MassvTransfer
UNIVERSITY OF MAIDUGURI
v v
FACULTY OF ENGINEERING DEPART
v v v
MENT OF CHEMICAL ENGINEERING
v v v
COMPILED LECTURE NOTE
v v
CHE 309 TRANSPORv v
T PHENOMENON II
v v
[3 UNITS]
v
2015/2016 ACADEMIC SESSION
v v
,MassvTransfer
TRANSPORTv PHENOMENONv IIv (3v UNITS)
MASSv TRANSFER
COURSEv OUTLINE
➢ Molecularv Diffusion
➢ Fick’sv law
➢ Steadyv Statev Ratev Equations
➢ Boundaryv Layer
➢ Momentumv Equations
➢ Laminarv Layer
➢ Universalv Velocity
➢ Boundaryv LayervThickness
➢ Eddyv Diffusion
➢ Massv Transferv withv Chemicalv Reactions
➢ Interphasev Massv Transfer
➢ Whitmanv Thinv Filmv Theory
➢ Ticv lines
➢ Solutionsv in-termsv ofv dimensionlessv groups
➢ Condensationv andv Boiling
➢ Filmv andv Dropwisev Condensation
➢ Filmv Transferv Coefficient
➢ Condensationv Number
➢ Boilerv Design
➢ DifferentvPhasev ofv Boiling
➢ HeatvFluxv Relations
,MassvTransfer
MASSv TRANSFER
MOLECULARv DIFFUSIONv ANDv FICK’Sv LAW
Introduction
Thev variousv unitv operationsv canv bevclassifiedv into;
(1) Momentum transfer: Occurs in unit operations as fluid flow,
mixing,vsedimentationvandvfiltration.
(2) Massv transfer:v Occursv inv distillation,v adsorption,v drying,v liquid-
liquidv extractionvetc
(3) Heatvtransfer:vOccursvinvconvectivevandvconductivevtransfervofvheatvevaporation,vdi
stillationvandvdrying.
Question:v Mentionv tenv applicationsv ofv thev classifiedv unitv operationsv eachv (momentum,vm
assvandvheatvtransfer)
GeneralvMolecularv TransportvEquation
𝐷𝑟𝑖𝑣𝑖𝑛gv𝑓𝑜𝑟𝑐𝑒
Ratevofvavtransfervprocessv=v
𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑑𝑟
v v =v − v∫
𝑑𝑧
Forvmomentumv transfervequation
𝜇v𝑑(𝑉𝑧,v𝜌)
𝜙𝑧v =v−v
𝜌 𝑑𝑧
Forv heatv transferv equation
𝑞𝑧 𝑑(𝜌,v𝐶𝑝,v𝑇)
=v−𝛼
𝐴 𝑑𝑧
Forvmassv transferv equation
𝐽𝐴Z =v−𝐷𝐴𝐵 𝑑𝐶𝐴 - ------------------ (1)
𝑑𝑧
where 𝐽𝐴Zv=vmolarvfluxvinv𝐾𝑔𝑚𝑜𝑙/𝑚2.v𝑠
𝐷𝐴𝐵v =v molecularv diffusivityv 𝑚2/𝑠
𝐶𝐴v=vconcentrationv 𝐾𝑔𝑚𝑜𝑙/𝑚3
Question:vDefinevthevtermsvinvmomentumvandvheatvtransfervequationsvandvthevgeneralvmolec
ularvtransportvequation.
Thevkineticvtheoryvofvgasesvgivesvusvavgoodvphysicalvofvindividualvmoleculesvinvfluids.vBecau
sevofvtheirvkineticvenergy,vthevmoleculesvarevinvrapidvrandomvmovementvoftenvcollidingv withv
eachv other.v Molecularv transportv orv molecularv diffusionv ofv av quantityv suchvasvavmomentu
m,vheatvorvmassvoccursvinvavfluidvbecausevofvthesevrandomvmovementsvofvindividualvmolecu
le.vEachvmoleculevcontainingvthevquantityvbeingvtransferredvmovesvrandomlyvinvallvdirection
svandvthesevarevfluxesvinvallvdirections.vHence,vifvtherevisvavconcentrationvgradientvofvthevpro
perty,vtherevwillvbevavnetvfluxvofvthevquantityvfromvhighvtovlowvconcentration.vThisvoccursvbec
ausevequalvnumbersvofvmoleculesvdiffusevinveachvdirectionvbetweenvthevhighvconcentrationv
andvlowvconcentration.
, MassvTransfer
Equationv (1)v isv thev Fick’sv lawv forv molecularv transportv ofv massv inv av fluidv orv solidv for
constantv totalv concentrationv invthevfluidv i.e.
𝑑𝐶𝐴
𝐽𝐴Zv =v−𝐷𝐴𝐵
𝑑𝑧
ThevDiffusionvCoefficient
Fick’sv lawv proportionality,v 𝐷𝐴𝐵,v isv knownv asv thev diffusionv coefficient.v It’sv fundamentalvdim
ensionsvmayvbevobtainedvfrom
𝑑𝐶𝐴
𝐽𝐴Zv=v−𝐷𝐴
v 𝑀v 𝑑𝑧
𝐽 𝑑𝑧 ( )𝐿 𝐵
𝐿2
𝐿2𝑇v
𝐷𝐴𝐵 =v−vv 𝐴𝑍 v
=v v 𝑀
v
=
𝑑𝐶𝐴 𝑇
𝐿3
𝐷𝐴𝐵v′sv dimensionsv arev identicalv tov thev fundamentalv dimension’sv ofv thev otherv transportvpro
perties;v kinematicv viscosity,v V,v andv thermalv diffusivity,v 𝛼v orv it’sv equivalentv ratio,
2 2
𝐾/ .v Thev massv diffusivityv hasv beenv reportedv inv 𝑐𝑚 /𝑠v (cgs).v Thev SIv unitsv arev 𝑚 /𝑠
𝜌𝐶𝑃
𝑓𝑡2
whichv isv av factorv 104 smaller.v Inv thev Englishv system /
ℎ𝑟 arev commonlyv used.
Conversionv betweenv thesev systemsv involvesv thev simplev relations.
𝐷𝐴𝐵(𝑐𝑚2/𝑠)
=v104
𝐷𝐴𝐵(𝑚2/𝑠)
𝐷𝐴𝐵(𝑓𝑡2/ℎ𝑟)
=v 3.87
𝐷𝐴𝐵 (𝑐𝑚2/𝑠)v
𝐷𝐴𝐵v(𝑓𝑡2/ℎ𝑟)
=v3.875v×v104
𝐷𝐴𝐵(𝑚2/𝑠)
Thev diffusionv coefficientv dependsv uponv thev pressure,v temperaturev andv compositionv ofvt
hevsystem.vAsvonevmightv expectv fromv considerationsv ofv thevmobilityvofv thev molecules,vthe
v diffusionv coefficientsv arev generallyv highestv forv gasesv (inv thev rangev ofv 5v×v10
−6v tov 1v×v1
0 𝑚 /𝑠),vthanvforvliquidsv(invthevragevofv10 v tov10 v 𝑚 /𝑠)v whichvarevhighervthanvthevvalue
−5 2 −10 −9 2
sv reportedvforv solidsv (invthevrangevofv 10−14v tov10−10v 𝑚2/𝑠)
Example:vMixturevofvHevandv𝑁2v gasvisvcontainedvinvavpipevatv298Kvandv1atmvtotalvpressurev
whichvisvconstantvthroughout.vAtvonevendvofvthevpipevatvpointv1,vthevpartialvpressurev𝑃𝐴1v ofvH
evisv0.60atmvandvatvthevothervendv0.2mv(20cm),v𝑃𝐴2v=v0.20atm.v CalculatevthevfluxvofvHevatvst
eadyvstatevifv𝐷𝐴𝐵v ofvthevHev–
vmixturev isv0.687v×v10
−4𝑚2/𝑠v(0.687𝑐𝑚2/𝑠).v UsevS.Ivandv c.g.svunit
Solution
Forvsteadyv state,v thevfluxv 𝐽𝐴Zv isv constant.v Also,v 𝐷𝐴𝐵v forv av gasv isv constant
𝑑𝐶
𝐽𝐴Z =v−𝐷𝐴𝐵 𝐴
𝑑𝑧
Integratingv willv yield
UNIVERSITY OF MAIDUGURI
v v
FACULTY OF ENGINEERING DEPART
v v v
MENT OF CHEMICAL ENGINEERING
v v v
COMPILED LECTURE NOTE
v v
CHE 309 TRANSPORv v
T PHENOMENON II
v v
[3 UNITS]
v
2015/2016 ACADEMIC SESSION
v v
,MassvTransfer
TRANSPORTv PHENOMENONv IIv (3v UNITS)
MASSv TRANSFER
COURSEv OUTLINE
➢ Molecularv Diffusion
➢ Fick’sv law
➢ Steadyv Statev Ratev Equations
➢ Boundaryv Layer
➢ Momentumv Equations
➢ Laminarv Layer
➢ Universalv Velocity
➢ Boundaryv LayervThickness
➢ Eddyv Diffusion
➢ Massv Transferv withv Chemicalv Reactions
➢ Interphasev Massv Transfer
➢ Whitmanv Thinv Filmv Theory
➢ Ticv lines
➢ Solutionsv in-termsv ofv dimensionlessv groups
➢ Condensationv andv Boiling
➢ Filmv andv Dropwisev Condensation
➢ Filmv Transferv Coefficient
➢ Condensationv Number
➢ Boilerv Design
➢ DifferentvPhasev ofv Boiling
➢ HeatvFluxv Relations
,MassvTransfer
MASSv TRANSFER
MOLECULARv DIFFUSIONv ANDv FICK’Sv LAW
Introduction
Thev variousv unitv operationsv canv bevclassifiedv into;
(1) Momentum transfer: Occurs in unit operations as fluid flow,
mixing,vsedimentationvandvfiltration.
(2) Massv transfer:v Occursv inv distillation,v adsorption,v drying,v liquid-
liquidv extractionvetc
(3) Heatvtransfer:vOccursvinvconvectivevandvconductivevtransfervofvheatvevaporation,vdi
stillationvandvdrying.
Question:v Mentionv tenv applicationsv ofv thev classifiedv unitv operationsv eachv (momentum,vm
assvandvheatvtransfer)
GeneralvMolecularv TransportvEquation
𝐷𝑟𝑖𝑣𝑖𝑛gv𝑓𝑜𝑟𝑐𝑒
Ratevofvavtransfervprocessv=v
𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑑𝑟
v v =v − v∫
𝑑𝑧
Forvmomentumv transfervequation
𝜇v𝑑(𝑉𝑧,v𝜌)
𝜙𝑧v =v−v
𝜌 𝑑𝑧
Forv heatv transferv equation
𝑞𝑧 𝑑(𝜌,v𝐶𝑝,v𝑇)
=v−𝛼
𝐴 𝑑𝑧
Forvmassv transferv equation
𝐽𝐴Z =v−𝐷𝐴𝐵 𝑑𝐶𝐴 - ------------------ (1)
𝑑𝑧
where 𝐽𝐴Zv=vmolarvfluxvinv𝐾𝑔𝑚𝑜𝑙/𝑚2.v𝑠
𝐷𝐴𝐵v =v molecularv diffusivityv 𝑚2/𝑠
𝐶𝐴v=vconcentrationv 𝐾𝑔𝑚𝑜𝑙/𝑚3
Question:vDefinevthevtermsvinvmomentumvandvheatvtransfervequationsvandvthevgeneralvmolec
ularvtransportvequation.
Thevkineticvtheoryvofvgasesvgivesvusvavgoodvphysicalvofvindividualvmoleculesvinvfluids.vBecau
sevofvtheirvkineticvenergy,vthevmoleculesvarevinvrapidvrandomvmovementvoftenvcollidingv withv
eachv other.v Molecularv transportv orv molecularv diffusionv ofv av quantityv suchvasvavmomentu
m,vheatvorvmassvoccursvinvavfluidvbecausevofvthesevrandomvmovementsvofvindividualvmolecu
le.vEachvmoleculevcontainingvthevquantityvbeingvtransferredvmovesvrandomlyvinvallvdirection
svandvthesevarevfluxesvinvallvdirections.vHence,vifvtherevisvavconcentrationvgradientvofvthevpro
perty,vtherevwillvbevavnetvfluxvofvthevquantityvfromvhighvtovlowvconcentration.vThisvoccursvbec
ausevequalvnumbersvofvmoleculesvdiffusevinveachvdirectionvbetweenvthevhighvconcentrationv
andvlowvconcentration.
, MassvTransfer
Equationv (1)v isv thev Fick’sv lawv forv molecularv transportv ofv massv inv av fluidv orv solidv for
constantv totalv concentrationv invthevfluidv i.e.
𝑑𝐶𝐴
𝐽𝐴Zv =v−𝐷𝐴𝐵
𝑑𝑧
ThevDiffusionvCoefficient
Fick’sv lawv proportionality,v 𝐷𝐴𝐵,v isv knownv asv thev diffusionv coefficient.v It’sv fundamentalvdim
ensionsvmayvbevobtainedvfrom
𝑑𝐶𝐴
𝐽𝐴Zv=v−𝐷𝐴
v 𝑀v 𝑑𝑧
𝐽 𝑑𝑧 ( )𝐿 𝐵
𝐿2
𝐿2𝑇v
𝐷𝐴𝐵 =v−vv 𝐴𝑍 v
=v v 𝑀
v
=
𝑑𝐶𝐴 𝑇
𝐿3
𝐷𝐴𝐵v′sv dimensionsv arev identicalv tov thev fundamentalv dimension’sv ofv thev otherv transportvpro
perties;v kinematicv viscosity,v V,v andv thermalv diffusivity,v 𝛼v orv it’sv equivalentv ratio,
2 2
𝐾/ .v Thev massv diffusivityv hasv beenv reportedv inv 𝑐𝑚 /𝑠v (cgs).v Thev SIv unitsv arev 𝑚 /𝑠
𝜌𝐶𝑃
𝑓𝑡2
whichv isv av factorv 104 smaller.v Inv thev Englishv system /
ℎ𝑟 arev commonlyv used.
Conversionv betweenv thesev systemsv involvesv thev simplev relations.
𝐷𝐴𝐵(𝑐𝑚2/𝑠)
=v104
𝐷𝐴𝐵(𝑚2/𝑠)
𝐷𝐴𝐵(𝑓𝑡2/ℎ𝑟)
=v 3.87
𝐷𝐴𝐵 (𝑐𝑚2/𝑠)v
𝐷𝐴𝐵v(𝑓𝑡2/ℎ𝑟)
=v3.875v×v104
𝐷𝐴𝐵(𝑚2/𝑠)
Thev diffusionv coefficientv dependsv uponv thev pressure,v temperaturev andv compositionv ofvt
hevsystem.vAsvonevmightv expectv fromv considerationsv ofv thevmobilityvofv thev molecules,vthe
v diffusionv coefficientsv arev generallyv highestv forv gasesv (inv thev rangev ofv 5v×v10
−6v tov 1v×v1
0 𝑚 /𝑠),vthanvforvliquidsv(invthevragevofv10 v tov10 v 𝑚 /𝑠)v whichvarevhighervthanvthevvalue
−5 2 −10 −9 2
sv reportedvforv solidsv (invthevrangevofv 10−14v tov10−10v 𝑚2/𝑠)
Example:vMixturevofvHevandv𝑁2v gasvisvcontainedvinvavpipevatv298Kvandv1atmvtotalvpressurev
whichvisvconstantvthroughout.vAtvonevendvofvthevpipevatvpointv1,vthevpartialvpressurev𝑃𝐴1v ofvH
evisv0.60atmvandvatvthevothervendv0.2mv(20cm),v𝑃𝐴2v=v0.20atm.v CalculatevthevfluxvofvHevatvst
eadyvstatevifv𝐷𝐴𝐵v ofvthevHev–
vmixturev isv0.687v×v10
−4𝑚2/𝑠v(0.687𝑐𝑚2/𝑠).v UsevS.Ivandv c.g.svunit
Solution
Forvsteadyv state,v thevfluxv 𝐽𝐴Zv isv constant.v Also,v 𝐷𝐴𝐵v forv av gasv isv constant
𝑑𝐶
𝐽𝐴Z =v−𝐷𝐴𝐵 𝐴
𝑑𝑧
Integratingv willv yield
