The Flow of Non-Newtonian Fluids Down Inclines

J. Non-Newtonian Fluid Mech. 136 (2006) 64–75

The flow of non-Newtonian fluids down inclines

I.D. Sutalo ∗, A. Bui, M. RudmanCommonwealth Scientific and Industrial Research Organisation (CSIRO), Manufacturing and Infrastructure Technology,

PO Box 56, Highett, Vic. 3190, Australia

Received 25 October 2004; received in revised form 17 August 2005; accepted 20 February 2006

Abstract

Equipment units (e.g. tray column heaters) in many chemical engineering and mineral processing industries involve the flow of non-Newtonianfluids down inclined plates. When designing these equipment units the non-Newtonian fluid flows often are not fully understood and so the designsare not properly optimised. In this study the flow down a series of inclined plates was experimentally and numerically investigated to betterunderstand the flow for various fluids and to validate a computational fluid dynamics (CFD) model.

In the experimental rig there were a series of consecutive plates inclined at 45◦. An optically clear polymer solution was used to simulate a yieldpseudo-plastic material and allowed flow visualisation to be undertaken of the flow. The fluid film thickness was observed to decrease down theconsecutive plates. Experiments were also carried out using a yield pseudo-plastic mineral slurry and the results were found to be qualitativelys

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K

1

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0d

imilar.An analytical model was developed to calculate the fluid film layer thickness on the first plate and a CFD model was used to compute the flow

own a series of flat plates. The CFD model employed a homogeneous multiphase model and surface-sharpening algorithm. The CFD modelccurately predicted the fluid film thicknesses and flow patterns. The validated CFD model can now be used with confidence as a design tool.

2006 Elsevier B.V. All rights reserved.

eywords: Inclined plates; Pseudo-plastic; Yield stress; Non-Newtonian

. Introduction

Non-Newtonian flows down inclined plates occur in manyquipment units in many chemical engineering and mineralrocessing industries. There is very little existing literature onodelling of splash column geometry, and most of it is related

o tray columns used in the petro-chemical industry for distil-ation (Fair [1], Porter et al. [2] and Wohlhuter et al. [3]). Thisork is also relevant to tailings disposal in minerals applica-

ions and the way the deposits are laid down. In these columns,he plates (or trays) are generally horizontal, often have perfora-ions and may have seal dams to hold back the liquid on the tray.n Wohlhuter et al. [3] a computational model was presentedhat allowed calculation of the surface profile of the flow oversimplified horizontal baffle. Flows down inclined planes wereumerically investigated in the work by Ruyer-Quil and Man-eville [4,5]. They used a simplified model of the flows basedn the so-called boundary layer approximation. This approxima-

∗ Corresponding author. Tel.: +61 3 9252 6343; fax: +61 3 9252 6240.

tion allowed most flow variables to be eliminated and yieldeda one-equation model governing the effective dynamics of thelocal film thickness. In none of the above papers were non-Newtonian fluids considered. An asymptotic technique has alsobeen applied to numerically analyze the stability of flows downan inclined plane (see Huang and Khomami [6]). It is worth not-ing that all the above-mentioned numerical approaches can onlyprovide an approximate solution for the flows over a single plane.When the flow path is more complex with a number of platespresent, the full CFD modelling approach is more appropriate.However, CFD modelling of the flows over inclined planes is adifficult task owing to the presence of a free boundary.

Only a few studies have experimentally investigated the flowof non-Newtonian fluids down inclined plates or planes. Astaritaet al. [7] measured the film thickness of non-Newtonian fluids infully developed laminar flow down inclined plates for a range offlow rates and plate angles. Therien et al. [8] measured the filmthickness for power law fluids flowing down inclined plates andcompared the results with an analytical expression with goodagreement. Sylvester et al. [9] compared experimental and pre-dicted film thicknesses for power law fluids flowing down a

E-mail address: [email protected] (I.D. Sutalo). vertical wall in the laminar and wavy fully developed regimes.

377-0257/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2006.02.011

mailto:[email protected]

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I.D. Sutalo et al. / J. Non-Newtonian Fluid Mech. 136 (2006) 64–75 65

Narayana Murthy and Sarma [10,11] modelled the flow ofa power law fluid down an inclined plate using integral anal-ysis and included the effect of interfacial drag. Later Tekic etal. [12] included pressure gradient and surface tension. Ander-sson and Irgens [13] used the integral method to simulate theflow of power law fluids down vertical walls. Entrance effectswere considered by Andersson and Irgens [13], Tekic et al. [12],Narayana Murthy and Sarma [10].

Paslay and Slibar [14] developed a relationship between filmthickness and flow rate for a Bingham fluid down an inclinedplane, a similar relationship was developed by Chaturani andUpadhya [15] for a fluid with coupled stress. Uhlherr et al.[16] used flow down an inclined plate to determine the yieldstress of Bingham fluids (two Ultrez 10 solutions and twosuspensions). De Kee et al. [17] developed analytic velocityprofile and flow rate relationships for viscoplastic fluids downinclined plates using the Exponential (DDK), Herschel-Bulkleyand Casson fluid models. They carried out yield stress mea-surement experiments on two suspensions: milk of lime andketchup. They measured film thickness and flow rate from whichthey determined the yield stress using the Exponential DDKequation.

Andersson and Shang [18] developed a mathematical modelof accelerating power law fluid flow down an inclined plane.They later included heat transfer from the inclined plate to thefluid (Shang and Andersson [19]).

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2. Experimental rig

In the experimental rig there were a series of four consecutiveplates inclined at 45◦ as shown in Fig. 1. Each inclined platewas 381 mm long. The second and third plates were removableto allow the model to be converted from a four plate to a twoplate system. The model was built from acrylic to allow flowvisualisation. The model width was 50 mm. The material waspumped using a low shear positive displacement pump (MONOAC12 M) into the dam on the top right of the model where it flowsover the weir and down the inclined plates into the underflowtank, where it is again pumped back to the top dam. The flowrate was measured using a magnetic flow meter calibrated foreach material.

The materials used in these experiments were Ultrez 10 solu-tion and mineral slurry. The optically clear Ultrez 10 solutionwas used to simulate a yield pseudo-plastic material and allowedflow visualisation to be undertaken. The mineral slurry wasan industrial yield pseudo-plastic material. The rheologies (i.e.

Fig. 1. Schematic of the inclined plate physical model with a series of fourinclined plates. The width of the model is 50 mm (all dimensions are in mm).

Liu and Mei [20,21] studied the slow flow of a Bingham fluidn an inclined plane. They developed a simplified mathematicalodel to describe the flow and compared the results with exper-

ments they carried out with Kaolinite. The comparison wasood. Di Federico [22] extended the model to simulate unsteadyaminar flow of a Herschel-Bulkley fluid down an inclinedlate.

Piau [23] mathematically modelled the flow of Herschel-ulkley and Bingham fluids down an inclined plane. Balmforthnd Craster [24] developed a thin-layer theory for flow of a Bing-am plastic and Herschel-Bulkley fluid flow down an inclinedlane to eliminate inconsistency seen in earlier studies in theredicted velocity fields.

Coussot and Proust [25] and Wilson and Burgess [26] mod-lled unconfined spreading of mudflow. Coussot and Proust [25]ompared their predictions with experiments carried out withingham and Herschel-Bulkley model fluids of kaoline. Theybtained good results for the fluid depth, but overestimated theateral extent by 30%.

The previous approaches did not attempt to employ a fullFD method to model complex non-Newtonian free-surfaceow down a series of inclined plates. Most of the previous stud-

es were also not validated experimentally. In this investigationow visualisation and fluid film layer thickness measurementsere undertaken in an experimental model to better understandon-Newtonian flow down a series of inclined plates. CFD sim-lations were carried out and compared with experimental andnalytical work to provide additional insights into the flow char-cteristics, and provide a validated CFD model that can be usedo investigate flow of yield pseudo-plastic materials in moreomplex geometries than simple inclined plates.

66 I.D. Sutalo et al. / J. Non-Newtonian Fluid Mech. 136 (2006) 64–75

Table 1Rheology parameters for Ultrez 10 solutions and mineral slurry

Material Yield stress, τy (Pa) Consistency, K (Pa sn) Flow index, n Density (kg m−3) Static yield stress (Pa)

Ultrez 10 solution 0.1 wt.% 3.86 9.04 0.413 1000 6.8Ultrez 10 solution 0.15 wt.% 4.12 13.75 0.412 1000 19Mineral slurry 71.3 16.6 0.322 1320 101

shear stress versus shear rate) of the materials were measuredusing a Bohlin rheometer CVO50 with cone and plate geometryfor the Ultrez 10 solutions and cup and bob geometry for themineral slurry. The shear stress versus shear rate relationshipsfor the materials was reasonably well described by the Herschel-Bulkley model curve fit:

τ = τy + Kγn, (1)

where τy is the yield stress (dynamic), K the consistency, γ theshear rate and n is the flow index.

The rheological details for two different Ultrez 10 solutionsand mineral slurry are presented in Table 1. The static yieldstress of these materials were measured using a vane and gavehigher values than the dynamic yield stress values. This is dueto the different strain histories during the measurements andresults from the breakdown and reforming of the structure inthe materials. The static value is relevant, for example, in thestart-up of a pipeline and the dynamic value is relevant once thefluid is moving. It is the dynamic value that is used in the CFDmodelling. The errors in the viscosity measurements were lessthan ±1%. The maximum error bars to the fits of the rheology ofthe high viscosity mineral slurry were up to ±4 Pa in the shearstress which corresponds up to ±0.4 mm in the predictions fromthe analytical model detailed in Section 4.

3

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tion was far more difficult as the model quickly became coveredwith slurry. An estimate of the later thickness had to be made bystudying surface patterns on the face of the model. The accuracyof such measurements is unlikely to be better than ±2 mm andmay be influenced by other factors such as slurry on the facesof the model coalescing with the main stream on the plate. Forthe mineral slurry the velocities were 0.36–0.95 m s−1 whichcorrespond to a Re of 1.5–9.2.

4. Analytical model

The plates in the inclined plate model are modelled as sim-ple inclined rectangular plates. Fig. 2 shows a film falling downan inclined plate under the influence of gravity. The film has athickness δ and a width (into the page) of W. Assuming steady-state (i.e., acceleration is finished) a momentum balance on aslice of the film �x allows an expression for the momentum fluxdistribution and the shear stress to be determined. In the calcula-tions it is assumed that the flow is laminar without rippling (i.e.,no entrance and exist disturbances on the length modelled, L).Other relevant parameters are gravity (g) and the angle from thevertical (β). In the velocity distribution calculations the film isseparated into two layers. In the top layer (x ≤ H) there is plugflow due to the yield stress of the fluid and in the bottom layer

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. Experimental technique

For each experiment flow visualisation of the flow in thenclined plate model was carried out using photography andideo. The liquid film thickness over each of the inclined platesere measured with a ruler. A number of experiments were

un with Ultrez 10 solutions of 0.1 and 0.15 wt.%. Results foroth were similar and usually only those for the 0.15 wt.%re shown here. Generally, three different flow rates were usedhich were 18.8, 37.3 and 56.5 l min−1. For the 0.15 wt.% Ultrez0 solution these flow rates correspond to mean velocities of.42–0.98 m s−1 and Re of 3–13 where the Reynolds number isefined as follows:

e = ρ < uz > δ

µB, (2)

here µB is the effective viscosity

B = τy + Kγn

γ. (3)

The uncertainty in the measured film thickness for the Ultrez0 solution is approximately ±1 mm. The main reason for thencertainty is a result of the film thickness that is not uniformcross the plate. For the industrial mineral slurry flow visualisa-

ig. 2. Flow of a viscous non-Newtonian fluid film down an inclined plate underhe influence of gravity.


the film has a velocity gradient in the x-direction. As a result ofthe fluid yield stress the film thickness has to be greater than Hfor the film to move.

The rheology of the material is described by the Herschel-Bulkley model shown in Eq. (1). From the momentum balancethe momentum flux distribution for the falling film is:

τxz = ρgx cos β. (4)

When x = H, the stress is just equal to the yield stress and wefind that the minimum film thickness that will flow is:

H = τy

ρg cos β. (5)

Substitution of the Herschel-Bulkley model equation into themomentum flux distribution equation gives:

τy + Kγn = ρgx cos β, (6)

where the shear rate, γ , can be written in terms of the velocitygradient as

γ = −duz

dx. (7)

The velocity distribution across the sheared part of the film isdetermined by integrating across the film and using the bound-amfi

f

u

ifv

Q

Dtoaaot

5. Numerical modelling method

A CFD model was developed to compute the free-surfaceflow of a yield pseudo-plastic material, and applied specificallyto the experimental geometry in order to provide a level of val-idation. Because it was not possible to experimentally measurethe velocity profiles in the experiment, the only means of val-idation available is to compare the position of the free surfaceprediction to that observed in the experiment. The surface posi-tion is related to an integration of the velocity field in time, so itprovides a reasonably rigorous validation, although not as com-plete or robust as velocity measurements.

The simulations were carried out in two dimensions usingthe commercial CFD software CFX-4.4. The code is based on asemi-implicit numerical solver and is capable of modelling theheat and mass transfer associated with multiphase flows in com-plex geometries. It uses a block-structured quadrilateral mesh,and details of the software implementation may be found in theuser manual [28]. The total number of grid cells was approxi-mately 20,000 and the grid resolution was set higher along thepath of liquid propagation and lower at other (air-filled) parts ofthe domain.

In the study described here, the CFX-4 homogeneous multi-phase flow model was used to describe the free-surface flow. Inthis model, the entire domain of the splash column is modelledas a single fluid which has a varying density and rheology. TheisTrtbdmtnttdrtNtdotgp

wdbttitv

ry conditions uz = 0 at x = δ and τxz = τy at x = H. After someanipulation we find that the velocity distribution across thelm is given by

uz = K−1/n

[n

(n + 1) (ρg cos β)

] [(δρg cos β − τy

)(n+1)/n]

0 ≤ x ≤ H (8)

or the top plug flow layer, and

z = K−1/n

[n

(n + 1) (ρg cos β)


)(n+1)/n

− (xρg cos β − τy

)(n+1)/n]

H ≤ x ≤ δ, (9)

n the sheared layer near the plate (bottom layer). This expressionor uz can be integrated over x to obtain an expression for theolumetric flow rate, Q

= K−1/nW

[n

(n + 1)(ρg cos β)2


)(n+1)/n]

×[(

n + 1

2n + 1

) (δρg cos β − τy

) + τy

]. (10)

e Kee et al. [17] have previously also derived the final equa-ions. However, there was an error in the paper in their derivationf the velocity profile equation in the bottom layer. Coussot [27]lso derived the same equations, although they were presented inform where the effect of K, angle and τy were less obvious. Inur study the analytical model is extensively used to investigatehe effect of different plate angles and rheological properties.

nterface position is defined to be the location where the den-ity of the fluid changes from the density of gas to that of liquid.his approach is quite common and has been used extensively to

eliably model gas/liquid systems (e.g. [29]), and has the advan-ages that the exact position of the interface does not need toe known a priori. The CFX homogeneous multiphase modeloes not explicitly track the interface (as in volume-trackingethods, e.g. [30]) but instead uses flux-limited mass advection

echniques coupled to a surface sharpening procedure to mitigateumerical diffusion at the free surface. This procedure smoothshe true interface across 2–3 computational cells. The surfaceension force is modelled in CFX-4 using the continuum methodeveloped by Brackbill et al. [31]. The CFX-4 user-FORTRANoutines were employed to describe the non-Newtonian proper-ies of the flow, and allowed a smooth switching between theewtonian viscosity of air to the Herschel-Bulkley rheology of

he liquid. Because of a limitation of the CFX-4 software whichoes not allow simultaneous modelling of non-Newtonian rhe-logy and turbulence, the air flow in the domain was assumedo be laminar. Due to the high viscosity of the liquid phase, anyas-phase turbulence is likely to be unimportant in the liquidhase.

The initial conditions are set to be air throughout the domainith zero velocity except at the lower boundary of the inflowam where the material type (density and viscosity) was set toe liquid with a uniform vertical velocity profile chosen to matchhe volumetric flow rate used in the experiment (normalised byhe thickness of the experimental rig). A uniform outflow veloc-ty boundary condition was chosen at the base of the domaino exactly match the inflow condition and ensured that the totalolume was conserved. Simulations were time-stepped from the


initial condition and as the dam filled, it overflowed and liquidran down the first plate, dropping onto the second plate, etc.until reaching the base of the domain where it exited the under-flow. Ultimately a steady flow solution was obtained that wascompared to the experimental results.

One modification to the experimental domain was required inthe computational model. In the experiments, the flow of fluid offthe plates usually occurred as a stream with a width less than thatof the experimental model. Consequently, there was a continuouspressure pathway between all of the gas spaces in the model (i.e.the gas zones between the plates). In the two-dimensional CFDmodel, the liquid in effect completely fills this third dimensionand there is no continuous pressure pathway between these gasspaces. This can lead to errors in the flow solution. For example,at start-up, the liquid path as it drops off the first plate willtrap a fixed quantity of air in the top part of the model. Thisliquid configuration may not be the steady-state solution but thevolume of gas in the top of the model is now set at this initial

Fofp

Fig. 4. Flow of 0.15 wt.% Ultrez 10 solution in the inclined plate model withfour plates at 56.5 l min−1.

ig. 3. Grid sensitivity study: (a) superimposition of the predicted film thicknessbtained from a fine and a coarse grid. The two lines lay on top of each otheror almost the entire profile. Only the fine grid is shown. (b) Close-up of theredicted film thickness along the plate obtained from a fine and a coarse grid.

Fig. 5. Close-up of flow running off the first plate and falling onto the secondplate. The view is from in front and above. Clearly seen is the ‘heel’ that isformed upstream (on the left) of the second plate. Also seen is the ‘necking’ ofthe Ultrez 10 solution as it flows off the first plate.


Table 2Film layer thickness for 0.15 wt.% Ultrez 10 solution flowing at 56.5 l min−1

Analytical model Experimental

Plate 1 Plate 1 Plate 2 Plate 3

Distance from plate tip (mm) 100 200 100 100Film thickness (mm) 19.2 19.5–20 19.5–20 17.5–19.5 17.0–18.5

incorrect value. However, volume conservation ensures that thisincorrect configuration cannot change unless the excess gashas a chance to break through the liquid stream. If it can-not do this, then the liquid configuration will remain physi-cally incorrect. To ameliorate this issue, channels were placedthrough the plates at their junctions with the walls of themodel, and this allowed a continuous pressure pathway between

all of the gas spaces without adversely affecting the liquidflow.

5.1. Grid resolution study

One of the key issues in all computational fluid dynamicsmodelling is the effect of grid resolution on the solution. Grid

Fs

ig. 6. Flow of 0.15 wt.% Ultrez 10 solution in the inclined plate model with two plide view of the jet hitting the bottom plate. Clearly seen is the reduced size of the flu

ates at 56.5 l min−1. On the left is the front view and on the right is a close-upid jet that has a width of approximately one third of the width of the model.


resolution was considered here using a simplified domain con-sisting of just one plate, and several simulations were run todetermine an acceptable resolution at which to undertake thefull computations. Local grid refinement was applied with thefinest grid elements located near the top surface of the plate.The cell sizes in the fine grid case are approximately half thesize of that in the course grid case. In the fine grid, the small-est thickness of the grid elements was approximately 1.2 mm. Inthe course grid, the smallest thickness of the elements was about2 mm. Both the grids had 52 grid elements along the length ofthe plate (i.e., grid element lengths were 0.019L).

The comparison of the liquid layer thicknesses obtained onfine and coarse grids is shown in Fig. 3a and b. The predictionsfrom the two grid resolutions coincide well which indicates littlegrid sensitivity for the two chosen grid resolutions. The differ-ence between fine and coarse grid results was in the range 0–4%in the middle section of the plate. At the beginning and the endof the plate, the maximum difference in film thickness is 6%.In the following sections, the fine grid was used in a numericalparametric study while the coarser grid was used in the largerdomain modelling.

6. Results and discussion

6.1. Experimental modelling of Ultrez 10 solution flowd

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Fig. 7. Flow of the mineral slurry in the inclined plate model with four plates at58 l min−1.

the combined effects of acceleration and surface tension. Fig. 5shows the curved nature of the film as it hits the plate below. Thiscurvature is a result of the flow in the edges of the film movingslower than that in the middle (because of side wall friction)and hence not travelling as far horizontally. The film thicknessfor 0.1 wt.% Ultrez 10 solution at a flow rate of 56.5 l min−1

measured on the first plate was 14.5–15 mm which compareswell to the film thickness value of 15.4 mm determined by theanalytical model.

The reduced film thickness on the second and subsequentplates is probably a result of the increased energy (caused by

Ff

own inclined plates

Flow visualisation images of the flow of 0.15 wt.% Ultrez0 solution at a flow rate of 56.5 l min−1 down a series of fournclined plates are shown in Figs. 4 and 5. The flows for thewo lower flow rates are similar, with the horizontal distance thetream reaches being further with higher flow rate. The flow ofltrez 10 solution is quite steady except for some small vari-

tion resulting from the pump and occasional entrainment ofir bubbles. The film layer thickness on each plate is measured00 mm from the end of the plate (as well as 200 mm from thend on Plate 1) and results are compared to the analytical modelredictions using Eq. (10). As can be seen in Tables 2 and 3, theesults compare fairly well on the first plate, and become pro-ressively worse on subsequent plates. Note that the plate has toe long enough to ensure equilibrium, non-acceleration state iseached without entrance and exit effects.

As stated earlier, the film shape is not uniform across thelates. This is particularly true on the second and subsequentlates on which the film shape is influenced significantly by theall from the plate above. Instead of flowing off the plate asuniform film across the entire width of the model, the fluidlm ‘necks’ (i.e. forms a thinner jet) most probably because of

able 3redicted and experimentally measured film thicknesses for 0.15 wt.% Ultrez0 solution at 56.5 l min−1 at 100 mm from the end of the plate

Plate 1 Plate 2 Plate 3

xperimental (mm) 19.5–20 17.5–19.5 17–18.5FD numerical (mm) 21 18.1 17.1nalytical model (mm) 19.2 – –
ig. 8. Close-up of the mineral slurry flow running off the second plate and
alling onto the third plate.


Fig. 9. Numerical predictions of the flow in an inclined plate model.

acceleration of the flow free fall between the plates) that theUltrez 10 has on impact that must be dissipated by viscousdissipation. It is worth noting that the path of the flow on thesubsequent plates may not be long enough so that the film thick-ness can be stabilised and the entrance effect can be completelyeliminated. There is no indication of turbulence or any signifi-cant unsteadiness, thus the dissipation can only occur as a resultof increased shear which in turn reduces the viscosity, and hencefilm thickness.

Also undertaken in the physical model was the removal of themiddle two plates to allow a longer fall distance for the Ultrez10 as it came off the first plate as shown in Fig. 6. The filmthickness on the bottom plate is 15.5 mm, which is less again thanthat measured in the four plate model. The additional fall heightallows 3D effects to occur. In this case, the necking observedpreviously also plays a significant role, with the jet becomingeven smaller in cross-section than previously, with a width ofapproximately one third of the model width. This is expectedbecause the fluid is falling further, moving more rapidly and inorder to maintain the same volumetric flow rate, its cross-sectionmust reduce.

As the flow rate was increased from 18.8 to 56.5 l min−1 theimpinging point increased from being approximately 210 mmdown the plate from the far side wall to approximately 40 mmdown the plate from the far side wall.

6.2. Experimental modelling of mineral slurry flow downinclined plates

For the mineral slurry at the highest flow rate of 58 l min−1

in the 4-plate model the flow sheet almost completely filled themodel width after Plate 1, while after Plates 2 and 3 it completely

Fig. 10. Comparison of the film thickness predicted using the analytical a
nd numerical models for varying (a) τy, (b) K, (c) n and (d) angle.


filled the model width (refer to Fig. 7). There was also moresplashing at the high flow rate compared to the low flow rates.The stream leaving the plates was fairly steady in the case offour 45◦ plates since the stream leaving a plate hit the next platebefore it had a chance to start moving from side to side. Fig. 8shows that the stream leaving the second plate completely fillsthe model width and that a ‘heel’ forms upstream (on the right)of the third plate.

At the low flow rate of 18.8 l min−1 there was necking and theslurry sometimes had free surface ripple patterns flowing alongthe plate. These patterns remained during the transit along thefirst plate and are indicative of a lack of shearing at the freesurface due to the low flow rate and high yield stress – i.e. theslurry was effectively ‘solid’ in a small surface region. Also,roll waves described by Coussot [32] may cause the surfacedisturbances.

6.3. Computational fluid dynamics (CFD) modelling

6.3.1. Numerical modelling of start-up of flow downinclined plates

The CFD modelling of the flows under investigation is com-plicated owing to the presence of free surfaces and associatedsteep changes of fluid properties across these interfaces. In addi-

tion to numerical oscillations caused by property jumps at thegas–liquid interfaces, the sharpness of the interfaces is difficultto maintain using low-order discretisation schemes and/or lowcomputational grid resolution. For the flows under considera-tion, non-Newtonian characteristics of the flows may also causefurther difficulties in the modelling as numerical convergenceand correct choice of non-Newtonian flow model/parametersbecome more difficult. However, a simulation test (see Fig. 9)shows the two-dimensional (2D) CFD model can satisfacto-rily predict the behaviour of the liquid film propagating alongthe inclined plates with very little numerical diffusion at theliquid–gas surface.

6.3.2. Comparison of analytical and numerical solutionsfor non-Newtonian flows of different properties and underdifferent conditions

Fig. 10 shows the effect on film thickness of varying τy, K,n and plate angle. The plate angle was defined as 90 − β. Therewas very good agreement between the analytical and numericalmodel predictions (usually less than 4% difference) for realisticrheology conditions. In the analytical model the typical valuesof the rheology parameters used in the Herschel-Bulkley modelcurve were set at τy = 5 Pa, K = 15 Pa sn, n = 0.4. The flow ratewas set to 60 l min−1. The film thickness increased for increasing

F
ig. 11. Comparison of the film thickness predicted using the analytical and numerica l models for various angles and for varying (a) τy, (b) K, (c) n and (d) flow rate.


Fig. 12. Comparison between the numerical predictions and experimental results of the 0.15 wt.% Ultrez 10 solution flowing at 56.5 l min−1 in the inclined platemodel with four plates. The black contour lines represent the free surface profile as predicted by the numerical model, on the right side the predicted free surfaceprofile is superimposed on the experimental results.

τy, K, and n, and decreasing plate angle. In the investigation,various angles between 1◦ and 90◦ were investigated when thefollowing variables were varied: τy = 5–100 Pa, K = 10–30 Pa sn,n = 0.3–0.7, and flow rate from 20 to 60 l min−1.

Fig. 11 shows the effect on the film thickness of changing therheology and for different angles. As expected increasing τy, K,n and flow rate increases the film thickness; the increases in filmthickness were larger for smaller plate angles. For angles lessthan 30◦ the film thickness increases significantly when the plateangle is decreased compared to an equivalent reduction in plateangle at higher plate angles. What is interesting is that doublingthe flow rate only causes a relatively small change in the filmthickness.

6.3.3. Simulation of the Ultrez 10 solution flow in theinclined plate physical model

The CFD model was used to simulate the flow of the0.15 wt.% Ultrez 10 solution in the inclined plate model geome-try with four plates at 45◦ and the results are presented in Fig. 12.The black contour lines represent the predicted free surface. Asseen in Fig. 12 the agreement between CFD predictions andexperimental results is quite good. The predicted angle that thejet falls off the second plate is not in as good agreement with theexperimental results as the jet angle coming off the first plate. Itis believed that the primary reason for this difference is that in the2D CFD model, the gas space between the first and second plates(oaif Fig. 13. Numerical predictions of velocity field in the inclined plate model.

on the right of the jet) cannot communicate with the gas spacesn the left. Thus, if the jet tries to wobble, it will create eithern overpressure or vacuum that will force it back to the shapet has in the image. In the experiment, the surface tension wasound to narrow and round the jet during its free-fall between the


Fig. 14. Comparison between the numerical predictions and experimental results of the mineral slurry flowing at 58 l min−1 in the inclined plate model with fourplates. The black contour lines represent the free surface profile as predicted by the numerical model, on the right side the predicted free surface profile is superimposedon the experimental results.

plates and jet did not fully fill the space between front and backplates of the model. As a result, air can move freely between theleft and right side of the jet. Therefore, this problem is expectedto be eliminated by carrying out three-dimensional (3D) CFDmodelling.

The experimental and predicted film thicknesses are pre-sented in Table 3 and the agreement is very good consideringthat the CFD model is strictly two-dimensional, whereas theexperiment has some noticeable three-dimensional effects. TheCFD prediction indicates that the flow is not strictly the samewhen moving from one plate to another. The thickness of thefluid layer is highest at the top plate and the decrease of thefluid layer thickness can be seen in the subsequent plates. Thisreduction of the fluid layer thickness is possibly caused by theacceleration of the fluid during its free falling between the plates.The fluid acceleration is also seen to give rise to significant varia-tion of the fluid layer thickness along the second and third platesas the fluid layer is thinnest at the higher ends of the plates (nearthe jet impingement place) and gradually recovers to a higherthickness at the other ends of the plates. As the length of theplates is relatively short, the fluid layer on the second and thirdplates may never be able to restore to its ‘stabilised’ thicknesswhich is predicted by the analytical model above. As can be seenfrom Table 3, this reduction of the fluid layer thickness is alsocaptured by experimental measurements, but to a lesser extent.

Fig. 13 shows the velocity field in the model. Acceleration ofttt

the fluid layer. The ‘heels’ behind the waterfalls are stationary.The lack of flow inside the ‘heels’ is noticeable and the heelsare kept steady in time by the balance of buoyancy force andpressure increase resulted from the impact of the flow on theplate.

6.3.4. Simulation of the mineral slurry flow in the inclinedplate physical model

In Fig. 14 the predicted flow of mineral slurry in the fourplate model is compared to the experimental results. In this casethe numerical convergence of the model was extremely slow,but the higher grid resolution and increased flow rate resulted ina physically more accurate simulation result. The experimentalresults for this case were not as easy to measure as the modelvery quickly became covered in slurry. The predicted andexperimentally measured slurry layer thicknesses are comparedin Table 4. The agreement between the CFD and analyticalpredictions and experimental measured data on the first plateis reasonably good. However the CFD model underpredicts thefilm thickness on subsequent plates, where the experimental

Table 4Predicted and experimentally measured film thicknesses for mineral slurry at58 l min−1 at 100 mm from the end of the plate

Plate 1 Plate 2 Plate 3

ECA

he air is observed near the edges of the plates. The flow insidehe fluid layer is mostly invariable along the plates except forhe plate top ends, resulting in a mostly constant thickness of

xperimental (mm) 20–21 21–22 22–23FD numerical (mm) 20 17.5 16nalytical model (mm) 19.9 – –


measurement suggested larger layer thicknesses. The errorswere larger in the mineral slurry experiments due to the opacityof the slurry and due to splash making measurements of thefilm thickness significantly less accurate.

7. Conclusions

The flow of yield pseudo-plastic materials down inclinedplates have been investigated experimentally, analytically andnumerically. The present study showed that the analytical modelaccurately predicted the film thickness on the first plate, but wasless accurate on subsequent plates. As a design tool the analyt-ical model provides a good first estimate of the film thicknesson the first plate. It can also be used to test the effect of varyingvarious slurry properties and operating conditions on the filmthickness. The impinging point from the inclined plate was fur-ther at higher flow rates. When the four inclined plate system wasconverted to the two plate system the larger fall distance causedfurther necking of the jet leaving the plate and the film thicknesswas lower on the second plate. As far as the authors are awarethis is the first time an attempt to employ the full CFD method tomodel complex non-Newtonian free-surface flow down a seriesof inclined plates has been made. The use of the CFD methodprovides additional insights into the flow under investigationwhere the complex flow path and transient nature of the flowmay restrict the applicability of an analytical model. The fluidlatoef

R

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ayer thickness predicted using the CFD model was in reason-bly good agreement with the experimental results especially onhe first plate. The CFD predictions also indicate the reductionf fluid layer thickness on subsequent plates which can be partlyxplained by the acceleration of the fluid during its free fallingrom one plate to another.

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The Flow of Non-Newtonian Fluids Down Inclines

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