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Chapter 6 Chapter 6 Principles of Diffusion Principles of Diffusion
and Mass Transfer and Mass Transfer Between PhasesBetween Phases
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1.THEORY OF DIFFUSION1.THEORY OF DIFFUSION• Diffusion Diffusion is the movement, under the influence of a is the movement, under the influence of a
physical stimulus, of an individual component physical stimulus, of an individual component through a mixturethrough a mixture
• The most common The most common cause of diffusioncause of diffusion is a is a concentration gradientconcentration gradient of the diffusing component.of the diffusing component.
• E.g., The process of dissolution of E.g., The process of dissolution of ammonia into ammonia into waterwater: (1)A concentration gradient in the gas phase : (1)A concentration gradient in the gas phase causes ammonia to diffuse to the gas-liquid causes ammonia to diffuse to the gas-liquid interface; (2)Ammonia dissolves in the interface; interface; (2)Ammonia dissolves in the interface; (3)A gradient in the liquid phase causes ammonia (3)A gradient in the liquid phase causes ammonia to diffuse into the bulk liquid.to diffuse into the bulk liquid.
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• A A concentration gradientconcentration gradient tends to move the tends to move the component in such a direction as to component in such a direction as to equalize equalize concentrations and destroy the gradientconcentrations and destroy the gradient..
• If the If the two phases are in equilibriumtwo phases are in equilibrium with each with each other, diffusion, or other, diffusion, or mass transfer fluxesmass transfer fluxes is is equal to equal to zerozero..
• Other causes of diffusion: Other causes of diffusion: activity gradient (reverse activity gradient (reverse osmosisosmosis); ); temperature gradient (thermal diffusion); temperature gradient (thermal diffusion); application of an external force field (forced application of an external force field (forced diffusion, e.g., centrifugediffusion, e.g., centrifuge, , etc).etc).
• Two kindsTwo kinds of diffusion caused by concentration of diffusion caused by concentration gradient : gradient : molecular diffusionmolecular diffusion （（分子扩散）分子扩散） and and eddy diffusioneddy diffusion （（涡流扩散）涡流扩散） . [. [e.g., diffusion e.g., diffusion process of ink in the stagnant or agitated waterprocess of ink in the stagnant or agitated water…]…]
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• Mass transfer driving forcesMass transfer driving forces• E.g., absorption or stripping process: Gas-liquid E.g., absorption or stripping process: Gas-liquid
phases phases are notare not in in equilibriumequilibrium with each other with each other..
Absorption columnAbsorption column
Driving force
Driving force
Driving forces:Driving forces:
)(
)(
xx
yy
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Absorption columnAbsorption column
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Absorption column
Driving force
Driving force
Driving forces:
)(
)(
cc
pp
•Question: Can we use (p-c) or (y-x) as mass transfer Question: Can we use (p-c) or (y-x) as mass transfer driving force? Compare mass transfer driving forces driving force? Compare mass transfer driving forces with heat transfer driving force?with heat transfer driving force?
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(1)Comparison of diffusion and heat transfer(1)Comparison of diffusion and heat transfer
db
dcDJ
dy
dTkq
dy
du
AABA
Momentum transferMomentum transfer
Heat transferHeat transfer
Mass transferMass transfer
)()(
)(
AiAgAAicA
wh
PPkcckJ
TThq
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(2)Diffusion quantities(2)Diffusion quantities （（扩散通量）扩散通量）1.Velocity 1.Velocity uu, length/time., length/time.
2.Flux across a plane 2.Flux across a plane NN, mol/area, mol/area••time.time.
3.Flux relative to a plane of zero velocity 3.Flux relative to a plane of zero velocity JJ, , mol/areamol/area••time.time.
4.Concentration 4.Concentration c c and molar density and molar density MM, mol/volume , mol/volume
(mole fraction may also be used).(mole fraction may also be used).
5.Concentration gradient 5.Concentration gradient dc/dbdc/db, where , where bb is the length is the length of the path perpendicular to the area across which of the path perpendicular to the area across which diffusion is occurring.diffusion is occurring.
Appropriate subscriptsAppropriate subscripts are used when needed.are used when needed.
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(3)Velocities in diffusion(3)Velocities in diffusion （（扩散速率）扩散速率）
•VelocityVelocity without qualification refers to the velocity without qualification refers to the velocity relative to the interface between the phases and is that relative to the interface between the phases and is that apparent to an observer at rest with respect to the apparent to an observer at rest with respect to the interface.interface.
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(4)Molal flow rate, velocity , and flux(4)Molal flow rate, velocity , and flux
•Where Where
MM=molar density of the mixture=molar density of the mixture
N= total molar flux in a direction perpendicular to a N= total molar flux in a direction perpendicular to a stationary planestationary plane
uu00= volumetric average velocity= volumetric average velocity
)1.17(0uN M
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•For components A and B crossing a stationary For components A and B crossing a stationary plane, the molal fluxes areplane, the molal fluxes are
)2.17(AAA ucN
)3.17(BBB ucN •By definition there is no net volumetric flow across By definition there is no net volumetric flow across the reference plane moving at the volume-average the reference plane moving at the volume-average velocity u0, although in some cases there is a net velocity u0, although in some cases there is a net molar flow or a net mass flow. molar flow or a net mass flow.
)4.17()( 00 uucucNJ AAAAA
=Diffusion flux of component A in the mixture=Diffusion flux of component A in the mixtureAJ
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)5.17()( 00 uucucNJ BBBBB
=Diffusion flux of component B in the mixture=Diffusion flux of component B in the mixtureBJ
•Fick’s first law of diffusion for a binary mixtureFick’s first law of diffusion for a binary mixture （（二元混合物）二元混合物） ::
)7.17(
)6.17(
db
dcDJ
db
dcDJ
BBAB
AABA
=diffusivity of component A in its mixture with =diffusivity of component A in its mixture with component B, mcomponent B, m22/s/s
=molar concentration gradient, mol/m=molar concentration gradient, mol/m44
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(5)Relations between diffusivities(5)Relations between diffusivities•For For ideal gases,ideal gases, and for diffusion of A and B in a and for diffusion of A and B in a gas at constant temperature and pressure,gas at constant temperature and pressure,
)11.17(
0
0
BAAB
BBA
AABBABA
BABA
BA
MBA
MBA
DDdb
dcD
db
dcDJJJJ
dndnConstRT
PVnnnand
db
dc
db
dc
ddcdcRT
Pcc
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•For For liquidliquid with with same mass density [kg/msame mass density [kg/m33]], ,
•For no For no volume flowvolume flow across the reference plane, the across the reference plane, the sum of the volumetric flows due to diffusion is zero. sum of the volumetric flows due to diffusion is zero. The volumetric flow rate is the molar flow rate times The volumetric flow rate is the molar flow rate times the molar volume M/the molar volume M/ and and
•SubstitutingSubstituting((代入代入 )) Eq.(17.13) into Eq.(17.14) givesEq.(17.13) into Eq.(17.14) gives
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•A common form of the diffusion equation gives the A common form of the diffusion equation gives the total flux of component Atotal flux of component A::
Where Dv=volumetric diffusivity, mWhere Dv=volumetric diffusivity, m22/h, cm/h, cm22/s/s
For gasesFor gases, ,
)17.17(
)16.17.(,/, 0
db
dyDNyN
becomesEqNuyc
AMvAA
MAMA
For liquidFor liquid, if , if MM =Constant, =Constant,
)17.17(
)16.17.(,/, 0
bdb
dxDNxN
becomesEqNuxc
AMvAA
MAMA
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(6)Interpretation of diffusion equations(6)Interpretation of diffusion equations 扩散方程扩散方程•The The vector nature vector nature of the fluxes and concentration of the fluxes and concentration gradients must be understood, since these quantities gradients must be understood, since these quantities are characterized by directions and magnitudes.are characterized by directions and magnitudes.
•The sign of the gradient is opposite to the direction of The sign of the gradient is opposite to the direction of the diffusion flux, since diffusion is in the direction of the diffusion flux, since diffusion is in the direction of lower concentrations. lower concentrations. 物质物质 AA的浓度梯度在方向上的的浓度梯度在方向上的变化与扩散通量相反，变化与扩散通量相反，表示扩散是沿物质浓度降低方表示扩散是沿物质浓度降低方向进行的向进行的
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(7)Equimolal diffusion(7)Equimolal diffusion((等分子扩散等分子扩散 ))
Zero convective flow and equimolal Zero convective flow and equimolal counterdiffusioncounterdiffusion （ （ 等分子反扩散等分子反扩散 ） ） of A and Bof A and B, as , as occurs in the diffusive occurs in the diffusive mixing of two gasesmixing of two gases and in and in the diffusion of A and B in the vapor phase for the diffusion of A and B in the vapor phase for distillations that have constant molal overflow.distillations that have constant molal overflow.
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Fig.17.1(a) Fig.17.1(a) Component A and B Component A and B diffusing at diffusing at same same molal (equimolal) molal (equimolal) ratesrates in in opposite opposite directionsdirections [Like the [Like the case of diffusion of case of diffusion of A and B in the vapor A and B in the vapor phase for phase for distillations that distillations that have have constant molal constant molal overflowoverflow].].
Note that for Note that for equimolal diffusion, equimolal diffusion, NNAA=J=JAA..
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•Assuming a constant flux NAssuming a constant flux NAA and zero total flux and zero total flux
(N=0), integrating Eq.(17.17) over a film thickness B(N=0), integrating Eq.(17.17) over a film thickness BTT, ,
oror
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•The concentration The concentration gradient for A is gradient for A is linear linear in the film, in the film, and the gradient and the gradient for B has the same for B has the same magnitude but the magnitude but the opposite sign, as opposite sign, as shown in Figure shown in Figure 17.1a.17.1a.
[From Eq.(17.17),[From Eq.(17.17),
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(8)One-component mass transfer (one-way diffusion)(8)One-component mass transfer (one-way diffusion) Fig.17.1(b) Fig.17.1(b) Component A Component A diffusing, diffusing, component B component B stationary with stationary with respect to interface. respect to interface. [Like the case of [Like the case of diffusion of solute A diffusion of solute A from gas phase into from gas phase into liquid phase in liquid phase in absorption process.]absorption process.]
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(8)One-component mass transfer (one-way diffusion)(8)One-component mass transfer (one-way diffusion)
•When only component A is being transferred, the When only component A is being transferred, the total fluxtotal flux to or away from the interface to or away from the interface N is the same N is the same as Nas NAA,, and Eq.(17.17) becomes and Eq.(17.17) becomes
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•Rearranging and integrating, we haveRearranging and integrating, we have
•Equation(17.24) can be rearranged by using the Equation(17.24) can be rearranged by using the logarithmic mean of 1-ylogarithmic mean of 1-yAA for easier comparison with for easier comparison with Eq.(17.19) for equimolal diffusion. The logarithmic Eq.(17.19) for equimolal diffusion. The logarithmic mean of 1-ymean of 1-yAA is is
)25.17()]1/()1ln[()]1/()1ln[(
)1()1()1(
AiA
AAi
AiA
AiALA yy
yy
yy
yyy
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Combining Eqs(17.24) and (17.25) givesCombining Eqs(17.24) and (17.25) gives
[Comparing one-way diffusion[Comparing one-way diffusion in the Chinese textbook,in the Chinese textbook,
]
1)1(
1
,,
)( 21
漂流因数factordriftp
P
p
P
y
RT
PBzDD
Here
ppp
P
RTz
DN
Bm
BmLA
MTv
AABm
A
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•Comparing Eq.(17.26) with Eq.(17.19),the flux of Comparing Eq.(17.26) with Eq.(17.19),the flux of component A for a given concentration difference is component A for a given concentration difference is therefore therefore greater for one-way diffusion than for greater for one-way diffusion than for equimolal diffusion.equimolal diffusion.
•[Example17.1.][Example17.1.]
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•For diffusion in liquid,For diffusion in liquid,
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2.PREDICTION OF DIFFUSIVITIES2.PREDICTION OF DIFFUSIVITIES
•Diffusivities are Diffusivities are physical propertiesphysical properties of fluids. of fluids.
•Diffusivities are best estimated by Diffusivities are best estimated by experimental experimental measurementsmeasurements, or from published correlations. , or from published correlations.
•The factors influencing diffusivities are The factors influencing diffusivities are temperature, temperature, pressure, and compositionspressure, and compositions for a given for a given fluidfluid..
•Fick’s first law of diffusion for a binary mixture:Fick’s first law of diffusion for a binary mixture:
)7.17(
)6.17(
db
dcDJ
db
dcDJ
BBAB
AABA
(1)Diffusion in gases(1)Diffusion in gases
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•Assume concentrations of Assume concentrations of component A in the component A in the two layerstwo layers with distance of the molecular with distance of the molecular mean free path mean free path of a binary of a binary mixture are cmixture are cA1A1 and c and cA2A2, ,
respectively, the diffusion flux is respectively, the diffusion flux is
•Therefore,Therefore,
•WhereWhere =average molecular velocity, m/s=average molecular velocity, m/s
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P
TDTu
P
Tv
5.15.0,
•A more rigorous approach based on modern kinetic A more rigorous approach based on modern kinetic theory allows for the different sizes and velocities of the theory allows for the different sizes and velocities of the molecules and the mutual interactions as they approach molecules and the mutual interactions as they approach one another [one another [Eq.(17.28) for binary diffusion.Eq.(17.28) for binary diffusion.]]
•The collision integral The collision integral D D decreases with increasing decreases with increasing temperature, which makes Dtemperature, which makes DAB AB increase with more than increase with more than the 1.5 power of the absolute temperature. the 1.5 power of the absolute temperature.
]1000~300[][ 75.18.1~7.1 KTTTDv
•For diffusion in air,For diffusion in air,
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•In general, In general, influence of concentrations for diffusion influence of concentrations for diffusion in gases can be neglected,in gases can be neglected, and and
v
v
vBA
DP
DT
DMorM ,
•[*Diffusion in small pores]([*Diffusion in small pores]( 自学自学 ))
•Example 17.2. [p.520] Example 17.2. [p.520] ((自学自学 ))
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•(2)Diffusion in liquids(2)Diffusion in liquids
•Diffusivities in liquids are generally Diffusivities in liquids are generally 4 to 5 orders of 4 to 5 orders of magnitude smallermagnitude smaller than in gases at atmospheric than in gases at atmospheric pressure.pressure.
smDliquidsIn
smDgasesIn
v
v
/10~10,
/10~10,2109
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•Influence of pressure for diffusion in liquids can be Influence of pressure for diffusion in liquids can be neglected.neglected.
•Diffusivities for Diffusivities for dilute liquid solutions dilute liquid solutions can be can be calculated from Eq.(17.31)[Empirical correlationcalculated from Eq.(17.31)[Empirical correlation].].
vvv DDTT
D
,[,
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•Other empirical correlations for diffusivities:Other empirical correlations for diffusivities:
•For For dilute aqueous solutions of non-electrolytesdilute aqueous solutions of non-electrolytes, using , using Eq.(17.32).Eq.(17.32). （（自学）自学）•For dilute solutions of completely ionized univalent For dilute solutions of completely ionized univalent electrolytes,using Nernst equation(17.33). electrolytes,using Nernst equation(17.33). （（自学）自学）•(3)Schmidt number Sc(3)Schmidt number Sc
•Sc is analogous to the Prandtl number.Sc is analogous to the Prandtl number.
k
c
ck
DDSc
p
p
vv
)]/([Pr
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•For gases, Sc is independent of pressure when the For gases, Sc is independent of pressure when the ideal gas law appliesideal gas law applies, since the viscosity is independent , since the viscosity is independent of pressure, and the effects of pressure on of pressure, and the effects of pressure on and Dv and Dv cancel. cancel. Temperature has only a slight effectTemperature has only a slight effect on Sc on Sc because because and and Dv both change with about TDv both change with about T0.7~0.80.7~0.8..
•For liquids,For liquids, Sc decreases markedly with increasing Sc decreases markedly with increasing temperaturetemperature because of the decreasing viscosity and because of the decreasing viscosity and the increase in the diffusivity.the increase in the diffusivity.
•UnlikeUnlike the case for binary gas mixtures the diffusion the case for binary gas mixtures the diffusion coefficient for a dilute solution of A and B is not the coefficient for a dilute solution of A and B is not the same for a dilute solution of B in A. i.e., same for a dilute solution of B in A. i.e., DDABAB≠≠DDBABA
[[Comparing with eq.(17.15)?]Comparing with eq.(17.15)?]
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•EXAMPLE 17.3. [p.522]EXAMPLE 17.3. [p.522]
•From eq.(17.31), it is apparent that From eq.(17.31), it is apparent that unlikeunlike the case the case for binary gas mixturesfor binary gas mixtures, the diffusion coefficient for a , the diffusion coefficient for a dilute solution of A and B is not the same for a dilute dilute solution of A and B is not the same for a dilute solution of B in A. solution of B in A.
•But, from But, from EXAMPLE 17.3., the diffusivitiesEXAMPLE 17.3., the diffusivities of of benzene in toluene and toluene in benzenebenzene in toluene and toluene in benzene have only have only a slight difference, and in this case the conclusion of a slight difference, and in this case the conclusion of Eq.(17.15) DEq.(17.15) DABAB=D=DBA BA is still effective is still effective
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(4)Turbulent diffusion(4)Turbulent diffusion
•In a turbulent stream the moving eddies transport In a turbulent stream the moving eddies transport matter from one location to another, just as they matter from one location to another, just as they transport momentum and heat energy.transport momentum and heat energy.
WhereWhere =molal flux of A, =molal flux of A, relative to phase as a relative to phase as a wholewhole, caused by turbulent action, caused by turbulent action
=eddy diffusivity=eddy diffusivity
The total molal flux, relative to the entire phase, The total molal flux, relative to the entire phase, becomesbecomes
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•The eddy diffusivity is The eddy diffusivity is not a parameter of physical not a parameter of physical propertyproperty, it depends on the , it depends on the fluid propertiesfluid properties but also but also on the velocity and position in the flowing streamon the velocity and position in the flowing stream ..
