Computational transport processes for micromixers* - Micromixers Fundamentals Design and Fabrication - page 85

D V ¼ D x D y (3.48a)

A e ¼ A w ¼ D y (3.48b)

A n ¼ A s ¼ D x : (3.48c)

The diffusion coefficients at the interface in Eqn (3.47), i.e., G s , are determined using

a harmonic mean approach. Upon substitution of Eqns (3.45) and (3.46) into Eqn (3.43) , we have
r P F P r P F

G e , G w , G n , and

P D

o

V

t
þF e F P þðF P F E ÞkF e ; 0

D

k F w F P þðF P F W ÞkF w ; 0 k

(3.49)

þF n F P þðF P F N ÞkF n ; 0 kF s F P þðF P F S ÞkF s ; 0 k
¼ D e ðF E F P ÞD w ðF P F W ÞþD n ðF N F P ÞD s ðF P F S ÞþðS C þ S P F P ÞDV

or, upon rearrangement,

r P F

P DV
D

r P F P D V
D t þð

F e F w þ F n F s ÞF P

t

þðF P F E ÞkF e ; 0 k þðF P F W ÞkF w ; 0 k

(3.50)

þðF P F N ÞkF n ; k

¼ D e ðF E F P ÞD w ðF P F W ÞþD n ðF N F P ÞD s ðF P F S ÞþðS C þ S P F P ÞDV:

0 k þðF P F S ÞkF s ; 0

To enforce continuity, the continuity equation has to be considered in the discretization of the general
transport equation. For this purpose, the continuity equation (Eqn 3.3 ) is similarly integrated over the
CV to give

ðr P r P ÞD

V

þ F e F w þ F n F s ¼ 0 (3.51a)

D t

and, upon multiplication by F P ,
r P F P D r P F P D

V V

t þð

F e F w þ F n F s ÞF P ¼ 0 : (3.51b)

D t D

Subtracting equation (3.51b) from Eqn (3.50) results in

r P F P D r P F

o

V

P D

V

D t D t

þðF P F E Þk F e ; 0 k þðF P F W Þk F w ; 0 k

(3.52)

þðF P F N Þk F n ; 0 k þðF P F S Þk F s ; 0 k

¼ D e ðF E F P Þ D w ðF P F W Þþ D n ðF N F P Þ D s ðF P F S Þþð S C þ S P F P ÞD V

and, upon rearrangement, gives the final discretized equation for the general transport equation in
a compact form as

a P F P ¼ a E F E þ a W F W þ a N F N þ a S F S þ b (3.53a)