# Chapter 5 : Fundamentals of Flow¶

## Example 5.1 Page No : 121¶

In :
import math

# Variables :
m = 2000.;			#litre or kg(1litre water  = 1kg)
M = m/60;			#kg/s
p = 4.5;			#bar
p = p*10**5;			#N/m**2
g = 9.81;			#consmath.tant
w = g*1000;			#N/m**3

# Calculations
H = p/w;			#m
Power = M*g*H/1000;			#kW

# Results
print "Power required in kW : %.f"%Power

Power required in kW : 15


## Example 5.2 Page No : 122¶

In :
import math

# Variables :
v1 = 400.*10**-3;			#m/s
d1 = 300./1000;			#meter
d2 = 450./1000;			#meter

# Calculations and Results
A1 = math.pi*d1**2/4;			#m**2
A2 = math.pi*d2**2/4;			#m**2
Q1 = A1*v1*100;			#litres/sec(1m**3 = 1000litres)
print "Discharge of pipe in litres/sec : %.2f"%Q1
v2 = (Q1/100)/A2;			#m/s(Q1 = Q2)
print "Mean velocity of flow in m/s : %.3f"%v2

#Answer of discharge is wrong in the book.

Discharge of pipe in litres/sec : 2.83
Mean velocity of flow in m/s : 0.178


## Example 5.3 Page No : 123¶

In :
import math

# Variables :
PotentialHead = 2;			#meter of fluid
print "Potential Head is ",(PotentialHead)," meter of fluid."
v = 5.;				#m/s
g = 9.81;			#constant

# Calculations and Results
VelocityHead = v**2/2/g;			#m
print "Velocity Head is ",round(VelocityHead,3)," meter of fluid."

w = g*1000;			#N/m**3
S = 0.8;			#sp. gravity of fluid
p = 200;			#kPa
PressureHead = p*10**3/w/S;			#meter of fluid
print "Pressure Head is ",round(PressureHead,3)," meter of fluid."

TotalHead = PotentialHead+VelocityHead+PressureHead;			#meter of fluid
print "Total Head is ",round(TotalHead,3)," meter of fluid."

Potential Head is  2  meter of fluid.
Velocity Head is  1.274  meter of fluid.
Pressure Head is  25.484  meter of fluid.
Total Head is  28.758  meter of fluid.


## Example 5.4 Page No : 124¶

In :
import math

# Variables :
p = 0.8/10**-4;			#kg/m**2
datumH = 4.;			#meter
v = 0.8;			#m/s
g = 9.81;			#consmath.tant

# Calculations
VelocityH = v**2/2/g;			#m
w = 1000;			#kg/m**3
PressureH = p/w;			#meter of fluid
TotalH = datumH+VelocityH+PressureH;			#meter of fluid

# Results
print "Total Energy is ",round(TotalH,4)," meter."

Total Energy is  12.0326  meter.


## Example 5.5 Page No : 125¶

In :
import math

# Variables :
D1 = 800./1000;			#m**2
D2 = 600./1000;			#m**2
p1 = 100.;			#kPa
p2 = 40.;			#kPa
v1 = 4000.*10**-3;			#m/s
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
Z1 = 4.;			#meter
Z2 = 7.;			#meter
rho = 1.;			#sp. gravity
g = 9.81;			#constant

# Calculations
PHeadA = p1/rho/g;			#meter of fluid
PHeadB = p2/rho/g;			#meter of fluid
v2 = A1*v1/A2;			#m/s
VHeadA = v1**2/2/g;			#meter
VHeadB = v2**2/2/g;			#meter
E1 = Z1+PHeadA+VHeadA;			#meter
E2 = Z2+PHeadB+VHeadB;			#meter

# Results
if E1>E2:
print "Total Energy at A(",round(E1,4)," meter) is greater than total energy at B(",round(E2,4)," meter). \
\nFlow of water is from A to B."
else:
print "Total Energy at B(",(E2)," meter) is greater than total energy at A(",(E1)," meter). Flow of water is from B to A."

Total Energy at A( 15.0092  meter) is greater than total energy at B( 13.6548  meter).
Flow of water is from A to B.


## Example 5.6 Page No : 130¶

In :
import math

# Variables :
D1 = 1.25;			#meter
D2 = 0.625;			#meter
slope = 100.;
L = 300.;			#/meter
g = 9.81;			#consmath.tant
Z12 = L/slope;			#meter
Q = 100.;			#litres/sec
Q = Q*10**-3;			#m**3/sec

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
v1 = Q/A1;			#m/s
v2 = Q/A2;			#m/s
p1 = 100.;			#kN/m**2
#Higher End :
w = 9.81;			#kN/m**3
Phead = p1/w;			#meter
Vhead = v1**2/2/g;			#meter
#Lower End :
w = 9.81;			#kN/m**3
#Phead = p1/w;			#meter
Vhead = v2**2/2/g;			#meter
p2 = (Z12+v1**2/2/g+p1/w-v2**2/2/g)*w;			#kN/m**2(By Bernoulli's theorem)

# Results
print "Pressure at the lower end in kN per m**2 : %.2f"%p2

Pressure at the lower end in kN per m**2 : 129.38


## Example 5.7 Page No : 132¶

In :
import math

# Variables :
Z1 = 0.;			#meter
Z2 = 5.;			#meter
Q = 300.*10**-3;			#m/s
D1 = 0.3;			#meter
D2 = 0.6;			#meter
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
v1 = Q/A1;			#m/s
v2 = Q/A2;			#m/s
p1 = 100.;			#kN/m**2
p2 = 600.;			#kN/m**2
g = 9.81;			#consmath.tant

# Calculations and Results
Vhead11 = v1**2/2/g;			#meter
Vhead22 = v2**2/2/g;			#meter
Phead11 = p1/g;			#meter
Phead22 = p2/g;			#meter
E1_11 = Z1+Vhead11+Phead11;			#meter
E2_22 = Z2+Vhead22+Phead22;			#meter

if E1_11>E2_22:
print "Energy at section 1-1(",round(E1_11,3)," meter) is greater than energy at section 2-2(",(E2_22)," meter). Flow of water is from section 1-1 to 2-2."
HeadLoss = E1_11-E2_22;			#meter
print "Head Loss in meter : ",HeadLoss
else:
print "Energy at section 2-2(",round(E2_22,3)," meter) is greater than energy at section 1-1(",round(E1_11,3)," meter). Flow of water is from section 2-2 to 1-1."
HeadLoss = E2_22-E1_11;			#meter
print "Head Loss in meter : %.3f"%HeadLoss

Energy at section 2-2( 66.219  meter) is greater than energy at section 1-1( 11.112  meter). Flow of water is from section 2-2 to 1-1.
Head Loss in meter : 55.108


## Example 5.8 Page No : 133¶

In :
import math

# Variables :
D = 400./1000;			#meter
v1 = 20.;			#m/s
Z1 = 28.;			#meter
Z2 = 31.;			#meter
p1 = 4./10**-4;			#kg/m**2
p2 = 3./10**-4;			#kg/m**2
g = 9.81;			#consmath.tant
w = 1000.;			#kg/m**3

# Calculations
Vhead1 = v1**2/2/g;			#meter
Phead1 = p1/w;			#meter
Vhead2 = Vhead1;			#meter
Phead2 = p2/w;			#meter
E1 = Z1+Vhead1+Phead1;			#meter
E2 = Z2+Vhead2+Phead2;			#meter
HL = E1-E2;			#meter

# Results
print "Loss of head between P & Q in meter : ",HL


Loss of head between P & Q in meter :  7.0


## Example 5.9 Page No : 134¶

In :
import math

# Variables :
Z1 = 0.;			#meter
Z2 = 4.;			#meter
rho = 0.8;			#sp. gravity

# Calculations and Results
Q = 250.*10**-3;			#m/s or cumec
D1 = 250./1000;			#meter
D2 = 500./1000;			#meter
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
v1 = Q/A1;			#m/s
v2 = Q/A2;			#m/s
p1 = 0.1*10**3;			#N/m**2
p2 = 0.06*10**3;			#N/m**2
g = 9.81;			#consmath.tant
Vhead1 = v1**2/2/g;			#meter
Phead1 = p1/rho/g;			#meter
Vhead2 = v2**2/2/g;			#meter
Phead2 = p2/rho/g;			#meter
H1 = Z1+Vhead1+Phead1;			#meter
H2 = Z2+Vhead2+Phead2;			#meter
if H1>H2 :
print "Total head at A(",round(H1,3)," meter) is greater than total head at B(",round(H2,3)," meter). Flow will take place from A-B."
HeadLoss = H1-H2;			#meter
print "Head Loss in meter : %.3f"%HeadLoss
else:
print "Total head at B(",(H2)," meter) is greater than total head at A(",(H1)," meter). Flow will take place from B-A."
HeadLoss = H2-H1;			#meter
print "Head Loss in meter : ",HeadLoss


Total head at A( 14.064  meter) is greater than total head at B( 11.728  meter). Flow will take place from A-B.
Head Loss in meter : 2.336


## Example 5.10 Page No : 135¶

In :
import math

# Variables :
Q = 200.*10**-3;			#m**3/s
D1 = 250./1000;			#meter
D2 = 200./1000;			#meter
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
v1 = Q/A1;			#m/s
v2 = Q/A2;			#m/s
Z1 = 2.;			#meter
Z2 = 8.;			#meter
g = 9.81;			#consmath.tant
w = 1000;			#kg/m**3

# Calculations
p1 = w*(Z1-v1**2/2/g);			#kg/m**2
p2 = v1**2/2/g*w+p1+Z2*w-v2**2/2/g*w-4*w;			#kg/m**2(by Bernolli's theorem)
p1 = p1*g;			#N/m**2
p2 = p2*g;			#N/m**2

# Results
print "Pressure intensity at point P in N/m**2 : ",p1
print "Pressure intensity at point Q in N/m**2 : ",p2

# Note :Answer in the book is not accurate.

Pressure intensity at point P in N/m**2 :  11319.768636
Pressure intensity at point Q in N/m**2 :  38595.7632715


## Example 5.11 Page No : 137¶

In :
import math

# Variables :
slope = 1./10;
Z1 = 0.;			#meter
Z2 = 40.*slope;			#meter
p1 = 1.5/10**-4;			#kg/cm**2
v2 = 4.1;			#m/s
D1 = 600./1000;			#meter
D2 = 300./1000;			#meter

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
v1 = A2*v2/A1;			#m/s
g = 9.81;			#consmath.tant
w = 1000;			#kg/m**3
p2 = (p1/w+v1**2/2/g+Z1-v2**2/2/g-Z2)*w;			#kg/m**2(by Bernolli's theorem)
p2 = p2*10**-4;			#kg/cm**2
Q1 = A1*v1;			#m**3/sec
Q1 = Q1*1000;			#litre/sec

# Results
print "Pressure intensity at point Q in kg/cm**2 : %.3f"%p2
print "Discharge of pipe in litres/sec : %.3f"%Q1

#Answer in the book is not accurate. calculation for A1 & A2 is wrong.

Pressure intensity at point Q in kg/cm**2 : 1.020
Discharge of pipe in litres/sec : 289.812


## Example 5.12 Page No : 143¶

In :
import math

# Variables :
D1 = 180./1000;			#meter
D2 = 90./1000;			#meter
g = 9.81;			#gravity consmath.tant
S = 0.8;			#sp. gravity of oil
Sm = 13.6;			#sp. gravity of mercury
x = 300./1000;			#meter
K = 0.97;			#coeff. of meter

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
C = A1*A2*math.sqrt(2*g)/math.sqrt(A1**2-A2**2)
h = x*(Sm/S-1);			#meter of oil
Q = K*C*math.sqrt(h);			#m**3/sec
Q = Q*1000;			#litre/sec

# Results
print "Discharge of oil in litres/sec : %.2f"%Q

Discharge of oil in litres/sec : 61.85


## Example 5.13 Page No : 145¶

In :
import math

# Variables :
D1byD2 = 1/0.7;
D1 = 320./1000;			#meter
D2 = 320.*0.7/1000;			#meter
g = 9.81;			#gravity consmath.tant
Q = 30.6/60;			#m**3/sec

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
C = A1*math.sqrt(2*g)/math.sqrt((D1byD2)**4-1);
h = 1.2;			#meter of water
K = Q/C/math.sqrt(h);			#Coeff. of meter

# Results
print "Coefficient of meter : %.3f"%K

#Answer in the book is wrong.

Coefficient of meter : 2.325


## Example 5.14 Page No : 146¶

In :
import math

# Variables :
D1 = 320./1000;			#meter
D2 = 224./1000;			#meter
g = 9.81;			#gravity consmath.tant
Q = 25000./1000./60;			#m**3/sec

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
C = 0.4984;			#venturi consmath.tant
K = 0.92;			#Coeff. of meter
h = (Q/K/C)**2
S = 1;			#sp. gravity
Sm = 13.6;			#sp. gravity
x = h/(Sm/S-1);			#meter of water

# Results
print "Deflection in  manometer(mm) : %.1f"%(x*1000)

Deflection in  manometer(mm) : 65.5


## Example 5.15 Page No : 146¶

In :
import math

# Variables :
D1 = 120./1000;			#meter
D2 = 120*0.55/1000;			#meter
g = 9.81;			#gravity consmath.tant
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
Q = 30./1000;			#m**3/sec

# Calculations
C = A1*math.sqrt(2*g)/math.sqrt((D1/D2)**4-1);			#venturi consmath.tant
K = 0.94;			#Coeff. of meter
h = (Q/K/C)**2;			#meter
Z1 = 0;			#meter
Z2 = 0.3;			#meter
S = 0.79;			#sp. gravity
w = 1000*S;			#kg/m**3
delta_p = (h+Z1-Z2)*w;			#kg/m**2
delta_p = delta_p*g;			#N/m**2

# Results
print "Pressure difference in N/m**2 : %.3f"%delta_p

#answer is wrong in the book.

Pressure difference in N/m**2 : 28903.473


## Example 5.16 Page No : 147¶

In :
import math

# Variables :
D1 = 160./1000;			#meter
D2 = 60./1000;			#meter
g = 9.81;			#gravity consmath.tant
S = 0.8;			#sp. gravity
Sm = 13.6;			#sp. gravity of mercury
Q = 0.05;			#m**3/sec
K = 0.98;			#Coeff. of meter

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
C = A1*math.sqrt(2*g)/math.sqrt((A1/A2)**2-1);			#venturi consmath.tant
h = (Q/K/C)**2;			#meter
x = h/(Sm/S-1);			#meter

# Results
print "Deflection in meter : %.2f"%x

Deflection in meter : 1.02


## Example 5.17 Page No : 149¶

In :
import math

# Variables :
D1 = 200./1000;			#meter
D2 = 100./1000;			#meter
x = 220./1000;			#meter
g = 9.81;			#gravity consmath.tant
K = 0.98;			#Coeff. of meter
S = 1.;			#sp. gravity
Sm = 13.6;			#sp. gravity of mercury

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
C = A1*math.sqrt(2*g)/math.sqrt((A1/A2)**2-1);			#venturi consmath.tant
h = x*(Sm/S-1);			#meter
Q = K*C*math.sqrt(h);			#m**3/sec
Q = Q*1000;			#litres/sec

# Results
print "Rate of flow in litres/sec : %.f"%Q

Rate of flow in litres/sec : 59


## Example 5.18 Page No : 150¶

In :
import math

# Variables :
D1 = 40./100;			#meter
D2 = 15./100;			#meter
x = 25./100;			#meter
g = 9.81;			#gravity consmath.tant
K = 0.98;			#Coeff. of meter
S = 1.;			#sp. gravity
Sm = 13.6;			#sp. gravity of mercury

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
C = A1*A2*math.sqrt(2*g)/math.sqrt(A1**2-A2**2);			#venturi consmath.tant
h = x*(Sm/S-1);			#meter
Q = K*C*math.sqrt(h);			#m**3/sec
Q = Q*1000*3600;			#litres/hour

# Results
print "Flow of water in litres/hour : %.f"%Q

#Answer in the book is wrong.

Flow of water in litres/hour : 495043


## Example 5.19 Page No : 151¶

In :
import math

# Variables :
D1 = 15./100;			#meter
D2 = 7.5/100;			#meter
g = 9.81;			#gravity consmath.tant
p1 = 4*g*10**4;			#N/m**2
p2 = 1.5*g*10**4;			#kg/cm**2
w = 9.81;			#kg/m**2

# Calculations
A1 = math.pi*D1**2/4;			#m**2
A2 = math.pi*D2**2/4;			#m**2
v1BYv2 = A2/A1;
#v1**2/2/g+p1/w = v2**2/2/g+p2/w
#v1**2 = v2**2-50*g
v2 = math.sqrt(50*g/(1-v1BYv2**2));			#m/s
Q = A2*v2;			#m**3/sec
Q = Q*1000;			#litres/sec

# Results
print "Flow of water in litres/sec : %.f"%Q
#Answer is wrong in the book.

Flow of water in litres/sec : 101


## Example 5.20 Page No : 152¶

In :
import math

# Variables :
D1 = 20./100;			#meter
D2 = 15./100;			#meter
A1 = math.pi/4*D1**2;			#m**2
A2 = math.pi/4*D2**2;			#m**2

# Calculations and Results
v1 = 2;			#m/s
v2 = A1*v1/A2;			#m/s
print "Velocity at another section in m/s : %.2f"%v2
FlowRate = A1*v1;			#m**3/s
FlowRate = FlowRate*1000;			#litres/s
print "Flow Rate in litres/sec : %.1f"%FlowRate

#Answer of velocity in the book is not accurate.

Velocity at another section in m/s : 3.56
Flow Rate in litres/sec : 62.8


## Example 5.21 Page No : 153¶

In :
import math

# Variables :
rd = 0.75;			#relative density
D = 12.5/100;			#meter
p = 1.;			#bar
p = p*1.02;			#kg/cm**2

# Calculations
p = p*9.81*10**4/1000;			#kPa
g = 9.81;			#gravity consmath.tant
w = g*rd;			#N/m**3
pH = p/w;			#meter
Z = 2.5;			#meter
Et = 20;			#Nm
v = math.sqrt((Et-p/w-Z)*2*g);			#m/s
A = math.pi/4*D**2;			#m**2
Q = A*v;			#m**3/sec
Q = A*v*1000;			#litres/sec

# Results
print "Flow Rate of oil in litres/sec : %.f"%Q

Flow Rate of oil in litres/sec : 107


## Example 5.22 Page No : 154¶

In :
import math

# Variables :
rd = 0.75;			#relative density
d1 = 0.3;			#meter
d2 = 0.1;			#meter
Q = 50./1000;			#m**3/sec

# Calculations
A1 = math.pi/4*d1**2;			#m**2
A2 = math.pi/4*d2**2;			#m**2
v1 = Q/A1;			#m/s
v2 = A1*v1/A2;			#m/s
p1 = 200;			#kN/m**2
p2 = 100;			#kN/m**2
w = 9.81;			#kN/m**3
g = 9.81;			#gravity consmath.tant
Z1 = 0;			#meter
Z2 = Z1+p1/w+v1**2/2/g-p2/w-v2**2/2/g;			#meter

# Results
print "Z in meter : %.2f"%Z2

Z in meter : 8.15


## Example 5.23 Page No : 155¶

In :
import math

# Variables :
D1 = 300./1000;			#meter
D2 = 150./1000;			#meter
Q = 50./1000;			#m**3/sec

# Calculations
A1 = math.pi/4*D1**2;			#m**2
A2 = math.pi/4*D2**2;			#m**2
delpBYw = 3;			#p1/w-p2/w = 3;			#m
v1BYv2 = A2/A1;
Z1 = 0;			#meter
Z2 = 0;			#meter
g = 9.81;			#gravity consmath.tant
#HeadLoss = 1/8*v**2/2/g
#Z1+p1/w+v1**2/2/g = Z2+p2/w+v2**2/2/g+HeadLoss
v2 = math.sqrt((Z1-Z2+delpBYw)/(1./2/g-v1BYv2**2/2/g+1/8/2/g));			#m/s
Q = A2*v2;			#m**3/s
Q = Q*1000;			#litres/sec

# Results
print "Discharge in pipe in litres/sec : %.1f"%Q

# note : rounding off error

Discharge in pipe in litres/sec : 140.0