Chapter 3 Introduction to Mass Transfer

3.1 Basic Concepts3.1.1 Diffusion and convection

Mass transfer refers to the transfer of a species in the presence of aconcentration gradient of the species. Under a concentration gradient mass transfer can occur by either diffusion or convection.

Diffusion refers to the mass transfer that occurs a stationary solid or fluid in which a concentration gradient exists. In contrast, convection refers to masstransfer that occurs across a moving fluid in which a concentration gradient exists.

Consider the dissolution of a sugar cube in water, the concentration of sugar moleculesis significant only in the vicinity of the cube.By stirring the water with a spoon to create forced convection, sugar molecules are transferred to the bulk water much faster.

11

3.1.2 Fick’s law of diffusion3.1.2.1 One-dimensional

Consider the diffusion of species A through a thin sheet of thickness L, as shown. Let wA be the mass fraction of species A in the sheet. A steady-state concentration profile wA(y) is established in the sheet.

A diffusion flux, jAy, is defined as the amount of material diffused per unit area per unit time and can be expressed by Fick’s law of diffusion

dydwDj A

AAy [3.1.1]

Where is the mass density of the solution, DA the diffusion coefficient of species A in the solution, and wA the mass (or weight) fraction of species A.

Minus sign and unit. 2

3.1.2.2 Three-dimensional

In the case of three-dimensional diffusion, the diffusion equation can be expressed as follows

dxdwDj A

AAx dydwDj A

AAy dzdwDj A

AAz [3.1.2] [3.1.3] [3.1.4]

Note that is has been assumed that the material is isotropic, that is, and DA are direction-independent.

Eqs. [3.1.2] through [3.1.4] can be expressed in a vector form as follows:

AAA wD j [3.1.5]

jA is the mass diffusion flux vector. If the mass density is constant, Eq.[3.1-5] can be written as follows:

AAA D j [3.1.6]

Where A =wA is the mass of species A per unit volume of the solution, or the mass concentration of species A.

3

Fick’s law of diffusion can also be written as

[3.1.7]A AA

j cD x Aj Where is the molar diffusion flux vector, c the molar density of the

solution, and xA the mole fraction of species A.

If the molar density of the equation is constant, Eq. [3.1-7] can be written as:

Where cA =cxA is the moles of species A per unit volume of the solution, that is, the molar concentration of species A. In a dilute solution the molar density of the solution c is essentially constant.

3.1.3 Thermal diffusion (thermal diffusion describes the tendency for species to diffuse under the influence of a temperature gradient; this effect is quite small, but devices can be arranged to produce very steep temperature gradients so that separations of mixtures are effected)

In a nonisothermal system, spatial temperature variations can induce the so-called thermal diffusion, and Fick’s law of diffusion can be modified as follows:

[3.1.8]

[3.1.9]1j ( ln )A A A T A BD T

A AAj D c

4

( ) ( ) ( ) ( )j j j j jp g TA A A A A

where , p, g, and T refer to concentration gradient, pressure gradient, gas diffusion and thermal diffusion, respectively.

(Transport phenomena, Bird et al., pp 567-8)

Dividing Eq. [3.1.9] by MA, the molecular weight of species A,

)ln1(j TcccD BATAAA

Where T and T are thermal diffusion factors based on mass and molar concentrations, respectively. The two factors are related to each other through T = MBT, where MB is the molecular weight of species B.

3.1.4 Diffusion boundary layer

Consider a fluid approaching a flat plate in the direction parallel to the plate, as shown in Fig. 3.1-3. the plate is coated with a material containing species A, which has a limited solubility in the fluid. The composition of theapproaching fluid is wA∞ and that of the fluid at the plate surface is wAS, both of which are constant.

Because of the effect of diffusion, the concentration of the fluid in the region near the plate is affected by the coating, varying from wAS at the plate surface to wA∞ in the stream. This region is called the diffusion or concentration boundary layer.

[3.1.10]

5

The thickness δc of the concentration boundary layer is taken as the

distance from the plate surface at which the dimensionless concentration(wA-wAS)/(wA∞ -wAS) or (wAS-wA)/(wAS-wA∞) levels off to 0.99. In practice it is

usually specified that wA = wA∞ and at y = δc .0/ ywA

With increasing distance from the leading edge of the plate, the effect of diffusion penetrates farther into the stream and the boundary layer grows in thickness. The effect of diffusion is significant only in the boundary layer. Beyond it the concentration is uniform and the effect or diffusion is no longer significant.

6

3.1.5 Mass transfer flux

Let vA and vB be the velocities of species A and B with respect to stationary coordinates, respectively. These species velocities result from both the bulk motion of the fluid at velocity v and the diffusion of the species super imposed on the bulk motion. The mass flux of species A with respect to stationary coordinates, specially, nA = AvA, can be considered to result from a mass flux due to the bulk motion of the fluid, Av, and a mass flux due to the diffusion superimposed on the bulk motion, jA. In other words,

[3.1-11]

For a binary system consisting of species A and B, which is the focus of the present chapter, the velocity v is a local mass average velocity defined by

[3.1-12]

Substituting Eq.[3.1-12] into Eq. [3.1-11]

[3.1-13]

n v v jA A A A A

A A BA

v + v v + v n +nv v v+

A B B A B B AA B B

A B

w w

n v j v j (n n ) jA A A A A A A B Aw w 7

Similarly, the molar flux of species A with respect to stationary coordinates, specially, , can be considered to result from a molar flux, , due to the bulk motion of the fluid at velocity v and a molar flux due to the diffusion superimposed on the bulk motion, . In other words,

AA vn Ac vAc

An v v jA A Ac c [3.1-14]

Where the local molar average velocity

[3.1-15]

Substituting Eq. [3.1-15] into Eq. [3.1-14], we have

[3.1-16]

Aj

n v j (n +n ) jA A A A A B Ax c x

v v v v n nv ( ) v x vA A B B A A B B A BA A B B

A B

c c c c xc c c c

8

3.1.6 Mass transfer coefficient

Consider fluid flow over a flat plate as shown in Fig. 3.1-3. The mass diffusion flux across the solid/liquid interface is

[3.1-17]

This equation cannot be used to calculate the diffusion flux when the concentration gradient is an unknown. A convenient way to avoid this problem is to introduce a mass transfer coefficient.

[3.1-18]

At the solid/liquid ( or liquid/gas) interface the mass transfer coefficient km is defined by

[3.1-19]

Where A0 and A∞ are the mass concentrations of species A in the fluid at the interface and in the bulk (or free-stream) fluid, respectively. If is constant, Eq. [3.1-18] can be rewrite as

00

AAy Ay

y

wj Dy

00Ay m A Ayn k w w

00Ay m A Ayn k

00Ay m A Ayn k w w

9

[3.1-20]

If the solubility of species A in the fluid is limited so that vy is essentially zero at the interface, the following approximation can be made in view of Eq. [3.1-11]

0 0Ay Ayy yn j

Substituting Eqs [3.1-17] and [3.1-19] into Eq. [3.1-20], we have

[3.1-21]

[3.1-22]

For fluid flow through a pipe such as that shown in Fig. 3.1-4, we can write

Where the average concentration is defined as follows

0 0Ay A Ay y

mAS A AS A

j D w yk

w w w w

,,

A AAr r R r Rm

AS A avAS A av

D w rjk

w ww w

10

[3.1-24]

Similar equations on the molar basis can be written for flow over a flat plate

and for flow through a tube

[3.1-25]

Where the average concentration is defined as follows

[3.1-26]

0 0Ay A Ay ym

AS A AS A

j D x yk

c x x x x

,,

Ay A Ar R r Rm

AS A avAS A av

j D x rk

x xc x x

,A A AA A A

A avav

A

cx vdA cx vdA cx vdAx

m cv AcvdA

[3.1-23],

A A AA A AA av

avA

w vdA w vdA w vdAw

m v AvdA

Notice that the numerator is the species mass flow rate.

11

3.1.7 Diffusion in solids

3.1.7.1 Diffusion mechanisms

Vacancy diffusion and interstitial diffusion are the two most frequently encountered diffusion mechanisms in solids, although other mechanisms have also been proposed.

In vacancy diffusion an atom in a solid jumps from a lattice position of the solid into a neighboring unoccupied lattice site or vacancy, as illustrated in Fig. 3.1-5a. At temperature above absolute zero all solids contain some vacancies; the higher the temperature,the more the vacancies. The atom can continue to diffuse through the solid by a series of exchanges with vacancies that appears to be adjacent to it from time to time.

12

Vacancy diffusion is usually the diffusion mechanism for substantially solid solutions. In such materials the solute atoms, which are comparable to the solvent atoms in size, substitute the solvent atoms at their lattice sites. Examples of substitutional solid solutions are Cu-Zn alloy and Au—Ni alloy.

In interstitial diffusion an atom in a solid jumps from an interstitial site of the lattice to a neighboring one, as illustrated in Fig. 3.1-6a. The atom can continue to diffuse through the solid by a series of jumps to neighboring interstitial sites that are unoccupied.

Interstitial diffusion is the diffusion mechanism for interstitial solid solutions. In such materials the solute atoms, which are significantly smaller than the solvent atoms, occupy the interstitial sites of the lattice. The most well know example of interstitial solid solution is the iron-carbon alloy, specially, carbon steel, in which the small carbon atoms occupy the interstitial sites of the iron lattice.

13

3.1.7.2 Diffusion coefficients

/0

Q RTD D e [3.1-27]

In the so-called self-diffusion experiment, a solute A in the form of a radio-active isotope, such as 63Ni, is allowed to diffuse through the lattice of a non-radioactive solid of the same material, Ni. The diffusion coefficient DA is known as the self-diffusion coefficient, in view of the absence of a chemical composition gradient as the driving force for diffusion.

In practical situations, however, diffusion usually occurs under the influence of a chemical composition gradient, such as the diffusion of carbon in steel from a higher-carbon-concentration region to the lower one. This diffusion coefficient DA is known as the intrinsic diffusion coefficient. The so-called interdiffusion coefficient D is often used to describe situations involving the interdiffusion of two different chemical species, such as Au into Ni and Ni into Au as in an Au-Ni diffusion couple.

3.1.7.3 Effect of temperature The diffusion coefficient has been observed to increase with increasing temperature according to the following Arrhenius equation:

Where D: diffusion coefficient; D0: a proportional constant; Q: the active energy R: the gas constant; T: absolute temperature 14

As illustrated in Fig. 3.1-5b, a significant energy barrier has to be overcome before an atom can jump from one lattice site to a neighboring one by vacancy diffusion.

Similarly, as illustrated in Fig. 3.1-6b, a smaller but still significant energy barrier has to be overcome before an interstitial atom can jump from one interstitial site to a neighboring one by interstitial diffusion.

Tables 3.1-3 and 3.1-4 list the experimental data of D0 and Q for sub-stitutional diffusion and interstitial diffusion in some materials. As shown, Q is significantly lower for interstitial diffusion than for substitutional diffusion.

15

As shown in Table 3.1-3, Q is smaller for substitutional self diffusion in body-center-cubic iron than in face-center-cubic iron. Since atoms are more loosely packed in a bcc structure than an fcc structure, Q is smaller in bcc iron than in fcc iron. For the same reason, Q is also smaller for interstitial diffusion of C, N, and H in bcc iron than in fcc iron. Fig. 3.1-7 shows some diffusion coefficients as a function of temperature.

16

3.2 Species overall mass-balance equation3.2.1 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.

in out wallA A A A [3.2-1]

(1) (2) (3)

Rate of Rate of Rate of species A species A in species A out accumulation by mass inflow by mass outflow

Rate of other species A transfer to system from surround

(4) (5)

Rate of species A generation

ings in system

[3.2-3]

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:

17

Term 1: Mass of species A in the control volume

dA

The rate of change in the mass of species A in Ω

dt A

Terms 2&3: Mass flow rate of species A through a differential area dA

dAA A nv

Term 4: Species A goes into the control volume from the surrounding other than terms (2) and (3):

dAA A nj

Term 5: The mass production rate of species A in Ω:

drA

(int ) ; ( )Aegral M overall

d(int ) ; ( ) term (1)dt Aegral M overall

A in A out A wallA A A

( ) -( ) +( ) ( ) terms (2 & 3)vdA vdA vdA overall

(int );egral

(int ) ; J ( ) term (4)Aegral overall

(int ) ; R ( ) term (5)Aegral overall18

Substituting the integral form of terms (1) through (5) into Eq. [3.2.3], we have

A A A AA Ad dA dA r d

t

v n j n [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 , the local molar average velocity v, and the molar production rate of A per unit volume . This equation is

A A A AA A Ac d c dA dA r d

t

v n j n [3.2-5]

Note that the molar convection and diffusion terms can be combined into one through

AjAr

An v jA Ac

19

( ) ( )

AA A A AA Ain out

A in A out A A

dM w vdA w vdA J Rdtm m J R

[3.2-6]

, ,

, ,( ) ( )

AA av av A av av A Ain out

A av in A av out A A

dM w v A w v A J Rdtmw mw J R

[3.2-7]

Now substituting the overall form of terms (1) through (5) into Eq.[3.2-3] and with A =wA.

Where MAΩ is the mass of species A in the control volume; that is dwA

When the convective mass transfer at the wall has been neglected, substituting Eq. [3.1-23] into Eq.[3.2-6], we obtain

Where 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 (= )JA: mass transfer rate of species A into control volume from surroundings by diffusion

Ar

20

( ) ( )

AA A A AA Ain out

A in A out A A

dM cx vdA cx vdA J Rdtm m J R

[3.2-9] , ,

, ,( ) ( )

AA av av A av av A Ain out

A av in A av out A A

dM cx v A cx v A J Rdtmx mx J R

[3.2-8]

Equations similar to [3.2-6] and [3.2-7] can be derived on the molar basis, i.e.,

and

Where 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(= )JA: mass transfer rate of species A into control volume from surroundings by diffusion

Ar

21

Example 3.2-2 Diffusion through composite foil

Basic Eq. :

Given: No convection nor chemical reactions Find: Steady-state diffusion flux jH

, ,

, ,( ) ( )

AA av av A av av A Ain out

A av in A av out A A

dM w v A w v A J Rdtmw mw J R

1 2( ) ( ) 0H H Hz z

J Aj Aj Sol.:

1 2constant

=constant

H H Hz z A

H H H HA B C

j j j

j j j j

H H HH H H H

A B C

dw dw dwD D D Jdz dz dz

22

1 2 2 1

2 3 3 2

3 4 4 3

( )( )

( )( )

( )( )

HH H

H A

HH H

H B

HH H

H c

jw w z zDjw w z zDjw w z zD

[3.2-26]

[3.2-27]

[3.2-28]

Adding Eqs. [3.2-26] through [3.2-28], we have

2 2

3 2 4 32 11 4

1

3 2 4 32 11 4

1

1/ 2 1/ 23 2 4 32 1

( ) ( ) ( )

( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

H H HH A H B H C

H H HH A H B H C

H p H high H lowH A H B H C

z z z zz zw w jD D D

or

z z z zz zj w wD D D

z z z zz zj K p pD D D

23

Example 3.2-3 Diffusion of gas through tube wall

Basic Eq. :

Given: No convection nor chemical reactions Steady-state

Sol.: no chemical reaction RA=0 consider a C.V. of length dz along the longitudinal direction

( ) ( ) 0A in A A out Amx m x dx J

, ,

, ,( ) ( )

AA av av A av av A Ain out

A av in A av out A A

dM cx v A cx v A J Rdtmx mx J R

0( ) ( ) /ASA A A A A

cJ Ddz D Ddz D k x ll

Substituting Eq. [3.2-35] into [3.2-34], we have

0

0

(ln )

integrating ln or exp(- )L

A A AA

A

A A A A AAL A

A

dx DD kd x dzx mlx DD k L DD k Lx xx ml ml

24

3.3 Species differential mass-balance equation3.3.1 Derivation

The species integral mass-balance equation can be written as follows

[3.3-1]

The surface integrals in Eq. [3.3-1] can be converted into volume integrals using the Gauss divergence theorem

[3.3-2]

[3.3-3]Substituting Eq. [3.3-2] and [3.3-3] into Eq. [3.3-1]

[3.3-4]

The integrand, which is continuous, must be zero since the equation must hold for any arbitrary region Ω. Therefore,

AA A AA A

d dA dA r dt

v n j n

A AAdA d

v n v

A AAdA d

j v n j

0AA A Ar d

t

v j

25

[3.3-5]

[3.3-6]

Fick’s law of diffusion, according to Eqs. [3.1-5] and [3.1-7], is

[3.3-7]

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 , the local molar average velocity , and the molar production rate of A per unit volume . This equation is

AjArv

Noting that from Eq. [3.1-14]An v +jA Ac

Noting that from Eq. [3.1-11]An v +jA A

[3.3-8]

(variable properties)AA A A A Ar r

t

v j n

(variable properties)AA A A A A

c c r rt

v j n

A A AD w j

A A AcD x j26

[3.3-9]

[3.3-10]

[3.3-11]

[3.3-12]

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;

Let us now consider the case of incompressible fluids. From the continuity Equation and so . Since for constant and DA, Eq. [3.3-9] reduces to

v = 0 A A A( v) = v + v =vA

Eqs. [3.3-11] or [3.3-12] is the species differential mass-balance equation or the species continuity equation.

AA A A AD w r

t

v

AA A A A

c c cD x rt

v

2A (constant and D )A

A A A AD rt

v

2A (constant and D )A

A A A Ac c D c rt

v

27

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:

[3.3.14]Where (cA1 –cA0) is characteristic concentration difference.

If no chemical reaction occurred, Eq. [3.3.12] reduces to

[3.3.21]

[3.3.20]

Substituting Eq. [3.3-13] through [3.3-19] (please see the book for details) intoEq. [3.3-20]

0

1 0

A AA

A A

c cc

c c

(dimensionless concentration)

2AA A A

c c D ct

v

1 0 1 0

21 02

1

1

A A A A A A

A A A A

V c c c V c c cL t L

D c c cL

v

28

Multiplying Eq. [3.3-21] by L/[V (cA1 –cA0)]

[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:

[3.3.25]

[3.3.23]

[3.3.24]

[3.3.26]

Continuity:

Momentum:

Energy:

Species:

2A AA A

c Dc ct LV

v

0 v

21 1Re gp

t Fr

v v v v e

2 21 1Re Pr P T

T T T Tt e

v

2 21 1Re P

AA A A

S

c c c ct Sc e

v

29

where Re, Fr, and Pr are defined in the previous chapter, the new dimensionless parameters defined in this chapter are listed in the following:

[3.3.30]

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:

[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)

A

ScD

(Schmidt number = )A

viscous diffusivity species diffusivity D

Re PrTLVPe

(thermal Peclet number = )

v 1 0

1 0

convection heat transport C V T T conduction heat transport k T T / L

ReSA

LVPe ScD

(solutal Peclet number = )

A1 A0

A A1 A0

convection species transport V c c diffusion species transport D c c / L

30

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 con-centration wAf. A boundary conditions consistent with Eq. [3.1-21] is as follows: (case 4)

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)

[3.3-33]

Ay AA m A Af

j wD k w wy

31

32

3.3.4 Solution procedureThe 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

33

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 constantAnanlysis: Stationary:vx=vy=vz, one-dimensional problem (varies in x-dir only)Sol:

*

* * * *2

A (constant and D )AA A A A

cc D c r

t

v

* * * * * * *

* *

2 2 2

x y z 2 2 2v v vA A A A A A AA A

c c c c c c cD r

t x y z x y z

No chemical reaction * =0Ar

The governing Eq. becomes * *

*

2

2A A

A

c cD

t x

34

* *

*

with initial condition ( ,0) 0 and ( , ) 0

the mass conservation

M [3.3-38]

A A

A

c x c t

and

c dx

35

*

*

*

**

2

2

Refer Case P p. 640

I exp( )4

I is an arbitrary constant which can be determined using Eq. [3.3-38]

exp( )42

AA

AAA

xcD tt

M xcD tD t

Example 3.3-2Given: Two interstitial alloys with concentration of cA1 and cA2 at t=0Find: Concentration profile cA(x,t)Assume: Overall molar density c and intrinsic diffusion coefficient DAare constantAnalysis: Symmetric concentration with respect to the interface concentration cAS, due to constant c and DA, cAS = (cA1+cA2)/2 =constantSol: Basic Eq.

B.Cs.& I.Cs.:

The solution

2

2A A

Ac cDt x

2 2( ,0) (0, ) ( , )A A A AS A Ac x c c t c c t c

2

[3.3 51] erf(x)=erf(-x).4

A AS

A AS A

c c xerfc c D t

The solution can be used to determine the intrinsic diffusion coefficient DA from experiment 36

Example 3.3-3Given: Initial composition of cA1 and cA2 in solids 1 and 2, respectively. The concentration profile after annealing is shownFind: Intrinsic diffusion coefficient DA

Analysis: Overall molar concentration c is constant and no bulk motion and chemical reactionSol:

A AA

c cDt x x

Basic Eq.

Combine x and t so that cA can be expressed as a function of a single variable (combination of variables), let 1

2

x

t

32

12

A A Ac c cxt t t

12

1A A Ac c cx x t

Substituting the above two equations into governing Eq. 37

12

AA A

dcdc d Dd

B.Cs.

2

1

at

at A A

A A

c c x

c c x

Eq. (3.3-57) can be integrated as follows:

11

1

12

1 2 /

A

AA A A

A

A A

A

cA A A

A A A Acc c c

c

A Ac cA c

dc dc dcdc D D Dd d d

or

D dcdc d

Because

11

22

0 at and

0 at

A AA A

A AA A

dc dc c cd ddc dc c cd d

38

12 1

1 02

A

AA A

cA A

A A Acc c

dc dcdc D Dd d

Express Eq. 3.3-61 & 3.3-62 in terms of x rather than η

1/ 2

1/ 2 1/ 2

A A A

A A A

dc dc dcdx td dx d dx

dc t xdc t xdc

1

1 12 ( / )

A

A AA

c

A Ac cA c

D xdct dc dx

2

1

0A

A

c

Acand xdc

[3.3-63]

[3.3-64]

Eqs. [3.3-63] and [3.3-64] were used to determine the diffusion coefficient,and was called Boltzman-Matano Method.

39

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

Analysis: steady-state no convection within the sphere

no concentration gradients in the θand ψdirections

0Act

rv v = v 0 2

2 0A Ac c

Sol: The species continuity eq. (Eq. [c] of Table 3.3-2] can be reduced to

2 12

1 AA

A

c k ad r cr dr r D

B.Cs

A AS0 at r=0, c c at r=RAcr

Concentration distribution 1 where br br

AbR bR

AS A

c k aR e e bc r e e D

40