Terminal velocity formula for spheres in a viscous fluid

TERMINAL VELO CITY FORMULA FOR SPHERES IN A VISCOUS FLUID

R.E. Slot

Report no. 4 - 83

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

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TERMINAL VELOCITY FORMULA FOR

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SPHERES IN A VISCOUS FLUID

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R.E. Slot

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Report no. 4 - 83

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LaboratoryDepartment ofof Fluid Mechanicscivil Engineering

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Delft University of Technology

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Contents page

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Introduction

General Velocity Formula Fall Velocity Formula

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Sedimentation Diameter Cd - Re diagram Conclusions References 6 7 7 8 9

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'-1 -INTIWDUCTION

Various attempst have been made to develop a general expression for the

terminal velocity of spheres in a viscous fluid (Stokes, Prandtl, Oseen, Rubey, etc.: see Bogardi,

1974

and'Vanoni,

1975).

All of these formulae

show a lack of accuracy and/or are restricted to a relatively small range

of Reynolds numbers.

Since sieving is,a time consuming method for determining partiele s~ze the interest 1n the use of settling tubes has increased as a mean of

particle-size analysis. A measure of the S1ze of an irregular shaped

partiele is the sedimentation diameter defined as the diameter of the

sphere that has the same density and the same terminal settling veloeity

as the given partiele in the same sedÏ1nentation fluid. It is therefore

very convenient to have a general and accurate expression for the

con-version from settling velocity to sedimentation diameter.

,

In the folLowi.ngnote new formulae for the terminal velocity as weIL as the sedimentation diameter are derived having a high aceuracy and a wide

range of validity.

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_

G

EHERAL

V

ELOCITY

fORHULA

For small Reynolds numbers there exists a linear relationship between

the acting force and the terminal velocity of a sphere (Stokes),

whereas for large Reynolds numbers this relationship is quadratic (Newton) .

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.-2

-and the terminal veloeity w may be written as

(1)

from which foll.ows

w (2)

where a and

S

are functions of the physieal quantities involved.

These quantities are the density pand the kinematic viseosity v of the fluid, the diameter d of the sphere as weIl as the aeting force F.

The funetional relationship between the terminal velocity and the

above-mentioned physi.caI quantities can be wri.tten as

w:= f(F,p,v,d). (3)

From dirnensional analysis it follows that w ~ f(_F__ cl (4) )

.

Frorn eqs. 2 and 4 it follows (5) "

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.-3 -and ) (6)

where fa and fS are functions bf the dimensionless quantity F/pv2 (the factor 3n has been added for convenience).

. .

Now the aim is to find relatively simple functions fa_and fS which wili fit the.ex~erimental results. Simply assuming f =1, it turns out

a

that a simple function fS can be found wh ich satisfactorily fits the experimental data. This function is of the form

-1/3

) (7)

1 )

where Co and Cl are constants For the calculation of Co and Cl it

1S neeessary to express fS as a funetion of the relevant physieal

quantities. Substituting eq. 5 (f =1) and eq. 6 into eq. 1. it a is foul1dthat (8) whi.chcan be rewri.tten as f

S

1 24 24 ( Cd - Re ) (9) "

where Cd is the drag eoeffieient (deL: :F=Cd~pw2brd2)and Re=wd/v (Reynolds number). So fS ean be interpreted as a new drag coefficient proportional to the difference between the real drag coefficient and

the drag coefficient according to the Stokes equation.

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The ~eQson for the exact value +]/3 for the exponent is the possibility to

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4-Using eq. 9 fS has been calculated from experimental data (Allen, 1900; Liebster, 1927; Gibbs et al, 1971) and plotted versus (F/pv2)-1/3.

Fi~. shows that this relationship is almost linear for large Reynolds numbers (small values of (F/pv2)-1/3). Although for small Reynolds

numbers the scatter in the experimental data is rather large, it appears that this scatter will also average out at a more or less linear relationship

(it will be showu hereafter that for small Reynolds numbers a relative

error in fa has less influence on the relative error in the velocity

w than for large Reynolds numbers).

In addition to the experimental data ~n Fig. 1 two curves have been

drawn from tabuLated data (Schiller et al, 1933 and Weast, 1975) which,

of course, should be the average of the experimental data. Nevertheless the deviation between both curves is obvious. The tabulated data from

Schiller seem to be the best fit to the experimental data. For this

reason and for convenience these numerical data (Table I) have been

used for thc calculation of the constants Co and Cl in eq.

7.

Substituting eqs. 5 (f =1) and 6 into eq. 2 the velocity becomes

a v [-w=~~ -I + (I 2df

S

(10)

4

F

+-~, -3n 2 pv

For the relative error ~n w due to a small relative error ~n fS it

follows from eq. 10 t:.w w ( 1 1) where " kf S ₍₁₂₎ f

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,-5 -with k=4F/3np

v

2=C

d

Re2

/

6

.

The values of f are g1ven 1n Table I, except

.for Re=O.1 because eq. 11 does not hold here due to the large deviation

between the Schiller data and eq. 7. Table I shows that the values of

f for small Re are .much smaller than for large Re, sa the influence of the scatter of fS (see Fig. I) on the value of w is much less for small Re.

The values of Co and Cl are chosen such that the absolute value of

the largest rel~tive error in w (according to eq. I I) is minimum in the range Re=O.2 to Re=2000. The relative error in fS has been calculated as the quotient of fS according to eqs. 7 and 9 respectively minus ane.

It 1S found that Co=O.OI25 and CI=O.348 are optimum in this sense and far

these values the relative error 1n w 1S given in Table 1.

The relative error 1n w is less that 2% for Reynolds numbers ranging from

Re;:::O.2 to Re=2000, however, for Re=O.1 the error is 3%. On the other hand the accuracy of the Schiller table is not kno,vn. With regard

to t.lie experirnentaldata (see Fig. I) the steep descent of the Schiller curve does not seem reaListi.c, Ta gi.vean impression of the dependence of fS on measuring errors two curves have been drawn 1n Fig. I which

i.ndicate the error band of fS due to allerror <1% in w, F, p, \)as weIl,

as d (see eq. 8).

It has to be emphasized that the optimum values of Ca and Cl are based on the Schiller tabie; they are not necessarily optimum with respect

to the experimental data.

A

further discussion in this respect and

about different functions fa and fS' however, is useless as long as the

accuracy of the experimental data is not an order of magnitude higher.

Using Re=wdZv, eq. 10 ean be writ ten as

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(13)

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.-6 -where fS=O.OI2S+0.348(F/pv2)-1/3. In fig. 2 Re has been plotted versus

F/pv2 according to eq. 13 in addition with experimental data and

various other formulae.

FALL VELOCITY FOilllULA

When a sphe re is settling 1.11 a'fLui.d the acting force F is caused

by th e field of gravitation

(14)

where p

=

uensity of sphere anel g=acceleration of gravity. Af ter

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substit.uti.on of eq, lL. into eq , 10 the fall velocity formula becomes

w

.:»:

[-1

+ (

1

+

2

t,gd3fa)

~J-2df (3 9v2 IJ

(15)

where I:::.=(ps-p)/pand f13=0.0125+0.348 (6v2/1Tl:::.g1/3d3). This fall velocity

formula is of a similar form as the one given by Gibbs et al.(the

diameter d figures in the same way, but v, I:::. and g do not). The

objection to the Gibbs formula ~s that it contains dimensional "constants" 2)~

The validity of this formula will not be general; it will only hold under the conditions of Gibbs' experirnents (that is for certain values of v and 6.). So the calculated fall veloeities tabulated by Gibbs for various water temperatures (i.e. viscosities) cannot be entirely correct.

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2)

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SEDlMENT1I.TION.DIAHETER

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Sometimes fall velocity analyses are used to obtajn the sedimentation diameter of the partiüles~ Knowing the fall velocity w the sedimentation

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diameter can be solved from eq. 15 g~v~ng

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( V

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v?

I/ 3)} ~

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- + C (--) w . I 1TD.g (lei)

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F'roru eqs. 7 and 9 it fo Ll.owsthat the relationship between the drag

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coefficient Cd and the Reynolds numb er Re is given by an algebraic

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equation of the fourth degree in Cd

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B ( 17)

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3 2

where A=24(Co+I/Re) and B=(48CI) /1TRe with Co=0.0125 and CI=0.348.

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Eq. 17 has an analytical solution in terms of radicals (Reichardt, 1968),

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so ~n principle Cd is solvable although the solution is somewhat

complicated. In fig. 3 Cd has been plotted versus Re according to eq. 17

together with experiroental data and various other formulae.

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As is generally known, for Re>2000 (Newton region) the experimental

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Cd-value is almost constant (about 0.4). In this region eq. 17 deviates

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from the experimental data because the calculated Cd-value approaches a somewhat lower value (24C =0.3) for a much larger Reynolds number

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(about 105).

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-8-CONCLUSIONS

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relatively simple analytical expression

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been derived for the

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dteenrsmityinal vpaenlocity w od kinematic vf sphiscosityere withv, dudiamee to an ater d in a viscting forcouscefF.luid with

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This expression ~s

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21/3

where f

S

=0.012s+0.348(F/pV,) . The accuracy of this formula is better than 2% (as far as it can be checked against experimental data

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with_{to 2000}erro_. rs of the same order of magnitude) for Reynolds numbers up

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In settling tube analyses, where the acting force is caused by

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the field of gravitation, the terminal fall velocity is given by eq. IS. It is possibl~ to rewrite this equation as an explicit expression for

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the_Furth(se_ed_rmoreimentatio_{with know}n) diameter of a_{n Reynolds number}(non-sphe_Rerica_{(or veloci}l) partic_{ty w)}le (eq. 16)._{the drag}

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coefficient Cd (or acting force F) can be calculated.

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- -9-Rl~FERENCES

Allen, H.S., 1900. The Motion of a Sphere 1n a Viscous Fluid. Phil. Mag.,

vol. 50, no. 5: 323-338 and 519-534.

Bogardi,

J

.

,

1974. Sediment Transport 1n Alluvial Streams. Akadémiai

Kiadó, B~dapest: 55-BO.

Gibbs, R.J., Hatt.hews, J:.1.Da.nd Link, D.A., 1971. The Relationship between

Sphere Size and Settling Velocity. J. of Sed. Petr., vol. 41, no , I: 7-1B.

Liebster, H., 1927. Ueber den Hiderst~lndvan Kugeln. Ann. der Physik ,

vol. 82, no. 4: 541-562.

Reichardt, H., 1968. Kleine Enzyklopädie/l1athematik. VEB Verlag Enzyklopädie,

Leipzig: 108-113.

Schiller, L. und Naumann, A., 1933. Ueber die grundlegenden Berechnungen

bei der Schwerkraftaufbereitung. Zeitschrift des Vereines deutscher Ingenieure, vol. 77, no. 12: 318-320.

Vanani, V.A., 1975. Sedimentation Engineering. ASCE, New York; chap. Ir.

Weast, ·R.C., 1975. Handbaak of Chemistry and Physi.cs. CRC Press, Ohio:

table 5-26.

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·7 .6

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1'\

SchilIer toble \ \ \

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C>Allon Iporoffine inonlline, oir bubble In water'

• Liebsltr !stetl Inwoter l • Glbbs!quorlz In wc terl o ·5 _{o_} o ·3 -G' _ o _ -Hondbeek olCh.cnd Ph.• """ 0~----r----.~2---.3r----.~4---.5r---t~0~~;~~--'-=r---~~~-.9r---~1-(-F~~ ~- PViJ 10 2 .1-Re

Fig.l. f

S

=(Cd-24/Re)/24 versus the dimensionless quantity (F/pv2)-1/3

Re

t

10'

o AII.n Iparolline inonilin., air bubbl .. In wolorl

• U.b,ter Ist ••1in wot.rl • Gibbs !quortz In wol.rl 10 1Ó2~----~-----r----~----~------r---~----~ 10-1 1 10 102 103 104 10S 10&...E.. -PV2

dimensionle~s quantity F/pv2

Fig.2. Reynolds number Re versus the Cd

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103 for spheres 10

,,

,,

.r~ o~

....

o AII"n (porollin. incniline,oir bubble in woter'

• Llsbst..rIslul In wotHI • Gibbs (quartz-In water' - - - __ 0500n " "-

....

'~--...-; - - - - - -Rubey 1

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Re

_Cd CdRe 2 0.1 241 2.41 0.2 126 5.038

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.tI 64.8 10.37 0.6 44.3 1'5.92 0.8 33.8 21.67 1.0 27.6 27.6 2.0 14.9 59.6 4.0 8.32 133.3 6.0 6.06 218 8.0 4.89 313 10.0 4.15 415 20.0 2.62 1 044 30.0 2.04 1 840 40.0 1.74 2 776 "'-50.0 1.54 3 840 60.0 1.40 5 040 70.0 1.30 6 345 80.0 1.21 7 765 90.0 1.J5 9 290 100 I.09 10 920 200 0,806 32 200 300 0.684 61 550 400 0.612 97 900 500 0.563 140 700 600 0.526 189 400 700 0.498 243 800 80,0 0.488 310 000 1 000 0,1162 462 000 2 000 0.410 1 644 000 Schiller table

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--

-

-

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-( ,,2)-1/3 f·

F/p\)

B

f 6H/W(%) 1.019 0.OLd67 ----- -3.0 0.7966 0.2500 0.04544 -0.75 0.6262 0.2000 0.06898 -1.0 0.5428 0.1792 0.08837 -1.2 O.LI898 0.1583 0.1012 -1.4 0.4519 0.1500 0.1154 -1.5 0.3496 0.1208 O. 1629 -1.7 0.2673 0.09667 0.2182 -1.8 0.2269 0.08583 0.2536 -1.7 0.2011 0.07875 0.2788 -1.3 0.1831 0.07292 0.2966 -1.3 0.1346 0.05917 0.3512 -0.35 0.1114 0.05167 0.3782 0.42 0.09716 0.04750 0,3957 0.83 0.08720 0.04417 0.4076

1.1

0.07965 0.04167 0.4167 1.5 0.07376 0.03988 0.4239 1.6 0.06896 0.03792 0.4293 1.8 0.06496 0.03681 0.4343

1.9

0.06155 0.03542 0.4382 2.0 0.04292 0.02858 0.4598 1.8 0.03459 0.02517 0.4689 1.2 0.02963 0.02300 0.4742 0.38 0.02626 0.02146 0.4777

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i 0.02378 0.02025 0.4802 -1.2 0.02186 0.01932 0.4822 -2.0 0.02018 0.01908 0.4841 -1.5 0.0'1766 0.01825 0.4867 -1.0 0.01157 0.01658 0.4926 0.30 calculated c;lata "

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