Flow In Circular Pipes Objective To measure the pressure drop in the straight section of smooth, rough, and packed pipes as a function of flow rate. To correlate this in terms of the friction factor and Reynolds number. To compare results with available theories and correlations. To determine the influence of pipe fittings on pressure drop To show the relation between flow area, pressure drop and loss as a function of flow rate for Venturi meter and Orifice meter.
Transcript
Flow In Circular PipesObjective
To measure the pressure drop in the straight section of smooth, rough, and packed pipes as a function of flow rate.
To correlate this in terms of the friction factor and Reynolds number.
To compare results with available theories and correlations.
To determine the influence of pipe fittings on pressure drop
To show the relation between flow area, pressure drop and loss as a function of flow rate for Venturi meter and Orifice meter.
APPARATUS
Pipe NetworkRotametersManometers
Theoretical DiscussionFluid flow in pipes is of considerable importance in process.
•Animals and Plants circulation systems.
•In our homes.
•City water.
•Irrigation system.
•Sewer water system
Fluid could be a single phase: liquid or gases
Mixtures of gases, liquids and solids
NonNewtonian fluids such as polymer melts, mayonnaise
Newtonian fluids like in your experiment (water)
Theoretical DiscussionLaminar flow
To describe any of these flows, conservation of mass and conservation of momentum equations are the most general forms could be used to describe the dynamic system. Where the key issue is the relation between flow rate and pressure drop.
If the flow fluid is:
a. Newtonian
b. Isothermal
c. Incompressible (dose not depend on the pressure)
d. Steady flow (independent on time).
e. Laminar flow (the velocity has only one single component)
Laminar flow
Navier-Stokes equations is govern the flow field (a set of equations containing only velocity components and pressure) and can be solved exactly to obtain the Hagen-Poiseuille relation
.
Vz(r)
In
Pz
r+dr rBody force due to gravity
Flow If the principle of conservation of momentum is applied to a fixed volume element through which fluid is flowing and on which forces are acting, then the forces must be balanced (Newton second law)
Pz+dz
Pz+dz
Laminar flowContinue Forces balance
+dFz r=2πrτzr r
dz
−dFz r +dr=2π(r +dr)τzr r+dr
dz 1…Shear forces
+pz2πrdr
−pz+dz2πrdr 2….Pressure
€
ρg2πrdrdz 3…..Body force
Sum of forces
in the z-direction
⎡
⎣ ⎢
⎤
⎦ ⎥ =
Rate of change of momentum
in the z-direction
⎡
⎣ ⎢
⎤
⎦ ⎥
Laminar flowContinue
Momentum isMass*velocity (m*v)Momentum per unit volume is*vz
Rate of flow of momentum is*vz*dQdQ=vz2πrdrbutvz = constant at a fixed value of r
+ρvz(v2πrdr) z −ρvz(v2πrdr) z+dz =0Laminar flow
Laminar flowContinue
€
2πrτzr rdz−2π(r +dr)τzr r +dr
dz+pz2πrdr−pz+dz2πrdr+ρg2πrdrdz=0
€
τ=μdvzdr
€
Q= 2πvzdr0
R∫ =πR4
8μΔpL
€
Δp=pz=0 −pz=L +ρgL
Hagen-Poiseuille
Turbulent flow
When fluid flow at higher flowrates, the streamlines are not steady and straight and the flow is not laminar. Generally, the flow field will vary in both space and time with fluctuations that comprise "turbulence
For this case almost all terms in the Navier-Stokes equations are important and there is no simple solution
P = P (D, , , L, U,)
uz
úz
Uz
average
ur
úr
Ur
average
p
P’
p
average
Time
Turbulent flow
All previous parameters involved three fundamental dimensions,
Mass, length, and time
From these parameters, three dimensionless groups can be build
€
ΔPρU 2 =f (Re,
LD
)
€
Re=ρUD
μ=
inertiaViscous forces
€
Re=ρUD
μ=
inertiaViscous forces
Friction Factor for Laminar Turbulent flows
From forces balance and the definition of Friction Factor
€
ΔP ×Ac =τ ×S×L
Ac
S=rh =
14
D
τ =ΔP2L
R
€
f =τ
1/2ρU2
For Laminar flow(Hagen - Poiseuill eq)
€
Q=πr4
8μΔPL
ΔPL
=8μUR2
€
f =ΔP RρU 2L
€
f =ΔPL
RρU 2 =
8μρUR
=16Re
For Turbulent Flow
€
f =ΔPL
D2ρU 2 =0.079Re−0.25
Ac: cross section area of the pip
S: Perimeter on which T acts (wetted perimeter)
Rh hydraulic radius
Surface Roughness
Additional dimensionless group /D need to be characterize
Thus more than one curve on friction factor-Reynolds number plot
Fanning diagram or Moody diagram
Depending on the laminar region.
If, at the lowest Reynolds numbers, the laminar portion corresponds to f =16/Re Fanning Chart
or f = 64/Re Moody chart
Friction Factor for Smooth, Transition, and Rough Turbulent flow
€
1
f= 4.0 * log Re* f[ ] − 0.4
Smooth pipe, Re>3000
€
1
f= 4.0 * log
D
ε+ 2.28
Rough pipe, [ (D/)/(Re√ƒ) <0.01]
€
1
f= 4.0 * log
D
ε+ 2.28 − 4.0 * log 4.67
D /ε
Re f+1
⎡
⎣ ⎢
⎤
⎦ ⎥
Transition function for both smooth and rough pipe
The equations for empty pipe flow do not work with out considerable modification
Ergun Equation
€
f ≡ΔPDpε
3
LρUo2(1−ε)
=150(1−ε)μ
UoDpρ+1.75
Reynolds number for a packed bed flow as
€
Re=UoDpρ
(1−ε)μ
Dp is the particle diameter, is the volume fraction that is not occupied by particles
This equation contains the interesting behavior that the pressure drop varies as the first power of Uo for small Re
and as Uo2 for higher Re.
Flow
Dp
A
Energy Loss in Valves
€
Ev = KU 2
2
hv =Δp
ρ= Kv
U 2
2g= 2 f
Leq
D
U 2
g
€
E v = KU 2
2
hv =Δp
ρ= Kv
U 2
2g= 2 f
Leq
D
U 2
g
Function of valve type and valve position The complex flow path through valves can
result in high head loss (of course, one of the purposes of a valve is to create head loss when it is not fully open)
Ev are the loss in terms of velocity heads
Friction Loss Factors for valves
Valve K Leq/D
Gate valve, wide open 0.15 7
Gate valve, 3/4 open 0.85 40
Gate valve, 1/2 open 4.4 200
Gate valve, 1/4 open 20 900
Globe valve, wide open 7.5 350
Energy Loss due to Gradual Expansion
€
EE =KE
U1 −U2( )2
2
€
EE =KE
U1 −U2( )2
2
€
EE =KE
U22
2β −1( )
2
β =A2
A1
€
EE =KE
U22
2β −1( )
2
β =A2
A1
angle ()
00.10.20.30.40.50.60.70.8
0 20 40 60 80
KE
A2
A1
Sudden Contraction (Orifice Flowmeter)
Orifice flowmeters are used to determine a liquid or gas flowrate by measuring the differential pressure P1-P2 across the orifice plate
€
Q=Cd A2
2(p1 −p2)ρ(1−β2)
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1/2
€
Q=Cd A2
2(p1 −p2)ρ(1−β2)
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1/2
0.60.650.7
0.750.8
0.850.9
0.951
102 105 106 107
Re
Cd
Reynolds number based on orifice diameter Red
P1 P2
dD
Flow
103 104
Venturi Flowmeter
The classical Venturi tube (also known as the Herschel Venturi tube) is used to determine flowrate through a pipe. Differential pressure is the pressure difference between the pressure measured at D and at d
D d Flow
Boundary layer buildup in a pipe
Pipe Entrance
v vv
Because of the share force near the pipe wall, a boundary layer forms on the inside surface and occupies a large portion of the flow area as the distance downstream from the pipe entrance increase. At some value of this distance the boundary layer fills the flow area. The velocity profile becomes independent of the axis in the direction of flow, and the flow is said to be fully developed.
Pipe Flow Head Loss(constant density fluid flows)
Pipe flow head loss is proportional to the length of the pipe proportional to the square of the velocity
(high Reynolds number) Proportional inversely with the diameter
of the pipe increasing with surface roughness independent of pressure Total losses in the pipe system is
obtained by summing individual head losses of roughness, fittings, valves ..itc
Pipe flow head loss is proportional to the length of the pipe proportional to the square of the velocity
(high Reynolds number) Proportional inversely with the diameter
of the pipe increasing with surface roughness independent of pressure Total losses in the pipe system is
obtained by summing individual head losses of roughness, fittings, valves ..itc
Pipe Flow Summary
The statement of conservation of mass, momentum and energy becomes the Bernoulli equation for steady state constant density of flows.
Dimensional analysis gives the relation between flow rate and pressure drop.
Laminar flow losses and velocity distributions can be derived based on momentum and mass conservation to obtain exact solution named of Hagen - Poisuille
Turbulent flow losses and velocity distributions require experimental results.
Experiments give the relationship between the fraction factor and the Reynolds number.
Head loss becomes minor when fluid flows at high flow rate (fraction factor is constant at high Reynolds numbers).
Images - Laminar/Turbulent Flows
Laser - induced florescence image of an incompressible turbulent boundary layer
Simulation of turbulent flow coming out of a tailpipe
