VISCOUS FLOW IN CONDUITS When we consider viscosity in conduit flows, we must be able to quantify the losses in the flow 87-351 Fluid Mechanics [ physical

VISCOUS FLOW IN CONDUITS

When we consider viscosity in conduit flows, we must be able to quantify the losses in the flow

87-351 Fluid Mechanics

[ physical interpretation: what are we doing today? ]

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

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

Who Cares!?

The complexity of turbulent flows typically necessitate the use of extensive experimental data and empirical formulae


The pressure drop and head loss in a pipe are dependent on the wall shear stress, tw, between the fluid and pipe surface

[ introduction to the moody diagram ]

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

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

- (1)

Though pipe roughness, e, is not a factor for laminar flow, it is included for the accommodation of turbulent flow



The figure represents the flow in the viscous sub-layer for rough and smooth walls




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


- (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

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.,


VISCOUS FLOW IN CONDUITS[ introduction to the moody diagram ]

- (4)

Where f is now a function of two dimensionless terms, the Reynolds number, Re, and the relative roughness, e/D

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

- (3)

- (5)



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)



Here is the friction factor, f as a function of Reynolds number and relative roughness (e/D) for round pipes



Following are some typical values for pipe wall roughnesses



Now, we recall our energy equation for steady incompressible flow

- (6)

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

- (4)

- (7)

(7) is referred to as the Darcy Weisbach equation, valid for any fully developed, steady, incompressible pipe flow (horizontal or otherwise)



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


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

- (7)

The Colebrook expression is valid for the entire non-laminar range of the Moody diagram

- (8)


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)


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

then we write

- (10)

- (11)


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


VISCOUS FLOW IN CONDUITS[ minor losses: entrance flow conditions ]


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



The sudden expansion loss mechanism can actually be evaluated analytically



Here

- (12)

- (13)

- (14)

and, we know - (15)

which can be rearranged as

- (16)

of course in this development87-351 Fluid Mechanics

VISCOUS FLOW IN CONDUITS[ example 1: determination of pressure drop ]

GIVEN:

REQD: Determine the pressure at (1) if a: no losses considered, b: just major losses considered, c: all losses considered

Water flows at 60oF from the basement to the second floor under the following conditions



SOLU:

1. Let us write the energy equation for this flow

- (E1)

from which we can rearrange for p1 as

- (E2)

a: [NO LOSSES CONSIDERED]



SOLU:

2. With no losses hL goes to 0, so

- (E3)

or

- **(ans a:)

**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

a: [NO LOSSES CONSIDERED]



SOLU:

- (E1)

and we can compute hL from (11) (D-W)

- (E4)

b: [ONLY MAJOR LOSSES CONSIDERED]

1. We of course still apply (E1)



SOLU:

- (E5)

and we can compute hL from (11) (D-W)

- (E4)

b: [ONLY MAJOR LOSSES CONSIDERED] 2. From given data we assemble

going to the Moody, we write

3. Applying (E1), we simplify to



SOLU:

- (E5)

which becomes

b: [ONLY MAJOR LOSSES CONSIDERED]

- **(ans b:)

**NB- (ans b:)

of this pressure drop, 10.6 psi is due to pipe friction



SOLU:

- (E6)

which becomes

c: [MAJOR and MINOR LOSSES CONSIDERED]

1. Applying (E1), we simplify to



SOLU:

- (E7)


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:



SOLU:

- (E8)


3. Now we pick up the loss coefficients for all the minor losses in the system



SOLU: c: [MAJOR and MINOR LOSSES CONSIDERED]

4. Thus, summing from a: and b:

- **(ans c:)




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


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VISCOUS FLOW IN CONDUITS When we consider viscosity in conduit flows, we must be able to quantify the losses in the flow 87-351 Fluid Mechanics [ physical