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3.MASS-TRANSFER THEORIES3.MASS-TRANSFER THEORIES

(1)Mass Transfer Coefficient(1)Mass Transfer Coefficient

•For steady-state mass transfer through a stagnant For steady-state mass transfer through a stagnant layer of fluid layer of fluid , , mass transfer rate can be predicted by mass transfer rate can be predicted by following equations:following equations:

(1)Equimolal diffusion(1)Equimolal diffusion

oror

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(2)One-component mass transfer (one-way diffusion) (2)One-component mass transfer (one-way diffusion)

•More common used type of equations:More common used type of equations:

Analogous to heat transfer,Analogous to heat transfer,

Heat transfer rate=(Heat transfer coefficient) Heat transfer rate=(Heat transfer coefficient) (Heat transfer driving force)(Heat transfer driving force)

Mass transfer rate=(Mass transfer coefficient) Mass transfer rate=(Mass transfer coefficient) (Mass transfer driving force)(Mass transfer driving force)

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•Definition of mass transfer coefficient: Definition of mass transfer coefficient: The rate of The rate of mass transfer per unit area per unit concentration mass transfer per unit area per unit concentration difference, usually based on equal molal flows.difference, usually based on equal molal flows.

Other forms of mass transfer equations:Other forms of mass transfer equations:

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ThereforeTherefore, ,

=mass transfer coefficient based on molal =mass transfer coefficient based on molal concentration driving forceconcentration driving force

=gas phase mass transfer coefficient based on the =gas phase mass transfer coefficient based on the partial pressure driving forcepartial pressure driving force

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=gas phase mass transfer coefficient based on the =gas phase mass transfer coefficient based on the mole fraction differencesmole fraction differences

=liquid phase mass transfer coefficient based on =liquid phase mass transfer coefficient based on the mole fraction differencesthe mole fraction differences

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•Relations between mass transfer coefficients:Relations between mass transfer coefficients:

)38.17(RT

Pkkk

cc

Jcc

J

yy

Jk

yc

ccMy

AAi

AM

M

A

M

Ai

A

AAi

Ay

MAA

Similarly, in liquid phase, Similarly, in liquid phase,

)39.17(M

kkk

cc

Jcc

J

xx

Jk

xc

xccMx

AAi

AM

M

A

M

Ai

A

AAi

Ax

MAA

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Here, Here,

=density of liquid, kg/m=density of liquid, kg/m33

=average molecular weight of liquid =average molecular weight of liquid

Significance of kc: fromSignificance of kc: from

)42.17(T

vc B

Dk

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•For steady-state equimolal diffusion in a stagnant For steady-state equimolal diffusion in a stagnant film, mass transfer coefficient kc is the film, mass transfer coefficient kc is the molecular molecular diffusivity diffusivity divided by the thickness of the stagnant divided by the thickness of the stagnant layerlayer （（ BBTT ）） . .

•When we are dealing with When we are dealing with unsteady-state diffusion unsteady-state diffusion or diffusion in or diffusion in flowing streamsflowing streams （（对流）对流） , , Eq.(17.42) can Eq.(17.42) can still be used to give an still be used to give an effective film thicknesseffective film thickness B BTT

from known values of kc and Dv.from known values of kc and Dv. 对于对流传质，对于对流传质，（（ 17.4217.42 ）式有效）式有效 , , BBTT 为有效膜厚度为有效膜厚度

)42.17(T

vc B

Dk

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(2)Film Theory(2)Film Theory

•Analogous to convective heat transfer,Analogous to convective heat transfer,

Heat transfer rate q:Heat transfer rate q:

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Illustrational diagram Illustrational diagram of of wetted wall towerwetted wall tower

)42.17(T

vc B

Dk

•The basic concept of the The basic concept of the film theory is that the film theory is that the resistance to diffusion resistance to diffusion can be considered can be considered equivalent toequivalent to （（相等于）相等于） that in that in a stagnant film of a a stagnant film of a certain thicknesscertain thickness

•Then, Then, BBTT

膜理论基本概念是传质阻力相等于膜理论基本概念是传质阻力相等于 停滞膜厚度停滞膜厚度

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•The implication is that the coefficient The implication is that the coefficient kc varies with the kc varies with the first power of Dv, which is rarely truefirst power of Dv, which is rarely true, but this does not , but this does not detract from the value of the theory in many detract from the value of the theory in many applications. The film theory is often used as a basis for applications. The film theory is often used as a basis for complex problems of multi-component diffusioncomplex problems of multi-component diffusion （（多组分扩散）多组分扩散） or diffusion plus chemical reaction.or diffusion plus chemical reaction.

•The value of The value of BBTT depends on the diffusivity Dv and not depends on the diffusivity Dv and not

just on flow parameters, such as Reynolds numberjust on flow parameters, such as Reynolds number. The . The concept of an effective film thickness is useful, but values concept of an effective film thickness is useful, but values of Bof BTT must not be confused with the actual thickness of must not be confused with the actual thickness of

the laminar layerthe laminar layer （（层流底层）层流底层） ..

)42.17(T

vc B

Dk

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•Effect of one-way diffusionEffect of one-way diffusion

•When only one component A is diffusing through a When only one component A is diffusing through a stagnant film, the rate of mass transfer for a given stagnant film, the rate of mass transfer for a given concentration difference is greater than if component concentration difference is greater than if component B is diffusing in the opposite direction. B is diffusing in the opposite direction.

)43.17(1)(

1

)1(

1

LBLAA

A

yyJ

N

Where, Where,

=molal flux of one-way diffusion=molal flux of one-way diffusion

=molal flux of equimolal diffusion=molal flux of equimolal diffusion

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•Some times the mass transfer coefficient for one-way Some times the mass transfer coefficient for one-way transfer is transfer is denoted denoted by kby kcc’ or k’ or kyy’, then’, then

)44.17(1)(

1

)1(

1

LBLAy

y

c

c

yyk

k

k

k

)45.17()( AAiyA yykN

)46.17()1(

)(

LA

AAiyA

y

yykN

Because the correction is small compared to the uncertainty in Because the correction is small compared to the uncertainty in the diffusivity and the mass-transfer coefficient.the diffusivity and the mass-transfer coefficient.

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(3)Boundary Layer Theory(3)Boundary Layer Theory

•When diffusion through a stagnant fluid film ,When diffusion through a stagnant fluid film ,

•When diffusion takes place in a thin boundary layer When diffusion takes place in a thin boundary layer near a surface where the fluid is in laminar flownear a surface where the fluid is in laminar flow ,,

•For For boundary layer flowsboundary layer flows, no matter what the shape of , no matter what the shape of the velocity profile or value of the physical properties, the velocity profile or value of the physical properties, the transfer rate cannot increase with the 1.0 power of the transfer rate cannot increase with the 1.0 power of the diffusivitythe diffusivity, as implied by the film theory., as implied by the film theory.

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(4)Penetration Theory(4)Penetration Theory （（渗透理论） 渗透理论） and and Surface Renewal TheorySurface Renewal Theory 表面更新理论表面更新理论•When the boundary layer becomes turbulent or When the boundary layer becomes turbulent or separation occurs, penetration theory and surface separation occurs, penetration theory and surface renewal theory apply, andrenewal theory apply, and

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(5) Two-Film Theory(5) Two-Film Theory

•Basic viewpoints:Basic viewpoints:

•1)On two sides of the interface, there exist two 1)On two sides of the interface, there exist two effective films of certain thickness, component A effective films of certain thickness, component A passes through these two film by molecular diffusion.passes through these two film by molecular diffusion.

•2)At the interface, the gas is in equilibrium with 2)At the interface, the gas is in equilibrium with liquid.liquid.

•3)The concentration gradients in the two bulk phases 3)The concentration gradients in the two bulk phases equal to zero. equal to zero. （（ 11 ）） 接触的两相流体间存在相界面，界面两侧各有一个接触的两相流体间存在相界面，界面两侧各有一个很薄的停滞层，组分很薄的停滞层，组分 AA 以分子扩散方式通过此两膜层。（以分子扩散方式通过此两膜层。（ 22 ）） 相界面处，气、液相界面处，气、液两相达到平衡（界面上不存在阻力）。 （两相达到平衡（界面上不存在阻力）。 （ 33 ））两滞流膜外的气液相主体中，流体两滞流膜外的气液相主体中，流体充分湍动，物质浓度均匀。充分湍动，物质浓度均匀。

17[x[xAiAi is in equilibrium with y is in equilibrium with yAiAi]]

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•In the two-film theory, the rate of mass transfer to In the two-film theory, the rate of mass transfer to the interface is set equal to the rate of the transfer the interface is set equal to the rate of the transfer from the interface:from the interface:

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LetLet

=overall mass transfer coefficient in gas phase=overall mass transfer coefficient in gas phase

=overall mass transfer driving force=overall mass transfer driving force

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•To get Ky in terms of kx and ky, To get Ky in terms of kx and ky,

)55.17(1

r

yy

r

yy

r

yy

KAAiAiAAA

y

)57.17(11

,,

)56.17()()(

1

xyy

AAAiAi

AAix

AAi

AiAy

AiA

y

k

m

kK

mxymxy

xxk

yy

yyk

yy

K

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==overall resistance to mass transferoverall resistance to mass transfer

== resistance to mass transfer in the gas film resistance to mass transfer in the gas film

==resistance to mass transfer in the liquid filmresistance to mass transfer in the liquid film

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•Similarly, letSimilarly, let

=overall mass transfer coefficient in liquid phase=overall mass transfer coefficient in liquid phase

=overall mass transfer driving force=overall mass transfer driving force

WhereWhere

•We can getWe can get

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==overall resistance to mass transferoverall resistance to mass transfer

== resistance to mass transfer in the liquid film resistance to mass transfer in the liquid film

==resistance to mass transfer in the gas filmresistance to mass transfer in the gas film

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•Gas film “controls” and Liquid film “controls”Gas film “controls” and Liquid film “controls”

•WhenWhen ""11

,1

controlsfilmGaskKk

m

k yyxy

•WhenWhen ""11

,11

controlsfilmLiquidkKmkk xxyx