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()
( )
( )
xx
y
y
zz
xx x
zz z
yy
yy y
yzn
n
xzn
n
xyn
n
αα
α
=
= +Δ
=
==
+
Δ
ΔΔ
−
+ΔΔ
−
−
Time Dependent & 2D Component Transfer
Component mass/molar balance for homogeneous media
Chemical
reaction
Mass influx of A
(inward arrows)
Mass outflux of A
(outward arrows)
Total mass flux of A
Mass/moles
Influx of A
Mass/moles
outflux of A
Mass/mole
accumulation of A
Rate of disappear/appear
of A due to reactions
Mass/moles
Influx of A
Mass/moles
outflux of A
Rate of disappear/appear
of A due to reactions
,
ii
i
i
i
nu
D
u
D
x
x
ω
ρ
ρω
∂
∂
=−
≡
−
∂
∂
Rate of conversion of A due to chemical reactions per unit volume
r
Mass balance of
α
over a rectangular volume element
Total net mass flux
Converting the balance into a partial differential form
Rate of conversion
xyz
t
∂
ΔΔΔ
∂
Mass accumulation
.
y
xz
n
nn
rr
tx
y
z
∂
⎛⎞
∂∂
∂
+
+
+
≡−∇
+
⎜⎟
∂
∂
⎝⎠
n
rx
y
z
Δ
z
x
y
1,2,3,.
..,
N
=
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View Full Document Time Dependent & 2D Component Transfer
Component mass/molar balance for homogeneous media
Adding up N equations
()
(
)
(
)
11
..
.
NN
t
αα
α
ρ
ωρ
==
∂
⎡⎤
=−
∇=
−
∇
=
−
∇
⎢⎥
∂
⎣⎦
∑∑
nu
u
For molar balance
.
c
R
t
∂
= −∇
+
∂
N
*
1
.
N
c
cR
t
=
∂
=− ∇
+
∂
∑
u
For constant c and
ρ
Adding up N equations
[]
1
1
*
*
1
.
N
N
x
cx
R
x
R
t
R
c
=
=
∂
⎛⎞
+∇
=
−
∇
+
−
⎜⎟
∂
⎝
⇒
⎠
∑
∑
uj
u
[] ()
.
.0
.
r
t
ω
ρω
∂
+
−
∇
+
∂
∇
⎝⎠
=⇒
u
.
1,2,3,.
..,
y
xz
n
nn
rr
tx
y
z
N
∂
∂∂
∂
+
+
+
≡−∇
+
=
∂
∂
n
*
=
+
Nu
J
Time Dependent & 2D Component Transfer
Dimensionless analysis of component balances without reaction
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This note was uploaded on 09/01/2011 for the course PGE 312 taught by Professor Peters during the Fall '08 term at University of Texas at Austin.
 Fall '08
 Peters

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