Chapter 10 : Hydroelecric power plant

Ex: 10.1 Pg: 709

In [51]:
from math import sqrt,cos,pi
from __future__ import division
#Input data
P=4000#Power in kW
N=400#Speed in r.p.m
h=200#Head in m
e=90#Efficiency in percent
d=1.5#Diameter in m
vd=10#Percentage decrease in velocity
a=165#Angle with which jet is deflected in degrees

#Calculations
V1=sqrt(2*9.81*h*(e/100))#Velocity in m/s
Vb=(3.14*d*N)/60#Velocity in m/s
nn=((2*(1-((e/100)*cos(pi/180*a)))*(V1-Vb)*Vb)/V1**2)*100#Efficiency in percent
p=(P/(nn/100))#Power developed in kW
pj=(p/2)#Power developed per jet in kW
dx=sqrt((pj*8)/(3.14*V1**3))#Diameter of each jet in m

#Output
print " (a) the efficiency of the runner is %3.2f percent \n (b) the diameter of each jet is %3.4f m"%(nn,dx)

 (a) the efficiency of the runner is 93.17 percent 
 (b) the diameter of each jet is 0.1614 m

Ex: 10.2 Pg: 710

In [3]:
from math import sqrt
#Input data
P=6000#Power in kW
h=300#Net head availabe in m
N=550#Speed in r.p.m
rd=(1/10)#Ratio of jet diameter to wheel diameter
nh=0.85#Hydraulic efficiency 
Cv=0.98#Coefficient of velocity
f=0.46#Speed ratio
d=1000#Density in kg/m**3

#Calculations
V1=Cv*sqrt(2*9.81*h)#Velocity in m/s
Vb=f*sqrt(2*9.81*h)#Velocity in m/s
Q=((P*10**3)/(nh*d*9.81*h))#Discharge in m**3/s
D=((Vb*60)/(3.14*N))#Diameter in m
d=(D/10)#Diameter of jet in m
n=(Q/((V1*(3.14/4)*d**2)))#Number of jets

#Output
print " (a) the number of jets are%3.0f \n (b) diameter of each jet is %3.3f m \n (c) diameter of the wheel is %3.2f m \n (d) the quantity of water required is %3.1f m**3/s"%(n,d,D,Q)

 (a) the number of jets are  3 
 (b) diameter of each jet is 0.123 m 
 (c) diameter of the wheel is 1.23 m 
 (d) the quantity of water required is 2.4 m**3/s

Ex: 10.3 Pg: 711

In [52]:
from math import sqrt
from __future__ import division
#Input data
P=10#Capacity in MW
h=500#Head in m
Ns=10#Specific speed of the turbine
on=80#Overall efficiency in percent
Cv=0.98#Coefficient of velocity
x=0.46#Speed of the bucket wheel to the velocity of jet
da=1000#Density in kg/m**3

#Calculations
N=(Ns*h**(5/4))/sqrt(P*10**3)#Speed in r.p.m
V=(Cv*sqrt(2*9.81*h))#Velocity in m/s
Vb=(x*V)#Speed of bucket wheel in m/s
D=((60*Vb)/(3.14*N))#Diameter in m
d=sqrt((P*10**6)/((on/100)*(3.14/4)*da*V*9.81*h))#Diameter in m

#Output
print " Diameter of jet is %3.3f m \n Diameter of bucket wheel is %3.2f m"%(d,D)

 Diameter of jet is 0.183 m 
 Diameter of bucket wheel is 3.61 m

Ex: 10.4 Pg: 711

In [5]:
from math import sqrt
#Input data
Cv=0.97#Coefficient of velocity
f=0.45#Friction coefficient
h=0.85#Head in m
d=1000#Density in kg/m**3
n=1#For a single jet turbine

#Calculations
Ns=((60/3.14)*(f*sqrt(2*9.8))*sqrt(n*(3.14/4)*Cv*sqrt(2*9.8)*9.8*h))#Specific speed in terms of d/D 

#Output
print "The specific speed of a single jet Pelton wheel is about %3.0f (d/D) where d and D represent the jet and bucket wheel diameters respectively"%(Ns)

The specific speed of a single jet Pelton wheel is about 202 (d/D) where d and D represent the jet and bucket wheel diameters respectively

Ex: 10.5 Pg: 712

In [53]:
from math import cos,pi
#Input data
n=4#Number of jets
d=60#Diameter of each jet in mm
a=165#Angle in degrees
v=45#Speed of the bucket wheel in m/s
de=1000#Density in kg/m**3

#Calculations
v1=(2*v)#Jet velocity in m/s
Q=(3.14/4)*(d/1000)**2*v1#Discharge in m**3/s
P=(1-cos(pi/180*a))*(v1**2/4)*Q*de*10**-3#Power developed in kW
P4=(P*4)#For four jets in kW
nd=((1-cos(pi/180*a))/2)*100#Maximum efficiency in percent

#Output
print " Velocity of the jet for maximum efficiency is %3.0f m/s \n Power developed is %d kW \n Hydraulic efficiency is %3.1f percent"%(v1,P4,nd)

 Velocity of the jet for maximum efficiency is  90 m/s 
 Power developed is 4050 kW 
 Hydraulic efficiency is 98.3 percent

Ex: 10.6 Pg: 713

In [12]:
from math import atan, degrees
#Input data
v=20#Peripheral velocity in m/s
vw=17#Velocity of whirl in m/s
vr=2#Radial velocity in m/s
Q=0.7#Flow in m**3/s
hn=80#Hydraulic efficiency in percent
d=1000#Density in kg/m**3

#Calculations
H=((vw*v)/(9.81*(hn/100)))#Head on the wheel in m
P=(d*Q*9.81*H*(hn/100)*10**-3)#Power generated in kW
al=180-degrees(atan(vr/vw))#Angle of guide vanes in degrees
bl=degrees(atan(vr/(v-vw)))#Inlet blade angle in degrees

#Output
print " Head on the wheel is %3.1f m \n The power generated by the turbine is %3.0f kW \n Eit angle of guide vanes is %3.2f degrees and Inlet blade angle is %3.1f degrees"%(H,P,al,bl)

 Head on the wheel is 43.3 m 
 The power generated by the turbine is 238 kW 
 Eit angle of guide vanes is 173.29 degrees and Inlet blade angle is 33.7 degrees

Ex: 10.7 Pg: 714

In [54]:
from math import atan,tan,pi,sqrt,degrees
#Input data
od=1.5#Outer diameter in m
id=0.75#Inner diameter in m
h=150#Head in m
P=14000#Power in kW
Ns=120#Specific speed 
vw2=0#Velocity in m/s
a=(11+(20/60))#Angle in degrees
hn=92#Hydraulic efficiency in percent

#Calculations
N=(Ns*h**(5/4))/sqrt(P)#Speed in rpm
vb1=(3.14*od*N)/60#velocity in m/s
vw1=(((hn/100)*9.81*h)/vb1)#velocity in m/s
vf1=(tan(pi/180*a)*vw1)#Velocity in m/s
vf2=vf1#Velocity in m/s
b1=degrees(atan(vf1/(vb1-vw1)))#Angle in degrees
b1x=(b1-int(b1))*60#For output
vb2=(vb1/2)#Velocity in m/s
b2=degrees(atan(vf1/(vb2-vw2)))#Angle in degrees
b2x=(b2-int(b2))*60#For output

#Output
print " Inlet blade angle is %3.0f degrees %3.0f minutes \n Outlet blade angle is %3.0f degrees %3.0f minute"%(b1,b1x,b2,b2x)

 Inlet blade angle is  35 degrees  38 minutes 
 Outlet blade angle is  17 degrees  16 minute

Ex: 10.8 Pg: 715

In [55]:
from math import sqrt,pi,atan,degrees
#Input data
h=70#net head in m
N=700#speed in rpm
o=85#over all efficiency in %
P=350#shaft power in kW
he=92#hydraulic efficiency in %
fr=.22#flow ratio
b=.1#breadth ratio
s=2#outer diameter in terms of inner diametre
#Calculations
vf1=fr*sqrt(2*9.81*h)#velocity in m/s
q=(P/(9.81*h*(o/100)))#discharge in m**3/s
d1=sqrt(q/(.94*b*vf1*3.14))#diameter in metre
b1=d1*b#breadth in metre
d2=d1/2#diametre in metre
vb1=(3.14*d1*N)/60#velocity in m/s
vw1=((he/100)*9.81*h)/vb1#velcity in m/s
a=degrees(atan(vf1/vw1))#angle in degrees
bet=degrees(atan(vf1/(vw1-vb1)))#angle in degrees
vb2=(d2/d1)*vb1#velocity in m/s
bet2=degrees(atan(vf1/vb2))#angle in degrees

#Output
print " (a)the guide vane angle is %3.1f degrees \n (b)the runner vane angle at inlet is %3.1f degrees and outlet is %3.2f degrees \n (c)the diametres of the runner at inlet is %3.1f metre and outlet is %3.2f metre\n (d)the width of the wheel at inlet is %3.2f metre"%(a,bet,bet2,d1,d2,b1)

 (a)the guide vane angle is 13.3 degrees 
 (b)the runner vane angle at inlet is 26.6 degrees and outlet is 41.72 degrees 
 (c)the diametres of the runner at inlet is 0.5 metre and outlet is 0.25 metre
 (d)the width of the wheel at inlet is 0.05 metre

Ex: 10.9 Pg: 717

In [20]:
from math import ceil,sqrt,cos,pi
#Input data
n=4#Number of units
P=70000#Power in kVA
f=50#Frequency in Hz
p=10#No.of pair of poles
h=505#Gross head in m
tn=94#Transmission efficiency in percent
po=260#Power in MW
e=91#Efficiency in percent
nn=0.98#Nozzle efficiency
Cv=0.98#Coefficient of velocity
x=0.48#Vb=0.48 V
dd=25#Nozzle diameter is 25% bigger than jet diameter
a=165#Angle of buckets in degrees
de=99.75#Discharge efficiency in percent

#Calculations
N=(120*f)/(p*2)#Synchronous speed in r.p.m
nh=((tn/100)*h)#Net head in m
pt=(po*10**3)/n#Power developed per turbine in kW
ip=(pt/(e/100))#Input water power in kW
Q=(ip/(9.81*nh))#Discharge in m**3/s
Qj=(Q/n)#Discharge per jet in m**3/s
V1=Cv*sqrt(2*9.81*nh)#Velocity in m/s
d=sqrt((4/3.14)*(Qj/V1))#Diameter of jet in m
nd=(1+(dd/100))*d#Nozzle tip diameter in m
Vb=(x*V1)#Velocity in m/s
D=((Vb*60)/(3.14*N))#Pitch circle diameter of the wheel in m
Ns=((N*sqrt(po*10**3))/nh**(5/4))#Specific speed
jr=(D/d)#Jet ratio
nob=(jr/2)+15#Number of buckets
nobb=ceil(nob)#Rounding off to next integer
W=((V1-Vb)*(1-(nn*cos(pi/180*a)))*Vb)/9.81#Workdone per kg in kg.m/kg
nth=((W/nh)*de)#Hydraulic efficiency in percent

#Output
print " (a) the discharge of the turbine is %3.2f m**3/s \n (b) the jet diameter is %3.3f m \n (c) the nozzle tip diameter is %3.3f m \n (d) the pitch circle diameter of the wheel is %3.2f m \n (e) the specific speed is %3.2f \n (f) the number of buckets on the wheel are %3.0f \n (g) the workdone per kg of water on the wheel is %3.2f kg.m/kg \n (h) the hydraulic efficiency is %3.0f percent"%(Q,d,nd,D,Ns,nobb,W,nth)

 (a) the discharge of the turbine is 15.34 m**3/s 
 (b) the jet diameter is 0.227 m 
 (c) the nozzle tip diameter is 0.284 m 
 (d) the pitch circle diameter of the wheel is 2.89 m 
 (e) the specific speed is 69.04 
 (f) the number of buckets on the wheel are  22 
 (g) the workdone per kg of water on the wheel is 443.02 kg.m/kg 
 (h) the hydraulic efficiency is  93 percent

Ex: 10.10 Pg: 718

In [56]:
from math import sin,cos,degrees,atan,sqrt,pi
#Input data
gh=35#Gross head in m
md=2#Mean diameter in m
N=145#Speed in rpm
a=30#Angle in degrees
oa=28#Outlet angle in degrees
x=7#Percentage of gross head lost
y=8#Reduction in relative velocity in percent

#Calculations
H=((100-x)/100)*gh#Net haed in m
V1=sqrt(2*9.81*H)#Velocity in m/s
Vb=(3.14*md*N)/60#Velocity in m/s
b1=degrees(atan((V1*sin(pi/180*a))/((V1*cos(pi/180*a))-Vb)))#Angle in degrees
Vr1=((V1*sin(pi/180*a))/sin(pi/180*b1))#Velocity in m/s
Vr2=((100-y)/100)*Vr1#Velocity in m/s
Vw1=(V1*cos(pi/180*a))#Velocity in m/s
Vw2=(Vb-(Vr2*cos(pi/180*oa)))#Velocity in m/s
E=((Vb*(Vw1-Vw2))/9.81)#Workdone in kg.m/kg
nb=(E/gh)*100#Hydraulic efficiency in percent

#Output
print " Blade angle at inlet is %3.0f degrees \n Hydraulic efficiency is %3.0f percent"%(b1,nb)

 Blade angle at inlet is  62 degrees 
 Hydraulic efficiency is  81 percent

Ex: 10.11 Pg: 719

In [26]:
from math import sqrt
#Input data
P=10000#Power in kW
h=12#Head in m
Nr=2#Speed ratio
Fr=0.65#Flow ratio
x=0.3#Diameter of hub is 0.3 times the eternal diameter of the vane 
on=94#Overall efficiency in percent

#Calculations
Q=(P/(9.81*h*(on/100)))#Discharge in m**3/s
Vr1=(Fr*sqrt(2*9.81*h))#Velocity in m/s
Ab=(Q/Vr1)#Area of flow in m**2
D=sqrt(((Ab*4)/3.14)/(1-x**2))#Diameter of runner in m
Vb=(Nr*sqrt(2*9.81*h))#Velocity in m/s
N=((Vb*60)/(3.14*D))#Speed in rpm
f=50#Taking frequency as 50 Hz
p=(120*50)/N#Number of poles
N1=(120*f)/int(p)#Speed in rpm
Ns=(N1*sqrt(P))/h**(5/4)#Specific speed

#Output
print " (a) the speed is %3.1f rpm \n (b) the diameter of the runner is %3.2f m \n (c) the specific speed is %3.0f"%(N1,D,Ns)

 (a) the speed is 166.7 rpm 
 (b) the diameter of the runner is 3.56 m 
 (c) the specific speed is 746

Ex: 10.12 Pg: 721

In [29]:
from math import sqrt
#Input data
P=10000#Power in kW
h=25#Head in m. In textbook it is given wrong as 2 m
N=135#Speed in rpm
h1=20#Head in m

#Calculations
Ns=((N*sqrt(P))/h**(5/4))#Specific speed
N1=sqrt(h1/h)*N#Speed in rpm
P2=P/(h/h1)**(3/2)#Power in kW

#Output
print " Specific speed is %3.1f \n Normal speed is %3.1f rpm \n Output under a head of %d m is %3.0f kW"%(Ns,N1,h1,P2)

 Specific speed is 241.5 
 Normal speed is 120.7 rpm 
 Output under a head of 20 m is 7155 kW

Ex: 10.13 Pg: 721

In [32]:
from math import ceil
#Input data
Q=175#Discharge in m**3/s
h=18#Head in meter
N=150#Speed in rpm
oe=82#Overall efficiency in percent
Ns1=460#Maximum specific speed
Ns2=350#Maximum specific speed
d=1000#Density in kg/m**3

#Calculations
P=(d*Q*9.81*h*(oe/100)*10**-3)#power in kW
P1=((Ns1*h**(5/4))/N)**2#Power in kW
n1=P/P1#No.of turbains
P2=((Ns2*h**(5/4))/N)**2#Power in kW
n2=ceil(P/P2)#No.of turbains

#Output
print "The number of turbines in \n (a) Francis turbine are%3.0f \n (b) Kaplan turbine are %d"%(n1,n2)

The number of turbines in 
 (a) Francis turbine are  2 
 (b) Kaplan turbine are 4

Ex: 10.14 Pg: 722

In [34]:
from math import sqrt
#Input data
Ns=210#Specific speed 
P=30#Power in MW
N=180#Speed in rpm
Q=0.6#Discharge in m**3/s
h=4.5#Head in m
e=88#Efficiency in percent
d=1000#Density in kg/m**3

#Calculations
Pm=(d*Q*9.81*h*(e/100)*10**-3)#Power in kW
Nm=(Ns*h**(5/4))/sqrt(Pm)#Speed in rpm
Hp=((N*sqrt(P*1000))/Ns)**(4/5)#Head in m
Dpm=(Nm/N)*sqrt(Hp/h)#Scale ratio
Qp=(P*10**6)/(d*9.81*Hp*(e/100))#Discharge in m**3/s

#Output
print " Speed is %3.0f rpm \n Power is %3.3f kW \n Scale ratio is %3.3f \n Flow through the turbine is %3.1f m**3/s"%(Nm,Pm,Dpm,Qp)

 Speed is 285 rpm 
 Power is 23.309 kW 
 Scale ratio is 5.518 
 Flow through the turbine is 63.6 m**3/s

Ex: 10.15 Pg: 723

In [36]:
from math import sqrt
#Input data
x=1/5#Scale model
h=1.5#Head in m
P=5#Power in kW
N=450#Speed in rpm
h1=30#Head in m

#Calculations
N1=(x*N)/sqrt(h/h1)#Speed in rpm
Ns=(N*sqrt(P))/h**(5/4)#Specific speed
P1=((Ns*h1**(5/4))/N1)**2#Power in kW

#Output
print " Speed is %3.0f rpm \n Power is %3.0f kW"%(N1,P1)

 Speed is 402 rpm 
 Power is 11180 kW

Ex: 10.16 Pg: 723

In [38]:
from math import sqrt
#Input data
h=19#Head in m
Q=3#Flow rate in m**3/s
N=600#Speed in rpm
h1=5#Head in m

#Calculations
N1=N/sqrt(h/h1)#Speed in rpm
Q1=Q/sqrt(h/h1)#Discharge in m**3/s

#Output
print " Speed of the turbine is %3.1f rpm \n Maximum flow rate is %3.1f m**3/s"%(N1,Q1)

 Speed of the turbine is 307.8 rpm 
 Maximum flow rate is 1.5 m**3/s

Ex: 10.17 Pg: 724

In [39]:
from math import ceil
#Input data
Q=350#Discharge in m**3/s
h=30#Head in m
e=87#Turbine efficiency in percent
f=50#Frequency in Hz
p=24#Number of poles
Ns1=300#Specific speed
Ns2=820#Specific speed
d=1000#Dnsity of water in kg/m**3

#Calculations
N=(120*f)/p#Speed in rpm
P=d*Q*9.81*h*(e/100)*10**-3#Power in kW
P1=((Ns1*h**(5/4))/N)**2#Power in kW
n1=P/P1#No.of turbines
P2=((Ns2*h**(5/4))/N)**2#Power in kW
n2=ceil(P/P2)#No.of turbines

#Output
print "Least number of machines required if using \n (a) Francis turbines are %3.0f \n (b) Kaplan turbines are %3.0f"%(n1,n2)

Least number of machines required if using 
 (a) Francis turbines are  13 
 (b) Kaplan turbines are   2

Ex: 10.18 Pg: 725

In [41]:
#Input data
h=27#Head in m
A=430#Area in sq.km
R=150#Rainfall in cm/year
pr=65#Percentage of rainfall utilised
pe=95#Penstock efficiency in percent
te=80#Turbine efficiency in percent
ge=86#Generator efficiency in percent
lf=0.45#Load factor
d=1000#Density of water in kg/m**3

#Calculations
Q=A*10**6*(R/100)*(pr/100)#Discharge in m**3 per year
Qs=(Q/(365*24*3600))#Quantity of water per second in m**3
P=(pe/100)*(te/100)*(ge/100)*d*Qs*9.81*h*10**-3#Power in kW
plc=(P/lf)#Peak load capacity in kW
C=(plc/(2*(ge/100)))#Capacity of each unit in kW

#Output
print " (a) Power developed is %3.0f kW \n (b) As the available head is low, Kaplan turbines are suggested.\n Two turbines each of 3000kW capacity may be installed."%(P)

 (a) Power developed is 2302 kW 
 (b) As the available head is low, Kaplan turbines are suggested.
 Two turbines each of 3000kW capacity may be installed.

Ex: 10.19 Pg: 725

In [50]:
%matplotlib inline
from matplotlib.pyplot import plot, subplot, title,show,xlabel,ylabel
#Input data
q=[0,30,25,20,0,10,50,80,100,110,65,45,30]#Mean discharge in millions of cu.m per month respectively
h=90#Head in m
n=86#Overall efficiency in percent

#Calculations
Qm=(q[(1)]+q[(2)]+q[(3)]+q[(4)]+q[(5)]+q[(6)]+q[(7)]+q[(8)]+q[(9)]+q[(10)]+q[(11)]+q[(12)])/12#Mean discharge in millions m**3/s
Q=[0,30,30,25,25,20,20,0,0,10,10,50,50,80,80,100,100,110,110,65,65,45,45,30,30,0]#Discharge(million m**3/month) on y-axis
y=[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12]#Months on x-ais
D=[0,110,100,90,80,70,60,50,40,30,25,20,10,0]#Discharge per month in million m**3
pt=[0,8.3,16.7,25,25,25,33.3,41.7,50,66.7,75,83.3,91.7,100]#Percentage time 
Po=((Qm*10**6*9.81*h*(n/100))/(30*24*3600*1000))#Power developed in MW

#Output
#subplot(131)
plot(y[1:],Q[1:])#Graph Discharge(million m**3/month) vs Month
title("Discharge(million m**3/month) vs Month")
xlabel("Months")
ylabel("Discharge(million m**3/month)")
show()
#subplot(133)
plot(pt[1:],D[1:])#Graph percentage time vs Discharge(million m**3/month)
title("percentage time vs Discharge(million m**3/month)")
xlabel("percentage time")
ylabel("Discharge(million m**3/month)")
show()
print "Power developed is %3.2f MW"%(Po)

Power developed is 13.79 MW