5 Dilute systems. 5.1 Weight, drag and Particle Reynolds number

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1 5 Dilue sysems This chaper considers he behaviour of a single paricle suspended in a fluid. In pracice, he equaions and principles described are used o undersand how a number of paricles behave, provided ha he concenraion is sufficienly low enough o ensure ha he behaviour of he paricle under consideraion is no significanly inerfered wih by he presence of oher paricles. Applicaions of he principles covered include paricle size analysis by sedimenaion mehods; where he seling rae is relaed o he size of he paricle, and he indusrial process of clarificaion by sedimenaion: in which paricles are removed from a fluid sream by allowing sufficien ime for he paricles o sele. 5.1 Weigh, drag and Paricle Reynolds number All forces mus reduce o Newon's basic equaion F = ma Forces eiher cause paricle moion in a fluid, or resis i. A force balance can be wrien using all he forces described, or some of hese. The easies force o appreciae is he paricle weigh, bu his is jus one example of a field force. The paricle weigh is he produc of is mass and he graviaional acceleraion. Paricles are usually oo small o weigh; hence he paricle diameer is used o calculae he volume, which is hen muliplied by he densiy o give he mass. Thus, for a spherical paricle, he paricle weigh is (in Newons) πx s g ρ (5.1) However, he paricle will experience an upward force, in accordance Archimedes principle, which numerically is πx ρg (5.) Hence, combining equaions (5.1) and (5.) provides he buoyed paricle weigh πx ( ρ s ρ)g (5.) Before considering oher field forces i is illusraive o conduc a simple force balance o see he applicaion of his approach. In a fluid, paricle weigh will cause an acceleraion ha will be resised by fluid drag. When he fluid drag force is equal o he paricle weigh he moion will be uniform; i.e. no longer acceleraing and he paricle will aain is erminal seling velociy. Fluid drag force comes from a suiable soluion o he Navier-Sokes equaion. However, his has only been achieved analyically under condiions of no urbulence wihin he fluid; i.e. sreamlines of fluid flowing pas he paricle, as Archimedes principle Saes ha when a body is wholly or parially immersed in a fluid i experiences an uphrus equal o he weigh of he fluid displaced Fig. 5.1 Flow sreamlines in fluid around a sphere Fig. 5. Sreamlines and urbulences in a fluid around a sphere a higher Re

2 4 Dilue sysems Galileo (154-14) Galileo is credied wih dropping differen sized balls from he op of he leaning ower of Pisa o show ha hey fell a he same rae. This ignores air drag and Galileo knew beer han his, he had worked on wind fricion. Noe ha equaion (5.5) is no valid for balls in air, bu (5.11) is generally valid. exercise 5.1 Deermine he maximum paricle size a which Sokes law should be applied for he daa: solid densiy: 500 kg m liquid densiy: 1000 kg m viscosiy: Pa s i.e. use Re =0. and subsiue equaion (5.5) in o (5.). Fig. 5. The drag coefficien or fricion facor plo for single spherical paricles illusraed in Figure 5.1. Under hese condiions Sokes drag expression is valid F = πuxµ (5.4) D which can be combined wih equaion (5.) o provide an expression for erminal seling velociy (U ), called Sokes law U x ( ρ s ρ) g = (5.5) 18µ In equaion (5.4) he drag force is relaed o he paricle velociy (U), for all values of velociy, whereas in equaion (5.5) we are referring o he final (erminal) seling velociy of he paricle in a saic fluid afer he period of acceleraion (U ). Clearly, he seling rae of a paricle is a funcion of is size, solid densiy and physical properies of he suspending fluid. Equaion (5.5) is only valid when he degree of urbulences wihin he fluid is negligible, see Figure 5.. This is measured by The Paricle Reynolds Number xuρ Re' = (5.) µ where he hreshold for sreamline flow pas he paricle is believed o be abou 0.. The Paricle Reynolds Number measures he raio of inerial o viscous forces wihin he fluid; hence i is he fluid properies ha should be used in i: fluid densiy and viscosiy. A Paricle Reynolds Numbers greaer han 0. he degree of urbulence becomes more significan leading o an addiional fluid drag force due o form drag. Hence, he erminal seling velociy will be lower han ha prediced by Sokes law, equaion (5.5), which considers only viscous drag around he paricle. In common wih fluid flow in pipes, i is possible o correlae a fricion facor wih Reynolds number for he case of fluid flow pas a single spherical paricle. This is illusraed in Figure 5.. The fricion facor is he shear sress in a plane a righ angles o he direcion of moion a he paricle surface (R) divided by he fluid densiy and relaive velociy beween he paricle and fluid squared. The drag force is he produc of he shear sress and he paricle area, which is he projeced area o he fluid flow (A p ). In paricle seling i is usual o use a drag coefficien (C d ), raher han fricion facor, hese are relaed as follows f R Cd = = (5.7) ρu where he fluid drag (F D ) is F D = RA p (5.8) The projeced area for a sphere is A p π = 4 x (5.9)

3 Fundamenals of Paricle Technology 47 Combining equaions (5.7) o (5.9) provides F D C d U π = ρ 4 x (5.10) Equaion (5.10) can be equaed wih (5.) o provide a generally valid equaion for he erminal seling velociy U ( ρ ρ) s gx = (5.11) ρc d However, equaion (5.11) can only be used o predic erminal seling velociy if a value of he drag coefficien is known, see Figure 5.. In he sreamline flow region we know ha 1 C d = (5.1) Re' which can be subsiued ino equaion (5.11) ogeher wih (5.) o provide equaion (5.5). Hence, he drag coefficien and Sokes law approach o paricle seling are compaible. However, for Paricle Reynolds numbers greaer han 0. no single and simple analyical funcion, equivalen o equaion (5.1), can be used. Many correlaions have been suggesed; all of hem are valid over a resriced range of Paricle Reynolds numbers. An alernaive approach comes from considering he drag coefficien furher ( ρ s ρ) g Cd = x (5.1) ρu Equaion (5.1) conains boh seling velociy and paricle size and canno be used o give he drag coefficien from a diameer because he seling velociy is also required. However, muliplying by he Paricle Reynolds number squared resuls in C d Re' = ( ρ ρ) g x U ρ ( ρ s ρ) gρ s ρu x µ = µ x (5.14) The erm in he square brackes conains neiher paricle diameer, nor seling velociy. Likewise, dividing Paricle Reynolds number by he drag coefficien gives xu ρ ρu = = d µ s Re' C ( ρ ρ) gx ( ρ ρ) s ρ U g µ Equaions (5.14) and (5.15) can be wrien as C d Re' = P H x and Re' U C = Q d H (5.15) where boh P H and Q H are no dependen upon paricle size or seling velociy. The fricion facor correlaion, Figure 5., can hen be redrafed in hese erms o give Figure 5.4. So, for he purposes of deermining he seling velociy from a given paricle diameer, equaion (5.14) provides a value for C d Re, Figure 5.4 is hen used o find Re /C d and equaion (5.15) o provide he seling velociy. Fig. 5.4 Modified drag and Reynolds number plo

4 48 Dilue sysems In pracice Figure 5.4 is no very easy o use: i is a logarihmic plo and he resoluion reduces as a decade is approached. Thus i may be easy o read off an unambiguous value a 11, bu difficul o read off a value a 90. To overcome his problem a se of ables was produced by Heywood, correlaing log 10(P H x) agains log 10(U /Q H ) and vice-versa, see he Appendix. Thus, in order o deermine he seling velociy from a paricle diameer, log 10(P H x) is firs calculaed and used o deermine log 10(U /Q H ) from The Heywood Tables. This value is hen ani-logged and muliplied by Q H o give he velociy. Clearly, his procedure is only worh he exra compuaional effor when he seling is a Paricle Reynolds numbers greaer han 0.. See he box on page 50 for an example of how o use he ables. The main advanage of he Heywood Tables approach, over empirical correlaions beween he Paricle Reynolds number and a derived funcion of he drag coefficien, is ha i is valid for all Paricle Reynolds numbers. I is also possible o implemen on a compuer and is available via he Inerne a: An alernaive popular correlaion, using he Paricle Reynolds number and The Archimedes (Ar) number, is 0.5 ( Ar ) 0.5 Re' =.798 (5.1) valid for <Re <0 000, where The Archimedes number is closely relaed o equaion (5.14) and is ( ρ ρ) s gρ x Ar = (5.17) ρ µ Thus, o deermine a seling velociy he Archimedes number is deduced from equaion (5.17), followed by he Paricle Reynolds number by equaion (5.1) and hence he velociy from (5.). 5. Oher forces on paricles Field forces oher han he graviaional include πx cenrifugal: F = ( ρ s ρ) rω (5.18) elecrical, hermophoreic (due o a emperaure gradien), hermal creep (due o greaer loss of molecules from he hoer side of a paricle), and phoophoreic (due o a ligh inensiy gradien). The cenrifugal field force is considered furher in Chaper 8, elecrical and hermophoreic forces in 14 and colloidal forces in 1. The inerial force (F i ) is he rae of change of momenum π du Fi = x ρ s (5.19) d noe ha he produc of volume and paricle densiy has been used for mass, assuming a spherical paricle.

5 Fundamenals of Paricle Technology 49 The fluid drag force may be subjec o he mean free pah correcion, which is required when he paricle size is comparable o he mean free pah of he fluid. This is required because he paricles can slip beween he fluid molecules - effecively reducing he viscous drag. I is more prevalen when he fluid is a gas. The correcion o he drag coefficien is 1 C = C [ λ / x (5.0) d d(stokes) ] where λ is he mean free pah lengh of he gas. For air λ=0.1 µm (approximaely) so he error in assuming he Sokes drag erm is 17% for a 1 µm paricle and 170% for a 0.1 µm paricle seling in air. When paricles come o res on each oher, or a surface, here is a solids sress gradien, or reacion force. This force can be raionalised by considering when a paricle is a res a he base of a vessel, as i experiences no drag or ineria, bu sill possesses a weigh (field) force. This force mus be balanced as he paricle does no accelerae. The reacion force is due o a pressure, or sress gradien, exered from he vessel base. 5. Paricle acceleraion in sreamline flow In he derivaion of equaions (5.5) and (5.11) he drag force was equaed o he graviaional field force, o deermine he erminal seling velociy of he paricle. This simple force balance is only valid if he paricle inerial forces can be negleced. Therefore, he ime aken o reach he erminal seling velociy, or 99% of i, is a useful check on he validiy of he simple force balance used o derive hese equaions. A force balance of he apparen mass (buoyed mass), drag and ineria for a spherical paricle is πx du ( ρ s ρ) g - πµ xu mp = 0 (5.1) d where m p is he acual paricle mass, no buoyed mass. Equaion (5.1) can be rearranged o give du π ( ρ s ρ) x g / πµ xu = (5.) d mp bu π ( ρ s ρ) g = πµxu (5.) i.e. he equaion for he erminal seling velociy. Therefore, du xu = πµ (1 U / U ) (5.4) d mp We need o inegrae equaion (5.4) o find ime aken o reach a given velociy, or fracion of erminal seling velociy. The equaion can be rearranged o give 1/ U πµ x du = d 1 U / U m p Brownian moion When small paricles are suspended in liquids, hey are subjec o molecular bombardmen giving rise o Brownian moion. Hence, finely divided paricles may no sele. In pracice, paricles smaller han µm suspended in waer will sele slower han prediced by Sokes law and paricles less han 1 µm migh no sele a all.

6 50 Dilue sysems i.e. a mahemaical relaion of he form: f ' ( x) = ln[ f ( x)] f ( x) Therefore, = U = U πµ x [ ln(1 U / U )] U = 0 = m = 0 he consan of inegraion is zero. Hence, mp = ln(1 U / U ) πµ x where he acual mass of paricle is m p : (5.5) π m p = x ρ s On considering equaion (5.5) i should be apparen ha he paricle will never reach is erminal seling velociy: i asympoes o his value. However, mos small paricles ha are encounered wihin Paricle Technology will reach 99.9% of heir erminal seling velociy wihin a very shor acceleraion ime. See Figure 5.5 for an illusraion of his. Fig. 5.5 Time aken o reach 99.9% of erminal seling velociy for he condiions illusraed Fig. 5. Criical rajecory model for coninuous seling basin design 5.4 Seling basin design (Camp-Hazen) Figure 5. illusraes he principle behind coninuous seling basin design, using a recangular clarifier. The feed flow eners he vessel on he lef and plug flow condiions are assumed, wih reaed effluen leaving on he righ of he vessel. Whils inside he vessel paricles sedimen and if hey reach he base of he vessel, before being removed in he effluen, hen he paricles are assumed o sick o he base and be removed from he liquid. Hence, he vessel design requiremen is o allow sufficien residence ime wihin he vessel o provide adequae paricle removal. The design is based on he criical rajecory model: where a paricle size is seleced and a balance underaken equaing he ime aken for he paricle o sele he full basin heigh, and he residence ime wihin he basin assuming plug flow. The resuling simple vecor analysis of he rajecory is a sraigh diagonal line: saring a he op lef of he vessel and finishing a he boom righ. All paricles of his size will be colleced; as hose saring heir rajecory from furher down han he full vessel heigh (H) will follow a parallel rajecory o he criical and, herefore, reach he vessel base before he full vessel lengh (L). The ime aken o sele will be H s = (5.) U and he residence ime, assuming plug flow, is HWL r = (5.7) Q

7 Fundamenals of Paricle Technology 51 where W is he vessel widh (i.e. residence ime is vessel volume divided by volumeric flow rae). Equaing hese wo imes, cancelling and rearranging provides Q = LW (5.8) U The produc of he vessel lengh and widh is he plan area. Hence, for complee removal of paricles of a given size, he volume flow rae divided by he corresponding erminal seling velociy is equal o he plan area for he complee removal. If he plan area is oo small hen no all he paricles of he seleced size will be removed. In seling i is always he plan area ha is he imporan design parameer and no he vessel cross-secional area. This imporan fac will be me again and in all cases he provision of a oo small plan area will resul in incomplee seling, or paricle removal. Furher consideraion of equaion (5.8) and (5.5) provides an indicaion of he efficiency of removal of paricles smaller han he criical size. All paricles larger han he criical will sedimen ou in he available ime, and he fracion of smaller ones removed will be direcly proporional o he seling velociy assuming ha he feed flow is in fac uniformly disribued over he heigh of he vessel and no all enering he vessel a he op. Hence, paricles half he criical size will only be colleced wih 5% efficiency because he seling velociy is proporional o he paricle diameer squared. 5.5 Laboraory ess Pracical laboraory ess o deduce seling parameers for he design of indusrial clarifiers involve he shor ube and long ube ess. Figure 5.7 illusraes he long ube es, where he suspension is allowed o sele wihin he ube for a se ime and he conens above a sample poin are drained off. The concenraion above he sample poin is deermined by weighing and drying. For each heigh i is possible o plo he concenraion remaining in suspension agains inverse ime, as illusraed in Figure 5.8. I may be possible o exrapolae his plo o an inverse ime value of zero; which will represen he concenraion of unselable solids. These are fine paricles ha remain suspended due o molecular bombardmen, or colloidal repulsion forces. Assuming a plo similar o Figure 5.8 provides a suspended solids concenraion ha is accepable for an effluen from a coninuous clarifier, hen he required seling ime and heigh of he sample poin (measured downwards from he suspension op), are used in he following equaion for vessel plan area A = Q 1 H / E (5.9) A where E A is an area efficiency o ake ino accoun urbulences, poor flow disribuion, ec. wihin he vessel. Fig. 5.7 The long ube es Fig. 5.8 Resuls from long ube es

8 5 Dilue sysems 5. Summary The seling velociy of small paricles may be reliably obained from Sokes law. Larger paricles, however, do no obey Sokes law. Alernaive correlaions beween drag coefficien and Paricle Reynolds number do exis bu he seling velociy is a consiuen of he Paricle Reynolds number; hence he answer needs o be known before he appropriae equaion o use can be idenified! To overcome his problem Heywood published a se of ables ha can be used over a wide range of seling velociies and paricle sizes. The single paricle seling discussed in his chaper is widely used in engineering calculaions. For example, wihin a spray drier rajecory analysis is ofen performed using he drag coefficien and he difference in velociy beween he paricle and he gas is of use in mass ransfer calculaions. Single paricle seling also forms he basis for undersanding he behaviour of more concenraed dispersions, which is he subjec of he nex chaper. Heywood Tables (see Appendix) 4( ρ s ρ) ρg P H = µ and 1/ 1/ 4( ρ s ρ) µ g Q H = ρ Boh funcions are size and velociy independen. When calculaing he seling velociy given a paricle diameer (x) he value of log(p H x) is firs calculaed. Then he firs wo significan figures of log(p H x) are given by he firs column of he able, he second and hird come from he scale given a he op of he able (he firs row). The corresponding value of log(u /Q H ) is hen read or esimaed from he able and convered ino a value for U using he calculaed funcion Q H. 5.7 Problems 1. i). A solid and liquid has a specific graviies of.8 and 1.0, respecively and he liquid viscosiy is Pa s, he value of he funcion P H is.87x10 4 m 1, he value for Q H is (SI unis): a:.87x10 b:.87x10 4 c:.87 d: ii). The SI unis of he funcion Q H are: a: m 1 b: m s 1 c: m s d: s m 1 iii).use he Heywood Tables o complee he following: Paricle diameer (µm): log(p Hx): log(u /Q H ): Seling velociy* (m s 1 ) Sokes seling velociy (m s 1 ): * seling velociy using he Heywood Tables iv). Why should he Sokes seling velociies of he larger paricles always be greaer han hose found in pracice (and given by he Heywood Tables)? xu ρ v). The Paricle Reynolds is defined as Re = µ which should be below some hreshold for Sokes law o be applicable. The maximum paricle size a which Sokes law is applicable for he above sysem is (µm): a: 59 b: 5900 c: 17 d: 171

9 Fundamenals of Paricle Technology 5. i). See he coninuous seling basin on he righ. Wha will be he rajecory of paricles he same size as he criical paricle size, bu which sar heir descen from a heigh less han H? ii). The solid and liquid densiies are 900 and 1000 kg m, he viscosiy is Pa s and he criical paricle diameer is 50 µm, he erminal seling velociy (U ) is (m s 1 ): a:.x10 b:.x10 5 c:.x10 1 d: 0.5 iii). The Paricle Reynolds number is: a: 0. b: 19 c: 0.19 d: iv). An expression for he criical paricle residence ime verically ( v ) is (s): a: v = L / U b: v = H / U c: v = H / v d: v = L / U v). An expression for he criical paricle residence ime horizonally ( h ) is (s): a: h = LWH / Q b: h = H / Q c: h = L / U d: h = Q / LWH vi). An expression for LW is (SI unis): a: LW = U / Q b: LW = Q / U c: LW = HQ / U d: LW = Q / HU vii). Wha are he unis of LW, and wha does i represen? viii). If he volume flow rae ino he basin is 10 m min 1 he minimum seling area required o remove all paricles of he diameer given in Par (ii), and bigger, is (m ): a:.4 b: 4 c: 40 d: 400. i). An effluen conaining a mineral in suspension wih solid and waer densiies of 00 and 1000 kg m, respecively, is pumped ino a bach vessel 5 m high and lef for 0 minues prior o discharge ino a river. The viscosiy of waer is Pa s. The maximum paricle diameer ha will be in he discharge is (µm): a: 180 b: 5.4 c: 90 d:.4 In he Camp-Hazen seling basin model he feed o a basin is assumed o ener well mixed and disribued evenly over he full deph of he vessel. The criical rajecory is given by a paricle of a cerain (criical) size ha eners he basin a he op lef and has jus seled by he end of he basin lengh - boom righ. The criical rajecory will be a sraigh line. Paricles smaller han he criical size will also have a sraigh line rajecory bu one ha does no inercep wih he base by he ime he fluid elemen has reached he end of he basin. ii). The effluen solid has he following paricle size disribuion: Cumulaive mass undersize (%): Paricle diameer (µm): If he iniial concenraion of he effluen before seling was 0 mg l 1, he concenraion of solids below he size calculaed in your answer in Par (ii) (he criical size) is (mg l 1 ): a: 0 b: 4. c: 4. d: 5.8

10 54 Dilue sysems This represens a 'wors case' esimae of he concenraion in he effluen discharge afer seling, as i assumes ha no solids smaller han he criical size sele in he allowed ime. iii). Of he concenraion of solids below he criical size a considerable fracion will also have seled ou. The amoun seled ou a each paricle diameer is proporional o he raio of is seling velociy compared o he velociy of he criical paricle. For example, a paricle wih a seling velociy half ha of he criical paricle will ravel.5 m in he allowed 0 minues and, if we assume he suspension was homogeneous before seling, half of he solids a ha diameer will sele ou. Complee he following able: Diameer (µm): Fracion seled a size: 1.00 Fracion undersize: iv). Now, o esimae he amoun of maerial seled below he criical size a plo of fracion of paricles seling in allowed ime agains fracion of maerial undersize is made and he area under he curve is calculaed by graphical means. Plo hese on he lef. The area under he curve is: a: 0.1 b: 0.5 c: 0.5 This represens he fracion of he oal disribuion below he criical size bu which sill seles because he paricles sill reach he base of he vessel in 0 minues. Add his fracion o he fracion of maerial in he size disribuion above he criical size (which has all seled ou). See your answer o Par (ii) o help you find his. NB he nex quesion does no wan he fracion seled i asks for he effluen concenraion going o discharge. v). Hence, he concenraion in he effluen discharge is (mg l 1 ): a: 9.1 b: 4. c: 4. d: 18. vi). If he effluen discharge consen limi is 0 mg l 1, will you be able o discharge his suspension?