Loading...

PPT – Steady Incompressible Flow in Pressure Conduits PowerPoint presentation | free to download - id: 83f254-NjQyZ

The Adobe Flash plugin is needed to view this content

Steady Incompressible Flow in Pressure Conduits

Laminar and Turbulent Flow in Pipes

- Laminar
- Paths of Particles dont obstruct each other
- Viscous forces are dominant
- Velocity of fluid particles only changes in

magnitude - Lateral component of velocity is zero
- Turbulent
- Paths do intersect each other
- Inertial forces are dominant
- Velocity of fluid particles change in magnitude

and direction - Lateral components do exist.

Laminar and Turbulent Flow in Pipes

- If we measure the head loss in a given length of

uniform pipe at different velocities , we will

find that, as long as the velocity is low enough

to secure laminar flow, the head loss, due to

friction, is directly proportional to the

velocity, as shown in Fig. 8.1. But with

increasing velocity, at some point B, where

visual observation of dye injected in a

transparent tube would show that the flow changes

from laminar to turbulent, there will be an

abrupt increase in the rate at which the head

loss varies. If we plot the logarithms of these

two variables on linear scale or in other words,

if we plot the values directly on log-log paper,

we will find that, after passing a certain

transition region (BCA in Fig. 8.1), the lines

will have slopes ranging from about 1.75 to 2.

Laminar and Turbulent Flow in Pipes

- Thus we see that for laminar flow the drop in

energy due to friction varies as V, while for

turbulent flow the friction varies as Vn, where n

ranges from about 1.75 to 2. The lower value of

1.75 for turbulent flow occurs for pipes with

very smooth walls as the wall roughness

increases, the value of n increases up to its

maximum value of 2. - If we gradually reduce the velocity from a high

value, the points will not return along line BC.

Instead, the points will lie along curve CA. We

call point B the higher critical point, and A the

lower critical point. - However, velocity is not the only factor that

determines whether the flow is laminar or

turbulent. The criterion is Reynolds number.

Laminar and Turbulent Flow in Pipes

Reynolds Number

- Ratio of inertia forces to viscous forces is

called Reynolds number. - Where we can use any consistent system of

units, because R is a dimensionless number.

Significance of Reynolds Number

- To investigate the development of Laminar and

Turbulent flow - Investigate Critical Reynolds Number
- Develop a relationship between head loss (hL) and

velocity.

Critical Reynolds Number

- The upper critical Reynolds number, corresponding

to point B of Fig. 8.1, is really indeterminate

and depends on the care taken to prevent any

initial disturbance from effecting the flow. Its

value is normally about 4000, but experimenters

have maintained laminar flow in circular pipes up

to values of R as high as 50,000. However, in

such cases this type of flow is inherently

unstable, and the least disturbance will

transform it instantly into turbulent flow. On

the other hand, it is practically impossible for

turbulent flow in a straight pipe to persist at

values of R much below 2000, because any

turbulence that occurs is damped out by viscous

friction. This lower value is thus much more

definite than the higher one, and is the real

dividing point the two types of flow. So we

define this lower value as the true critical

Reynolds number.

Critical Reynolds Number

- It will be higher in a converging pipe and lower

in a diverging pipe than in a straight pipe.

Also, it will be less for flow in a curved pipe

than in a straight one, and even for a straight

uniform pipe it may be as low as 1000, where

there is excessive roughness. However, for normal

cases of flow in straight pipes of uniform

diameter and usual roughness, we can take the

critical value as - Rcrit

2000

Problem

- Q In a refinery oil (? 1.8 x 10-5 m2/s) flows

through a 100-mm diameter pipe at 0.50 L/s. Is

the flow laminar or turbulent? - Solution
- Q 0.50 Liter/s 0.0005 m3/s
- D 100 mm 0.1 m
- Q AV (pD2/4)V, V (4Q)/(pD2)
- V (4 x 0.0005)/(3.14 x 0.1 x0.1) 0.0637

m/s - R (DV)/? (0.1 x 0.0637)/(1.8 x 10-5)

354 - Since R lt Rcrit2000, the flow is

laminar.

Hydraulic Radius

- For conduits having non-circular cross sections,

we need to use some value other than the diameter

for the linear dimension in the Reynolds number.

The characteristic dimension we use is the

hydraulic radius, defined as - Rh A/P
- Where A is the cross sectional area of the

flowing fluid, and P is the wetted perimeter,

that portion of the perimeter of the cross

section where the fluid contacts the solid

boundary, and therefore where friction resistance

is exerted on the flowing fluid. For a circular

pipe flowing full, - Full-pipe flow Rh (p r2)/(2pr)

r/2 D/4

Friction Head Loss in Conduits

- This discussion applies to either laminar or

turbulent flow and to any shape of cross section. - Consider steady flow in a conduit of uniform

cross section A, not necessarily circular as

shown Fig. below. The pressures at sections 1 and

2 are p1 and p2, respectively. The distance

between the sections is L.

Friction Head Loss in Conduits

- For equilibrium in steady flow, the summation of

forces acting on any fluid element must be equal

to zero (i.e., SFma0). Thus, in the direction

of flow, - p1A p2A ?LAsina t0(PL) 0 (1)
- where we define t0, the average shear stress

(average shear force per unit area) at the

conduit wall. - Nothing that sina (z2 z1)/L and dividing

each term in eq. (1) by ?A gives, - p1A/(?A) p2A/(?A) ?LA(z2 - z1)/(?AL)

t0(PL)/(?A) - p1/? p2/? z2 z1 t0(PL)/(?A) .

(2) - From the left hand sketch of the Fig., we can

see that the head loss due to friction at the

wetted perimeter is - hf (z1 p1/?) (z2 p2/?) .. (3)

Friction Head Loss in Conduits

- The eq.(3) equation indicates that hf depends

only on the values of z and p on the centerline,

and so it is the same regardless of the size of

the cross-sectional area A. Substituting hf from

eq.(3) and replacing A/P by Rh in eq.(2), we get, - hf t0L/(Rh?)

(4) - This equation is applicable to any shape of

uniform cross section, regardless of whether the

flow is laminar or turbulent.

Friction Head Loss in Conduits

- For a smooth-walled conduit, where we can

neglect wall roughness, we might assume that the

average fluid shear stress t0 at the wall is

some function of ?, µ, V and some characteristic

linear dimension, which we will here take as the

hydraulic radius Rh. Thus - t0 f(?, µ, V, Rh)
- Using the pi theorem of dimensional analysis

to better determine the form of this

relationship, we choose ?, Rh and V as primary

variables, so that - ?1 µ ?a1 Rhb1 Vc1
- ?2 t0 ?a2 Rhb2 Vc2
- With the dimensions of the variables being

ML-1T-1 for µ, ML-1T-2 for t0, ML-3 for ?,

L for Rh, and LT-1 for V,

Friction Head Loss in Conduits

- the dimensions for ?1 are
- ?1 µ ?a1 Rhb1 Vc1
- M0L0T0 (ML-1T-1) (ML-3)a1 (L)b1 (LT-1)c1
- For M 0 1 a1
- For L 0 -1 3a1 b1 c1
- For T 0 -1 c1
- The solution of these simultaneous equations

is - a1 b1 c1 -1, from which
- ?1 µ ?-1 Rh-1 V-1
- ?1 µ /(? Rh V) R-1
- where (? Rh V)/µ is a Reynolds number with Rh

as the characteristic length.

Friction Head Loss in Conduits

- the dimensions for ?2 are
- ?2 t0 ?a2 Rhb2 Vc2
- M0L0T0 (ML-1T-2) (ML-3)a2 (L)b2 (LT-1)c2
- For M 0 1 a2
- For L 0 -1 3a2 b2 c2
- For T 0 -2 c2
- The solution of these simultaneous equations

is - a2 -1, c2 -2, b2 0, from which
- ?2 t0 ?-1 V-2
- ?2 t0 /(?V2)
- We can write ?2 ?( ?1-1), which results in
- t0 ? V2 ? (R)
- .

Friction Head Loss in Conduits

- Setting the dimensionless term ? (R) ½ Cf ,

this yields - t0 Cf ? V2/2
- Where Cf average friction-drag coefficient

for total surface (dimensionless) - Inserting this value of t0 and ? ?g, in

eq. (4), which is - hf t0L/(Rh?) , we get
- hf Cf (L/Rh)(V2/2g) (5)
- which can apply to any shape of smooth-walled

cross section. From this equation , we may easily

obtain an expression for the slope of the energy

line, - S hf / L Cf /Rh (V2/2g) (6)
- which we also know as the energy gradient.

Friction in Circular Conduits

- Head loss due to friction, hf Cf

(L/Rh)(V2/2g) - Energy gradient, S hf / L Cf /Rh (V2/2g)
- For a circular pipe flowing full, Rh D/4,

and - f 4Cf , where f is friction factor (also

some times called the Darcy friction factor) is

dimensionless and some function of Reynolds

number. - Substituting values of Rh and Cf into above

equations, we obtain (for both smooth-walled and

rough-walled conduits) the well known equation

for pipe-friction head loss, - Circular pipe flowing full (laminar or

turbulent flow) - hf f (L/D) (V2/2g) . (7)
- and hf /L S f /D (V2/2g) .. (8)

Friction in Circular Conduits

- For a circular pipe flowing full, by

substituting Rh r0/2, where r0 is the radius of

the pipe in the eq. (4), we get - hf t0L/(Rh?) 2t0L/(r0?)
- where the local shear stress at the wall, t0,

is equal to the average shear stress t0 because

of symmetry.

Friction in Circular Conduits

- The shear stress is zero at the center of the

pipe and increases linearly with the radius to a

maximum value t0 at the wall as shown in Fig.

8.3. This is true regardless of whether the flow

is laminar or turbulent. - From eq.(4) hf t0L/(Rh?), we have
- t0 hf (Rh?)/L, substituting eq.(7) and Rh

D/4 into this, we obtain - t0 f (L/D)(V2/2g)(D/4)(?/L)
- t0 (f /4) ? (V2/2g) or t0 (f /4) ?

(V2/2) where ??g - With this equation, we can compute t0 for

flow in a circular pipe for any experimentally

determined value of f.

Friction in Circular Conduits

- For laminar flow under pressure in a circular

pipe, - We may use the pipe-friction equation (7) with

this value of f as given by the above equation.

Problem

- Q Stream with a specific weight of 0.32 lb/ft3

is flowing with a velocity of 94 ft/s through a

circular pipe with f 0.0171. What is the shear

stress at the pipe wall? - Solution
- ? 0.32 lb/ft3
- V 94 ft/s
- f 0.0171
- g 32.2 ft/s2
- t0 ?
- t0 (f /4) ? (V2/2g)
- t0 (0.0171/4)(0.32)(94x94)/(2x32.2)
- t0 0.187 lb/ft2

Problem

- Q Stream with a specific weight of 38 N/m3 is

flowing with a velocity of 35 m/s through a

circular pipe with f 0.0154. What is the shear

stress at the pipe wall? - Solution
- ? 38 N/m3
- V 35 m/s
- f 0.0154
- g 9.81 m/s2
- t0 ?
- t0 (f /4) ? (V2/2g)
- t0 (0.0154/4)(38)(35x35)/(2x9.81)
- t0 9.13 N/m2

Problem

- Q Oil of viscosity 0.00038 m2/s flows in a 100mm

diameter pipe at a rate of 0.64 L/s. Find the

head loss per unit length. - Solution
- ? 0.00038 m2/s
- D 100 mm 0.1 m
- Q 0.64 L/s 0.00064 m3/s
- g 9.81 m/s2
- hf /L ?
- Q AV (pD2/4)V, V (4Q)/(pD2)
- V (4 x 0.00064)/(3.14 x 0.1 x 0.1)

0.0815 m/s - R (DV)/? (0.1 x 0.0815)/(0.00038)

21.45

Problem

- f 64 / R 64/21.45 2.983
- hf /L S f /D (V2/2g)
- hf /L (2.983/0.1)(0.0815x0.0815)/(2x9.81)
- hf /L 0.010 m/m

Friction in Non-Circular Conduits

- Most closed conduits we use in engineering

practice are of circular cross section however

we do occasionally use rectangular ducts and

cross sections of other geometry. We can modify

many of the equations for application to non

circular sections by using the concept of

hydraulic radius. - For a circular pipe flowing full, that
- Rh A/P (p D2/4)/(pD) D/4
- D 4 Rh
- This provides us with an equivalent diameter,

which we can substitute into eq. (7) to yield - hf f (L/4Rh)(V2/2g)

Friction in Non-Circular Conduits

- and when substitute into equation of Reynolds

number, we get - R (DV?)/µ (4RhV?)/µ (4RhV)/?
- This approach gives reasonably accurate

results for turbulent flow, but the results are

poor for laminar flow, because in such flow

viscous action causes frictional phenomena

throughout the body of the fluid, while in

turbulent flow the frictional effect occurs

largely in the region close to the wall i.e., it

depends on the wetted perimeter.

Entrance Conditions in Laminar Flow

- In the case of a pipe leading from a

reservoir, if the entrance is rounded so as to

avoid any initial disturbance of the entering

stream, all particles will start to flow with the

same velocity, except for a very thin film in

contact with the wall. Particles in contact with

the wall have zero velocity and with the slight

exception, the velocity is uniform across the

diameter.

Entrance Conditions in Laminar Flow

- As the fluid progresses along the pipe,

friction origination from the wall slows down the

streamlines in the vicinity of the wall, but

since Q is constant for successive sections, the

velocity in the center must accelerate, until the

final velocity profile is a parabola as shown in

Fig. 8.3. Theoretically, this requires an

infinite distance, but both theory and

observation have established that the maximum

velocity in the center of the pipe wall reach 99

of its ultimate value in a distance - Le 0.058 RD
- We call this distance the entrance length. For

a critical value of R 2000, the entrance length

Le equals 116 pipe diameters. In other cases of

laminar flow with Reynolds number less than 2000,

the distance Le will be correspondingly less in

accordance with the above equation.

Entrance Conditions in Laminar Flow

- Within the entrance length the flow is

unestablished that is the velocity profile is

changing. In this region, we can visualize the

flow as consisting of a central inviscid core in

which there are no frictional effects, i.e., the

flow is uniform, and an outer, annular zone

extending from the core to the pipe wall. This

outer zone increases in thickness as it moves

along the wall, and is known as the boundary

layer. Viscosity in the boundary layer acts to

transmit the effect of boundary shear inwardly

into the flow. At section AB the boundary layer

has grown until it occupies the entire cross

section of the pipe. At this point, for laminar

flow, the velocity profile is a perfect parabola.

Beyond section AB, for the same straight pipe the

velocity profile does not change, and the flow is

known as (laminar) established flow or (laminar)

fully developed flow.

Entrance Conditions in Laminar Flow

- The flow will continue as fully developed so

long as no change occurs to the straight pipe

surface. When a change occurs, such as at a bend

or other pipe fitting, the velocity profile will

deform and will require some more flow length to

return to established flow. Usually such fittings

are so far apart that fully developed flow is

common but when they are close enough it is

possible that established flow never occurs.