Transcript Document

3.2 Species overall mass-balance equation
3.2.1 Derivation
Consider an arbitrary stationary control volume Ω bounded by surface A
through which a moving fluid is flowing, as illustrated in Fig. 3.2-1. The control
surface A can be consider to consist of three different regions.
A  Ain  Aout  Awall
[3.2-1]
Consider the transfer rate of species A through dA shown in Fig. 3.2-2a.
As shown in Fig. 3.2-2b, the outward transfer and inward transfer rate are ja•ndA
and -ja•ndA, respectively.
Consider the mass conservation law for species A in the control volume shown in
Fig.3.2-1:
 Rate of
  Rate of
  Rate of


 
 

species
A

species
A
in

species
A
out

 
 

 accumulation   by mass inflow   by mass outflow 

 
 

(1)
(2)
 Rate of other species

  A transfer to system
 from surroundings

(4)
(3)
  Rate of species
 
  A generation
  in system
 


 [3.2-3]


(5)
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Term 1: Mass of species A in the control volume

 Ad (int egral ) ; M A (overall )

The rate of change in the mass of species A in Ω

d
(int
egral
)
;
M A (overall ) term (1)
 A d

dt
t 
Terms 2&3: Mass flow rate of species A through a differential area dA
   A v  ndA (int egral );
A
( A vdA)in -( A vdA)out +( A vdA) wall (overall ) terms (2 & 3)
A
A
A
Term 4: Species A goes into the control volume from the surrounding other than
terms (2) and (3):
  jA  ndA (int egral ) ; J A (overall ) term (4)
A
Term 5: The mass production rate of species A in Ω:


rAd (int egral ) ; R A (overall ) term (5)
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Substituting the integral form of terms (1) through (5) into Eq. [3.2.3], we have

 A d      A v  ndA   jA  ndA   rA d 


A
A

t
[3.2-4]
Note that the mass convection and diffusion terms can be combined into one
through nA =Av+jA
A similar equation can be derived on the basis of the molar density cA, the
molar flux j A , the local molar average velocity v, and the molar production rate of
A per unit volume rA . This equation is

cA d     cA v  ndA   jA  ndA   rA d 


A
A
A
t
[3.2-5]
Note that the molar convection and diffusion terms can be combined into one
through n A  c A v  jA
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Now substituting the overall form of terms (1) through (5) into Eq.[3.2-3] and with
A =wA.

 
dM A
   wAvdA    wAvdA
A
A
in
dt
 (mA )in  (mA )out  J A  RA

out
 J A  RA
Where MAΩ is the mass of species A in the control volume; that is
[3.2-6]


wAd
When the convective mass transfer at the wall has been neglected, substituting
Eq. [3.1-23] into Eq.[3.2-6], we obtain
dM A
   wA,av vav Ain    wA,av vav Aout  J A  RA
dt
 (mwA,av )in  (mwA,av )out  J A  RA
Where
[3.2-7]
MAΩ: mass of species A in control volume((=wAΩ=MwA if uniform wA)
M: mass flow rate at inlet or outlet (=vAVA)
RA:mass generation rate of species A in control volume (= rA )
JA: mass transfer rate of species A into control volume from
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surroundings by diffusion
Equations similar to [3.2-6] and [3.2-7] can be derived on the molar basis, i.e.,

 
dM A
  cxA vdA   cxA vdA
A
A
in
dt
 (mA )in  (mA )out  J A  RA

out
 J A  RA
[3.2-8]
and
dM A
  cxA,av vav Ain   cxA,av vav Aout  J A  RA
dt
 (mxA,av )in  (mxA,av )out  J A  RA
Where
[3.2-9]
MAΩ: mass of species A in control volume(=cxAΩ=MxA if uniform cxA)
m: mass flow rate at inlet or outlet (=cvavA)
RA:mass generation rate of species A in control volume(= rA )
JA: mass transfer rate of species A into control volume from
surroundings by diffusion
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3.3 Species differential mass-balance equation
3.3.1 Derivation
The species integral mass-balance equation can be written as follows
 A
 t d   A  A v  ndA  A jA  ndA   rAd 
[3.3-1]
The surface integrals in Eq. [3.3-1] can be converted into volume integrals using the
Gauss divergence theorem

A

 A v  ndA    A vd 
[3.3-2]
j v  ndA   jAd 
[3.3-3]

A A

Substituting Eq. [3.3-2] and [3.3-3] into Eq. [3.3-1]
  A





v



j

r
A
A
A d   0
  t

[3.3-4]
The integrand, which is continuous, must be zero since the equation must hold for
any arbitrary region Ω. Therefore,
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 A
    A v    jA  rA    n A  rA (variable properties)
t
Noting that n A   A v +jA from Eq. [3.1-11]
[3.3-5]
By following a similar approach, an equation can be derived based on the basis
of molar density cA (moles A per unit volume), the molar diffusion flux j A , the local
molar average velocity v , and the molar production rate of A per unit volume rA .
This equation is
cA
   cA v    jA  rA    n A  rA (variable properties)
t
[3.3-6]
Noting that n A  c A v + jA from Eq. [3.1-14]
Fick’s law of diffusion, according to Eqs. [3.1-5] and [3.1-7], is
jA    DAwA
[3.3-7]
jA  cDAxA
[3.3-8]
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Substituting Eqs. [3.3-7] and [3.3-8] into Eqs. [3.3-5] and [3.3-6], respectively, the
following equations can be obtained;
 A
     A v       DAwA   rA
t
cA
    cA v      cDAx A   rA
t
[3.3-9]
[3.3-10]
Let us now consider the case of incompressible fluids. From the continuity
Equation  v = 0 and so   (  A v) = v A +A  v =v A. Since
for constant  and DA, Eq. [3.3-9] reduces to
 A
 v  A  DA 2  A  rA (constant  and D A )
t
c A
 v c A  DA 2 c A  rA (constant  and D A )
t
[3.3-11]
[3.3-12]
Eqs. [3.3-11] or [3.3-12] is the species differential mass-balance equation or the
species continuity equation.
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3.3.2 Dimensionless form
Besides the dimensionless parameters listed in Sections 1.5.2 and 2.3.2, we
Includes a new parameter for concentration:
cA

c A  c A0

c A1  c A0
(dimensionless concentration)
[3.3.14]
Where (cA1 –cA0) is characteristic concentration difference.
If no chemical reaction occurred, Eq. [3.3.12] reduces to
c A
 v c A  DA 2 c A
t
[3.3.20]
Substituting Eq. [3.3-13] through [3.3-19] (please see the book for details) into
Eq. [3.3-20]
V 
1 2 

  1


c
c

c

V
v
c
c

c

D
 A1 A0  A 2  cA  cA1  cA0  [3.3.21]
A0  
A
  A  A1
L t
L
L
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Multiplying Eq. [3.3-21] by L/[V (cA1 –cA0)]
cA
DA 2 
  
 v  cA 
 cA

t
LV
[3.3.22]
By combining Eq.[3.3-22] with Eqs. [2.3-18] through [2.3-24], the following
equations can be obtained for mass transfer in forced convection:
Continuity:
Momentum:
  v   0
v
1 2  1

 
 
 v  v   p 
 v 
eg

t
Re
Fr
Energy:
T 
1
1 2 

 
2 
 v  T 
 T 
 T

t
Re Pr
P eT
Species:
cA
1
1 2 

 
2 

v

c


c

 cA
A
A

t
Re Sc
P eS
[3.3.23]
[3.3.24]
[3.3.25]
[3.3.26]
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where Re, Fr, and Pr are defined in the previous chapter, the new dimensionless
parameters defined in this chapter are listed in the following:

viscous diffusivity 
[3.3.30]
Sc 
(Schmidt number = species diffusivity D )
DA
A
By combining Eq.[3.3-22] with Eqs. [2.3-18] through [2.3-24], the following
equations can be obtained for mass transfer in forced convection:
PeT  Re Pr 
LV

(thermal Peclet number =
LV
PeS  Re Sc 
DA
(solutal Peclet number =
convection heat transport  C v V  T1  T0 
)
conduction heat transport k  T1  T0  / L
convection species transport V  c A1  c A0 
diffusion species transport D A  c A1  c A0  / L
[3.3.31]
[3.3.32]
)
3.3.3 Boundary conditions
Boundary conditions commonly encountered in mass transfer are summarized
and listed:
1. At the plane or axis of symmetry, the concentration gradient in the transverse
direction is zero, (case 1)
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2. A wall in contact with a fluid or the surface of a solid or fluid may be kept at a
given solute concentration, (case 2)
3. A wall in contact with a fluid or the surface of a solid or liquid may allow no
penetration, evaporation, or reactions, (case 3)
4. The free surface of a fluid may be
exposed to a gas of solute concentration wAf. A boundary conditions
consistent with Eq. [3.1-21] is as
follows: (case 4)
wA
  DA
 km  wA  wAf 

y
jAy
[3.3-33]
5. For two phase in perfect contact
with each other, the concentration
and the diffusion flux are both
continuous across the interface,
that is, they are the same
on both sides of the interface, (case 5)
30
31
3.3.4 Solution procedure
The purpose of the species equation is to determine the concentration distribution,
the step-by step procedure for solving the species continuity equation are listed in
Fig. 3.3-3
32
Example 3.3-1
Given: Initial A* (in moles):M (at t=0)
Find: CA*(x,t) and diffusion flux jA*x
Assume:Overall molar concentration c and
the self-diffusion coefficient DA*x are constant
Ananlysis: Stationary:vx=vy=vz, one-dimensional
problem (varies in x-dir only)
Sol:
33
Example 3.3-2
Given: Two interstitial alloys with concentration
of CA1 and CA2 at t=0
Find: Concentration profile CA(x,t)
Assume: Overall molar density c and intrinsic
diffusion coefficient DAare constant
Ananlysis: Symmetric concentration with
respect to the interface concentration CAS,
due to constant c and DA, CAS = (CA1+CA2)/2
=constant
Sol:
34
Example 3.3-3
Given: Initial composition of CA1 and CA2 in
solids 1 and 2, respectively. The concentration
profile after annealing is shown
Find: Intrinsic diffusion coefficient DA
Analysis: Overall molar concentration c is constant
and no bulk motion and chemical reaction
Sol:
35
Example 3.3-6
Problem: A chemical species A diffuses
from a gas phase into a porous catalyst
sphere of radius R in which it is
converted into species B.
Given: Concentration of A at the surface
of the sphere is CAS, A is consumed
according to rA = -k1aCA, constant DA
Find: Steady-state concentration
distribution of A in the sphere
Sol:
36