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

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516.6

Radiant Energy Transport in Absorbing Media

507

tive term in Eq. 16.6-2, since the photons move independently of the local material veloc-

ity. Note further that the term

appears with opposite signs in Eqs. 16.6-1 and

indicating that a net gain of radiant energy occurs at the expense of molecular energy.

Equation 16.6-2 can also be written for the radiant energy within a frequency range

dv:

This expression is obtained by differentiating Eq. 16.6-2 with respect to

For the purpose of simplifying the discussion, we consider

steady-state nonflow

system in which the radiation travels only in the positive

direction. Such a system can

be closely approximated by passing a collimated light beam through a solution at tem-

peratures sufficiently low that the emission by the solution is unimportant. (If emissions

were important, it would be necessary to consider radiation in all directions.) These are

the conditions commonly encountered in spectrophotometry. For such a system, Eqs.

16.6-1 and 2 become

In order to use these equations, we need information about the volumetric absorption

rate

For a unidirectional beam a conventional expression is

in which

is known as the extinction coeficient. Basically, this states that the rate of pho-

ton absorption is proportional to the concentration of photons.

A monochromatic radiant beam of frequency

focused parallel to the z-axis, passes through

an absorbing fluid. The local rate of energy absorption is given by mad:), in which

m,,

the

Absorption

extinction coefficient for radiation of frequency v. Determine the distribution of the radiant

Monochromatic

flux q!)(z) in the system.

Radiant Beam

SOLUTION

We neglect refraction and scattering of the incident beam. Also, we assume that the liquid is

cooled so that re-radiation can be neglected. Then Eq. 16.6-5 becomes for steady state

Integration with respect to z gives

This is Lambert's law of absorption: widely used in spectrophotometry. For any given pure ma-

terial,

m,,

depends in a characteristic way on

The shape of the absorption spectrum is there-

fore a useful tool for qualitative analysis.

Lambert,

Photometria,

Augsburg

(1760).

508

Chapter 16 Energy Transport by Radiation

QUESTIONS FOR DISCUSSION

PROBLEMS

The "named laws" in this chapter are important. What is the physical content of the laws as-

sociated with the following scientists' names: Stefan and Boltzmann, Planck, Kirchhoff, Lam-

bert, Wien?

How are the Stefan-Boltzmann law and the Wien displacement law related to the Planck

black-body distribution law?

Do black bodies exist? Why is the concept of a black body useful?

In specular (mirrorlike) reflection, the angle of incidence equals the angle of reflection. How

are these angles related for diffuse reflection?

What is the physical significance of the view factor, and how can it be calculated?

What are the units of q'",

qt',

and qf)?

Under what conditions is the effect of geometry on radiant heat interchange completely ex-

pressible in terms of view factors?

Which of the equations in this chapter show that the apparent brightness of a black body with

a uniform surface temperature is independent of the position (distance and direction) from

which it is viewed through a transparent medium?

What relation is analogous to Eq. 16.3-2 for an ideal monatomic gas?

Check the dimensional consistency of Eq. 16.3-9.

16A.1. Approximation of a black body by a hole in

sphere. A thin sphere of copper, with its internal surface

highly oxidized, has a diameter of 6 in. How small a hole

must be made in the sphere to make an opening that will

have an absorptivity of 0.99?

Answer:

Radius

0.70 in.

16A.2. Efficiency of a solar engine.

device for utilizing

solar energy, developed by ~bbot,' consists of a parabolic

mirror that focuses the impinging sunlight onto a Pyrex tube

containing a high-boiling, nearly black liquid. This liquid is

circulated to a heat exchanger in which the heat energy is

transferred to superheated water at 25 atm pressure. Steam

may be withdrawn and used to run an engine. The most effi-

cient design requires a mirror 10 ft in diameter to generate

2 hp, when the axis of the mirror is pointed directly toward

the sun. What

the overall efficiency of the device?

Answer:

15%

16A.3.

Radiant heating requirement. A shed is rectangu-

lar in shape, with the floor 15 ft by 30 ft and the roof 7.5 ft

above the floor. The floor is heated by hot water running

through coils. On cold winter days the exterior walls and

roof are about -10°F. At what rate must heat be supplied

through the floor in order to maintain the floor temperature

at 75"F? (Assume that all surfaces of the system are black.)

16A.4. Steady-state temperature of a roof. Estimate the

maximum temperature attained by a level roof at

45"

north

latitude on June 21 in clear weather. Radiation from sources

Abbot, in

Solar Energy Research

(F.

Daniels and

Duffie, eds.), University of Wisconsin Press, Madison (1955),

pp.

91-95; see also

U.S.

Patent No.

2,460,482

(Feb.

1,1945).

other than the sun may be neglected, and a convection heat

transfer coefficient of 2.0 Btu/hr

fi?

may be assumed.

maximum temperature of 100°F may be assumed for the

surrounding air. The solar constant of Example 16.4-1 may

be used, and the absorption and scattering of the sun's rays

by the atmosphere may be neglected.

(a) Solve for a perfectly black roof.

(b)

Solve for an aluminum-coated roof, with an absorptiv-

ity of 0.3 for solar radiation and an ernissivity of 0.07 at the

temperature of the roof.

16A.5. Radiation errors in temperature measurements.

The temperature of an air stream in a duct is being mea-

sured by means of a thermocouple. The thermocouple

wires and junction are cylindrical, 0.05 in. in diameter, and

extend across the duct perpendicular to the flow with the

junction in the center of the duct. Assuming a junction emis-

sivity

0.8, estimate the temperature of the gas stream

from the following data obtained under steady conditions:

Thermocouple junction temperature

500°F

Duct wall temperature

300°F

Convection heat transfer coefficient

from wire to air

50 Btu/hr.

fi?

The wall temperature is constant at the value given for 20

duct diameters upstream and downstream of the thermo-

couple installation. The thermocouple leads are positioned

so that the effect of heat conduction along them on the

junction temperature may be neglected.

16A.6. Surface temperatures on the Earth's moon.

(a) Estimate the surface temperature of our moon at the

point nearest the sun by a quasi-steady-state radiant en-

ergy balance, regarding the lunar surface as a gray sphere.

Problems

509

Neglect radiation and reflection from the planets. The

solar constant at Earth is given in Example 16.4-1.

(b)

Extend part (a) to give the lunar surface temperature as

function of angular displacement from the hottest point.

16B.1. Reference temperature for effective emissivity.

Show that, if the emissivity increases linearly with the tem-

perature,

Eq.

16.5-3 may be written as

Q12

e?dl(G

2-3

(16B.1-1)

in which ey is the emissivity of surface 1 evaluated at a ref-

erence temperature

given by

16B.2. Radiation across an annular gap. Develop an ex-

pression for the radiant heat transfer between two long,

gray coaxial cylinders

and

Show that

where

is the surface area of the inner cylinder.

16B.3. Multiple radiation shields.

(a)

Develop an equation for the rate of radiant heat transfer

through a series of

very thin, flat, parallel metal sheets,

each having a different emissivity e, when the first sheet is

at temperature T, and the nth sheet is at temperature

T,.

Give your result in terms of the radiation resistances

for the successive pairs of planes. Edge effects and conduc-

tion across the air gaps between the sheets are to be ne-

glected.

(b) Determine the ratio of the radiant heat transfer rate for

identical sheets to that for two identical sheets.

(c)

Compare your results for three sheets with that ob-

tained in Example 16.5-1.

The marked reduction in heat transfer rates produced

a number of radiation shields in series has led to the use

multiple layers of metal foils for high-temperature insu-

lation.

16B.4. Radiation and conduction through absorbing

media. A glass slab, bounded by planes

0 and

infinite extent in the

and

directions. The tempera-

tures of the surfaces at

0 and

are maintained at To

and T,, respectively. A uniform monochromatic radiant

beam of intensity

q('

in the

direction impinges on the face

Emission within the slab, reflection, and incident

radiation in the negative

direction can be neglected.

(a)

Determine the temperature distribution in the slab, as-

suming

and

to be constants.

(b) How does the distribution of the conductive heat flux

depend on

m,?

16B.5.

Cooling of a black body in vacuo.

thin black

body of very high thermal conductivity ha: a volume

surface area

density

and heat capacity

C,.

At time

this body at temperature TI is placed in a black enclo-

sure, the walls of which are maintained permanently at

temperature T2 (with T2

TI).

Derive an expression for the

temperature T of the black body as a function of time.

16B.6. Heat loss from an insulated pipe.

Schedule 40

two-inch horizontal steel pipe (inside diameter 2.067 in.,

wall thickness 0.154 in.;

26 Btu/hr

. F)

carrying

steam is insulated with

in. of 85% magnesia

0.35

Btu/hr

and tightly wrapped with a layer of clean

aluminum foil

0.05).

The pipe is surrounded by air at

atm and 80°F and its inner surface is at 250°F.

(a) Compute the conductive heat flow per unit length,

Q"""~'/L, through the pipe wall and insulation for as-

sumed temperatures, To, of 100°F and 250°F at the outer

surface of the aluminum foil.

(b) Compute the radiative and free-convective heat losses,

Q(~~~)/L

and Q""""~/L, for the same assumed outer surface

temperatures

To.

(c)

Plot or interpolate the foregoing results to obtain the

steady-state values of

and Q"""~'/L

Q(rad)/L

Q(~O""' /L.

16C.1.

Integration of the view-factor integral for a pair

of disks (Fig. 16C.1). Two parallel, perfectly black disks of

radius

are placed a distance H apart. Evaluate the view-

factor integrals for this case and show that

in which

R/H.

I I

Fig. 16.C-1. Two perfectly

+R+

black disks.

16D.1. Heat loss from a wire carrying an electric cur-

rent.3 An electrically heated wire of length

loses heat to

the surroundings by radiative heat transfer. If the ends of

the wire are maintained at a constant temperature To, ob-

tain an expression for the axial variation in wire tempera-

ture. The wire can be considered to be radiating into a

black enclosure at temperature To.

Christiansen,

Wiedernann's

Ann.

Physik,

19,267-283

(1883);

see also

Jakob,

Heat

Transfer,

Vol.

11,

Wiley, New

York

(19571,

14.

Carslaw and

Jaeger,

Conduction

Heat

Solids,

2nd edition, Oxford University Press (1959),

pp.

154-156.

This Page Intentionally Left Blank

Part

Three

Mass

Transport

This Page Intentionally Left Blank

Chapter

Diffusivity

and

the Mechanisms

Mass Transport

Fick's law of binary diffusion (Molecular Mass Transport)

Temperature and pressure dependence of diffusivities

Theory of diffusion in gases at low density

Theory of diffusion in binary liquids

Theory of diffusion in colloidal suspensions

Theory of diffusion of polymers

Mass and molar transport by convection

Summary of mass and molar fluxes

The Maxwell-Stefan equations for multicomponent diffusion in

gases at low density

In Chapter

we began by stating Newton's law of viscosity, and in Chapter

we began

with Fourier's law of heat conduction. In this chapter we start by giving Fick's law of dif-

fusion, which describes the movement of one chemical species

through a binary mix-

ture of

and

because of a concentration gradient of

The movement of a chemical species from a region of high concentration to a region

of low concentration can be observed by dropping a small crystal of potassium perman-

ganate into a beaker of water. The KMnO, begins to dissolve in the water, and very near

the crystal there is a dark purple, concentrated solution of KMnO,. Because of the con-

centration gradient that is established, the KMnO, diffuses away from the crystal. The

progress of the diffusion can then be followed by observing the growth of the dark pur-

ple region.

517.1

we give Fick's law for binary diffusion and define the diffusivity

BAB

for the

pair

A-B.

Then we discuss briefly the temperature and pressure dependence of the diffu-

sivity. After that we give a summary of the theories available to predict the diffusivity

for gases, liquids, colloids, and polymers. At the end of the chapter we discuss the trans-

port of mass of a chemical species by convection, thus paralleling the treatments in

Chapters

and

for momentum and heat transfer. We also introduce molar units and

the notation needed for describing diffusion in these units. Finally, we give the

Maxwell-Stefan equations for multicomponent gases at low densities.

Before starting the discussion we establish the following conventions. For multicom-

ponent diffusion, we designate the species with lower-case Greek letters

. . .

and

their concentrations with the corresponding subscripts. For

bina

diffusion

we use the

capital italic letters

and

For self-difusion (diffusion of chemically identical species)

514

Chapter 17 Diffusivity and the Mechanisms of Mass Transport

we label the species

and

A*.

The "tagged" species

may differ physically from

virtue of radioactivity or other nuclear properties such as the mass, magnetic moment,

spin.' The use of this system of notation enables one to see at a glance the type of system

to which a given formula applies.

517.1

FICK'S LAW OF BINARY DIFFUSION

(MOLECULAR MASS TRANSPORT)

Consider a thin, horizontal, fused-silica plate of area

and thickness

Suppose that ini-

tially (for time t

both horizontal surfaces of the plate are in contact with air, which

we regard as completely insoluble in silica. At time t

the air below the plate is sud-

denly replaced by pure helium, which is appreciably soluble in silica. The helium slowly

penetrates into the plate by virtue of its molecular motion and ultimately appears in the

gas above. This molecular transport of one substance relative to another is known as

dif-

fusion (also known as mass diffusion, concentration diffusion, or ordinary diffusion). The air

above the plate is being replaced rapidly, so that there is no appreciable buildup of he-

lium there. We thus have the situation represented

Fig. 17.1-1; this process is analo-

gous to those described in Fig. 1.1-1 and Fig. 9.1-1 where viscosity and thermal

conductivity were defined.

In this system, we will call helium "species

and silica "species

B."

The concentra-

tions will be given by the "mass fractions"

and

w,.

The mass fraction

is the mass

helium divided by the mass of helium plus silica in a given microscopic volume element.

The mass fraction

is defined analogously.

Thickness

slab

fused silica

(substance

Fig.

17.1-1.

Build-up to the

steady-state concentration pro-

file for the diffusion of helium

(substance

through fused sil-

ica (substance B). The symbol

stands for the mass fraction

helium, and

w,,

is the solubility

of helium in fused silica, ex-

pressed as the mass fraction.

See

Figs. 1.1-1 and 9.1-1 for the anal-

ogous momentum and heat

transport situations.

Large

O,=O UA

W~~

Stejskal and

Tanner,

Chem. Phys.,

42,288-292 (1965);

Stilbs,

Puog.

NMR

Spectuos,

19,

145 (1987);

T. Callaghan and

StepiSnik,

Adv.

Map. Opt. Reson.

19,325388 (1996).

g17.1

Fick's Law of Binary Diffusion (Molecular Mass Transport)

515

For time

less than zero, the mass fraction of helium,

w,,

is everywhere equal to

zero. For time

greater than zero, at the lower surface,

the mass fraction of helium

is equal to

w,,.

This latter quantity is the solubility of helium in silica, expressed as mass

fraction, just inside the solid. As time proceeds the mass fraction profile develops, with

w,,

at the bottom surface of the plate and

at the top surface of the plate. As

indicated in Fig. 17.1-1, the profile tends toward a straight line with increasing

At steady state, it is found that the mass flow w,, of helium in the positive

direc-

tion can be described to a very good approximation by

That is, the mass flow rate of helium per unit area (or

mass flux)

is proportional to the

mass fraction difference divided by the plate thickness. Here

is the density of the sil-

ica-helium system, and the proportionality factor

9lAB

is the

difusivity

of the silica-he-

lium system. We now rewrite

Eq.

17.1-1 for a differential element within the slab:

Here wA,/A has been replaced by jAy, the

molecular mass flux

of helium in the positive

direction. Note that the first index,

designates the chemical species (in this case, he-

lium), and the second index indicates the direction in which diffusive transport is taking

place (in this case, the

direction).

Equation 17.1-2 is the one-dimensional form of

Fick's

first

law

diffusion.'

It is valid

for any binary solid, liquid, or gas solution, provided that jAy is defined as the mass flux

relative to the mixture velocity

v,.

For the system examined in Fig. 17.1-1, the helium is

moving rather slowly and its concentration is very small, so that

is negligibly different

from zero during the diffusion process.

In general, for a binary mixture

Thus v is an average in which the species velocities, v, and

v,,

are weighted according to

the mass fractions.

This

kind of velocity is referred to as the

mass average velocity.

The species

velocity

is not the instantaneous molecular velocity of a molecule of

but rather the

arithmetic average of the velocities of all the molecules of

within a tiny volume element.

The mass flux j,, is then defined, in general, as

The mass flux of

is defined analogously. As the two chemical species interdiffuse there

is, locally,

shifting of the center of mass in the

direction if the molecular weights of

and

differ. The mass fluxes j4 and

jBy

are so defined that jAy

jBy

In other words,

the fluxes jA, and jBy are measured with respect to the motion of the center of mass. This

point will be discussed in detail in ss17.7 and

If we write equations similar to

Eq.

17.1-2 for the

and

directions and then com-

bine all three equations, we get the vector form of Fick's law:

Fick,

Ann.

der

Physik,

94/59-86

(1855).

Fick's second law, the diffusional analog of the heat

conduction equation in

Eq.

11.2-10,

is given in

Eq.

19.1-18.

Adolf

Eugen

Fick

(1829-1901)

was a medical

doctor who taught in Ziirich and Marburg, and later became the Rector of

the

University of Wiirzburg.

He postulated the laws of diffusion by analogy with heat conduction, not by experiment.

516

Chapter 17 Diffusivity and the Mechanisms of Mass Transport

similar relation can be written for species

P~BA~WB

(17.1-6)

It is shown in Example 17.1-2 that

9,,

9,,. Thus for the pair

A-B,

there is just one dif-

fusivity; in general it will be a function of pressure, temperatu~e, and composition.

The mass diffusivity 9,,, the thermal diffusivity

k/pC,, and the momentum dif-

fusivity (kinematic viscosity)

p/p all have dimensions of (length)'/time. The ratios

of these three quantities are therefore dimensionless groups:

cpEl.

The Prandtl number: Pr=-=-

(17.1-7)

The Schmidt number:'

El.

sc=-=-

p9Afl

The Lewis number:'

These dimensionless groups of fluid properties play a prominent role in dimensionless

equations for systems in which competing transport processes occur. (Note: Sometimes

the Lewis number is defined as the inverse of the expression above.)

In Tables 17.1-1,2,3, and

some values of

9,,

in cm2/s are given for a few gas, liq-

uid, solid, and polymeric systems. These values can be converted easily to m2/s by

mul-

tiplication by

lop4.

Diffusivities of gases at low density are almost independent of

w,,

increase with temperature, and vary inversely with pressure. Liquid and solid diffusivi-

ties are strongly concentration-dependent and generally increase with temperature.

There are numerous experimental methods for measuring diffusivities, and some of

these are described in subsequent

chapter^.^

For gas mixtures, the Schmidt number can range from about 0.2 to

as can be seen in

Table 17.1-1. For liquid mixtures, values up to 40,000 have been ~bserved.~

Up to this point we have been discussing isotropic fluids, in which the speed of dif-

fusion does not depend on the orientation of the fluid mixture. For some solids and

structured fluids, the diffusivity will have to be a tensor rather than a scalar, so that

Fick's first law has to be modified thus:

in which AAD is the (symmetric) dimsivity te~sor.~,~ According to this equation, the mass

flux is not necessarily collinear with the mass fraction gradient. We do not pursue this

subject further here.

These groups were named for: Ernst Heinrich Wilhelm Schmidt (1892-1975), who taught at

the universities in Gdansk, Braunschweig, and Munich (where he was the successor to Nusselt);

Warren Kendall Lewis (1882-1975), who taught at

MIT

and was a coauthor of a pioneering textbook,

W. H. Walker, W.

Lewis, and W.

McAdams,

Principles of Chemical Engineering,

McGraw-Hill,

New York (1923).

For an extensive discussion, see W.

Wakeham, A. Nagashima, and

Sengers,

Measurement

of the Transport Properties of Fluids: Experimental Thermodynamics, Vol.

IlI,

CRC Press, Boca Raton, Fla.

(1991).

A. Shaw and

Hanratty,

AIChE Journal,

23,28-37,160-169 (1977);

Harriott and

Hamilton,

Chem. Eng. Sci.,

20,1073-1078 (1965).

For flowing polymers, theoretical expressions for the diffusion tensor have been derived using

kinetic theory; see

Ottinger,

AIChE Journal,

35,279-286 (1989), and

F. Curtiss and

B. Bird,

Adv. Polym. Sci.,

1-101 (1996), §§6 and

15.

Glicksman,

Diffusion in Solids: Field Theory, Solid State Principles, and Applications,

Wiley,

New York (2000).