EROSION OF PLANE BED BY SAND SLURRY CURRENT IN PIPE

J. Hydrol. Hydromech., 52, 2004, 3, 156–161

EROSION OF PLANE BED BY SAND SLURRY CURRENT IN PIPE VÁCLAV MATOUŠEK Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands, mailto: [email protected]

If a sand slurry current flows over a plane bed the thickness of the bed remains constant providing that equilibrium is established between the vertical fluxes that govern the exchange of particles between the slurry current and the sand bed. These solids fluxes are the settling flux and the erosion (pick-up) flux through the top of the bed. In the literature, relations to quantify the erosion flux from a sand bed consider the water current above an erodible bed. The effect of the presence of solid particles in the current is not taken into account. However, it has been observed that the presence of solid particles tends to suppress pick-up. This phenomenon is called hindered erosion. At present, the knowledge of the process of hindered pick-up is very limited although it is suspected that the hindered pick-up plays an important role in many practical situations. The paper discusses the results of the plane-bed test recently carried out in a 150-mm pipe of the Laboratory of Dredging Engineering of Delft University of Technology. Measurements of the thickness of the bed, the concentration profiles in the pipe cross section, the mean velocity, and the pressure drop allowed us to evaluate the conditions governing the vertical exchange of solids at the top of the stationary bed for currents of different solids concentrations. KEY WORDS: Pick-up, Hindered Erosion, Suspension Flow, Bed Friction. Václav Matoušek: EROZE PÍSKOVÉHO LOŽE PROUDEM HYDROSMĔSI V POTRUBÍ. Vodohosp. Čas., 52, 2004, 3; 8 lit., 6 obr. Při proudění písčité hydrosměsi nad nepohyblivým ložem v tlakovém potrubí dochází k výměně pískových částic mezi proudem a ložem. Tloušťka lože zůstává neměnná, pokud při výměně existuje rovnováha mezi vertikálním proudem částic ze suspenze do sedliny (sedimentační proud) a opačně (erozní proud). V literatuře existující modely pro erozní proud jsou navrženy pro situace, kdy eroze je způsobena prouděním vody nad ložem, vliv přítomnosti unášených částic v proudu tedy není uvažován. V minulosti však již bylo pozorováno, že přítomnost suspendovaných částic může vést k zmenšení schopnosti proudu sbírat částice ze dna. Tento jev se nazývá rušená eroze. V současnosti jsou ještě poznatky o rušené erozi velmi omezené, přestože se dá předpokládat, že rušená eroze hraje významnou roli v mnoha situacích vyskytujících se v praxi. Tento příspěvek pojednává o výsledcích nedávných měření proudění písčité směsi nad nepohyblivým ložem v kruhovém potrubí průměru 150 mm v Laboratoři Dredging Engineering Technické univerzity v Delftu. Měřeny byly tloušťky lože, koncentrační profily nad ložem, střední rychlosti a tlakové ztráty v potrubí. Tato měření umožnila vyhodnocení vertikálního erozního proudu částic z povrchu lože pro různé koncentrace unášených částic ve směsi proudící nad ložem a vlivu koncentrací na erozi. KLÍČOVÁ SLOVA: sběr částic erozí (pick-up), rušená eroze, proud suspenze, dnové tření.

1. Introduction If a slurry current flows over a plane bed the thickness of the bed remains constant providing that the equilibrium is established between the vertical fluxes that govern the exchange of particles between the slurry current and the granular bed. These solids fluxes are the settling flux and the erosion (pick-up) flux through the top of the bed. The lit156

erature presents equations for the erosion flux from a sand bed, but the proposed equations (e.g. van Rijn, 1984) are based on and thus valid for water flow above the bed. The effect of the presence of solid particles in the current that causes the pick-up is not taken into account. The laboratory tests with a silt-suspension current above the bed in an inclined open channel (Winterwerp et al., 1992) showed that the presence of solid particles tends to

Erosion of plane bed by sand slurry current in pipe

suppress the pick-up. This effect is called hindered erosion. In natural open channels, the flow of concentrated sediments above the bed most often occurs at extreme flow rates (floods). In slurry pipes, however, the flow of concentrated suspensions above the bed is much more common. Therefore, a slurry pipe is a suitable conduit for an investigation of the effect of the solids concentration in suspension on the erosion process in currents above a granular bed. 2. Experimental work The aqueous flow of the narrow-graded 0.2–0.5 mm sand (d50 = 0.37 mm) was tested in the pipe circuit of the Laboratory of Dredging Engineering of the Delft University of Technology. The pipe has an internal diameter 150 mm. The long straight horizontal pipe of the circuit is composed of the 14meter long pipe in front of the measuring section, the 3-meter long measuring section and the 0.7meter long pipe (including the 0.4-meter long clear Perspex section) behind the measuring section. The test was carried out for mean volumetric concentration of solids 0.36–0.38 in the pipe circuit. The test runs covered a wide range of solids concentrations in a suspension current above the bed, the Cv1 values varied from 0.06 to 0.32 approximately. Different constant average velocities in the pipe were obtained by changing the rpm of the centrifugal pump and, for the lowest velocities, by changing the opening of the control valve mounted to the circuit. The purpose of the experimental work was to collect data for a wide range of particle mobility numbers, θb, in the pipe with the stationary bed. The mobility number reads θb =

τb

( ρs − ρ f ) gd50

.

One test run was done at extremely low velocity (the control valve was almost closed) and at this velocity dunes were observed. All the other test runs were carried out at higher velocities at which the plane bed developed (1.5 < θb < 23 approximately). The plane-bed tests are analyzed in this paper. A series of test runs started at a specific low value of mean velocity and the velocity was increased in steps. When a higher velocity value was set, a new situation was established in which the top of the bed found its new steady position. The stationary bed became thinner than it was at the

previous lower velocity (Fig. 1). The mean velocity in the test circuit fluctuated slightly, which accounted for the small variation in velocity at a given (considered constant) bed thickness. During the tests the pressure differential and the mean velocity of slurry were measured. The position of the bed was observed in the 0.5-meter long Perspex pipe mounted to the circuit behind the measuring section. For certain velocities concentration profiles across the discharge area were measured using a radiometric density meter. More details about the tests and their results are given in Matousek (2004). That paper shows the measured variation of the pressure drop with the thickness of the bed and the average velocity of the slurry flow above the bed.

Fig. 1. Relation between the bed thickness and Vm1 for slurry flow of Cvi = 0.36–0.38. Obr. 1. Závislost rychlosti Vm1 na tloušťce lože při proudĕní směsi o koncentraci Cvi = 0.36 – 0.38.

3. Bed friction 3.1 Bed shear stress The bed shear stress, τb, was determined from the measured parameters and the observed thickness of the bed, using the Wilson’s method (Pugh and Wilson, 1999). This method is based on the analogy of pipe-wall friction in the water flow through the entire cross sectional area of the pipe and in the water flow through the wall-associated discharge area above the stationary bed. Thus the method assumes that traveling solid particles contribute to flow friction exclusively via the bed shear stress. This is a simplification, as particles traveling above the stationary bed can also contribute to pipe-wall friction through either dispersive wall shear stress 157

V. Matoušek

(collisions) or liquid-like wall shear stress (increased density of the carrier due to the presence of suspended particles). The accuracy of the above method for the determination of the bed shear stress would be lower for flows with a considerable portion of suspended particles. For each new installed thickness of the bed (the bed becomes thinner than in the previous situation, see Chapter 2), the hydraulic radius associated with the bed, Rhb, is higher, resulting in a higher bed shear stress, τb. The τb acts at the top of the bed that is lower than in the previous situation (Fig. 2). Thus the higher τb occurs in a flow with the higher amount of solid particles travelling above the stationary bed.

velocity, ks = Nikuradse’s equivalent sand roughness). Summer’s data cover a relatively narrow range of particle mobility numbers (0.8 < θb < 5 approximately) range. Our tests show that there is a clear correlation between ks/d and θb also in flow regimes with larger θb (Fig. 3).

Fig. 3. Bed roughness for the Nikuradse’s friction equation. Obr. 3. Drsnost dna v Nikuradseho rovnici pro dnový součinitel tření. 0.78 § I · The bed friction-law λb = 0.87 ¨ m ¸ pro© Ss − 1 ¹

Fig. 2. Relation between bed shear stress and bed thickness in the slurry flow above the stationary bed. Obr. 2. Závislost dnového smykového napětí na tloušťce lože při proudĕní směsi nad nepohyblivým ložem.

posed by Nnadi (1992) as valid for Im/(Ss-1) > 0.0167 provides a reasonable prediction only in flows with a thick shear layer [Im/(Ss-1) > 0.08 approximately]. Much smaller λb values than predicted by the model are required for flows with mobility parameter values not far above 1 (Fig. 4).

3.2 Bed friction coefficient Values of the bed friction coefficient, λb, are determined from measured quantities using 8gRhb I m λb = . The data (for the 0.2–0.5-mm sand Vm21 vt = 0.62 , vt = terminal settling with g ( S s − 1) d velocity of particle, d = particle diameter) show the same trend as found by Sumer et al (1996) for the vt acryl particles ( = 0.72 ) processed by g ( S s − 1) d the Nikuradse’s resistance relation for a rough V 14.8Rhb boundary m1 = 2.46ln (u* = bed shear u* ks 158

Fig. 4. Bed friction coefficient λb measured (+) and predicted by Nnadi (line). Obr. 4. Dnový součinitel tření λb měřený (+) a určený z Nnadiiny rovnice (-).

Erosion of plane bed by sand slurry current in pipe

4. Hindered erosion 4.1 Mean solids concentration in slurry current The density meter accurately determines both the mean spatial concentration in the entire pipe crosssection, Cvi, (obtained by integrating a measured concentration profile) and the local concentration within the stationary bed, Cv,bed. The thickness of the stationary bed is determined visually in the Perspex section with sufficient accuracy. From the Cvi, Cv,bed and the position of the top of the bed it is possible to determine the mean concentration of solids in the slurry current above the stationary bed, Cvi A − Cv ,bed Abed [–]. Cv1 = A − Abed 4.2 Near-bed concentration The situation is steady providing that the settling flux equals the erosion (pick-up) flux at the top of the bed. The settling flux is determined by using the near-bed concentration, cva, which is the local concentration at a certain position above and near the top of the bed. Results from our radiometric density meter are not suitable for the direct determination of the cva values. Owing to the rather thick beam of the radiometric density meter the values of the concentration at different positions above the bed must be considered mean values over the height of approximately 2.5 cm rather than local concentration values at the particular position. However, by using a combination of the visually observed positions of the top of the stationary bed and the measured concentration profiles it is possible to estimate localconcentration values round the top of the bed. The comparison of the estimated values with models from literature suggests that the function by Zyserman & Fredsøe (1994), 1.5

cva =

0.152 (θb − 0.045 )

, can be consid1.5 0.46 + 0.331(θb − 0.045 ) ered suitable to estimate the near-bed concentration in a slurry pipe. Comparison of the near-bed concentrations estimated using the measured local concentrations and predicted by the Zyserman & Fredsøe function is shown in Fig. 5.

Fig. 5. Comparison of the near-bed concentrations estimated from the measured local concentrations (+) and predicted by the Zyserman & Fredsøe function (line). Obr. 5. Porovnání koncentrací těsně nad ložem z měřených lokálních koncentrací (+) a určených z rovnice Zysermana a Fredsøe (-).

4.3 Pick-up by slurry current A measure of pick-up is given by the erosion rate ª kg º (or pick-up rate), E « » , the mass of particles ¬ m2 s ¼ picked up per unit area of the top of the bed per time. In the equilibrium situation (characterized by no variation of the bed thickness) the erosion rate ª kg º equals the settling rate S = ρsvthcva « » . Several ¬ m 2s ¼ erosion-rate models are summarized in van Rijn (1984). Van Rijn (1984) carried out laboratory tests in a laboratory flume equipped with a sediment lift, the objective being to quantify the amount of solids the water current picks up from an erodible bed under different conditions (different current velocities and particle fractions). The simple pick-up function, based on his tests, suggests that the erosion flux varies with θb1.5. The function reads Φ p = 0.00033d*0.33T 1.5 and Φp is the dimen-

sionless pick-up rate defined as E Φp = (Einstein, 1950), ρ s S s − S f gd50

(

)

d* – the dimensionless particle parameter d* = d50 3

( Ss − S f ) g ν2

and T is the dimensionless 159

V. Matoušek

transport-stage parameter u 2 − u*2cr θb − θb,cr . T= * = θb,cr u*2cr Our laboratory data, collected in the 150-mm pipe for the 0.37-mm sand flows of different solids concentrations above an erodible bed, indicate that the erosion rate tends to decrease if the current above the bed is loaded with solid particles. The testing of the van Rijn function using our data shows that the van Rijn’s T-exponent (1.5 in van Rijn's original equation) tends to vary with the concentration of solids in the suspension current (see Fig. 6). According to our tests the exponent value already drops below 1 at a very low concentration (0.06 approximately) and decreases further with the increasing concentration to the value lower than 0.7 for the highest solids concentration obtained in the suspension current during the tests (Cv1 round 0.32). The presented test has shown a trend in the relation between the erosion rate and the solids concentration in the current above the bed. However, more tests must be carried out to collect a representative database on ground of which a more general formula for the hindered erosion rate could be found.

Van Rijn’s pick-up function) for water flow above the plane bed for flows that carry particles. The effect of the particle mobility number, θb, on the vertical erosion flux decreases with the increasing solids concentration in the flow over the bed. The Van Rijn’s pick-up function for water currents assume that the erosion rate varies with θb1.5. Our data suggest that the exponent tends to drop from theoretical 1.5 for no particles in the current above the bed (Cv1 = 0) to less than 0.7 for the current of high concentrated slurry (Cv1 = 0.32). The Nikuradse’s friction law for a rough boundary with the modified roughness number provides a good prediction of the bed friction within a wide range of values of the particle mobility number. Acknowledgments. The laboratory tests would not have been possible without the financial support of the Boskalis Westminster Dredging Company. Their support was highly appreciated, as also was the assistance during the experiments of students Jelte Kymmell and Gijs Velter. List of symbols A – cross sectional area of pipe [m2], Abed – cross sectional area of stationary bed [m2], cbed – mean spatial volumetric concentration of solids in stationary bed [–], Cv,bed – near-bed volumetric concentration of solids [–], Cvi – mean spatial volumetric solids concentration in pipe cross section [–], Cv1 – mean spatial volumetric solids concentration in discharge area [–], d – particle diameter [m], d50 – median particle diameter [m], d* – particle parameter [–], E – erosion (pick-up) rate [kg m2 s-1], g – acceleration of gravity [m s-2], – hydraulic gradient of slurry flow in pipe [–], I m ks – Nikuradse's equivalent sand roughness [m], – hydraulic radius of area associated with bed [m], R hb S – settling rate [kg m2 s-1], – relative density of liquid, ρf /ρf [–], S f

Fig. 6. Variation of the T-exponent in the Van Rijn’s pick-up function with mean volumetric concentration of particles in current above the bed. Obr. 6. Zmĕna exponentu pro parametr T ve van Rijnovĕ rovnici v závislosti na střední objemové koncentraci částic v proudu nad ložem.

5. Conclusions The test confirms the existence of the hindering effect of solids concentration in flow on the pick-up caused by erosion from the top of the plane bed. It is necessary to adapt a pick-up function (e.g. the 160

S s T u* u*,cr vt vth Vm1 Φp ν θb θb,cr λb ρf ρs

relative density of solids, ρs /ρf [–], transport-stage parameter [–], bed-shear velocity [m s-1], critical bed-shear velocity according to Shields [m s-1], terminal settling velocity of solid particle [m s-1], hindered settling velocity of solid particle [m s-1], average velocity of current above bed [m s-1], dimensionless pick-up rate [–], kinematic viscosity [m2 s-1], particle mobility number (Shields number) [–], critical particle mobility number according to Shields [–], – Darcy-Weisbach friction coefficient for bed [–], – density of liquid [kg m-3], – density of solid particle [kg m-3],

– – – – – – – – – – –

Erosion of plane bed by sand slurry current in pipe τb

– bed shear stress [kg m-1 s-2].

REFERENCES EINSTEIN H.A., 1950: The bed-load function for sediment transportation in open channel flow. US Department of Agriculture, Technical Bulletin, No. 1026. MATOUŠEK V., 2004: Medium-sand flow over plane stationary bed in 150-mm pipe. Proc. 16th Int. Conf. Hydrotransport, BHR Group, Cranfield, UK, 561–569. NNADI F.N., 1992: Bed-load transport at high shear stress: with application to friction in rivers and sand waves.[PhD Thesis.] Queen’s University. PUGH F.J., WILSON K.C., 1999: Velocity and concentration distributions in sheet flow above plane beds. J. Hydraul. Engng, 125, 2, 117–125. RIJN L.C. VAN, 1984: Sediment pick-up functions. J. Hydraul. Engng, 110, 10, 1494–1502. SUMER B.M., KOZAKIEWICZ A., FREDSØE J., DEIGAARD R., 1996: Velocity and concentration profiles in sheet-flow layer of movable bed. J. Hydraul. Engng, 122, 10, 549–558. WINTERWERP, J.C., BAKKER, W.T., MASTBERGEN, D.R., ROSSUM, H. VAN, 1992: Hyperconcentrated sandwater mixture flows over erodible bed. J. of Hydraul. Eng, 118, 11, 1508–1525. ZYSERMAN J.A., FREDSØE J., 1994: Data analysis of bed concentration of suspended sediment. J. Hydraul. Engng, 120, 9, 1021–1042. Received 12. July 2004 Scientific paper accepted 28. July 2004

EROZE PÍSKOVÉHO LOŽE PROUDEM HYDROSMĔSI V POTRUBÍ Václav Matoušek Příspěvek se zabývá vlivem unášených částic na schopnost proudu erodovat nepohyblivé pískové dno. V praxi se s tímto vlivem můžeme setkat jak v říčních korytech (zvláště za povodní, kdy proud unáší velké množství splavenin), tak v potrubích (při hydraulické dopravě sypanin). Vliv proudu směsi na erozi lože byl zkoumán v laboratorním potrubí pro hydrosměs písku zrnitosti 0,2 – 0,5 mm o střední objemové koncentraci v trubním okruhu 0,36 – 0,38. Testy pokryly tokové situace v rozmezí Shieldsových čísel 1,5 až 23 se středními objemovými koncentracemi proudu směsi nad ložem od 0,06 do 0,32. Prvním problémem při vyhodnocování proudění nad erodibilním dnem je určení součinitele tření pro povrch dna. Problém je jen větší, pokud musí být součinitel určen v širokém rozmezí Shieldsových čísel. Měření ukázala, že pro námi testované situace bylo možno tření téměř v celém rozmezí Shieldsových čísel dobře reprodukovat modifikovanou Nikuradseho rovnicí pro drsné

rozhraní, v níž relativní drsnost byla nahrazena parametrem závislým na Shieldsově čísle. Pro vyhodnocení erozního proudu na základě měření je vedle dnové třecí rychlosti důležité správné určení místní koncentrace částic těsně nade dnem. Tuto koncentraci lze odhadnout použitím existujícího teoretického modelu. Porovnání hodnot erozního proudu interpretovaných z našich měření s teoretickými hodnotami získanými z van Rijnova erozního modelu potvrzuje přítomnost rušivého vlivu suspendovaných částic na erozní proud. Vliv dnové třecí rychlosti na velikost erozního proudu klesá s rostoucí koncentrací částic unášených tokem způsobujícím erozi. Příspěvek nabízí způsob modifikace van Rijnova modelu tak, aby byl model schopen reprodukovat námi měřený trend, nicméně obecnější řešení problému rušené eroze vyžaduje další experimentální a teoretický výzkum. Seznam symbolů A – Abed – cbed – Cv,bed – Cvi

–

Cv1 – d d50 d* E g Im ks Rhb

– – – – – – – –

S Sf

– –

Ss T u* u*,cr vt vth Vm1 Φp ν θb θb,cr λb ρf ρs τb

– – – – – – – – – – – – – – –

plocha příčného průřezu potrubí [m2], plocha příčného průřezu nepohyblivého lože [m2], střední objemová koncentrace částic v loži [–], lokální objemová koncentrace částic těsně nad ložem [–], střední objemová koncentrace částic v příčném průřezu potrubí [–], střední objemová koncentrace částic v příčném průřezu průtočné plochy [–], průměr částice [m], střední průměr částice [m], parametr charakterizující částici [–], erozní proud [kg m2 s-1], tíhové zrychlení [m s-2], hydraulický gradient pro proud směsi [–], Nikuradseho ekvivalentní písková drsnost [m], hydraulický poloměr části průtočné plochy spojené se dnem [m], sedimentační proud [kg m2 s-1], relativní hustota kapaliny, ρf /ρf [–], relativní hustota částice, ρs /ρf [–], parametr charakterizující pohyb splavenin [–], dnová třecí rychlost [m s-1], kritická dnová třecí rychlost podle Shieldse [m s-1], usazovací rychlost částice [m s-1], rušená usazovací rychlost částice [m s-1], střední rychlost směsi nad ložem [m s-1], bezrozměrný erozní proud [–], kinematická viskozita [m2 s-1], Shieldsovo číslo [–], kritické Shieldsovo číslo [–], Darcy-Weisbachův součinitel tření na povrchu lože [–], hustota kapaliny [kg m-3], hustota částice [kg m-3], smykové napětí na povrchu lože (dnové smykové napětí) [kg m-1 s-2].

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