Bird R.B., Stewart W.E., Lightfoot E.N. Transport Phenomena

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522.2 Analytical Expressions for Mass Transfer Coefficients

677

Then, when the characteristic area is chosen to be the area of the interface

WL,

we see

that

Sh,

1.128(~e~c)"~ (22.2-2)

This equation expresses the Sherwood number (the dimensionless mass transfer coeffi-

cient) in terms of the Reynolds number and the Schmidt number, with Re defined in

terms of the maximum velocity

v,,,

in the film and the film length

The Reynolds num-

ber could also be defined in terms of the average film velocity with a different numerical

coefficient.

Similarly, for the dissolution of a slightly soluble material

from the wall into a

falling liquid film of pure

we can put Eq. 18.6-10 into the form of Eq. 22.1-3 as

follows:

Then, using the definition

~g6/p

given just after

Eq.

18.6-1 and the expression

for the maximum velocity in the film in Eq. 2.2-19, we find the Sherwood

follows:

In this instance we have not only the Reynolds number and Schmidt number

but also the ratio of the film length to the film thickness.

number as

(22.2-4)

appearing,

These two problems-gas absorption by a falling film and the dissolution of a solid

wall into a falling film-illustrate two important situations. In the first problem, there is

no velocity gradient at the gas-liquid interface, and the quantity ReSc appears to the

$-power in the expression for the Sherwood number. In the second problem, there is

velocity gradient at the solid-liquid interface, and the quantity ReSc appears to the

in the Sherwood number expression.

Mass Transfer for Flow Around Spheres

Next we consider the diffusion that occurs in the creeping flow around a spherical gas

bubble and around a solid sphere of diameter

This pair of systems parallels the two

systems discussed in the previous subsection.

For the gas absorption from a gas bubble surrounded by a liquid in creeping flow,

we can put Eq. 20.3-28 in the form of Eq. 22.1-5 thus:

The Sherwood number is then

Here the Reynolds number is defined using the approach velocity

of the fluid (or, al-

ternatively, the terminal velocity of the rising bubble).

678

Chapter

Interphase Transport

Nonisothermal Mixtures

For the creeping flow around a solid sphere with a slightly soluble coating that dis-

solves into the approaching fluid, we may modify the result in Eq. 12.4-34 to get

This result may be rewritten in terms of the Sherwood number as

As in the preceding subsection we have ReSc to the :-power for the gas-liquid system

and ReSc to the $-power for the liquid-solid system.

Both Eq. 22.2-6 and Eq. 22.2-8 are valid only for creeping flow. However, they are

not valid in the limit that Re goes to zero. As we know from Problem 10B.l and Eq. 14.4-

if there is no flow past the solid sphere or the spherical bubble, Sh,

2. It has been

found that a satisfactory description of the mass transfer all the way down to Re

can

be obtained by using the simple superpositions: Sh,,

0.6415(~e~c)'/~ and Sh,

0.991 (~eSc)'/~ in lieu of Eqs. 22.2-6 and

Mass Transfer

Steady, Nonseparated Boundary

Layers on Arbitrarily Shaped Objects

For systems with a fluid-fluid interface and no velocity gradient at the interface,

found the mass flux at the surface to be given by Eq. 20.3-14:

The local Sherwood number is

in which the constant, 1

/G,

is equal to 0.5642 and Re

lovop/p.

Similarly for systems with fluid-solid interfaces and a velocity gradient at the inter-

face, the mass flux expression is given in Eq. 20.3-26 as

The analogous Sherwood number expression is

where the numerical coefficient has the value 0.5384. In these equations

and v, are

characteristic length and a characteristic velocity that can be chosen after the shape

the body has been defined. Here again we see that the on ReSc appears in the

fluid-fluid system, and the $-power on ReSc appears in the fluid-solid system-

regardless of the shape. The radicands of the Sherwood number expressions are

dimensionless.

g22.3

Correlation of Binary Transfer Coefficients in One Phase

679

Mass

Transfer in the

Neighborhood of a Rotating Disk

For a disk of diameter

coated with a slightly soluble material

rotating with angular

velocity

in a large region of liquid

the mass flux at the surface of the disk is inde-

pendent of position. According to Eq. 19D.4-7 we have

This may be expressed in terms of the Sherwood number as

Here the characteristic velocity in the Reynolds number is chosen to be

DO.

522.3

CORRELATION OF BINARY TRANSFER

COEFFICIENTS IN ONE PHASE

In this section we show that correlations for binary mass transfer coefficients at low

mass-transfer rates can be obtained directly from their heat transfer analogs simply

change of notation. These correspondences are quite useful, and many heat transfer cor-

relations have, in fact, been obtained from their mass transfer analogs.

To illustrate the background of these useful analogies and the conditions under

which they apply, we begin by presenting the diffusional analog of the dimensional

analysis given in 514.3. Consider the steadily driven, laminar or turbulent isothermal

flow of a liquid solution of

in the tube shown in Fig. 22.3-1. The fluid enters the

tube at

with velocity uniform out to very near the wall and with a uniform inlet

composition

XAl

From

the tube wall is coated with a solid solution of

and

which dissolves slowly and maintains the interfacial liquid composition con-

stant at

xAO.

For the moment we assume that the physical properties

and

9AB

are

constant.

The mass transfer situation just described is mathematically analogous to the heat

transfer situation described at the beginning of 514.3. To emphasize the analogy, we pre-

sent the equations for the two systems together. Thus the rate of heat addition by con-

duction between 1 and

in Fig. 14.3-1 and the molar rate of addition of species

Nozzle

Fluid

enters

with uniform

composition

XA,

;

composition

XAb2

-k

assumed to

small

Fig.

22.3-1.

Mass transfer in

pipe with a soluble wall.

680

Chapter 22 Interphase Transport in Nonisothermal Mixtures

diffusion between 1 and 2 in Fig. 22.3-1 are given by the following expressions, valid for

either laminar or turbulent flow:

heat transfer:

Equating the left sides of these equations to hl(.rrDL)(To

TI) and kxl(.rrDL)(xAo

xA1) re-

spectively, we get for the transfer coefficients

heat transfer:

h,(t)

.rrDL(To

TI)

/2w(+k$r=JR

dB dz

/L/2*(+c9AB$ir=J~d~dz

mass transfer:

kXl(f)

xA,)

We now introduce the dimensionless variables

r/D,

z/D,

To)/(Tl

To),

and

(x,

xAO)/(xAl

xAO) and rearrange to obtain

heat transfer:

hlD

/""["(-g

NU,@)

2nL/Do dr

?=+

mass transfer:

kXlD

lo2"

(-5

Shl(t)

-------

&JAB 2vLID

-,=?

Here

Nu is the Nusselt number for heat transfer without mass transfer, and Sh is the

Sherwood number for isothermal mass transfer at small mass-transfer rates. The Nus-

selt number is a dimensionless temperature gradient integrated over the surface, and

the Sherwood number is a dimensionless concentration gradient integrated over the

surface.

These gradients can, in principle, be evaluated from Eqs. 11.5-7,

and 9 (for heat

transfer) and Eqs. 19.5-8, 9, and 11 (for mass transfer), under the following boundary

conditions (with ir and

defined as

514.3 and with time averaging of the solutions

the flow is turbulent):

velocity and pressure:

0, ir

forO~?<f

atr=,,ir=O for5

at?=Oand i=O,@

temperature:

concentration:

The boundary condition in Eq. 22.3-8, on the velocity at the wall, is accurate for the heat-

transfer system and also for the mass-transfer system provided that xA&WAO

WBO) is

small; the latter criterion is discussed in 5322.1 and 8. No boundary conditions are

needed at the outlet plane,

L/D, when we neglect the d2/dz2 terms of the consenra-

tion equations in the manner of 34.4 and 514.3.

522.3 Correlation of Binary Transfer Coefficients

One Phase

681

If we can neglect the heat production by viscous dissipation in Eq. 11.5-9 and if there

is no production of

by chemical reaction as in Eq. 19.5-11, then the differential equa-

tions for heat and mass transport are analogous along with the boundary conditions. It

follows then that the dimensionless profiles of temperature and concentration (time

smoothed, when necessary) are similar,

F(F,

8,i,

Re, Pr);

F(?,

O,i,

Re, Sc)

(22.3-14,15)

with the same form of

in both systems. Thus, to get the concentration profiles from the

temperature profiles, one replaces

and Pr by Sc.

Finally, inserting the profiles into Eqs. 22.3-5 and 6 and performing the integrations

and then time-averaging give for

forced convection

Nu,

G(Re, Pr, LID);

Shl

G(Re, Sc, LID)

(22.3-1 6,171

Here

is the same function in both equations. The same formal expression is obtained

for Nu,, Nul,, Nu,,, as well as for the corresponding Sherwood numbers. This important

analogy permits one to write down a mass transfer correlation from the corresponding

heat transfer correlation merely by replacing Nu by Sh, and Pr by Sc. The same can be

done for any geometry and for both laminar and turbulent flow. Note, however, that to

get this analogy one has to assume (i) constant physical properties, (ii) small net mass-

transfer rates, (iii) no chemical reactions, (iv) no viscous dissipation heating,

(v)

no ab-

sorption or emission of radiant energy, and (vi) no pressure diffusion, thermal diffusion,

or forced diffusion. Some of these effects will be discussed in subsequent sections of this

chapter; others will be treated in Chapter 24.

For

free

convection

around objects of any given shape, a similar analysis shows that

Nu,

H(Gr, Pr); Sh,

H(Gr,, Sc) (22.3-18,191

Here

is the same function in both cases, and the Grashof numbers for both processes

are defined analogously (see Table 22.2-1 for a summary of the analogous quantities for

heat and mass transfer).

To allow for the variation of physical properties in mass transfer systems, we extend

the procedures introduced in Chapter 14 for heat transfer systems. That is, we generally

evaluate the physical properties at some kind of mean film composition and tempera-

ture, except for the viscosity ratio

pb/pO.

We now give three illustrations of how to "translate" from heat transfer to mass

transfer correlations:

Forced Convection Around Spheres

For forced convection around a solid sphere, Eq. 14.4-5 and its mass-transfer analog are:

Nu,

0.60 ~e~/~ Pr1/3; Sh,,,

0.60 Sc'l3 (22.3-20,21)

Equations 22.3-20 and 21 are valid for constant surface temperature and composition, re-

spectively, and for small net mass-transfer rates. They may be applied to simultaneous

heat and mass transfer under restrictions (i)-(vi) given after Eq. 22.3-17.

Forced Convection along

Flat

Plate

As another illustration of the use of analogies, we can cite the extension of Eq. 14.4-4 for

the laminar boundary layer along a flat plate, to include mass transfer:

682

Chapter 22 Interphase Transport in Nonisothermal Mixtures

The Chilton-Colburn j-factors, one for heat transfer and one for diffusion, are defined as1

The three-way analogy in Eq. 22.3-22 is accurate for Pr and Sc near unity (see Table

12.4-1) within the limitations mentioned after Eq. 22.3-17. For flow around other objects,

the friction factor part of the analogy is not valid because of the form drag, and even for

flow in circular tubes the analogy with

ifio,

is only approximate (see 514.4).

The

Chilton-Colburn Analogy

The more widely applicable empirical analogy

a function of Re, geometry, and boundary conditions

(22.3-25)

has proven to be useful for transverse flow around cylinders, flow through packed beds,

and flow in tubes at high Reynolds numbers. For flow in ducts and packed beds, the

"approach velocity"

has to be replaced by the interstitial velocity or the superficial ve-

locity. Equation 22.3-25 is the usual form of the

Chilton-Colburn analogy.

It is evident

from Eqs. 22.3-20 and

21,

however, that the analogy is valid for flow around spheres

only when

and Sh are replaced by (Nu

and (Sh

2).

It would be very misleading to leave the impression that all mass transfer coeffi-

cients can be obtained from the analogous heat transfer coefficient correlations. For mass

transfer we encounter a much wider variety of boundary conditions and other ranges

the relevant variables. Non-analogous behavior is addressed in gs22.5-8.

spherical drop of water, 0.05 cm in diameter, is falling at a velocity of 215 cm/s through

dry, still air at

atm pressure with no internal circulation. Estimate the instantaneous rate

from

evaporation from the drop, when the drop surface is at

70°F and the air (far from the

Freely Falling

Drop

drop) is at

140°F. The vapor pressure of water at 70°F is 0.0247 atm. Assume quasi-

steady state conditions.

SOLUTION

Designate water as species

and air as species

The solubility of air in water may be ne-

glected, so that

WBO

Then assuming that the evaporation rate is small, we may write Eq.

22.1-3 for the entire spherical surface as

The mean mass transfer coefficient,

k,,,

may be predicted from Eq. 22.3-21 in the assumed ab-

sence of internal circulation.

The film conditions needed for estimating the physical properties are obtained

follows:

Chilton

and

Colburn,

Ind.

Eng.

Chem.,

26,1183-1187

(1934).

s22.3 Correlation of Binary Transfer Coefficients in One Phase

683

In computing

xAfr

we have assumed ideal gas behavior, equilibrium at the interface, and com-

plete insolubility of air in water. The mean mole fraction,

xAf,

of the water vapor is sufficiently

small that it can be neglected in evaluating the physical properties at the film conditions:

3.88

lop5

g-moles/cm3

1.12

g/cm3

1.91

lod4

g/cm

s (from Table 1.1-1)

%AB

0.292 cm2/s (from

Eq.

17.2- 1)

When these values are used in

Eq.

22.3-21 we get

Sh,

0.60(63)~/~(0.58)'/~

5.96

and the mean mass transfer coefficient is then

C~AB

(3.88

1OP5)(O.292) (5.96)

kxm

Sh,

0.05

Then from

Eq.

22.3-26 the evaporation rate is found to be

This result corresponds to a decrease of 1.23

lop3

cm/s in the drop diameter and indicates

that a drop of this size will fall a considerable distance before it evaporates completely.

In this example, for simplicity, the velocity and surface temperature of the drop were

given. In general, these conditions must be calculated from momentum and energy balances,

as discussed in Problem 22B.1.

EXAMPLE

22.3-2

The Wet

and

Bulb

We next turn to a problem for which the analogy between heat and mass transfer leads to a sur-

prisingly simple and useful,

approximate, result. The system, shown in Fig. 22.3-2, is a pair of

thermometers, one of which is covered with a cylindrical wick kept saturated with water. The

Psychrometer

wick will cool by evaporation into the moving air stream and for steady operation will ap-

proach an asymptotic value known as the

wet bulb temperature.

The bare thermometer, on the

other hand, will tend to approach the actual temperature of the approaching air, and this value

is called the

dry

bulb temperature.

Develop an expression for determining the humidity of the air

from the wet and dry bulb temperature readings neglecting radiation and assuming that the

re-

placement of the evaporating water has no significant effect on the wet bulb temperature mea-

surement.

Problem 228.2 we will see how radiation can be taken into account.

SOLUTION

For simplicity, we assume that the fluid velocity is high enough that the thermometer read-

ings are unaffected by radiation and by heat conduction along the thermometer stems, but

not so high that viscous dissipation heating effects become significant. These assumptions are

usually satisfactory for glass thermometers and for gas velocities of 30 to 100 ft/s. The dry

bulb temperature is then the same as the temperature

of the approaching gas, and the wet

bulb temperature is the same as the temperature

of the outside of the wick.

Let species

be water and species

be air. An energy balance is made on a system that

contains a length

of the wick (the distance between planes

and 2 in the figure). The rate of

684

Chapter 22 Interphase Transport in Nonisothermal Mixtures

Dry-bulb Wet-bulb

thermometer thermometer

Surface

Wick saturated

Fig.

22.3-2.

Sketch of a wet-bulb and dry-bulb psychrometer installation. It is assumed that no heat or mass

moves across plane 2.

heat addition to the system by the gas stream is h,(nDL)(T,

To). Enthalpy also enters via

plane

at a rate WAIHAl in the liquid phase and leaves at the mass transfer surface at a rate

w~~J~~, both of these occurring at a temperature To. Hence the energy balance gives

since the water enters the system at plane

at the same rate that it leaves as water vapor at

the mass transfer interface

To a very good approximation,

HA,

may be replaced

A&,,,

the molar heat of vaporization of water.

From the definition of the mass transfer coefficient

in which WBo

as in the preceding example. Combination of Eqs. 22.2-32 and 33 gives then

Then using the definitions of Nu, and Sh,, and noting that

p?p

ct,,

we may rewrite

Eq.

22.3-34 as

Because of the analogy between heat and mass transfer, we can expect that the mean Nusselt

and Sherwood numbers will be of the same form:

Nu,

F(Re)Prn;

Sh,

F(Re)Scn

(22.3-36,37)

where F is the same function of Re in both expressions. Therefore, knowing the dry and wet

bulb temperatures and the mole fraction of the water vapor adjacent to the wick (x,,), we can

calculate the upstream composition

XA,

of the air stream from

The exponent

depends to a slight extent on the geometry, but is not far from

and the

quantity (Sc/PrI1-" is not far from unity.* Furthermore, the

wet

bulb temperature is seen to be

somewhat different equation, with

0.56, was recommended for measurements in air by

Bedingfield and

B. Drew,

Ind.

Eng.

Chem.,

42,1164-1173

(1950).

522.3 Correlation of Binary Transfer Coefficients in One Phase

685

independent of the Reynolds number under the assumption introduced in Eqs. 22.3-36 and

37. This result would also have been obtained by using the Chilton-Colburn relations, which

would give

directly.

The interfacial gas composition

xAo

can be accurately predicted, at low mass-transfer

rates, by neglecting the heat and mass transfer resistance of the interface itself (see s22.4

for further discussion of this point). One can then represent

xAO

by the vapor-liquid

equilibrium relationship:

relation of this kind will hold for given species

and

if the Iiquid is pure

as assumed

above

commonly used approximation of this relationship is

in which

PA,",,

is the vapor pressure of pure

at temperature

To.

This relation assumes tacitly

that the presence of

does not alter the partial pressure of

at the interface, and that

and

form an ideal gas mixture.

If an air-water mixture at

atm pressure gives a wet bulb temperature of 70°F and a dry

bulb temperature of 140°F, then

pA,vap

0.0247 atm

xA0

0.0247, from

Eq.

22.3-40

6.98 Btu/lb-mole.

at 105"F, the film temperature

A&,

18,900 Btu/lb-mole at 70°F

0.58 (see Example 22.2-1)

0.74, from Eq. 9.3-16

Substitution into Eq. 22.3-37, with

then gives

From this the mole fraction of water in the approaching air is

Since we assumed that the film concentration was

0 as a first approximation, we

could go back and make a second approximation by using an average film concentration

of i(0.0247

0.0033)

0.0140 in the physical property calculations. The physical proper-

ties are not known accurately enough here to justify recalculation.

The calculated result in Eq. 22.3-43 is in only fair agreement with published humidity

charts, because these are typically based on the adiabatic saturation temperature rather than

the wet bulb tem~erature.~

EXAMPLE

22.3-3

Mass Transfer in

Creeping

Flow

Through

Many important adsorptive operations, from purification of proteins in modern biotechnol-

ogy to the recovery of solvent vapor by dry-cleaning establishments, occur in dense particu-

late beds and are typically carried out in steady creeping flow-that is, at Re

D,vg/p

20.

Here D, is the effective particle diameter and

is the superficial velocity, defined as volumet-

Packed

Beds

ric flow rate divided by the total cross section of the bed (see 56.4). It follows that the dimen-

sionless velocity

v/vo

will have a spatial distribution independent of the Reynolds number.

Detailed information is available only for spherical packing particles.

A. Hougen,

Watson, and R.

Ragatz,

Chemical

Process

Principles,

Part

2nd edition,

Wiley,

New

York,

(1954),

120.

686

Chapter 22 Interphase Transport in Nonisothermal Mixtures

SOLUTION

Using the dimensional analysis discussion at the beginning of this section, predict the

form of the steady-state mass transfer coefficient correlation for creeping flow.

The dimensional analysis procedure in s19.5 may be used, with

as the characteristic length

and

the characteristic velocity. Then, from Eq. 19.5-11, we see that the dimensionless con-

centration depends only on the product ReSc, in addition to the dimensionless position coor-

dinates and the geometry of the bed.

The most extensive data are for creeping flow at large Peclet numbers. Experimental data

on the dissolution of benzoic acid spheres

water4 have yielded the result

1 09

Sh,

(R~sc)"~

ReSc

(22.3-43)

where

is the volume fraction of the bed occupied by the flowing fluid. Equation 22.3-43 is

reasonably consistent with the relation

Sh,

0.991(Re~c)~'~

(22.3-44)

which incorporates the creeping flow solution for flow around an isolated sphere5

1) (see

5522.2b). This suggests that the flow pattern around an isolated sphere is not much different

from that around

sphere surrounded by other spheres, particularly near the sphere surface

where most of the mass transport takes place.

No reliable data are available for the limiting behavior at very low values of ReSc,

but

numerical calculations for a regular packing6 predict that the Sherwood number asymptoti-

cally approaches a constant near 4.0 if based on a local difference between interfacial and bulk

compositions.

Behavior within the solid phase is far more complex, and no simple approximation is

wholly trustworthy. However, experiments to date7 show that where intraparticle mass trans-

port is described by Fick's second law, one can use the approximation

where

kc,

is the effective mass transfer coefficient within the solid phase and

9,,

is the diffu-

sivity of

in the solid phase. The equation is for "slow" changes in the solute concentration

bathing the particle. This is an asymptotic solution for a linear change of surface concentra-

tion with time: and has been justified9 by calculations. For a Gaussian (bell-shaped) concen-

tration wave, "slow" means that the passage time (temporal standard deviation) of the wave

is long relative to the particle diffusional response time, which is of the order of

D;/69,,.

Fick's second law must be solved with the detailed history of surface concentration when this

inequality is not satisfied.

In packed beds, as with tube flow, one must keep

mind the fact that there will be

nonuniformities in the concentration as a function of the radial coordinate. This was dis-

cussed in 514.5 and s20.3.

Wilson and

Geankopolis,

Ind.

Eng.

Chem.

Fundamentals,

5,9-14 (1966).