APPLICATION OF ERGUN EQUATION TO COMPUTATION OF CRITICAL SHEAR VELOCITY SUBJECT TO SEEPAGE

Size: px

Start display at page:

Download "APPLICATION OF ERGUN EQUATION TO COMPUTATION OF CRITICAL SHEAR VELOCITY SUBJECT TO SEEPAGE"

Emil Thornton
5 years ago
Views:

1 Citatio: Cheg, N. S. (). Applicatio of Ergu equatio to computatio of critical shear velocity subject to seepage. Joural of Irrigatio ad Draiage Egieerig, ASCE. 9(4), APPLICATION OF ERGUN EQUATION TO COMPUTATION OF CRITICAL SHEAR VELOCITY SUBJECT TO SEEPAGE Nia-Sheg Cheg Abstract The Ergu equatio that is widely used i the chemical egieerig is geeralised for applyig to descriptio of seepage through the sedimet boudaries i this study. Comparisos with the measuremets of the hydraulic gradiet for flows through packed beds show that the Ergu equatio may ot be suitable for the trasitioal zoe where both viscous ad iertial effects are of the same order of magitude. The geeralised Ergu equatio is the used to evaluate the critical shear velocity for icipiet sedimet motio subject to seepage. The computed results are fially compared to the experimetal data ad those predicted usig the previously proposed approach. Itroductio Seepage occurs across porous boudaries ecoutered i atural rivers ad artificial irrigatio caals. Depedig o whether it icreases or decreases the flow rate i the ope chael, seepage ca be idetified as ijectio or suctio. I the presece of seepage, the structural features of the ope chael flow ca be modified cosiderably. This icludes variatios i the average velocity profiles, turbulece itesities ad boudary shear stresses, as reported by Cheg ad Chiew (998a), who coducted a experimetal observatio of the ijectio effects o ope chael flows. Whe sedimet trasport occurs o a porous bed, the bed particles experiece a additioal hydrodyamic force due to seepage. To determie the seepage force exertig o iterfacial bed particles, Marti ad Aral (97) experimetally observed Asst. Professor, School of Civil ad Evirometal Egieerig, Nayag Techological Uiversity, Nayag Aveue, Sigapore cscheg@tu.edu.sg

2 effects of ijectio ad suctio o the agle of repose of sedimet particles. Their results were icoclusive i resolvig whether the seepage force exerted o the top layer of particles was smaller tha that o the particles several layers below the bed surface. Watters ad Rao (97) measured the drag ad lift forces o a sphere which was placed at four typical positios o the bed. They foud that whether the forces icrease or decrease is geerally depedet o the relative positio of the sphere to its eighbours. However, i particular, the total drag force i the presece of ijectio always reduced regardless of the positio of the sphere. Their measuremets were affected appreciably by the Reyolds umber because of the use of highly viscous fluid i their study. Recetly, Cheg ad Chiew (999) preseted a aalytical result for the threshold coditio of sedimet trasport by icludig the additioal hydrodyamic effect due to ijectio. Their derivatio yields that the relative critical shear velocity ca be associated with the relative seepage velocity i the followig form: m u *c v () s u *oc = v sc where u *c = critical shear velocity for the coditio of the bed ijectio, v s = seepage velocity, u *oc = critical shear velocity for v s =, v sc = critical seepage velocity for the coditio of fluidisatio, ad m = expoet. With the dimesioless void scale L * defied as ρ g L* = μ () ε d ε where ρ = desity of fluid, g = gravitatioal acceleratio, μ = dyamic viscosity of fluid, ε = porosity, ad d = diameter of particle, the expoet m ca be evaluated usig the followig empirical equatio: * * +.5L m = +.75L It is oted that i the derivatio made by Cheg ad Chiew (999), a expoetial fuctio was used to represet the relatioship betwee the hydraulic gradiet ad seepage velocity. This approach was subsequetly discussed by Nive (), who reported that the expoetial relatioship could be idirectly derived by performig a series of umerical computatios based o the followig biomial fuctio, i.e., the ()

3 Ergu equatio: μ i = 5 ρgd ε ( ε ) ε v vs +.75 v s gd ε v (4) where i = hydraulic gradiet of seepage, d v = ϕd = diameter of the equivalet-volume sphere, ad ϕ = sphericity of particle. For atural sedimets, ϕ.8, ad for agular particles, such as crushed sads, ϕ (or <).6. Eq. (4) was origially give by Ergu (95) as a extesio of Darcy law ad has widely bee used i the chemical egieerig (Churchill 988). This equatio implies that the eergy loss ca be computed simply by summig up the two compoets, oe caused by the viscous effect ad the other due to the iertial effect (Nive ). Similar summatios ca also be foud i other studies. For example, some studies o the settlig velocity of particles suggest that the total drag could be obtaied by addig the Stokes term to that caused by the iertial flow (Chie ad Wa 999). Such approaches ofte yield relatively simple formulatios of the physical pheomea. However, the relevat predictios may ot agree well with measuremets for the trasitioal zoe where both viscous ad iertial effects are of the same order of magitude (Cheg 997). I this study, a alterative relatioship betwee the hydraulic gradiet ad seepage velocity is first derived. This leads to a geeralised Ergu equatio, which ca be used for differet flow regimes icludig the trasitioal zoe. This equatio is the applied to evaluatio of the critical shear velocity for the icipiet motio of sedimet particles subject to ijectio. The computed results of the critical shear velocity are fially compared with the previous aalysis ad experimetal data. Derivatio It is well kow that if the flow through the porous medium is very slow, its equatio of motio ca be give by Darcy s law, which idicates that the hydraulic gradiet, i, is liearly proportioal to the seepage velocity, v s : i = K v s (5) where K= coefficiet of permeability. Sice ot icludig the effect of the kietic eergy of the flow, Darcy s law may be ivalid, for example, for a porous medium

4 comprised of coarse graular materials where the kietic eergy of the flow may be sigificat. Therefore, extesive attempts have bee made to develop o-liear relatioships betwee the seepage velocity ad hydraulic gradiet for the trasitioal ad iertial regios. Typically, these relatioships ca be expressed either i the biomial form (i s = av s +bv s, where a ad b = coefficiets) or expoetial form (i s = cv m s, where m = ~ ad c =coefficiet), as metioed previously. Eve beig differet from each other, both the biomial ad expoetial relatioships have bee proposed o the same groud that if the viscous effect is sigificat, Darcy s law applies ad if the iertial force is domiat, the hydraulic gradiet is proportioal to the square of the velocity. Give the same properties of fluid ad particles, the above argumet ca also be formulated as follows: d l i d l vs d l i d l vs (6) = for small v s (7) = for large v s Furthermore, (6) ad (7) ca be combied i the form: d l i d l v s + αv = + αv s s where α = coefficiet ad = expoet. The iterpolatio betwee the two extreme coditios give by (8) is ot uique. However, it is preferred here because it is mathematically simple. Itegratio of (8) with respect to v s yields (8) i = β v + ( α ) / s v s (9) where β = coefficiet. It is oted that for =, (9) reduces to the biomial fuctio. Evaluatio of the two coefficiets α ad β ca be made by comparig (9) ad (4) for two limitig coditios, i.e., oe domiated by the viscous effect ad the other by the iertial effect. For these two coditios, two differet forms of (9) ca be obtaied, respectively: i = βv s for small v s () ad 4

5 i = βα v for large v s () / s Correspodigly, (4) ca also be simplified, respectively, to i ( ) μ ε () = 5 v s for small v s ρgdvε ε () i =.75 v s for large v s gd ε v By comparig () with () ad () with (), oe gets ( ε ) μ β = 5 ρgd ε βα / v v ε =.75 gd ε (4) (5) Substitutig (4) ito (5) leads to.75 ρdv α = 5 μ ( ε ) With (4) ad (6), (9) ca be rewritte as ( ε ) μ i = 5 ρgd ε v.75 ρdv vs + 5 μ ( ε ) v s (6) (7) Eq. (7) is idetical to (4) for =. Therefore, it ca be referred to as a geeralized Ergu equatio. I additio, usig the dimesioless parameters L * give by () where d is replaced with the equivalet diameter d v, ad the seepage Reyolds umber defied as ρvsdv Rs = μ ( ε ) where v s /(- ε) represets the average iterstitial velocity, (7) ca further be expressed as (9) 5.75 i = R s + Rs L* 5 Eq. (9) is plotted i Fig. as il * agaist R s for several -values. The figure shows (8) 5

6 that for the itermediate R s -values, for example, R s = ~, the relatioship of il * ad R s chages obviously with the -value. To determie possible variatios i the -value, (9) is further plotted i Fig. agaist laboratory measuremets. Seve sets of experimetal data are therefore used, which were collected previously by Mitz ad Shubert (957) for flows through various packed beds. The graular materials selected for the experimet icluded steel bearigs (d =.646 cm, ρ s /ρ= 7.4, where ρ s = desity of particles ad ρ = desity of fluid), coal particles (d =.97~.779 cm, ρ s /ρ=.66) ad quartz gravels (d =.665 cm, ρ s /ρ =.64). I computig L * ad R s, the sphericity ϕ is take as. for the steel bearigs,.8 for the quartz gravels ad.6 for the coal particles (Nive ad refereces cited therei). Fig. shows that each series of data ca be represeted well with (9) by choosig suitable -value. For all the seve cases, the -values vary from.6 to.8. It is iterestig to ote that they are all less tha., rather tha equal to. as implied by the Ergu equatio. Such differeces may ot be easily observed if the experimetal results are preseted o the logarithmic scale, for example, i terms of the Reyolds umber as foud i may previous studies. Evaluatio of Critical Shear Velocity for Icipiet Sedimet Motio As a alterative to (), the relative critical shear velocity, u *c /u *oc, ca also be expressed i terms of the relative hydraulic gradiet, i/i c, as follows (Cheg ad Chiew 999): u *c i () u *oc = i c where i c = critical hydraulic gradiet for quick sad or fluidisatio, which depeds o the porosity ad relative desity of particles, i.e. i c ρs = ρ ( ε ) By applyig (9) for the case of i = i c, i c ca be related to the critical seepage velocity, v sc, as () 6

7 () 5.75 ic = R sc + Rsc L* 5 where R sc = ρv sc d v /[μ(-ε)] = critical seepage Reyolds umber. Substitutig (9) ad () ito () gives u u *c *oc R = R s sc.75 + Rs Rsc 5 () Eq. () idicates that the shear velocity ratio reduces with icreasig seepage Reyolds umber, for R s R sc. Compariso of Eq. () with Eq. () For the compariso purpose, () ca be chaged to m u *c R (4) s u *oc = R sc ad () ca be re-writte as R s 5 u*c R R + s sc.75r (5) sc u = *oc R sc R sc Eq. (4) shows that u *c /u *oc depeds o R s /R sc ad m that is a fuctio of L *, while (5) suggests that the relatioship of u *c /u *oc ad R s /R sc varies with R sc if is kow. I the computatios made subsequetly, for each R sc, L * is first evaluated usig () ad the m is computed usig (). It ca be see from () that L * ca be computed oly if R sc, i c ad are kow. As discussed i the previous sectio, the rage of the -values derived from the measuremets is limited ad for simplificatio, ca be take as.7 as a average. O the other had, as show i (), i c varies depedig o the porosity, ε, ad relative desity of the particles, ρ s /ρ. For example, for the seve sets of data used i Fig., the computatios with () give that i c =. ~. for the coal particles, 7

8 .86 for the quartz gravels, ad.56 for the steel spheres. Fig. shows differet relatioships betwee the relative shear velocity ad seepage Reyolds umber, which are computed usig (4) ad (5), respectively, by settig i c =. ad., ad R sc = 5,, 6, ad. The results suggest that (5) agree with (4) for small i c -values, ad they differ from each other if i c is icreased. Geerally, the critical shear velocity predicted with (5) is larger tha that with (4). Compariso with Measured Critical Shear Velocities The critical shear velocity for the icipiet sedimet motio subject to ijectio was experimetally observed with a glass-sided horizotal flume, which is 7.6 m log,. m wide ad.4 m deep. The relevat iformatio was detailed previously by Cheg ad Chiew (999), so oly a summary is provided here. As sketched i Fig. 4, a sedimet recess i the flume was prepared as a seepage zoe ad the ijectio flow itroduced from its bottom. Three uiform sedimets with media grai diameters of.6,. ad.95 mm, respectively, were used as the bed materials. The particular sedimet particles, which were used i the seepage zoe, were also glued to the impermeable bed to furish a cosistet surface roughess throughout the chael for each test. The icipiet motio of bed particles was observed accordig to the criteria of weak movemet as described i Vaoi (975). The velocity profiles, which were measured at the middle sectio of the seepage zoe uder the threshold coditio, were the applied to the computatio of the critical shear velocities. This ca be doe coveietly usig the modified logarithmic law, which was proposed by Cheg ad Chiew (998b) by icludig the seepage effect o the velocity profile i ope-chael flows. Fig. 5 is a plot of the computed critical shear velocities i the presece of ijectio agaist the measured results. The agreemet of the computatios with the measuremets is reasoably good cosiderig that the laboratory observatios associated with the icipiet sedimet motio are largely subjective. However, it is foud that for the same coditios used i the experimets, i particular, for the test with the sad of.95 mm diameter, the predictios usig the preset method [Eq. (5)] are slightly larger tha those obtaied usig the approach proposed by Cheg ad Chiew (999), i.e., () or (4). These differeces are highlighted i Fig. 6. The deviatios from the best fit lie seem systematic for the differet size particles. Such a 8

9 result would be expected if the weak movemet defiitio has a iheret scale bias. Additioal computatios by varyig the -value, for example, from.7 to. idicate that the so-iduced decease i the predicted shear velocity is isigificat, whe compared the marked scatter of the measured results. Coclusios The relatioship of the hydraulic gradiet ad seepage velocity ca be expressed i the form of biomial fuctios, for example, the Ergu equatio that is widely used i the chemical egieerig. This equatio is geeralised i this study, i particular, for icludig possible variatios i the characteristics of flows i the trasitioal zoe, where viscous ad iertial effects are i the same order of magitude. The geeralised Ergu equatio is the used to derive the formula for computig the critical shear velocity for the threshold coditio of bed sedimet subject to seepage. The computed results are foud to be i good agreemet with the measuremets coducted i ope chael flows with the bed ijectio, ad the predictios obtaied with the approach that was proposed previously based o a expoetial relatioship of the hydraulic gradiet ad seepage velocity. Ackowledgemets The writer is grateful to Robert K. Nive, School of Civil Egieerig, The Uiversity of New South Wales at the Australia Defece Force Academy, Caberra, Australia, ad two aoymous reviewers for their commets that improved this mauscript. Appedix I: Refereces.. Cheg, N. S. (997). Simplified settlig velocity formula for sedimet particle. J. Hydr. Egrg., ASCE, (), Cheg, N. S., ad Chiew, Y. M. (998a). Turbulet ope-chael flow with upward seepage. J. Hydr. Res., IAHR, 6(), Cheg, N. S., ad Chiew, Y. M. (998b). Modified logarithmic law for velocity distributio subjected to upward seepage. J. Hydr. Eg., ASCE, 4(). 4. Cheg, N. S., ad Chiew, Y. M. (999). Icipiet sedimet motio with upward 9

10 seepage. J. Hydr. Res., IAHR, 7(5), Chie, N., ad Wa, Z. (999). Mechaics of Sedimet Trasport. ASCE Press. 6. Churchill, S. W. (988). Viscous Flows: The Practical Use of Theory, Butterworths Series i Chemical Egieerig, Butterworth Publishers, U. K. 7. Ergu, S. (95). Fluid flow through packed colums. Chemical Egieerig Progress, 48, Marti, C. S., ad Aral, M. M. (97). Seepage force o iterfacial bed particles. J. Hydr. Div., ASCE, 97(HY7), Mitz, D. M., ad Shubert, S. A. (957). Hydraulics of graular materials. Traslated by Y. J. Hui ad H. M. Ma, Water Resources Press, Beijig, P. R. Chia (i Chiese).. Nive, R. K. (). Discussio o Icipiet sedimet motio with upward seepage by Cheg, N.-S. & Chiew, Y.-M. J. of Hydr. Res., 8(6), Nive, R. K. (). Physical isight ito the Ergu ad We & Yu equatios for fluid flow i packed ad fluidised beds. Chemical Egieerig Sciece, 57, Vaoi, V. A. (975). Sedimetatio Egieerig, ASCE Maual, No. 54, ASCE Press.. Watters, G. Z., ad Rao, M. V. P. (97). Hydrodyamic effects of seepage o bed particles. J. Hydr. Div., ASCE, 97(), Appedix II. Notatio The followig symbols are used i this paper: a = coefficiet; b = coefficiet; c = coefficiet; d = diameter of particle; d v g i i c K m = diameter of the equivalet-volume sphere; = gravitatioal acceleratio; = hydraulic gradiet of seepage; = critical hydraulic gradiet for fluidisatio; = coefficiet of permeability; = expoet; = expoet;

11 L * R s R sc u *c = dimesioless void dimesio; = seepage Reyolds umber; = critical seepage Reyolds umber; = critical shear velocity for the coditio of seepage; u *oc = critical shear velocity for v s = ; v s v sc α β ε μ ρ ρ s ϕ = (superficial) seepage velocity; = critical seepage velocity for the coditio of fluidisatio; = coefficiet; = coefficiet; = porosity; = dyamic viscosity of fluid; = desity of fluid; = desity of particles; ad = sphericity of particle.

12 CAPTION FOR FIGURES FIG.. Fig.. FIG.. Fig. 4. FIG. 5. FIG. 6. Geeralised Ergu Equatio i Terms of il * ad R s for Various Expoets. Comparisos of Geeralised Ergu Equatio to Experimetal Results Give by Mitz ad Shubert (957) Relatioships of Relative Shear Velocity ad Seepage Reyolds Number Experimetal Set-up for Observig Icipiet Sedimet Motio Subject to Ijectio Compariso of Computed Critical Shear Velocities with Experimetal Data Compariso of Critical Shear Velocities Computed Usig (4) ad (5), Respectively

13 .E+9.E+8.E+7 = E+6 il*.e+5.e+4.e+.e+.e+.e-.e+.e+.e+.e+.e+4.e+5 R s Fig..

14 i () d =.97cm ε =.54 ν =.64cm /s =.8 measuremets computed curve 4 5 v (cm/s) i 5 4 () d =.8cm ε =.56 ν =.4cm /s = v (cm/s) Fig.. 4

15 i.5.5 () d =.46cm ε =.5 ν =.cm /s = v (cm/s) i (4) d =.5cm ε =.54 ν =.76cm /s = v (cm/s) Fig.. 5

16 i (5) d =.779cm ε =.54 ν =.675cm /s =.78.5 i v (cm/s) (6) d =.665cm ε =.47 ν =.7cm /s = v (cm/s) Fig.. 6

17 i (7) d =.646cm ε =.446 ν =.7cm /s = v (cm/s) Fig.. 7

18 u*c/u*oc Eq. (4) (i =.) Eq. (5) (a) R sc = c u*c/u*oc R s Eq. (4) (i = ) Eq. (5) R sc = R s c (b) Fig.. 8

19 Perforated Pipes Perforated Plate Flow Meter From Submersible Pump Fig. 4. 9

20 .5 computed u*c (cm/s) measured u *c (cm/s) d =.95 mm..6 Fig. 5.

21 u*c computed with eq (5) (cm/s) d =.95 mm u *c computed with eq (4) (cm/s) Fig. 6.

On the Blasius correlation for friction factors

On the Blasius correlation for friction factors O the Blasius correlatio for frictio factors Trih, Khah Tuoc Istitute of Food Nutritio ad Huma Health Massey Uiversity, New Zealad K.T.Trih@massey.ac.z Abstract The Blasius empirical correlatio for turbulet