VISCOUS FLOW IN CONDUITS
[ physical interpretation: what are we doing today? ]



When we consider viscosity in conduit flows, we must be able to
quantify the losses in the flow
The magnitude of these losses will vary significantly depending on many
factors, including whether the flow is laminar or turbulent
For most practical purposes, the Reynolds number is such that conduit
flows that serve us in everyday life are turbulent

The complexity of turbulent flows typically necessitate the use of
extensive experimental data and empirical formulae

Who Cares!?
 our knowledge of how to quantify losses in conduit flows allows us to optimize
performance and efficiency in contained flows from water and oil pipelines, to
chemical networks, air supplies, and the conduit network in the human body
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]



The pressure drop and head loss in a pipe are dependent on the wall
shear stress, tw, between the fluid and pipe surface
In turbulent flow the shear stress is a function of fluid density, this is not the
case in laminar flow, where it is only dependent on the viscosity, m
We can consider the pressure drop, p, in a steady, incompressible
turbulent flow, in a horizontal pipe of diameter, D to be expressed as
- (1)
 here pressure drop is a function of fluid velocity, V, pipe diameter, D, pipe
length, l, pipe surface roughness height, e, fluid viscosity m, and fluid density, r

Though pipe roughness, e, is not a factor for laminar flow, it is included
for the accommodation of turbulent flow
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]

The figure represents the flow in the viscous sub-layer for rough and
smooth walls
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]


The factors laid out in (1), are in fact a complete list of influencing
parameters on pressure drop, that is to say that other factors such as
surface tension, and vapor pressure, etc. do not affect the pressure for
the conditions we have assumed, i.e., steady, incompressible,
horizontal, and round pipe
Recalling our dimensional analysis, the number of variables in this
problem, k=7, and the number of basic dimensions, m=3, we therefore
expect to see 4 dimensionless groups i.e.,
- (2)

We discover that this function representation can be simplified by
assuming that pressure drop is proportional to pipe length, this we
arrive at through our knowledge of many experiments (not by
dimensional analysis), so we factor l/D out
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]
- (3)

We can rearrange (3) to construct a term pD/(lrV2/2) that we will refer
to as the friction factor, f, i.e., we write (3) now as
- (4)

Where f is now a function of two dimensionless terms, the Reynolds
number, Re, and the relative roughness, e/D
- (5)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]

For laminar, fully developed flow the value of f is only dependent on the
Reynolds number, i.e., f=64/Re (no dependence on e/D)

For turbulent flow, there is not as yet an analytical solution for the
friction factor, f; rather, results for f are summarized from experiments
on the Moody Diagram (or an equivalent curve fitting formula)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]

Here is the friction factor, f as a function of Reynolds number and relative
roughness (e/D) for round pipes
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]

Following are some typical values for pipe wall roughnesses
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]

Now, we recall our energy equation for steady incompressible flow
- (6)
- (4)

When we talk about flow with a constant diameter, D, (i.e., constant
velocity, V1=V2, and horizontal, z1=z2, then p=p1-p2=ghL), we can re-write
(6), (combined with (4)) to get
- (7)

(7) is referred to as the Darcy Weisbach equation, valid for any fully
developed, steady, incompressible pipe flow (horizontal or otherwise)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ introduction to the moody diagram ]
- (7)

In (7), f is had from the Moody diagram, or may be more conveniently
calculated through an expression that is valid for some portion of the
diagram

The Colebrook expression is valid for the entire non-laminar range of the
Moody diagram
- (8)

Of course, the implicit dependence on f requires an iterative solution (not
an issue with programmable calculators or computers)

We must remember to exercise caution when utilizing either Colebrook’s
expression or the Moody diagram, as results can only be taken with
assurances of a 10% accuracy
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses
]

Expressions like Darcy Weisbach’s (7), are useful in computing headloss
over long sections of straight pipe

Typical pipe networks however contain many bends, tees, joints, and
valves

The flow will experience losses though
such sections (mostly due to losses
associated with changes in flow geometry
and direction—momentum losses)

Considering the overall head loss in the
system, the losses associated with these
sections are usually minor, thus their being
termed “minor” losses (relative to the
more significant friction losses incurred
over long stretches of straight pipe)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses
]

Theoretical evaluations of the losses through each valve and fitting in a
system are as yet not plausible, therefore the head loss data for such
components has been determined by experiment

The most common method for determining
the head losses or pressure drops across
these elements is to specify a loss
coefficient, KL, defined as
- (9)

where
- (10)

then we
write
- (11)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses: entrance flow conditions
]

What typically happens (the essence of vena contracta):

The flow separates from the corner (basically it can’t make the turn)

The max velocity at (2) will be greater than (3) and as a result the
pressure drops, if the flow could put the brakes on and convert that saved
velocity to pressure, you would have an ideal pressure distribution—and
no losses, this is not what happens
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses: entrance flow conditions
]
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses: exit flow conditions
]

Here the entire kinetic energy of the exiting fluid, V1, is dissipated through
viscous effects as the incoming stream mixes with ambient water and
eventually comes to rest (V2=0), thus exit losses from (1) to (2) are
typically equal to one velocity head (V2/2g), or KL=1
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses: exit flow conditions
]

The sudden expansion loss mechanism can actually be evaluated analytically
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ minor losses: exit flow conditions
]

Here
- (12)
- (13)

and, we know

which can be rearranged as
- (15)
- (16)
- (14)

of course in this development
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
GIVEN: Water flows at 60oF from the basement to the second floor under
the following conditions
REQD:
Determine the pressure at (1) if a: no losses considered, b: just
major losses considered, c: all losses considered
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
a: [NO LOSSES
CONSIDERED]
1. Let us write the energy equation for this flow
- (E1)
 from which we can rearrange for p1 as
- (E2)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
a: [NO LOSSES
CONSIDERED]
**NB- (ans a:)
we note here how 8.67 psi
of the pressure drop is due
to change in elevation and
2.07 psi is due to increase
in kinetic energy
2. With no losses hL goes to 0, so
- (E3)
 or
- **(ans a:)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
b: [ONLY MAJOR
LOSSES CONSIDERED]
1. We of course still apply (E1)
- (E1)
 and we can compute hL from (11) (D-W)
- (E4)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
b: [ONLY MAJOR
LOSSES CONSIDERED]
2. From given data we assemble
 going to the Moody, we write
3. Applying (E1), we simplify to
- (E5)
 and we can compute hL from (11) (D-W)
- (E4)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
b: [ONLY MAJOR
LOSSES CONSIDERED]
**NB- (ans b:)
of this pressure drop, 10.6 psi
is due to pipe friction
- (E5)
 which becomes
- **(ans b:)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
c: [MAJOR and MINOR
LOSSES CONSIDERED]
1. Applying (E1), we simplify to
- (E6)
 which becomes
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
c: [MAJOR and MINOR
LOSSES CONSIDERED]
2. Here, the 21.3 psi is due to elevation change, kinetic energy change,
and the major losses we have accounted for from a: through b:
- (E7)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
c: [MAJOR and MINOR
LOSSES CONSIDERED]
3. Now we pick up the loss coefficients for all the minor losses in the
system
- (E8)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
SOLU:
c: [MAJOR and MINOR
LOSSES CONSIDERED]
4. Thus, summing from a: and b:
- **(ans c:)
87-351 Fluid
VISCOUS FLOW IN CONDUITS
[ example 1: determination of pressure drop ]
c: [MAJOR and MINOR
LOSSES CONSIDERED]
Let us examine the behaviour of the pressure through the system, note
that not all losses are irreversible (like friction and momentum loss),
losses due to elevation and velocity changes are reversible
87-351 Fluid