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APPENDIX CContinuity Equation
C.1 RECTANGULAR COORDINATES
The following form of the continuity or total mass-balance equation in rectangularcoordinates is expressed in terms of the mass density ρ, which can be nonconstant,and mass-average velocity components ui :
∂ρ
∂t+ ∂
∂x(ρux) + ∂
∂y(ρuy) + ∂
∂z(ρuz) = 0 (C.1-1)
For the special case of an incompressible fluid or fluid having a constant massdensity, equation (C.1-1) simplifies to
∂ux
∂x+ ∂uy
∂y+ ∂uz
∂z= 0 (C.1-2)
The following form of the continuity or total mass-balance equation in rectangularcoordinates is expressed in terms of the molar density c, which can be nonconstant,and molar-average velocity components ui :
∂c
∂t+ ∂
∂x(cux) + ∂
∂y(cuy) + ∂
∂z(cuz) = G (C.1-3)
where G is the total molar generation rate per unit volume. For the special case ofa fluid having a constant molar density, equation (C.1-3) simplifies to
∂ux
∂x+ ∂uy
∂y+ ∂uz
∂z= G (C.1-4)
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.
486
SPHERICAL COORDINATES 487
C.2 CYLINDRICAL COORDINATES
The following form of the continuity or total mass-balance equation in cylindricalcoordinates is expressed in terms of the mass density ρ, which can be nonconstant,and mass-average velocity components ui :
∂ρ
∂t+ 1
r
∂
∂r(ρrur) + 1
r
∂
∂θ(ρuθ ) + ∂
∂z(ρuz) = 0 (C.2-1)
For the special case of an incompressible fluid or fluid having a constant massdensity, equation (C.2-1) simplifies to the following:
1
r
∂
∂r(rur) + 1
r
∂uθ
∂θ+ ∂uz
∂z= 0 (C.2-2)
The following form of the continuity or total mass-balance equation in cylindricalcoordinates is expressed in terms of the molar density c, which can be nonconstant,and molar-average velocity components ui :
∂c
∂t+ 1
r
∂
∂r(crur ) + 1
r
∂
∂θ(cuθ ) + ∂
∂z(cuz) = G (C.2-3)
where G is the total molar generation rate per unit volume. For the special case ofa fluid having a constant molar density, equation (C.2-3) simplifies to
1
r
∂
∂r(rur ) + 1
r
∂uθ
∂θ+ ∂uz
∂z= G (C.2-4)
C.3 SPHERICAL COORDINATES
The following form of the continuity or total mass-balance equation in sphericalcoordinates is expressed in terms of the mass density ρ, which can be nonconstant,and mass-average velocity components ui :
∂ρ
∂t+ 1
r2
∂
∂r(ρr2ur) + 1
r sin θ
∂
∂θ(ρuθ sin θ) + 1
r sin θ
∂
∂φ(ρuφ) = 0 (C.3-1)
For the special case of an incompressible fluid or fluid having a constant massdensity, equation (C.3-1) simplifies to
1
r2
∂
∂r(r2ur) + 1
r sin θ
∂
∂θ(uθ sin θ) + 1
r sin θ
∂uφ
∂φ= 0 (C.3-2)
488 APPENDIX C
The following form of the continuity or total mass-balance equation in sphericalcoordinates is expressed in terms of the molar density c, which can be nonconstant,and molar-average velocity components ui :
∂c
∂t+ 1
r2
∂
∂r(cr2ur ) + 1
r sin θ
∂
∂θ(cuθ sin θ) + 1
r sin θ
∂
∂φ(cuϕ) = G (C.3-3)
where G is the total molar generation rate per unit volume. For the special case ofa fluid having a constant molar density, equation (C.3-3) simplifies to
1
r2
∂
∂r(r2ur ) + 1
r sin θ
∂
∂θ(uθ sin θ) + 1
r sin θ
∂uφ
∂φ= G (C.3-4)
