Computational Fluid Mechanics and Heat Transfer Third Edition_17 pot

Table 11.1 Typical diffusion coefficients for binary gas mix-tures at 1 atm and dilute liquid solutions [11.4].. Pressure gradients and body forces acting unequally on the different species

Table 11.1 Typical diffusion coefficients for binary gas

mix-tures at 1 atm and dilute liquid solutions [11.4]

where k T is called the thermal diffusion ratio and is generally quite small.

Thermal diffusion is occasionally used in chemical separation processes

Pressure gradients and body forces acting unequally on the different

species can also cause diffusion Again, these effects are normally small

A related phenomenon is the generation of a heat flux by a concentration

gradient (as distinct from heat convected by diffusing mass), called the

diffusion-thermo or Dufour effect.

In this chapter, we deal only with mass transfer produced by

concen-tration gradients

11.4 Transport properties of mixtures6

Direct measurements of mixture transport properties are not always able for the temperature, pressure, or composition of interest Thus, wemust often rely upon theoretical predictions or experimental correlationsfor estimating mixture properties In this section, we discuss methodsfor computingD im , k, and µ in gas mixtures using equations from ki-

avail-netic theory—particularly the Chapman-Enskog theory [11.2,11.8,11.9]

We also consider some methods for computingD12in dilute liquid tions

solu-The diffusion coefficient for binary gas mixtures

As a starting point, we return to our simple model for the self-diffusioncoefficient of a dilute gas, eqn (11.32) We can approximate the averagemolecular speed,C, by Maxwell’s equilibrium formula (see, e.g., [11.9]):

where k B = R ◦ /N Ais Boltzmann’s constant If we assume the molecules

to be rigid and spherical, then the mean free path turns out to be

where d is the effective molecular diameter Substituting these values

ofC and  in eqn (11.32) and applying a kinetic theory calculation that

6 This section may be omitted without loss of continuity The property predictions

of this section are used only in Examples 11.11 , 11.14 , and 11.16 , and in some of the end-of-chapter problems.

Figure 11.6 The Lennard-Jones

potential

in the bulk properties of the gas The Chapman-Enskog kinetic theory

takes all these factors into account [11.8], resulting in the following

M A + M1

B where the units of p, T , and D ABare atm, K, and m2/s, respectively The

functionΩAB

D (T ) describes the collisions between molecules of A and B.

It depends, in general, on the specific type of molecules involved and the

temperature

The type of molecule matters because of the intermolecular forces

of attraction and repulsion that arise when molecules collide A good

approximation to those forces is given by the Lennard-Jones

intermolec-ular potential (see Fig.11.6.) This potential is based on two parameters,

a molecular diameter, σ , and a potential well depth, ε The potential well

depth is the energy required to separate two molecules from one another

Both constants can be inferred from physical property data Some values

are given in Table 11.2 together with the associated molecular weights

(from [11.10], with values for calculating the diffusion coefficients of

wa-ter from [11.11])

Table 11.2 Lennard-Jones constants and molecular weights of

selected species

Species σ (Å) ε/kB(K) M

kgkmol



Species σ (Å) ε/kB(K) M

kgkmol

aBased on mass diffusion data.

bBased on viscosity and thermal conductivity data.

An accurate approximation toΩAB

D (T ) can be obtained using the

Len-nard-Jones potential function The result is

ΩAB

D (T ) = σ2

ABΩD

kBT ε AB where, the collision diameter, σ AB, may be viewed as an effective molecu-

lar diameter for collisions of A and B If σ A and σ Bare the cross-sectional

diameters of A and B, in Å,7 then

The collision integral,ΩD is a result of kinetic theory calculations lations based on the Lennard-Jones potential Table11.3gives values of

calcu-7 One Ångström (1 Å) is equal to 0.1 nm.

ΩD from [11.12] The effective potential well depth for collisions of A

Equation (11.42) indicates that the diffusivity varies as p −1and is

in-dependent of mixture concentrations, just as the simple model indicated

that it should The temperature dependence ofΩD, however, increases

the overall temperature dependence of D AB from T 3/2, as suggested by

eqn (11.39), to approximately T 7/4

Air, by the way, can be treated as a single substance in Table 11.2

owing to the similarity of its two main constituents, N2and O2

Example 11.3

ComputeD ABfor the diffusion of hydrogen in air at 276 K and 1 atm

Solution. Let air be species A and H2 be species B Then we read

D AB = (1.8583 × 10 −7 )(276) 3/2

(1)(3.269)2(0.8822)

21

Limitations of the diffusion coefficient prediction Equation (11.42) is

not valid for all gas mixtures We have already noted that concentration

gradients cannot be too steep; thus, it cannot be applied in, say, the

interior of a shock wave when the Mach number is significantly greater

than unity Furthermore, the gas must be dilute, and its molecules should

be, in theory, nonpolar and approximately spherically symmetric

Reid et al [11.4] compared values ofD12calculated using eqn (11.42)

with data for binary mixtures of monatomic, polyatomic, nonpolar, and

polar gases of the sort appearing in Table11.2 They reported an average

absolute error of 7.3 percent Better results can be obtained by using

values of σ AB and ε ABthat have been fit specifically to the pair of gases

involved, rather than using eqns (11.40) and (11.41), or by constructing

a mixture-specific equation forΩAB

D (T ) [11.13, Chap 11]

The density of the gas also affects the accuracy of kinetic theory

pre-dictions, which require the gas to be dilute in the sense that its molecules

interact with one another only during brief two-molecule collisions Childs

and Hanley [11.14] have suggested that the transport properties of gases

are within 1% of the dilute values if the gas densities do not exceed the

following limiting value

ρmax= 22.93M (σ3Ωµ ) (11.43)

Here, σ (the collision diameter of the gas) and ρ are expressed in Å and

kg/m3, andΩµ—a second collision integral for viscosity—is included in

Table11.3 Equation (11.43) normally gives ρmaxvalues that correspond

to pressures substantially above 1 atm

At higher gas densities, transport properties can be estimated by a

variety of techniques, such as corresponding states theories, absolute

reaction-rate theories, or modified Enskog theories [11.13, Chap 6] (also

see [11.4, 11.8]) Conversely, if the gas density is so very low that the

mean free path is on the order of the dimensions of the system, we have

what is called free molecule flow, and the present kinetic models are again

invalid (see, e.g., [11.15])

Diffusion coefficients for multicomponent gases

We have already noted that an effective binary diffusivity, D im, can be

used to represent the diffusion of species i into a mixture m The

pre-ceding equations for the diffusion coefficient, however, are strictly

appli-cable only when one pure substance diffuses through another Different

equations are needed when there are three or more species present

If a low concentration of species i diffuses into a homogeneous ture of n species, then
J j

If a mixture is dominantly composed of one species, A, and includes

only small traces of several other species, then the diffusion coefficient

of each trace gas is approximately the same as it would be if the othertrace gases were not present In other words, for any particular trace

species i,

Finally, if the binary diffusion coefficient has the same value for eachpair of species in a mixture, then one may show (Problem 11.14) that

D im = D ij, as one might expect

Diffusion coefficients for binary liquid mixtures

Each molecule in a liquid is always in contact with several neighboringmolecules, and a kinetic theory like that used in gases, which relies ondetailed descriptions of two-molecule collisions, is no longer feasible.Instead, a less precise theory can be developed and used to correlateexperimental measurements

For a dilute solution of substance A in liquid B, the so-called dynamic model has met some success Suppose that, when a force per molecule of F A is applied to molecules of A, they reach an average steady speed of v A relative to the liquid B The ratio v A /F A is called the mobil- ity of A If there is no applied force, then the molecules of A diffuse

hydro-as a result of random molecular motions (which we call Brownian tion) Kinetic and thermodynamic arguments, such as those given by

mo-Einstein [11.16] and Sutherland [11.17], lead to an expression for the

dif-fusion coefficient of A in B as a result of Brownian motion:

D AB = kBT (v A /F A ) (11.46)Equation (11.46) is usually called the Nernst-Einstein equation.

To evaluate the mobility of a molecular (or particulate) solute, we

may make the rather bold approximation that Stokes’ law [11.18] applies,

even though it is really a drag law for spheres at low Reynolds number

Here, R A is the radius of sphere A and β is a coefficient of “sliding”

friction, for a friction force proportional to the velocity Substituting

eqn (11.47) in eqn (11.46), we get

This model is valid if the concentration of solute A is so low that the

molecules of A do not interact with one another.

For viscous liquids one usually assumes that no slip occurs between

the liquid and a solid surface that it touches; but, for particles whose size

is on the order of the molecular spacing of the solvent molecules, some

slip may very well occur This is the reason for the unfamiliar factor in

parentheses on the right side of eqn (11.47) For large solute particles,

there should be no slip, so β
→ ∞ and the factor in parentheses tends

to one, as expected Equation (11.48) then reduces to8

The most important feature of eqns (11.48), (11.49a), and (11.49b)

is that, so long as the solute is dilute, the primary determinant of the

groupDµ T is the size of the diffusing species, with a secondary

depen-dence on intermolecular forces (i.e., on β) More complex theories, such

8 Equation ( 11.49a ) was first presented by Einstein in May 1905 The more general

form, eqn ( 11.48 ), was presented independently by Sutherland in June 1905

Equa-tions ( 11.48 ) and ( 11.49a) are commonly called the Stokes-Einstein equation, although

Stokes had no hand in applying eqn ( 11.47 ) to diffusion It might therefore be argued

that eqn ( 11.48) should be called the Sutherland-Einstein equation.

Table 11.4 Molal specific volumes and latent heats of

vapor-ization for selected substances at their normal boiling points

Substance V m (m3/kmol) h fg (MJ/kmol)

solute-solvent pair, with the only exception occuring in very high ity solutions Thus, most correlations of experimental data have usedsome form of eqn (11.48) as a starting point

viscos-Many such correlations have been developed One fairly successfulcorrelation is due to King et al [11.21] They expressed the molecular size

in terms of molal volumes at the normal boiling point, V m,A and V m,B, andaccounted for intermolecular association forces using the latent heats of

Figure 11.7 Comparison of liquid diffusion coefficients

pre-dicted by eqn (11.50) with experimental values for assorted

which has an rms error of 19.5% and for which the units ofD AB µ B /T are

kg·m/K·s2 Values of h fg and V mare given for various substances in

Ta-ble11.4 Equation (11.50) is valid for nonelectrolytes at high dilution, and

it appears to be satisfactory for both polar and nonpolar substances The

difficulties with polar solvents of high viscosity led the authors to limit

eqn (11.50) to values ofDµ/T < 1.5×10 −14kg·m/K·s2 The predictions

of eqn (11.50) are compared with experimental data in Fig.11.7 Reid et

al [11.4] review several other liquid-phase correlations and provide an

assessment of their accuracies

The thermal conductivity and viscosity of dilute gases

In any convective mass transfer problem, we must know the viscosity ofthe fluid and, if heat is also being transferred, we must also know its

thermal conductivity Accordingly, we now consider the calculation of µ and k for mixtures of gases.

Two of the most important results of the kinetic theory of gases are

the predictions of µ and k for a pure, monatomic gas of species A:

µ A =2.6693 × 10 −6 3M A T

σ A2Ωµ

(11.51)and

in kelvin, and σ A again has units of Å

The equation for µ A applies equally well to polyatomic gases, but

k A must be corrected to account for internal modes of energy storage—chiefly molecular rotation and vibration Eucken (see, e.g., [11.9]) gave asimple analysis showing that this correction was

k =



9γ − 5 4γ



for an ideal gas, where γ ≡ c p /c v You may recall from your

thermo-dynamics courses that γ is 5/3 for monatomic gases, 7/5 for diatomic

gases at modest temperatures, and approaches unity for very complexmolecules Equation (11.53) should be used with tabulated data for c p;

on average, it will underpredict k by perhaps 10 to 20% [11.4]

An approximate formula for µ for multicomponent gas mixtures was

developed by Wilke [11.22], based on the kinetic theory of gases He troduced certain simplifying assumptions and obtained, for the mixtureviscosity,

#

j =1 x j φ ij

(11.54)

The analogous equation for the thermal conductivity of mixtures was

developed by Mason and Saxena [11.23]:

#

j =1

x j φ ij

(11.55)

(We have followed [11.4] in omitting a minor empirical correction factor

proposed by Mason and Saxena.)

Equation (11.54) is accurate to about 2 % and eqn (11.55) to about 4%

for mixtures of nonpolar gases For higher accuracy or for mixtures with

polar components, refer to [11.4] and [11.13]

Example 11.4

Compute the transport properties of normal air at 300 K

Solution. The mass composition of air was given in Example11.1

Using the methods of Example11.1, we obtain the mole fractions as

xN 2 = 0.7808, xO 2 = 0.2095, and xAr= 0.0093.

We first compute µ and k for the three species to illustrate the use

of eqns (11.51) to (11.53), although we could simply use tabled data

in eqns (11.54) and (11.55) From Tables11.2and11.3, we obtain

Species µcalc(kg/m ·s) µdata(kg/m ·s)

N2 1.767 × 10 −5 1.80 × 10 −5

O2 2.059 × 10 −5 2.07 × 10 −5

Ar 2.281 × 10 −5 2.29 × 10 −5

where we show data from AppendixA(TableA.6) for comparison We

then read c p from AppendixAand use eqn (11.52) and (11.53) to getthe thermal conductivities of the components:

Species c p (J/kg ·K) kcalc(W/m ·K) kdata(W/m ·K)

To compute µ m and k m, we use eqns (11.54) and (11.55) and the

tabulated values of µ and k Identifying N2, O2, and Ar as species 1,

2, and 3, we get

φ12= 0.9894, φ21= 1.010

φ13= 1.043, φ31= 0.9445

φ23= 1.058, φ32= 0.9391 and φ ii = 1 The sums appearing in the denominators are

When they are substituted in eqns (11.54) and (11.55), these valuesgive

µ m,calc = 1.861 × 10 −5 kg/m ·s, µ m,data = 1.857 × 10 −5 kg/m ·s

k m,calc = 0.02596 W/m·K, k m,data = 0.02623 W/m·K

so the mixture values are also predicted within 0.3 and 1.0%, tively

respec-Finally, we need c p m to compute the Prandtl number of the

mix-ture This is merely the mass weighted average of c p, or #

i m i c p i,

and it is equal to 1006 J/kg ·K Then

Pr= (µc p /k) m = (1.861 × 10 −5 )(1006)/0.02596 = 0.721.

This is 1% above the tabled value of 0.713 The reader may wish to

compare these values with those obtained directly using the values

for air in Table 11.2 or to explore the effects of neglecting argon in

the preceding calculations

11.5 The equation of species conservation

Conservation of species

Just as we formed an equation of energy conservation in Chapter 6, we

now form an equation of species conservation that applies to each

sub-stance in a mixture In addition to accounting for the convection and

diffusion of each species, we must allow the possibility that a species

may be created or destroyed by chemical reactions occuring in the bulk

medium (so-called homogeneous reactions) Reactions on surfaces

sur-rounding the medium (heterogeneous reactions) must be accounted for

in the boundary conditions

We consider, in the usual way, an arbitrary control volume, R, with a

boundary, S, as shown in Fig.11.8 The control volume is fixed in space,

with fluid moving through it Species i may accumulate in R, it may travel

in and out of R by bulk convection or by diffusion, and it may be created

within R by homogeneous reactions The rate of creation of species i is

denoted as ˙r i (kg/m3·s); and, because chemical reactions conserve mass,

the net mass creation is ˙r =#r˙i = 0 The rate of change of the mass of

species i in R is then described by the following balance:

Figure 11.8 Control volume in a

fluid-flow and mass-diffusion field

This species conservation statement is identical to our energy tion statement, eqn (6.36) on page293, except that mass of species i has

conserva-taken the place of energy and heat

We may convert the surface integrals to volume integrals using Gauss’stheorem [eqn (2.8)] and rearrange the result to find:

∂ρ i

∂t + ∇ · (ρ i v)
= −∇ ·
j i + ˙ r i (11.58)

We may obtain a mass conservation equation for the entire mixture bysumming eqn (11.58) over all species and applying eqns (11.1), (11.17),and (11.22) and the requirement that there be no net creation of mass:

so that

∂ρ

∂t + ∇ · (ρ
v) = 0 (11.59)

This equation applies to any mixture, including those with varying

den-sity (see Problem6.36)

Incompressible mixtures For an incompressible mixture, ∇ ·
v = 0

(see Sect.6.2or Problem11.22), and the second term in eqn (11.58) may

therefore be rewritten as

∇ · (ρ i v)
=
v · ∇ρ i + ρ i ∇ ·

 v

=0

=
v · ∇ρ i (11.60)

We may compare the resulting, incompressible species equation to the

incompressible energy equation, eqn (6.37)

Dρ i

Dt = ∂ρ ∂t i +
v · ∇ρ i = −∇ ·
j i + ˙ r i (11.61)

ρc p DT

In these equations: the reaction term, ˙r i, is analogous to the heat

gener-ation term, ˙q; the diffusional mass flux,
j i, is analogous to the heat flux,

q; and dρ i is analogous to ρc p dT

We can use Fick’s law to eliminate
j i in eqn (11.61) The

result-ing equation may be written in different forms, dependresult-ing upon what

is assumed about the variation of the physical properties If the

prod-uct ρ D im is independent of (x, y, z)—if it is spatially uniform—then

eqn (11.61) becomes

D

Dt m i = D im ∇2m i + ˙ r i /ρ (11.62)

where the material derivative, D/Dt, is defined in eqn (6.38) If, instead,

ρ and D im are both spatially uniform, then

Dρ i

Dt = D im ∇2ρ i + ˙ r i (11.63)The equation of species conservation and its particular forms may

also be stated in molar variables, using c i or x i , N i , and J i ∗ (see

Prob-lem11.24.) Molar analysis sometimes has advantages over mass-based

analysis, as we discover in Section11.7

Figure 11.9 Absorption of ammonia into water.

Interfacial boundary conditions

We are already familiar with the general issue of boundary conditionsfrom our study of the heat equation To find a temperature distribution,

we specified temperatures or heat fluxes at the boundaries of the domain

of interest Likewise, to find a concentration distribution, we must

spec-ify the concentration or flux of species i at the boundaries of the medium

of interest

Temperature and concentration behave differently at interfaces At

an interface, temperature is the same in both media as a result of theZeroth Law of Thermodynamics Concentration, on the other hand, need

not be continuous across an interface, even in a state of thermodynamic

equilibrium Water in a drinking glass, for example, shows discontinouschanges in the concentration of water at both the glass-water interface onthe sides and the air-water interface above In another example, gaseousammonia is absorbed into water in some types of refrigeration cycles Agas mixture containing some particular mass fraction of ammonia willproduce a different mass fraction of ammonia just inside an adjacentbody of water, as shown in Fig.11.9

To characterize the conditions at an interface, we introduce

imagi-nary surfaces, s and u, very close to either side of the interface In the

ammonia absorption process, then, we have a mass fraction mNH3,s on

the gas side of the interface and a different mass fraction mNH3,u on the

liquid side

In many mass transfer problems, we must find the concentration

dis-tribution of a species in one medium given only its concentration at the

interface in the adjacent medium We might wish to find the

distribu-tion of ammonia in the body of water knowing only the concentradistribu-tion of

ammonia on the gas side of the interface We would need to find mNH3,u

from mNH3,s and the interfacial temperature and pressure, since mNH3,u

is the appropriate boundary condition for the species conservation

equa-tion in the water

Thus, for the general mass transfer boundary condition, we must

specify not only the concentration of species i in the medium adjacent

to the medium of interest but also the solubility of species i from one

medium to the other Although a detailed study of solubility and phase

equilibria is far beyond our scope (see, for example, [11.5, 11.24]), we

illustrate these concepts with the following simple solubility relations

Gas-liquid interfaces For a gas mixture in contact with a liquid mixture,

two simplified rules dictate the vapor composition When the liquid is

rich in species i, the partial pressure of species i in the gas phase, p i,

can be characterized approximately with Raoult’s law, which says that

where p sat,i is the saturation pressure of pure i at the interface

temper-ature and x i is the mole fraction of i in the liquid When the species i is

dilute in the liquid, Henry’s law applies It says that

p i = H x i for x i  1 (11.65)

where H is a temperature-dependent empirical constant that is tabulated

in the literature Figure11.10shows how the vapor pressure varies over

a liquid mixture of species i and another species, and it indicates the

regions of validity of Raoult’s and Henry’s laws For example, when x iis

near one, Raoult’s law applies to species i; when x iis near zero, Raoult’s

law applies to the other species

If the vapor pressure were to obey Raoult’s law over the entire range of

liquid composition, we would have what is called an ideal solution When

x i is much below unity, the ideal solution approximation is usually very

poor

tabulated values of µ and k Identifying N2, O2, and Ar as species...

conserva-taken the place of energy and heat

We may convert the surface integrals to volume integrals using Gauss’stheorem [eqn (2.8)] and rearrange the result to find:

∂ρ... conservation equation for the entire mixture bysumming eqn (11.58) over all species and applying eqns (11.1), (11.17) ,and (11.22) and the requirement that there be no net creation of mass:

so that