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CBE 150A – Transport Spring Semester 2014 Friction Losses Flow through Conduits Incompressible...
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Transcript of CBE 150A – Transport Spring Semester 2014 Friction Losses Flow through Conduits Incompressible...
CBE 150A – Transport Spring Semester 2014
Friction LossesFlow through Conduits
Incompressible Flow
CBE 150A – Transport Spring Semester 2014
Goals
• Calculate frictional losses for laminar and turbulent flow through circular and non-circular pipes
• Define the friction factor in terms of flow properties
• Calculate the friction factor for laminar and turbulent flow
• Define and calculate the Reynolds number for different flow situations
• Derive the Hagen-Poiseuille equation
CBE 150A – Transport Spring Semester 2014
Flow Through Circular Conduits
Consider the steady flow of a fluid of constant density in fully developed flow through a horizontal pipe and visualize a disk of fluid of radius r and length dL moving as a free body. Since the fluid posses a viscosity, a shear force opposing the flow will exist at the edge of the disk
CBE 150A – Transport Spring Semester 2014
Balances
Mass Balance
→222111 SVSV 21 VV
CBE 150A – Transport Spring Semester 2014
Balances
4
2
12
DppFw
Momentum Balance
gw FFSpSpVVm 22111122
HorizontalSSS 1212
CBE 150A – Transport Spring Semester 2014
Momentum Balance (contd)If we imagine that the fluid disk extends to the wall, Fw is just due to the shear stress τw acting over the length of the disk.
Equating and solving for p over a length of pipe L.
ww LDF )(
wD
Lp
4
CBE 150A – Transport Spring Semester 2014
Mechanical Energy Balance
fh
pzg
VW
2
ˆ2
H o r iz o n t a lW 120ˆ
p
h f
CBE 150A – Transport Spring Semester 2014
Viscous Dissipation (Frictional Loss) Equation
Combining the Momentum and MEB results:
• Applies to laminar or turbulent flow• Good for Newtonian or Non-Newtonian fluids• Only good for friction losses as result of wall shear.
Not proper for fittings, expansions, etc.
wf D
Lh
4
CBE 150A – Transport Spring Semester 2014
The Friction Factor
w is not conveniently determined so the dimensionless friction factor is introduced into the equations.
headvelocitydensity
stressshearwall
Vf w
22
CBE 150A – Transport Spring Semester 2014
Fanning Friction Factor
• Increases with length• Decreases with diameter• Only need L, D, V and f to get friction loss• Valid for both laminar and turbulent flow• Valid for Newtonian and Non-Newtonian fluids
–
24
2V
D
Lfh f
CBE 150A – Transport Spring Semester 2014
Calculation of f for Laminar Flow
wD
Lp
4
22
2rR
Ru w
x
First we need the velocity profile for laminar flow in a pipe. We’ll rely on Chapter 8 for that result.
2221 1
4 R
r
L
Rppux
Recall our earlier result: LRppw 221
CBE 150A – Transport Spring Semester 2014
Laminar FlowFind Bulk Velocity (measurable quantity).
S
dSuV S
2
02
2
R
drur
R
drru
R
S
R
w drrrRR 0
323
24maxw uR
V
CBE 150A – Transport Spring Semester 2014
Reynolds Number
forceviscous
forceinertiaVDN
Re
Osbourne Reynolds (1842-1912)
CBE 150A – Transport Spring Semester 2014
Laminar Flow
4
22
RVand
Vf ww
RVf
8
←Laminar Flow←Newtonian FluidRe
f16
CBE 150A – Transport Spring Semester 2014
Hagen-Poiseuille (Laminar Flow)
Recall again:
RL
ppw 2
21
Use: Measurement of viscosity by measuring p and q through a tube of known D and L for Laminar flow.
L
DppV
32
221
L
ppRRVSVq
82142
CBE 150A – Transport Spring Semester 2014
Turbulent FlowWhen flow is turbulent, the viscous dissipation effects cannot be derived explicitly as in laminar flow, but the following relation is still valid.
24
2V
D
Lfh f
The problem is that we can not write a closed form solution for the friction factor f. Must use correlations based on experimental data.
CBE 150A – Transport Spring Semester 2014
Friction FactorTurbulent Flow
For turbulent flow f = f( Re , k/D ) where k is the roughness of the pipe wall.
Note, roughness is not dimensionless. Here, the roughness is reported in inches.
MaterialRoughness, k
inches
Cast Iron 0.01
Galvanized Steel 0.006
Commercial SteelWrought Iron
0.0018
Drawn Tubing 0.00006
CBE 150A – Transport Spring Semester 2014
How Does k/D Affect f(Text Figure 13.1)
CBE 150A – Transport Spring Semester 2014
Friction FactorTurbulent Flow
As and alternative to Moody Chart use Churchill’s correlation:
16
16
9.0
121
23
12
37530
27.07
1ln457.2
182
ReB
DkReA
BARef
CBE 150A – Transport Spring Semester 2014
Friction FactorTurbulent Flow
A less accurate but sometimes useful correlation for estimates is the Colebrook equation. It is independent of velocity or flow rate, instead depending on a combined dimensionless quantity
.Re f
f
Dk
f Re
255.1
7.3log4
1
CBE 150A – Transport Spring Semester 2014
Flow Through Non-Circular Conduits
Rather than resort to deriving new correlations for the friction factor, an approximation is developed for an ‘equivalent’ diameter Deq with which to calculate the Reynolds number and the friction factor.
pHeq LSRD 44
where:• RH = hydraulic radius• S = cross-sectional area• Lp = wetted perimeter
Note: Do not use Deq to calculate cross-sectional area or for laminar flow situations.
CBE 150A – Transport Spring Semester 2014
ExamplesCircular Pipe
DRR
RDeq
2
24
2
Rectangular Ducts
WH
WH
WH
WHDeq 2
24
CBE 150A – Transport Spring Semester 2014
Example 1
Water flows horizontally at a rate of 600 gal/min through 400 feet of 5 in. diameter Schedule 40 cast-iron pipe. Find the average (bulk) velocity and the pressure drop.
400 ft.
5 in.
600 GPM
CBE 150A – Transport Spring Semester 2014
Text Appendix M
CBE 150A – Transport Spring Semester 2014
10 Minute ProblemMy father is installing a sprinkler system at his lake house. The pump pulls water from the lake through a feed line and delivers 12 GPM to the sprinkler system distribution line at a point in the front yard. For the sprinkler system to operate properly, the pressure at the branch point must be 90 psig. What horsepower pump does he buy ?
40 ft.
25 ft.
10 ft.Tubing lengths:Lake to pump suction – 50 ft.Pump to distribution line – 150 ft.