Reconstruction of and Numerical Sensitivity Analysis on Wilson Model for Hydraulic Transport of Solids in Pipelines.

S.A. Miedema[1], E.J. van Riet[2] and V. Matouek[3]

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 of discussion.

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 the behaviour of the mathematical model. The computation has been completed out using the MathCad document described by the authors in (van Riet et al., 1995).

GEOMETRY OF TWO-LAYER MODEL

A schematic cross section of a pipe is illustrated in figure 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 (Wilson, 1984).

(7)

FORCE BALANCE TO DETERMINE THE MAXIMUM DEPOSIT VELOCITY CURVE AND THE RESISTANCE CURVE FOR FULLY-STRATIFIED FLOW

The Wilson model uses the following important parameters for the slurry pipeline design and operation:

The maximum deposit velocity V_aMDV (at the MDV curve). V_aMDV is the maximum average velocity of slurry flow in a pipe at which a stationary bed still occurs. The MDV curve depicts V_aMDV 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 coordinates.

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 (Wilson, 1970, 1974, 1976). 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 the Nikuradse friction equation for turbulent flow in a hydraulically-rough pipe (Wilson, 1984):

(8)

The MathCad document solves a set of the model balance equations.

The equilibrium average velocity V_eq in the upper layer is obtained by solving V₁ in the force balance while the following quantities are kept constant:

Bed height

Bed velocity

Physical properties of the fluid and solids.

V_eq is velocity V₁ for which the force balance is found by iteration in the MathCad document.

The 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 (Wilson, 1976).

(9)

Shear stress ₁₂ is calculated for velocity equal to the difference between the 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 the normal force exerted by the bed against the pipe wall multiplied by the mechanical friction coefficient m (Wilson, 1970, Wilson et al, 1992).

(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 (Wilson, 1976, 1978, Wilson et al., 1992), 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 (C_b, , S_s) 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 on the resistance curve (i-V_a curve for constant C_del) 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 C_del required by the constructed resistance curve.

The resistance curve computed is presented in the same plot as a nomograph in the literature (Wilson et al., 1992).

The following dimensionless parameters are used in the nomograph:

• relative velocity V_a/V_max
• relative concentration C_del
• 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 V_max.

The MDV curve and the resistance curve are plotted in figure 2. This figure is a product of the MathCad document described in (van Riet et al., 1995).

Figure 2: Non-dimensional MDV curve and resistance curve (fully stratified flow).

THE INCORPORATION OF SUSPENSION, HETEROGENEOUS MODEL

An adaptation of the two-layer model has been proposed (Wilson, 1976) 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 figure 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 (Wilson et al., 1980). This change in the physical properties of flow should explain a significant decrease of the V_max with decreasing particle size (for particles smaller than approximately 0.7 mm) provided by the curve of the demi McDonald nomograph (V_max=f(d, D, S_s)) (Wilson et al., 1978, 1992). Although this decreasing trend can be produced by a numerical simulation of the model (van Riet et al., 1995), it appears impossible to reproduce such a large drop in the V_max 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 a packed granular bed and water above the bed is called the 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.

THE THREE-LAYER MODEL

Publications (Nnadi et al., 1995, Wilson et al., 1966, 1984, 1990, 1995, 1995) deal with a description of the shear on the bed-fluid interface. Originally it was assumed (Wilson, 1984) that the hydraulic roughness of the interface is equal 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 D_eq was one of the variables.

Later Wilson and Nnadi (1990) derived that the hydraulic diameter can be cancelled from the equations and that the friction factor at the bed surface depends only on i/(S_s-1) providing the following relationship

(17)

Rb should be determined using a method from (Wilson, 1966). An application of the eq. (7) has led to the following semi-empirical formula expressing a friction law for sheet flow (Wilson et al., 1990)

(18)

revised in (Wilson et al., 1995) as

(19)

Eq.(17) has been also implemented in the general friction equation for a rough-wall boundary, that is expressed as:

(20)

Empirical constants in eq. (20) have been determined by a calibration of eq. (20) by the experimental data. Different constants have been published for different data (characterised here by different ):

• equation published in (Nnadi et al., 1995) for =24

(21)

• equation published in (Wilson, 1995) for =18

(22)

The equations (19, 21, 22) give similar f₁₂ values but the eq. (18) differs.

When the recently published value =14 (Wilson et al., 1995) 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 the form of a fit function (Wilson et al., 1992, 1995). The three-layer model outputs have shown that V_max is not dependent on the particle diameter when the friction law for sheet flow is used for the interface between layers

(24)

Wilson and Pugh (1995) have recommended to use this equation instead of the curve in the demi McDonald nomograph when the value of V_max obtained from the demi McDonald nomograph exceeds that from the fit function.

The three-layer model has been tested in the MathCad document (van Riet et al., 1995). The V_max outputs for various friction equations are compared with the fit function in figure 3. The following input parameters to the model are used: =0.4, r=10-5, C_b=0.6, S_s=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.

The fit function eq. (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 V_max of 15 - 20%.

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 be numerically analysed in the MathCad document. Examples of the analysis are presented in figures 2 and 3. Issues from an extensive testing are generalized 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 equal to the velocity of the sliding bed (Wilson, 1976). Then, the graph given in (Wilson et al., 1992) can be reproduced by the outputs of the two-layer model as shown in figure 2.

Wilson and Brown (1982) have later published 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 in the computational model, the resistance curve no longer has a horizontal asymptote as shown in figure 4.

Figure 4: Non-dimensional MDV curve and resistance curve from the model with implemented viscous friction f₂ according to (Wilson et al., 1982) (fully-stratified flow).

Thus, an implementation of this method is not appropriate for the two-layer model. An absence of the horizontal asymptote in figure 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 the ratio V_eq/V₂ increases with increasing V_a 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 C_del), the bed height must increase with increasing V_a. A thicker granular bed provides more resistance and so a higher pressure gradient exists in a pipe.

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 p_ex vs. V_a/V_max (see figures 2 and 4) is based on the fully-stratified flow pattern. It is assumed that no particles are delivered until the average velocity in a pipe exceeds 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.

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	m^2
A1	cross-sectional area of upper layer	m^2
A2	cross-sectional area of lower layer	m^2
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	mm
D	inside pipe diameter	m
Deq	equivalent hydraulic diameter	m
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	N
F2	driving force to contact layer due to pressure gradient	N
F2d	mechanical friction force of contact layer against pipe wall	N
F2v	viscous friction force between lower layer and pipe wall	N
g	gravitational acceleration	m/s^2
i	hydraulic gradient	-
L1	perimeter of pipe between upper layer and pipe wall	m
L12	perimeter of interface between upper layer and lower layer	m
L2	perimeter of pipe between lower layer and pipe wall	m
MDV	maximum deposit velocity	m/s
P	pressure gradient for mixture flow	Pa
Pclear	pressure gradient for clear water flow	Pa
Pex	relative excess pressure gradient	-
Pplug	pressure gradient for plug flow	Pa
r	absolute roughness of flow boundary	mm
Rb	hydraulic radius associated with bed	m
Re2	Reynolds number	-
Sf	relative density of fluid	-
Ss	relative density of solids	-
V	average velocity in waterway	m/s
Va	average slurry velocity in full cross-sectional area of pipe	m/s
VaMDV	value of Va at limit of deposition	m/s
Veq	average velocity in upper layer for which force balance is found	m/s
Vmax	maximum value of VaMDV	m/s
V1	average velocity in upper layer	m/s
V2	average velocity in lower layer	m/s
	angle defining position of surface of real/virtual interface
	angle defining position of surface of contact-load layer
s	thickness of the shear layer	mm
	dynamic viscosity of fluid	Pa.s
	von Karman constant
	mechanic friction coefficient of solids against pipe wall	-
	density of fluid	kg/m3
	shear stress at waterway boundary	Pa
1	shear stress between upper layer and pipe wall	Pa
2	shear stress between granular bed and pipe wall	Pa
12	shear stress at stratified-flow interface	Pa
	angle of internal friction of particles (dynamic)

REFERENCES

Nnadi, F. N. and Wilson, K.C. (1995). "Bed-load Motion at High Shear Stress: Dune Washout and Plane-bed Flow". Journal of Hydraulic Engineering, ASCE, 121(3).

Riet van, E.J., Matousek, V. and 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.

Wilson, K.C. (1966). "Bed-load Transport at High Shear Stress". Journal of the Hydraulic Division, ASCE, 92(HY6).

Wilson, K.C. (1970). "Slip Point of Beds in Solid-liquid Pipeline Flow". Journal of the Hydraulic Division, ASCE, 96(HY1).

Wilson, K.C. (1974). "Coordinates for the Limit of Deposition in Pipeline Flow". Proceedings Hydrotransport 3, BHRA, Cranfield, UK.

Wilson, K.C. (1976). "A Unified Physically Based Analysis of Solid-liquid Pipeline Flow". Proceedings Hydrotransport 4, BHRA, Cranfield, UK.

Wilson, K.C. and Judge, D.G. (1978). "Analytically Based Nomographic Charts for Sand-water Flow". Proceedings Hydrotransport 5, BHRA, Cranfield, UK.

Wilson, K.C. and 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).

Wilson, K.C. and Brown, N.P. (1982). "Analysis of Fluid Friction in Dense-phase Pipeline Flow". The Canadian Journal of Chemical Engineering, 60.

Wilson, K.C. (1984). "Analysis of Contact-load Distribution and Application to Deposition Limit in Horizontal Pipes". Journal of Pipelines, 4.

Wilson, K.C. and 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.

Wilson, K.C., Addie, G.R. and Clift, R. (1992). "Slurry Transport Using Centrifugal Pumps". Elsevier Applied Science, London.

Wilson, K.C. and 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.

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.

Keywords: Hydraulic transport, two-layer model, slurry pipeline.

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[1] Dr.ir. Sape A. Miedema, Delft University of Technology, Faculty of Mechanical Engineering and Maritime Technology, Chair of Dredging Technology, Postbus 5034, 2600 GA Delft, the Netherlands.

Email: s.a.miedema@wbmt.tudelft.nl

[2] Ir. Egbert J. van Riet, Shell Research BV Rijswijk, Postbus 60, 2288 GD Rijswijk, the Netherlands.

[3] Dr.ir. Vaclav Matousek, Delft University of Technology, Faculty of Mechanical Engineering and Maritime Technology, Chair of Dredging Technology, Postbus 5034, 2600 GA Delft, the Netherlands.