3 Mass Transfer Rate Laws
Transcript of 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!
12/3
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!
2/12/1
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.
2/1
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