Analytic model of laminar-turbulent transition velocity for Bingham plastics

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ANALYTIC MODEL OF LAMINAR-TURBULENT TRANSITION FOR BINGHAM PLASTICS

K. C. Wilson1* and A. D. Thomas2

1. Queen’s University, Department of Civil Engineering, Kingston, ON, Canada K7L 3N6

2. Slurry Systems Pty Ltd., 7 Cantwell Road, Lochinvar NSW 2321, Australia

It is often desirable to operate industrial pipelines transporting non-Newtonian materials near the transition from laminar to turbulent ﬂ ow. For the commonly used Bingham plastic model, the Hedström technique overestimates turbulent ﬂ ow friction losses because it does not take account of viscous-layer thickening. In the present paper, the Wilson-Thomas model is applied to predict the transition point for Bingham plastics. Laminar and turbulent friction losses are calculated to show that conditions at transition depend only on the Hedström number. The results are approximated by simpliﬁ ed ﬁ t functions. Comparison with existing empirical correlations and experimental data from various sources shows satisfactory agreement.

Il est souvent préférable d’utiliser les pipelines industriels transportant des matériaux non newtoniens près de la transition entre l’écoulement laminaire et l’écoulement turbulent. Pour le modèle classique des ﬂ uides de Bingham, la technique d’Hedström surestime les pertes de friction de l’écoulement turbulent parce qu’elle ne prend pas en compte l’épaississement de la couche visqueuse. Dans le présent article, on applique le modèle de Wilson-Thomas pour prédire le point de transition pour des ﬂ uides de Bingham. Les pertes de friction laminaires et turbulentes sont calculées et montrent que les conditions lors de la transition dépendent uniquement du nombre d’Hedström. Les résultats sont exprimés sous forme approximative par des fonctions de calage simpliﬁ ées. La comparaison avec des corrélations empiriques et des données expérimentales provenant de diverses sources montre un accord satisfaisant.

Keywords: non-Newtonian ﬂ ow, turbulent ﬂ ow, viscous-layer thickening, transition, pipelines

numbers or friction factors that could be substituted into equations for Newtonian pipe ﬂ ow, rather than considering turbulent mechanisms.

INTRODUCTION T

he ﬂ ow of non-Newtonian materials is of major commercial importance in many industries. In pipeline ﬂ ow of such materials it is often desirable (for reasons of stability and economics) to operate near the transition from laminar to turbulent ﬂ ow.

Over the decades, many formulas or “rheologic models” have been developed to represent non-Newtonian rheograms. It has been found that models with only two rheological parameters (one degree of complexity more than the basic Newtonian model) are of the greatest value from an engineering viewpoint. The most widely used in practice is the well-known Bingham or yield-plastic model, which is analyzed in the present paper.

The turbulent ﬂ ow of Newtonian ﬂ uids in pipes is adequately understood for engineering purposes, although details may still be subject to uncertainty. In the simple models in common engineering use, the effect of viscosity is conﬁ ned to a thin viscous sublayer near the pipe wall. Thus it is logical to expect that the main effect of non-Newtonian viscous parameters should also be in the sublayer. Making use of a suggestion made by Lumley (1973, 1978) to account for drag reduction caused by long-chain molecules, the authors of the present paper proposed some years ago that the viscous sublayer for non-Newtonian turbulent ﬂ ows is generally thicker than for equivalent Newtonian ﬂ ows and put forward a quantitative model to calculate this effect (Wilson and Thomas; 1985; Thomas and Wilson, 1987). This model has been widely cited in the textbook literature, including

* Author to whom correspondence may be addressed. E-mail address: [email protected]

For laminar ﬂ ow in circular pipes, the analysis of the ﬂ ow of a Bingham plastic is a classical case that has been known for decades; but turbulent ﬂ ow, and the transition between the two types, has been more resistant to analysis. The early work of Hedström (1952), which will be considered below, is found to give estimates of turbulent friction gradient that are often much too conservative. Many subsequent workers have simply attempted to devise expressions for non-Newtonian Reynolds

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At this point it is useful to introduce a dimensionless parameter developed by Hedström (1952)—the Hedström number He, deﬁ ned as:

Shook and Roco (1991), Chhabra and Richardson (1999), and Wilson et al. (2006). An extended version of the model will be used below to analyze turbulent ﬂ ow and the laminar-turbulent transition for pipeline ﬂ ows of Bingham plastics.

2 (5)

He = ρ τB D2/ ηB

LAMINAR FLOW OF NON-NEWTONIAN MATERIALS

The next step is to consider the ratio of V to the shear velocity U*, which is equal to √[τo/ρ], or θ0.5(τB/ρ)0.5. For a Bingham plastic this ratio is given by Equation (4) as:

V/U* = DτBθ ρ0.5/(8ηB(τB))0.5θ 0.5) [1- 4/(3θ) + 1/(3θ4)] (6)

Together with Equation (5), this expression simpliﬁ es to:

For a plastic material, the rheogram does not pass through the origin, and no strain rate occurs until the shear stress exceeds some yield value. As noted above, the simple two-parameter model called the Bingham plastic is often employed to quantify this behaviour. Its rheogram is linear, passing through the Bingham yield stress τB at du/dy = 0 and having a slope ηB that is called “plastic viscosity,” or “tangent viscosity,” thus:

V/U* = (He)0.5 θ0.5 [1- 4/(3θ)+1/(3θ4)]/8 (7)

τ τ η = + du/dy B B (1)

TURBULENT FLOW OF NON-NEWTONIAN MATERIALS

where u is local velocity and y is distance from the wall. This behaviour is shown schematically in Figure 1.

As shown on this ﬁ gure, the viscosity μ, i.e., the ratio τ/(du/ dy) represents the slope of a straight line passing through the origin. For a Bingham plastic, it is convenient to use a reduced stress variable θ, deﬁ ned as τ/τB (for pipe ﬂ ow θ = τo/τB, where τo is the shear stress at the pipe wall). From the geometry of Figure 1, it is found that:

μ = ηB θ/(θ -1) (2)

In order to understand and analyze non-Newtonian turbulent ﬂ ow, some sort of ﬂ ow model is required. One approach is to begin with the expression for the mean velocity of Newtonian turbulent ﬂ ow in a smooth-walled pipe. For tests of turbulent ﬂ ow of a non-Newtonian material, all quantities except μ will be available for each data point. Therefore, the equation can be solved for the viscosity in each case; the viscosity thus determined is called the equivalent turbulent-ﬂ ow viscosity, and given the symbol μeq. The equation is then written:

V = 2.5 U* ln(ρ D U*/μeq) (8)

Substitution of the Hedström number (analogous to the derivation of Equation (7) gives the ratio V/U* as:

V/U* = 2.5 ln[(He)0.5θ 0.5(ηB /μeq )] (9)

If the material being tested had been a Newtonian ﬂ uid, the rheogram would have been a straight line through the origin. As noted by Wilson and Thomas (1985), a measure of the departure from Newtonian behaviour is the area ratio α, i.e., the ratio of the stippled area beneath the rheogram of Figure 1 to the triangular area for a Newtonian ﬂ uid having the same value of τ. For a Bingham plastic this ratio is given in terms of θ by:

α = (θ + 1)/θ (3)

At this point a model must be developed for predicting μeq from the rheogram. An early model was that of Hedström (1952) who simply set μq equal to ηB. However, as demonstrated below, this model does not give good predictions for turbulent ﬂ ow.

For the case of laminar ﬂ ow in a circular pipe, the shear stress varies linearly from τo at the wall to zero at the centre line and, by Equation (2), μ can be found at any point in the ﬂ ow. Integration gives the velocity at any location, and a further integration can be used to determine the discharge and hence the mean velocity V, as shown in any rheology text. The result is:

V = D τo /(8 ηB)[1- 4/(3θ) + 1/(3θ4)] (4)

An adequate model of the turbulent ﬂ ow of non-Newtonians must begin from Newtonian turbulent ﬂ ow, which itself is a complex phenomenon. However, all that is usually required is a reasonable approximation to the velocity distribution. The model developed by the authors (Wilson and Thomas, 1985; Thomas and Wilson, 1987) was based on the simple ‘engineering’ proﬁ le in which the effect of ﬂ uid viscosity is conﬁ ned to a thin sublayer which extends from the wall to a distance δ given by:

δ μ ρ 11.6 / U ≈ * (10)

where μ and ρ are the viscosity and density, respectively, of the Newtonian ﬂ uid and U* is the shear velocity at the wall, as deﬁ ned previously. Within the sublayer the velocity proﬁ le is taken to follow the linear relation, which incorporates the usual no-slip wall condition:

u = τoy/μ (y < δ) (11)

In the main ﬂ ow it is assumed that all momentum transfer takes place by turbulent mixing, which is an inertial process rather than a viscous one. This gives rise to the well-known logarithmic velocity proﬁ le in this region. As the velocity Figure 1. Schematic rheogram for Bingham plastic

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gradient in the viscous sublayer is much steeper than that in the logarithmic zone, the sublayer has an inﬂ uence on mean velocity which is disproportionate to its small thickness, and thus it is the sublayer that determines the inﬂ uence of viscosity on the friction factor for Newtonian turbulent ﬂ ow.

At large values of y the inertial macro-eddies are much bigger than the dissipative micro-eddies, and between them is a whole range of turbulent eddy sizes, shown shaded in Figure 2. As the wall is approached, the range of possible eddy sizes shrinks until the size of the largest and smallest eddies are equal, indicating the elimination of turbulence. At this point y equals δ, the thickness (in a statistical sense) of the viscous sublayer.

The dashed line of Figure 2 shows that increasing the size of the dissipative micro-eddies leads to an increase in the thickness of the viscous sublayer. It is known that the micro-eddy size is increased by long-chain molecules, and non-Newtonian rheological properties can produce a similar effect, as shown by Wilson (1989) and Wang and Larsen (1994). If other quantities are unaffected the thickened sublayer will, in turn, produce a higher mean velocity for the same wall shear stress, giving a lower friction factor.

If a non-Newtonian ﬂ uid is to be substituted for the Newtonian one it is expected that in the logarithmic zone the momentum transport is still inertial in nature. The velocity gradient in this zone was not affected by Newtonian viscosity, and it would be expected that the same should apply for non-Newtonian rheological properties. (In the part of the ﬂ ow nearest the pipe axis, some change in the velocity proﬁ le results from nonzero τB, and this will be incorporated below as a term denoted Ω.) Within the sublayer, which occupies such a small portion of the ﬂ ow that variations in shear stress and velocity gradient are negligible, laminar conditions are approached. Here a linear velocity increase equivalent to Equation (11) should also apply for a non- Newtonian ﬂ uid.

Two questions arise at this point. The ﬁ rst concerns the value of μ that applies in Equations (10) and (11), and the second refers to the coefﬁ cient that speciﬁ es the thickness of the viscous sublayer (for Newtonian ﬂ ows this is 11.6, as given in Equation (10), but the value is larger for non-Newtonians, as discussed below). The ﬁ rst question can be resolved noting that the “secant” slope μ is equivalent to the Newtonian viscosity in that it is the ratio of τ to du/dy. Thus, Equation (10) should still apply in principle for non-Newtonians, provided μ is understood in this sense and evaluated at τ = τo.

In the predictive model for non-Newtonians developed by the authors (Wilson and Thomas, 1985; Thomas and Wilson, 1987) it was noted that the interaction of the eddies in turbulent ﬂ ow causes them to increase in axial length, often very rapidly. This “vortex stretching” process produces abrupt increases in strain rates, and in the viscous dissipation of the smallest eddies. As the energy available for dissipation is ﬁ xed by the “turbulent energy cascade,” the result is an increase in the micro-eddy size. The model uses the ratio of the integrals under the non- Newtonian and Newtonian rheograms—denoted by α, and deﬁ ned above in connection with Figure 1—to estimate the size increase of the micro-eddies. From the relationship shown on the ﬁ gure, it follows that the thickness of the sublayer should also be multiplied by a factor equal to α, while that of the superposed logarithmic layer is reduced by (α - 1). The result of these thickness changes is a displacement of the velocity proﬁ le beyond the sublayer. This displacement can be expressed formally by introducing an equivalent viscosity µsl into the wellknown logarithmic velocity proﬁ le, giving:

u/U* = 2.5 ln(ρyU*/µsl) + 5.5 (12)

The evaluation of µsl is based on the combination of the thickened sublayer, proportional to µα, and the reduced logarithmic layer, dependent on (α - 1). The result is:

The remaining question concerns the thickness of the viscous sublayer. Wilson and Thomas (1985) proposed that this can be related to a conceptual model proposed by Lumley (1973, 1978) to explain the phenomenon of drag reduction in aqueous ﬂ ows, wherein small quantities of certain long-chain molecules were added and a substantial reduction in frictional pressure drop was observed. These substances act to increase the size of the smallest, dissipative, turbulent eddies. Figure 2, which illustrates Lumley’s model, graphs the distance from the wall, y, as the ordinate and representative eddy sizes as the abscissa. As indicated in the ﬁ gure, the size of the large eddies (the macroscale of turbulence) is directly proportional to the distance from the wall, while the size of the smallest eddies (the dissipative, or Kolmogorov, scale) does not change much with this distance.

sl = e μ μ α α − − 4 64 1. ( ) (13)

where the coefﬁ cient 4.64 represents the product of the sublayer coefﬁ cient of 11.6 and von Kármán’s coefﬁ cient of 0.4. For a Bingham plastic, μ and α are given in terms of θ by Equations (2) and (3), and Equation (13) becomes:

μsl = ηB[(θ + 1)/(θ - 1)]e-4.64/θ (14)

Substitution of Equation (14) into Equation (9) gives the following expression for the ratio of V to U* for turbulent ﬂ ow of a Bingham plastic:

V/U* = 2.5 ln[(He)0.5] + 2.5 ln[θ 0.5(θ - 1)/(θ + 1)] + 11.6/θ - Ω (15)

The ﬁ nal term in Equation (15) takes account of any blunting of the velocity proﬁ le near the pipe centre line where τ < τB. As introduced by Wilson and Thomas (1985), Ω depends only on θ, and can be written:

Figure 2. Eddy scales in turbulent ﬂ ow (from Wilson and Thomas, 1985)

Ω = - 2.5 ln[(θ - 1)/θ] - 2.5(θ + 0.5)/θ2 (16)

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Whether such blunting actually occurs remains a moot point, reference may be made to Xu et al. (1993).

below the line of μeq = ηB, indicating that μeq, and hence μsl, are substantially less than ηB. This ﬁ nding implies that there is indeed a signiﬁ cant thickening of the sublayer for turbulent ﬂ ows of Bingham plastics.

A third point illustrated by Figure 3 is that the turbulent ﬂ ow of a Bingham plastic tends to show remarkably small variations in f for large changes in ﬂ ow velocity. In addition, this “plateau” value of f is smaller for the larger pipe, which follows from the present model as an effect of Hedström number. This behaviour of f is quite distinct from the wall-roughness effect shown on the Stanton-Moody diagram for Newtonian pipe ﬂ ow, since the latter lies above the smooth pipe line, not below it.

REMARKS ON LAMINAR-TURBULENT TRANSITION

It should also be noted that the V/U* can be written √(8/f) where f is the Darcy-Weisbach friction factor. Thus, f for a turbulent ﬂ ow of a Bingham plastic can be determined directly from Equation (15) (with Equation (16) substituted in, if required). Values of f determined in this way are plotted in Figure 3 against Hedström’s version of the Reynolds number (i.e. ρVD/ηB). The parameters τB and ηB were evaluated to match a 7% kaolin slurry tested by Thomas (Wilson and Thomas, 1985) in the three pipe sizes noted. It is worth noting that the basic similarity of the results for the three pipes sizes studied supports the use of the no-wall-slip condition. This slurry had a rheogram that was very close to Bingham plastic behaviour. The ﬁ gure also shows Thomas’ data for these experiments, and the line of Hedström’s hypothesis that μeq = ηB.

Figure 3 illustrates several basic points. The ﬁ rst is the good general agreement between the values calculated from the model and the experimental data. In this regard, noteworthy agreement for various Bingham plastics has been found by other authors including Xu et al. (1993). The second point is that, for turbulent ﬂ ow, both calculated and observed values of f fall signiﬁ cantly

Figure 4 displays the kaolin data used for Figure 3, plotting both laminar and turbulent ﬂ ow on a logarithmic graph of pressure gradient versus ﬂ ow velocity V. The transitions shown in this ﬁ gure are distinguished by a clear change of slope when the ﬂ ow shifts from laminar to turbulent, marking a distinct intercept of the two individual curves.

This type of intercept is typically found when the material is strongly non-Newtonian (i.e., has a large value of the Hedström number). On the other hand, at sufﬁ ciently low Hedström numbers [below about 1700, as will be shown below] the behaviour begins to resemble that of a Newtonian ﬂ uid, where transition occurs when the Reynolds number ρVD/μeq has a certain value, usually taken as ρVTD/μeq ≈ 2100. For this case, it is useful to know the friction factor, which is found, by iteration, to be fT ≈ 0.049, equivalent to VT/U* = 12.76. The latter value can be substituted directly into the left-hand side of Equation (15) to obtain the required condition in terms of He and θ. For engineering purposes, it is generally more convenient to use the Hedström-type Reynolds number ρVTD/ηB, and on this basis the transition condition for low Hedström numbers can be closely approximated by the ﬁ t equation:

ρVTD/ηB = 2100/{1.0 + 8.3(10-8)[Log10(He)]13} [He < 1700] (17)

Figure 3. Logarithmic plot of f versus ρVD/ηB

For cases like those described by Figure 4, typical of higher Hedström numbers, the laminar-turbulent intercept can be obtained by equating Equations (7) and (15), which have a common left-hand side, a single parameter He, and a variable θ. First, He is set to a series of values and then, for each one, θ is varied until the desired equality is achieved. This value of θ can then be substituted back into the equations to obtain VT/U*, where VT is the ﬂ ow velocity at transition. As shown below, this in turn gives the friction factor at transition, fT, and the ratios ρVTD/ηB and VT/[(τB/ρ)0.5], the last of which is also denoted by VTrel. Comparison with experimental results will be presented in the following section.

ANALYTIC MODEL AND ITS OUTPUT

Figure 4. Logarithmic plot of pressure gradient versus V

The basic relationships for the present analytic model of laminarturbulent transition for pipeline ﬂ ow of Bingham plastics have been presented in the previous sections as Equations (7) and (15) (with the substitution of Equation (16)) plus the condition for ρVD/μeq = 2100 (given by Equation (17)). As noted, the lefthand side of Equations (7) and (15) equals √(8/f), and hence the solution of these equations can be reduced to a plot of f at transition (fT) as a function of Hedström number He. This plot is

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see Tuft (1978). Wasp (2005) is a personal communication and the publications by Kenchington (1978) and Kazanskij et al. (1978) are listed in the References below.

shown as Figure 5. As noted above, the condition that ρVD/ μeq ≈ 2100 is equivalent to f of 0.049. This forms an upper limit to fT in Figure 5, applying to all values of He < 1700.

A particularly useful way of displaying the model output and the data is shown as Figure 8, which again uses the logarithm of He on the abscissa, but now has as ordinate VT/[(τB/ρ)0.5]. The latter quantity, called the relative transition velocity VTrel, equals

An alternative method of plotting the result of the model is shown in Figure 6, which also has the Hedström number on the abscissa, but has on the ordinate the ratio ρVTD/ηB, which is equal to the product [VT/U*](He)0.5θ0.5, and can readily be calculated from the output of the model. For He > 105, the model output is virtually identical to the equation:

ρVTD/ηB = 25(He)0.50 [He >105] (18)

In the intermediate range between the applicability of Equations (17) and (18), the output of the model can be approximated by the equation:

ρVTD/ηB = 80(He)0.40 [1700 < He <105] (19)

The next step is to compare the lines of Figure 6 to available experimental data. First, however, it is worth stressing that experimental data have not been used in deriving the lines on Figure 6, which represent the predictions of the Wilson-Thomas model based on sublayer thickening. (The solid lines are the model predictions and the dotted lines are the approximating functions.)

The corpus of data which is plotted as Figure 7 (using the same axes as Figure 6) comes from many sources. For the points marked “loam clay,” “kaolin,” “magnetite concentrate,” “minus 1 mm coal” and “clay + ﬁ ne sand,” see Thomas (1978). For the zinc concentrate see Weston et al. (1978), and for the silica ﬂ our

Figure 7. Logarithmic plot of ρVTD/ηB versus He, showing data

Figure 5. Logarithmic plot of fT versus He

Figure 6. Logarithmic plot of ρVTD/ηB versus He, showing correlations and ﬁ t lines

Figure 8. Plot of VTrel versus Log(He)

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NOMENCLATURE

[VT/U*]θ0.5 and again can readily be calculated from the model output. For values of He between 1700 and 105, VTrel can be approximated by 80(He)-0.10 (equivalent to Equation (19)).

It is particularly noteworthy that for all values of He greater than about 105 the relative transition velocity is effectively equal to a constant value, predicted by the Wilson-Thomas model to be 25, i.e.;

VT/(τB/ρ)0.5 = 25 [He ≥105] (20)

D internal pipe diameter (m) f friction factor (Darcy-Weisbach) (-) fT value of f at laminar-turbulent transition (-) He Hedström number (see Equation (5)) (-) u local velocity at distance y from wall (m/s) U* shear velocity (√[τo/ρ]) (m/s) V mean (ﬂ ow) velocity (m/s) VT value of V at laminar-turbulent transition (m/s) VTrel relative transition velocity VT/(τB/ρ)0.5 (-) y distance from pipe wall (m)

Greek Symbols

Comparison may be made with correlations developed by other workers. One of the best-known is that of Thomas (1963), which is equivalent to a quadratic equation linking ρVTD/ηB and He. As shown elsewhere (Wilson et al. 2006, p. 75), this predicts a value of VT/(τB/ρ)0.5 = 30 at He = 104, dropping to 19 at He ≥ 108. Another early proposal was that of Hanks (1963).

α area ratio of rheogram (-) δ thickness of viscous sublayer (m) ηB Bingham plastic (tangent) viscosity (Pa.s) θ shear stress ratio τo/τB (-) μ secant viscosity (Pa.s) μeq equivalent viscosity for turbulent ﬂ ow (see Equation (8)) (Pa.s) μsl equivalent viscosity for viscous sublayer (see Equation (13)) (Pa.s) ρ density of material (kg/m3) τ shear stress (Pa) τB Bingham (yield) shear stress (Pa) τo shear stress at pipe wall (Pa) Ω effect of τB on velocity-proﬁ le blunting (see Equation (16))

Although references to this proposal are still found in the literature, it was shown by Govier and Aziz (1972), Venton (1982) and Wilson and Thomas (1985) that Hanks’ proposed transition velocity seriously under-predicts experimental determinations. Later, Malin (1997) applied both k-ε and k-ω modelling to turbulent pipe ﬂ ow of Bingham plastics. His analysis correctly predicts that the laminar line for f passes through the Hedström line and then drops below it (like the behaviour shown in Figure 3). However, it has the turbulent line snapping back to the Hedström line at higher He; thus failing to duplicate the plateau in f which is observed experimentally (and predicted by the present model). In this instance, it appears that the more complex turbulent models have not yet matched that of simple sublayer thickening.

REFERENCES

Chhabra, R. P. and J. F. Richardson, “Non-Newtonian Flow

in the Process Industries,” Butterworth-Heinemann, Oxford, U.K. (1999). Govier, G. W. and K. Aziz, “The Flow of Complex Mixtures in

A more recent publication, that of Slatter and Wasp (2000), is based directly on the correlation of experimental points, using straight ﬁ t lines on a logarithmic plot of ρVTD/ηB versus He. For the range 1.7 x 103 < He < 1.5 x 105, their result is equivalent to ρVTD/ηB = 155 He0.35 (which plots rather near Equation (19) above), and for He > 1.5 x 105 they obtained VT/(τB/ρ)0.5 = 26 (which is remarkably close to Equation (20)).

Pipes,” Van Nostrand Reinhold, New York, NY, U.S. (1972) Hanks, R. W., “The Laminar-Turbulent Transition for Fluids

with a Yield Stress,” AIChE J. 9(3), 306–309 (1963). Hedström, B. O. A., “Flow of Plastics Materials in Pipes,”

CONCLUSION

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Pseudoplastic Slurries in Pipe Flow,” Proc. Hydrotransport 5, BHR Group Ltd., Cranﬁ eld, Bedford, U.K. (1978), pp. C3-35– C3-48. Kenchington, J. M., “Prediction of Pressure Gradient in Dense

Phase Conveying,” Proc. Hydrotransport 5, BHR Group Ltd., Cranﬁ eld, Bedford, U.K. (1978), pp. D7-91–D7-102. Lumley, J. L., “Drag Reduction in Turbulent Flow by Polymer

Non-Newtonian pipeline ﬂ ows are of importance in many industries, and for reasons of stability and economics it is often desirable to operate near the laminar-turbulent transition. Although the laminar side of the transition was solved decades ago, the turbulent side has been less well understood. For the commonly-used two-parameter Bingham plastic model, the old Hedström technique overestimates friction losses. The authors of the present paper showed that this occurs because the Hedström model does not take account of viscous-layer thickening.

Additives,” J. Macromol. Sci., Polym. Rev. 7, 263–290 (1973). Lumley, J. L., “Two-Phase Flow and Non-Newtonian Flow,”

in Turbulence, P. Bradshaw, Ed., Topics in Applied Physics, Vol. 12, Springer-Verlag, Berlin, Chap. 7 (1978). Malin, M. R., “The Turbulent Flow of Bingham Plastic Fluids

in Smooth Circular Pipes,” Int. J. Heat Mass Transfe 24(6), 793–804 (1997). Shook, C. A. and M. C. Roco, “Slurry Flow Principles and

Practice,” Butterworth-Heinemann, Boston, MA, U.S. (1991). Slatter, P. T. and E. J. Wasp, “The Laminar/Turbulent Transition

The Wilson-Thomas model, which does account for viscouslayer thickening, has now been applied to the prediction of the transition point for Bingham plastics. Laminar and turbulent friction losses are expressed in terms of common parameters and variables, and equated at the transition point. On iterating to eliminate the stress ratio, it is found that the conditions at transition depend only on the Hedström number. The calculated relationships have been approximated by simpliﬁ ed ﬁ t functions valid over speciﬁ c ranges, and compared to existing empirical correlations and to experimental data from various sources, with satisfactory results.

in Large Pipes,” Proc. 10th Int. Conf. on Transport and Sedimentation of Solid Particles, AUV, Wrocław, Poland (2000), pp. 389–399.

It is of particular interest that for He > 105 the dimensionless transition velocity VTrel predicted by the authors’ model has a constant value, equal to 25.

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Manuscript received November 25, 2005; revised manuscript received February 9, 2006; accepted for publication February 9, 2006.