CHEN10031
List of Equations and Information Pipe flow : Re <2100 :
Lamina flow
Turbulent How
All symbols have their usual meanings. Re > 10000 :
Transitional How
velocityL Density
2100 Re 210000 :
Reynolds number: Diameter
determine nature of fluid 𝐷𝑢𝜌 ashle
𝐷
RH𝐻 𝑢𝜌 Open channel :
Inertia force Re = =
Re = 𝜇 𝜇
Viscous force & &
viscosity
Viscosity: shear stress Kinematic viscosity : v
=
↓ d𝑢
des velociet e
< shear rate) : 𝜏 = 𝜇𝛾̇ = 𝜇 * For Newtonian liquid
from the wall d𝑦
Laminar flow: Pressure difference
L
Diameter
𝜋|ΔP|𝐷4 &
𝑄=
128𝜇𝐿 -
length of
pipe
|Δ𝑃|𝐷2
𝑢̅ =
32𝜇𝐿
& viscosity
Conservation of mass:
𝜌1 𝑢1 𝐴1 = 𝜌2 𝑢2 𝐴2 = 𝑀̇ Re
= A :Th e
head (required fluid to flowl for
Bernoulli’s equation: prop
&
𝑃1 𝑢12 𝑃2 𝑢22
+ + ℎ1 + Δℎ𝑝 = + + ℎ2 + Δℎ𝑓 & loss due to viscosity
𝜌1 𝑔 2𝑔 𝜌2 𝑔 2𝑔
to KE
loss related
Friction losses: Friction
shear stress
Fanning R
Factory 4𝐿 𝑢2
𝑢2
To a
f =
Darcy Equation
- Δℎ𝑓 = 𝑓 =𝐾 >
- losses in
fitting
0 .
Spur 𝐷 2𝑔 2𝑔
16 f E
Laminar regime surface roughness =
𝑓=
~
Re
Re
Colebrook
1 𝜀/𝐷 1.11 6.9
equation >
- Turbulent regime = −3.6 log (( ) + )
√𝑓 3.7 Re
Pipe networks:
Δℎ = 𝑅 ∗ 𝑄 2
𝑅∗ =
∑𝑖 𝐾𝑖 ::: *
2𝑔𝐴2 R
↑
Series 𝑅𝑇∗ = ∑ 𝑅𝑖∗ => Ri +
Ros "
...
Resistance to flow
𝑖
1 1
RF QY =
RY Q Y Parallel =∑ => ...
√𝑅𝑇∗ 𝑖
√𝑅𝑖∗
Force balance:
− ∯ 𝜌V ⋅ (V ⋅ n)dΓ − ∯ 𝑃ndΓ + ∯ τ ⋅ ndΓ − ∯ 𝜌𝑔𝑧 ⋅ ndΓ + F𝑟 = 0
Γ Γ Γ Γ
Momentum F Pressure I
I ShearF ↑ Reaction Force
F
Garity
· A ,
U, -
PAzus , A , -P , Az
P -
Force required to hold pipework in
place
<
Stationary Vertical Column
! Through
nonzle Paper continues/…
-Momentum F-Pressure F =
Er
-
Pressure F =
Gravity F
)
? E = -
BA(u ,
- u ,
P
, -
Pc =
PgCzz -
z
,
/
Pgch
(
-P =
(
I
, CHEN10031
Transient forces: Water Hammer
change in
P when value is
closing in
𝜌𝐿𝑢 d
Full Δ𝑃 = relation with time
𝑡
1 of fluid (Pa)
1 1 𝐾 𝐷 −2
K = bulk modulus of
elasticity
Speed of sound limited Δ𝑃 = 𝑢𝐾 2 𝜌2 (1 + ) E : bull modulus of elasticity of
pipe material
𝐸 𝑡𝑤 tw wall thickness
pipe
:
Pumps: I I
>
-
-
&
𝑃2 − 𝑃1 − 𝑢22 𝑢12 u
+ 2ki c (
ki
=
+
(system head) pump head >
- Δℎ𝑝 = + ℎ2 − ℎ1 + + Δℎ𝑓 i zy
𝜌𝑔 2𝑔
power
needed to run a
pump >
-
𝑊̇𝑅 =
𝑄𝜌𝑔Δℎ 𝑝
= ( +
[K) 80 since u
=
:
𝜂 efficiency
&
Vapor pressure
of
purped fluid
𝑃𝑜 𝑃𝑜 - abs P et suction side
Net positive suction head available>
-
NPSH-A = + ℎ𝑜 − Δℎ𝑓,𝑜 − Po = resevoir
𝜌𝑔 𝜌𝑔 ho = reservoir liquid level relative to prop centraline
↳ head loss due to friction on suction side
Hydraulic radius:
𝐴𝑐 𝐷𝐻 -
hydraulic diameter
𝑅𝐻 = =
𝑃 4
Area and perimeter of different channels:
Square 𝐴𝑐 = 𝑑𝑏
𝑃 = 𝑏 + 2𝑑
Trapezoidal 𝐴𝑐 = 𝑑(𝑏 + 𝑧𝑑)
𝑃 = 𝑏 + 2𝑑 (1 + 𝑧 2 )1/2
Triangular 𝐴𝑐 = 𝑧 𝑑 2
𝑃 = 2𝑑 (1 + 𝑧 2 )1/2
b = base, d = normal depth, z = side slope of channel (as z:1)
Velocity in a steady, uniform open channel:
𝑢 = 𝐶(𝑅𝐻 𝑆)1⁄2
Chezy coefficient:
↓2𝑔 1/2
𝐶=( )
𝑓
friction factor
Manning coefficient:
&
fanning
1 1/6
𝐶= 𝑅
𝑛 𝐻
coefficient nature of channel and its surface
Manning roughness
* depends
.
-
on
Compound channels:
5/3
1 𝐴𝑐
𝐾=
𝑛 𝑃2/3
𝑄𝑇 = ∑ 𝐾𝑖 𝑆 1/2
𝑖
Open Channel Bernoulli’s equation (d = normal depth, h = channel height):
𝑢12 𝑢22
𝑑1 + + ℎ𝑜,1 = 𝑑2 + + ℎ𝑜,2 + Δℎ𝑓
2𝑔 2𝑔
Paper continues/…
List of Equations and Information Pipe flow : Re <2100 :
Lamina flow
Turbulent How
All symbols have their usual meanings. Re > 10000 :
Transitional How
velocityL Density
2100 Re 210000 :
Reynolds number: Diameter
determine nature of fluid 𝐷𝑢𝜌 ashle
𝐷
RH𝐻 𝑢𝜌 Open channel :
Inertia force Re = =
Re = 𝜇 𝜇
Viscous force & &
viscosity
Viscosity: shear stress Kinematic viscosity : v
=
↓ d𝑢
des velociet e
< shear rate) : 𝜏 = 𝜇𝛾̇ = 𝜇 * For Newtonian liquid
from the wall d𝑦
Laminar flow: Pressure difference
L
Diameter
𝜋|ΔP|𝐷4 &
𝑄=
128𝜇𝐿 -
length of
pipe
|Δ𝑃|𝐷2
𝑢̅ =
32𝜇𝐿
& viscosity
Conservation of mass:
𝜌1 𝑢1 𝐴1 = 𝜌2 𝑢2 𝐴2 = 𝑀̇ Re
= A :Th e
head (required fluid to flowl for
Bernoulli’s equation: prop
&
𝑃1 𝑢12 𝑃2 𝑢22
+ + ℎ1 + Δℎ𝑝 = + + ℎ2 + Δℎ𝑓 & loss due to viscosity
𝜌1 𝑔 2𝑔 𝜌2 𝑔 2𝑔
to KE
loss related
Friction losses: Friction
shear stress
Fanning R
Factory 4𝐿 𝑢2
𝑢2
To a
f =
Darcy Equation
- Δℎ𝑓 = 𝑓 =𝐾 >
- losses in
fitting
0 .
Spur 𝐷 2𝑔 2𝑔
16 f E
Laminar regime surface roughness =
𝑓=
~
Re
Re
Colebrook
1 𝜀/𝐷 1.11 6.9
equation >
- Turbulent regime = −3.6 log (( ) + )
√𝑓 3.7 Re
Pipe networks:
Δℎ = 𝑅 ∗ 𝑄 2
𝑅∗ =
∑𝑖 𝐾𝑖 ::: *
2𝑔𝐴2 R
↑
Series 𝑅𝑇∗ = ∑ 𝑅𝑖∗ => Ri +
Ros "
...
Resistance to flow
𝑖
1 1
RF QY =
RY Q Y Parallel =∑ => ...
√𝑅𝑇∗ 𝑖
√𝑅𝑖∗
Force balance:
− ∯ 𝜌V ⋅ (V ⋅ n)dΓ − ∯ 𝑃ndΓ + ∯ τ ⋅ ndΓ − ∯ 𝜌𝑔𝑧 ⋅ ndΓ + F𝑟 = 0
Γ Γ Γ Γ
Momentum F Pressure I
I ShearF ↑ Reaction Force
F
Garity
· A ,
U, -
PAzus , A , -P , Az
P -
Force required to hold pipework in
place
<
Stationary Vertical Column
! Through
nonzle Paper continues/…
-Momentum F-Pressure F =
Er
-
Pressure F =
Gravity F
)
? E = -
BA(u ,
- u ,
P
, -
Pc =
PgCzz -
z
,
/
Pgch
(
-P =
(
I
, CHEN10031
Transient forces: Water Hammer
change in
P when value is
closing in
𝜌𝐿𝑢 d
Full Δ𝑃 = relation with time
𝑡
1 of fluid (Pa)
1 1 𝐾 𝐷 −2
K = bulk modulus of
elasticity
Speed of sound limited Δ𝑃 = 𝑢𝐾 2 𝜌2 (1 + ) E : bull modulus of elasticity of
pipe material
𝐸 𝑡𝑤 tw wall thickness
pipe
:
Pumps: I I
>
-
-
&
𝑃2 − 𝑃1 − 𝑢22 𝑢12 u
+ 2ki c (
ki
=
+
(system head) pump head >
- Δℎ𝑝 = + ℎ2 − ℎ1 + + Δℎ𝑓 i zy
𝜌𝑔 2𝑔
power
needed to run a
pump >
-
𝑊̇𝑅 =
𝑄𝜌𝑔Δℎ 𝑝
= ( +
[K) 80 since u
=
:
𝜂 efficiency
&
Vapor pressure
of
purped fluid
𝑃𝑜 𝑃𝑜 - abs P et suction side
Net positive suction head available>
-
NPSH-A = + ℎ𝑜 − Δℎ𝑓,𝑜 − Po = resevoir
𝜌𝑔 𝜌𝑔 ho = reservoir liquid level relative to prop centraline
↳ head loss due to friction on suction side
Hydraulic radius:
𝐴𝑐 𝐷𝐻 -
hydraulic diameter
𝑅𝐻 = =
𝑃 4
Area and perimeter of different channels:
Square 𝐴𝑐 = 𝑑𝑏
𝑃 = 𝑏 + 2𝑑
Trapezoidal 𝐴𝑐 = 𝑑(𝑏 + 𝑧𝑑)
𝑃 = 𝑏 + 2𝑑 (1 + 𝑧 2 )1/2
Triangular 𝐴𝑐 = 𝑧 𝑑 2
𝑃 = 2𝑑 (1 + 𝑧 2 )1/2
b = base, d = normal depth, z = side slope of channel (as z:1)
Velocity in a steady, uniform open channel:
𝑢 = 𝐶(𝑅𝐻 𝑆)1⁄2
Chezy coefficient:
↓2𝑔 1/2
𝐶=( )
𝑓
friction factor
Manning coefficient:
&
fanning
1 1/6
𝐶= 𝑅
𝑛 𝐻
coefficient nature of channel and its surface
Manning roughness
* depends
.
-
on
Compound channels:
5/3
1 𝐴𝑐
𝐾=
𝑛 𝑃2/3
𝑄𝑇 = ∑ 𝐾𝑖 𝑆 1/2
𝑖
Open Channel Bernoulli’s equation (d = normal depth, h = channel height):
𝑢12 𝑢22
𝑑1 + + ℎ𝑜,1 = 𝑑2 + + ℎ𝑜,2 + Δℎ𝑓
2𝑔 2𝑔
Paper continues/…
