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Pressure drops and flow patterns in sand-mixture pipes

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The paper discusses a relation between the flow friction and the flow pattern in a pipe transporting a sand-water mixture. The study is based on the laboratory experiments that were carried out in a 150-mm pipe for flows of various sand fractions and
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  Pressure drops and flow patterns in sand-mixture pipes V. Matousek  * Delft University of Technology, Chair of Dredging Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Accepted 8 November 2001 Abstract The paper discusses a relation between the flow friction and the flow pattern in a pipe transporting a sand–water mixture. Thestudy is based on the laboratory experiments that were carried out in a 150-mm pipe for flows of various sand fractions and differentflow patterns. Three pipe inclinations (horizontal, vertical and  35   descending pipes) were used to establish different flow patterns.During the tests the flow patterns varied from fully stratified to fully suspended. In the fully stratified flow all particles weretransported in a granular bed sliding over the bottom of a pipe or in a shear layer linked to the granular bed (all particles weretransported as the contact load). In the fully suspended flow no bed was present in a pipe and all particles were dispersed within acarrying liquid (all particles were transported as the suspended load). In many cases the flow was partially stratified, i.e. a proportionof solid particles occupied a granular bed at the bottom of a pipe and the rest of the particles were dispersed within a carrying liquidabove the bed. The tested solids were three narrowly graded fractions of sand ( d  50  ¼ 0 : 12, 0.37 and 1.85 mm) and two sorts of mixedsand (blended from two narrowly graded fractions: the 0.37-mm sand þ the 0.12-mm sand or the 1.85-mm sand þ the 0.12-mmsand). The distribution of solids across the pipe cross-section was measured to identify the flow pattern. Integral flow parameters(mean mixture velocity, delivered concentration of solids, differential pressure) were measured to determine the flow friction forvarious flow conditions.The paper presents a survey of selected results from a large database collected during a long measuring campaign. It shows theway the measured solids distributions across a pipe cross-section were analyzed to distinguish between the contact load and thesuspended load and compares the frictional pressure drops in mixtures of different flow patterns. The measurements confirmed thatflows with a higher degree of flow stratification exhibit considerably higher friction than flows with a lower degree of stratification.For the horizontal flow of the mixture of the volumetric concentration 0.12–0.13 at velocities slightly above the deposition-limitthreshold (around 3 m/s) the frictional pressure drop in the 0.12-mm sand mixture was approximately two thirds of that in the 0.38-mm sand mixture and less than one half of that in the 1.85-mm sand mixture. The addition of the 0.12-mm sand (the solidsconcentration 0.13–0.15) to the mixture of the concentration 0.12–0.13 of either the 0.37-mm sand or the 1.85-mm sand reduced thesolids effect on the total frictional pressure drop to approximately one half at velocities slightly above the deposition-limit threshold.The vertical flows exhibit slightly less solids friction than the horizontal non-stratified flows. For the mixtures of concentration 0.26of both the 0.37-mm sand and the mixed sand blended of the 0.37-mm sand and the 0.12-mm sand, the solids effect on the totalfrictional pressure drop in vertical flow was approximately two thirds of that in a non-stratified horizontal flow. A comparison of theexperimental data with the predictive models for the frictional pressure drop suggests that more reliable models are available forstratified flows than for non-stratified flows.   2002 Elsevier Science Inc. All rights reserved. Keywords:  Slurry flow; Concentration distribution; Flow friction 1. Introduction It is well known that friction losses in pipeline flowsof sand–water mixtures are strongly dependent on theflow pattern developed in a pipeline. Sand–water mix-tures are settling mixtures. If the mean velocity of amixture in a horizontal pipe is low they form a granularbed at the bottom of a pipe. The bed is stationary atvelocities below the deposition-limit threshold and slidesover the pipeline wall at velocities above this threshold.In principle, the velocity range within which the contactbed occurs in the mixture flow depends on the pipe size,the particle size and the mean concentration of solidsin a mixture flow. A contact bed is a major contributor Experimental Thermal and Fluid Science 26 (2002) 693–702www.elsevier.com/locate/etfs * Tel.: +31-15-278-3717; fax: +31-15-278-2492. E-mail address:  v.matousek@wbmt.tudelft.nl (V. Matousek).0894-1777/02/$ - see front matter    2002 Elsevier Science Inc. All rights reserved.PII: S0894-1777(02)00176-0  to solids friction in mixture flow. Sand particles thatdo not contribute to a contact bed are suspended inthe carrying liquid. The mechanism of friction of sus-pended particles is different from that of contact-bedparticles.To determine the friction mechanism governing theflow it is essential to have insight into the internalstructure of a mixture flow and reasonable progress canbe achieved if concentration profiles are known. Anextensive experimental programme was carried out atDelft University of Technology, the aim of which was toprovide information that would lead to greater under-standing of the mechanisms governing the pipeline flowof sand–water mixtures. Concentration profiles and in-tegral flow parameters (mean mixture velocity, deliveredconcentration of solids, hydraulic gradient) were mea-sured for mixture flows of various types of sand in a 150-mm pipe. 2. Experimental apparatus and procedure The 150-mm circuit (Fig. 1) in the laboratory of theChair of Dredging Technology of Delft University of Technology consists of a 24-m long test loop that can beinclined from horizontal to vertical positions, an 18-mlong vertical U-tube, the connecting pipes and the sumptank by means of which solids are introduced into thepipeline and in which solids are stored at the end of eachexperimental run. During measurements the tank can bebypassed. The entire pipeline circuit has a diameter of 150-mm and is 65-m long. A centrifugal pump driven bya 164 kW diesel engine with variable speed serves thesystem.Both measuring sections of the test loop are 3-m longand are equipped with a differential pressure transmitterand a radiometric density meter. Measuring sections areplaced in the straight pipes in such way that the mixtureflow structure in the sections is not affected by bends andother sources of flow disturbance. The measuring sec-tions start 6.55 m (i.e. in the 44 times  D  distance) behindthe last bend. The vertical U-tube also contains 3-m longmeasuring sections in both the ascending and descend-ing limbs of the U-tube. A flow meter is installed in thedescending limb behind the end of the measuring sec-tion. A 0.5-m long Plexiglass observation section wasmounted in the test loop. For the tests reported in thispaper an another 0.5-m Plexiglass observation sectionreplaced the bed velocity meter section (item 14 in Fig.1) in the test loop. Nomenclature  A 2  cross-sectional area of bed, m 2 c v  local volumetric concentration of solids inpipe cross-section C  vb  mean volumetric solids concentration inbed C  vd  delivered mean volumetric solids concentra-tion in pipe cross-section d   particle diameter, m d  50  mass-median particle diameter, m D  pipe diameter, m  F  N  normal intergranular force against pipe wall,N  F  W  submerged weight of granular bed, N  g   gravitational acceleration, m/s 2  I  l  frictional head loss (hydraulic gradient) forflow of liquid  I  m  frictional head loss (hydraulic gradient) forflow of mixture L  length measured along pipe, m m  Richardson–Zaki exponent O b  perimeter of the top of bed, m O 2  perimeter of pipe in contact with granularbed, m P   mean pressure, Pa Re  Reynolds number u * shear velocity for liquid flow, m/s v t  terminal settling velocity of a single particle,m/s V   mean velocity in pipe cross-section, m x  distance along pipe, Pa  y  vertical distance from pipe wall defining aposition in pipe cross-section, m Greek symbols b  angle defining a position of the top of gran-ular bed, deg  s  solids dispersion coefficient, m 2 /s k d  bulk linear concentration l l  dynamic viscosity of liquid, Pas l s  mechanical friction coefficient of solidsagainst pipe wall q l  density of liquid (water), kg/m 3 q m  density of mixture, kg/m 3 q s  density of solids (sand), kg/m 3 s b ; l  liquid shear stress at the top of bed, Pa s w ; l  liquid shear stress at pipe wall, Pa s w ; m  mixture shear stress at pipe wall, Pa s 2 ; l  liquid shear stress at pipe wall in contact withbed, Pa s 2 ; s  solids shear stress at pipe wall in contact withbed, Pa x  pipe-inclination angle, deg 694  V. Matousek / Experimental Thermal and Fluid Science 26 (2002) 693–702  The feeding of the pipeline circuit with solids at thebeginning of an experimental run is achieved by liftingthe funnel of the sump tank to open the tank outletfor the sediment deposited on the bottom of the tank.When the tank inlet is closed and water circulates in thepipeline circuit through a bypass, there is no flowthrough the tank and the solids from the bottom of the tank flow only by gravity to the circuit. Feedingis slow and steady, and this prevents the creation of unsteady mixture-flow conditions in the circuit. Con-stant delivered concentration is reached along the entirecircuit. The temperature of the water in the circuit wasmaintained within a narrow range by regulation of thegland-water flow rate at the centrifugal pump. A smallamount of gland water always entered the system in thecentrifugal pump and the same amount of water left thesystem via the overflow at the edge of the sump tank.Pressure differences over the 3-m long measuringsections are measured by Fisher–Rosemount differentialpressure transmitters (Model 1151DP) with errors lessthan   3%. The instruments appeared to be reliable andstable in all conditions that occurred in the circuit dur-ing the tests. The Krohne magnetic-inductive flow me-ter, Altometer TIV 50, is used to measure the meanmixture velocity in the laboratory circuit. The instru-ment has a maximum error of    1%. Mounting the in-strument on a vertical pipe allows the interpretation of the measured velocity as the mixture velocity becausethe slip velocity between phases is considered, and alsoverified by experiments, to be negligible in the verticalpipe for all solids tested during the experiments. Fur-thermore, the accuracy of measurement in a vertical pipeis not affected by distortion of velocity distribution inthe pipe cross-section.The inverted, vertically mounted, U-tube is used asthe counter-flow meter to determine the mixture densityin the pipeline. This device is often used in labora-tory and field installations because it is simple to con-struct and operate. Differential pressure is measuredby Fisher–Rosemount differential pressure transmitters(Model 1151DP) over the equally long sections in theascending and the descending limbs of the vertical U-tube. Pressure drop due to friction is considered inde-pendent of solids concentration and equal in both pipesections. Averaging measured differential pressures fromboth sections eliminates the influence of wall shear stress(and thus of friction) and the average pressure dropcan be attributed to the hydrostatic pressure exerted bya mixture column in a pipe section. The calculatedmixture density of the mixture column is the averagemixture density for both limbs of a U-tube. It is inter-preted as the average spatial concentration in a verticalU-tube. When the absolute value of slip velocity is as-sumed identical in the ascending and descendinglimbs, the slip effect is also eliminated by averaging themeasured differential pressures. The mean deliveredconcentration in a pipeline is then obtained by thecounter-flow meter. The method described by Moffat [1]was used to estimate the experimental uncertainty of thedelivered concentration and the frictional pressure drop Fig. 1. Laboratory circuit with the 150-mm pipe. V. Matousek / Experimental Thermal and Fluid Science 26 (2002) 693–702  695  in the vertical U-tube. For the assumed constant tem-perature of the mixture, the error analysis estimates theconcentration uncertainty 3.6% and the pressure dropuncertainty 4.7%.The local value of mixture density in a pipeline cross-section is sensed by a radiation density meter BertholdLB 367 with a Cs-137 source. A special support, in whichthe Berthold radiometric density meter is mounted, en-ables vertical positioning of a radioactive source anda transmitter in a pipeline cross-section. The radiationbeam is collimated by a hole in a shield of a lead linedchamber locking a radioactive source. The radiationbeam is directed horizontally in the pipeline cross-sec-tion. By traversing the beam in a vertical direction acrossthe pipeline cross-section the chord-averaged densityprofiles are measured. Values for mixture density areconverted to values of the local spatial volumetric con-centration of the solids in the pipeline cross-section. Atwo-point calibration was conducted in the pipeline witha beam directed to the center of the pipeline cross-sec-tion. The change in radiation intensity was measured fora water-filled pipeline and a water-filled pipeline withglass plates of known volume and specific gravity. Thespecific gravity of the glass was very similar to that of sand and gravel. The results were processed by the in-strument software and the absorption coefficients weredetermined automatically. Furthermore, the instrumentwas calibrated for a water-filled pipeline at each verti-cal position in the measuring pipe cross-section inwhich it was also planned to measure mixture density. Toeliminate the influence of pipe wall wear on the values of measured local concentrations, the instruments were re-calibrated for each position several times during the longperiod of the experimental work. The estimated maxi-mum error of the measuring system is  4%.The tested solids were three narrowly graded frac-tions of sand (see Table 1) and two sorts of mixed sand.The mixed sand sorts were blended from different pro-portions of the two narrowly graded sand fractions: the0.38-mm sand þ the 0.12-mm sand or the 1.85-mmsand þ the 0.12 mm sand. 3. Flow patterns 3.1. Aqueous mixtures of medium and fine sand fractions The fine-sand mixtures ( d  50  ¼ 0 : 12 mm) exhibitedconcentration profiles that indicated no contact bed forvelocities higher than the deposition-limit threshold ina horizontal pipe. The concentration profiles could besuccessfully approximated across the entire pipelinecross-section by the Rouse–Schmidt turbulent-diffusionmodel with the implemented hindered settling effect e s d c v d  y   ¼ v t ð 1  c v Þ m c v  ð 1 Þ in which the Richardson–Zaki exponent  m ¼ 4 : 7 ð 1 þ 0 : 15  Re 0 : 687p  Þ = ð 1 þ 0 : 253  Re 0 : 687p  Þ  and  Re p  ¼ v t d  q l = l l .The concentration profiles in horizontal flows of themedium-sand mixture ( d  50  ¼ 0 : 37 mm) could be ap-proximated by the turbulent-diffusion model with theimplemented hindered settling effect too, but here themodeled profiles were linked to the contact bed occu-pying the bottom of the pipe. The vertical position atwhich a turbulent-suspension profile could be linked tothe contact bed was dependent on the mean velocity of mixture and mean concentration of solids in a pipe flow.The analysis of the measured profiles [2] showed that thesolids dispersion coefficient  e s  can be considered con-stant across the turbulent-suspension flow for both thefine and the medium sand flows, see Fig. 2a.The mixed flows of fine sand and medium sand ex-hibited considerably lower stratification than the flow atthe same concentration with medium sand alone. Themixed medium-fine flows exhibited concentration gra-dients that were very similar to that for a fine-sand flowat the same solids concentration (Fig. 2a). Presumably,the flow of mixed medium and fine sand at the highvelocity (6 m/s in Fig. 2a) could not develop fully in therelatively short straight pipeline section in front of themeasuring location of the laboratory loop. The flowmight have been more stratified if the straight sectionpipeline had been longer. 3.2. Aqueous mixtures of coarse and fine sand fractions The concentration gradients in coarse-sand flows( d  50  ¼ 1 : 85 mm) were steeper than that predicted by theturbulent-diffusion model (Eq. (1)). The concentrationprofiles above the contact bed were due to shearing of the top of the bed rather than turbulent support of theparticles. The particles were too coarse and heavy to besuspended by turbulent eddies developed in the carrier.The pressure gradient over the length of the pipelineproduced shear stress high enough to shear the top of the contact bed and led to the development of a shearlayer with the characteristic shape of a concentrationprofile [3].The flows of mixed fine sand and coarse sand ex-hibited considerably lower degrees of stratification thanthe coarse sand alone. However, the mixed coarse-fineflows exhibited a thickness of the contact bed similar tothat of coarse flows without a fine addition (see Fig. 2b).It is possible that the presence of fine particles prevented Table 1Tested sand fractionsMaterial Particle sizerange (mm) d  50  (mm)  d  10  (mm)  d  90  (mm)Fine sand 0.1–0.2 0.12 0.09 0.22Medium sand 0.2–0.5 0.37 0.24 0.51Coarse sand 1.6–2.7 1.85 1.65 2.70696  V. Matousek / Experimental Thermal and Fluid Science 26 (2002) 693–702  a certain small portion of coarse particles from reachingthe contact bed in the straight pipe in front of themeasuring location. The calculated settling velocity of a1.85 mm particle drops from 266 to 115 mm/s if theparticle settles in fine-sand mixture instead of water.Nevertheless, as indicated hypothetically by the turbu-lent-diffusion model (Fig. 2b), the effects of buoyancyand hindered settling should not considerably changethe concentration gradient of coarse particles if they aretransported in the fine-mixture carrier instead of water. 3.3. Deformation of concentration profile near the pipewall  The measurement of local concentration of solidsnear the bottom of a pipeline revealed the interestingphenomenon of repelling solid particles from the flowboundary under the suitable flow conditions. The dropin the local concentration near the pipeline wall ob-served in certain flow situations can be the result of theliquid lift acting on particles travelling near the pipelinewall or of the particle dispersion due to sporadic inter-particle contacts. In essence, both the liquid lift and thedispersion due to sporadic contacts act on large particlesin a region of the steep velocity gradient of carryingliquid (typically the near-wall region) and of the highconcentration of solids.During our tests the phenomenon of repelling solidparticles from the pipe wall was observed in both hori-zontal and vertical flows. In horizontal flows the repel-ling mechanism acted on coarse sand particles if thelocal concentration was higher than   0.4 and mean Fig. 2. (a) Concentration profiles measured and predicted at mean velocity 6.0 m/s. The  e s = u ¼ 0 : 0115 for the medium sand and 0.009 for the finesand. ( M ): medium-sand mixture  C  vd  ¼ 0 : 34, ( N ): medium sand  C  vd  ¼ 0 : 12 þ fine sand  C  vd  ¼ 0 : 23, ( O ): medium-sand mixture  C  vd  ¼ 0 : 34, ( – ):turbulent-diffusion model (Eq. (1)). (b) Concentration profiles measured and predicted at mean velocity 6.0 m/s. The  e s = u ¼ 0 : 0115 for the coarsesand. (  ): coarse-sand mixture  C  vd  ¼ 0 : 13, ( j ): coarse sand  C  vd  ¼ 0 : 13 þ fine sand  C  vd  ¼ 0 : 21, ( – ): turbulent-diffusion model (Eq. (1)).Fig. 3. (a, b) Concentration profiles with the drop in local concentration near the pipeline wall in a horizontal flow at velocity 6.0 m/s. ( M ): medium-sand mixture  C  vd  ¼ 0 : 43, ( N ): medium sand  C  vd  ¼ 0 : 25 þ fine sand  C  vd  ¼ 0 : 18, (  ): coarse-sand mixture  C  vd  ¼ 0 : 24, ( j ): coarse sand C  vd  ¼ 0 : 25 þ fine sand  C  vd  ¼ 0 : 10. V. Matousek / Experimental Thermal and Fluid Science 26 (2002) 693–702  697
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