3 Mass Transfer Rate Laws

8/3/2019 3 Mass Transfer Rate Laws

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3 Introduction to Mass Transfer


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OverviewOverview

ThermodynamicsThermodynamics

heat and mass transferheat and mass transfer

chemical reaction ratechemical reaction rate

theory(chemical kinetics)theory(chemical kinetics)


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Rud iments of Mass Transfer Rud iments of Mass Transfer

Op en a bottle of perfume in theOp en a bottle of perfume in the

center of a roomcenter of a room--- ---mass transfermass transferMolecular p rocesses(e.g.,Molecular p rocesses(e.g.,collisions in an ideal gas)collisions in an ideal gas)

turbulent p rocesses.turbulent p rocesses.


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Mass Transfer Rate Laws Mass Transfer Rate Laws

F icks law of Diffusion

one dimension:

dxdY

DmmY m A AB B A A A Vdddd!dd )(

Mass flow of species A perunit area

Mass flow of species Aassociatedwith bulk flow

per unit area

Mass flow of s peciesA associated withmolecular diffusionper unit area


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The mass flux is defined as the massflowrate of species A per unit area

perpendicular to the flow:

The units are kg/s-m 2

DAB is a property of the mixture and has

units of m2/s, the binary diffusivity.

Amm A A /!dd


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I t means that species A is transporte d bytwo means: the first term on the right-han d -si d e representing the transporte d of

A res ul ting from the b ulk motion of the f lu i d, an d the secon d term representing

the d iff u sion of A s u perimpose d on thebulk f l ow.


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I n the abseence of diffusion, we obtain theobvious result that

where is the mixture mass flux. Thediffusion flux adds an additional

component to the flux of A:

Aspeciesof fluxBulk )( |dd!dddd!dd mY mmY m A B A Am dd

diff.A,mA,speciesof fluxlDiffusiona dd|dxdY

D A AB V


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An analogy between the diffusion of massand the diffusion of heat (conduction ) can

be drawn by comparing Fouriers law of conduction:

dxdT k Q x !dd


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The more general ex pression

where the bold symbols re p resent vectorquantities. In many instants, the molar

form of the above equation is useful:

A AB B AY DY dddd!dd V)( mmm AA

A AB A x D x dddd!dd V)( BAA NNN


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W here is the molar flux (kmol/s-m 2) of species A, x A is the mole fraction, and c isthe mixture molar concentration (kmol mix /m3)The meanings of bulk flow and diffusionflux become clearer if we express the total

mass flux for a binary mixture as the sum of the mass flux of species A and the mass fluxof species B:

A N dd


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B A mmm dddd!dd

Mixturemassflux

Speies Amassflux

Species Bmassflux


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For one dimension:

or dx

dY DmY dx

dY DmY m B BA B A AB A V V dddd!dd

dxdY

Ddx

dY DmY Y m B BA

A AB B A V V dd!dd )(


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For a binary mixture , Y A+Y B=1 , thus,

0!dx

dY Ddx

dY D B AB A

AB V V

Diffusionalflux of sp ecies A

Diffusionalflux of sp ecies B


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In general, overall mass

conservation required that: !dd 0,diff im


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This is called ordinary diffusion.

N ot binary mixture;

thermal diffusion

pressure diffusion.


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Mo l ecul ar basis of Diff u sion Mo l ecul ar basis of Diff u sionK inetic theory of gases: Consider a stationary(no bulk flow ) plane layer of a binary gasmixture consisting of rigid, nonattractingmolecules in which the molecular mass of each species A and B is essential equal. Aconcentration (mass-fraction ) gradient exists in

the gas layer in the x-direction and issufficiently small that the mass-fractiondistribution can be considered linear over adistance of a few molecular mean free paths, P ,as illustrated in Fig 3. 1


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Average mo l ecul ar properties d erive d

from k inetic theory:

P

WT

P

T

32

occurscollisionnextwhere planetocollisionlastof planefromdistancelar perpendicuAvearage

)(2

1 pathfreeMean

)(4

1

areaunit per moleculesAof frequencycollisionW all

)8

(moleculesAspecies

of speedmean

2

2/1

!|

!|

!|dd

!|

a

V n

vV n

Z

mT k

v

tot

A A

A

B


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W here k B is Boltzmanns constant;

m A the mass of a single A molecular,

n A/V is the number of A molecular perunit volume,

n tot /V is the total number of molecules

per unit volumeW is the diameter of both A and Bmolecules.


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Assuming no bulk flow for simplicity, thenet flux of A molecules at the x-plane is

the difference between the flux of Amolecules in the positive x-direction andthe flux of A molecules in the negative x-direction:

which, when expressed in terms of thecollision fre uenc , becomes

dir x Adir x A A mmm dddd!dd )(,)(,


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dddd!dd

molecule

single

Mass

areaandunit time per

aat x planefrom

goriginatinat x plane

crossingmolecules

Aof Number

molecule

single

Mass

areaandunit time per

a-at x planefrom

goriginatinat x plane

crossingmolecules

Aof Number

Aspecies

of flux

mass Net

)( )( Aa x A Aa x A A m Z m Z m


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W e can use the definition of density( V| m

tot/V

tot) to relate Z

Ato the mass

fraction of A molecules:

vY vm

mnm Z A

tot

A A A A V V

4

1

4

1!!dd


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S ubstituting the above Equation into theearly one, and treating the mixture densityand mean molecular speeds as constantsyields

)(4

1,, a x Aa x A A Y Y vm !dd V


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W ith our assumption of a linear concentration distribution

S olving the above equation for theconcentration difference and substitutinginto equation 3. 1 4, we obtain our finalresult:

3/42,,,,

Pa x Aa x Aa x Aa x A A Y Y

a

Y Y

dxdY

!!


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dx

dY vm A A

3

P V!dd

C om paring the above equation with the first equation,we define the binary diffusivity D AB as

3Pv

D AB !


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U sing the definitions of the mean molecular speed and mean free path, together with the

ideal-gas equation of state P

V=nk BT , thetemperature and pressure dependence of D ABcan easily be determined

or P T mT k D A

B AB 2

2/13

3

)(32 WT!

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wP

T D AB


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Thus, we see that the diffusivity depends

strongly on temperature ( to the 3/2 power ) and inversely with pressure. The mass flux of species A, however, depends on the product

V D AB,which has a square-root temperaturedependence and is independent of pressure:

I n many simplified analyses of combustion processes, the weak temperature dependenceis neglected and V D is treated as a constant.

2/102/1 T P T D AB

!w V


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C omparison with Heat

C onductionTo see clearly the relationship betweenmass and heat transfer, we now apply

kinetic theory to the transport of energy.W e assume a homogeneous gas consistingof rigid nonattracting molecules in which atemperature gradient exists. Again, thegradient is sufficiently small that thetemperature distribution is essentiallylinear over several mean free paths, as

illustrated in Fig. 3.2.


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The mean molecular speed and mean free path have the same definitions as given in

Eqns. 3. 1 0a and 3. 1 0c, respectively;however, the molecular collision frequencyof interest is now based on the totalnumber density of molecules, ntot /V , i.e.,

vV n

Z tot

!|dd4

1

areaunit per frequencycollisionwallAverage


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I n our no-interaction-at-a-distance hard-sphere model of the gas, the only energy

storage mode is molesular translational, i.e.,kinetic, energy. W e write an energy balance at the x-direction is the difference between the kinetic energy flux associatedwith molecules traveling from x-a to x andthose traveling form x+a to x

a xa x x ke Z ke Z Q dddd!dd )()(


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S ince the mean kinetic energy of amolecule is given by

the heat flux can be related to thetemperature as

T k vmke B23

2

1 2 !!

)(23

a xa x B T T Z k Q dd!dd


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The temperature difference in Eqn 3.22relates to the temperature gradientfollowing the same form as Eqn. 3. 15 i.e.,

S ubstituting difference in Eqn. 3.22employing the definition of Z and a, we

obtain our final result for the heat flux:

a

T T

dx

dT a xa x

2!

dxdT

vV n

k Q B x P)(2

1!dd


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Comparing the above with Fouriers law of heat conduction (Eqn. 3. 4) , we can identify

the thermal conductivity k as

Expressed in terms of T and molecular mass and size, the thermal conductivity is

PvV n

k k B )(2

1!

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43

3

T mk

k B

!

WT


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The thermal conductivity is thus proportional to the square-root of temperature,

as is the V D AB product. For real gases, thetrue temperature dependence is greater.

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T k w


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S pecies Conservation S pecies Conservation

C onsider the one-

dimensional control volumeof F ig. 3.3, a p lane layer ( xthick.


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The net rate of increase in the mass of Awithin the control volume relates to the

mass fluxes and reatction rate as follows:V m Am Am

dt

dm A x x A x A

cv A ddddddd! ( ][ ][ ,

Rates of increase of mass of Awithincontrolvolume

Massflow of A intothecontrol

volume

Mass flowof A out of the controlvolume

Mass produtionrate of species A

by chemicalreaction


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is the mass production rate of species A per unit volume (kgA/m3-s). I n Chapter 5 ,we specifically deal with how to determine

. Recognizing that the mass of A withinthe control volume is m A ,cv=Y amcv=Y A VV cvand that the volume V cv=A ( x, Eqn. 3.2 8 can be written:

Am ddd

Am ddd


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Dividing through by A ( x and taking thelimit as ( xp 0, Eqn. 3.29 becomes

x Am xY DmY A

xY

DmY At Y

x A

A x x A

AB A

x A

AB A A

(dddx

xdd

xxdd!

xx

(

(][

][)(

V

V V

A A

AB A A m

xY

DmY xt

Y dddx

xddxx

!x

x][

)( V

V


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O r, for the case of steady flow where

Equation 3.3 1 is the steady-flow, one-dimensional form of species conservationfor a binary gas mixture, assuming species

diffusion occurs only as a result of concentration gradients; i.e., only ordinarydiffusion is considered. For themultidimensional case, Eqn. 3.3 1 can begeneralized as

0/)( !xx t Y A

V

0][ !ddddddx

dY DmY

dxd

m A AB A A V


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0 !dddddd A A mm

N et rate of p roduction of species A by

chemicalreaction, perunit volume

N et flow of species Aout of

controlvolume, perunit volume


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S ome application

The stefan Problem:Consider liquid A, maintained at a fixedheight in a glass cylinder as illustrated inFig. 3. 4.


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Mathematically, the overall conservationof mass for this system can be expressed as

S ince = 0, then

B A mm xm dddd!!dd constant)(

Bm dd

costant)( !dd!dd xmm A


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Equation 3. 1 now becomes:

Rearranging and separating variables, weobtain

dxdY DmY m A AB A A A Vdd!dd

A

A

AB

A

Y dY

dx Dm

!dd

1 V


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Assuming the product V D AB to be constant,Eqn. 3.36 can be integrated to yield

where C is the constant of integration.W ith the boundary condition:

C Y x Dm

A AB

A !dd

]1ln[ V

i A A Y xY ,)0( !!


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W e eliminate C and obtain the followingmass-fraction distribution after removingthe logarithm by exponentiation:

]exp[)1(1)( , AB

Ai A A

D

xmY xY

V

dd!


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The mass flux of A, , can be found byletting Y A(x=L )= YA, in Eqn. 3.39. Thus,

Am dd

]1

1ln[

,

,

i A

A AB A

Y

Y

L

Dm !dd g

V


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From the above equation, we see that themass flux is directly proportional to the

product of the density and the massdiffusivity and inversely proportional to thelength, L . L arger diffusivities thus producelarger mass fluxes.

To see the effects of the concentrations atthe interface and at the top of the varyingY A ,i , the interface mass fraction, from zero

to unity.


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P hysically, this could correspond to anexperiment in which dry nitrogen is blownacross the tube outlet and the interface massfraction is controlled by the partial pressureof the liquid, which, in turn, is varied bychanging the temperature. Table 3. 1 shows

that at small values of Y A ,i , the dimensionlessmass flux is essentially proportional to Y A ,i ,For Y A , I greater than about 0. 5 , the mass flux

increases very rapidly.


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Table 3.1 Effect of interface

mass fraction on mass fluxY A ,i )//( L Dm AB A Vdd0 00.0 5 0.0 51 30.1 0 0. 1 05 40.20 0.223 1

0.5 0 0.693 10.90 2.3030.99 6.90 8


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Liquid-Vapor Interface

BoundaryC

onditions

3 Mass Transfer Rate Laws

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