In [4]:

```
# Resultant force on the elbow
from math import *
# Given
Q = 0.3 # Water flow rate in m**3/s
d1 = 0.3 # diameter at inlet in meters
A1 = pi*d1**2/4 # inlet area in m**2
d2 = 0.15 # diameter at outlet in m
A2 = pi*d2**2/4 # area at outlet in m**2
P1 = 175*10**3 # inlet pressure in kN/m**2
P2 = 160*10**3 # Pressure at outlet in kN/m**2
F1 = P1*A1 # Force at inlet
F2 = P2*A2 # Force at outlet
rho = 1000 # density of water in kg/m**3
V1 = Q/A1 # inlet velocity in m/s
V2 = Q/A2 # Velocity at outlet in m/s
theta = 45*pi/180 # angle in deg
# Soultion
# Applying the X momentum equation we get
Rx = F1 - F2*cos(theta)-rho*Q*(V2*cos(theta)-V1)
# Applying the Y momentum equation
Ry = F2*sin(theta)+rho*Q*(V2*sin(theta)-0)
R = sqrt(Rx**2+Ry**2)
print "Resultant force on the elbow = ",round(R,2),"N"
a = atan(Ry/Rx)*180/pi
print "Angle of resultant force = ",round(a,4),"deg"
```

In [7]:

```
# Force exerted by the jet on the vane
from math import *
# Given
V1 = 80 # Velocity in ft/s
A1 = 0.1 # area in ft**2
g = 32.2 # Acceleration due to gravity in ft/s**2
rho = 1.94 # density in lb/ft**3
a = pi/3 # angle of pipe bend
# Solution
Q = A1*V1 # Total discharge in m**3
# Applying bernoullis at point 1 and 2
V2 = sqrt((2*g*V1**2/(2*32.2))-3*2*g)
# Pressure at the end of the section are atmospheric and hence 0
# momentum equation in X direction
Rx = -(rho*Q*(V2*cos(a)-80))
# momentum equation in Y direction
Ry = (rho*Q*(V2*sin(a)-0))
R = sqrt(Rx**2+Ry**2)
print "Resultant force = ",round(R,0),"lbs"
ang = atan(Ry/Rx)*180/pi
print "Angle of resultant force = ",round(ang,4),"deg"
```

In [10]:

```
# Force needed to hold the Y position
# Given
from __future__ import division
from math import *
Q1 = 0.5 # discharge from pipe 1 in m**3/s
Q2 = 0.3 # discharge from pipe 2 in m**3/s
Q3 = 0.2 # discharge from pipe 3 in m**3/s
d1 = 0.45 # diameter of pipe 1 in m
d2 = 0.3 # diameter of pipe 2 in m
d3 = 0.15 # diameter of pipe 3 in m
A1 = pi*d1**2/4 # area in m**2
A2 = pi*d2**2/4 # area in m**2
A3 = pi*d3**2/4 # area in m**2
P1 = 60*10**3 # Pressure at point 1 in kPa
gma = 9810
g = 9.81 # acceleration due to gravity in m/s**2
rho = 1000 # density in kg/m**3
# Solution
V1 = Q1/A1
V2 = Q2/A2
V3 = Q3/A3
P2 = gma*((P1/gma) + V1**2/(2*g) - V2**2/(2*g))
P3 = gma*((P1/gma) + V1**2/(2*g) - V3**2/(2*g))
F1 = P1*A1
F2 = P2*A2
F3 = P3*A3
Rx = rho*(Q2*V2*cos(pi/6)-Q3*V3*cos(pi/9)-0)+F3*cos(pi/9)-F2*cos(pi/6)
Ry = rho*((Q2*V2*sin(pi/6)+Q3*V3*sin(pi/9)-Q1*V1))+F3*sin(pi/9)-F2*sin(pi/6)-F1
R = sqrt(Rx**2+Ry**2)
a = atan(Ry/Rx)*180/pi
print "Resultant Force = ",round(R,0),"N"
print "Angle with horizontal = ",round(a,1),"deg with horizontal"
```

In [11]:

```
# Normal force on the plate
# Given
from math import *
d = 2 # diameter in inches
A = pi*d**2/(4*144) # Area of jet
V = 100 # velocity of jet in ft/s
Q = A*V # dischargge in ft**3/s
gma = 62.4 # mass
g = 32.2 # acceleration due to gravity in ft/s**2
# Solution
Rx = (gma*Q*V)/g # horizontal force required to keep plate in position
print "Normal force on the plate = ",round(Rx,0),"lbs"
```

In [12]:

```
# Force on the plate ; work doen per second; efficiency
# Given
from math import *
D = 0.075 # diameter in m
A =pi*D**2/4 # area of jet
V =15 # velocity of jet in m/s
w = 9810 # specific weight
g = 9.81 # acceleration due to gravity in m/s^2
# Solution
Q =A*V # Discharge in m**3/s
Vp = 10 # velocity of plate in m/s
Rx = w*Q*(V-Vp)/g # force in X direction
print "Force on the plate = ",round(Rx,2),"N"
W = Rx*Vp
print "Work done per second = ",round(W,1),"N.m/s"
Eff = 2*(V-Vp)*Vp/V**2
E = 100*Eff
print "Efficiency = ",round(E,1),"%"
```

In [13]:

```
# Force exerted on the plate
# Given
from math import *
d = 3 # diameter in inches
A = pi*d**2/(4*144) # Area of jet
Q = 2 # discharge in ft**3/s
rho = 1.94 # density in lbs/ft**3
# Solution
V = Q/A # velocity in ft/s
alpha = pi/6 # inlet vane angle
bta = pi/6 # outlet vane angle
Rx = rho*Q*(V*cos(bta)+V*cos(alpha)) # force in X direction
Ry = rho*Q*(V*sin(bta)-V*sin(alpha)) # force in Y direction
print "Force exerted in X direction = ",round(Rx,1),"lbs"
print "Force exerted in Y direction = ",round(Ry,1),"lbs"
```

In [15]:

```
# Angle of blade tips at inlet and exit ; work done on the vane; efficiency of the vane
# Given
from math import *
V1 =40 # velocity in m/s
Vp = 20 # velocity of the plate in m/s
alpha = pi/6 # inlet vane angle
bta = pi/9 # outlet vane angle
g = 9.81
# Solution
V1x = V1*cos(alpha)
Vw1 = V1x;
V1y = V1*sin(alpha)
dV = V1x - Vp
theta = atan(V1y/dV)*180/pi
Vr1 = V1y/sin(theta*pi/180)
Vr2 = Vr1
# from trial and error we get the blade tip angle at inlet and outlet
print "a ) Angle of blade top at inlet and Outlet, Phi = 4 deg"
phi = 4*pi/180
V2 = Vr2*sin(phi)/sin(bta)
V2w = V2*cos(bta)
W = (V2w+V1x)*Vp/g
print "b ) Work done per N of fluid per second = ",round(W,2),"N.m"
Eff = (1 - (V2/V1)**2)*100
print "c ) Efficiency = ",round(Eff,2),"%"
```

In [16]:

```
# Thrust on the plane; propeller efficiency ; theoretical horsepower ; pressure difference acros the blades
# Given
from math import *
v = 220 # velocity in ft/s
d = 6 # diameter of the propeller
Q = 12000 # discharge in ft**3/s
mf = 0.0022 # mass flow rate in slugs/ft**3
# Solution
V1 = v*5280/3600 # velocity in ft/s
V = Q/(pi*d**2/4) # velocity in ft/s
V4 = 2*V-V1
F = mf*Q*(V4-V1) # thrust on the plane
print "a - Thrust on the plane = ",round(F,1),"lbs"
Eff = V1/V # efficiency
E = Eff*100
print "b - Theoretical efficiency = ",round(E,0),"%"
Thp = F*V1/(500*Eff)
print "c - Theoretical horsepower required = ",round(Thp,0),"hp"
dP = mf*(V4**2-V1**2)/2
print "d - Pressure difference across blades = ",round(dP,2),"lbs/ft**3"
```

In [ ]:

```
```