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Chapter 3 Introduction to Mass Transfer

3.1 Basic Concepts 3.1.1 Diffusion and convection

Mass transfer refers to the transfer of a

species in the presence of a concentration

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

mass transfer 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 molecules Is

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.

3.1.2 Ficks law of diffusion 3.1.2.1

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 Ficks law of

diffusion

3.1.1

Where r 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.

3.1.2.2 Three-dimensional

In the case of three-dimensional diffusion,

the diffusion equation can be expressed as

follows

3.1.2

3.1.3

3.1.4

Note that is has been assumed that the

material is isotropic, that is, r and DA are

direction-independent.

Eqs. 3.1.2 through 3.1.4 can be expressed

in a vector form as follows

3.1.5

If the mass density r is constant, Eq.3.1-5 can

be written as follows

3.1.6

Where rA rwA is the mass of species A per

unit volume of the solution, or the mass

concentration of species A.

Ficks law of diffusion can also be written

as

3.1.7

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

3.1.8

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 Ficks law of diffusion

can be modified as follows

3.1.9

and dividing by MA, the molecular weight of

species A,

3.1.10

Where aT and aT are thermal diffusion factors

based on mass and molar concentrations,

respectively. The two factors are related to each

other through aT MBaT, 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

the Approaching fluid is wA8 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 wA8 in the stream. This region is called the

diffusion or concentration boundary layer.

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.

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 rAvA, can be

considered to result from a mass flux due to the

bulk motion of the fluid, rAv, 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

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,

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

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.

At the solid/liquid ( or liquid/gas)

interface the mass transfer coefficient km is

defined by

3.1-18

Where rA0 and rA8 are the mass concentrations of

species A in the fluid at the interface and in

the bulk (or free-stream) fluid, respectively. If

r is constant, Eq. 3.1-18 can be rewrite as

3.1-19

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

3.1-20

Substituting Eqs 3.1-17 and 3.1-19 into Eq.

3.1-20, we have

3.1-21

For fluid flow through a pipe such as that shown

in Fig. 3.1-4, we can write

3.1-22

Where the average concentration is defined as

follows Notice that the numerator is the

species mass flow rate.

3.1-23

Similar equations on the molar basis can be

written for flow over a flat plate

3.1-24

and for flow through a tube

3.1-25

Where the average concentration is defined as

follows

3.1-26

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.

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-n alloy and AuNi 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.

3.1.7.2 Diffusion coefficients

In the so-called self-diffusion experiment, a

solute A in the form of a radio- active isotope,

such as Ni, 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-concentratio

n 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

3.1-27

Where D diffusion coefficient D0 a

proportional constant Q the active energy

R the gas constant T absolute temperature

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- stitutionaldiffusion

and inte4rstitial diffusion in some materials. As

shown, Q is significantly lower for interstitial

diddusion than for substitutional diffusion.

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.

Example

Example

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