSPH=62#Difference between the sum of static and potential heads at entrance to runner and at exit from runner in m
W=12250#Power given to runner in kW
Q=12#Flow rate of water from the turbine in m**3/s
H=115#Net head from the turbine in m
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
U1=(3.1415*D*N)/60#Runner tip speed in m/s
Cx1=(W*10**3)/(dw*Q*U1)#Absolute inlet velocity in m/s    as flow is radial at outlet Cx2=0  in m/s as Cx2=0 as zero whirl at outlet
a1=degrees(atan(Cr1/Cx1))#Guide vane angle in degree
C1=(Cr1**2+Cx1**2)**(1/2)#Inlet velocity in  m/s
b1=degrees(atan(Cr1/(Cx1-U1)))#Runner blade entry angle in degree
dHr=dSPH+(((C1**2)-(C2**2))/(2*g))-(U1*Cx1/g)#Loss of head in the runner in m

#output
print '(a)\n    (1)Guide vane angle at inlet is %3.1f degree\n    (2)Inlet absolute velocity of water at entry to runner is %3.1f m/s\n(b)Runner blade entry angle is %3.1f degree\n(c)Total Loss of head in the runner is %3.2f m'%(a1,C1,b1,dHr)

(a)
    (1)Guide vane angle at inlet is 16.3 degree
    (2)Inlet absolute velocity of water at entry to runner is 33.8 m/s
(b)Runner blade entry angle is 84.8 degree
(c)Total Loss of head in the runner is 13.50 m

Ex 9.11 Page 420

In [11]:
from math import sin, tan, degrees
#input data
D1=0.9#External diameter of the turbine in m
D2=0.45#Internal diameter of the turbine in m
N=200#Speed of turbine running in rpm
b1=0.2#Width of turbine at inlet in m
Cr1=1.8#Velocity of flow through runner at inlet in m/s
Cr2=Cr1#Velocity of flow through runner at outlet in m/s
a11=10#Guide blade angle to the tangent of the wheel in degree
a22=90#Discharge angle at outlet of turbine in degree
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
C1=Cr1/sin(a11*pi/180)#Absolute velocity of water at inlet of runner in m/s
Cx1=Cr1/tan(a11*pi/180)#Velocity of whirl at inlet in m/s
U1=(3.1415*D1*N)/60#Runner tip speed at inlet in m/s
Wx1=Cx1-U1#Inlet whirl velocity component in m/s
W1=(Wx1**2+Cr1**2)**(1/2)#Relative velocity at inlet in m/s
b11=degrees(atan(Cr1/Wx1))#Runner blade entry angle in degree
U2=(3.1415*D2*N)/60#Runner tip speed at exit in m/s
b22=degrees(atan(Cr2/U2))#Runner blade exit angle in degree
b2=D1*b1/D2#Width of runner at outlet in m
Q=3.1415*D1*b1*Cr1#Discharge of water in turbine in m**3/s
m=dw*Q#Mass of water flowing through runner per second in kg/s
V2=Cr2#Velocity of water at exit in m/s 
H=(U1*Cx1/g)+(V2**2/(2*g))#Head at the turbine inlet in m
W=m*U1*Cx1*10**-3#Power developed in kW
nH=(U1*Cx1/(g*H))#Hydraulic efficiency

#output
print '(a)Absolute velocity of water at inlet of runner is %3.3f m/s\n(b)Velocity of whirl at inlet is %3.3f m/s\n(c)Relative velocity at inlet is %3.3f m/s\n(d)\n    Runner blade entry angle is %3.2f degree\n    Runner blade exit angle is %3.2f degree\n(e)Width of runner at outlet is %3.1f m\n(f)Mass of water flowing through runner per second is %3.f kg/s\n(g)Head at the turbine inlet is %3.3f m\n(h)Power developed is %3.3f kW\n(i)Hydraulic efficiency is %0.2f %%'%(C1,Cx1,W1,b11,b22,b2,m,H,W,nH*100)

(a)Absolute velocity of water at inlet of runner is 10.366 m/s
(b)Velocity of whirl at inlet is 10.208 m/s
(c)Relative velocity at inlet is 1.963 m/s
(d)
    Runner blade entry angle is 66.47 degree
    Runner blade exit angle is 20.91 degree
(e)Width of runner at outlet is 0.4 m
(f)Mass of water flowing through runner per second is 1018 kg/s
(g)Head at the turbine inlet is 9.972 m
(h)Power developed is 97.925 kW
(i)Hydraulic efficiency is 98.34 %

Ex 9.12 Page 423

In [12]:
#input data
P=330#Power output from the turbine is kW
H=70#Head of operating turbine in m
N=750#Speed of the turbine in rpm
nH=0.94#Hydraulic efficiency
n0=0.85#Overall efficiency
FR=0.15#Flow ratio 
BR=0.1#Breadth ratio
D1D2=2#Ratio inner and outer diameter of runner
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
Cr1=FR*(2*g*H)**(1/2)#Flow velocity at inlet in m/s
Q=(P*10**3)/(dw*g*H*n0)#Discharge at outlet in m**3/s
D1=(Q/(nH*3.1415*BR*Cr1))**(1/2)#Runner inlet diameter in m
b1=BR*D1#Height of the runner vanes at inlet in m
U1=(3.1415*D1*N)/60#Runner tip speed at inlet in m/s
Cx1=(nH*g*H)/(U1)#Velocity of whirl at inlet in m/s
a11=degrees(atan(Cr1/Cx1))#Guide blade angle in degree
b11=degrees(atan(Cr1/(Cx1-U1)))#Runner vane angle at inlet in degree
D2=D1/D1D2#Runner outlet diameter in m
U2=(3.1415*D2*N)/60#Runner tip speed at outlet in m/s
Cr2=Cr1#Flow velocity at outlet in m/s
b22=degrees(atan(Cr2/U2))#Runner vane angle at outlet in degree
b2=D1*b1/D2#Width at outlet in m

#output
print '(a)Flow velocity at inlet is %3.2f m/s\n(b)Discharge at outlet is %3.3f m**3/s\n(c)Runner inlet diameter is %3.3f m\n(d)Height of the runner vanes at inlet is %3.4f m\n(e)Guide blade angle is %3.2f degree\n(f)    Runner vane angle at inlet is %3.2f degree\n       Runner vane angle at outlet is %3.2f degree\n(g)Runner outlet diameter is %3.4f m\n(h)Width at outlet is %3.4f m\n(i)Runner tip speed at inlet is %3.2f m/s\n(j)Velocity of whirl at inlet is %3.f m/s'%(Cr1,Q,D1,b1,a11,b11,b22,D2,b2,U1,Cx1)

(a)Flow velocity at inlet is 5.56 m/s
(b)Discharge at outlet is 0.565 m**3/s
(c)Runner inlet diameter is 0.587 m
(d)Height of the runner vanes at inlet is 0.0587 m
(e)Guide blade angle is 11.23 degree
(f)    Runner vane angle at inlet is 48.23 degree
       Runner vane angle at outlet is 25.75 degree
(g)Runner outlet diameter is 0.2934 m
(h)Width at outlet is 0.1174 m
(i)Runner tip speed at inlet is 23.05 m/s
(j)Velocity of whirl at inlet is  28 m/s

Ex 9.13 Page 424

In [13]:
#input data
H=30#Working head of the turbine in m
D1=1.2#Inlet wheel diameter in m
D2=0.6#Outlet wheel diameter in m
b11=90#Vane angle at entrance in degree
a11=15#Guide blade angle in degree
Cx2=0#Velocity of whirl at inlet in m/s
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
U11=1/tan(a11*pi/180)#Runner tip speed at inlet in m/s in terms of Cr1
Cr1=(H/((U11**2/g)+(1/(2*g))))**(1/2)#Flow velocity at inlet in m/s
Cr2=Cr1#Flow velocity at outlet in m/s
U1=Cr1*U11#Runner tip speed at inlet in m/s  
N=(60*U1)/(3.1415*D1)#Speed of the wheel in rpm
U2=(3.1415*D2*N)/60#Runner tip speed at inlet in m/s 
b22=degrees(atan(Cr2/U2))#Vane angle at exit in degree

#output
print '(a)Speed of the wheel is %3.2f rpm\n(b)Vane angle at exit is %3.2f degree'%(N,b22)

(a)Speed of the wheel is 268.27 rpm
(b)Vane angle at exit is 28.19 degree

Ex 9.14 Page 425

In [14]:
#input data
D1=0.6#Internal runner diameter in m
D2=1.2#External runner diameter in m
a11=15#Guide blade angle in degree
Cr1=4#Flow velocity at inlet in m/s
Cr2=Cr1#Flow velocity at outlet in m/s
N=200#Speed of the turbine in rpm
H=10#Head of the turbine in m
a22=90#Discharge angle at outlet in degree
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
U1=(3.1415*D1*N)/60#Runner tip speed at inlet in m/s 
U2=(3.1415*D2*N)/60#Runner tip speed at outlet in m/s 
Cx1=Cr1/tan(a11*pi/180)#Velocity of whirl at inlet in m/s
Wx1=Cx1-U1#Inlet whirl velocity component in m/s
b11=degrees(atan(Cr1/Wx1))#Vane angle at entrance in degree
b22=degrees(atan(Cr2/U2))#Vane angle at exit in degree
Wm=U1*Cx1#Work one per unit mass flow rate in W/(kg/s)     as Cx2=0 in m/s
nH=(U1*Cx1/(g*H))#Hydraulic efficiency 

#output
print '(a)\n    Inlet vane angle is %3.2f degree\n    Outlet vane angle is %3.2f degree\n(b)Work done by the water on the runner per kg of water is %3.2f W/(kg/s)\n(c)Hydraulic efficiency is %0.2f %%'%(b11,b22,Wm,nH*100)

(a)
    Inlet vane angle is 24.83 degree
    Outlet vane angle is 17.66 degree
(b)Work done by the water on the runner per kg of water is 93.79 W/(kg/s)
(c)Hydraulic efficiency is 95.61 %

Ex 9.15 Page 427

In [15]:
#input data
H=23#Net head across the turbine in m
N=150#Speed of the turbine in rpm
P=23#Power developed by the turbine in MW
D=4.75#Blade tip diameter in m
d=2#Blade hub diameter in m
nH=0.93#Hydraulic efficiency
n0=0.85#Overall efficiency
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
dm=(D+d)/2#Mean diameter of the turbine in m
Pa=(P*10**6)/n0#Power available in MW
Q=(Pa/(dw*g*H))#Flow rate in the turbine in m**3/s
Um=(3.1415*dm*N)/60#Rotor speed at mean diameter in m/s
Pr=Pa*nH*10**-6#Power given to runner in MW
Cx1=Pr*10**6/(dw*Q*Um)#Velocity of whirl at inlet in m/s    as Cx2=0 in m/s
Ca=Q/((3.1415/4)*(D**2-d**2))#Axial velocity in m/s
b11=180-degrees(atan(Ca/(Um-Cx1)))#Inlet blade angle in degree
Wx2=Um#Outlet whirl velocity component in m/s
b22=degrees(atan(Ca/Wx2))#Outlet blade angle in degree

#output
print '(a)The inlet blade angle at mean radius is %3.1f degree\n(b)The outlet blade angle at mean radius is %3.1f degree'%(b11,b22)

(a)The inlet blade angle at mean radius is 156.1 degree
(b)The outlet blade angle at mean radius is 17.2 degree

Ex 9.16 Page 429

In [16]:
#input data
P=9100#Power developed by the turbine in kW
H=5.6#Net head available at the turbine in m
SR=2.09#Speed ratio
FR=0.68#Flow ratio
n0=0.86#Overall effiiciency of the turbine
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3
DbD=1/3#Ratio of diameter of the boss to diameter of the runner

#calculations
U1=SR*(2*g*H)**(1/2)#Runner tip speed at inlet in m/s
Cr1=FR*(2*g*H)**(1/2)#Flow velocity at inlet in m/s
Q=(P*10**3)/(n0*dw*g*H)#Discharge through the turbine in m**3/s
D=(Q*4/(3.1415*Cr1*((1**2)-(DbD**2))))**(1/2)#Diameter of the runner in m
N=(U1*60)/(3.1415*D)#Speed of the the turbine in rpm
Ns=(N*(P)**(1/2))/(H)**(5/4)#Specific speed 
#output
print '(a)Diameter of the runner of the turbine is %3.2f m\n(b)Speed of the turbine is %3.1f rpm\n(c)The specific speed is %3.2f'%(D,N,Ns)

(a)Diameter of the runner of the turbine is 6.22 m
(b)Speed of the turbine is 67.3 rpm
(c)The specific speed is 744.71

Ex 9.17 Page 430

In [17]:
#input data
H=20#Head developed over the turbine in m
P=11800#Power developed by turbine in kW
D=3.5#Outer diameter of the runner in m
Db=1.75#Hub diameter in m
a11=35#Guide blade angle in degree 
nH=0.88#Hydraulic efficiency 
n0=0.84#Overall efficiency
Cx2=0#Velocity of whirl at outlet in m/s
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
Q=(P*10**3)/(n0*g*H*dw)#Discharge of turbine in m**3/s
Cr1=Q/((3.1415/4)*(D**2-Db**2))#Flow velocity at inlet in m/s
Cx1=Cr1/tan(a11*pi/180)#Velocity of whirl at inlet in m/s
U1=(nH*H*g)/(Cx1)#Runner tip speed at inlet in m/s
Wx1=U1-Cx1#Inlet whirl velocity component in m/s
b11=180-degrees(atan(Cr1/-Wx1))#Runner inlet angle in degree
Cr2=Cr1#Flow velocity at outlet in m/s      for a kaplan turbine
U2=U1#Runner tip speed at outlet in m/s     for a kaplan turbine
b22=degrees(atan(Cr2/U2))#Runner outlet angle in degree 
N=(U1*60)/(3.1415*D)#The speed of the turbine in rpm

#output
print '(1)\n    (a)The runner inlet angle is %3.2f degree\n    (b)The runner outlet angle is %3.1f degree\n(2)The speed of the turbine is %3.2f rpm'%(b11,b22,N)

(1)
    (a)The runner inlet angle is 101.33 degree
    (b)The runner outlet angle is 39.2 degree
(2)The speed of the turbine is 66.49 rpm

Ex 9.18 Page 432

In [18]:
#input data
N=50#Speed of the turbine in rpm
d=6#Runner diameter of the turbine in m
Ae=20#Effective area of flow in m**2
b11=150#The angle of the runner blades at inlet in degree
b22=20#The angle of the runner blade at outlet in degree
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
U1=(3.141*d*N)/60#Runner tip speed at inlet in m/s
U2=U1#Runner tip speed at outlet in m/s
Cr2=U2*tan(b22*pi/180)#Flow velocity at outlet in m/s
Cr1=Cr2#Flow velocity at inlet in m/s
Q=Ae*Cr1#Discharge by the  turbine in m**3/s
Cx1=U1-(Cr1/(tan((180-b11)*pi/180)))#Velocity of whirl at inlet in m/s
P=dw*g*Q*(U1*Cx1/g)*10**-3#Theoretical Power developed in kW
C2=Cr2#Absolute outlet velocity in m/s
H=(U1*Cx1/g)+(C2**2/(2*g))#Net head across the turbine in m
nH=(U1*Cx1/g)/(H)#Hydraulic efficiency

#output
print '(a)Discharge of the turbine is %3.1f m**3/s\n(b)Theoretical Power developed is %3.2f kW\n(c)Hydraulic efficiency is %0.2f %%'%(Q,P,nH*100)

(a)Discharge of the turbine is 114.3 m**3/s
(b)Theoretical Power developed is 10421.35 kW
(c)Hydraulic efficiency is 84.80 %

Ex 9.19 Page 433

In [19]:
#input data
D=8#Outer diameter of the turbine in m
Db=3#Inner diameter of the turbine in m
P=30000#Power developed by the turbine in kW
nH=0.95#Hydraulic efficiency
N=80#Speed of the turbine in rpm
H=12#Head operated by the turbine in m
Q=300#Discharge through the runner in m**3/s
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
U1=(3.1415*D*N)/60#Runner tip speed at inlet in m/s
U2=U1#Runner tip speed at outlet in m/s     as flow is axial
Cr1=Q/((3.1415/4)*(D**2-Db**2))#Flow velocity at inlet in m/s
Cr2=Cr1#Flow velocity at outlet in m/s      as flow is axial
b22=degrees(atan(Cr2/U2))#The angle of the runner blade at outlet in degree
Cx1=(nH*g*H)/U1#Velocity of whirl at inlet in m/s
b11=180-degrees(atan(Cr1/(U1-Cx1)))#The angle of the runner blade at inlet in degree
nM=(P*10**3)/(dw*g*Q*(Cx1*U1/g))#Mechanical efficiency
n0=nM*nH#Overall efficiency

#output
print '(a)Blade angle at\n    inlet is %3.2f degree\n    outlet is %3.2f degree\n(b)Mechanical efficiency is %0.1f %%\n(c)Overall efficiency is %0.1f %%'%(b11,b22,nM*100,n0*100)

(a)Blade angle at
    inlet is 167.04 degree
    outlet is 11.71 degree
(b)Mechanical efficiency is 89.4 %
(c)Overall efficiency is 84.9 %

Ex 9.20 Page 434

In [20]:
#input data
P=11500#Rated power of the turbine in kW
H=4.3#Average head of the turbine in m
n0=0.91#Overall efficiency of the turbine 
DbD=0.3#Ratio of Diameters of runner boss and runner
SR=2#Speed ratio
FR=0.65#Flow ratio
g=9.81#Acceleration due to gravity in m/s**2
dw=1000#Density of water in kg/m**3

#calculations
U=SR*(2*g*H)**(1/2)#Runner tip speed in m/s
Cr=FR*(2*g*H)**(1/2)#Flow velocity in m/s
Q=(P*10**3)/(n0*dw*g*H)#Discharge of the turbine in m**3/s
D=((4*Q)/(Cr*3.1415*(1**2-DbD**2)))**(1/2)#Runner diameter in 
N=(60*U)/(3.1415*D)#Speed of the turbine in rpm 

#output
print '(a)Runner diameter of the turbine is %3.2f m\n(b)Operating speed of the turbine is %3.1f rpm'%(D,N)

(a)Runner diameter of the turbine is 8.38 m
(b)Operating speed of the turbine is 41.9 rpm