3 Fluid flow in porous media

Transcription

1 Fluid flw in prus media In Chapter we cnsidered hw t represent a particle size distributin by, where pssible, a single term that is representatie f all the particle sizes. This term may then be used fr mdelling, design r simply t understand a prcess within Particle Technlgy. One such example is in the fluid flw thrugh a prus medium, r prus media (plural). There are a number f practical applicatins f fluid flw, including filtratin, flw in a packed clumn, permeatin f water, r il, within the matrix f a prus rck, etc. Befre discussing the cnsequences f ur chice f a single alue t represent the distributin, and the apprpriate mdelling equatins, we must define the cmmnly used terms.. Definitins By definitin, a prus medium cnsists f pres between sme particulate phase, cntained within a essel, r sme cntrl lume, as illustrated in Figure.. The fluid flw rate thrugh the bed is Q (m s ) and the bed crss sectinal area is A (m ). Thus the superficial (r empty tube) elcity U 0 is the ttal flw rate diided by the crss sectinal area. The existence f the particles within the bed will reduce the area aailable fr fluid flw; i.e. t presere fluid cntinuity with the entering superficial flw the fluid will hae t squeeze thrugh a smaller area; hence the elcity within the bed (U interstitial elcity) will be greater than the superficial. In Particle Technlgy calculatins it is the lume fractin that is mst imprtant, and nt the mass fractin. The lume fractin f slids present (i.e. lume slids in bed diided by ttal bed lume) is usually referred t simply as the lume cncentratin, r slids fractin, and the remaining fractin is that f the ids. The id fractin is als called the idage and the bed prsity. It is imprtant t realise that, in liquid systems, the ids are usually filled with liquid and nt t assume that the bed cnsists f just slids and air. The prsity is usually an istrpic prperty (i.e. the same in all directins); hence, the interstitial elcity is simply related t the superficial elcity by the fllwing expressin, which cmes frm a cnsideratin f fluid cntinuity. U U = (.) Clearly, the resistance t fluid flw thrugh the prus medium is related t the amunt f particles present, r lume cncentratin, but it is cnentinal t wrk in terms f bed prsity. At ne extreme, when the bed is full f slids (prsity is zer pssible with cubic particles placed carefully within the bed) the resistance is infinite. At the ther, when n slids are present and the prsity is unity, the interstitial elcity will be the same as the superficial elcity. The resistance t fluid flw gies rise t a pressure drp in Fig.. Illustratin f fluid flw thrugh a prus medium and cnsideratin f the lume fractins present exercise. Using cntinuity: i.e. Q = cnstant, deduce equatin (.)

2 Fluid flw in prus media Measuring prsity Fr a dry pwder in air: fill a weighed measuring cylinder t a graduatin, gently ibrate and reweigh. The bed mass er lume will gie the bulk density (ρ b ) f the pwder: ρ b = ( ) ρs + ρ As the fluid is a gas, its cntributin t the bulk density is minimal and = ρ ρ b / s the fluid (). Pressure is nt a ectr quantity, but a pressure gradient with respect t distance (/L) is. The pressure decreases in the directin f the fluid elcity, hence the pressure gradient shuld be negatie: (-/L). Hweer, fr the sake f breity, we will adpt the term pressure difference, which is a scalar quantity, hence the negatie symbl will nt be used in the fllwing text. The prsity f a packed bed f material depends strngly upn the nature f the particles and hw the bed has been treated. Incmpressible glass beads, including marbles, pack t a prsity f abut 45%, but beds f alumina particles (nt catalyst pellets) ften reach prsities f 75%. Bilgical material, such as yeast cells, may frm a packed bed with prsities f 90%, r higher. Therefre, it is dangerus t make the cmmn assumptin that packed beds are 50% slids and 50% ids. It is a simple parameter t measure, s lng as the slid and fluid densities are knwn; see the bx n the left.. Flw regimes On page the flw Reynlds number was stated. In a prus medium, as with all fluid flw prblems, we need t cnsider energy lsses frm the fluid due t iscus and frm drags. The frmer can be simply referred t as laminar flw (lw Re) whereas turbulent flw has additinal drag due t eddies in the fluid within the prus medium. The Mdified Reynlds number is used t determine the flw regime f the fluid within the prus medium. Mdificatin t the fluid elcity term (u) and the characteristic linear dimensin (d) are required. When cnsidering flw within the bed the apprpriate elcity is the interstitial, hence u = U, which can be related t the superficial elcity by equatin (.). The characteristic linear dimensin was deduced by Kzeny and is the lume pen t the fluid flw diided by the surface area er which it must flw (i.e. prduct f lume f slids and specific surface area per unit lume) AL d = = (.) AL( ) S ( ) S Thus, equatins (.) and (.) used in the Reynlds number expressin gie the Mdified Reynlds number (Re ) U ρ Re = = (.) ( ) S µ ( ) S µ Cnceptually, the number still represents the rati f inertial t iscus frces in the fluid and prides a means t assess when the inertial effects becme significant. The cnentinally applied threshld t indicate significant turbulence is, whereas fr the flw Reynlds number (page ) the cnentinal threshld is abut 000. It is imprtant t nte that the density term in equatin (.) is the density f the fluid: the turbulences described are that f the fluid, the particles d nt me in a packed bed.

3 Fundamentals f Particle Technlgy. Darcy s law and the Kzeny-Carman equatin Darcy s law, and the Kzeny-Carman equatin, are alid fr laminar flw (Re <); i.e. fr iscus drag by the fluid n the surface f the particles within the bed. The analgy with electrical flw is shwn in Figure.. A cell, r pump, prides the driing ptential and the flw, in either system, depends n the resistances in the circuit. Fr tw equal resistances, the ptential, r pressure, is equally diided between the tw. In fluid flw, high resistance is prided by high fluid iscsity (treacle is mre difficult t pump than air) and by lw permeability (k) f the bed. A alue f zer permeability wuld gie rise t infinite resistance bth fr electrical and fluid flw. Darcy s law is µ dv = (.4) L k dt A where V is the lume f fluid flwing in time t. Fr a gien bed length the pressure drp will rise linearly with lume flw rate, r fluid elcity, nting that Q dv U = = (.5) A dt A as illustrated in Figure.. The permeability f a packed bed can be measured in this way, using the gradient frm such a plt. The permeability is ften assumed t be a cnstant in a packed bed, prided the particle packing is als unifrm within the bed. It shuld, therefre, be an intrinsic prperty f a material. Hweer, in rder t use equatin (.4) fr design, e.g. t specify a pump required t pass liquid thrugh a bed at a desired flw rate, we need a methd fr predicting the permeability f the bed. This is prided by the Kzeny-Carman equatin. The Kzeny-Carman equatin was deried frm the Hagen- Piseuille equatin fr laminar flw f a fluid in a circular channel µ = u (.6) L d where d is the channel diameter. The deriatin assumed that flw in a prus medium can be represented as flw thrugh many parallel channels and equatins (.) and (.) were used t represent the equialent channel diameter and t cnert between the fluid elcity within the channel and the superficial. Hence, substituting these equatins in (.6) and cllecting the cnstants tgether in a single term called The Kzeny cnstant (K), which includes a factr relating the trtuus flw channel length t the measured bed depth gies the Kzeny-Carman equatin K ( ) S K ( ) S dv = µ U = µ (.7) L dt A Darcy bsered the law during the 850 s by mnitring pressure drp er sand filters at Dijn, France. Fig.. Analgy between fluid and electrical flws The flw rate (current r fluid) is prprtinal t the driing ptential: ltage in Ohms law and pressure gradient in a fluid. The cnstant f prprtinality is resistance (R) which fr a fluid is iscsity diided by bed permeability (k). Fig.. Graphical representatin f Darcy s law fr a bed f fixed erall length

4 4 Fluid flw in prus media Permeability Cmparing equatins (.4) and (.7) the permeability is k = K ( ) S hence the SI unit is m. The Kzeny cnstant is ften 5, but there is much experimental eidence t suggest that K = f ( ) i.e. the Kzeny cefficient is a functin f prsity. Cmparisn f equatins (.4) and (.7), results in the cnclusin that the Kzeny-Carman equatin is simply a subset f Darcy s law, with an analytical expressin fr permeability. There are many alternatie expressins fr permeability, but the Kzeny apprach is the mst frequently encuntered. In many instances the Kzeny cnstant has a alue clse t 5, but this is nt uniersally true. Inspectin f equatin (.7) shws that the permeability, r inerse resistance t fluid flw, is dependent upn the bed prsity (r slids cncentratin) and the specific surface area per unit lume f the particles within the bed. This is lgical because the higher the bed surface area the greater the iscus drag f the fluid n the particles. Equatin (.4) can be substituted int (.7) t pride a ersin f the equatin in terms f particle size 6K ( ) = µ U (.8) L xs where x S is the Sauter mean diameter fr the particle distributin. On page 6 and 7, the questin which particle diameter t use t represent the distributin? was asked and Figure. prided an example distributin. The x 50 and x S fr this distributin are 9. and 6.4 µm, respectiely. The cnsequence f using an inapprpriate equialent spherical diameter can be illustrated by using bth f these diameters in equatin (.8) and cmparing the results. Under identical flw cnditins, the rati f pressure drps calculated by the Sauter mean t that by the median particle size is :; i.e. there is 00% difference between the pressure drps calculated by these diameters; yet bth are equialent spherical diameters fr the same particle size distributin. Clearly, the Sauter mean is the mst apprpriate diameter t use, as indicated in equatin (.8), but it is a calculated alue and the median is quick t read ff the cumulatie distributin cure. Hence, there is great temptatin t use the median, but it wuld predict nly 50% f the likely pressure drp. Fig..4 The frictin factr plt fr fluid flw thrugh prus media.4 Frictin factr When turbulences within the fluid flwing thrugh the prus medium becme significant, i.e. Mdified Reynlds numbers greater than, additinal drag terms t the iscus nes quantified in the last sectin becme imprtant. In fluid flw thrugh pipes and channels a frictin factr was deduced t represent this regin and Carman extended the analgy with pipe flw t cer bth flw regins in prus media. The prus media frictin factr is illustrated in Figure.4 and the methd used t relate the shear stress at the surface f the slids, t the pressure drp, fllws (the same apprach fr flw in pipes is included in the bx erleaf fr cmparisn).

5 Fundamentals f Particle Technlgy 5 R (.9) Equatin (.9) is the frictin factr and R is the shear stress, r drag frce per unit area, n the particle surface. A frce balance at the particle surface can be cnstructed as fllws. surface area f particles= S ( ) LA (m ) drag frce = R. particle surface area (N) and pressure drp n fluid= P (N m ) frce by fluid= PA (N) Equating the tw frces and rearranging gies R = (.0) S ( ) L Nte that equatin (.0) is the analgue f that prided fr pipe flw in the bx. Finally, expanding int a frictin factr, equatin (.9), tgether with equatin (.) gies R = (.) S ( ) L Alternatiely, the pressure drp per unit length is R S ( ) = (.) L where the bracketed term is the frictin factr. S, gien a flw rate, hence superficial elcity, it is pssible t calculate the Mdified Reynlds number frm equatin (.) and the frictin factr frm Figure.4. This can then be used in equatin (.) t pride the pressure drp, r gradient, under cnditins f laminar r turbulent flw thrugh the prus medium..5 Carman and Ergun crrelatins The frictin factr plt, with Reynlds number, fr fluid flw thrugh prus media is a smther functin than that fund in pipe flw. This is due t the smth increase in turbulences within the bed as flw rate increases. Thus the frictin factr plt can be represented by just ne, r tw, empirical cures. The Carman crrelatin is generally used fr slid bjects frming a bed R = + (.) Re Re 0. The Ergun crrelatin is fr hllw bjects, such as packing rings R 4.7 = (.4) Re The frm f bth equatins is similar: with a crrectin term added t the laminar flw term t accunt fr resistance due t turbulences. Frce balance n a pipe wall (fr pure fluid n particles) frce n wall: Rπ d. δx frce n fluid: δ P πd 4 cmbine and integrate: d RL = r 4 d R = 4 L c.f. equatin (.0) Laminar flw Carman crrelatin withut the turbulent crrectin is R 5 = Re Using equatin (.0) and (.) prides =... S ( ) L K ( ) Sµ... = S, K ( ) S = µ U L i.e. equatin (.7)

6 6 Fluid flw in prus media This is illustrated in the bx, which shws that the Carman crrelatin reduces t the Kzeny-Carman equatin, with K=5, when the turbulent crrectin term is drpped. In practice, when perfrming flw calculatins with Mdified Reynlds numbers greater than, equatins (.) r (.4) are used t determine the frictin factr rather than Figure.4, and equatin (.) is used t calculate the pressure drp. Packing arrangements Unifrm spheres, packed tgether in a regular pattern, ary frm a crdinatin number f 6, fr simple cubic packing, t fr hexagnal clsepacked; which is the clsest pssible packing fr unifrm spheres. The slids cncentratin fr these arrangements aries frm 0.54 t 0.740, respectiely (prsities f t 0.60). Randmly packed spheres hae a slid cncentratin f 0.50 t In thery, with a size distributed slids the finer particles culd fit inside the gaps between the larger particles priding een higher slid packing. In practice, slid cncentratins much lwer than the regular packed arrays are fund. It is always safest t measure cncentratin, r prsity, as described n page..6 Cncentratins by mass and lume Slid cncentratin by lume fractin (C) was illustrated under Figure. and is simply the lume f slids present diided by the ttal bed lume. It is numerically equal t unity minus bed prsity. In mst f the fllwing chapters, it is mre cnenient t wrk in terms f slid cncentratin rather than prsity. Hweer, labratry analyses ften pride slid cncentratin by mass. Fr example, taking a sample f a filter cake cntaining water: weighing, drying and then weighing the dried cake will pride the cncentratin by mass if the last mass is diided by the first. Hence, cnersin between the tw different types f cncentratins is frequently required. Cnsideratin f what the slid cncentratin by lume fractin means leads t the fllwing expressin lume slids C = lume slids + lume fluid If a sample has resulted in a cncentratin by mass (C w ), and the ttal sample mass is M, then the lumes present can be deduced if the densities f the slid and fluid are knwn by Cw M / ρ C = s Cw M / ρs + ( Cw ) M / ρ (.5) Diiding thrugh by the lume f slids gies C = ( Cw ) ρ (.6) + s Cw ρ A similar argument can be applied t cnert frm cncentratin by lume fractin t mass fractin..7 Summary In this chapter we hae seen the imprtance f specific surface: it defines the surface area that is present within a prus medium; and it is the frictin f fluid flwing er that area that causes a pressure drp. Finer particles pride a higher surface area per unit lume than carser nes, therefre, a higher flw resistance. Turbulences within the fluid inside the bed, at higher flw rates, cause an additinal flw resistance, r pressure drp. Thus, a calculated pressure drp by a laminar flw equatin, Darcy s law r Kzeny- Carman, will always underestimate the true pressure drp if significant turbulence is present. Often the temptatin t use Kzeny-

7 Fundamentals f Particle Technlgy 7 Carman, rather than the prcedure described in Sectins.4 and.5, is t great een when the Mdified Reynlds number is high. This will lead t an under-design, r specificatin, fr equipment such as pumps and fans. Priding anther example f the failure in design discussed in the Frward. Een under cnditins f laminar flw, the use f the median size t respresent the size distributed slids rather than the Sauter mean diameter can cause significant errrs, as shwn in Sectin.. Finally, a brief cnsideratin f particle packing arrangements has been included, but in mst prcesses the packing arrangement is randm and nt structured. Hence, the safest prcedure fr the analysis f a flw thrugh prus media prblem is t cnduct experiments t deduce characteristics such as the permeability and if it aries with flw; a pssibility if the finer particles becme transprted within the bed r if suspended slids within the fluid depsit inside the bed. The latter is depth filtratin, which is cered in the next chapter..8 Prblems. (i) A pwder is cntained in a essel t frm a cylindrical plug 0.8 cm in diameter and cm lng. The pwder density is.5 g cm and.0 grams f pwder was used t frm the plug. The prsity inside the plug f pwder is (-): a: 0.58 b: 0.4 c: 0.75 d: 0.5 (ii). Air was drawn thrugh the plug at a rate f 6.6 cm per minute. A mercury manmeter was used t measure the pressure drp during this prcess: a pressure drp f 60 mm Hg was recrded. The specific graity f mercury is.6, thus the pressure drp acrss the plug was (Pa): a: 80 b: 800 c: 8000 d: (iii). The superficial gas elcity in (ii) was (m s ): a: 0.00 b:.65x0 5 c:.x0 6 d: (i).the iscsity f the air was.8x0 5 Pa s, using the Kzeny- Carman equatin, the specific surface area per unit lume f the pwder was (m - ): a: 0 b:.0x0 c: 5.5x0 5 d:.x0 6 (). The Sauter mean diameter f the pwder was (µm): a: 600 b: c: d: 5.0 (i). The air density was. kg m, the Mdified Reynlds Number f the system was (-): a: 0.0 b: 8.4x0 0 c: 4.6x0 4 d: Equatin summary Under laminar flw: K ( ) S = µ U L where K is the Kzeny cnstant At high alues f the 'Mdified Reynlds Number' (Re >): U ρ Re = µ ( ) S turbulent cnditins pertain. A pressure drp gien a flw rate can still be deduced, but first the Mdified Reynlds number is required, then the frictin factr using say the Carman crrelatin: R = + Re Re 0. A frce balance n the surface f the slids and n the fluid gies: RS LA( ) = A where L is the bed height r depth. Gien a alue fr the shear stress n the slids (R) calculated frm the Carman crrelatin then the pressure drp () can be calculated frm the frce balance.

8 8 Fluid flw in prus media (ii). Cmment n whether yur use f the Kzeny-Carman equatin was alid r nt:. (i). A cylindrical in exchange bed cmpsed f spherical particles mm in diameter packed at a bed idage f 0.45 is t be used t deinise a liquid f density and iscsity 00 kg m and Pa s respectiely. The design flw rate is 5 m hur and the bed height and diameter are and 0. m respectiely, using the Kzeny-Carman equatin the pressure drp is (Pa): a: b:.x0 7 c: 4400 d: (ii). The Mdified Reynlds Number is: a:.9 b: 4.78 c: 90 d: (iii). Cmment n yur use f the Kzeny-Carman equatin: (i). The interstitial liquid elcity inside the bed is (m s ): a: 0.00 b: c: 0.08 d: (). Using the Carman crrelatin the shear stress n the in exchange beads is (Pa): a: 7. b:.5 c:.5 d: 7.0 (i). Hence the dynamic pressure drp er the bed is (kpa): a: 84 b: 0 c: 99 d: 50 (ii). Why is the answer t (i) different t that in (i)? (iii). If the liquid has a datum height equal t the psitin at the base f the in exchange essel and, therefre, needs raising t the tp f the clumn befre it enters the in exchange bed the additinal pressure drp t effect this, i.e. the static pressure drp er the bed, is (kpa): a:.6 b:.6 c: 6 d: 0.