Mass Transfer. Dispersion. Lecture 13, , Dr. K. Wegner

Transcription

1 Mass Tansfe Dispesion Letue 3, 3..7, D. K. Wegne

2 Dispesion to dispese to spead widely. Dispesion is the at o poess of dispesing, of distibuting one substane (small volume fation) in anothe (lage volume fation).. Dispesion is the state of being dispesed. Examples: Paints: pigments ae dispesed in wate o solvent Fog, louds: wate doplets ae dispesed in ai Sooting andle: soot patiles in ai. Pollution of ives, ai,... Speading of diseases, hamstes,... Mass Tansfe Dispesion 3-

3 Examples Dust Clouds fom Afia Global tavel of dust aies miobes aoss oeans and ontinents. Mass Tansfe Dispesion 3-3

4 Pollen Clouds Atmosphei Clouds In 6, bih pollen fom Denmak taveled aoss the Noth Sea to England. Wate doplets dispesed in ai appea as fog/louds (and they eate ainbows) Mass Tansfe Dispesion 3-4

5 Suspensions and Emulsions Paint, fo instane, is a dispesion of pigments in a solvent. Homogeneous dispesion of the pigments in the fluid as well as longtem stability of the dispesion (shelf-life) ae impotant. Mass Tansfe Dispesion 3-5

6 Soot, smoke Condensation plumes fo 5 ft and 5 ft staks in Salem (MA), showing the omplex themal stutue in the lowe atmosphee. Mass Tansfe Dispesion 3-6

7 Dispesion on the miosopi level involves diffusion (of doplets, patiles, moleules,...). On the maosopi level dispesion is govened by fluid dynamis (lamina, tubulent flow, eddy fomation). Goal: To undestand how e.g. plumes emitted by industial failities and ships behave. The mehanisms ausing the mixing and dilution e.g. of the plumes ae hydodynami. Moeove, the govening equations ae vey omplex and an exat solution is impossible, espeially if the vaiability, e.g. of the weathe is inluded. Mass Tansfe Dispesion 3-7

8 The dispesion equation It is impossible to alulate the atual onentation at a etain point fo a etain time but we ty to alulate the aveage onentation. This has been done befoe fo moleules diffusing in a medium. We ty to apply the mathematis applied to diffusion to dispesion. Let's stat with a athe simple poblem: A point soue only ative at time t, e.g. an emission pulse of smoke fom a stak. Mass Tansfe Dispesion 3-8

9 Deay of a pulse (see Chapte, Fik's nd law) The onentated solute oiginally loated at z diffuses as the Gaussian pofile shown. B.C.: t : M A δ(z) δ(z): Dia ft. z : t > : z : z z : Mass Tansfe Dispesion 3-9

10 Fik's nd law: t D z The onentation pofile is (Chap. ): M A 4πDt e z 4Dt whee M is the mass of the solute and A the oss-setional aea ove whih diffusion (dispesion) is ouing. Fo patiles with onentation N (#/m 3 ) eleased at t and z : n t D p n z and n N 4πD t p e z 4Dt with the patile diffusivity D p, see "Diffusion oeffiients" Mass Tansfe Dispesion 3-

11 As we have seen in Chapte, the Fouie numbe fo mass tansfe an indiate fo an unsteady state poblem how fa (how long) mass tansfe has oued. Assume D t Fo L Aoding to mass tansfe by diffusion in gases with D -5 m /s, the width of the peak oughly should be afte hou: L D t 5 m s 36s. m Howeve, measuements show that afte about h a plume has spead km! Use the dispesion oeffiient E instead of the diffusion oeffiient D! The dispesion oeffiient must be detemined expeimentally and typially has diffeent values in diffeent dietions. Mass Tansfe Dispesion 3-

12 Dispesion in one o moe dimensions The soue of the plume is at (,) and wind is blowing at speed v in the x-dietion. We want to know the aveage pollutant onentation at (x, y). 4π E y ( x / v ) e y 4E y t Based on this geneal equation fo one-dimensional dispesion in y dietion, a vaiety of onentation pofiles have been deived. Mass Tansfe Dispesion 3-

13 Conentation pofiles fo fee dispesion Mass Tansfe Dispesion 3-3

14 Smoke plume of a foest fie follows the dispesion patten of this pulse deay Foest fie in southen Fane (Le Boulou) nea the Spanish bode in July 7. Vaation home of the letue was x 3 km away fom the point soue. v 6 km/h. Photo: L'Indépendant (J. Gallado) Mass Tansfe Dispesion 3-4

15 Dispesion oeffiients... - ae athe independent of hemisty (no mateial popety) - ae stongly dependent on flow (veloity, lam. / tub.) - depend on position - have diffeent values in diffeent dietions - typially must be detemined expeimentally Mass Tansfe Dispesion 3-5

16 In 95, five muskats ("Bisamatte") esaped in Bohemia. The animals quikly spead ove Euope, as shown in the Figue. Define a dispesion oeffiient simila to the diffusion oeffiient (A. Einstein): Et Skellam, J.G. (95) Biometika 38, 96. Mass Tansfe Dispesion 3-6

17 Example : Chemial spill A ontaine with hemials beaks, aidently eleasing its ontent into a ive flowing at v.6 km/h. When expets of the envionmental potetion ageny aive to take wate samples, the maximum onentation of 86 ppm is loated km downsteam the elease point. 5 m fom the maximum, the onentation is 4 ppm. a) How lage is the dispesion oeffiient? b) What will the maximum onentation be 5 km downsteam? v.6 km/h t initial spill t, s km: max 86 ppm Mass Tansfe Dispesion 3-7

18 Solution: Stat with a mass balane fo a setion of ive dz aound the maximum onentation, moving with veloity v: d dt with z j j E E z t z Bounday onditions: t : M δ(z) A z : t > : z : z z : Mass Tansfe Dispesion 3-8

19 Conentation pofile: Deay of a pulse M A 4πEt measuedmaximum onentation e z 4Et a) So, 5 m away fom the maximum: 4 ppm 86 ppm e ( 5m) km 36s 4E.6km h h m E 7 (ompae diffusion oeffiients: -5 m /s) s km b) Note:, max ~, thus ( 5km) 86ppm 34ppm t 5km Mass Tansfe Dispesion 3-9

20 Example : Tubulent flow in gas pipeline A 3 km long pipeline with m diamete is used to tanspot diffeent gases fom a stoage aea to a hemial eato. The gas veloity is 5 m/s. Afte swithing fom gas A to B, how muh will the gases mix? Fist hek the ondition of flow: kg m 5.m ρ v d 3 Re m s -6 η Pa s 5' Tubulent flow! We'll teat lamina pipe flow aftewads! Mass Tansfe Dispesion 3-

21 Plug flow pofile B v 5m/s z A Assumption: Well mixed adially but onentation hanges in axial dietion. Mass balane fo point at the intefae moving with v 5 m/s: E B.C.: t, z > : t z t >, z : t >, z : whee is the aveage onentation of the two gases. Mass Tansfe Dispesion 3-

22 The solution is simila to the geneal semi-infinite slab solution: ef z 4Et The dispesion oeffiient fo tubulent dispesion in pipelines is appoximately: d E v,thus in this ase E 5 m /s Let us onside a onentation hange signifiant fo and lose. z 4Et ef().84, thus This is the ase fo z 4.5 m s 3 m m.5 s 4 m Mass Tansfe Dispesion 3-

23 Example 3: Lamina flow Taylo dispesion A typial example fo dispesion is the spead of a solute pulse in steady lamina flow. R v z Injetion of solute pulse We an do an analysis using the following assumptions: The solution is dilute The flow is always lamina, no fition (no axial hange in veloity) Mass tanspot is by axial onvetion and adial diffusion only Mass Tansfe Dispesion 3-3

24 Goal: Pedition of the dispesion oeffiient Let us look at the solute mass balane of a ing-shaped egion with dimensions Δ and Δ z, while the adius of the ing is : t n θ n z θ z ( n ) + () This simplifies to: t ( j ) v z z () Veloity pofile fo lamina tube flow: v z ( ) v R aveage veloity v max (3) Mass Tansfe Dispesion 3-4 v

25 Mass Tansfe Dispesion 3-5 Eq. () an be tansfomed by substituting Fik s law and eq. (3) into: z R v D t (4) Bounday onditions: t, all z: ) (z R M δ π t >, : R :

26 Mass Tansfe Dispesion 3-6 One solution to this equation is: ζ + 4 D R v 4 (6) Let us sit on a efeene fame moving with veloity v to obseve the ation happening on the z-axis: o R t z v ζ R Substitute and So eq. (4) beomes: ζ R v D (5)

27 New appoah with futhe simplifiation: Assuming that the adial vaiations in ae small elative to the axial ones (esp. fo fast diffusion), we an wok with the adial aveage: R π (,z)d d πr Now we think of this onentation as a solute diffusing in the moving efeene fame and ewite the mass balane: t j ( ζr ) whee ζ R z v t (7) (8) Hee j (the aveage flux in flow dietion) is equal to: R j π (v π z v )( ) d R (9) Mass Tansfe Dispesion 3-7

28 Eq. (8) an be witten as: ( tv R ) τ ( j v ) ζ ζ 4 Inseting eq. (6) and using eq. (7) we have thus: d τ v R 48D ζ The bounday onditions fo the new equation ae simila to the ones fo the "deaying pulse" poblem, namely: τ >, ζ τ, ζ : ± : ; M πr 3 δ( ζ) τ >, ζ : ζ () Mass Tansfe Dispesion 3-8

29 The solution to this poblem is a Gaussian uve: M ( πr 4πE t ) e (zv t) 4E t () The dispesion oeffiient is given by (Taylo, 953): v R E v R 48D () Note that E depends invesely on the diffusion oeffiient! Rapid (adial) diffusion leads to small (axial) dispesion. Taylo, Si G. (953), "Dispesion of soluble matte in solvent flowing slowly though a tube", Po. Royal So. A 9, 86-3 Mass Tansfe Dispesion 3-9

30 v R E v R 48D adial diffusion Rapid (adial) diffusion leads to small (axial) dispesion. Mass Tansfe Dispesion 3-3

31 Mass Tansfe Dispesion 3-3 Coetion fo axial diffusion: Taylo-Ais Dispesion + + z D z v t axial diffusion 4E t t) v (z e E t 4 ) R M ( π π also yields but now + 48D R v D E Ais, R (958), "On the dispesion of a solute in a fluid flowing though a tube", Po. Royal So. A 35,

32 Belongia B.M. and Baygents, J.C. (997): "Measuements on the Diffusion Coeffiient of Colloidal Patiles by Taylo-Ais Dispesion" Moving sample Zone UV Absobane by patiles in the sample zone as ft of time Patile-fee aie solution Pessue Gadient Belongia B.M., Baygents, J.C. (997), J. Colloid Intefae Si. 95, 9-3. Mass Tansfe Dispesion 3-3