APPENDIX BGeneralized Form of theTransport Equations

B.1 CONTINUITY EQUATION

The following form of the continuity or total mass-balance equation in general-ized vector notation is expressed in terms of the mass density ρ, which can benonconstant, and mass-average velocity vector �u:

∂ρ

∂t= −∇ · ρ �u (B.1-1)

The following form of the continuity or total molar-balance equation in general-ized vector notation is expressed in terms of the molar density c, which can benonconstant, and molar-average velocity vector �u:

∂c

∂t= −∇ · c �u + G (B.1-2)

where G is the total molar generation rate per unit volume.

B.2 EQUATIONS OF MOTION

The following form of the equations of motion in generalized vector–tensor notationallows for nonconstant physical properties, a body force due to a gravitational field,and an unspecified viscous stress tensor τ :

ρ∂ �u∂t

+ ρ �u · ∇ �u = −∇P − ∇ · τ + ρ �g (B.2-1)

Equation (B.2-1) can be applied to non-Newtonian fluids if the appropriate con-stitutive equation relating the viscous stress to the rate of strain is known. The

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.

482

THERMAL ENERGY EQUATION 483

viscous stress tensor τ follows the sign convection for the force on a fluid particlediscussed in Appendix A and for a Newtonian fluid is given by

τ = −(

κ − 2

3µ

)(∇ · �u)δ − µ[∇�u + (∇�u)†] (B.2-2)

where µ is the shear viscosity, κ the bulk viscosity, δ the second-order identitytensor, and † denotes the transpose of a second-order tensor. For the special case ofan incompressible Newtonian fluid with constant viscosity, equation (B.2.1) whencombined with equation (B.2-2) simplifies to

ρ∂ �u∂t

+ ρ �u · ∇ �u = −∇P + µ∇2�u + ρ �g (B.2-3)

B.3 EQUATIONS OF MOTION FOR POROUS MEDIA

The following form of the equations of motion in generalized vector–tensor notationfor flow through porous media is based on Brinkman’s empirical modification ofDarcy’s law and assumes a body force due to a gravitational field, and an incom-pressible fluid having constant viscosity µ and permeability kp; ��u denotes the

superficial velocity based on considering the porous media to be homogeneous1:

0 = −∇P − µ

kp

��u + µ∇2 ��u + ρ �g (B.3-1)

B.4 THERMAL ENERGY EQUATION

The following form of the thermal energy equation in generalized vector–tensornotation allows for nonconstant physical properties, energy generation, and con-version of mechanical to internal energy by means of viscous dissipation, which isexpressed in terms of an unspecified viscous stress tensor τ :

ρCv

∂T

∂t+ ρCv �u · ∇T = ∇ · (k∇T ) − T

∂P

∂T

∣∣∣∣ρ

(∇ · �u) − (τ : ∇�u) + Ge (B.4-1)

where Cv is the heat capacity at constant volume, k the thermal conductivity, andGe the energy generation rate per unit volume. Equation (B.4-1) can be applied tonon-Newtonian fluids if the appropriate constitutive equation relating the viscousstress to the rate of strain is known. For the special case of an incompressibleNewtonian fluid with constant thermal conductivity k and for which the viscousstress tensor is given by equation (B.2-2), equation (B.4-1) simplifies to

ρCp

∂T

∂t+ ρCp �u · ∇T = k∇2T + µ[∇�u + (∇�u)†] : ∇�u + Ge (B.4-2)

where Cp is the heat capacity at constant pressure.

1H. C. Brinkman, Appl. Sci. Res., A1, 27–34, 81–86 (1947).

484 APPENDIX B

B.5 EQUATION OF CONTINUITY FOR A BINARY MIXTURE

The following form of the equation of continuity or species-balance equation ingeneralized vector–tensor notation for component A in a binary system allows fornonconstant physical properties and is expressed in terms of the mass concentrationρA and the mass flux vector �nA:

∂ρA

∂t+ ∇ · �nA = GA (B.5-1)

where GA is the mass rate of generation of component A per unit volume. Themass flux vector is given for a binary system by Fick’s law of diffusion in theform

�nA = ωA(�nA + �nB) − ρDAB∇ωA = ρA�u − ρDAB∇ωA (B.5-2)

in which ωA is the mass fraction of component A, DAB the binary diffusion coef-ficient, and �u the mass-average velocity. The form of the species-balance equationgiven by equation (B.5-1) is particularly useful for describing mass transfer inincompressible liquid and solid systems for which the mass density ρ is constant.For the special case of an incompressible Newtonian fluid with a constant binarydiffusion coefficient DAB , equation (B.5-1) when combined with equation (B.5-2)simplifies to

∂ρA

∂t+ �u · ∇ρA = DAB∇2ρA + GA (B.5-3)

The following form of the species-balance equation in generalized vector–tensornotation for component A in a binary system allows for nonconstant physicalproperties and is expressed in terms of the molar concentration cA and the molarflux vector �NA:

∂cA

∂t+ ∇ · �NA = GA (B.5-4)

where GA is the molar generation rate of component A per unit volume. Themolar flux vector is given for a binary system by Fick’s law of diffusion in theform

�NA = xA( �NA + �NB) − cDAB∇xA = cA�u − cDAB∇xA (B.5-5)

in which xA is the mole fraction of component A and �u is the molar-averagevelocity. The form of the species-balance equation given by equation (B.5-4) 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 alsoused to describe reacting systems for which the rate of generation of species is

EQUATION OF CONTINUITY FOR A BINARY MIXTURE 485

dictated by the reaction stoichiometry in terms of molar concentrations. For thespecial case of an incompressible Newtonian fluid with a constant binary diffusioncoefficient DAB , equation (B.5-4) when combined with equation (B.5-5) simpli-fies to

∂cA

∂t+ �u · ∇cA = DAB∇2cA + GA (B.5-6)