Formula Sheets: Flow Through Pipes | Fluid Mechanics for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Flow Through Pipes F ormula Sheet for GA TE
Civil Engineering
Fluid Mechanics: Flow Through Pipes
1. Types of Flow
• Reynolds Number (Re) : Determines flow type (laminar , tr ansitional, turbu-
lent)
Re=
?VD
µ
=
VD
?
where ? = density , V = velocity , D = pipe diameter , µ = dynamic viscosity , ? =
kinematic viscosity .
– Laminar: Re<2000
– Tr ansitional: 2000= Re=4000
– Turbulent: Re>4000
2. Continuity Equation
• F or incompressible flow:
A
1
V
1
=A
2
V
2
=Q
whereA = cross-sectional area,V = velocity ,Q = discharge (flow r ate).
3. Bernoulli’ s Equation
• F or steady , incompressible flow (along a streamline):
p
1
?g
+
V
2
1
2g
+z
1
=
p
2
?g
+
V
2
2
2g
+z
2
+h
L
wherep = pressure,g = gr avitational acceler ation,z = elevation,h
L
= head loss.
4. Major Losses (Frictional Losses)
• Darcy-W eisbach Equation :
h
f
=f
L
D
V
2
2g
where h
f
= head loss due to friction, f = friction factor , L = pipe length, D =
diameter ,V = vel ocity .
1
Page 2


Flow Through Pipes F ormula Sheet for GA TE
Civil Engineering
Fluid Mechanics: Flow Through Pipes
1. Types of Flow
• Reynolds Number (Re) : Determines flow type (laminar , tr ansitional, turbu-
lent)
Re=
?VD
µ
=
VD
?
where ? = density , V = velocity , D = pipe diameter , µ = dynamic viscosity , ? =
kinematic viscosity .
– Laminar: Re<2000
– Tr ansitional: 2000= Re=4000
– Turbulent: Re>4000
2. Continuity Equation
• F or incompressible flow:
A
1
V
1
=A
2
V
2
=Q
whereA = cross-sectional area,V = velocity ,Q = discharge (flow r ate).
3. Bernoulli’ s Equation
• F or steady , incompressible flow (along a streamline):
p
1
?g
+
V
2
1
2g
+z
1
=
p
2
?g
+
V
2
2
2g
+z
2
+h
L
wherep = pressure,g = gr avitational acceler ation,z = elevation,h
L
= head loss.
4. Major Losses (Frictional Losses)
• Darcy-W eisbach Equation :
h
f
=f
L
D
V
2
2g
where h
f
= head loss due to friction, f = friction factor , L = pipe length, D =
diameter ,V = vel ocity .
1
• Friction F actor (f ) :
– Laminar flow: f =
64
Re
– Turbulent flow: Use Moody’ s chart or Colebrook-White equation:
1
v
f
=-2 log
10
(
?/D
3.7
+
2.51
Re
v
f
)
where? = pipe roughness.
• Pressure Drop :
?p=f
L
D
?V
2
2
• Pumping Power :
P =
Q?gh
f
?
where? = pump efficiency .
5. Minor Losses
• Head loss due to fittings, bends, valves, etc.:
h
m
=K
V
2
2g
whereK = loss coefficient (depends on fitting type).
• Co mmonK values:
– Sudden contr action: K =0.5
(
1-
A
2
A
1
)
– Sudden expansion: K =
(
1-
A
1
A
2
)
2
– Entr ance: K˜0.5 , Exit: K˜1.0
• T otal head loss:
h
L
=h
f
+
?
h
m
6. Chezy’ s F ormula
• F or open channel or pipe flow:
V =C
v
RS
whereC = Chezy’ s coefficient, R = h ydr aulic r adius (A/P ),S = slope of energy
gr ade line.
• Relation with Manning’ sn :
C =
1
n
R
1/6
wheren = Manning’ s roughness coefficient.
2
Page 3


Flow Through Pipes F ormula Sheet for GA TE
Civil Engineering
Fluid Mechanics: Flow Through Pipes
1. Types of Flow
• Reynolds Number (Re) : Determines flow type (laminar , tr ansitional, turbu-
lent)
Re=
?VD
µ
=
VD
?
where ? = density , V = velocity , D = pipe diameter , µ = dynamic viscosity , ? =
kinematic viscosity .
– Laminar: Re<2000
– Tr ansitional: 2000= Re=4000
– Turbulent: Re>4000
2. Continuity Equation
• F or incompressible flow:
A
1
V
1
=A
2
V
2
=Q
whereA = cross-sectional area,V = velocity ,Q = discharge (flow r ate).
3. Bernoulli’ s Equation
• F or steady , incompressible flow (along a streamline):
p
1
?g
+
V
2
1
2g
+z
1
=
p
2
?g
+
V
2
2
2g
+z
2
+h
L
wherep = pressure,g = gr avitational acceler ation,z = elevation,h
L
= head loss.
4. Major Losses (Frictional Losses)
• Darcy-W eisbach Equation :
h
f
=f
L
D
V
2
2g
where h
f
= head loss due to friction, f = friction factor , L = pipe length, D =
diameter ,V = vel ocity .
1
• Friction F actor (f ) :
– Laminar flow: f =
64
Re
– Turbulent flow: Use Moody’ s chart or Colebrook-White equation:
1
v
f
=-2 log
10
(
?/D
3.7
+
2.51
Re
v
f
)
where? = pipe roughness.
• Pressure Drop :
?p=f
L
D
?V
2
2
• Pumping Power :
P =
Q?gh
f
?
where? = pump efficiency .
5. Minor Losses
• Head loss due to fittings, bends, valves, etc.:
h
m
=K
V
2
2g
whereK = loss coefficient (depends on fitting type).
• Co mmonK values:
– Sudden contr action: K =0.5
(
1-
A
2
A
1
)
– Sudden expansion: K =
(
1-
A
1
A
2
)
2
– Entr ance: K˜0.5 , Exit: K˜1.0
• T otal head loss:
h
L
=h
f
+
?
h
m
6. Chezy’ s F ormula
• F or open channel or pipe flow:
V =C
v
RS
whereC = Chezy’ s coefficient, R = h ydr aulic r adius (A/P ),S = slope of energy
gr ade line.
• Relation with Manning’ sn :
C =
1
n
R
1/6
wheren = Manning’ s roughness coefficient.
2
7. Flow Through Par allel Pipes
• Head loss is same across par allel pipes:
h
f
=h
f1
=h
f2
=...
• T otal flow r ate:
Q=Q
1
+Q
2
+...
8. Hydr aulic and Energy Gr ade Lines
• Hydr aulic Gr ade Line (HGL) :
HGL=z+
p
?g
• Energy Gr ade Line (EGL) :
EGL=z+
p
?g
+
V
2
2g
• EGL slopes downward due to head losses; HGL ma y rise or fall due to changes
in pressure or elevation.
9. Pipe Networks
• Hardy-Cross Method : Iter ative method to solve for flow distribution in pipe
networks.
• Continuity at junctions: Sum of inflows = Sum of outflows.
• Head loss balance in loops:
?
h
f
=0 .
3