A Theoretical Description and Numerical Sensitivity Analysis on Wilson's Model for Hydraulic Particle Transport in Pipelines.

E.J. van Riet M.Sc.

V. Matouek M.Sc.

S.A. Miedema Ph.D.

ABSTRACT

The Wilson model for hydraulic particle transport in pipelines is a widely used model. The theoretical background of the model has been published piece by piece in a large number of articles over the years. The variety of information provided in these publications makes the model difficult to reconstruct.

A good understanding of the model is necessary to be able to extend or adapt the model. A description of the theory of the model and the results obtained from solving the equations of equilibrium of the model numerically has been presented here. The calculation results show some peculiarities of the publications and provide subjects for discussion.

Keywords: Hydraulic particle transport, slurry, pipeline.

INTRODUCTION

This paper contains an overview of the theory of the Wilson model for hydraulic particle transport in pipelines as published by Wilson in a large number of articles over the years. Results obtained from numerical calculation of the model are presented. The presented results are intended to provide insight in the behaviour of the mathematical model. The calculations have been carried out with the aid of a Mathcad document that has been described by the authors in [20]. This Mathcad document is available on floppy disk at the authors upon request.

The paper has been sub-divided in five paragraphs:

1 The basic equations for flow and geometry.

2 The force balance for the calculation of maximum deposit velocity and

resistance curves in the fully stratified flow.

3 Incorporation of suspension in the balance (the heterogeneous model).

4 The three layer model.

5 Discussion and conclusions

1. THE BASIC EQUATIONS FOR FLOW AND GEOMETRY

In this paragraph, the basic equations and the variables describing the geometry are listed.

The cross section of the pipe with a particle bed as defined in the Wilson two layer model has been illustrated by fig.1. (For an explanation of the differences between the fully stratified case and the heterogeneous case is referred to paragraphs 2 and 3.)

Figure 1: Terminology for the fully stratified and the heterogeneous two layer model.

The geometry has been defined by the following equations.

The cross sectional lengths:

The cross sectional surface areas:

The hydraulic diameter as function of the bed height [8] is equal to four times the cross sectional area divided by the wetted perimeter:

The Nikuradze rough wall equation for turbulent flow has been used to describe the shear stress on the walls [3,5,11]:

2. THE FORCE BALANCE TO DEFINE THE MDV AND RESISTANCE CURVES FOR FULLY STRATIFIED FLOW

The two most important results of the Wilson model are:

The maximum deposit velocity (MDV). This is the maximum average slurry velocity at which a stationary bed can just exist.

The resistance curves. These curves, depict the pressure drop as a function of the flowrate in a pipe at a constant delivered concentration of solids.

The MDV and the resistance curves are calculated from a force balance of four main forces, acting on a stationary or moving bed of particles in contact with the bottom of a pipe [1,2,4,5,6,13]. These forces are (per unit length of the pipe):

1: A driving shear force on the bed surface which is calculated from the Nikuradze equation multiplied by an empirical constant. Wilson originally assumed this constant to be equal to 2 [5,6].

While calculating in this equation, the velocity difference between the upper and lower layer has to be used.

2: A driving force caused by the pressure gradient times the bed cross sectional area. The pressure gradient is:

The force is:

3: A resisting dry friction force between bed and pipe wall equal to the normal force the bed exerts on the pipe wall multiplied by the dry friction coefficient m [2,13].

4: A viscous friction force between the bed and the pipe wall:

Wilson originally assumed the viscous friction between bed and wall to be equal to the friction of clear water at the same average velocity as the sliding bed [6].

The equilibrium average upper layer velocity V_eq is calculated by solving V₁ from the force balance while keeping the next quantities constant:

The bed height.

The bed velocity.

The physical properties of the fluid and the particles.

The force balance is:

This balance can only be solved numerically because of the implicit set of equations. The V₁ for which the force balance is in equilibrium is V_eq.

The relative delivered solids concentration at equilibrium is hereafter obtained from:

Resistance Curves

A point of the resistance curves can be numerically calculated by solving the force balance while taking the next variables constant:

The bed velocity.

Physical properties of the fluid and particles.

The bed height can now be varied in a numerical iteration procedure until the next two criteria have been satisfied simultaneously:

The force balance is in equilibrium.

The calculated delivered concentration equals the one specified by the desired resistance curve.

By assuming an array of bed velocities, the whole resistance curve can be calculated. The numerical procedure to solve this with the aid of Mathcad has been described by the authors in [20].

Wilson has published dimensionless graphs. To be able to reproduce these graphs, the excess pressure gradient relative to plug flow and the average slurry velocity relative to V_max have to be calculated.

The relative average velocity is easily obtained by averaging the cross sectional areas multiplied by the velocities of the upper and lower layer. Hereafter, this velocity has to be divided by V_max.

The dimensionless excess pressure gradient is defined as:

In this equation, , The pressure gradient of equivalent clear water flow.

And , the pressure gradient for equivalent plug flow

MDV Line

A point of the MDV-line is calculated by solving the force balance for a given bed height while taking V₂=0. The whole curve is obtained when the balance is solved for an array of bed heights. The maximum of the MDV-line gives V_max.

When the equations above are numerically solved with the aid of the method described in [20], the excess pressure gradient of the MDV- and resistance curve are obtained as a function of the dimensionless velocity. The curves are shown in fig 2:

Figure 2: Dimensionless MDV- and resistance curves (fully stratified).

3. THE INCORPORATION OF SUSPENSION; HETEROGENEOUS MODEL

The two layer model has been extended for suspension of particles in the flow. Suspension of particles causes an increase of the specific density and viscosity of the upper layer [7]. This change in the physical properties of the upper layer should explain the significant decrease of the MDV at decreasing particle size below about 0.7mm as shown by the well known demi McDonald [9,13]. Although a decrease can be numerically calculated with the computer model as described in [20], it appeared impossible to reproduce such a strong effect as the demi McDonald shows.

The heterogeneous method is, to the authors' knowledge no longer in use and will therefore not be further explained here. For a more thorough explanation is referred to [20].

Nowadays is recommended to use the three layer model instead of the heterogeneous model for small particles. The three layer model is described in the next paragraph.

4. THE THREE LAYER MODEL.

An improved, theoretically based, model for the description of the shear stress on the bed-fluid interface [8,10,11,17,18] has been published by Wilson in the years from 1984 until the present. This theory has been derived based on the assumption that between the upper layer and the packed bed, a 'shear layer' is present.

Because of the presence of this interface layer, this model is also referred to as the 'three layer model'. The expression that was found however, is suitable for incorporation in the force balance of the two layer model.

The relation to determine the shear stress on the virtual surface of a sand size particle bed has been evolving over the years. Originally was proposed [8] that the hydraulic roughness equals half shear layer thickness and that the shear layer thickness is a function of the shear stress. In this theoretically derived implicit set of equations, the hydraulic diameter was one of the variables.

Wilson and Nnadi later derived that the hydraulic diameter can be cancelled from the equations and that the friction factor at the bed surface only depends on i/(S_s-1). This derivation has been carried out with the aid of a method described by Wilson to calculate the interface associated part of the hydraulic radius [1]. The derivation has been based on the following relation which has been described in [19]:

An thorough explanation of this relation has not been found by the authors in any of the references. Therefore, only the result of the work is discussed here. Readers interested in the subject are referred to [1,18,19]. By evaluating experimental data and this theory, Wilson has derived the constants in the next commonly used rough wall equation:

Wilson has published a number of different versions of this equation and some approximation functions of it. The different constants in these versions have been derived by changing as can be derived from the equations.

(=24⁰ , published in [19])

(=18⁰ , published in [18])

(fit function, published in [17])

(fit function, published in [15])

The first three equations exhibit more or less the same behaviour (Within 20% for f) but that the fourth equation behaves different. This large difference cannot be explained by the authors.

The most recently published value of is 14^o [17]. When this value is incorporated, the equation can be written as:

Based on analytical results obtained from the three layer model, Wilson has derived a fit function to describe V_max mainly as a function of the pipe diameter [13,17].

The fit function is:

Here, f_f is the hydraulic friction factor for fluid alone.

This equation is only valid for 'sand size' particles. 'Sand size' is not well defined. Wilson recommends to use this equation instead of the demi McDonald under a number of circumstances; as a rule of thumb is given [17]:

Use the fit function if the value of V_max obtained from the demi McDonald exceeds the V_max from the fit function.

Use the demi McDonald in other cases.

The authors have simulated the two layer model with their Mathcad computer model [20]. The published hydraulic bed-fluid friction equations have been incorporated herein. The results from this analysis and the behaviour of the fit function have been depicted in fig.3. The authors have used the following values as input during the evaluation of the model: =0.4, r=10^-5, C_b=0.6, S_s=2.65 and f_f=according Nikuradze.

Figure 3: Maximum deposit velocity output of the two layer model for the published interface friction equations.

Legend:

drawn curve: Fit function for Vmax published in [17]

diamonds: Two layer model output with equation published in Prague [18]

squares: Two layer model output with fit function published in Prague [17]

crosses: Two layer model output with equation published in 1995 [19]

plusses: Two layer model output with =14 degrees (derived by the authors)

Based on these results, the following remarks can be made:

The fit function predictions are obviously well reconstructed with all the mentioned interface friction equations. The best result is obtained with the equation that was published in [18] (see above).

A decrease of from 29 to 14 degrees causes a decrease in MDV of 15 to 20%.

5. DISCUSSION AND CONCLUSIONS

The theoretical background and the equations of the fully stratified and heterogeneous Wilson model have been described. The set of equations has been solved numerically. The results that are obtained give rise to some comment on the model. This comment specifically concerns:

The viscous bed-wall friction calculation.

The horizontal asymptotes of the resistance curves

The zero delivered concentration at the MDV-line

The empirical constant in the bed-fluid interface shear stress calculation

The viscous bed-wall friction and horizontal asymptote of resistance curves

Originally, the viscous bed wall friction was assumed to be equal to the friction of water at the same average velocity. In case this assumption is made, the graph given in [13] can be reproduced with the two layer model as has been shown in fig. 2.

Wilson however, has published an improved friction description [14]. He compared the viscous friction of a sliding bed to the friction capsules encounter in a pipe. According to this, the viscous friction factor and wall shear stress change to:

If Reynolds<335:

If Reynolds>335:

And:

The Reynolds number here is:

With these equations implemented into the model, the next dimensionless graph (fig. 4) is the result:

Figure 4: Dimensionless MDV and Resistance curve for improved viscous bed friction model. (fully stratified model).

Note that the resistance lines no longer have horizontal asymptotes.

The absence of the horizontal asymptote in fig.4 can be explained from the following. With increasing velocity, the increase of hydraulic lower layer-wall friction given by this expression is higher than the increase of upper layer-wall friction. This implies that (for a given bed height) V₁/V₂ increases at increasing average velocity. This results in a decrease of the delivered concentration. To maintain a constant delivered concentration, the bed height has to increase with increasing average velocity. As a result of this, the relative pressure loss in the pipe will increase.

The zero delivered concentration at the MDV line

If the bed in the pipe is stationary and the upper layer velocity is high enough, solids are delivered. This is caused by suspension of particles in the flow (heterogeneous model) or from the particles travelling in the shear layer [3,8,10,12] between the upper and lower layer. These effects have not been taken into account in the original model. If these are taken into account, the resistance lines for low delivered concentration should cross the MDV-line.

Empirical constant in interface friction factor calculation

Numerical simulations have shown that the empirical constant that Wilson uses to adapt the Nikuradze equation for the interface friction factor does not reproduce the demi McDonald. This constant should be taken equal to 2.75 to match the fully stratified part of the demi McDonald curve. In the heterogeneous part of the curve, an even higher constant will produce the values to meet the demi McDonald.

NOMENCLATURE

A Cross sectional surface area

A₁ Cross sectional surface area upper layer

A₂ Cross sectional surface area lower layer

C_b Solids concentration in loose packed bed

C_c Relative contact load concentration

C_del Relative delivered solids concentration

C_is Relative in situ solids concentration

C_sr Relative particle concentration in upper layer

D_eq Equivalent hydraulic diameter

d Particle diameter

f Hydraulic friction factor

f_f Hydraulic friction factor for equivalent water flow

F₁ Force between upper layer and pipe wall

F₂ Force between lower layer and pipe wall

F₁₂ Force between upper layer and lower layer

F_2d Dry friction force between lower layer and pipe wall

F_2v Viscous friction force between lower layer and pipe wall

g Gravity

L₁ Circumpherential length upper layer - pipe wall

L₂ Circumpherential length lower layer - pipe wall

L₁₂ Interfacial length upper layer - lower layer

n Parameter in Vocadlo equation

P Pressure gradient

P_clear Pressure gradient at equivalent clear water flow

P_ex Dimensionless excess pressure gradient

P_MDV Pressure gradient at MDV

P_plug Pressure gradient at equivalent plug flow

V Velocity

V_a Average velocity over full pipe cross section.

V_aMDV Average velocity at MDV

V_eq Equilibrium velocity upper layer

V_max Maximum MDV

V_t Threshold velocity

V_s Settling velocity

V₁ Average velocity upper layer

V₂ Average velocity lower layer

r Hydraulic roughness wall

S_f Relative specific density fluid

S_s Relative specific density fluid

y Half shear layer thickness

Virtual bed height angle

Bed height angle

Dynamic friction angle

Shear stress

Dry friction coefficient particles - pipe wall

Dynamic viscosity fluid

Dynamic viscosity slurry according to Vocadlo

Density water

REFERENCES

Articles written by K.C. Wilson et al.:

1 Bed-load transport at high shear stress.

ASCE J. of hydr. div. Vol.92 No.Hy6 Nov. 1966.

2 Slip point of beds in solid-liquid pipeline flow
ASCE J. of hydr. div. Vol 96 N0. HY1 Jan. 1970

3 A formula for the velocity required to initiate particle suspension in pipeline flow.
Proc. Hydrotransport 2, 1972 BHRA, Cranfield

4 Slip-model correlation of dense two-phase flow.
Proc. Hydrotransport 2, 1972 BHRA, Cranfield

5 Co-ordinates for the limit of deposition in pipeline flow.
Proc. Hydrotransport 3, 1974 BHRA, Cranfield

6 A unified physically based analysis of solid-liquid pipeline flow.
Proc. Hydrotransport 4, 1976 BHRA, Cranfield

7 New techniques for the scale-up of pilot plant results to coal slurry pipelines.

J. of Powder & Bulk solids Tech. 4, 1980, 1 pag.15-22

8 Analysis of contact-load distribution and application to deposition limit in horizontal pipes.
J. of pipelines 4 1984 pag. 171-176

9 Effect of solids concentration on deposit velocity.
J. of pipelines 5 1986 pag.251-257

10 Dispersive force modelling of turbulent suspension in hetergeneous slurry flow

Can. J. of Chem Eng. Vol.66 Oct.1986

11 Evaluation of interfacial friction for pipeline transport models.
Proc Hydrotransport 11, 1988 BHRA Cranfield.

12 Behaviour of mobile beds at high shear stress.
Proc. Coastal Eng. 1990.

13 Slurry transport using centrifugal pumps

Elsevier science publ. LTD. 1992, ISBN 1-85166-745-8

14 Analysis of fluid friction in dense phase pipeline flow.
Can. J. of chem. eng. Vol.60 Febr. 1982.

15 Behaviour of Mobile Beds at high Shear Stress
Proceedings Coastal eng. 1990

16 Analysis of Bed Load Motion at high Shear Stress
Journal of Hydraulic Engineering Vol. 113, No.1, Jan. 1987

17 Real and Virtual Interfaces in Slurry Flows

Proc. 8th International Conference Transport and Sedimentation of Solid Particles, 24-26 Jan. 1995, Prague.

18 Contact Load and Suspended Load in Pipes and Open Channels

Proc. 8th International Conference Transport and Sedimentation of Solid Particles, 24-26 Jan. 1995, Prague.

Other:

19 Bed-Load Motion at High Shear Stress: Dune Washout and Plane-Bed Flow
F. N. Nnadi, K.C. Wilson
Journal of Hydraulic Engineering Vol.121 No.3 March 1995.

20 A Reconstruction of and Sensitivity Analysis on the Wilson Model for Hydraulic Particle Transport.
E.J. van Riet, V. Matousek, S.A. Miedema.
Proc. 8th International Conference Transport and Sedimentation of Solid Particles, 24-26 Jan. 1995, Prague.