Theoretical
Description and Numerical Sensitivity Analysis on Wilson Model for Hydraulic
Transport of Solids in Pipelines.
* E.J. van Riet M.Sc.
V. Matouek M.Sc.
S.A. Miedema Ph.D.
ABSTRACT
The Wilson model for the hydraulic
transport of solids in pipelines is a widely used model. A theoretical
background of the model has been published piece by piece in a number of
articles over the years. A variety of information provided in these
publications makes the model difficult to reconstruct.
A good understanding of the model structure
is inevitable for the user who wants to extend or adapt the model to specific
slurry flow conditions. An aim of this article is to summarise the model theory
and submit the results of the numerical analysis carried out on the various
model configurations. The numerical results show some differences when compared
with the nomographs presented in the literature as the graphical presentations
of the generalised model outputs. Model outputs are sensitive on a number of
input parameters and on a model configuration used. A reconstruction of the nomographs
from the computational model outputs is a subject to discussion.
Wilsonuv
model pro hydraulickou dopravu sypanin v potrubich je siroce pouzivanou
vypocetni pomuckou. Teoreticky podklad modelu byl postupne v prubehu let
zverejnovan v serii publikaci. Na zaklade techto publikaci je vsak tezke zrekonstruovat
model v jeho originalni matematicke podobe.
Pritom
spravne pochopeni struktury dvouvrstveho modelu je nezbytne pro jeho adaptaci
do ruznych podminek proudeni smesi v potrubi. Cilem clanku je shrnout
teoreticky aparat modelu a navrhnout zpusob numerickeho reseni teoretickych
rovnic modelu v jeho ruznych konfiguracich. Porovnani vysledku numerickeho
reseni s nomogramy zverejnenymi v publikacich Wilsona a jeho spolupracovniku
ukazuje na odlisnosti. Vystupy
modelu jsou zavisle na rade vstupnich parametru a na konfiguraci modelu. Rekonstrukce
nomogramu dle vystupu vypocetniho modelu je predmetem diskuse o zpusobu
pouzivani modelu.
Keywords: Hydraulic
transport, two-layer model, slurry pipeline.
INTRODUCTION
This article contains an overview of a
theory for the Wilson two-layer model as it has been published in a number of articles over the years.
Results are presented from the model computation. The results provide an
insight to a behaviour of the mathematical model. The computation has been
carried out using the MathCad document described by the authors in [2]. This
MathCad document is available on a floppy disk at the authors upon request.
1. GEOMETRY OF TWO-LAYER MODEL
A schematic cross section of a pipe is
illustrated in Fig.1 as it is defined in the two-layer model for the
fully-stratified flow and for the heterogeneous (partially-stratified) flow.
Figure
1: Schematic cross-section for two-layer model.
The geometry of the pipe cross section is
defined by the following equations.
The cross-sectional
perimeters:
(1)
(2)
(3)
The cross-sectional
areas:
(4)
(5)
(6)
The
equivalent hydraulic diameter of
the non-circular waterway section above the bed is a function of the bed height
[10]
(7)
2. FORCE BALANCE TO DETERMINE THE MDV CURVE AND
THE RESISTANCE CURVE FOR FULLY-STRATIFIED FLOW
The Wilson model provides following
parameters important for the slurry pipeline design and operation:
the
maximum deposit velocity VaMDV (at the MDV curve). VaMDV
is the maximum average velocity of slurry flow in a pipe at which a stationary
bed still occurs. The MDV curve depicts VaMDV as a function of the
bed height in a pipe.
the
friction loss (at the resistance curve). This curve depicts the pressure drop
as a function of the flow rate in a pipe for slurry of the constant delivered
concentration of solids.
Both curves can be plotted in one system of
co-ordinates.
The MDV curve and the resistance curve are
calculated from a force balance of four main forces (per unit length of the
pipe) acting on the stationary or moving bed, which is formed by particles in
mutual contact and contact with a pipe wall [4,5,6]. The force balance is
written for forces and shear stresses averaged over the perimeters of flow
boundaries.
The shear stresses on the flow boundaries
are determined using Nikuradse friction
equation for turbulent flow in a hydraulically-rough pipe [10]
(8)
The MathCad document solves a set of the
model balance equations.
The equilibrium average velocity Veq
in the upper layer is obtained by solving V1 in the force balance
when the next quantities are kept constant:
the
bed height
the
bed velocity
the
physical properties of the fluid and solids.
Veq is velocity V1
for which the force balance is found by iteration in the MathCad document.
Following procedure is used for a model
computation:
1. The driving
shear force on the bed surface is calculated using the Nikuradse equation
multiplied by an empirical constant for shear stress on the bed surface. This
constant was originally assumed to be
equal to the value 2 [6].
(9)
Shear stress 12 is calculated for velocity
equal to the difference between velocity in the upper and in the lower layer.
2. The driving force
caused by the pressure gradient over a pipe section of a unit length is
determined from the pressure gradient
(10)
and the driving force is
(11)
3. The resisting mechanical
friction force between bed and pipe wall is determined; this is the normal
force exerted by the bed against the pipe wall multiplied by the mechanical
friction coefficient m [4,12].
(12)
4. The viscous friction force between the bed and
the pipe wall is calculated
(13)
5. The force
balance is
(14)
6. The relative
delivered concentration of solids in slurry flow is determined as
(15)
Relative delivered concentration is a ratio
of the absolute delivered concentration and concentration of solids in a
loose-packed bed.
Wilson and his co-workers have published the
nomographs [6,7,12] - the tools to predict the slurry flow parameters without
handling the computational two-layer model. The nomographs are based on the
computational model outputs. A comparison of the nomographic values with those
from a computational model is of interest since it is not always clear for
which slurry characteristics (as Cb, , Ss) and model configuration
the nomographs are proposed. The outputs of the computational model have been
found very sensitive to the input parameters and a chosen model configuration.
The
resistance curve
Any point of the resistance curve (i-Va
curve for const. Cdel) is obtained by a numerical solution of the
force-balance equations for the following conditions:
constant
bed velocity
constant
physical properties of the fluid and solids.
The bed height is a variable in a numerical
iteration procedure. The bed height is determined for which two criteria are
satisfied simultaneously:
the
force balance in pipe section is found
the
calculated delivered concentration equals the Cdel required by the
constructed resistance curve.
The resistance curve computed is presented
in the same plot as a nomograph in the literature [12].
Following diamesionless parameters are
handled in the nomograph:
the relative
velocity Va/Vmax,
the relative
concentration Cdel,
the relative
excess pressure gradient which is defined as
(16)
when 
is the pressure
gradient of equivalent clear water flow and 
is the pressure
gradient for equivalent plug flow.
The
MDV curve
Any point of the MDV curve is obtained by
solving the force balance for a given bed height and V2=0. The curve
is produced by solving the balance for an array of bed heights. A maximum at
the MDV curve gives Vmax.
The MDV curve and the resistance curve are plotted
in Fig 2. This figure is a product of the MathCad document described in [2].
Figure
2: Non-dimensional MDV curve and resistance curve (fully-stratified flow).
3. THE INCORPORATION OF SUSPENSION;
HETEROGENEOUS MODEL
An adaptation of the two-layer model has
been proposed [6] for the partially-stratified flow, i.e. flow in which a part
of transported solid particles is suspended in the stream above the bed (see
Fig. 1). Suspension of particles due to carrier turbulence causes an increase
in the density (and viscosity at the highest concentrations) of mixture flow in
the upper layer [8]. This change in the physical properties of flow should
explain a significant decrease of the Vmax with decreasing particle
size (for particles smaller than approximately 0.7 mm) provided by the curve of
the demi McDonald nomograph (Vmax=f(d, D, Ss)) [7,12].
Although this decreasing trend can be produced by a numerical simulation of the
model [2], it appears impossible to reproduce such a large drop in the Vmax
values as the demi McDonald nomograph gives.
Wilson's (and his co-workers')
investigation of the sheet flow has led to a further development in a structure
of the two-layer model. Description of the flow in the shear layer, i.e. of the
bed-load motion at high shear stress, has provided a new formulation of the
friction law for an interface between bed and waterway.
A transition zone between packed granular
bed and waterway above the bed is called shear layer. The model may be called
'three layer model' when the shear layer is implemented to its structure. At
present the shear layer effect on the model structure is expressed only by an
implementation of the new interfacial friction law to the two-layer model so
not by changes in the model geometry.
4. THE THREE-LAYER MODEL
Publications [1,3,10,11,13,14] deal with a
description of the shear on the bed-fluid interface.
Originally it was assumed [10] that the
hydraulic roughness of the interface equals to one half of the shear layer
thickness. The shear layer thickness is a function of the shear stress at the
real/virtual interface. Thus shear stress was determined from a theoretical
implicit equation in which the hydraulic diameter Deq was one of the
variables.
Later Wilson & Nnadi [11] derived that
the hydraulic diameter can be cancelled from the equations and that the
friction factor at the bed surface depends only on i/(Ss-1)
providing the following relationship
(17)
Rb should be determined using a method
from [3]. An application of the eq. (7) has led to the following semi-empirical
formula expressing a friction law for sheet flow [11]
(18)
revised in [13] as
(19)
Eq.(17) has been also implemented to the
general friction equation for rough-wall boundary
(20)
Empirical constants in the eq. (20) have
been determined by a calibration of the eq. (20) by the experimental data.
Different constants have been published for different data (characterised here
by different ):
equation published in [1](for =240)
(21)
equation published in [14](for =180)
(22)
The equations (19, 21, 22) give similar f12
values but the eq. (18) differs.
When the recently published value =14o [13] for a tested material
is used in the eq. (20) the following equation can be written
(23)
Recently, Wilson has proposed a correction
of the demi McDonald nomograph based on analytical results from the three-layer
model. This has a form of a fit function [12,13]. The three-layer model outputs
have shown that Vmax has not been dependent on the particle diameter
when the friction law for sheet flow has been used for the interface between
layers
(24)
Wilson & Pugh [13] have recommended to
use this equation instead of the curve in the demi McDonald nomograph when the
value of Vmax obtained from the demi McDonald nomograph exceeds that
from the fit function.
The three-layer model has been tested in
the MathCad document [2]. The Vmax outputs for various friction
equations are compared with the fit function in Fig. 3. Following input
parameters to the model are used: =0.4, r=10-5, Cb=0.6,
Ss=2.65 and ff according to Nikuradse.
Figure
3: Maximum deposit velocity. Comparison of the fit function with the outputs of
the three-layer model for various interface-friction equations.
Legend:
line: fit function, eq. (24)
diamonds: output of the three-layer model with eq.
(22)
squares: output of the three-layer model with
eq. (19)
crosses: output of the three-layer model with
eq. (21)
plusses: output of the
three-layer model with eq. (23)
The fit function (24) matches reasonably the
three-layer model outputs for all tested friction equations. The best fit is
reached by the eq. (22). A decrease in from 29 to 14 degrees causes a decrease in
Vmax of 15 - 20%.
5. DISCUSSION AND CONCLUSIONS
The theoretical background of the Wilson
model for fully-stratified flow, heterogeneous flow and stratified flow with a
shear layer has been examined. Model configurations can numerically analysed in
the MathCad document. Examples of the analysis are presented on Fig. 2 and Fig.
3. Issues from an extensive testing are generalised to the following remarks
regarding a configuration and an application of the computational model and the
nomographs.
1. The viscous bed-wall friction and
horizontal asymptote of resistance curves
It
was assumed originally that viscous friction between bed and pipe wall was that
for clear water at the pipe wall for the average velocity equalled to velocity
of the sliding bed [6]. Then the graph given in [12] can be reproduced by the
outputs of the two-layer model as shown in Fig. 2.
Wilson & Brown have later published [9]
a method for a determination of viscous friction between sliding granular bed
and pipe wall. They compared the viscous friction between a sliding bed and a
pipe wall to the friction between a capsule and a pipe wall. According to their
analysis the viscous friction factor
and wall shear stress should be
determined according to the following procedure.
If 
< 335 then
(25)
If Re2 > 335 then
(26)
The shear stress is
(27)
When this method is implemented to the
computational model, the resistance curve no longer has a horizontal asymptote as
shown in Fig. 4.
Figure
4: Non-dimensional MDV curve and resistance curve from the model with
implemented viscous friction f2 according to [9] (fully-stratified
flow).
Thus an implementation of this method is
not appropriate for the two-layer model. An absence of the horizontal asymptote
in Fig.4 can be explained from the following. The proposed method provides
higher viscous shear stress between bed and pipe wall than is that for fluid.
Therefore ratio Veq/V2 increases with increasing Va
when the slurry flow is simulated for a given bed height. This results in a
decrease in the delivered concentration because all solids are delivered by the
lower layer according to the model structure. To maintain a constant delivered
concentration (as required by a resistance curve of constant Cdel),
the bed height must increase with increasing Va. Thicker granular
bed provides more resistance and so higher pressure gradient in a pipe.
2. The cross section between the MDV curve
and the resistance curve - zero delivered concentration at the MDV curve
A determination of the MDV curve and the
resistance curve in the plot pex vs. Va/Vmax
(Fig.2, 4) is based on the fully-stratified flow pattern. It is assumed that no
particles are delivered until the average velocity in a pipe prevail the
critical value determined by the MDV curve. In most real flow situations some portion of solids is delivered also at
the average velocities below the critical value for which granular bed starts to
slide. This is caused by a suspension of particles due to high fluid velocity
in the upper layer and/or by a development of a shear layer at the top of a
granular bed. Therefore the resistance curves for the low delivered
concentrations should cross the MDV curve.
3. Empirical constant for a determination
of the friction factor at the layers interface
Numerical simulations have shown that the
multiplication coefficient proposed for the Nikuradse equation to determine the
interfacial friction factor does not reproduce the demi McDonald curve. The
coefficient equal to 2.75 (instead of 2.00) provides model outputs matching the
demi McDonald curve for particle sizes for which the fully-stratified flow is
expected (approx. d > 0.7 mm). Even higher value of the coefficient would
have to be used to reproduce the demi McDonald curve for heterogeneous flow (a
curve section for approx. d < 0.7 mm).
NOMENCLATURE
A cross-sectional
area of pipe
A1 cross-sectional area of upper layer
A2 cross-sectional area of lower layer
B empirical
coefficient
volumetric concentration of
solids in shear layer
Cb volumetric concentration of solids in the
loose-packed bed
Cdel relative delivered concentration of
solids
d particle
diameter
D pipe
diameter
Deq equivalent hydraulic diameter
ff Darcy-Weisbach friction factor for fluid flow
f12 Darcy-Weisbach friction factor at
stratified-flow interface
f2 Darcy-Weisbach friction factor for bed flow
F12 driving force on the surface of contact layer
F2 driving force to contact layer due to pressure
gradient
F2d mechanical friction force of contact layer against
pipe wall
F2v viscous friction force between lower layer and
pipe wall
g gravitational
acceleration
i hydraulic
gradient
L1 perimeter of
pipe between upper layer and pipe wall
L12 perimeter of interface between upper layer and
lower layer
L2 perimeter of pipe between lower layer and pipe
wall
P pressure
gradient for mixture flow
Pclear pressure gradient for clear water flow
Pex relative excess pressure gradient
Pplug pressure gradient for plug flow
r absolute roughness of flow boundary
Rb hydraulic radius associated with bed
Re2 Reynolds number
Sf relative density of fluid
Ss relative density of solids
V average
velocity in waterway
Va average slurry velocity in full cross-sectional
area of pipe
VaMDV value of Va at limit of deposition
Veq average velocity in upper layer for which force
balance is found
Vmax maximum value of VaMDV
V1 average velocity in upper layer
V2 average velocity in lower layer
angle
defining position of surface of real/virtual interface
angle
defining position of surface of contact-load layer
s thickness of the shear layer
dynamic
viscosity of fluid
von
Karman constant
mechanic
friction coefficient of solids against pipe wall
density
of fluid
shear
stress at waterway boundary
1 shear stress between upper layer and pipe wall
2 shear stress between granular bed and pipe wall
12 shear stress at stratified-flow interface
angle
of internal friction of particles (dynamic)
Abbreviation:
MDV maximum
deposit velocity
REFERENCES
[1]
Nnadi, F. N. & Wilson, K.C. (1995). Bed-load motion at
high shear stress: dune washout and plane-bed flow, Journal of Hydraulic Engineering, ASCE, 121(3), 267-73.
[2] Riet van, E.J., Matousek, V. & Miedema, S.A. (1995). A reconstruction of and sensitivity
analysis on the Wilson model for hydraulic particle transport, Proc. 8th Int. Conf. on Transport and
Sedimentation of Solid Particles, Prague, Czech republic, pp. A8-1-9.
[3]
Wilson, K.C. (1966).
Bed-load transport at high shear stress, Journal
of the Hydraulic Division, ASCE, 92(HY6),
49-59.
[4] Wilson, K.C. (1970). Slip point of beds in solid-liquid pipeline flow, Journal of the Hydraulic Division, ASCE,
96(HY1), 1-12.
[5] Wilson, K.C. (1974). Co-ordinates for the limit of deposition in pipeline
flow, Proceedings Hydrotransport 3,
BHRA, Cranfield, UK, pp. E1-1-13.
[6] Wilson, K.C. (1976). A unified physically-based analysis of solid-liquid
pipeline flow, Proceedings Hydrotransport
4, BHRA, Cranfield, UK, pp. 1-16.
[7]
Wilson, K.C. & Judge, D.G. (1978). Analytically-based nomographic charts for sand-water flow, Proceedings Hydrotransport 5, BHRA,
Cranfield, UK, pp. A1-1-12.
[8]
Wilson, K.C. & Judge, D.G. (1980). New techniques for scale-up of pilot-plant results to coal
slurry pipelines, Journal of Powder &
Bulk Solids Technology, 4(1),
15-22.
[9]
Wilson, K.C. & Brown, N.P. (1982). Analysis of Fluid Friction in dense-phase pipeline flow, The Canadian Journal of Chemical Engineering,
60, 83-6.
[10] Wilson, K.C. (1984). Analysis of contact-load distribution and application
to deposition limit in horizontal pipes, Journal
of Pipelines, 4, 171-6.
[11]
Wilson, K.C. & Nnadi, F.N. (1990). Behaviour of mobile beds at high shear stress, Proc. 22nd Int. Conf. on Coastal Engrg.,
ASCE, New York, N.Y., Vol. 3, pp. 2536-41.
[12]
Wilson, K.C., Addie, G.R. & Clift, R. (1992). Slurry
Transport Using Centrifugal Pumps. Elsevier Applied Science, London.
[13]
Wilson, K.C. & Pugh, F.J.
(1995). Real and virtual interfaces in slurry flows, Proc. 8th Int. Conf. on Transport and Sedimentation of Solid Particles,
Prague, Czech republic, pp. A4-1-10.
[14]
Wilson, K.C. (1995).
Contact load and suspended load in pipes and open channels, Proc. 8th Int. Conf. on Transport and
Sedimentation of Solid Particles, Prague, Czech republic, pp. B1-1-16.