Friction losses in turbulent flow (Fanning Equation).pdf

Fluid mechanics

1. Friction losses in Turbulent
flow(Fanning equation)

2. Design Equations for Laminar and Turbulent
Flow in Pipes
Velocity Profiles in Pipes
•One of the most important applications of fluid flow is the flow of
fluids inside circular conduits, pipes, and tubes.
•Often, the flow behavior of fluids in these types of conduits is
dependent on the size (e.g., diameter) of the conduits.
•Of those sizes, Schedule 40 pipe is a common standard. As a
comparison, Schedule 80 pipe has a thicker wall and will withstand
about twice the pressure of Schedule 40 pipe.

3. •The schedule number of a pipe refers to pipe thickness, or the difference
between the outer and inner diameters.
•For example, 1-inch Schedule 40 and Schedule 80 pipes have the same
outer diameter, but will have different inner diameters due to the different
pipe thicknesses.
•Since both have the same outside diameter, they can use the same fittings.
Sizes of tubing are generally given by the outside diameter and wall
thickness.
•When a fluid is flowing in a circular pipe and the velocities are measured at
different distances from the pipe wall to the center of the pipe, it has been
shown that for both laminar and turbulent flow, the fluid in the center of
the pipe is moving faster than the fluid near the walls.

4. Fig 5.1.1 is a plot of the relative distance from the center of the pipe versus the
fraction of maximum velocity v/v
max, where v is local velocity at the given radial
position and v
max the maximum velocity at the center of the pipe.

5. •In many engineering applications, the relation between the average
velocity v
avin a pipe and the maximum velocity v
maxis useful, since in
some cases only the v
maxat the center point of the tube is measured.
•Hence, from only one point measurement, this relationship between
v
maxand v
avcan be used to determine v
av
•In Fig. 5.1-2, experimentally measured values of v
av/v
maxare plotted as
a function of the Reynolds numbers Dv
avρ/μ and Dv
maxρ/μ.

7. Pressure Drop and Friction Loss in Laminar
Flow
Pressure drop and loss due to friction.
•When the fluid in a pipe is flowing at steady-state in the laminar flow
regime, then, for a Newtonian fluid, the shear stress is given by Eq.
??????
��=−??????
�??????
??????
��

•where τ
yz= F/A and is the shear stress or force per unit area exerted
in the y-direction by fluid flowing in the z-direction in Newtons/m
2
(N/m
2
), which is rewritten for change in radius drrather than
distance dy, as follows: ??????
??????�=−??????
�????????????
�??????
•Using this relationship and making a shell momentum balance on the
fluid over a cylindrical shell, the Hagen–Poiseuilleequation for
laminar flow of a liquid in a circular tube is obtained.

8. •∆??????
� =??????
1−??????
2� =
32????????????(??????2−??????1)
??????
2
where p
1is upstream pressure at point 1,
N/m
2
; p
2is pressure at point 2; v is average velocity in tube, m/s; D is
inside diameter, m; and (L
2–L
1) or ΔL is length of the straight tube, m.
•One of the uses of this Eq. is in the experimental measurement of the
viscosity of a fluid by measuring the pressure drop and volumetric flow
rate through a tube of known length and diameter.
•The quantity (p
1–p
2)
for Δp
fis the pressure loss due to skin friction
caused by the flowing fluid. Then, for constant ρ, the friction loss F
fis
•??????
�=
??????
1−??????
2�
�
=
?????? ??????
??????�
or
??????
??????�
this is the mechanical energy loss due to skin
friction for the pipe in N · m/kg of fluid.

10. Use of friction factor for friction loss in laminar
flow.
•A common parameter used in laminar and especially in turbulent flow
is the Fanning friction factor, f,
•Definition: The drag force per wetted surface unit area (shear stress
τ
sat the surface) divided by the product of density times velocity
head, or
1
2
�??????
2
. Thus, the drag force is Δp
ftimes the cross-sectional
189 area πR
2
and the wetted surface area is 2πRΔL.
•Hence, the relation between the pressure drop due to friction and f is
as follows for both laminar and turbulent flow.

11. •�=
????????????
????????????
2
2
=
∆??????
??????�??????
2
/2�??????∆??????
????????????
2
2
=
∆??????
?????? ??????
2∆??????
/
�??????
2
2
rearranging this equation and using
D=2R
Δ??????
�=4��
∆??????
??????
??????
2
2
(SI)
Δ??????
�=4��
∆??????
??????
??????
2
2�??????
(English)
??????
�=
Δ??????
�
�
= 4�
∆??????
??????
??????
2
2
(SI)
??????
�=
Δ??????
�
�
= 4�
∆??????
??????
??????
2
2�??????
(English)

12. •For laminar flow only, combining Eqs.
•∆??????
� =??????
1−??????
2� =
32????????????(??????2−??????1)
??????
2
and Δ??????
�=4��
∆??????
??????
??????
2
2
yields the
following expression for the friction factor,
• �=
16
N
Re
•The above equations for laminar flow are valid for Reynolds numbers
up to around 2100.
•Beyond that, at a N
Revalue above 2100, Eqs. do not hold for turbulent
flow.

13. •Problem: Assume the same known conditions as in previous Example
except that the velocity of 0.275 m/s is known and the pressure drop
Δp
fis to be predicted. Use the fanning friction factor method.

14. •When the fluid is a gas and not a liquid, the Hagen–Poiseuilleequation
∆??????
� =??????
1−??????
2� =
32????????????(??????2−??????1)
??????
2
can be written as follows for laminar
flow:
??????=
�??????
4
�(??????
1
2
−??????
2
2
)
128(2????????????)??????(�
2−�
1)
??????=
�??????
4
�
��(??????
1
2
−??????
2
2
)
128(2????????????)??????(�
2−�
1)
where m = kg/s, M = molecular weight in kg/kg mol, T = absolute
temperature in K, and R = 8314.3 N · m/kg mol · K.
In English units, R = 1545.3 ft · lbf /lb mol · °R.

15. Pressure Drop and Friction Factor in Turbulent
Flow
•In turbulent flow, as in laminar flow, the friction factor also depends on the
Reynolds number.
•However, it is not possible to theoretically predict the Fanning friction
factor f for turbulent flow as was shown previously for laminar flow. The
friction factor must be determined empirically (experimentally), and it not
only depends upon the Reynolds number but also on the pipe’s surface
roughness.
•In laminar flow, the roughness has essentially no effect. Dimensional
analysis can also be used to show the dependence of the friction factor on
these factors. A large number of experimental data on friction factors for
both smooth pipes and pipes with varying degrees of equivalent roughness
have been obtained and correlated.

16. •For design purposes, to predict the friction factor f and, hence, the
frictional pressure drop for round pipes, the friction-factor chart can
be used. It is a log–log plot of f versus N
Re. The friction factor f is then
used in Eqs. to predict the Δ??????
�
Δ??????
�=4��
∆??????
??????
??????
2
2
(SI)
Δ??????
�=4��
∆??????
??????
??????
2
2�??????
(English)
??????
�=
Δ??????
�
�
= 4�
∆??????
??????
??????
2
2
(SI)
??????
�=
Δ??????
�
�
= 4�
∆??????
??????
??????
2
2�
??????
(English)

21. Streamlines
Introduction
• When a fluid flows through a pipe
or over a solid object, the velocity
of the fluid varies depending on
position.
•One way of representing variation
in velocity is streamlines, which
follow the flow path.
•Constant velocity is shown by
equidistant spacing of parallel
streamlines as shown in Figure.
Constant fluid velocity
•Constant fluid velocity

22. Streamlines
•The velocity profile for slow-
moving fluid flowing over a
submerged object is shown in
Figure (b);
•reduced spacing between the
streamlines indicates that the
velocity at the top and bottom
of the object is greater than at
the front and back.
Steady flow over a submerged object.

23. Streamlinesshowonlytheneteffectoffluidmotion;althoughstreamlinessuggestsmooth
continuousflow,fluidmoleculesmayactuallybemovinginanerraticfashion.
Theslowertheflowthemorecloselythestreamlinesrepresentactualmotion.Slowfluid
flowisthereforecalledstreamlineorlaminarflow.
Infastmotion,fluidparticlesfrequentlycrossandrecrossthestreamlines.Thismotionis
calledturbulentflowandischaracterisedbyformationofeddies.

24. Laminar and Turbulent Flow
•The type of flow occurring in a fluid moving in a channel is important for
many fluid dynamics problems.
•When fluids flow through a closed channel of any cross section (e.g.,
circular or rectangular), either of two distinct types of flow can be
observed, according to the conditions present.
•These two types of flow can commonly be seen in a flowing open stream
or river. When the velocity of the fluid is relatively slow, the flow patterns
are smooth.
•However, when the velocity is quite high, an unstable pattern is
observed. In this case, eddies, or small packets of fluid particles, are
present, moving in all directions and at all angles to the normal direction
of flow.

25. •The first type of flow, commonly observed at low velocities, where the
layers of fluid seem to slide by one another without eddies or swirls being
present, is called laminar flow. Newton’s law of viscosity holds for fluids in
this regime.
•The second type of flow, commonly observed at higher velocities, where
eddies are present giving the fluid a fluctuating nature, is called turbulent
flow.
•The existence of laminar and turbulent flows is most easily visualized by
the experiments of Reynolds, which are shown in the following figure.
•In his experiments, water was allowed to flow at a constant flowrate
through a transparent pipe.
•A thin, steady stream of dyed water was introduced from a fine jet, as
shown, and its flow pattern observed. At low rates of water flow, the dye
pattern was regular and formed a single line or stream similar to a thread,
as shown in Fig.a.

27. •There was no lateral mixing of the fluid, and it flowed in streamlines
down the tube.
•By putting in additional jets at other points in the pipe cross section,
it was shown that there was no mixing in any parts of the tube and
that the fluid flowed in straight, parallel lines.
•This type of flow is called laminar or viscous flow.
•As the velocity was increased, it was found that at a definite velocity
the thread of dye became dispersed and the pattern was very erratic,
as shown in Fig. b.
•This type of flow is known as turbulent flow. The velocity at which the
flow changes from laminar flow to turbulent flow is known as the
critical velocity.

28. •where N
Reis the Reynolds number,
•D is the diameter in m, ρ is the fluid density in kg/m
3
, μ is the fluid
viscosity in Pa · s, and
•v is the average velocity of the fluid in m/s (where average velocity is
defined as the volumetric rate of flow divided by the cross-sectional
area of the pipe).
•Units in the cgssystem are D in cm, ρ in g/cm3 , μ in g/cm · s, and v
in cm/s.
•In the English system D is in ft, ρ is in lbm/ft3 , μ is in lbm/ft· s, and v
is in ft/s.
•The instability of the flow that leads to disturbed or turbulent flow is
determined by the ratio of the kinetic or inertial forces (i.e., the
numerator) to the viscous forces (i.e., the denominator) in the fluid
stream.
•The inertial forces are proportional to ρv
2
and the viscous forces to
μv/D, and the ratio ρv
2
/(μv/D) is the Reynolds number.

29. For a straight circular pipe, when the value of the Reynolds number is
less than 2100, the flow is always laminar.
When the value is over 4000, the flow will be turbulent, except in very
special cases. In between—called the transition region—the flow can
be viscous or turbulent, depending upon the apparatus details, which
cannot be predicted.

30. •Reynolds Number for Milk Flow. Whole milk at 293 K having a
density of 1030 kg/m3 and viscosity of 2.12 cpis flowing at the rate of
0.605 kg/s in a glass pipe having an inside diameter of 63.5 mm.
•a. Calculate the Reynolds number. Is this turbulent flow?
•b. Calculate the flow rate needed in m3 /s for a Reynolds number of
2100 and velocity in m/s.

31. Assignment 1
Microfluidics. In this chapter, we have described how fluid velocity can greatly affect
whether a fluid is flowing in the laminar or turbulent flow regime. However, other
properties may affect fluid behavior.
Consider the area of microfluidics—a branch of fluid dynamics that deals with fluids
being transported in small chambers, often on the micro-or nano-scale. For these
systems, what flow regime usually dominates—laminar or turbulent flow?Explainyour
reasoning.
Rate of Drawing Blood. You have recently visited your doctor for a physical
examination and are required to have blood work performed. Although blood is a
complex fluid, we will assume it’s a Newtonian fluid with a kinematic viscosity to be 3 ·
10–6 m2 /s. If a 7-gauge syringe (ID = 0.15 in) is used to draw blood, what is the
maximum allowable flowrate possible, such that the fluid flow remains in the laminar
flow regime?

32. Boundary layer Concept
•In most practical applications, fluid flow occurs in the presence of a
stationary solid surface, such as the walls of a pipe or tank.
•That part of the fluid where flow is affected by the solid is called the
boundary layer.
•consider flow of fluid parallel to the flat plate shown in Figure .
•Contact between the moving fluid and the plate causes formation of a
boundary line. Above the boundary layer, fluid motion is the same as
if the plate were not there. The boundary layer grows in thickness
from the leading edge until it develops its full size. Final thickness of
the boundary layer depends on the Reynolds number for bulk flow.

35. •When fluid flows over a stationary object, a thin film of fluid in contact with the
surface adheres to it to prevent slip-page over the surface.
•Fluid velocity at the surface of the plate in Figure 7.3 is therefore zero.
•When part of a flowing fluid has been brought to rest, the flow of adjacent fluid
layers is slowed down by the action of viscous drag. This phenomenon is illus-
tratedin Figure 7.3(b).
•Velocity of fluid within the boundary layer, u, is represented by arrows; u is zero
at the surface of the layer beginning at the leading edge and developing on both
plate.
•Viscous drag forces are transmitted upwards through the top and bottom of the
plate. Figure 7.3 shows only the upper stream; fluid motion below the plate will
be a mirror image of that above. As indicated by the arrows in Figure 7.3(a), the
bulk fluid velocity in front of the plate is uniform and of magnitude u
B.
•The extent of the boundary layer is indicated by the brokenContact between the
moving fluid and the plate causes formation of a boundary line. Above the
boundary layer, fluid motion is the same as if the plate were not there.
•The boundary layer grows in thick-ness from the leading edge until it develops its
full size. Final thickness of the boundary layer depends on the Reynolds number
for bulk flow.

36. •When fluid flows over a stationary object, a thin film of fluid in contact
with the surface adheres to it to prevent slip-page over the surface.
•Fluid velocity at the surface of the plate in Figure 7.3 is therefore zero.
When part of a flowing fluid has been brought to rest, the flow of adjacent
fluid layers is slowed down by the action of viscous drag.
•This phenomenon is illustrated in Figure 7.3(b). Velocity of fluid within the
boundary layer, u, is represented by arrows; u is zero at the surface of the
plate.
•Viscous drag forces are transmitted upwards through the fluid from the
stationary layer at the surface. The fluid layer just above the surface moves
at a slow but finite velocity; layers further above move at increasing
velocity as the drag forces associated with the stationary layer decrease.
•At the edge of the boundary layer, fluid is unaffected by the presence of
the plate and the velocity is close to that of the bulk flow, u
B.
•The magnitude of u at various points in the boundary layer is indicated in
Figure 7.3(b) by the length of the arrows in the direction of flow. The line
connecting the heads of the velocity arrows shows the velocity profile in
the fluid

37. •A velocity gradient, i.e. a change in velocity with distance from the
plate, is thus established in a direction perpendicular to the direction
of flow.
•The velocity gradient forms as the drag force resulting from
retardation of fluid at the surface is transmitted through the fluid.
•Formation of boundary layers is important not only in determining
characteristics of fluid flow, but also for transfer of heat and mass
between phases.

38. Boundary-Layer Separation

39. •What happens when contact is broken between a fluid and a solid
immersed in the flow path?
•As an example, consider a flat plate aligned perpendicular to the
direction of fluid flow, as shown in Figure 7.4.
•Fluid impinges on the surface of the plate, and forms a boundary
layer as it flows either up or down the object.
•When fluid reaches the top or bottom of the plate its momentum
prevents it from making the sharp turn around the edge. As a result,
fluid separates from the plate and proceeds outwards into the bulk
fluid. Directly behind the plate is a zone of highly decelerating fluid in
which large eddies or vortices are formed. This zone is called the
wake. Eddies in the wake are kept in rotational motion by the force of
bordering currents.

40. •Boundary-layer separation such as that shown in Figure 7.4 occurs
whenever an abrupt change in either magnitude or direction of fluid
velocity is too great for the fluid to keep to a solid surface.
•It occurs in sudden contractions, expansions or bends in the flow
channel, or when an object is placed across the flow path.
Considerable energy is associated with the wake; this energy is taken
from the bulk flow.
•Formation of wakes should be minimized if large pressure losses in
the fluid are to be avoided; however, for some purposes such as
promotion of mixing and heat transfer, boundary-layer separation
may be desirable.