Method of Molerus and Wellman to Compute the Pressure Drop of Slurry Transport in Horizontal Pipes

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~vf;(

tH

Del

ft

Department of Civil Engineering

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Delft University ofTechnology

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Method of Molerus and Wellman to Compute the Pressure Drop of Slurry Transport in Horizontal Pipes

by

X.Q. Yu

Report Nr. 17-85

Delft University of Technology Department of Civil Engineering Laboratory of Fluid Mechanics

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1

Method of Molerus and Wellmann to Compute the Pressure Drop of Slurry Transport

in Horizontal Pipes

X. Q. Vu

The Administration of North-West Electrical Power System Xian, China

Report No. 17 - 85

Delft University of Technology Department of Civil Engineering Laboratory of Fluid Mechanics

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Abstract

In the literaturey many methods for the calculation of

pipeline hydraulic transport of solids have been published. Their degree of exactness and applicability is variabie and therefore it is hard to the designer to make a convincible decision regarding their practical use.

A new concept for the computation of slurry hydraulic transport in horizontal pipes proposed by O. Molerus and P. Wellmann based on numerous measurement data seems to provide a practicle method to predict the head loss in slurry pipeline transport.

In this report the new method is introduced and a number of new experimental results to verify the new concept. Furthermore some suggestions are presented te impreve the new methed.

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List of Nomenclature

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c

v internal diameter of pipe absolute pipe roughness mean partical diameter

refers to the mid-point of the grading eurve density of solid

single particie, fall velocity density of fluid

gravitational acceleration average suspension velocity

volumetrie coneentration

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D k d p dSO p

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v C va V rel

volumetrie eoneentration of veloeity v a

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slip velocity

non-dimensional factor in the expression for the additional pressure drop whieh is~independent of the eoneentration of solids

modified non-dimensional factor in the expression for the additional pressure drop whieh is related to the eoneentration of solids

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~L ~p/~L ~Pf

Frp

Fr% ~P m tube length

pressure gradient

<i

m

fluid part of the pressure drop

partiele Fraude number defined by equation (5) Froude number defined by equation (7)

total pressure drop

additional part of the pressure drop

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Introduction

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Tc design a pipeline for slurry transport the designer has to estimate the head loss occurring under various operating conditions, since this determines finally the dimensions of the piping and pumping system. In this connection such a choice must be made that the solution is the most economic one.

There is a lot of literature related to the estimation of the pressure drop in a slurry pipeline. Many authors claim to present a general method holding under almost all hydraulic circumstances occurring in practice. However, such general

methods for the determination of the pressure drop in a pipeline transporting a slurry, generally do not hold for all ranges of particle size, tube diameter or concentration, and the designer's problem therefore is not equivalent to the problem of designing water lines. Having a special problem, it is often required to conduct a series of tests in which the problem is studied more specifically (ref. 1,2,11>.

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The general approach to the subject

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The flow behaviour of a solid-liquid suspension is governed by the properties of the slurry as weIl as the properties of flow. In general the following regimes are distinguished, namely homogeneous fiow, heterogeneous flow, saltation flow and flow with a stationary bed (ref. 4).

In homogeneous flow the solids are uniformly distributed across the pipe cross section. This type of flow is encountered with slurries of high solids concentration and fine particle size and at high velocities.

In heterogeneous flow, solids are not evenly distributed and a pronounced concentration gradient exists along the vertical axis of the pipe. This type of flow is encountered with slurries of

low concentration, large particle sizes, and at low velocities. In the saltation flow regime the turbulence of fluid is not sufficient to keep the particles in suspension for a prolonged time. The particles travel by making successive jumps downstream. This type of flow results when the particles are very large or the velocity of flow is low.

Flow with a stationary bed occurs when a stationary bed

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= ._j

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farms on the pipe bottom.

Achart showing the varieus flow regions is shown on the

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chart below. The methods to estimate friction lasses in the

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different flow regimes are different. It is necessary to give the conditiens when the transitien from one regime into another one

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Dccurs. Due to the complexity of the phenomenon, it is difficult

to predict the regime of the flow from independent variables,

such as particle diameter, pipe diameter, volumetric

concentration, mean velocity of the flow and density of the

solid. Most of the designing slurry pipelines for long distances

transportation of solids are therefore treated experimentally and

such invesigations mainly aim at development of 50me kind of the

prediction of pressure gradients versus flow conditions.

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Lamina Flow

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Newtonian Irransition Flowl

Turbulent Flow

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Lamina Flow

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Homogeneous

rl

Pseudo Plastic Fluid ITransition Flowl Flow

Turbulent Flow

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Lamina Flow

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Non-lBingham Plastic Fluid ITransition Flowl Newtonian

Turbulent Flow

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Lamina Flow

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Yi.el.dPseudo Plastic Flui4

ITransition Flowl ~Turbulent Flow

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Heterogeneous Flow

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2. Partially suspension flow:

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Saltation Flow

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3. Non-suspension flow:

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Flow with a stationary bedl

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The Npw Method

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At the University of Erlangen-Ntlrnberg, West-Germany, O.

Molerus and P. Wellmann developed a new concept for the

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calculation of the pressure drop with hydrotransport of solids in

horizontal pipes (ref. 1'. The results are presented using non-dimensional groups. From a pragmatic point of view, it is

undoubtedly a significant attempt to provide a prediction of head loss in slurry pipeline transportation. The idea and the computing process will be presented in the next paragraph.

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The Computing Process of the New Concept

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The pressure drop in a horizontal pipeline transporting a slurry is subdivided into two parts

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( 1)

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With the aid of the formula known as for single phase flow (ref. 13), the fluid part of the pressure drop is presented as

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(2)

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_{M-W assume that the additional part of the pressure drop~Pad} is caused by the slip between particles and fluid. Using

dimensional analysis using measurements found in literature and own measurements, the following figures and formulae are given by the author-s,

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Figures see Fig. 1

&

Fig. 2

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Formulae ~Pad =XI _C~)2 ~ CvCPs- Pf) g • Wf CV I/V)2 re Xo _{- CV}

_IvF

rel

{

Xo ~ X d ~ Xo + __L Cc - C ) dC v vo v (3)

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(4)

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for0 ~ C ~ C ) v vo (5) for CC ~ C ~ C ) vo v vmax

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CC L= regarded as 0.25 in the data) vo

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dXdCX r Frx (7)

*

0.1 Fr (see Fi g.2) (6)

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FrP

v

(8)

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(

~

-

1)g d Pf P

A complete description of their derivations is given in ref. 1.

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To calculate the pressure drop the following data are

needed: pipe diameter D, absolute roughness of pipe k, mean

particle diameter dp' density of solids Ps' density and kinematic

v racossit.vof carrying fluid Pf ~~v. To outline the idea, the

calculation procedure is shown as following charta

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Comment of the Chart:

1- from ref. 8 & 9

'? _{from equation (8)}

.L. •

"Z _{from equation (7)} "_

.

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4. from Fig. 1

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Data should be known:

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D d

_P

_s

_Pf

"

P

V

g r---___,,/ v

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Data for Design Parameter: .(Wf)<) 3 .Frp}~/rel)~(:lÇ)V . 0

fr* V 5 _{(dx )}

*

4

% (x ) 6P dC v _~(~) toL .C C C -C v v 'v v

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Comment of the Chart:

5. from Fi9 . 2

6. from equation (4)

7. from equation (5)

8. from equation (3), the additional part of pressure

drop is obtained.

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Thus with equation (1), the total pressure drop of the

mi;-~tureflow tlPm can be obtained. i.e.

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tiP

m tiL

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With the help of a computer, the dependenee of hydraulic

gradient against volumetrie eoncentration of solids with the average suspension velocity as parameter may be obtained. This

program is designated as program HY. In order to present the head

10ss as a funetion of the velocity directly the programs HYP1

&

HYP2 have been written. These programs are provided at the end of

paper as appendixes.

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The Comparison with Recent Measurements

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To eompare the new calculating method, some diagrams based on

experimental data of hydraulic gradient against velocity with the

fraetional volumetrie coneentration of solids as parameter have

been selected from recent literature (ref. 3,10,12).

It should be noted that the <New Concept> is mainly based on the use of moderate velocities such as in industrial

applications. Therefore the eomparison is aecordinlgly restricted

within a limited range of velocities, just as mentioned in the

<New Concept>. The following region will be excluded in the camparison. i.e. the region of high velocities causing usually high pressure drops, particle attrition as weIl as pipeline wear

such velocities are normally avoided in practice. The region of

very low velocities is not of interest for the practice. The two

regions mentioned are shown in Fig. 1.

In each figure, the lowest velocity is indicated with the

sign (+> (Fig. 3 to Fig. 20) which means that below this velocity

the data are out of region as plotted in Fig. land regarded as

not relevant to the correlation obtained by the authors. The comparisons are earried out for materials as follows.

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_{Coal (s}

₌

_1.42>

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Fig. 3 to 10, prediction from <New Concept> and experimental

results are depieted for different pipe diameters, particle

sizes and eoncentration of solids. In the case of small particle

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size d

=

0.14 mm) and small pipe diameter (D

=

8.07 cm) as weIl

p

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as not too high concentration~ up to 41.4% in Fig. 3, the

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correlation is in suitable agreement. However, as a general trend

the predicted values are somewhat higher than the experimental

results.

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Iron Ore (s

=

5.07)

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Figs. 11-18 are related to iron ore data, which are also

depicted for different pipe diameters, particle sizes and

concentrations of solids. ln the main region of practicle

velocity Cv

>

1.5 mIs), the correlation between the prediction

and experimental results of iron ore seems not better than that

of coal. However, in this case the predicted values are lower

than experimental results. Furthermore, the correlation becomes

poorer when the concentration is higher.

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_{Sand (s}

₌

_2.65)

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Figs. 19-23 are related to sand data which are depicted for

two different pipe diameters (D

=

78.8 mm

&

D

=

108 mm) different

particle sizes (d

=

0.15 ~ 1.25 mm) and concentrations up to C

=

P v

25%. In general, it can be concluded that the correlation bet ween

the prediction and the experimental result is satisfactory, but

when the concentration is high the predicted values are higher

than those of the experiments (See Fig. 21).

Compared with the results for coal and iron the agreement

between theory and experiments in the case of sand is more

satisfactory. This may be caused by the fact that the <New

Concept> is mainly based on flow of sand-water mixtures.

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As there are 50 many methods for the calculation of the

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pressure drop in hydraulic pipeline transport of solids proposed

in literature (ref. 11), a comparison between them is shown in

Figs. 24 and 25 in which some results for a speeific case are

presented. The transport of a sand-water mixture is considered (D

=

200 mm, s

=

2.65, Cv

=

1O'ï.., d

=

O.5 mm & d= 1.5 mm) and a

plot is made of the pressure drop versus veloeity.

A general observation is that the results obtained from the

various methods differ mutually very much (ref. 11).

Comparing the results obtained by the <New Concept> two

points can be noted. The prediction coincides with none of the

other predictions and it is lower than all other predictions.

Therefore it is doubtful as to whether the <New Concept> is a

real improvement with regard to the other methods.

Nevertheless, the predicted trends do not look unrealistic

and taking account of the fact that the problem under

consideration is governed by so many parameters, such as the pipe

diameter, roughness, fluid density and viscosity, particle

density, size distribution and shape, particle eoefficient of

restitution, solids mass flow rate, fluid mass flow rate and

gravitational acceleration, by which the friction head loss

gradient should be affected, this new method can be regarded as a useful attempt to provide a physical basis to the pheriomena

observed.

Some points for further discussion and research are as follows.

1. The value of mean particle diameter dp can be computed

in different way, but the authors do not indicate which definition they use.

2. The method applies to particle size distributions which

are not too broad. It is necessary to give the limit of the

particle size distribution up to which the method is applicable.

3. At last it should be questioned as to whether it is not

better to develop methods based on fix density of the solid material, such as sand, coal, iron ore etc. even though it will lead to more correlation figures such as Fig. 1.

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11 Ackr;(JvJledgement

I wish to express my sineere gratitute to Prof. Kalkwijk,

J.P.Th. of Delft University of Technology who taak many times to

make constructive suggestion and useful advice, and ta ir.

FontiJn, H.L. ir. Slot, R.E. and Mrs. Capel, T. from whom I

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F:eferences

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<" "_"

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4.

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6.

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7.

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1

..L •

o

.

Molet-usand P. Wellman: "A new concept for the

calculation of pressure drop with hydraulic transport of solids in horizontal pipes". Chemical Engineering Science, vol.30, No.10, pp.1623-1632, 1981.

Babcock, H.A.: "Hetet-ogeneousflow of heterogeneous solids".

paper 8 of "Advances in solid-liquid flow in pipes and its application". ed. I. Zandi, Pergamon Press, 1971.

S. Olba and K. Ota: "Analysis a slurry pressure drop by

various fluid models". Hydrotransport 9, paper 02, B.H.R.A. Fluid Engineering, Oct. 1984.

T.L. Thompson and T.C. Aude: "Slurry pipeline design, research and e~·:perience".Journalof Pipelines, 1 (1981), 25-44.

J. Boothroyde, B.E.A. Jacobs: "Coarse particle hydraulic transport". Hydrotransport 6, paper El, B.H.R.A. Fluid Engineering, Sept. 1979.

K.C. Wilson and W.E. Watt: "Influence of particle diameter on the turbulent support of solids in pipeline flow".

Hydrotransport 3, paper 01, B.H.R.A. Fluid Engineering, May 1974.

K.C. Wilson: "Oeposition-Limit nomograms for particles of various densities in pipeline flow". Hydrotransport 6, paper Al, B.H.R.A. Fluid Mechanics, Sept. 1979.

E.J. Wasp: "Solid-Liquid Flow Slurry Pipeline

Transportation". series on Bulk Materials Handling, vol.1,

(1975/1977), no.4, pp.33-37.

Q.L. Tong: "The theoretical basic of two phase flow". North-East University of Technology, China.

:0.

N. Hisamitsu,

Y.

Shoji and S. Kosugi: "Effect of added 8.

9.

fine particles on flow properties of settling slurries". Hydrotransport 5, paper 03, B.H.R.A. Fluid Engineering, May 1978.

11. M. Sasic and P. Marjanovic: "On the methods for

calculation of hydraulic transport and their reliability in practice". Hydrotransport 5, paper A5, B.H.R.A. Fluid Engineering, May 1978.

E.J. Wasp, T.C. Aude, R.H. Seiter, T.L. Thompson: " Hetero-Homogeneous Solid/Liquid Flow in the Turbylent Regime". 12.

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13. 13

paper 13 of "Ad'vances in Solid-Liquid Flo~~Jin Pipes and

It's Application". pp.199-210, ed. Iraj Zandi.

living Granet P.E.: "FILüd Mechanics of Enginee ...ing

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0 10 7 '5

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("~elL

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(i7' '5

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2 (i2

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7 S

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2

.r3

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57

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2 ~ 7

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'5 0

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14 .2 .,2 ~r •g.D11.If. .rl • f 90 ':2 '\ --- ~ 'Cf'

-~~~~~~~~~~~~~~~~~J~~~~$~

--- --- --- __ ~'f~

---~!'~--- ---~!'~---~!'~---~!'~--- --- - - --- --- - - __ .l'!'"!._ ~--- ---- --- - - __ - la!,:_ 2alli'

---:

::: :.:-.:---_-:::

=-:~::':::~-_J~:"

--- i-!l!>__ 2.-Ó!>

---;:.ii~-·

-_

-_

-

---

---

---~---_--.:::.:

-_-~

_;~~-_:

--- ---- --- ---- -- - - __ .1•..!P~_

---,._y;_"--- !i.!'~~_.

---

---

---.l!L_

--- ----

---

-

---- --- -

---30 50 60 70 80 20 110 130 _{140 Fr 'IS{} p

Fig. 1.State diagram for hydraulic transport.

(After O.Molerus - 1981;ref.l)

'0·",.--- 1 dX' cIC" 102 '0 ~ 0 r.

...

1wJ/"') 0 12 27 26~0

•

2~ 27 26~ " .0 n 26~ .. t7'!i 101 26~ 0 _{10 27 2BO} • 175- 1~ 26~0

·

120 n H~O + 175- ~2 26~

..

J2 ~ H~

•

32 lOl ~2~0 D )2 20!! ~2~

•

32 )1~ ~2~ 85200 160 1210

•

·4 10 + io .7 10

Fig.2 lncrease d,a'de, in the nondimension ..1.additional prevsure drop a\ arunction of Fr"

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15

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Coal dso=O·14mm 5=142 O=00807m k=0028mm Tw=14.3"C Cv (0/0) o 26.7 o 32.1 A 355 6 41.4 b 53.6

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v.;

(mis)

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F~g.3 Pressure drop data for coal slurry.

{dSO=O.14mm, D=O.0807mJ (After S.Oba - 1984iref.3)

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l000·.---~

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Coal d50=0.14mm 5 = 1.42 E O=01053m k=O.040mm

-

Cv(0'.) Tw=14.6"C

Ë

_{o 28.3} Ol D 32.4 x: A 38.1 __J 6 43.3

--o, _{6 50.5} "q

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V

m (mis)

F1g.4 Pressure drop data for coal slurry

IdSO=O.14mm, O=O.1053rn)

(After S.Oba - 1984iref.3)

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16

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1000~---

__

Coal dso=O.22mm 5=1.42 0=O·0807m k=0.039mm CvC.'.) Tw=11·0·C 0269 el 32.1 6 37.4 6 42.6 ia 49.3

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-Ol ::r::: -.

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10

V

m (mIs)

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Fig.S Pressure drop data for coal slurry

(d50=O.22mrn, D=O.0807m) (After S.Oba - 1984iref.3)

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1000~---

~

Coal d!tO=O22mm 0=0.1053 m Cv(·'.) o 27.1 13 32 1 A 38.1 6 44.8 B 51.2 5=1.42 k=O.046mm Tw=12.6"C

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1.0

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_v.«

(mIs) 10

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Fig.6 Pressure drop data tor coa~ slurry (d50=O.22mrn, D=O.1053m)

(After S.Oba - 1984iref.3)

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17

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_l000

_,

_r---

_~

Coal d\O=O38mm 0=0.0807 m Cv(·'.) 028.1 11 32.9 • 39.3 é 42.9 6 47.4

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5= 1.42 k=0.062 mm Tw=14.5"C E ~

-Ol ~

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m (mis)

10

Fig.8 Pressure drop data tor coa~ slurry (d50=O.38~, D=O.lOS3m)

(After S.Oba - 1984;ref.3)

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l0001r---

~

Coal dso=0.38mm 0= 0.1053 m Cv(·'.) ei) 27.9 D 32.1 A 37.9 6 42.9 é 48.8 5=1.42 k=0.056mm Tw=13.8·C

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10~--~----~~~~~

__~ __~ __~~

01

1

.

0

10

V

m (m/ s)

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Fig.7 Pressure drop data for coai slurry

(d50=O.3~, D=O.0807m) (After S.Oba - 1984;ref.3)

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18

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1000r---__--

~

Coal dso=0.70 mm 0=0.0807 m CV (·I.) e 31.4 Cl 371 A 41.9

e

57.4 5=1.42 k=0.101 mm Tv.= 28 9·C E

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-Ë

-Ol !IC ....I

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-Q_ ~ A A

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10

V

m (mIs)

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Fig.9 Pressure drop data for coal slurry(dSO=O.70mm, D=O.0807m)

(After S.Oba - 1984;ref.3)

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lQOOr---,

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10~--~----~~~~--~~~~~~~

0.1

10

V

m (mis) Coal dso=O.70 mm 0= 0.1053m

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E ;:--E

-

Jl

-Q_

"'"

Cv(·I.) e 30.7 A 36.9 6 43.6 B 49.8 5=1.42 k=O·087mm Tw=28.5"C

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Fig.l0 Pressure drop data for coal slurry

(dSO=O.70rnrn, D=O.1053rn)

(After S.Oba - 1984;ref.3)

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19

1000

r---

--n Ir on OrE' dso=24)J 5=507 E _{0=0-0807 m k=0.084 mm}

i

_Cv(·'.) Tw= 7 9·C

_""

Ol o 17_9 ::.:: B 21.3 __J I:J. 256

-a...

0 308 ~ di _37.1 Vm(m/s)

10

Fl.g.ll _{Pressure drop data for iron slurry}

(dSO=24jA. , 0=0. 0807m)

(After S.Oba - 1984:ref.3)

1000

r--- __

ï 5=5.07 k=O.064 mm Tw= 8.2"C

a...

"'I

100

Iron OrE' d.,o=24)J. 0= 0.1053 m Cv (.,.) o 17 9 D 21.3 A 25.7 e 30.9 ti 37.0 __J

-V

m (mis)

Fig.l2 Pressure drop data for iron slurry

IdSO=24).l I 0=0.1053m)

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20

l000r--- __

Iron Ore d'W)=35.LL 0=O.0807m c"C·'.) ct 18.9 • 22.4 A 28.5 é 35.0

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~

-

E Ol :.r:::

-I

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5=5.07 k=0·134mm Tw=33.2"C

V

m (rnrs )

Fig.l3 Pressure drop data for iron slurry

(d50=35~ , D=O.0807m)

(Af ter S.Oba - 1984;ref.3)

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1000~---~

I ron Ort> dso=35,1.l 0=0.1053 m C,,( .,.) G 18.7 1122.3 A 28.7 6 35.6

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5 = 5.07 k= 0·105 mm Tw=33.9·C

lO

10

V

m (mis)

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Pressure drop data for iron slurry

(d50=35~ , D=O.1053rn)

(Af ter S.Oba - 1984;ref.3)

Fig.14

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21

1000

Iron Or e d~=45 )J. ₅₌₅₀₇ E _{0=0.0 B07 m} k=O 186 mm

--

_{Tw= 22.6"C} È

--

Cv(·'.) o 1 7 1 0\ X 11 20.9 _J A 27.1

-

ti 32 9 o, ~ _{lil 39.4}

10~--~----~~~~~

__

-L ~ __ '_~

0

.

1

_lO

V

m (mIs)

10

Fig.lS Pressure drop data for iron slurry ldSO=4S,IA, 0=0. OS07m)

(After S.Oba - 1984iref.3)

1000~---

~~

,

Iron OrE' d~=45.lL 5=5.07 "10 E D=0.1053m _{k=0.136 mm} <::>

--'1

Tw=23.5't

--

Ol _CvC·'.) x 0 _18.3 _J _a 22 2

-

6 26.5 o, ~

100

ó 32.8 é 40.0

10

V

m (mIs)

Fig.16 Pressure drop data tor iron slurry (dSO=4S)(, D=O.l053mJ

I

22

I

I

1000r---~

~

I

Iron Ore d!>O=68;t 0=0.0807 m 5 = 5.07

I

I

E Cv (el.)

-Ë

-

_{o 19.1} Cl _{B 23 3} ~ A 30.1 ...J 637.3

-Q.. ~

I

I

I

10~--~----~~~~----~--~--~~~

0

.

1

i

.

o

10

Vm (mIs)

I

Fig,l7 Pressure drop data for iron slurry

td50=68~ , D=O,0807m) (After S.Oba - 1984;ref.3)

I

_1000~---~

I

I

i

E ...J

-Q.. "J I ron Ore d!>O=68)L 0=0.1053m Cv( ·1.) o 19.0 B 24 9 A 30.7 6 38.0 5 = 5.07 k=0·145mm Tw=l8.4"C

I

I

I

I

I

V

m (mIs)

I

Fig.18 Pressure drop data for iron slurry

(dSO=6SP. , D=O.10S3rn)

(After S.Oba - 1984iref.3)

I

---10

3

10

3

8

8

...

6

_{silica sand CA)}

:t

silica sand (C)

~ D=78.8 mm D=78.8 mm ... dp=O.45 mm 0' _~

i

4

dp=O.15 mm _0' K=1.5*E-5 m

3

K=1.5*E-5 m

-<

3

-

-

I

ES

21-

&JII2

'

2

'"

a

."

E-4

e

10

2 E-4

10

2 ~

8

!

8

AA ~

9.5%-G-I

~

6

~

6

N Vol .~

OCv

•

8.5 "

~

_OCv

_-

_{9.5 •}

=

4

À

Cv • 20.0 "

~

4

6.

Cv - 10.5 •

{/l ~

3

OCv

• 25.0 "

Fi3

:[

o c~

'

- 15.5 •

~

2

V Cv • 20.0 •

10

0.7 1

2

3 4

MEAB

VELOCITY

V

m

6 810

(m/s)

10

0.7 1

MEAN

2

'

~4

6 8 10

VELocljy

V

m (mIs)

fi~. 19

Pressure gradient vs. velocity

tor solid

,A

,

8lurry

(After N.Hisamitsu - 1978;ref.10)

Fig.20

Pressure t.radientvs. velocity

tor 801id

.

C slurry

I

I

24

I

2 :3 4 5 6

I

1.0 0.8 0.6

I

c:::> 0.2 ; i

₁₁

_I I

1

0

I

1f/:l.H. i .IL, ---I

I

O-r- \"; f:I'" :"1',,'')

I

! h- l" ,/1. \. _:' I

.

- , '.!".) ~. '.~J ! _I I a-_r-- L I 1 t,1':",-1":~j.: ,"_ ,.,"" " ..: I

'

V

I -, !IJ..; \ J- ," ':'" i --; I. I.

o

Á

o

-

r+: 1:-; '.":.1 1.0.=4.25 Inches

=

1(,,( "'~

L

I

I

I

I ID-IQ..

VI'

...

_~,

IC. I 1~7o:I_,nu?~

f

.L

04 ~

5% SANO

~

f'

C EAR WATE

! 02

L

V

.01 j_

I

0.4 '"

=

I

V) 0.1 V) ~ 0.0 C> ~ 0.0

=

I

O.

I

O.

o

I

2 _{4 6 8 10} VELOClTY - Ft.:Sec. 20

I

(After E.J.Wasp - ref.12)

I

1.00 2 3 4- 5 6 0.80 0.60

I

I

_{1 1}

CORKEL\TlON- __

J-_r+ . \;f'ERI~~E:-'TAL poe,T5

IJ,- 1.'.:('.·OL.5).\:-':0 •- 5::('·OL.)$.-\:-':0 O- r-Ilhs..-rv.d Dc-r-osirionVclocltv

V

.'i()!;~.; _{\'d-Ft./S....-c•} --'; 10.54

oj

0 r- is 7.2S 1.0.=4.25 Inches=108mIfl

li

n._ _...l!~tjl rt>!• _"'..::

-

_'

_T

_-

_I ; ~ '11.

..

;~~V ~ 5%SI ~O-J

.

.

~~

L

CLEAR ~ ER 4

V

2

j_

V

01 _L

I

0.4

I

~ 0.2 C>

I

_...,

..., 0.1 ~ 0.0 C> ~ 0.0

=

I

0.0

I

0.0

I

o

.

2

I

4 6 8 10 VELOCITY - Ft.!Sec.

(After E.J.Wasp - ref.12) 20

I

I

In 40 m SA"D "A" MEDIA:-'; DIAMETER =0.26MM FIG.21 x o

5% Predicted by new methOI 15% Predicted by new meth

-SA!':D"B"

MEDIA:-i DIAMETER =0.66MM

FIG. 22

x

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

25 1.0 0.8 0.6 t 3 ...5 ,. 0.4 Ift 'u !

₁

11. ,

..

iO-I-- CORREL.\TIO'---EXP:':RI\IL,-"T \1 PUI'-JS

ID-I-- 0-6- 15s-'"~ '\'"OL':UI..).) 5.\'.:D5.\:--:D

V

Ohscrved Depestnon Yclccity .-:Sohds \'d-F<./5<c•

A

5 F\.2i

~o-

I-- IS 8.77 1.0.=4.25 Inches =J'lIJ 111m

77

1%SAH~.>V_~I'" Ift • oÓ; ,

..

5% SAND_f:--l~~

_/

'8

-...

/ '" .u _~v

/

4 ~ c-ILEAR WATIR 2

.L

V

0' ..L ~ 0.2

-

...

:z::

....

'" 0.1

'"

~ 0.0 Q ~ 0.0 :z:: 0.0 0.0

o.

2 4 6 8 10 VELOCITY - Ft.!Sec. 20

(After E.J.Wasp - ref.12)

m SAND"C" MEDIANDIAMETER =1.23MM x

•

FIG.23 5% Predicted by new method 15% Predicted by new method 4D

X OURANO Il. ZANDI 0.20

0

GORJUNOV

•

KRIEGEL

•

JUFIN

IJ

NEW METHOD

I

I

I

I

I

I

I

I

I

I

I

I

.

I

I

I

I

I

I

I

I

I

,.,

·

-

'10

.

15

"0

1!

Ol

.,

L. :;, en en

.,

L. Q, 0.10 0.05 26 0= 200nm

s=

2.65 d=0.5mm Cy

=

10%

o

1 ~O ~ 2 J 4 5

<I> - mean.

ve

t:

oclty

er flow. V (mis)

FIG.24

6

(Af ter M.Sasic - 1978iref.ll)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

27

~30r---

__

0.10 ...---.---..---..._.---.---

_e

...

--~

,..

I ~ . r»

~-

_c

.

•

"Cl 11 &0.20

•

...

::J

'"

'"

•

...

Q. 0.15 0= 200rnm d=15nwn s= 2.65 _C,,=10% X OURAND

A ZAND

I

.• JUFIN

•

KRIEGEL

o

60RJUNOV ~ NEW METHOD

me~n velC?city of f~ow. V (mis)

FIG.25

I

I

I

I

I

28 APPENDIX

*

**

PROGRAM---HY

***

10 PR INT 11

*

**

BAS IC F'HOGPAt1FDf;: NEW CONCEPT CALCULA1ION DF P

RESSURE DROP

**

.

jj-20 PRINT 30 READ G,D,PS,DS,PF,NU,MU. 40 PF<I:NT 11 ( 1) FLUID PART F'RES:3 URE DROP Pl" 50 INPUT V,K 60 RN = D

*

V / NU 70 FO = (20000

*

K / D + 1E6 / RN ....0..33333 80 F ; 0.005 + 0.005

*

FO 90 F = INT (F

*

10000 + 0.5) I 1 0000

I

I

I

100 _{Pl - F}

*

_PF

*

_V ••••• "-1 i ...:.:. .I C2 .l( DJ 110 Pl - INT (Pl

*

1000 + 0.5) ! 1000 120 PRINT 111.)="; V 130 PRINT 140 PRINT TAB( 10);"Fa-.j=:";RNTAB; ( 20);"F=";F; TAB( 30);"F'l::-.:";F'

I

1

I

150 160 170 PRINT PRINT PRINT " PARTICLE F{üL V

I

180 PRINTILOCITY W" 190 Bl

=

(PS PF) 200 82

=

(PS PF) 210 W = 54.5

*

DB 220 W

=

INT (W

*

/ 1 / t1U / PF ....2

*

Bl 1000 -I 0.~,)

I

I

I

I

000 230 RN

=

W

*

DB

*

PF ! MU 240 RN

=

INT (RN

*

1000 + 0.5) I 1000 IF RN

>

1 GOTO 280 PRINT "RN=";RN,"Lr4t'1.W=";~": GOTO 420 W = (3.33

*

OS

*

82) ~ 0.5 W = INT (W

*

1000 + 0.5) , 25() 260 270 280 290 ₁

I

000 300 RN

=

W

*

os

*

PF / MU 31(I RN; INT (RN

*

1000 +

o

,

5) / 1000 320 IF RN

<

1000 GOTD 350 3:30 PRINT "RN=";RN."TUF~. [;J=";~1J

I

340 GOTO 420 350 B3

=

(4

*

G

*

PF

*

OS 1) (:::. -1!' t'lU) 360 B3

=

INT (B3

*

1000 -I 0.0

I

1(,(iC) ~;7() F'F:IhIT iIE{2:::::::11;D::.~;!L.CJCi;::: FC)~:,. ~:-:IC:;

I

:380 I:'.jpUT S:,\l 390 W (RN

*

MU)

I

I

4j(; '+20 '130 440 ,-~~-.:.;(, ';1hO 470 480 4':-;0 5()() 51.o F'F.:I

r·rr

"F.:N:e:"; Fd'J. "TFU~iN. (LJ="" ~ [ij F'F.I N:T" ( ~:) i=POLiD t'lUM::"=:F~r '.) (r"[)) 1\ f"FINT ,.:.oF':J ["fT "V=-":.

v

,

"1,,')::11; ("I F'F - 1., .;; D'3 CJ) Cl • ~) FV ~ INT (FV

*

1000 + 0.5) l(;()n FW

=

W / (CPS ' PF - 11 ~ G 1_J,-·\

F

'

r

~

:I

rrr

FD :::= F·l!.! ~::: FD

=

INT (FD

*

lE9 + 0.5) / 1E"? F'F.: I

ur

"F\!::::", F'v', "FTi:=:'" :;FI) F'P

rnr

F'P I h!T "'~Hi-~·**·,,·L_DUf:: F DF: FIC:;...

TD f3ET 'Vl·~·)f'.:t-i·"··il1I r'G' I N'T F'h:I tfT " (4) COMPUTE AU. PRI f;E;I.JF;:E: r)F~iJF' F'~,2I1 5~30 Ph:I hl

r

~.5bO I rlF'UT '(:[ .:~7() FOP C

=

0 TO 0.25 STEP 0.05 P2

=

XO

*

L

*

(PS - PF)

*

G ( Ir) I-), ) P2

=

INT (P~ ~ 10000 + 0.5) 100(,\) F' oe F' i -I- P>: FF(INT TAB ( 2):: C~ li':iB( LO) ~i 2; Tr;B ( :'::;0):. F' NE\T FDR C - 0.3 TO o_~ STEP 0.0' P2 = X

*

C

*

(PS - PF)

*

G .::~ P2

=

INT CP2

*

1000 + 0.5) 10(H) 6 f=;(:0 F' :::: F'1 + F'':::' 6Ei'.".: F' =::0 II'·iT .:F

*

:lOOO : U~ '.," ()()() '. '., . ;-·.11:': \. ! _,::!.,-:, F';:.~J~h.lr

I

I

I

I

I

I

29 *** PROGRAM---HYPI

*

*

*

10 F'RUH "***BAS IC PF.:OGJ;:At1FOF; N

EW CONCEPT CALCULATION OF PR

ESSURE DROP***"

20 PRINT

30 READ G,D,PS,DS,PF,NU.MU,K

40 PRINT" (1) PARTICLE FALL VI

50 LOCITY W" PRINT BI = (PS -- PF) ./ MU 82

=

(PS - PF) ! F'F W - 54.5 1(. De' .. .-

_"*

_.

BI

~.

..;_

I

I

I

60 70 80 90 W

=

INT (W

*

1000 + 0.5) 10 00 100 RN

=

W

*

OS

*

PF ! MU 110 RN

=

INT (RN

*

1000 + 0.5) 1000 120 IF RN

>

1 GOTO 150 130 PRINT "RN=";RN,"LAt1 ~"'=";W 140 GOTO 290 150 W

=

(3.33

*

OS

*

82) ~ 0.5 160 W

=

INT (W

*

1000 + 0.5) / 1 000 170 RN

=

W

*

OS

*

PF / MU 180 RN

=

INT (RN

*

1000 + 0.5) / 1000 190 IF RN ::1000 GOTD 220 200 PRINT "RN=";RN,"TUR. W=";W 210 GOTO :::::90 220 B3

=

(4

*

G

*

PF

*

OS A 3

*

B 1) / C:!: -jf. t1U) 225 B3

=

INT (B3

*

1000 + 0.5) 1000

Z::;:O PRINT "B3:::::"; B:3, "LOCH:: FOR FIG

.3-2(REF.8)" 240 INPUT RN 250 W = (RN

*

MU) / (OS

*

PF) 260 W = INT (~'"

*

1000 + 0.5) / 1 000 270 PF~INT "F.:N11=; RN, "THAN. l!'):::::['-i "; 280 PRINT

I

I

I

I

I

I

I

I

I

I

I

I

>:'7'0 f'P I t~ T" (2:.) FFWUDE [-.tUt'1E:Er:: .~-: FLUID PRESSURE DROP;FV.FD.P~ " ~)(l(i PHJt-..!

r

"D'''''; D;" K"c'" ; f:. 31U P!={IN!"!Y3='''~DS:;I'\!Jc::",l!.i ~!·~::O PF.:H~T Tf:'JB( 2); '''Pi; TP.[-:( .~.); I1FIV'I!; -,.f~E: ( 1~.:j); I1F'I) I1~

-r

':::i E·:i: ::~t :;; "F'111; T?:lE: ( .:~8);" \ 11I 330 FOR V

=

100 TO 600 STEP 100 :~;40 FI-,) ::e \! _{( (F'f.:; j) .~. IY3} 3~.iCl F~'::~:::.:: 10000 2~;L.C Ft,.i :~ ~!j Di O. '.5 (F'S :370 F!) ::c: r:c(t...i 2 :3BO FD:c: It.IT (FD iri- :l 1:'=:";'+

O

.:::

'

.

i

1F_'-? 390 RE

=

D

*

V / NU 400

F

a

=

(20000

*

K _U _I lF.~. h: C:', j-', .~.-:~•.•~.•-::.•-;!" 1_.•" '.-' " -._'-,-' ',-' '_"._' F- :=-~ _{1III (F .~. 1!)()(i(j .}1_.. i', ,::::.., ...' " :J.(;('00 430 Pl - F

*

PF

*

V :.:.:: -,. !).i 'l4U F'1 10UO Jin (F'l .)j. 1000 + (."I.~,", /

i]./ (l _{G,D,PS,DS.PF,NU.MU,K}

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I