Ieperen, H.J., van

Wageningen Yield 

THE FALL VELOCITY OF GRAIN PARTICLES 
Publication 83 
H.J. van Ieperen 
Dept. of Hydraulics and Catchment Hydrology 
Agricultural University Wageningen, 1987 
2^MP 
CONTENTS 
1 INTRODUCTION 
2 FALL VELOCITY OF A SPHERE 
3 FALL VELOCITY OF GRAINS 
Page 
4 EQUATIONS FOR DRAG COEFFICIENT AND FALL VELOCITY 
5 POWER FUNCTIONS FOR THE TRANSITION REGION 
6 CONCLUDING REMARK 12 
LITERATURE 13 
LIST OF SYMBOLS 14 
APPENDIX 1: Derivation of drag coefficient at different formulas 
for the fall velocity. 15 
APPENDIX 2: Derivation of the equation for the fall velocity in 
the transition region. 17 
1 INTRODUCTION 
The fall or settling velocity of grain particles figures prominently in all 
sediment transport problems and although the concept is straight forward its 
precise evaluation or calculation is not. For implementation in computer 
models an adequate describtion of the fall velocity is required. In this 
note governing equations are presented to predict the fall velocity for 
laminar and turbulent motion and for the transition region. In addition 
simple power functions are used to express the relation between the drag 
coefficient and the particle Reynolds number. The latter relation is needed 
to obtain a direct solution for the fall velocity. 
2 FALL VELOCITY OF A SPHERE 
The fall velocity of a grain particle is a function of size, shape and 
density of the particle and viscosity of the fluid. It also depends on the 
extent of the fluid in which it falls, on the number of particles falling 
and on the level of turbulence intensity. Falling under influence of 
gravity, the particle will reach a constant velocity, the terminal velocity, 
when the drag force equals the submerged weight of the particle. For 
spherical particles the submerged weight is expressed as: 
G = (Ps - Pw> • \ * °3 • S <X> 
where, 
G - submerged weight [M L T"2] 
ps = density of sediment [M L"^] 
pw - density of water [M L"'] 
D = particle diameter [L] 
g = acceleration due to gravity [L T"^] 
The frictional drag force equals : 
FD - CD • \ * D2 • £^L (2) 
where, 
FD - drag force [M L T'2] 
Cj) - drag coefficient [1] 
w = particle fall velocity [L T"^] 
From Equation 1 and 2 the terminal fall velocity can be calculated as: 
W2 . * 1 g D E^lM (3) 
3 CD Pw 
The value of Cp depends on the Reynolds number of the settling particle: 
Re - ?—» (4) 
where, 
Re = Reynolds number [1] 
v = kinematic viscosity [1/ T"-*-] 
The relation between Cj) and Re holds three regions. For laminar flow the 
frictional resistance is only due to viscous forces and Cp varies inverse 
proportional to Re. 
CD = g (5) 
for Re < 0.5 and approximately for up to 1.0. At high Reynolds numbers the 
flow of water takes place under fully developed turbulent conditions. 
Compared with the eddying resistance the viscous forces are negligible and 
Cj) is approximately constant, or 
CD = 0.40 (6) 
for Re > 2000, and up to 105. 
For the transition zone an exact formula for Cj) cannot be given but for 
Reynolds numbers below 10^ (including the laminar flow region) a good 
approximation that can be used is (Huisman, 1973): 
24 . 3 
CD = Re- + jRe- + 0- 3 4 (7) 
3 FALL VELOCITY OF GRAINS 
So far a single spherical particle is considered in a fluid of infinite 
extent. For non-spherical particles with regular shape a correlation with 
the drag coefficient and a shape factor can be obtained. The shape factor is 
expressed as the ratio of the surface area of a sphere of the same volume as 
the particle to the actual surface area of the particle, and is also called 
"degree of true sphericity" 
tf 
AsN (8) 
where 
f = shape factor 
As - surface area of spherical particle 
A SN= surface area of nonspherical particle 
[1] 
[L2] 
[L2] 
Graf (1971) shows graphs of the settling velocity for various regular 
shapes, together with an average for irregular quartz grains (Figure 1). 
i ! 1 
i I i 
! ! 1 
For qua 
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\ 
\ ' ' \ i 
rtz(s5 = 2.65) at 20°C water 
Pettyjohn et al. data 
0 ¥ = 1,000 © Stokes' equation 
(T)V • 0.906 © Newton's equation 
© ¥ = 0.846 © Rubey's equation 
© ¥ = 0,806 © Dota from Mamak 
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Figure 1 Settling velocity versus particle diameter [After Graf et al. 
(1966)]. 
In case of a shape factor of 0.670 (tetahedron) the drag coefficient can be 
calculated as 
CD « i (») 
for the laminar region and 
CD = 2.00 (10) 
in case of turbulent motion. 
Expressions for the fall velocity from which the drag coefficient can be 
derived are given by Raudkivi (1976) and Van Rijn (1985) for laminar and 
turbulent motion while Zanke (1977) proposed an equation for the transition 
zone. Raudkivi also defines the transition zone by a number of points. The 
equations given by Raudkivi are: 
w ~ 663 D2 (11) 
where w is in mm/s and D in mm for D < 0.15 mm, and 
w « 135.5 DV2 (12) 
for D > 1.50 mm (Both equations at 20°C). If meter instead of millimeter is 
used as unit, and for comparitive and dimensional reasons the relative 
density, acceleration due to gravity and kinematic viscosity is 
incorporated, these Equations become: 
1 Ag D2 
w —
 a (D < 0.15 mm) (13) 24.42 v 
and 
w - 1.057 (AgD)1/2 (D > 1.5 mm) (14) 
respectively, with 
A _ PH - Pw (15) 
Pv 
where, 
A = relatively density [1] 
From Equation (3) and (13) it can be derived that in this case for laminar 
motion: 
CD - ^ (16) 
and, similar, for turbulent motion: 
CD = 1.193 (17) 
Van Rijn gives the same formulas as (13) and (14) but uses 18 and 1,1 as 
coefficients meaning that for laminar motion equation (5) is found and for 
tubulent motion: 
CD = 1.102 (18) 
The equation proposed bij Zanke and also used by Van Rijn for 0.1 < D < 1.0 
mm, yields: 
w = 10 V- [{l
 + 0.01 ^ Î } V 2 - 1] (19) 
Which corresponds to (see also appendix 1): 
cD - I (i + §2] 
This Equation results in a drag coefficient of 26.7/Re for small Reynolds 
number, a Cp value of 28 for Reynolds number equal to 1, and of 1.333 in 
case of high Reynolds numbers. This includes a discontinuity at the boundary 
of the transition region if used in combination with the mentioned equations 
for laminar and turbulent motion. In this way an inadequate description of 
the dependancy of the drag coefficient on the Reynolds number is obtained. 
4 EQUATIONS FOR DRAG COEFFICIENT AND FALL VELOCITY 
To deal with the problem mentioned, for laminar motion at Re < 1 the 
equation for the drag coefficient becomes: 
CD - g (2D 
For turbulent motion at Re > 250 a constant value of 1.15 is taken: 
CD - 1.15 (22) 
The limit of 250 is found using equation (14) with a particle diameter of 
0.15 mm and corresponds to the figures of Graf (1971). 
Adapting an equation like (20) but with different coefficients the formula 
for the transition region would yield: 
CD = 1.042 (1 + 2LJZ) (23) 
The accuracy of the coefficients is for computational reasons only. 
The corresponding equations for the particle fall velocity become for 
laminar motion (Re < 1): 
for turbulent motion (Re > 250): 
w = 1.077 (AgD)V2 (25) 
and for the transition region with 1 < Re < 250 
(see appendix 2): 
with a = 1.077 and ß - 25.87 this results in: 
5 {(l • ^ • 10-3 â ^ V 2 . ^ w = 12.94 ^ I + 7.399 "^ ^ 1 1 / 2 - lk (27) 
A comparison of the particle fall velocity at 20°C, calculated according to 
the different equations is given in table 1. 
Region 
laminar 
transition 
turbulent 
D 
mm 
0.01 
0.02 
0.04 
0.06 
0.08 
0.10 
0.15 
0.20 
0.40 
0.60 
0.80 
1.0 
1.5 
2.0 
4.0 
6.0 
8.0 
10 
20 
Raudkivi 
0.0663 
0.2652 
1.061 
2.387 
4.243 
6.63 
14.8 
20.4 
42.9 
65.2 
87.6 
110.0 
166.0 
190.2 
269.0 
329.5 
380.4 
425.3 
601.5 
w 
mm/s 
Van Rijn 
0.0899 
0.3598 
1.439 
3.238 
5.756 
8.994 
140.0 
171.4 
197.9 
279.9 
342.8 
395.9 
442.6 
625.9 
Zanke 
7.791 
16.24 
25.75 
59.27 
83.29 
102.0 
117.6 
equation (27) 
0.0771 
0.3084 
1.234 
2.775 
4.934 
7.709 
17.25 
26.57 
64.27 
92.00 
113.6 
131.6 
165.0 
193.8 
274.1 
335.7 
387.6 
433.3 
612.7 
Table 1 Comparison of particle fall velocity calculated in different way. 
5 POWER FUNCTIONS FOR THE TRANSITION REGION 
The results in the transition region are not very satisfactory, because a 
too high fall velocity is found, meaning that the drag coefficient is 
underpredicted. To avoid this an Equation like (7) could be tried out, but 
10 
the disadvantage of this formule is that no direct solution for the fall 
velocity can be found when it is substituted in Equation (3). A superior 
method, also suggested by Huisman (1973), is to subdivide the transition 
region and to approximate the drag coefficient by using formulae of the 
type: 
CD = gfp (28) 
Combining equation (3) and (28) yields: 
w2 = — 3 AgD (29) 
which is equivalent to : 
Re2 _ wfji _ * £ 4 AgDl 
or: 
a 3 v' 
a ReW = \ *$- (31) 
so that: 
w - £ P ? ^ -i2'ß (32) D [a 3 v2 J 
The upper and lower limit for the use of the different formulae can be based 
on numeric values of the righthand side of equation (31). In this way the 
choice for the right formula can made in advance while this is not possible 
if the boundaries are defined by the Reynolds number. The next equations are 
appropriate to predict the particle fall velocity: 
(Re < 1) 
C D - Ü 2 / 3 28 < f ^ 2 i < 960.1 (34) 
(1 < Re < 14.17) 
11 
r 18 
cD - R^I/2 
960 .1 < J ^ 4 < 6.903 • 104 
- 3 
(14.17 < Re < 245) 
(35) 
CD = 1.15 t ^ - > 6.903 • 104 
,2 ü (36) 
(Re > 245) 
The partiele fall velocity calculated according to equation (33) up until 
(36) is given in Table 2, showing very satisfactory results. 
Region 
laminar 
transition 
turbulent 
D 
nun 
0.01 
0.02 
0.04 
0.06 
0.08 
0.10 
0.15 
0.20 
0.40 
0.60 
0.80 
1.0 
1.5 
2.0 
w 
mm/s 
0.077 
0.308 
1.234 
2.775 
4.934 
7.709 
13.66 
19.57 
45.15 
67.73 
90.30 
112.9 
167.8 
193.8 
4 AgD3 
3 v2 
1 
2.159-10-2 
1.727-10-1 
1.382 
4.663 
1.105•10+1 
2.159-10+1 
7.285-10+1 
1.727-10+2 
1.382-10+3 
4.663-10+3 
1.105•10+4 
2.159•10+4 
7.285-10+4 
1.727-10+5 
Re 
1 
7.709-10-4 
6.168-10-3 
4.934-10-2 
1.665-10"1 
3.947-10-1 
7.709-10"1 
2.049 
3.914 
1.806-10+1 
4.064-10+1 
7.224-10+1 
1.129-10+2 
2.517-10+2 
3.875-10+2 
cD 
1 
3.632-10+4 
4.540-10+3 
5.675-10+2 
1.681-10+2 
7.094-10+1 
3.632-10+1 
1.736-10+1 
1.127•10+1 
4.236 
2.824 
2.118 
1.694 
1.150 
1.150 
Table 2 Particle fall velocity at 20°C. 
The dependancy of the drag coefficient on the Reynolds number according to 
Equation (33) up until (36) is shown in Figure 2 in comparison to other 
formulae. 
12 
10 
10 
10" 
^\ 
\ \y '•, \ \*\ \ \' 
N \ 
N 
> VNV 
N*' 
N, 
N 
v.# 
' s 
Raudkivi 
- — Van Rijn/Zanke 
Present author 
^ % 
"•» ^"s 
*Vsl*. 
... 
10" 10 Re 10" 
Figure 2 Drag coefficient versus Reynolds number. 
6 CONCLUDING REMARK 
The foregoing analysis gives an arbitrary solution to the problem of 
calculating the particle fall velocity. A refinement in the constants and 
powers in the proposed formulae can be accomplished by a more comprehensive 
study using additional literature and also experimental data for making a 
least square fit. 
13 
LITERATURE 
Graf, W.H. and E.R. Acaroglu (1966). Settling velocities of natural grains, 
Bull. Int. Ass. Sei. Hydrology, 11, no. 4. 
Graf, W.H. (1971). Hydraulics of Sediment Transport, Mc Graw-Hill Book 
Company, New York. 
Huisman, L. (1973). Sedimentation and Flotation, Department of Civil 
Engineering, Delft, University of Technology, Delft. 
Raudkivi, A.J. (1976). Loose Boundary Hydraulics, Pergamon Press, Oxford. 
Rijn, L.C. van (1984). Sediment transport, Part II: Suspended load 
transport. Journal of Hydraulic Engineering, Vol. 110, No. 11, November 
1984, ASCE. 
Zanke, U. (1977). Berechnung des Sinkgeschwindigkeiten von Sedimenten. Mitt. 
des Franzius - Institus für Wasserbau. Heft 46, Seite 243, Technical 
University, Hannover. 
14 
LIST OF SYMBOLS 
A SN= surface area of nonspherlcal particle 
As - surface area of spherical particle 
CD = drag coefficient 
D = particle diameter 
FQ = drag force 
g = acceleration due to gravity 
G = submerged weight 
Re = Reynolds number 
w = particle fall velocity 
A = relative density 
v = kinematic viscosity 
pw = density of water 
ps = density of sediment 
* — shape factor 
[L2] 
[L2] 
[1] 
[L] 
[M L I"2] 
[L I"2] 
[M L T'2] 
[1] 
[L T"1] 
[1] 
[L2 I'1] 
[M L"3] 
[M L"3] 
[1] 
15 
APPENDIX 1: Derivation of drag coefficient from different formulas for the 
fall velocity. 
Fall velocity equation: 
" CD 3 
w2 = J- | AgD (1) 
Laminar 
1 AgDs 
w - ^ ^ - (2) 
a v 
<"*«>- "-555 3 (3) 
°D - 1 ab <4) 
Turbulent motion: 
w = £ (AgD)1^ (5) 
- w2 - ß2 AgD (6) 
(1) & (6) - CD - | ^ (7) 
Transi t ion region: 
w = io£{[i + o.oi^i]1/2- i} 
(1) & (8) ^ - 10 <[[l + 0.01 | w2 CD ^ ] 1 / 2 - l } 
1 f 3 "11/2 
^ Re + 1 = | l + 0.01 I CD Re2J 
0.01 Re2 + 0.2 Re - 0.01 | CD Re 
(8) 
(9) 
2 
16 
Re2 + 20Re - - CD Re2 
CD thi) (10) 
17 
APPENDIX 2: Derivation of the equation for the fall velocity in the 
transition region. 
Equation for the drag coefficient: 
CD - a (l + f;] (1) 
Equation for the fall velocity: 
1_ 4 
CD 3 
(1) & (2) -> w2 - , Y . > 7 AgD (3) 
w2 = J- | AgD (2) 
w = 
or 
(4) 
w2 + 0 £ w - - | AgD = 0 (5) 
D a Ó 
The relevant solution of equation (5) yields 
I V2 &•{($* • j f ^ r 
»-ilh^t^r-1} 


The fall velocity of grain particles.

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