AN INTRODUCTION TO AQUEOUS HYDRAULIC CONVEYANCE OF SOLIDS IN PIPE LINES

,'AN INTRODUCTION TO AQUEOUS

HYDRAULIC CONVEY ANCE OF

SOLIDS IN PIPE LINES

_.

\.

C.

E. RIESEARCH RiI!POSt',No. 21

JUn, 1962

by: DR. ARTHUR BREBNER

A study sponsored by

THE

NÁTIONAL RESEARCH COUNCIL OF

CANADA

and

THE ONTARIO MINING

t

ASSOCtATION.

CIVIL ENGINEERING DEPARTMENT

QUEEN'S UNIVERSITT

AT

KINGSTON,

'ONTARIO

CIVIL ENGINEERING REPORT No. 21

AN INTRODUCTION TO AQUEOUS HYDRAULIC CONVEYANCE OF SOLIDS IN PIPE-LINES

by

ARTHUR BREBNER

SYNOPSIS

T'be report deals with the basic parameters and relationships involved inthe aqueous transport of solids in pipe -lines and is essentially a review ofwork done by others up to the present time.

Page

FOREWORD

ACKNOWLEDGEMENTS

LIST OF SYMBOLS 2

1. INTRODUCTION 3

2. TERMINOLOGY AND PARAMETERS AFFECTING THE PROBLEM OF

HYDRAULIC CONVEYING

a) Solid Particle Stz e , Shape and Specific Gravity Settling or Terminal Velocity

b) Definition of Concentration

c) Mode of Motion of Particles in Hydraulic Conveying

4 4 8 12 4

3. HEAD LOSSES IN HYDRAULlC CONVEYING 16

16 17 22 26 28 a) Horizontal Pipe

Head Losses for Homogeneous Suspension Heterogeneous Suspension

b) Vertical Pipe c) lnclined Pipe

4. PUMP CHARACTERlSTIC REQUIREMENTS 29

by pipe-line. The equipment consists essentially of a 400 ft. loop of4inch aluminum pipe with pumping, mixing and measuring facilities.

Two major areas are to be investigated, the conveyance of wood-chips whose specific gravity is approxirnately unity and the conveyance of mine produets of varying specific gravities upto about 5.

This report is essentially one giving background to the problem and is intended to acquaint the reader with the parameters in use and the basic work which has already been most ably done by other workers in this field. Subsequent reports will contain the specific resu.lts from tests on the materials mentioned above.

The problem of the aqueous hydraulic transport of solids in pipe-lines has two distinct facets. The first consists of the mathematical relationship involved in the head Io s s e s due to friction in transporting such s oIid s , The second consists in the technology of introducing such solids into the system and extracting them at the delivery end. This report deals essentially with the fi r st problem which may be stated in broad terms as follows:'

The water part of the mixture is the conveying medium, in much the same wayas an empty vehicle or s hipis the mode of transport, and the solids the "pay-load". All things being equal the greater the "pay-load" the more attractive is the proposition from an economie viewpoint. However, the type of "pay-load" or s oLid conveyed has a very marked effect on the economics of the situation. By type of "pay-load" is meant the size of the solid particle and its specific gravity, the shape (and associated with this its free fall or terminal velocity) and finally the ratio of amount of solid to amount of conveying water, that is the concentration. Moreover, the conveying vehicle or pipe-line has also an effect, its diameter and to a smaller extent its roughness being factors. Very small par t.icIes travel fully suspended in the water and the head losses associated with this type of homogeneous flow are fairly easily arrived at. Larger particles tend to settle out on the bottom of a horizontal pipe and the mechanism of conveyance and the associated head losses are much more complex. Associated with such solids is a minimum critical velocity below which the flowing water is incapable oftransporting the solids inmuch the same way as a r ive r in flood will sweep away large boulders it is incapable ofmoving under normal flow conditions •

The various facets affecting the head losses are dealt with separately in what follows and thei.r overall effect br ou ght together in a generalized relationship.

It must be emphasized that much of the basic empiricism ofaqueous conveyance of sol ids is knownand the majority of problern s in this field of engineering hydraulics are nowin areas of technique and practical engineering at the operatinglevel.

ACKNOWLEDGEMENTS

The author wishes to acknowledge gratefully the financial assistance of the National Research Council of Canada and the Ontario Mining Association and the help given in equipment by Aluminum Laboratories, Canadian Bechtel and Allis-Chalrners. The author also wishes tothank his colleagues Professor R.J. Kennedy and R.R. Faddick for their help and criticism in writ in g this report.

LIST OF SYMBOLS

Units

v

Volume of solid particle.

Specific weight of s oIid particle,

Specific weight of water.

ft3

lbl ft3 lbl ft3

Ww

Cd Dimensionless Coefficient of Dr a g,

A Projected area of the solid particle in a direction normal to Vs •

Free settling velocity related to still water of a single particle.

DimensionlesB Reynold st Number Kinematic viscosity. Particle diameter. ft2 ftl se c

z

ft

I

set: ft (or rnrn, 111icron, inch, etc,] CT Dimensionless concerrtrat ion by volume.

f3

Dimensionless funct ion,

Cd

Dimensionless "fictitious" Coefficient of Dr a g , sm Dimensionless specific gravity of mixture.

S Dimensionless specific gravity of sol id,

Shear stress.

Dynamic viscosity.

Yield value.

Coefficient of rigidity •

Limiting critical velocity Io r deposition.

lb. secl rtl

lbl ftl

lb. secl ft2 ftl sec K A dimensionle BB function or constant.

Dimensionless hold-up ratio.

Pressure.

Dimensionless friction factor or function.

D Internal Pipe diameter. ft

Mean mixture or water velocity in pipe.

Length of pipe over which head-loss occurs. Dimensionless hydraulic gradient.

4>

Afunction.

ftl sec ft v

1. INTRODUCTION

The pro btem of hydraulic conveying is essentially one of estimating the minimum energy required to transport a maximum amount of material at a specified rate between two points.

Transporting fluids by pipe-line, eg. oil , natural gas etc., is incontrovertibly the cheapest method of overland transport and there is good evidence to show that the conveyance of solids in a fluid medium through a pipe-line is also an attractive economic pr opo e it ion,

The commonest fluids available are air and water, the main hydraulic difference between them being the lower dynamic viscosity and density of air. From a practical viewpoint separ at ion of s olids and fluids at delivery is usually easier with an,air medium.

Since the fluid is acting as a tra ns por t medium only - - unless one cons ide r s , fo r example, a coal

-oil solid-fluid system where both can be processed at delivery - - it is obvious hom an economic viewpoint. that there will be an optimum amount of fluid flowing through the system. Essentially the operating cost of a hydraulic conveying system depends on the relationship between the rate of solid transport versus the power required to achieve such a r ate .

Moreland (1960) gives the following advantages made possible by conveying solids in pipe-lines: a) No Teturn of empty vehicles or ca r tons .

b) Less labour r equ ired .

c) Less dust and Iosses in transport. d) Less degradation.

e) Less equipment maintenance. f) Less pr odu ct contamination . g) Less space required.

On the other hand the many disadvantages are:

a) Pipe-lines which cannot be buried for maintenance purposes may be susceptibJe to Ire ezing

temperatures which greatly affect the Huid phase. b) Profile of unburied pipe-lines is influenced by topography.

c) Power failures may produce line blockages, thus requiring diesel or gasoline standby pumping

units or pipe-line drainage facilities. d) High initial cost.

e) Erosion of pipe by mater ia Is such as sand and co al. (Rotation of the pipe-Line may be required to partially offset th is, ]

f) Attrit ion of the transported material.

g) Long pipe-li.nesarcnot flexible and thus require an assured long-term source of supply and m arket.

h) Pipe-lines are only suitabJe for high capacity us e. There is a minimum velocity and an optimum economic solids concentration which can be.used.

i) Centrifugal pumps, which at present are the only pumps suitable for Ia rger particies , are relatively inefficient. When handling s Iur ries, efficiencies of greater than 700;. cannot normally be expected. Fo r long pipe-lines requiring high pressures the reciprocating posftive dis placement pump is the only practical solution at present and the wear on valves and cylinders is great.

i

)

Large volumes of water are usually required at the dispatching end and in some m.srances thls may involve the necessity of recirculating water.

k) Product must not be spoiled by contact with water; (this may be overcome by "packaging" the solids).

l) Asuitable slurry must be available for introduction into the pipe..l.ine- the preparation of this slurry sometimes involves considerable cost.

m) Insome cases (as for fine coal), the cost of separation of the product from the water at the receiving end is high.

2. TERMINOLOGY AND PARAMETERS AFFECTING THE PROBLEM OF HYDRAULIC CONVEYING

In this report only water transport systems will be referred to. The variables which are an

integral part ofthe aqueous hydraulic transport problem are as Iollows :

a) Solid Particle Size, Shape and Specific Gravity

The specific gravity may be greater than unity (sand, gilsonite, mine tailings etc.) or less than unity (unsaturated wood-chips etc.). One single parameter which repres e nts size, shape and specific gravity of the conveyed material and also represents the effect of the specific weight and

viscosity of the conveying water is the settling or terminal velocity of the particle in water. Since this

is a very important parameter in hydraulic conveying it will be examined more closely. Settling or Terminal Velocity

A5 a particIe falls (or rises if the specific gravity is less than the specific gravity of the surrounding water medium), it is subjected to at'iscou~drag by the water which opposes the gravity (or buoyant) force. Also, the pa r ticIe continuously displaces water as it falls (or rise s}, The drag

increases as the pa rticIe accelerates until an equilibrium state is reached when the gravity [or buoyant)

force is balanced by the dr ag force. The velocity at which this equilibrium state is attained is called the

free fall, settling or terminal velocity and is represented by the equation

(I)

where V

=

Volume of the solid par ticIe Ws

=

Specific weight of the solid particle

Ww =Specific weight of the Huid medium, in this case water Cd=Coefficient of Drag

A

=

Projected area of the solid particle in a direction normal to Vs Vs

=

Free setUing velocity relative to still water of a single particIe. F'or a single spherical particle of diameter d this may be re-written.

= Cd'--11"

i

._!!_j_2 4 2g that is 4 = _._. gd(s-I) 3 Cd (20)

for partieles heav ier than wate r

or gd ( 1 - s ) ( 2b)

for buoyant particles.

where s is the specific gravity of the solid particle with respect to water.

. . ~d

The Coeffic icnt (1f Drag, Cd ' 1S a function of NR ' the RevnoLds' Number, --y- i .y -5 2; is the kinematic vis cosity of the water, the value of

.y

at 60·F being about 1.2 x 10ft sec. [ully turbulent motion Cd is sensibly constant.

With In general Stokes Law 24

f,

0

.

69]

Cd =

N

RL1+O.15NR .. Schiller and Naumann 5 500 <NR< 2)( 10 Cd = 0.44

d

=

5 mmo (about 1/4 inch) have terminal veloeities which are directly proportional to

../d

i. e. VSI.,;gc;= constant, whereas partieles smaller in diameter than the order of d =50 microns

-6

(1 micron

=

lOmetres RI0.00004

proportional to d2 , i.e. vs/v'gd

inches) have terminal veIoc ities , in the laminar range ( Cd =24/NR),

C( d312

Examination of equations la and 2b shows that it is pos s ibIe for a relatively light padiele of

say specific grav ity s

=

1.5 to fall at a rate equal to the rate of ris e of a buoyant partiele, having the

same size, shape and surface texture, of s

=

0.5. HoweverjIt is impossible to find a buoyant particle

of sufficiently sm al l s value to rise at the same r ate of fall ofa particIe having an s value of 2.0

or greater.

F'o r noncsphericaI particles the vaIu e of Cdc an incor porate the shape effect assuming the

particle to be a sphe re ofdiameter d having the same volume as the non-spherical particle. Because of the difficulty in assigning mathematical values to shape for markedly non-spherical particles, eg.

wood-chips or metal turnings, laboratory tests must be performed to determine the value of Vs o r Cd;

the non-spherical particle may then be considered to have a mean diameter corresponding to a spherical

partiele having the same value of drag coefficient.

Figure 1 shows the settling velocities of spherical partieles of various sizes in water at normal tempe rature dedueed hom equations Z with appropriate values of Cd as shown.

These values wiil be redueed by wall effects if the partiele rises o r falls in a non-infinite

water medium and even more markedly redueed by eoneentration eHeets.

F'or non-spherical partieles, the ratio

settling velocity of non-spherical particle

RI dil

settling velocity of sphedcal partic Ie of same volume

where d is the diameter of a sphere of equal volume and t the greatest length of the non-sperieal partiele, is sensibly true if the greatest projeeted area of the particle is proportional to

,2

As the-art:l:!!fbe a::r't-'_ concentr ation of partic1es

increases, the particles themselves restriet the area through whieh the displaeed water flows, thus inereasing this velocity and deereasing the settling velocity. The settling velocity of a eoneentration of

partic1es is ealled the hindered settling velocity or the sedimentation velocity.

The reIat io nship between the hindered settling velocity affeeted by eoneentration of

partieles, and the settling velocity ofa single pa rtic Ie, has been shown by Maude and Whitmore (1958) to take the forrn

hinde red settli.ng velocity (3 )

where CT is the volume concentration, i.e, the ratio volume of solid partieles in a given volume of

wa teri-solid mixture and

(3

is a function of partiele shape, size distribution and Reynolds' Number given by Figure 2.

Equation 3 indieates that a 25% by volume concentration would have a hindered settling

velocity of befween 50% and 25% of the free settling velocity, Vs ' depending onthe particle size etc.

Sinee particle s ize is an important parameter in settling velocity it is felt judicious to

introduce at this time a eomparison between Tyler Mesh Sizes, U.S. Standard Mesh Sizes and the

o

..

>

>- I-U 0 .J W -:

>

<.!) Z .J

I-w

IJ)

q

u Ol>

..

~

..

o

o

(salpu!) S3H3HdS

.

:

W

'WVIO I'--~ _rt) "'",I'--r-,

r-,

r-,

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r-

r-~

~

I--r-. ...

,

r-,

_I'-...

...

r"-... I"<,

""

r-,

<, r-,<,

_K

_'"

...t'--,

<,

J"..

_"X

_~S

... <,

_r-,

_"'0

r-,

.<

o

·

"'~

.

"'o

0 0 0/

...

1':

0"'0 0",."'0 0

.

~~

\

~

Cl",

O·

t'-

è>i è>

e ~[\

_\.

°r

O

"

,'\. '\.

"'6'

.

"0 '\. I\. '\' '\.

o

1 0'/. I\.

_I

_\."

~

\

\ \

~~

"""'

0

/

_1\

.!'>_.p.

..r. '

_o. I\.

1\

_r\

_\

~~

/

1\

,

0 I-

>-ol- 1'\ 1'\ '\

Wz

" " "

a::::J

1

\

\ \ a::(/)

1

\

1\ \ ~et w

1

\

\ \\

a:: a:: w

\X

'

1-\

(!let ui~ \. \ a::

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Net 'Z 0 -V~ - _a::...J f---Z

o

(W:» S3H3HdS

s

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'WVIO

o

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o

z

(/) W

cr

LIJ J:

a.

(J) u,

o

>-...

Ü

o

...J LIJ

>

LIJ (/)

cr

cr

o

C> Z ...J

...

...

LIJ (/)

5

4.65

<,

~

\

~

2.32

(3

4 ( log SCALE)

3

2 -2

10

10

-I

10

10

4

10

5 ( log SCALE)

FIG

.

2

.

HINDER EO

SETTLING

VELOCITY

FUNCTION

VERSUS

REYNOLDS

NUMBER

T'yIer Mesh No. Opening in Opening in U.S. Starida rd Opening in Opening in

Inches Microns Mesh Inches Microns

400 0.0015 38 400 0.0015 37 200 0.0029 74 200 0.0029 74 100 0.0058 147 100 0.0059 149 65 0.0082 Z08 70 0.0083 210 60 0.0097 Z'i6 60 0.0098 250 35 0.0164 417 40 0.0165 420 20 0.0328 833 20 0.0331 840 10 0.065 1651 10 0.0787 2000 8 0.093 2362 ₈ 0.0937 2380 4 0.185 4699 6 O.1320 3360 3 0.263 6680 3 0.250 6350 1metre x 10-6 = 1micron: _1..50000 25.12.47 mmmm 2.00 50.8 mm 1mm = 1000microns 4.00 101.6 mm

Equations la and Zb may be re-written in the form v 4 1/2 ___:.s_ =

[-(S

-

I)]

...ïg(f 3 Cd or [ 4 ]1/2 -- (I-s) 3Cd

where vs/..;gais a form of Froude Nurn ber, Aplot of v~against d using the appropriate values of

Cd is given in Figure 3and a plot of

-te;;

against d in Figure 3A.

For non-spherical partic1es d is the nominal diaméter, measured as the mean of a lar ge

number of particles.

Equation 2a can be re-written as

~ = _1_

[~(S_I)]"2

=

_1_

.,/gd

v'Cd

3 ~ (2e)

where C~ is a so-called fictitious Drag Coefficient Ior a pa rticuIar particle of given shape, r oughrie ss

and specific gravity. The value of

Jc:;

is readily obtained by carrying out tests to find the value of cl

and Vs and 50 finding the value of

../ë!;

frorn the relationship 2c•

FOT pure spheres the value of Cd for various values of NR may beused to dete rrnine Cd

F'or example, for a sand, specific gravity 2.65, ~ = 1.48 ;,._.

...rCii VCd

In a later part of this report the dimensionless fictitious Drag Coefficient C~ a nd the

• Vs

Froude Number of settllng ~ will beused extensively.

v gd

b) Definition of Concentration

The concentration of solids transported at the delivery end, CT ' offh esystem may be

defined as

CT by weight _totsolid_{al w}wei_elgght (d_{ht of m}ry) d_lxtueli_revered_dehverein unit ti_{d m umt hm}me _e

OR

CT by volume solid volume (dry) delivered in unit time

total volume of rnrxtur e deHvered in un rt time

The latter relationship, that is, volume concentration is in more common usage and in

ensuing references to concentrations volume 'concentration shall be implied. The solid volume de Ii vered

multiplied by the specific weight of the solid gives the weight of solid delivered in unit time.

The specific gravity of the mixture, sm , is given by

weight of mixture

I

3

Sm

=

weight of equal,volume of water and since the specific weight of water is 6l.4 Ib ft

(volume of solid x s x 6l.4)

+

(volume of water x 6l.4)

(volume of rnixtu rex 62. 4)

(volume of sc"id x s)

+

(volume ofmixture - volume of solid) (volume of mlxture)

that is,

Sm = I + CT( s- I) (4)

and thus CT (byvolume) Sm - I

10-1 1 10 II III II1II I I III I d(mm) 10-2 10-1 1 I I I I I II II I I I I! I I I d (inches) 10 (/)

I-Z

:>

I-z

LU

I-en

(/)

z

o

u

a:::

o

u,

~

...

»:

.-

-v'"

_./

_~

V

-

_~

/v

~

/'"

V

1;'''' ",

~~

\

/

/'

V

...

~

\

/

...

,

./ ./ / ./ ./ '\ / V

...

,

1/ /' /'

,,

"

_/

_r

_/

_,/

_/

_/

_V

"

3 NR < 0.2

1\ "

'"

V/

/

~

V

_./

_V

\

~ NR>10

\

/

_V/

_1I

ti

lt'

./

~

."

V

V

V ~

v-II

l0 )(

/

V

/

-I

_/

_~

1

'"

,

1\ /

'-r

/

Vii

A

J

1/ Vlj

_'"

IJ' \

V

11

/

lL

11

~

_l

_/

.,

I

r

\

1I

/

II

l

j

V O' ,

\

V

J

c} 1\."

.1

V;

_lL

V

11

_1I

1I

~

'~

,

l

/

l

V

~

~

0) 1 _O' / / v2

=

4 1 / / / c} BAS EO ON _s -'--gd (5-1) 3 Cd

//

IL

_"

/ / /

I

/

'"

"

'L/

L

L

'"

WHERE Cd • { 24/NR

.

.

..

..

NR< 0.2

'j /

V

/

(24/NR)(I+.15N:9) '" 0.2<N I

,'IJ

V

/ 0.44 ... NR>500

V

V

_NR

₌

_(vsd)/,.,:

_YI'l1

I0-5 ft~se(

ti

, ,

-3 -2

, ,

,

10 10 I / 1/ / PARTICLE OIAM. d (ft) / / lil'

/

/

/

V

_J

FIG.

3

.

~

vs.

d

~I

rgd

en

en

LU _J Z ~ 10

z

LU ::ë

o

-10 R<500

-10 -I

o

2

0 ROUGH PLOT OF

./Cd

vs d 2 4 I FROM _VI

=

-

'

-gd(s

-

I)

3 Cd 0 24

\

\

-

FOR NR < 0.2

\

\

NR 8

\

\

WHERE Cd 24 .( I + 0.15"yR69)FOR 0.2< NR< 5

\

\

\

= 6 NR

\ \

_\

_{,_0.44} 5

\

\

\

500 <NR< 2·10 4

\

"\

\

f\

~

\

_r--\

\

r\.

2

-.

-,

".r-l'

.r

-I'

I

IS

.r

-l'

~6' 0"0 S

"",

I

'""

·S ...

_r-....

r-,

I

r-...

I'-r-,

I""'-

_~

_...

r--.

_~

---

-I--.

8

---=-

t---

t---

f---

I--~

v 00

o

.

O

.

0.67 0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 100 d(mm) 0.1" d (ft.) SIZE OF PARTICLE d

va r ious solid specific gravities. For example, a pipe-line system pumping 100/0hy volume of sand, s =l.65, has anaverage mixture specific gravity, sm of 1.165.

4

.

0

7

₇

₁

₇

₇

~

0

a

3

.

5

11

~

l-

a

en U

3

.

0

0 W l-a::

a

a..

2

.

5

-(/)

z

«

a:: l-C>

z

2

.

0

w

III 0 _J

a

(/) u,

1.5

a

>-

I->

«

a::

I

CT( 5 - I )

C> _Sm

=

₊

u I u,

_[CT

_(volume)

trom

_{0-1 ]}

u

1.0

w

a..

(/) 0.8 0.6 0.7 0.8 0.9 1.0 SPECIFIC 1.5

2

.

0

2.5

3

.

0

4

.

0

GRAVITY OF MIXTURE

c) Mode of Motion of particles in Hydraulic Conveying

It is reasonable to assume that the mode of transport of the particles is dir ect ly related to the orientation in a vertica l plane of the pipe-line. In a horizontalline heav ier than water particles will

tend, by gravity, to settle out and travel near the pipe invert wher eas in a verticaI line the particles

should be symmetrically dis tr ibute d a cr os s the pipe diameter. For an inclined line a combination of

horizontal and vertical modes might bereasonably expected. Ingeneral the twocornrnonest orientations will either be substantially horizontal or vertical as in a mine-shaft; a goodexample of the latter is thc

1363 ft. vertical lift of minus 1/4" copper or e at a Northe r n Rhode sian mine and of the for rne r the

lOB mile l ine carrying finely crus hed coal in Ohio.

Since horizontal conveying exhibits many more facets than vertical this orientation of Ii n

r-will nowbe dealt with fully be for e discussing verticaIlines.

Cons Ider a horizontal pi.pe.Hneconveying material particles of diameter cl anc1specific

gravity s at a concentration CT In a glass length of such a pipe-line several different modes of con -veyance are clearly visible with decreasing mixture velocities as {ollows (Newitt et al (1955».

1) Sus pended Flow: at high velocitie s the partieles move in a ful! y suspended 5tate. Depencling011

the size and specific gravity the suspension can move eit.her Iike a homogeneous suspension, eg. a soup or as a heterogeneous suspension having a concentration ina vertical plan normal to

the flow direction which is non-uniform. (With Inc rea s ingvelocity a heterogeneous suspension

tends to a homogeneous suspension. ) At Iower velocities in a heterogeneous suspension the vertica Iconcentration increases near the bottom (for S >I ) and the material there tends to move as a bed.

2) Suspended Flow with a Moving Bed: in the upper part of the pipe (for s > I )the particIes move

in heterogeneous suspension whilst onthe bottom of the pipe a layer of particles slide s or rolls

along at a uniform rate.

3) Suspended Flow with a Stationary Bed: as in (2)heterogeneous suspensions oc cu rs in tbe upper

part of the pipe whilst on the bottom a Iayer ofparticles slides over a stationar y dcposit by saltation.

3a) Saltation with a Stationary Bed: ri.ppIes of material move slowly forward in tbc direction of flow: 4) Blockage of Pipe - Isolated or Complete

[F'or dilute sus pensions of bleached sulphite pulp nurner ous expe rime ntcrs have found [ur ther mo.il'"

of travel , namely "plug flow" in var iou s Io r m s (Symposium on Flow of Suspensions, Nat ional

Research Council of Canada, 1956.)

As has been pointed out previously the settling velocity is a most important criterion in hydraulic conveying and for all practical purposes mixtures can be classified into twodistinct types,

"non-settling" and "settling" which are functions of this settling velocity. Mixtures with low settling velocities of the order of r oughly 0.005 ft. per second behave as "non-settling" pseudo-homogeneous fluids

at sensibly all velocities whereas mixtures with settling velocities greater than 0.005 ft. per second behavc

as "settling" hetet-ogeneous mixtures since even in very turbulent flow settling tends to take place in a

horizontal pipe.

Non-settling mixtures behave as pseudo-homogeneous fluids, the specific gravity and the so

-called viscosity or consistency of the mixture being the important parameters. Normally, non-settling

mixtures do not behave as Newtonian fluids but as plastic or pseudo-plastic fluid. (Ina Newtonian f luid ,

dv h . h . h dv

eg. water, 'T =.1'" dy w ere l" 1S t eumt s ear stress, _"... the dynamic viscosity and dy the velocity gradient. In a non-Newtonian fluid, eg. Bingham plastic, T- Ty

="'1

~~

where Ty is the

yield value and

"l

the coefficient of rigidity. )

Settling mixtures exhibit the modes of transport ~temized Irom 1to4 above and this report is

defined by:

Transition hom homogeneous to heterogeneous .. vH

Transition hom heterogeneous sus pension to suspended flowwith a moving

bed, between cases (1) and (2) vB

Transition hom heterogeneous suspended flow with a moving bed to

stationary bed and saltation, between cases (2) and (3) .Vc

Transition hom a stationary bed and saltation to pa.rtia Lor complete

blo cka ge, between cases (3a) and (4).

Knowledgeof the limiting velocity cha r acte rising the change f r orn the re gime of no

-deposition to the regime ofdeposition, namely Vc ' is very important since it corresponds to the economie operating point and permits a correct choice of pipe diameter. (Condolios et al, 1961). The var ious modes of flow are shown diagrammatically in Figures 5and 6 and typical values of the limiting velocity inTable IT.

C> Z (Ij <{

w

a:: U

z

CONSTANT PARTICLE SIZE

AND SPECIFIC GRAVITY

>- I-U

o

.J W

>

+ SALTATION

+

STATIONARY BED

w a:: ::> l-X

~

BLOCKAGE CONCENTRATION ( INCREASING)

FIG

.

5

C> Z (Ij <{ w a:: U

z

CONSTANT CONCENTRATION. PIPE DIAM.

AND SPECIFIC GRAVITY

W .J U l-a::

ft

STATIONARY BED WITH MOVING BED ::2 SALTATION <{

o

X <{ ::2

HOMOGENEOUS SUS"ENSION

MIXTURE VELOCITY

FIG

.

6

Material Pipe D Approx , Particle d Max. CT Limiting Velocity K= Vc Source 8. S.G. (inches) (mm) v ftl sec c .)2g0(s-1) Sand 6" 1/25 1mm 150/0 9.7 1. 35 Durand and (2.6) 10" 12.3 1. 35 Condolios IS" 16.5 1. 35 (1952) 36" 23.5 1. 35 Sand 1" 1/50 0.5 25 5 1.7 Turtle (2.6) 1"

l

is

3.2 20 5 1.7 (1952) 1" 3/16 4.S 15 4 1.3S pand 3" .24

_-

4.5 0.9 Spells (2.6) (1952) Coal 1" 1~16 1.6 35 3.5 2.4 Tu rtle (1. 4)

l

i

s

3.2 30 3 2.0 (1952) 1/16 4.S 20 3 2.0 Coal 6" 1/25 1 mm 15% 5.4 1.35 Dur and and (1. 5) 10" 7. 1 1.35 Condolios IS" 9.3 1. 35 (1952) 36" 13. 1 1.35 lPerspex 1" 1/50 0.5 40 2.5 2.5 Turtle (1. lS5) 1/16 1.6 30 2.5 2.5 (1952) 1/8 3.2 30 2.5 2.5 lMang. 1" 1(8 3.2 6 6 1.5 T'urtIe pxide (4.1) (1952)

TABLE

11.

TYPICAL

VALUES

OF LlMITING

VELOCITY

F'or values of d gr-eater than 1mmo (i.e. about 1/25 of an inch or 20 mesh) the limiting

velocity is apparently independent of concentradon and s ize, and depends largelyon D and s, Inview

of the difficulty in assessing the value of Vc by eye it is not surprising that there is some discrepancy amongst various sources for the value of Vc for particles greater than 1mmo However, it would

appear reasonable to say that, fo r such particles,

Yc = K..j2g0(s-l) (5)

where K is constant at about 1.5.

that is, Vc Rl 1.5.)2g0(s-1) (in consistent'units Ior d > rrnm ]

Since the neces s ar y minimum velocity to maintain flow without stationary deposit appears to

vary directly as the square root of the pipe s ize it follows that if, for example, the pipc size is increased

trom 3" to 12" the necessary limiting velocity is doubled.

For particles Ie s s than 1 mm. in size the relationship between Yc and the other parameters

is more complex. It has been suggested by Hughmark (1961) that the limiting velocity for particles below about 0.5 rnrn. in size decreases in a linear fashion with the particle diameter, that is K deercases in

a Liriea r fashion with decreasing par ticIe size. Durand and Condolios (1952) show a sirnilar decrease in the value of K(in K

=

Vc1.)2g0(s-1) )but show that below around 1mmo particle size the concentration

also has an effect. Grbe rt (1960) for a 0.2 mmo sand gives Vc Rl I.O"/2g0(5-1) for Cr> 7.5%. d) Hold-up and Slip

Ina homogeneous mixture the velocity of the mixture is the same as the velocity of the Huid

and solid phases. However , with decreasing mixture veloeities , a "settling" mixture flows as a heterogeneous suspension and there is a tendency for the fluid phase to slip past the solids ïn suspension

or, in other wor ds , there tends to be a "hold-up" of solids.

The ratio

Lîquidr'solid ratio (byvolume) delivered

hqUld! soha ratio (6y volume) at a grven s e ct ion

defined as solid volume at a given section

rnixture volume dehveTed and if C~ is m rxtur e volume at that given se ctron

then (6)

H the flow is a homogeneous one Cr = C~ ,that is, there is nei the r slip nor ho ld vup and the

hold-up ratio is unity. As the mixture flow becomes heterogeneous Cr bec~mes greate r than C~ and the hold-up ratio, HR ,tends to increase above unity, its final value tending to infinity when all the solids

remain stationary in the pipe and only liquid is delivered.

A plot of mean liquid velocity versus mean mixture velocity would appear as shown in

Figure 7, af ter Govier and Charles (1961), illustrating the differing important velocities and the changing

nature of HR •

CONSTANT CONCENTRATION CT' PIPE DIAM.,

PARTICLE DIAM. AND S.G. (.!) Z (/) cl IJJ a:: U Z

>- I-U

o

_.

IJJ

>

(/) :::> o ILI Z ILI Cl

o

::E

o

:I:

o

:::>

o

_.

MD(TURE VELOCITY Vm

FIG

.

7

(INCREASING)

The hold-up ratio, HR ' mayalso be plotted against the mean mixture velocity and would reveal a relationship as s hown in Figure 8.

(!)

z

I

c;;

c:(

LIJ _CONSTANT CONCENTRATIONI PIPE OIAM.,

0::

I

0 _{PARTICLE OIAM. ANO S. G.}

z

a::

·

1

J: LIJ

I

l-c:( 0:: n,

I

::> I

I

0 ..J 0 J:

I

I

UQUID VEl. --I MIXTURE VEL. UNITY

I

-i

-

i-I

_V

_c

I

_VB

I

_VH

MIXTURE VELOCITY

V

m

(lNCREASING)

FIG

.

8

3. HEAD LOSSES IN HYDRAULIC CONVEYING a) Horizontal Pipe

Let iw be the head Ios s in feet of wate r per foot length of pipe, that is the hydraulic grad ient,

for water only flowing in the system.

Then, by the Darcy-Weisbach relationship.

L y2 f-·_

o

2g head los s in feet of flowing Huid and since i

=....!!.b..

W L i_w = 2 f

...!..

._

y

_

w 0 2g (7) vD

where fw is a function of pipe roughness and Reynolds' Numbe:r, ---;r , and is dimensionless. At high values of NR the flow is usually fully developed turbulent flow so that fw is sensibly independent of NR

The value of fw for clean smooth pipe is approximately as IoLlows, eg. aluminum

N

R

104 2xl04 4xl04 6xl04 8xl04 105 2xl05 14xl05 6xl05 8xl05 106 f", 0.031 0.026 0.022 0.020 0.019 0.018 0.016-'0.014 0.013 0.013 0.012

4 for relationship between S , CT and Sm)' that is, the hydraulic gradient for a mixture flowing u nder

any mode, sa1tation, heterogeneous or homogeneous, in the system.

for homogeneous and heterogeneous mixtures will now be discussed separately. Head Loss "forHomogeneous Suspensions

Suspensions of fine particles may have a behaviour akin to a Newtonian Huid of specific

gravity sm and specific weight Wm

Then = f

~"L

m 0 2g (8)

where fm is the dimensionless friction factor for the mixture, this factor being a function of pipe rough

-ness and Reyno ld sr Number. The latter is a function, amongst other things, of viscosity. If the viscosity,

", , of the mixture does not differ too greatly frorrithat of water - and this wil! often be the case for

low concentrations - fm will be sensibly the same as fw at the same velocity inthe same pipe, internal

diameter D and relative roughness k/D . Thus

In words, for a homogeneous suspension, in theory the pressure drop express ed in feet of

homogeneous suspension of specific weight wm , is the same as the pressure drop for clear water expressed

in feet ofwater.

Let im be the pres s ur e gradient of the homogeneous mixture expr es eed as feet of water per

foot length of ptpe-Hne,

Then

But _{+ trom equation 4}

Thus im

(9)

That is, (s - I) _{for homogeneous Newtonian Flow}

(10 )

im- iw iw CT

a constant whose value is solely a simple function of the specific gravity of the solid in suspension. In wor ds , the dimensionless parameter _{is, in theory for homogeneous flow,}

Further, it implies that, for a given concentration

o

l

a given h?mogeneous mixture,

im =(constant) (iw)

w her ethe constant is in excess of unity. (For a 10%by volume concentration of sand, Specific Gravity

S =2.6, the constant is 1.16.) Since iw is sensibly proportional to y2 in turbulent flow,

Aplot of im versus v to log-log scales would indicate a theoretical behaviour as shown in Figure 9. Experimental results of Newitt et al (1955) conf'Jrrn the relationship ofequation 10for fine

particles of sand, mean diameter about 0.004 inches (between 100 and 200mesh) settiing velocity

t

I

Log -log PLOT OF

/

V

-

i

m

=

CT( 5-1)

iw.

_i

_w

/

1/

/

_/

~A!'/

-V

/

/

V

Y

/

V

/

/V

/

_V

~/

/

_V

Y

V

/

/

V

V

_/

~/

-,

_//

/

V

//

0

/

vy

V

~~ ~

~

Cl

/

V

t!) Z Cl)

«

w

0:: U Z W Q.

ä:

LL

o

::r:

~

t!) Z W

_.

~

o

o

LL 0:: W Q. 0:: W

~

«

3: LL

o

~

w

W LL

E

MEAN VELOCITY

v ft/sec

INCREASING

--F

1

G

.

9

.

log

-

log

PLOT

OF

i

m

VS v

.

[

i

m

=

Cds

-

l

)i

w

1-

iw]

Finer particles than these of Figure 10 did not confirm equation 10, indicating, inthis

instance, a non-Newtonian behaviour of the suspension akin to a plastic behaviour.

Larger particles tested by the same authors - Figure 11 iHustrates test res ults for a sand

having a mean diameter of 0.030 inches (20 mesh) and a settling velocity, Vs , of 0.37 ftl sec. - give

im =(constant) (i. i

differing results . Figure 10 indicate s that whereas Figure 11 iLlustrates a

convergence of im and iw with increasing velocity.

0

.

55

0

.

50

SAND

_s

=

2

.

64

Vs

=

0

.

032 ft/sec

d

r:::s

0

.

0043

i

n

(AFTER NEWITT ET AL)

0

.

35

0

.

30

t-=

IL ...

₀

_.

₂₅

0:

w

I-<t

3:

LL

0

.

20

0 I-IL

E

0

.

15

0

.

10

5

6

7

MEAN VELOCITY

8

9

10

v ft /se

c

FIG

.

10

Worstel' and Denny (1955) in Figures 12a and 12b.

Again, from Figure 12, a eonvergenee of im and iw is revealed, sueh a eonvergenee being

apparently a property of heterogeneous flow mixtures: thus equation 10would appeal' to be only

applieable to homogeneous mixtures of small particies. Howeve r , at high velocities the hydraulie

gr ad ie nt for heterogeneous flow beeomes sensibly parallel w ich the clear water line thus indieating that

u 4)

'"

...

--\

\

'\

~"

-.

>

u

4)

-\

1\

-,

-,

~

'"

..J

>-

-...

_«

_....

-

o~

r\

-

....

U

'

_{\ \ \}

\

"

~'"

q- ,.._ _{0 w} 0 cD ", ", _...J

[\

N

Ó Ó

....

W I-

_>

_{_}

\

\

1\

\

-,

r-,

_tilti ._>11

'"

_"0lZ

3ii

w _Z Z

_«

_{_}

\ \

\

"

[\

~~

~

0 0:: W

~

Z

«

WI-

~

\

\

_\

Cf) u,

-'" \J

1--0

«

-1\

~

\

,

\

\

\

V~~~

1

\

1

\

\

,

[\

<~'~

1

\

1\

I-"

\

,

\

,

\

\

-,

\

_

\

<,

1

\

11

\

,

~

\

~

0 It') (\J

~

0

~

~

~

0 0 0 Z I() ₀ 0 0 N

_-

-

I/)

Ei

0:: I-Z W u

z

/

0 u

J

I I I I I I I I I I I I I I

I/) 0 I() 0 I/)

I/) I/) q- q- ", Ó Ó Ó Ó Ó

o

", Ó I/) N Ó

o

N Ó I/) Ó

("l.:

U~31"M

.:1

0

0

1.:1)

w,

o

co

o

I/) LL

o

Ó

0.10

/

.//

/"

--""""",C

t e

A

~

77

V

,.

~

/

-

---::

v/

/7

v

~

/

---

---/

v~

/

/

~//Vo

-

<,

_i

"--'0%

' ----===

/

~

L~

·

I

T

<v~

<,

/

~

~~

~

.;;___

<;

5

0/0

/

/ 'v (a)

//

..É 1/ /

//

/

A _{1/2" COAl - 3" PIPE} .~

(AFTER WORSTER 8 OENNY)

/

I

0.08

LIJ

a..

0.0

a..

LL

o

t

0

.

04

a::

LIJ

a..

a::

LIJ I-oet :?; LL

0.02

o

._:

u, I E

0

.

0

1.5

2

345

MEAN VELOCITY v ft/sec

6

7

8

9

10

...: u,

0.05

OEPOSITION NO OEPOSITIO"

-:

CT =

15

0/0 _{OR HOlO-UP}

i

_{OR HOlO UP}

/

/

/

___.

»-:

_../"

-:

/

f--

-I

~/Vv.(

10%

/

---

/"

0

7

1-,--.,

/V

l{-;'

_

./

~

_~

_{L_}

/

_~

....

~~

I-

---z~~

~~

,

u

~<v

(b)

/

I 11) A E

V

.

~

11/2

"

COAl -

6"

PIPE

/

/

..} WORSTER 8(AFTER OENNY)

I 1

0

.

09

LIJ

a..

ä:

0.07

u,

o

a::

LIJ

a..

a::

LIJ

~

0.03

:;:

u,

o

I-u, E

0

.

0

1.5

2

3

4

5

6

7

8

9

10

MEAN VELOCITY v ft/sec

appreciable specific gravity, equation 10 may well hold true. (From Figure 12, at high velocities,

taking s

=

1.4 for 1/2" coal, at CT

=

100/0and v

=

8 ft/sec., Îm

=

1.14 i.. : equation 10 gives

i

m = 1.04

i...

Such a high velocity is uneconomic in practise and the relationship for i'm - iw

ÎIN CT homogeneous.

The velocity at which the flow is a homogeneous one, that is when the mixture gradient line is

is not as simple as indicated byequation 10 since the flow is heterogeneous and not

almost parallel to the clear water gradient line in a plot of Îm or i.. against mean velo city

called the "standard ve Ioci.tyn , (Spells, 1955).

Asimple relationship for VH suggested by Newitt et al (1955) is

v , is

38.6

V

D·vs (11)

This equation was derived from tests onparticles of various specific gravities andofsize up to 3/1611.

Differing authors give similar types of equation for the value of vH For any particular case the

numerical value for VH may vary quite an amount f r orn the value predicted by equation 11.

As a rough rule, mixtures which be have as homogeneous ones in even weak turbulent conditions

(as opposed to heterogeneous mixtures which tend to settle with a non-constant concentration across a vertical diameter in a horizontal pipe) are limited to mixtures of particles of Ie s s than 50microns. That is, mixtures of particles greater than about 50 microns will exhibit heterogeneous before showing

homogeneous behaviour with increasing velocity and do not follow the relationship ofequation 10 until the

velocity is probably much in excess of the economic velocity Vc The specific gravity of the particles

will obviously be a factor too since low speçific gravity particles of a given size can be kept in

homo-geneous suspension more readily than particles of high specific gravity. Thus fall velocity Vs is a factor in heterogeneous flow.

Heterogeneous Suspension

As seen frorn F'i gur esI 0 and 12 at velocities below the standard velocity vH for homogeneous

flowthe relationship of equation 10, namely

= (s - I) (10)

is no longer valid except for very small particIes ,

For velocities in excess of the critical velocity wher e the flow is a heterogeneous one, with or

without a moving bed, it would seem reasonable to amend equation 10 by introducing suitable additional parameters to the right-hand side of the equation whose values tend to unity at high velocities so that equation 10 again becomes applicable when the flow changes frorn heterogeneous to homogeneous .

Equation 10 may be re -wr itt.enas

where

111

,

is a simple function of s and CT (Jr homogeneous flow. It is agreed that

0

,

should be modified Ior heterogeneous flow by additional parameters. The question arises as towhat these

parameters might beo For obvious reasons these parameters ought to bedimensionless so that, using consistent units, any relationship should have fairlyuniversal application in any system ofunits.

*

~

results to corresponding analogous situations is the Froude Number NF Itwill be recalled

that this dimensionless Number appears in the pipe loss equation, namely

~

L

f Y 2 I y2

= -[-] = f---

-2.JgD 0 2g

A further measure which will also be of importanee is the size and shape of particle being conveyed. (Homogeneous flow has nothing to say about size, only specific gravity.) This can be taken care of by introducing either ~, another form of dimensionless Froude Number (See Figure 3)or

",9 d

by introducing

.;r:; ,

which is also dimensionless and is inextricably linked with both Ys and d .

(See Figure 3A), or by introducing

.R;;

from equation Ze.

It might reasonably be expected that an equation of the form

= (12)

would represent the typical result of Figure 11.

Experimental results of numerous authors would lead one to believe that the concentration term f,(Cr) is a linear one, at least within the range of tests reported. Hence equation 12 may be re-wr it.t en in

the form

= f:!s - I) , . (131

From than about Z mmo friction loss ratio

Figure 3 it is seen that

#

(

=

~)

tends to a constant value at values of d greater Thus it would appear that an incre~se in particle size over Z mm. has no effect on the

im - iw

Criw

Such a conclusion is borne out very well by Durand 's experimental wor k

on sand and gravel.

The question now arises as to the values to be assigned to the functions f2, f3 and f4 in

equation 13 anda s to whether they can be combined in any fashion since they are all dimensionless:

here one must rely solely on experimental evidence.

It would appear that the commonest method of expressing these functions, and one well-known

to hydr aulic enginee r s, is in the form, for instance

= Constant

v 0

( JgD)

where 'a' is a power determined experimentally as is the constant.

A relationship suggested byDurand and Condolios (1952)has the form

for sandand zr avel s =2.65. thatis s constant.

The Constant in the above equation, from tests carried out at the Neyrpic Grenoble Laboratories and the Loire Maritime Department is found to be 180 according to Gibert (1960) and 176 according to Worster (1952).

That is, for the no-deposit regime of heterogeneous flow

3 3/2 3/2

e 180

[_y'2Q_] [

~

]

= 180

[4 - ~]

v '" Cd v '" Cd

(14)

[

gO V] 3/2 180. __ .__5_

V2

./Od

(15)

Both equations 14and 15 are Ior sands and gravel of S.G. 2.65. However, Dur and and Condolios (1952) agreed that the effect ofspecific gravity could be incorporated in the above relationship in the form [ gO(s-1) I ]3/2 = Const .

~==~~

IJ

Ri"S=ï)

d [ gO(s-1) = Const 1J2 IJ 3/2

J9d;s-n]

gO gO(s-I) I I) (Ineffect -1J2 is replaced by 1J2 and

r:::r

C'by ~. vl..,d v C~(s - I)

Since (s - 1) is 1.65 for sand the r esu Itirrgequation for non-deposit he te rogeneou s flow becomes

[ 90(S-I) = 123 2 IJ ] 3/2 Ijs -.j7g==d~(s=:=tl) (16) 123 [ gO(s-1) 1J2 3/2 -.j';::Cd::='(=5=-=1=)] (17)

Re suLts for materials of s value ranging from 1.60 to 3.95 are in good agreement with the above formulae (Gibert 19(0). Equations 16 and 17 are shown on Figure 13.

Another relationship for heterogeneous flow proposed by Newitt et al (1955) takes the form

l.gO

Const. (5 -I) V

7

which when plotted to experimental results reduces to

v

(g 0)

1100 (5-1) ~

7

(18)

Comparing this latter equation with 16 (or 17) it is seen that

(1) is inversely proportional to the mean velocity offlow cubed in both equations

(2) in equation 16, IS propor tronal. to 03/2 whereas in 18 it is proportional to D

(3) in equat ion 16, im

_e

_-r,

- iw is proportional to (s - I)3/4 whereas in 18 it is proportional to (s - 1). (This difference is a minor one.)

(4) in equation 16, is proportional to

V 3/2

[~] whereas in 18 it is proportional to Vs • Since Newitt et al were testing in al" diameter pipe on1y it would appear to be reasonable to give credence to Durandts relationship Ior D insofar as the .a.tter tested many differing diameters. The discrepancies between the two expressions is not great except at low values of

accuracy is not too good.

.~ !I 1 t-E u

V

l23 1'-'\.

\.

\.

\.

_-

-\

\

im- iw

123 [OD('

-

I)

.

1

~3

t

r

\

_V

= = 123 gO(s-l) . v«

I\. ./' CTi

w

_y2 ,jC~(s-l) y2 .J'gd(s-I)

\:

\

[AFTER ,GJBERT,PONTS 8]

L\

CHAUSSEES 1960 p.340

\

1\

,

1\

\.

\

I\.

1\

Î\

1\

~

\

I

1\

100 10 10 100

or

.J'gd(S-I) gO(s-I) gO(s-l)

F

I

G

.

13

b) Vertical Pipe

Homogeneous mixtures can be treated as for horizontal pipes, the head losses being a simple function ofthe specific gravity of the mixture.

For heterogeneous mixtures at values of v greater than Vc ' the critical mean velocity for nodeposition, Durand and Condolios (1952)and Gtbe rt (1960) claim that the actual head losses are the same as with clear water.

Refer to Figure 14, which shows (a) a rising flow and (b)a descending flow using a water differential manometer. B L (0)

z

'

WATER

z~

AIR

z'

WATER FILTER

FIG

.

14

A L B (b )

The difference in pressure between A and B bas two parts, firstly the effect of a length L

of mixture S. G. Sm and secondly the friction loss between these points. If the f r ict iorrLos s is the same as that for clear water

where

(ft. of water) L sm + L iw for upward flow

- LSm+ L iw for downward flow

Itmayalso be readily shown that, from the manometer, considering upward flow,

that is But PA - Pa L + t.h WW PA - Pa = L Sm+ Liw Ww t.h = LSm + Liw - L t.h Liw

+

LISm - I1 t.h = Liw + LII - sml for upward flow i.e.

and for downward flow

These latter relationships make possible an easy method of measuring concentration since a knowledge of t.h, iw and L gives a value of Sm' Knowing Smand s , Cr may beobtained from

equation 4, narnely Sm = I

+

CrIs - 11.

The for e going relationship for head loss in vertical conveying, namely that it is the same as for clear water, is somewhat su rprisin g, The practical dïfficulties are fairly gr eat, however, in measuring over any appreciabie lengths in a vertical laboratory set-up and any effect of S.G, rnaybe masked at smal! concentrations. For example, the sm value of a 100/0 Cr of sand, s

=

2.65 is 1.165 whereas iw (in a 6 inch line), at say 8

ftl

sec., is about 0.03, a small percentage of the sm vaIue ,

When a concentration Cr is pumped vertically upa pipe, as in a mining operation, a distance

L , the following relationship may be obtained.

In an upwards vertical flow system the heavier than water particles will move downwards

relative to the upwards moving water with a velocity equal to the hindered settling velocity. As a con

-sequence the concentration in the pipe in vertical upwards flow will tend to be higher than the delivered concentration Cr. However, this effect of additional concentration will be small if v is a good deal gr e ate r than the hindered settling velocity. Ignoring this effect the following efficiency ofthe vertic al operation may be deduced.

Pressure at base of L feet of vertical pipe

=

wmL + Ww hL where wm and Ww are the

specific wei~hts of the mixture and the water respectively and hL is the water friction loss

( = f ~ < ; 9 ). The volume ofsolids r a.ised per second is Cr '11"402 .v and the useful wor k

donel sec. in raising the solids - the object of the operation ., is WsCr I""02/41 vL.

In a re-cireulating system, as a good approximation,the work done on the conveying water is

expended on friction on a length 2L, L up and L down.

Thus the wor k rll)lV~/sec. in circulating the water is

The hydraulic efficiency, ~H of the ra is in g operation is given by,

If the mechanical efficiency of the pump-motor set is f,p, then the overall efficiency of this mode of conveying is given by

overall efficiency E, [ S·CT ]

p s·C+_T2i _•

F'or example, if an ore of s

=

4 is delivered at ZOo/.concentration by volume with an i.

of O.OZ(6 ft/sec. in a 6 inch pipe) with a pump efficiency of 50% the overall efficiency is 47%.

However, this is an idealized situation, since in practice additional work would have to bedone

in introducing the solids against pr e s sur e at the bottom of the line. It would seem that in any charging

system an additionalloss would be sustained - 50adding a furthe r term to the denominator and thereby

reducin gthe overall efficiency. It may be that 30% is a more realistic overall efficiency: even at this

figure hydaulic hfting may be an economic and attractive alternative to conventional hoisting me thods.

c) Inclined Pipe

It was pointed out in the previous section that in a vertical situation that im ~ i.

However, in an inclined situation, the friction loss is greater than in a horizontal line according to

summarized replies on this question reported in a Hydraulics Institute questionnaire of April 196z

wberea s Gibert (1960) gives the expres sIon based (on horizontal pipes) of

Func!ion of

_.

y2

~

Z

gO cosOl

where '" is the inclination of the line ( Ol.= 0is horizontal). Such anexpression reve als a decreasing

value of im f r orn ct

=

0to ct

=

90· suchthat at d= 90· im ~ i•.

Ineffect the horizontal pipe-line relationship, equation 16, is amended to

[

gO(s-1) cos IX ]3/2

123 2· ~ .

v "Cd{S-1)

It may be that the upwards sloping pipe does requ ir e a greate r critical velocity Vc ' since

it is reasonable to assume that the gravity force also must be overcome with sliding bed mate rial.

F'ras er (1960) r e por rin g on the test work preceding the installation ofthe International Nickel Co. of

Canada tailings line found an increased friction head loss in uphill flowand a decreased friction loss in

downhill flow. Howeve r , in the view of Costantini (1961) "there should be no significant increase in

either minimum (i.e , critical) velocity or friction head loss up a sloping Iine;»

In view of the paucity of experimental evidence no general conclusions can be drawn regarding

end of a horizontal system per unit weight of material delivered, a) decreases with the concentration,

b) depends on the size of material being transported below a size of roughly 2 mm whereas at 2 mm and above it is at its maximum. The dependence on size of particle is such that

it decreases with size,

c) increases with increasing density of material being pumped.

The following figures of Condolios et al (1961) give some idea ofthe powe r s involved in

delivery at a rate of 100 Tons per hou r ofmaterial of s

=

2.65.

"to transport gr aveLl.arger than 2 mm diameter, provision must be made Ior power ofabout

6 H. P. per ton rnile;«

"for sand with a mean diameter of 0.4 mm, and included between 0.15 mm and 0.7 rnrn , the power required w ouldbe about 2.2 H.P. per ton miIe,«

"the power required would only be O.lZ H.P. per ton mile for raw cement paste with a mean diameter of the order of 40 to 50microns."

Based on these figures, if power is bought at $50 per H.P. per year, the cost per ton-mile

for overcoming the energy loss will vary hom 3'.5 cents per ton mile Ior largish rnaterial. to about 0.1

cents for sma11material on the basis of a c ontinuous pumping operat1on.

(For vertical conveyance it is sufficient to adopt ~bran sportvelocity v of about Z to 3

times the fall velocity vs ,The power consumption has already been indicated on page Z8 •)

For safety of operation agairrst jarnrningit is suggested that the pipe diameter be at least Z

to 3times the size of the greatest particle bein,gtransported. (This wil! obv iouslynot be applicable to

"packaged" or "sausage" type of plug flowofmaterIa Ls,)

From a power economy view point it wi11be ad varrta geous to work near a velocity vc'

However, with a constant-speed centrifugal pumpwith a flattish H v Q curve a slight drop in pump

speed or an iric r eas e in CT at the intake end will make it impossible for the two characteristics to

meet and there wi11 be atendency for the fIow r ate to diminish and deposition and jamming to occur,

This is illustrated in Figure 15 where conc entr ation C3 is incapable of being t.ransported and C

z

is

also in the unstable region since a fall in H due to a mornerrtary speed decrease couId cause asimilar

effect o r an increase in concentration start deposition and thus i.ncr ea s e head-Iosses.

From a practical viewpoint with centrifugal pumps itis.therefore preferabie to control the

feed rate of solids into the pump. The measurement of pressure drOp onthe initiallength of pipe on the delivery side of the pump is a good indication of the variation in concentration. An increase in pressure drop can be used as a sigr.a.Ito decrease the feed rate.

On long line s the us e of centrifugal pumps is not too practical at present since a single

-stage pump usually delivers a maximum of 150 ft. of head and the number of pumps in series becomes great.

F'or long lincs

iJ

:

volvin ghigh pres s,ures the reciprocating po~itive displa,cement pump is

the only practical answer at present. Since slIch a pump has a vertical H v Q characteristic this gives

a most sta ble'operation. (With a centrifugal pump itis preferably to work on the steeper fa11ingpart of

the H v Q curve for stability of operation. )

Starting and stopping of lines has been found to be best carried out - and the writer would fully concur with this from his experience in pumping wood-chips - by using water only. Further, the

pump speed s houId be increased or decreased gradua11y if possible to obviate water hammer, a difficult problem with one fluid alone but much more complex with mixtures. Ifnecessary pressurised surge chambers should be used to decrease water hammer effects.

For more detail regarding actual pumping insta11ations the reader is referred to the

Bibliography. Of particular interest to verticallifting in mining operations is the paper by Condolios et al (1961I.

W ..J <t U (/) C>

o

..J

o

<t W

::r:

>-et:: W

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..J W

o

et::

o

(/) (/)

o

..J

o

<t

w

::r:

MIXTURE DISCHARGE OR VELOCITY (LOG SCALE)

BIBLIOGR APHY

The following bibliography is far from complete and only contains references

which were readily available to the writer in Ouee nvs Urriver s ity Library. However, it is felt that

they do give an adequate coverage of the problem of hydraulic conveying both from a theoretical and

a practical viewpoint.

1. ANONYMOUS, "Advent of Big Coal Pipe-line Ne are rv, The Oil and Gas J'our na l, Nov. IJ, 1961,

pp 122-123.

2. ANONYMOUS, "12-Mile Pipe Bearing Trestle (Copper Cliff Installation)", Canadian Mining Journal,

Vol. 72, Dec. 1951, pp 63-64.

3. BINDER, R.C. and BUSHER, J.E., "A Study of Flow of Plastics Through Pi pe sv,

Trans. A.S .M.E., Vol. 68, June 1946, pp A101-AI05.

4. BLATCH, Miss N.S., "Flow of Sand and Water in Pipes Under Pressure - (Discussion of Paper

1036by Hazen and Ha r dy )v, Trans. A.S. C.E., Vol. 57, Dec. 1906, pp 400-408.

5. BOND, R. K., "Designing the Gilsonite Pipe-line", Chemical Engineering, Vol. 64, Oct. 1957,

pp 249-254.

6. BOUDALN, J., "Apparatus for Measuring Spatial Concentrations in a Current of Air and Water

Mixed", La Houille Blanche, May-June 1951, pp 406-410.

7. BRUCE, W.R., HODGSON, C.W., CLARK, K.A., "Hydraulic TransportationofOil- Sa nd Tailings in Small Diameter P'ipes!", Can. Inst. of Mining and Met., Vol. 55, pp 412-416,

1952.

8. COHEN DE LARA, G., "Equilibrium of a Mass Submitted to a Vertically Ascending Flow".

La Houille Blanche, March-April, 1955, pp 167-176.

9. COSTANTINI, R., "Basic Considerations for Long-distance Solids Pipe-lines in the Mineral

Industry" , AIME Transactions (Mining), Vol. 220, 1961, pp 261-270.

10. CONDOLLOS, E., COURTAlN, P., PARISET, E., "Transportation of Solids in Conduits _ Industrial

Application Po s s ibiLitiesu, The Engineering Journal (Canada), June J961,

pp 62-67.

11. DANEL, P., DURAND, R., CONDOLlOS, E., "Introduction a L'Etude de la Sa.Ltatic nv, La Houillc

Blanche, Dec. 1953, pp 815-829.

12. DURAND, R., "The Hydraulic Transportation of Gravel and Pebb1es in Pipes". La Houille Blanche,

Special B, 1951, pp 609-619.

13. DURAND, R., "The Hydraulic Transport of Solids in Pipes: Experimenta1 Studies Ior the Arrighi Power Plant Ash!!, La Houille Blanche, May-June 1951, pp384-393.

14. DURAND, R., "Ecoulements De Mixture en Conduites Verticales - Influence de la Densite etc.", La Houille Blanche, Special A, 1953, pp 124-131.

15. DURAND, R .• "Basic Rclatio ns hi ps of the Transportation of Solids in Pipes - Experirnental

Research", Proc. Minnesota International Hydraulics Convention, Sept. 1953. 16. DURAND" R. and COHEN DE LARA, G., "Settling Velocity of Sand Grains in Fluids in an Infinite

Medium", La Houille Blanche, May 1953, pp 254-259.

17. DURAND, R. and CONDOLlOS, E., "The Hydraulic Transport of Coal and Solid Materials in Pxpe an ,

Proc. of a Colloquium on the Hydraulic Transport of Coal, Nov. 1952, Scientific

Department, National Coal Board of Great Britain.

18. DURAND, R. and CONDOLlOS, E., "Concentration Measuring Instrument Ior Hydrau1ic Transport

a-tion Ins ta l.Iat io nso, La Houille BLanche, May 1953, pp 296-297.

19. FAIR BANK, L.C. andWILSON, W.E., "Pipe-line FlowofSolids in Sus pension!', A Symposium,