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This chapter deals with engineering relationships describing flow in a variety of geometries.

The treatment restricted to the so-called purely viscous or time-independent type of fluids for which the viscosity model describing the flow curve is already known.

However, subsequently a generalized treatment for the laminar flow of time-independent fluids in circular tubes is presented.

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Notwithstanding the existence of time-dependent and visco-elastic fluid behaviour, experience has shown that the shear rate dependence of the viscosity is the most p ysignificant factor in most engineering applications which invariably operate under steady state conditions.

Vi l ti b h i d t i ifi tlVisco-elastic behaviour does not significantly influence laminar flow through circular tubes.

Visco-elastic effects begin to manifest themselves for flow in non-circular conduits in the form of secondary flows and/or in pipe fittings due to sudden changes in the cross-fittings due to sudden changes in the crosssectional area available for flow thereby leading to acceleration/deceleration of a fluid element.

Even in these circumstances, it is often possible to develop predictive expressionspossible to develop predictive expressions purely in terms of steady-shear viscous properties.

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In the chemical and process industries, it is often required to pump fluids over long distances from storage to various processing units and/or from one plant site to another.

There may be a substantial frictional pressure loss in both the pipe line and in the individual unitsin both the pipe line and in the individual units themselves.

It is thus often necessary to consider the problems of: Calculating the power requirements for

pumping through a given pipe networkp p g g g p p The selection of optimum pipe diameterMeasurement and control of flow rate etc

Aside from circular pipes, one also encounters conduits f th ti d b d ith i lof other cross-sections and may be concerned with axial

flow in an annulus (as in a double-pipe heat exchanger), rectangular, triangular and elliptic conduits as employed in nuclear reactors and for extrusion through dies.

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The velocity profile established in a given flow situation strongly influences the heat and mass transfer processes.

Aside from circular pipes, one also encounters conduits of other cross-sections and may be concerned with axial flow in an annulus (as in a double-pipe heat exchanger), rectangular, triangularp p g ) g gand elliptic conduits as employed in nuclear reactors and for extrusion through dies.

L fl l Laminar flow in circular tubes

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Consider the laminar, steady, incompressiblefully developed flow of a time-independent fluid in a circular tube of radius, R , as shown in Figure 1.

Figure 1: Flow through a horizontal pipe

There is no angular velocityThe force balance in the z-direction on a fluid element situated at distance, r:

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The familiar linear shear stress distribution across the pipe cross-section

The shear stress being zero at the axis of the gtube, as shown in Figure below.

Equation (2) is applicable to both laminar and turbulent flow of any fluid

Since it is based on a simple force balance and no assumption has been made so far concerning the type of flow or fluid behaviour.

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For a power-law fluid:

(3)

where Vz is the velocity in the axial direction at radius r.

Combining equations (2) and (3) and the separation of variables leads to:of variables leads to:

(4)

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Integration with respect to r gives the expression for the velocity distribution:

(5)

At r =R: Vz =0

(6)

The velocity distribution in the pipe

(7)

The velocity profile may be expressed in terms of the average velocity, V, which is given by:

h Q i h l i fl f h li id

(8)

where Q is the volumetric flow rate of the liquid. On substitution for Vz from equation (7), and integration yields,

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Therefore:

(9)

Now the velocity distribution in the pipe may be rewritten in terms of the average velocity as:

(10)

The profile is flatter for a shear-thinning fluid (n <1)and sharper for a shear-thickening (n>1) fluid.

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The velocity is seen to be a maximum when r = 0, i.e. at the pipe axis. Thus:

(11)

The value of the centre line velocity drops from 2 33VThe value of the centre-line velocity drops from 2.33V to 1.18V as the value of the power-law index n decreases from 2 to 0.1. This ratio is 2 for a Newtonian fluid.

Rewriting eq. (9) in terms of the volumetric flow rate and the pressure gradient:

(12)

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For a given power-law fluid and fixed pipe radius, ΔpQn , i.e. for a shear-thinning fluid (n<1), the pressure gradient is less sensitive than for a Newtonian fluid to changes in flow rate. u d to c a ges ow ate.

The flow rate, on the other hand, shows a rather stronger dependence on the radius of pipe. For instance, for n=1, QR4 whereas for n= 0.5, QR5.

It is useful to rewrite equation (12) in dimensionless form by introducing a friction factor defined as:

w2

f1/ 2 V

where w

p D

L 4

and defining a suitable Reynolds number which will yield the same relationship as that for Newtonianyield the same relationship as that for Newtonian fluids, that is

16f

Re

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The wall shear stress in terms of the friction factor

2w

1V f (13)

2

The wall shear stress in terms of the pressure gradient

2p 4f V(14)

L D

Eq. 9

2n D 2f V DV (15)

3n 1 4m D 2

Which can be rearranged to give:

PL

16f (16)

Re

2 n nV DRe (17)

where the new Reynolds number RePL is defined by:

nn 1

Re (17)3n 1

8 m4n

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Example 3.1A polymer solution (density =1075 kg/m3) is being pumped at a rate of 2500 kg/h through a 25 mm inside diameter pipe. The flow is known to be laminar and the power-law constants for the solution are m=3 Pa sn andpower law constants for the solution are m 3 Pa s and n=0.5. Estimate the pressure drop over a 10 m length of straight pipe and the centre-line velocity for these conditions. How does the value of pressure drop change if a pipeHow does the value of pressure drop change if a pipe of 37 mm diameter is used?

Volumetric flow rate

SOLUTION

Substituting in eq. (12) and solving for pressure drop

Pipe radius

Substituting in eq. (12) and solving for pressure drop gives:

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Average velocity in pipe

SOLUTION

The centre-line velocity is obtained by using eq. (11)

For the same value of the flow rate, eq. (12) suggests that

Hence, the pressure drop for the new pipe diameter (37mm) can be estimated:

SOLUTION

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Bingham plastic and yield-pseudoplastic

fluidsfluids

A fluid with a yield stress will flow only if the applied stress (proportional to pressure gradient) exceeds the yield stress.

There will be a solid plug-like core flowing in the middle of the pipe where |τrz| is less than the yield stress, as shown in Figure below:

Schematic velocity distribution for laminar flow of a Bingham plastic fluid in a pipe

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The radius of the plug, Rp, will depend upon the

1. magnitude of the yield stress 2. the wall shear stress (i.e., the value of Δp/L)

From equation (2)

Bp0

w

R(18)

R

w

where τw is the shear stress at the wall of the pipe.

In the annular area Rp< r < R, the velocity will gradually decrease from the constant plug velocity to zero at the pipe wall.

The expression for this velocity distribution will nowThe expression for this velocity distribution will now be derived.

For the region Rp< r < R, the value of shear stress will be greater than the yield stress of the fluid.

The Bingham fluid model for pipe flow is given by:

B zrz 0 B

dV(19)

dr

The Bingham fluid model for pipe flow is given by:

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B zrz 0 B

dV

dr

rz

p r

L 2

Eq. (3):L 2

Bz 0

B

1 p rdV dr

L 2

Integration with respect to r

2B

z 0B

1 p rV r C (20)

L 4

2B

z 0B

1 p rV r C

L 4

zAt r R : V 0

2B0

B

1 p RC R (21)

L 4

B2 20

z 2B B

p R r rV 1 R 1 (22)

L 4 R R

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C learly, equation (22) is applicable only when |τrz|>|τ0

B| and r Rp. The corresponding velocity, Vzp , in the plug region (0r Rp) is obtained by substituting r = Rp in eq. (22) to give:to give:

The volumetric flow rate:

22p

zp p2B

Rp RV 1 (0 r R ) (23)

L 4 R

The volumetric flow rate:

(24)

Substituting from equations (22) and (23):

( )(25)

(26)

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Bp0

w

R

R

Eq. 26

(27)Further

simplification

(28)

It is useful to rewrite equation (28) in a dimensionless form by introducing the following definitions:

BB

VDRe (29)

Bingham Reynolds number

B

Friction factor w

2 2

D p Lf (30)

1 2 V 2 V

H d b2 B

0DH (31)

Hedstrom number 0

2B

He (31)

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2

QV

R

B B0 0

21 2 f V

Equation 28

Di idi b th id b V2 d li ht t

w 1 2 f V

(32)

Dividing both sides by ρV2 and slight rearrangement

(33)

(34)

This can now be rearranged to obtain the final form as:

(35)

(36)

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Friction factor as a function of the Reynolds number and Hedströmnumbers, as predicted by equation (36)

Many authors prefer to use another dimensionless parameter, namely, Bingham number, Bi, which is defined as:

B0 D

Bi (37)

B

Bi (37)V

It is easy to show that:

BHe Re BiEquation (36)

(38)

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Effect of Bingham number on the value of (f·ReB) as predicted by equation (38)

The Herschel-Bulkley fluid model

(39)(39)

(40)

(41)H0

w

where