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Fick'S (First) Law of Binary Diffusiona: - P Vwai Cartesian Coordinates (X

This document provides equations describing flux and continuity for fluids in different coordinate systems. It includes: 1) Fick's first law of binary diffusion, which describes flux as proportional to the gradient of concentration. 2) The equation of continuity, which states that the rate of change of density plus divergence of density times velocity is equal to zero. 3) The equation of energy for pure Newtonian fluids with constant density and thermal conductivity. 4) The equation of continuity for species in terms of flux, which equals the negative divergence of flux plus a rate of production term.

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ConsueloAndreaRiquelmeCarrasco
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45 views3 pages

Fick'S (First) Law of Binary Diffusiona: - P Vwai Cartesian Coordinates (X

This document provides equations describing flux and continuity for fluids in different coordinate systems. It includes: 1) Fick's first law of binary diffusion, which describes flux as proportional to the gradient of concentration. 2) The equation of continuity, which states that the rate of change of density plus divergence of density times velocity is equal to zero. 3) The equation of energy for pure Newtonian fluids with constant density and thermal conductivity. 4) The equation of continuity for species in terms of flux, which equals the negative divergence of flux plus a rate of production term.

Uploaded by

ConsueloAndreaRiquelmeCarrasco
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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846 Appendix B Fluxes and the Equations of Change

5B.3 FICK'S (FIRST) LAW OF BINARY DIFFUSIONa


[jA = -P%ABVWAI

Cartesian coordinates (x, y, z):

Cylindrical coordinates tr, 8 , ~ ) :

Spherical coordinates tr, 8 , 4 ) :

" To get the molar fluxes with respect to the molar average velocity, replace j,, p, and w, by J:, c, and x,.

5B.4 THE EQUATION OF CONTINUITYa


[ d p / d t + (V .pv) = 01

Cartesian coordinates (x, y, z):

Cylindrical coordinates (r, 8 , ~ ) :

Spherical coordinates (r, 8, 4):

3 + --
1 d (p?ur) +
dt y2 dr
--
r sin 8 d8
(pv, sin 8) + -- (pv+)= 0
r sin 8 dr$

" When the fluid is assumed to have constant mass density p, the equation simplifies to (V .v) = 0.
850 Appendix B Fluxes and the Equations of Change

sB.9 THE EQUATION OF ENERGY FOR PURE NEWTONIAN


FLUIDS WITH CONSTANTap AND k

Cartesian coordinates ( x , y, z):

Cylindrical coordinates (r, 0 , ~ ) :

Spherical coordinates (r, 8,4):

" This form of the energy equation is also valid under the less stringent assumptions k = constant and (d In p / d In T),,Dp/Dt = 0. The
assumption p = constant is given in the table heading because it is the assumption more often made.
The function @,, is given in sB.7. The term p @ ,is usually negligible, except in systems with large velocity gradients.

sB.10 THE EQUATION OF CONTINUITY FOR SPECIES a


IN TERMSaOF ja
[pDo,/Dt = -(V .j,) + r,]
Cartesian coordinafes ( x , y, z):

Cylindrical coordinafes (r, 8, z):

-- -

Spherical coordinates (r, 8,+):

do, dw, us d o , 1 d .
[La
Urn
dt fur-+--+---
dr Y r sm 8 a+ =
r2dr (r2j,r) + r sin 0 - ('"' sin 0) + Y sln 0 -
aia'i.m]+r,
d+ (8.10-3)

" To obtain the corresponding equations in terms of J,* make the following replacements:
Replace P v
sB.11 The Equation of Continuity for Species A in Terms of w, for Constant p%,, 851

gB.11 THE EQUATION OF CONTINUITY FOR SPECIES A


IN TERMS OF oAFOR CONSTANTapBAB

- -

Cartesian coordinates (x,y, 2):

Cylindrical coordinates (r, 19, z):

Spherical coordinates (r, O,+):


1
r2 sin 0 d o
(B.11-3)

" To obtain the corresponding equations in terms of x,, make the following replacements:
Replace P 0, v re
N
by c X, V* R, - x, 1 Rp
p=1

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