abul-4.qxd 2/28/02 9:16 AM Page 4.

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.5

sand particles with the same average diameter because the former is lighter than water
while the latter is 2.65 times heavier than water.

4-2 HOLD-UP

The previous section describes how different layers of solids move with different speeds,
from the bottom, coarser particles, to the finer particles at the top of the horizontal pipe.
The theory of hold-ups complicates this process, however. Hold-ups are due to velocity
slip of layers of particles of larger sizes, particularly in the moving bed flow pattern.
Newitt et al. (1962) conducted speed measurements of a slurry mixture in a horizontal
pipe. In the case of light Plexiglas pipe, zircon or fine sand did not result in local slip; par-
ticles and water moved at the same speed. However, for coarse sand and gravel, they ob-
served asymmetric suspension and a sliding bed. They also observed that in the upper lay-
ers of the horizontal pipe, the concentrations of larger particles were the same as for finer
solids, but were marked by differences in the magnitude of the discharge rate of the lower
layers.

4.3 TRANSITIONAL VELOCITIES

The four regimes of flow described in Section 4-1 can be represented by a plot of the
pressure gradient versus the average speed of the mixture (Figure 4-5). The transitional
velocities are defined as

V1: velocity at or above which the bed in the lower half of the pipe is stationary. In the
upper half of the pipe, some solids may move by saltation or suspension.

1.0
Ratio distanc e from bottom of pipe

0.8
to the inner diameter (y/D) I

0.6

0.4 C
v
7
3

0.2 4
0.0
0.0

0.10 0.1
0
0 0.05 0.1 0.15 0.20

Discharge solids concentration C y

FIGURE 4-3 Distribution of concentration of solids in a pipe versus average volumetric
concentration.
abul-4.qxd 2/28/02 9:16 AM Page 4.6

4.6 CHAPTER FOUR

V2: velocity at or above which the mixture flows as an asymmetric mixture with the
coarser particles forming a moving bed.
V3 or VD: velocity at or above which all particles move as an asymmetric suspension
and below which the solids start to settle and form a moving bed.
V4: velocity at or above which all solids move as a symmetric suspension.

Velocity (ft/se c)
0 5 10 15 20
Volumetric Concentration (%)

direction of flow
30

20

10

0
0 1 2 3 4 5 6
Velocity (m/s )

FIGURE 4-4 Simplified concept of particle distribution in a pipe as a function of volumetric

concentration and speed.

slurry
Pressure drop per unit of length

1
4

2
er

3
at
w

V1 V2 V Speed of flow
asymmetric flow

V4
symmetric flow

3
stationary bed
moving bed

FIGURE 4-5 Velocity regimes for heterogeneous slurry flows.

abul-4.qxd 2/28/02 9:16 AM Page 4.7

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.7

V3 is effectively the deposition velocity, often called in the past the Durand velocity
for uniformly sized coarse particles. It is no longer recommended that it be called the Du-
rand velocity, as tests in the last 20 years have led to new equations that include the ef-
fects of particle size and composition of the slurry. The magnitude of the velocity depends
on the volumetric concentration (Figure 4-7).

4-3-1 Transitional Velocities V1 and V2

The transitional velocity V1 is obviously not used for the operation of slurry lines. It may
be of interest in lab research, instrumentation, and monitor of start-up.
The transitional velocity V2 is determined individually from pressure measurements of
the pressure gradient. The main focus of the tests is to determine the height of the bed and
to derive a stratification ratio.
Wilson (1970) developed a model for the incipient motion of granular solids at V2. He
assumed a hydrostatic pressure exerted by the solids on the wall and proposed the follow-
ing equation:
1 P sin cos s
+ sin
Rw

L L 4 D tan i r

s(S 1) Cvb(sin cos )g

= (4-1)
2
where
(P/L)2 = pressure gradient at 2
= half the angle subtended at the pipe center due to the upper surface of the bed,
in radians
s = coefficient of static friction of the solid particles against the wall of the pipe
Rw = cross-sectional area of the bed divided by the bed width
r = angle of repose of the solid particles

100
Cumulative passing

10
(%)

0
0 10 100 1000
d 50

Particle mesh diameter ( m)

FIGURE 4-6 Concept of d50 by cumulative passing percentage versus particle size.
abul-4.qxd 2/28/02 9:16 AM Page 4.8

4.8 CHAPTER FOUR

S = ratio of density of solids to density of liquid

Cvb = volume fraction solids in the bed
(When USCS units are used, express density in slugs/ft3 rather than lbm/ft3).
For 0.7 mm (mesh 24) sand with water in a 90 mm (3.5 in) pipe, Wilson measured =
0.35 and concluded that the assumptions of hydrostatic distribution of the granular pres-
sure were correct.

4-3-2 The Transitional Velocity V3 or Speed for Minimum

Pressure Gradient
The transitional velocity V3 is extremely important because it is the speed at which the
pressure gradient is at a minimum. Although there is evidence that solids start to settle at
slower speeds in complex mixtures, operators and engineers often referred to transitional
velocity as the speed of deposition.
Durand and Condolios (1952) derived the following equation for uniformly sized sand
and gravel:
VD = V3 = FL{2 g Di[(s L)/L]}1/2 (4-2)
where
FL = is the Durand factor based on grain size and volume concentration
V3 = the critical transition velocity between flow with a stationary bed and a heteroge-
neous flow
Di = pipe inner diameter (in m)
g = acceleration due to gravity (9.81 m/s)
s = density of solids in a mixture (kg/m3)
L = density of liquid carrier
The Durand factor FL is typically represented in a graph for single or narrow graded
particles, as in Figure 4-7 after the work of Durand (1953). However, since most slurries

For single or narrow

CV = 15% graded slurries
CV = 10%
2.0 Based on Schiller
equation using d50
Durand Velocity Factor
FD

CV = 5%
C = 2%
1.0 V
CV = 15%
CV = 5%

0
0 1.0 2.0 3.0
Particle diameter (mm)

FIGURE 4-7 Durand velocity factor versus particle sizecomparison between the conven-
tional values for single graded slurries and Schillers equation using d50 for wide graded slurry.
abul-4.qxd 2/28/02 9:16 AM Page 4.9

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.9

are mixtures of particles of different sizes, this plot is considered too conservative. The
Durand velocity factor has been refined by a number of authors.
In an effort to represent more diluted concentrations, Wasp et al. (1970) proposed in-
cluding a ratio between the particle diameter and the pipeline diameter. Wasp proposed
the use of a modified factor F
L so that

S L

D
1/2 dp 1/6
L 2gDi
VD = V3 = F (4-3)
L i

Schiller and Herbich (1991) proposed the following equation for the Durand velocity
factor based on the d50 of the particles:
FL = {(1.3 C 0.125
v )[1 exp (6.9 d50)]} (4-4)
where d50 is expressed in mm.
Some reference books define a Froude number as Fr = FL 2. The particle size d50
is the statistically determined particle size below which half (or 50%) would be equal or
smaller to that set size. The following example illustrates the concept of d50.

Example 4-1
A sample of slurry is sieved for particle size. The data is collected in the laboratory (see
Table 4-1). Plot the data on a logarithmic graph and determine the d50.
Solution
The data is plotted in Figure 4-6; the d50 is determined to be 145 m.

Example 4-2
A slurry mixture has a d50 of 300 m. The slurry is pumped in a 30 in pipe with an
ID of 28.28. The volumetric concentration is 0.27. Using Equations 4-4 and 4-2, deter-
mine the speed of deposition for a sandwater mixture if the specific gravity of sand is
2.65.

Solution in SI Units
From Equation 4-4:
FL = (1.3 0.270.125)(1-exp (6.9 0.3))
FL = 1.1 0.8738
FL = 0.964
From Equation 4-2 the deposition velocity is
V3 = 0.964 (2 9.81 28.25 0.0254 1.65)1/2
V3 = 4.64 m/s

TABLE 4-1 Data for Example 4-1

Particle size (m) 425 300 212 150 106 75 53 45 38 38
Cumulative 97.2 87.1 68.3 51.3 35.9 20.5 14.5 11.8 10.8
passing (%)
abul-4.qxd 2/28/02 9:16 AM Page 4.10

4.10 CHAPTER FOUR

Solution in USCS Units

From Equation 4-4:
FL = (1.3 0.270.125)(1 exp (6.9 0.3))
FL = 1.1 0.8738
FL = 0.964
From Equation 4-2 the deposition velocity is
V3 = 0.964 (2 32.2 28.25/12 1.65)1/2
V3 = 15.25 ft/sec

Various curves have been published for the magnitude of FL. They are often plotted
for a single graded size and use difficult to read logarithmic scales. For the sake of accu-
racy, Table 4-2 tabulates the magnitude of FL between 0.08 mm < d50 < 5mm on the basis

TABLE 4-2 The Coefficient FL Based on Schillers Equation Using the d50 of the
Particles for Particles Between 0.080 and 5 mm for Volumetric Concentration up to
30%. FL = {(1.3 C v0.125)[1 exp(6.9 d50)]}
d50 (mm) CV = 0.05 CV = 0.10 CV = 0.15 CV = 0.20 CV = 0.25 CV = 0.30
0.08 0.379 0.414 0.435 0.451 0.464 0.474
0.10 0.446 0.486 0.511 0.530 0.545 0.557
0.12 0.503 0.549 0.577 0.599 0.616 0.630
0.14 0.554 0.604 0.635 0.658 0.677 0.693
0.16 0.598 0.652 0.686 0.711 0.731 0.748
0.18 0.636 0.693 0.729 0.756 0.777 0.795
0.20 0.669 0.730 0.768 0.796 0.818 0.837
0.25 0.735 0.801 0.843 0.874 0.898 0.919
0.30 0.781 0.852 0.896 0.929 0.955 0.977
0.35 0.814 0.888 0.934 0.968 0.995 1.018
0.40 0.837 0.913 0.961 0.996 1.024 1.048
0.45 0.854 0.931 0.980 1.015 1.044 1.068
0.50 0.866 0.944 0.993 1.029 1.058 1.083
0.55 0.874 0.953 1.002 1.039 1.069 1.093
0.60 0.880 0.959 1.009 1.046 1.076 1.101
0.65 0.884 0.964 1.014 1.051 1.081 1.106
0.70 0.887 0.967 1.017 1.055 1.084 1.109
0.75 0.889 0.969 1.020 1.057 1.087 1.112
0.80 0.890 0.971 1.021 1.059 1.089 1.114
0.85 0.891 0.972 1.023 1.060 1.090 1.115
0.90 0.892 0.973 1.023 1.061 1.091 1.116
1.00 0.893 0.974 1.0245 1.062 1.092 1.1172
1.5 0.8939 0.9748 1.0255 1.063 1.0931 1.1183
2 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
2.5 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
3.0 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
3.5 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
4.0 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
5.0 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
abul-4.qxd 2/28/02 9:16 AM Page 4.11

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.11

of Schillers equation. The magnitude of FL based on d50 is smaller than values published
in the literature for single graded slurry mixtures (lab mixtures using a uniform size of
particles). A number of authors have confirmed that this is the case (Warman Internation-
al Inc., 1990).
In order to compare the conventional magnitude of FL based on single and narrow
graded particles to the Schiller equation, both ranges of FL are plotted in Figure 4-7.
With a more complex approach that takes into account the actual viscosity of the slur-
ry mixture and the density of the particles, Gillies et al. (1999) developed an equation for
the Froude number F in terms of the Archimedean number (which we will discuss in Sec-
tion 4-4-5 for stratified coarse flows):
4
Ar = d 3 ( )g (4-5)
3L2 p L s L
To estimate the deposition velocity V3, Gilles et al. (1999) developed an equation for
the Froude number based on the Archimedean number:
Fr = aArb (4-6)
where
Fr = FL 2
For Ar > 540, a = 1.78, b = 0.019
For 160 < Ar < 540, a = 1.19, b = 0.045
For 80 < Ar < 60, a = 0.197, b = 0.4
For Ar < 80, the Wilson and Judge (1976) equation can be used, which expressed the
Froude number as

dp
) 2.0 + 0.30 log10
Fr = (2 (4-7)
DiCD
This correlation is useful in the range of 105 < (dp /DiCD) < 103.
To determine the drag coefficient, the actual density of the liquid should be used,
whereas the viscosity should be corrected for the presence of fines.

Example 4-3
Water at a viscosity of 0.0015 Pa s (0.0000313 slugs/ft-sec) is used to transport sand
with an average particle diameter of 300 m (0.0118 inch). The volumetric concentration
is 0.27. The pipes inner diameter is 717 mm (28.35). Using the Gilles equation (Equa-
tion 4-6), determine the deposition velocity if the specific gravity of sand is 2.65. Assume
CD = 0.45.
Solution in SI Units
d50 0.3
= = 0.4 103
Di 717
Iteration 1
Assuming CD > 103, by the Wilson and Judge correlation (Equation 4-7):

0.003
Fr = (2) 2.0 + 0.30 log10
0.717 0.45
Fr = 1.54
FL = Fr/2 = 1.54/2 = 1.09
abul-4.qxd 2/28/02 9:16 AM Page 4.12

4.12 CHAPTER FOUR

The specific gravity of the mixture is determined as:

Sm = Cv(Ss SL) + SL = 0.27 (2.65 1) + 1 = 1.446
VD = FL[2
gDi(
s/

L ]
1 = 4.82 m/s
Iteration 2
4 9.81 (3 104)3 1000 (1650)
Ar = = 258.98
3(1.5 103)2
for 160 < Ar < 540, a = 1.19, b = 0.045.
From equation 4.6:
Fr = aArb = 1.19 258.980.045 = 1.53
FL = F/2 = 1.53/2 = 1.082
VD = FL[2gDi(s/L 1)]0.5
VD = 1.082[2 9.81 0.717 (1.65)]0.5 = 5.21m/s

Solution in USCS Units

d50 0.00118
= = 0.4 103
Di 28.23
Iteration 1
Assuming CD > 103, by the Wilson and Judge correlation (Equation 4-7):

0.00118
Fr = (2) 2.0 + 0.30 log10
28.23 0.45
Fr = 1.54
FL = 1.54/2 = 1.09
The specific gravity of the mixture is determined as:
Sm = Cv(Ss SL) +SL = 0.27(2.65 1) + 1 = 1.446
VD = 1.09[2 32.2 (28.23/12) (2.65 1)]0.5
VD = 17.23 ft/sec
Iteration 2
The particles diameter is 0.984 103 ft
The density of water is 1.93 slugs/ft3
The density of sand is 5.114 slugs/ft3
Water dynamic viscosity is 0.0000313 slugs/ft-sec
4(0.984 103)3 1.93(5.114 1.93) 32.2
Ar = = 259
3(0.0000313)2
for 160 < Ar < 540, a = 1.19, b = 0.045.
From equation 4.6,
Fr = aArb = 1.19 258.980.045 = 1.53
FL = 1.53/2 = 1.082
VD = FL[2gDi(s/L 1)]0.5
VD = 1.082[2 32.2 2.35 (1.65)]0.5 = 17.1 ft/s
abul-4.qxd 2/28/02 9:16 AM Page 4.13

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.13

The solution by the Gilles equation is within the limits set by Schiller in Example 4-2.
In these two different examples, we applied two different formulae but obtained con-
sistent results. This demonstrates the sensitivity of approaches to equations derived from
empirical equations. It may be necessary sometimes try to solve a problem using two dif-
ferent equations, and to use common sense when similar results are obtained.
Table 4-3 presents values of the Archimedean number, the resultant magnitude of the
factor FL for particles d50 in the range of 0.08 to 50 mm for a specific gravity of 1.5,
which is typical of coal-based mixtures. Most coals may be pumped with different sizes
of particles as discussed in Chapter 11. The viscosity may be due to the presence of cer-

TABLE 4-3 The Coefficient FL Based on Gilles Equation for Particles Between 0.080
and 50 mm of Specific Gravity of 1.500 (e.g., Coal) as a Function of Viscosity
= 1 cP
_____________________ = 5 cP
_______________________ = 10 cP
_______________________
Archimedean Archimedean Archimedean
d50 (mm) number Ar FL number Ar FL number Ar FL
0.08 3.35 Eqn 4-7 0.13 Eqn 4-7 0.033 Eqn 4-7
0.10 6.54 Eqn 4-7 0.26 Eqn 4-7 0.065 Eqn 4-7
0.12 11.3 Eqn 4-7 0.45 Eqn 4-7 0.113 Eqn 4-7
0.14 17.9 Eqn 4-7 0.72 Eqn 4-7 0.18 Eqn 4-7
0.16 26.8 Eqn 4-7 1.07 Eqn 4-7 0.27 Eqn 4-7
0.18 38.1 Eqn 4-7 1.53 Eqn 4-7 0.38 Eqn 4-7
0.20 52.3 Eqn 4-7 2.1 Eqn 4-7 0.52 Eqn 4-7
0.25 102 0.89 4.1 Eqn 4-7 1.02 Eqn 4-7
0.30 177 1.062 7.1 Eqn 4-7 1.77 Eqn 4-7
0.35 280 1.084 11.2 Eqn 4-7 2.80 Eqn 4-7
0.40 419 1.104 16.75 Eqn 4-7 4.19 Eqn 4-7
0.45 596 1.420 23.8 Eqn 4-7 5.96 Eqn 4-7
0.50 818 1.43 32.7 Eqn 4-7 8.18 Eqn 4-7
0.55 1088 1.437 43.5 Eqn 4-7 10.9 Eqn 4-7
0.60 1413 1.445 56.51 Eqn 4-7 14.1 Eqn 4-7
0.65 1796 1.451 72 Eqn 4-7 18 Eqn 4-7
0.70 2243 2.457 89.7 0.84 22.4 Eqn 4-7
0.75 2579 1.463 110.4 0.914 27.6 Eqn 4-7
0.80 3348 1.469 134 0.99 33.5 Eqn 4-7
0.85 4016 1.474 161 1.058 40 Eqn 4-7
0.90 4768 1.478 191 1.066 48 Eqn 4-7
1.00 6540 1.487 262 1.081 65 Eqn 4-17
2.00 52320 1.547 2093 1.455 523 1.12
3.00 176580 1.583 7063 1.489 1765 1.45
4.00 418560 1.610 16742 1.514 4185 1.475
5.00 817500 1.63 32700 1.533 8175 1.494
6.00 1415640 1.647 56505 1.55 14126 1.51
8.00 3348480 1.674 133939 1.575 33485 1.534
10.00 6540000 1.696 261600 1.595 65400 1.554
20.00 5.23 107 1.764 2092800 1.66 523200 1.616
30.00 17.7 108 1.805 7063202 1.698 1765800 1.654
40.00 41.86 108 1.835 16742404 1.726 4185601 1.682
50.00 81.75 108 1.859 32700008 1.749 81750020 1.703
abul-4.qxd 2/28/02 9:16 AM Page 4.14

4.14 CHAPTER FOUR

tain fines, as with peat coals or degradation of the coal during pumping over long dis-
tances, or the use of a heavy medium such as magnetite at high concentration as a carrier
for coal in a water mixture.
Table 4-4 presents values of the factor FL for particles d50 in the range of 0.08 to 50
mm for a specific gravity of 2.65, which is typical of sand and tar-sand-based mixtures.
The largest particles are often found in tar sand applications, with some contribution of
the tar or oil to viscosity. In this table, there was no need to present the Archimedean
number, as this was demonstrated in the previous table.
Newitt et al. (1955) preferred to express the speed of transition between saltation
flow and heterogeneous flow in terms of the terminal velocity of particles (previously dis-
cussed in Chapter 3):
V3 = 17 Vt (4.8)
The reader should refer to Equation 3-18, which corrects the terminal velocity of a sin-
gle particle to a mass of particles at higher volumetric concentration. Although Equation
4-8 has served as the basis of many models, we will later discuss the recent corrections
proposed by Wilson et al. (1992).
The approach to obtain the magnitude of V3 is basically to conduct a test and measure
pressure drop per unit length of pipe. V3 is considered to occur at the minima, or the point
of minimum pressure drop. W. E. Wilson (1942) expressed the pressure gradient of non-
colloidal solids by referring to clean water and by proposing a correction to the
DarcyWeisbach equation (discussed in Chapter 2). He expressed the consumed power
due to friction by the following equation:

FIGURE 4-8 These taconite tailings must be pumped above a deposit velocity of 13 ft/s in
14 pipe due to the size of the particles.
abul-4.qxd 2/28/02 9:16 AM Page 4.15

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.15

TABLE 4-4 The Coefficient FL Based on Gilles Equation for Particles Between 0.080
and 50 mm of Specific Gravity of 2.65 (e.g., Sands and Oil Sands) as a Function of
Viscosity
d50 (mm) = 1 cP, FL = 5 cP, FL = 10 cP, FL
0.08 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.10 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.12 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.14 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.16 0.837 Eqn 4-7 Eqn 4-7
0.18 0.964 Eqn 4-7 Eqn 4-7
0.20 1.061 Eqn 4-7 Eqn 4-7
0.25 1.093 Eqn 4-7 Eqn 4-7
0.30 1.421 Eqn 4-7 Eqn 4-7
0.35 1.433 Eqn 4-7 Eqn 4-7
0.40 1.444 Eqn 4-7 Eqn 4-7
0.45 1.454 0.8 Eqn 4-7
0.50 1.462 0.906 Eqn 4-7
0.55 1.470 1.016 Eqn 4-7
0.60 1.478 1.065 Eqn 4-7
0.65 1.485 1.076 Eqn 4-7
0.70 1.491 1.087 Eqn 4-17
0.75 1.497 1.097 0.847
0.80 1.502 1.107 0.915
0.85 1.507 1.116 0.984
0.90 1.512 1.423 1.054
1.00 1.521 1.431 1.072
2.00 1.583 1.489 1.450
3.00 1.620 1.524 1.484
4.00 1.647 1.549 1.509
5.00 1.668 1.569 1.528
6.00 1.685 1.585 1.544
8.00 1.713 1.611 1.569
10.00 1.735 1.632 1.589
20.00 1.805 1.698 1.654
30.00 1.847 1.737 1.692
40.00 1.877 1.766 1.720
50.00 1.901 1.789 1.742

fDV CwVt
Hf = L + C1 (4-9)
2gDi V
where
Hf = head loss due to friction (in units of length)
fD = DarcyWeisbach friction factor
C1 = constant
Equation 4-9 may also be reexpressed as
Hf g fDV 2 C1CwVt g
=+ (4-10)
L 2Di V
abul-4.qxd 2/28/02 9:16 AM Page 4.16

4.16 CHAPTER FOUR

By differentiating this equation with respect to V, we obtain for the minimal value
2 fDV C1CwVt g
=
2Di V2
or
fDV C1CwVt g
=
Di V2
C1CwVt gDi
V 3 =
fD
at constant friction factor fD, or
[C1CwVt gDi]1/3
Vmin = (4-11)
f D1/3
The magnitude of the Darcy friction factor for water flow in rubber lined and HDPE
pipe was computed for pipes from 2 to 18 and results presented in Chapter 2.
Wilson (1942) defined a factor C3 to determine whether the particles will settle to
form a bed:
2Vt
C3 = (4-12)
(Hf fD gDi/L)1/2
If C3 > 1 most particles with a terminal velocity Vt will stay in suspension. If C3 1 most
particles with a terminal velocity Vt will settle out.
Whereas the equations of Newitt et al. (1955) and Wilson (1942) focused on the termi-
nal velocity, the work of Durand and Condolios (1952) focused on the drag coefficient for
sand and gravel. Zandi and Govatos (1967) and Zandi (1971) extended the work of Du-
rand to other solids and to different mixtures. They defined an index number as
V 2CD1/2
Ne = (4-13)
CvDi g(s/w 1)
At the critical value when Ne = 40, the flow transition between saltation and heteroge-
neous regimes occurs. This statement infers that when Ne < 40 saltation occurs, and when
Ne 40 heterogeneous flow develops. These results, based on a mixture of different par-
ticle sizes, did not apply to the work of Blatch (1906), who concentrated on particles of a
uniform size (sand 2030 mesh in water). Babcock (1967) reinterpreted this work and
demonstrated that for finely graded particles the transition occurred at an index number of
10. It is obvious that a complex mixture of particles of different sizes can increase the
magnitude of the transition index number.

Example 4-4
Tailings from a mine consist of solids at a volumetric concentration of 20%. The specific
weight of the solids is 4.2. The pipe diameter is 8 with a wall thickness of 0.375 and
rubber lining of 0.5. The particle Albertson shape factor is 0.7. The dynamic viscosity is
3 cP. The average d50 = 0.4 mm. Determine the speed of transition from saltation using
the Zandi approach as expressed by Equation 4-13.
Solution in SI Units
Pipe inner diameter Di = 8 2 (0.5 + 0.375) = 6.25 = 158.75 mm
abul-4.qxd 2/28/02 9:16 AM Page 4.17

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.17

Iteration 1
Let us first assume a transition from saltation at 3 m/s and let us determine the drag coef-
ficient of the particles in water at the stated dynamic viscosity:
Rep = 1000 0.0004 3/0.003 = 400
From Table 3.7, CD = 1.09.
The transition from saltation occurs when Ne = 40. From Equation 4-13, using SI units:
9 1.0
9
Ne =
0.2 0.15875 9.81 (4.2/1 1)
Ne = 9.43.
Iteration 2
Let us first assume a transition from saltation at 6 m/s and let us determine the drag coef-
ficient of the particles in water at the stated dynamic viscosity:
Rep = 1000 0.0004 6/0.003 = 800
From Table 3.7, CD = 1.15.
36 1.1
5
Ne =
0.2 0.15875 9.81 (4.2/1 1)
Ne = 39.
The transition from saltation therefore occurs at a speed of 6.1 m/s.
Solution in USCS Units
Iteration 1
Pipe diameter = 8 2 (0.375 + 0.5) = 6.25 = 0.521 ft
Let us first assume a transition from saltation at 10 ft/s and let us determine the drag coef-
ficient of the particles in water at the stated dynamic viscosity.
Particle size = 0.4 mm/304.7 mm = 1.3128 103 ft
= 0.003/47.88 = 6.265 105 lbf-sec/ft2
Density of water = 62.3 lbm/ft3/32.2 ft/sec = 1.935 slugs/ft3
1.935 slugs/ft3 1.3128 103 ft 10 ft/sec
Re =
6.265 105 lbf-sec/ft2 = 406
From Table 3.7, CD = 1.09.
100 1 .0
9
Ne =
0.2 0.5208 ft 32.2 ft/sec (4.2/1 1)
Ne = 9.73.
Iteration 2
Let us first assume a transition from saltation at 20 ft/s and let us determine the drag coef-
ficient of the particles in water at the stated dynamic viscosity:
Rep = 406 (20/10) = 804
abul-4.qxd 2/28/02 9:16 AM Page 4.18

4.18 CHAPTER FOUR

From Table 3.7, CD = 1.15.

202 1.15
Ne =
0.2 0.5208 ft 32.2 ft/sec (4.2/1 1)
Ne = 39.97.
The transition from saltation therefore occurs at a speed of 20 ft/sec.

4-3-3 V4: Transition Speed Between Heterogeneous and

Pseudohomogeneous Flow
For the transition to pseudohomogeneous flows, Newitt et al. (1955) expressed the speed
in terms of the terminal velocity of particles as
V4 = (1800 gDiVt)1/3 (4-14)
Refer to Chapter 3 and Equation 3-18 to calculate terminal velocity.
Govier and Aziz (1972) applied Newtons law (i.e., CD = 0.44) for particles immersed
in a fluid to Equation 4-14 to yield
4gdp
i (S 1)
V4 = 38.7D 1/3 1/6
(4-15)
3CD
Govier and Aziz (1972) analyzed the work of Spells (1955) on solid particles with a
diameter 80 m < dp < 800 m (mesh 180 < dp < 20) and derived the following equation:
V4 = 134CD0.816D 0.633
i V 1.63
t (4-16)
This equation was derived in USCS units with the diameter expressed in feet and the ve-
locity in feet per seconds.

Example 4-5
An ore with a specific gravity of 4.1 is to be pumped in a pseudohomogeneous regime in
a 24 in pipe with an ID of 22.23 in. The drag coefficient of the particles is assumed to be
0.44. The estimated flow rate is 12,000 US gpm. The particles have a sphericity of 0.72
and a diameter of 250 m. Solve for V4.
Solution in SI Units
12,000 3.785
Q = = 0.757 m3/s
60,000
Pipe ID = 22.25 0.0254 = 0.565 m
Cross-sectional area = 0.251 m2
Average speed of flow = 3.02 m/s
Sphericity = Asp/Ap = 0.72
dsp = 0.7
2 250 = 218 m

4 0.218 103 9.81 (4100 1000)
Vt =
3 0.44 1000
Vt = 0.142 m/s
abul-4.qxd 2/28/02 9:16 AM Page 4.19

HETEROGENEOUS FLOWS OF SETTLING SLURRIES 4.19

By Newitts equation (Equation 4.14):

V4 = (1800 9.81 0.565 0.142)1/3
V4 = 11.22 m/s
Alternatively using Equation 4.16:
Di = 1.854 ft
Vt = 0.466 ft/sec
V4 = 134C D0.816D 0.633
i V 1.63
t

V4 = 134 0.440.816 1.8540.633 0.4661.63 = 29.19 ft/sec or 8.9 m/s

Solution in USCS Units
Q = 12,000 0.002228 = 26.736 ft3/sec
Pipe ID = 22.25/12 = 1.854 ft
Cross-sectional area = 2.7 ft2
Average speed of flow = 9.9 ft/sec
Sphericity = Asp/Ap = 0.72
dsp = 0.7
2 250 = 218 m = 0.000715 ft

The density of water is 1.93 slugs/ft3
The density of solids is 7.913 slugs/ft3

4 0.000715 32.2 (7.913 1.93)
Vt =
3 0.44 1.93
Vt = 0.465 ft/s
By Newitts equation (Equation 4.14):
V4 = (1800 32.2 1.854 0.465)1/3
V4 = 36.83 ft/sec
Alternatively, using Equation 4.16:
V4 = 134C D0.816D 0.633
i V 1.63
t

V4 = 134 0.440.816 1.8540.633 0.4661.63 = 29.19 ft/sec

4-4 HYDRAULIC FRICTION GRADIENT OF

HORIZONTAL HETEROGENEOUS FLOWS

Having been able to determine the speed for transition from one regime to another, the
slurry engineer must determine the loss of head per unit length due to friction, called the
hydraulic friction gradient (Equation 2-24). The hydraulic friction gradient for the slurry
(im) is higher than the hydraulic friction gradient for an equivalent volume of water. Since
the first slurry pipelines were built, engineers and scientists have tried to correlate the
losses with slurry to those of an equivalent volume of water.
It was initially assumed that the friction losses would increase in proportion to the vol-