APPENDIX GEquation of Continuityfor a Binary Mixture

G.1 RECTANGULAR COORDINATES

The following form of the equation of continuity or species balance in rectangularcoordinates for component A in a binary system allows for nonconstant physicalproperties and is expressed in terms of the mass concentration ρA and the massflux components nAi :

∂ρA

∂t+ ∂nAx

∂x+ ∂nAy

∂y+ ∂nAz

∂z= GA (G.1-1)

where GA is the mass generation rate of component A per unit volume. The com-ponents of the mass flux are given for a binary system by Fick’s law of diffusionin the form

nAx = ωA(nAx + nBx) − ρDAB

∂ωA

∂x= ρAux − ρDAB

∂ωA

∂x(G.1-2)

nAy = ωA(nAy + nBy) − ρDAB

∂ωA

∂y= ρAuy − ρDAB

∂ωA

∂y(G.1-3)

nAz = ωA(nAz + nBz) − ρDAB

∂ωA

∂z= ρAuz − ρDAB

∂ωA

∂z(G.1-4)

in which ωA is the mass fraction of component A, DAB the binary diffusion coef-ficient, and ui the mass-average velocity component in the i-direction. The formof the equation of continuity for a binary mixture or species balance given byequation (G.1-1) is particularly useful for describing mass transfer in incompress-ible liquid and solid systems for which the mass density ρ is constant. For thespecial case of an incompressible fluid or fluid having a constant mass densityand a constant binary diffusion coefficient, equation (G.1-1) when combined withequations (G.1-2) through (G.1-4) simplifies to

∂ρA

∂t+ ux

∂ρA

∂x+ uy

∂ρA

∂y+ uz

∂ρA

∂z= DAB

∂2ρA

∂x2+ DAB

∂2ρA

∂y2+ DAB

∂2ρA

∂z2+ GA

(G.1-5)

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approachto Model Building and the Art of Approximation, By William B. KrantzCopyright 2007 John Wiley & Sons, Inc.

499

500 APPENDIX G

The following form of the equation of continuity or species balance in rect-angular coordinates for component A in a binary system allows for nonconstantphysical properties and is expressed in terms of the molar concentration cA and themolar flux components NAi :

∂cA

∂t+ ∂NAx

∂x+ ∂NAy

∂y+ ∂NAz

∂z= GA (G.1-6)

where GA is the molar generation rate of component A per unit volume. Thecomponents of the molar flux are given for a binary system by Fick’s law ofdiffusion in the form

NAx = xA(NAx + NBx) − cDAB

∂xA

∂x= cAux − cDAB

∂xA

∂x(G.1-7)

NAy = xA(NAy + NBy) − cDAB

∂xA

∂y= cAuy − cDAB

∂xA

∂y(G.1-8)

NAz = xA(NAz + NBz) − cDAB

∂xA

∂z= cAu − cDAB

∂xA

∂z(G.1-9)

in which xA is the mole fraction of component A. The form of the equation ofcontinuity for a binary mixture or species balance given by equation (G.1-6) isparticularly useful for describing mass transfer in gas systems for which the molardensity c is constant at a fixed temperature and pressure. This equation is also usedto describe reacting systems for which the generation rate of species is dictated bythe reaction stoichiometry in terms of molar concentrations. For the special case ofa fluid having a constant molar density and a constant binary diffusion coefficient,equation (G.1-6) combined with equations (G.1-7) through (G.1-9) simplifies to

∂cA

∂t+ ux

∂cA

∂x+ uy

∂cA

∂y+ uz

∂cA

∂z= DAB

∂2cA

∂x2+ DAB

∂2cA

∂y2+ DAB

∂2cA

∂z2+ GA

(G.1-10)

G.2 CYLINDRICAL COORDINATES

The following form of the equation of continuity or species balance in cylindricalcoordinates for component A in a binary system allows for nonconstant physicalproperties and is expressed in terms of the mass concentration ρA and the massflux components nAi :

∂ρA

∂t+ 1

r

∂

∂r(rnAr) + 1

r

∂nAθ

∂θ+ ∂nAz

∂z= GA (G.2-1)

where GA is the mass generation rate of component A per unit volume. The com-ponents of the mass flux are given for a binary system by Fick’s law of diffusionin the form

CYLINDRICAL COORDINATES 501

nAr = ωA(nAr + nBr) − ρDAB

∂ωA

∂r= ρAur − ρDAB

∂ωA

∂r(G.2-2)

nAθ = ωA(nAθ + nBθ ) − ρDAB

1

r

∂ωA

∂θ= ρAuθ − ρDAB

1

r

∂ωA

∂θ(G.2-3)

nAz = ωA(nAz + nBz) − ρDAB

∂xA

∂z= ρAuz − ρDAB

∂xA

∂z(G.2-4)

in which ωA is the mass fraction of component A, DAB the binary diffusioncoefficient, and ui the mass-average velocity component in the i-direction. Theform of the equation of continuity for a binary mixture or species balance givenby equation (G.2-1) is particularly useful for describing mass transfer in incom-pressible liquid and solid systems for which the mass density ρ is constant. Forthe special case of an incompressible fluid or fluid having a constant mass den-sity and a constant binary diffusion coefficient, equation (G.2-1) combined withequations (G.2-3) through (G.2-4) simplifies to

∂ρA

∂t+ ur

∂ρA

∂r+ uθ

1

r

∂ρA

∂θ+ uz

∂ρA

∂z= DB

1

r

∂

∂r

(r∂ρA

∂r

)

+ DB

1

r2

∂2ρ

∂θ2+ DAB

∂2ρA

∂z2+ GA

(G.2-5)

The following form of the equation of continuity or species balance in cylindricalcoordinates for component A in a binary system allows for nonconstant physicalproperties and is expressed in terms of the molar concentration cA and the molarflux components NAi :

∂cA

∂t+ 1

r

∂

∂r(rNAr) + 1

r

∂NAθ

∂θ+ ∂NAz

∂z= GA (G.2-6)

where GA is the molar generation rate of component A per unit volume. Thecomponents of the molar flux are given for a binary system by Fick’s law ofdiffusion in the form

NAr = xA(NAr + NBr) − cDAB

∂xA

∂r= cAur − cDAB

∂xA

∂r(G.2-7)

NAθ = xA(NAθ + NBθ ) − cDAB

1

r

∂xA

∂θ= cAuθ − cDAB

1

r

∂xA

∂θ(G.2-8)

NAz = xA(NAz + NBz) − cDAB

∂xA

∂z= cAuz − cDAB

∂xA

∂z(G.2-9)

in which xA is the mole fraction of component A and ui is the molar-averagevelocity component in the i-direction. The form of the equation of continuity fora binary mixture or species balance given by equation (G.2-6) is particularly use-ful for describing mass transfer in gas systems for which the molar density c

502 APPENDIX G

is constant at a fixed temperature and pressure. This equation is also used todescribe reacting systems for which the generation rate of species is dictatedby the reaction stoichiometry in terms of molar concentrations. For the specialcase of a fluid having a constant molar density and a constant binary diffu-sion coefficient, equation (G.2-6) combined with equations (G.2-7) through (G.2-9)simplifies to

∂cA

∂t+ ur

∂cA

∂r+ uθ

1

r

∂cA

∂θ+ uz

∂cA

∂z= DAB

1

r

∂

∂r

(r∂cA

∂r

)

+ DAB

1

r2

∂2cA

∂θ2+ DAB

∂2cA

∂z2+ GA

(G.2-10)

G.3 SPHERICAL COORDINATES

The following form of the equation of continuity or species balance in sphericalcoordinates for component A in a binary system allows for nonconstant physicalproperties and is expressed in terms of the mass concentration ρA and the massflux components nAi :

∂ρA

∂t+ 1

r2

∂

∂r(r2nAr) + 1

r sin θ

∂

∂θ(nAθ sin θ) + 1

r sin θ

∂nAφ

∂φ= GA (G.3-1)

where GA is the mass generation rate of component A per unit volume. The com-ponents of the mass flux are given for a binary system by Fick’s law of diffusionin the form

nAr = ωA(nAr + nBr) − ρDAB

∂ωA

∂r= ρAur − ρDAB

∂ωA

∂r(G.3-2)

nAθ = ωA(nAθ + nBθ ) − ρDAB

1

r

∂ωA

∂θ= ρAuθ − ρDAB

1

r

∂ωA

∂θ(G.3-3)

nAφ = ωA(nAφ + nBφ) − ρDAB

1

r sin θ

∂ωA

∂φ= ρAuφ − ρDAB

1

r sin θ

∂ωA

∂φ

(G.3-4)

in which ωA is the mass fraction of component A, DAB the binary diffusioncoefficient, and ui the mass-average velocity component in the i-direction. Theform of the equation of continuity for a binary mixture or species balance givenby equation (G.3-1) is particularly useful for describing mass transfer in incom-pressible liquid and solid systems for which the mass density ρ is constant. Forthe special case of an incompressible fluid or fluid having a constant mass den-sity and a constant binary diffusion coefficient, equation (G.3-1) combined withequations (G.3-2) through (G.3-4) simplifies to

SPHERICAL COORDINATES 503

∂ρA

∂t+ ur

∂ρA

∂r+ uθ

1

r

∂ρA

∂θ+ uφ

1

r sin θ

∂ρA

∂φ= DAB

1

r2

∂

∂r

(r2 ∂ρA

∂r

)

+ DAB

1

r2 sin θ

∂

∂θ

(sin θ

∂ρA

∂θ

)+ DAB

1

r2 sin2 θ

∂2ρA

∂φ2+ GA

(G.3-5)

The following form of the equation of continuity or species balance in sphericalcoordinates for component A in a binary system allows for nonconstant physicalproperties and is expressed in terms of the molar concentration cA and the molarflux components NAi :

∂cA

∂t+ 1

r2

∂

∂r(r2NAr) + 1

r sin θ

∂

∂θ(NAθ sin θ) + 1

r sin θ

∂NAφ

∂φ= GA (G.3-6)

where GA is the molar generation rate of component A per unit volume. Thecomponents of the molar flux are given for a binary system by Fick’s law ofdiffusion in the form

NAr = xA(NAr + NBr) − cDAB

∂xA

∂r= cAur − cDAB

∂xA

∂r(G.3-7)

NAθ = xA(NAθ + NBθ ) − cDAB

1

r

∂xA

∂θ= cAuθ − cDAB

1

r

∂xA

∂θ(G.3-8)

NAφ = xA(NAφ + NBφ) − cDAB

1

r sin θ

∂xA

∂φ= cAuφ − cDAB

1

r sin θ

∂xA

∂φ(G.3-9)

in which xA is the mole fraction of component A and ui is the molar-averagevelocity component in the i-direction. The form of the equation of continuity fora binary mixture or species balance given by equation (G.3-6) is particularly use-ful for describing mass transfer in gas systems for which the molar density c

is constant at a fixed temperature and pressure. This equation is also used todescribe reacting systems for which the generation rate of species is dictatedby the reaction stoichiometry in terms of molar concentrations. For the specialcase of a fluid having a constant molar density and a constant binary diffu-sion coefficient, equation (G.3-6) combined with equations (G.3-7) through (G.3-9)simplifies to

∂cA

∂t+ ur

∂cA

∂r+ uθ

1

r

∂cA

∂θ+ uφ

1

r sin θ

∂cA

∂φ= DAB

1

r2

∂

∂r

(r2 ∂cA

∂r

)

+ DAB

1

r2 sin θ

∂

∂θ

(sin θ

∂cA

∂θ

)+ DAB

1

r2 sin2 θ

∂2cA

∂φ2+ GA

(G.3-10)