Mass Transport 2: Fluids Outline

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Beryl Willis
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1 ass Transport : Fuids Outine Diffusivity in soids, iquids, gases Fick s 1st aw in fuid systems Diffusion through a stagnant gas fim Fick s nd aw Diffusion in porous media Knudsen diffusion ass Transfer coeffiecient Diffusion in chemica vapour deposition Stagnant boundary ayer Exampe: Growth rate for mass transport imited growth Summary 1

1 D 6 Diffusivity in soids Reca that diffusion in a soid is proportiona to the distance of an atomic jump (space between attice sites), and the frequency of jumps

2 1 D 6 Diffusivity in soids Reca that diffusion in a soid is proportiona to the distance of an atomic jump (space between attice sites), and the frequency of jumps Probabiity that nearest neighbour is a vacancy and probabiity of a jump Q e D D e Q Soids Diffusivity in soids Exponentia T dependence Q interstitia <Q substitutiona

3 Diffusivity in iquids ack of understanding of the structure of iquids makes prediction difficut. t is known that: - diffusivity is about the same for a iquids ( m /s) - the temperature dependence is weak activation energies are -16 kj/mo The Einstein reation Fdrag Fdiff Spherica partices with radius r diffuse due to a gradient in the density ρ. The diffusive force due to this gradient is counterbaanced by a drag force reated to the veocity v and the viscosity η. k T D 6r Diffusivity in iquids Soids Liquids Exponentia T dependence D= m s -1 Q interstitia <Q substitu tiona 3

4 Diffusivity in gases o Kirkenda effect The number of moecues moving in one direction must be counteracted by the same number of moecues moving in the opposite direction - otherwise a pressure gradient woud buid up. ~ D D D D However, buk fow must sti be considered when evauating diffusion data Diffusivity in gases Diffusion theories are based on kinetics modes of gases Sef-diffusivity of spherica atoms/moecues 3 1/ 3 / k T D 3 3 m Pd nterdiffusivity of unequa size spherica atoms/moecues and D 1/ 3 k m 1 m 1/ 3 / T d d P Gas-phase diffusivities are on the order of m=moecuar mass P=pressure T=temperature d=moecuar diameter

Soids Diffusivity in gases Liquids Gases Exponentia T dependence D=1-8 -1-9 D=1-3 -1-5 m s -1 m s -1 Q interstitia <Q substitu tiona ass transfer in fuids Fuid systems differ from soid in that the

5 Soids Diffusivity in gases Liquids Gases Exponentia T dependence D= D= m s -1 m s -1 Q interstitia <Q substitu tiona ass transfer in fuids Fuid systems differ from soid in that the buk motion of the fuid must be considered The moar fux is therefore defined as the sum of a diffusion part, for a component moving reative to the rest of the fuid, and a buk part, for the movement of the fuid reative to outside coordinates n one dimension: x J x, diff J x, buk 5

6 ass transfer in fuid systems onsider a fuid with n components oar concentration: oar average veocity: v x n i1 * 1 i n i1 v i n i1 ix i oar fux of component consists of two parts Fick s 1st aw J D x x D x v * x Diffusion uk motion Diffusion through a stagnant gas fim This is a situation where one gas-phase component diffuses through a gas which is essentiay stationary onsider a system where is evaporating into gas. ssume constant temperature and pressure, iquid eve x=, and concentration of at x= so assume equiibrium at x=, so the vapour pressure of is the same as the concentration When the system reaches a steady state, the vapour in the tube wi be stationary. However there wi be a concentration gradient of (from evaporation) so wi fow through in the tube. 6

7 Diffusion through a stagnant gas fim ote there is sti buk motion, since the motion of contributes to the average motion of the gas x, x oundary condition 1, constant, set by gas fow oundary condition, constant, set by vapour pressure v * x 1 n i1 v i ix acuate the buk motion: 1 1 x x x v v x Diffusion through a stagnant gas fim nsert the expression for gas veocity into the fux: x D D D x x x * x v x x x x D 1 x 7

8 Diffusion through a stagnant gas fim x D 1 x d x dx Steady state d dx d dx n D x x 1 c x c 1 Diffusion through a stagnant gas fim n 1 c x c 1 oundary conditions x x 1 n 1 x 1 n 1 8

9 Diffusion through a stagnant gas fim 1 n 1 x 1 n 1 f we note that 1- =, we see that there is a gradient of which exacty compensates the gradient of. tends to diffuse down the coumn due to this gradient, but this is compensated by the buk motion (no net motion of ) Diffusion through a stagnant gas fim We usuay want the mass-transfer rate at the iquid-gas interface: x x D 1 x D 1 1 n 1 o ore convenienty, this can be written: x x D n where n n / (og mean of ) 9

Diffusion through a stagnant gas fim We can aso simpify the soution for the case where is diute: D x x x D n x ote that in this case buk motion is negigibe, so this equation has the same form as: J D

10 Diffusion through a stagnant gas fim We can aso simpify the soution for the case where is diute: D x x x D n x ote that in this case buk motion is negigibe, so this equation has the same form as: J D x x Exampe To determine the diffusivity of n in gas phase, pure iquid n is hed in a chamber at 16 o, through which pure r fows. The n eve is cm beow the edge of the crucibe. The rate of evaporation with time is found to be.65 x 1-7 mo cm - s -1 Soution: We first note that the vapour pressure of n at this temperature is.3 atm (from a tabe). We assume the pressure at the edge is. To determine the diffusivity we need the equation for fux: x D Here we have noted that the concentration of n is very diute. n r x= = cm x = 1

11 x D Exampe We then rearrange the fux equation to express the diffusivity: D We then need to express the concentration of n at the surface: n P.3atm mo 1 1 V (.851atm L mo K )(1873K ) n 3 cm Therefore we find that: D n 7 1 (.651 mo cm s ).cm 1 3.3cm s mo cm Porous edia Knudsen diffusion There are often situations where materias diffuse through porous soids. Gases diffusing through porous soids foow one of two mechanisms, depending on the reationship between pore size and mean free path: Fickian diffusion λ<d Knudsen diffusion λ>d 11

12 Knudsen diffusion J K d wv dx w=probabiity factor depending on the shape of the pore ~ r 1/ 8 V V=verage moecuar speed is moecuar mass For a cyindrica tube with radius r D K 97r T cm s -1 Knudsen diffusion - tortuosity = tortuosity, reated to pore ength D eff D K = porosity, percentage of open pores 1

13 ass transfer in chemica vapor deposition The boundary ayer mode ass transport is assumed to occur ony by diffusion through a truy stagnant ayer. j D z 5 U 1/ The distance at which the gas veocity parae to the wa becomes 99% of its free-streem vaue U The mass transfer coefficient mass transfer coefficient for transfer of into or out of a phase is defined in terms of the diffusive contribution norma to the interface. j k With no forced convection k reduces to: D D j k onvection and turbuence are accounted for by the Sherwood number (Sh) D k Sh 13

14 ass transfer in chemica vapor deposition orrections Turbuence: Reynods number; at east one order of magnitude beow the aminar/turbuent transition for H, higher for r and but sti sufficianty ow. H fow is even aminar in the tubes of the gas handing system. Forced convection: egibe, precursors are too diute. atura convection: n cod wa reactors the heat gradient between the susceptor and the was can cause natura convection. For H and He this is negibe but for r and it has to be considered. Exampe: ass transfer in VD Exampe: To grow a thin fim of titanium meta, titanium tetraiodide (Ti ) gas fows through a reactor with a suitabe substrate and decomposes according to Ti ( g) Ti( s) ( g) The standard free energy change of this reaction is G 118T kj kmo -1 ssume the growth is mass transfer imited, determine an expression for the growth rate. Growth temperature is 15 K 1

15 Exampe: ass transfer in VD Ti ( g) Ti( s) ( g) To sove this probem, we first need to know the steady-state concentrations of the components in the system: Reaction thermodynamics dea gas aw G n( K ) K e Ti Ti P P Ti G Ti PK Exampe: ass transfer in VD We need to cacuate the equiibrium constant: G 118T kj kmo -1 K e R R (given) Ti PK R 7.51 P Ti kmo m kmo m -3 Ti kmo m

Exampe: ass transfer in VD -3 3 8.15 1 Ti kmo m -3 kmo m j Ti j j j Ti k k 6.51 kmo m -3 3 8.111 kmo m -3 We have fuxes both towards and away from the surface: Ti 6.

16 Exampe: ass transfer in VD Ti kmo m -3 kmo m j Ti j j j Ti k k 6.51 kmo m kmo m -3 We have fuxes both towards and away from the surface: Ti 6.51 k 3 Ti Ti 1 k Exampe: ass transfer in VD j 6 3 k.51 k j Ti k Ti Ti 1 k Obviousy the fux of is much ower. Therefore this is the rate-imiting step and determines the growth rate. Ti ( g) Ti( s) ( g 1 j j Ti ) For every moes of removed, we grow 1 moe of Ti meta R Growth is fim density, is atomic mass R Growth 1 j 1 k 16

17 Summary ontinuum approach: Fick s 1. aw: Fick s. aw: c J D x c c D t x x Substitutiona interstitia diffusion Q tomistic approach: D D exp rrhenius equation Kirkenda effect: movement of markers attice-fixed and distance-fixed frame of reference Summary Sefdiffusion coefficient: ntrinsic diffusion coefficient: nterdiffusion coefficient: Types of diffusion: D Vo D D D ~ D G n 1 n D 1 D D Surf 17

18 J D x Summary ass transfer in fuid systems odification of Fick s aws to account for buk motion Fick s 1st aw x D x v * x Fick s nd aw t D Diffision uk motion * t D v Diffusion through a stagnant gas fim x x Diffusivity Summary Soids D D e Q Liquids k T D 6r Gases D 1/ 3 k m 1 m 1/ 3 / T d d P 18

19 ass transfer in chemica vapor deposition on-stagnant boundary ayer pproximate paraboic veocity profie by a inear one J eff z.89 D Dhz v 1/ 3.89 h=reactor height Dhz v v =average veocity 1 / 3 19

Elements of Kinetic Theory

Elements of Kinetic Theory Eements of Kinetic Theory Diffusion Mean free path rownian motion Diffusion against a density gradient Drift in a fied Einstein equation aance between diffusion and drift Einstein reation Constancy of