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

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523.6

Use of the Macroscopic Balances to Solve Unsteady-State Problems

757

We begin by integrating Eq. 23.6-33, for

0, over the entire volume of the flow system

between planes

and 2:

The volume integrals may be converted into surface integrals by using the Gauss divergence

theorem to get

where S

SZ. The integral over the fixed surface Sf is zero according to Eq. 23.6-27

and 28, and the integrals over the inlet and outlet planes

and

can be simplified, so that

we get

Here we have made use of Eq. 23.6-29 to drop the diffusive terms at planes

and

If we as-

sume that

to'

is constant over a cross section, we may remove it from the integral, and then

we get

For an incompressible fluid, the volume rate of flow,

is constant, so that (u,)S,

(v&, and

That is,

I"'

evaluated at plane

is the same as

Po'

at plane 2, and at every point in the system.

It is standard notation to abbreviate this quantity as Mo, the zeroth (absolute) moment. Equa-

tion 23.6-38 is analogous to Eq. 23.1-2 for a steady-state system with no mass transport across

the walls. Next we evaluate

for the introduction of a mass

of tracer over a time interval

that is very small with respect to the mean tracer residence time

t,,,

V/Q:

The replacement, in the second step, of the upper limit by Q/V is permitted by the finite du-

ration of the tracer pulse. From the last two equations we then get

This provides the possibility of measuring blood flow rate from the mass of an injected tracer

and the value of

Go'

M,.

The latter can be obtained

means of a catheter inserted into the

blood vessel or by NMR techniques.

This simple formula6 was first introduced in 1829 and has been extensively used since

1897 for measurement of blood-flow rates,7 including cardiac output.' It is also widely used for

many environmental systems, such as rivers, and also for systems in the process industries.

Next we turn to Eq. 23.6-32, and integrate it over the volume of the flow system, once

again making use of the fact that the diffusive tern over the inlet and outlet is much smaller

than the convective term. This gives

Hering,

Zeits,

Physik,

3,85-126 (1829).

Stewart,

Physiol.

(London), 22,159-183 (1897).

Zierler,

Ann.

Biomed.

Eng.,

28,836-848 (2000).

758

Chapter 23 Macroscopic Balances for Multicomponent Systems

or, if

I'"'

is assumed constant over a cross-section,

-n(+

~')(r,

t)~)

((v~)s~$)

(v~)s~@))

(23.6-42)

Then, defining the quantity in parentheses on the left side as the volume average, we get

finally3

Now, if we set

1, we get the following:

If the tracer is injected as a

delta-function input,

so that

I',"

we can use the notation

I':'

(the first moment), and the last equation becomes

This result, used in conjunction with Eq. 23.6-40, has long been applied by cardiologists for

de-

termining blood volume. It has since found many other environmental and process-industry

calculations.

Higher moments have also proven useful, in particular the central moments

These are commonly applied for the special case of an impulsive tracer input. Then the nor-

malized second central moment, or variance, is

This is the square of the standard deviation, when the exit tracer profile is a Gaussian distrib-

ution. The third central moment is a measure of the asymmetry about ire,, and the fourth

measure of the kurtosis. In practice, the fourth moment is nearly impossible to determine

ac-

curately from experimental data, and obtaining even the third proves to be quite difficult.

Use of the second moment has found some very important applications in studying

tracer dynamics of biological tissue? and again the large literature in the medical field

has

been extended to many other applications. It is also interesting to note the additivity relation-

ships in serially connected systems. Thus

is invariant to the number of included subsys-

tems, and

M,,

p2,

and

are additive, but higher-order moments are not.

QUESTIONS

FOR

DISCUSSION

How are the macroscopic balances for multicomponent mixtures derived? How are they

re-

lated to the equations of change?

In Eq. 23.1-1, how are homogeneous and heterogeneous reactions accounted for? What is the

physical meaning of

w,,?

Give a specific example of a system in which the last term in Eq. 23.1-4 is zero.

In using Table

23.5-1

one normally specifies the directions of the streams (that is, whether they

are input or output streams). How could one proceed if the flow directions change with time?

Chinard,

Ann.

Biomed

Eng.,

28,849-859

(2000).

Problems

759

Summarize the calculation procedures for the enthalpy per unit mass,

in Eq. 23.3-1

and the partial molar enthalpy in Eq. 23.3-la. What are these quantities for ideal gas mixtures?

Review the derivation of the mechanical energy balance in s7.8. What would have to be

changed in that derivation, if one wishes to apply it to a nonisothermal, reacting mixture in a

flow system with no mass transfer surfaces?

To what extent does this chapter provide the background for studying unit operations, such

as absorption, extraction, distillation, and crystallization?

What changes would have to be made in this chapter to describe processes in a space ship or

on the surface of the moon?

PROBLEMS

23A.1. Expansion of a gas mixture: very slow reaction rate.

Estimate the temperature and velocity

of the water-gas mixture at the discharge end of the nozzle in Example

23.5-4

if the reaction

rate

is very slow.

Use the following data: loglo

-0.15, CprHz

7.217, Cp,CO,

12.995,

Cp,HZO

9.861, Cp,,,

7.932 (all heat capacities are in Btu/lb-mole

Is the nozzle exit

pressure equal to the ambient pressure?

Answers:

920K, 1726 ft/s; yes, the nozzle flow is subsonic.

23A.2. Height of a packed-tower absorber.

packed tower of the type described in Example 23.5-2

is to be used for removing 90% of the cyclohexane from a cyclohexane-air mixture by absorp-

tion into a nonvolatile light oil. The gas stream enters the bottom of the tower at a volumetric

rate of 363 ft3/min, at 30°C, and at 1.05 atm pressure. It contains 1% cyclohexane by volume.

The oil enters the top of the tower at a rate of 20 lb-mol/hr, also at 30°C, and it contains 0.3%

cyclohexane on a molar basis. The vapor pressure of cyclohexane at

30°C

is 121 mm Hg, and

solutions of it in the oil may be considered to follow Raoult's law.

(a)

Construct the operating line for the column.

(b)

Construct an equilibrium curve for the range of operation encountered here. Assume the

operation to be isothermal and isobaric.

(c)

Determine the interfacial conditions at each end of the column.

(d)

Determine the required tower height using Eq. 23.5-24 if

k!a

0.32 moles/hr

ft3,

kia

14.2 moles/hr

ft3, and the tower cross section

is 2.00

fi?.

(e)

Repeat part (d), using Eq. 23.5-25.

Answer:

(d)

ca. 62 ft;

(el

60 ft

23B.1. Effective average driving forces in a gas absorber.

Consider a packed-tower gas absorber of

the type discussed in Example 23.5-2. Assume that the solute concentration is always low and

that the equilibrium and operating lines are both very nearly straight. Under these conditions,

both

k;a

and

k:a

may be considered constant over the mass-transfer surface.

(a)

Show that

(YA

Y,,)

varies linearly with

YA.

Note that Y, is the bulk mole ratio of

the gas phase and

Y,,

is the equilibrium gas-phase mole ratio over a liquid of bulk composi-

tion

(see Fig. 22.4-2).

(b)

Repeat part (a) for

(Y,

YAO).

(c)

Use the results of parts (a) and (b) to show that

The overall mass transfer coefficient

is defined by Eq. 22.4-4. Note that this part of the

problem may be solved by analogy with the development in Example 15.4-1.

23B.2. Expansion of a gas mixture: very fast reaction rate.

Estimate the temperature and velocity of

the water-gas mixture at the discharge end of the nozzle in Example 23.5-4 if the reaction rate

may be considered infinitely fast. Use the data supplied in Problem 23A.1 as well as

the following: at 900K, loglo

-0.34;

3H2

+6340; HHz0(@

-49,378;

-16,636;

760

Chapter 23 Macroscopic Balances for Multicomponent Systems

ho,

-83,242 (all enthalpies are given in cal/g-mole). For simplicity, neglect the effect of

temperature on heat capacity, and assume that loglo

varies linearly with temperature

between 900 and 1000K. The following simplified procedure is recommended:

(a) It may be seen in advance that

will be higher than for slow reaction rates, and hence

greater than 920K (see Problem 23A.1). Show that, over the temperature range to be encoun-

tersd,

varies very nearly linearly with the temperature according to the expression

(dH/dT',,,

12.40 cal/gm-mol

(b)

Substitute the result in

(a)

into Eq. 23.5-41 to show that

937K.

(c)

Calculate

and

and show by use of Eq. 23.5-29 that

1750 ft/s.

Startup of a chemical reactor.

(a) Integrate Eq. 23.6-5 along with the given initial conditions to show that Eq. 23.6-8 cor-

rectly describes

MB,tot

as a function of time.

(b) Show that

and

in Eq. 23.6-9 are real and negative.

Hint:

Show that

(c)

Obtain expressions for

MA,t,,

and

Mc,tot

as functions of time.

Irreversible first-order reaction in a continuous reactor.

well-stirred reactor of volume

is initially completely filled with a solution of solute

in a solvent

at concentration

cAo.

time

0, an identical solution of

is introduced at a constant mass flow rate w. A small

constant stream of dissolved catalyst is introduced at the same time, causing

to disappear

according to an irreversible first-order reaction with rate constant

sec-'. The rate constant

may be assumed independent of composition and time. Show that the concentration of

the reactor (assumed isothermal) at any time is

in which

t;'

[(w/pV)

ky].

Mass and enthalpy balances in

adiabatic splitter. One hundred pounds of 40% by mass of

superheated aqueous ammonia with a specific enthalpy of 420 Btu/lb is to be flashed adiabati-

cally to a pressure of 10 atm. Calculate the compositions and masses of the liquid and vapor pro-

duced. For the purposes of this problem you may assume that at thermodynamic equilibrium

where

YNHi

and

XNH3

are the

mass

ratios of ammonia to water. The enthalpies of saturated liq-

uid and vapor at 10 atm may be assumed to be

Btu/lb of saturated vapor, and

Btu/lb of saturated liquid. Here

xNH3

and

yNH,

are

mass

fractions of ammonia.

Answer:

36.5 Ibs,yp

0.713,

877 Btu/lb,;

63.6 Ib,,

0.22,

157 Btu/lb,

Flow distribution in an ideal cascade. Determine the upflowing and downflowing stream

flows of individual stages for the ideal cascade described in Example 23.5-3. Express your re-

sults as fractions of the feed rate, and start from the bottom of the cascade. Use 12 stages as

the closest integer providing the desired separation. It is suggested that you begin by calculat-

ing the upflowing and downflowing stream compositions and then use the mass balances

below the feed plate and the corresponding balances above it. Use 10 stages with the bottoms

(W) composition equal to a mole fraction of 0.1.

Problems

761

23B.7.

Isotope separation and the value function. You wish to compare an existing isotope frac-

tionator that processes 50 moles/hr of a feed containing 1.0 mole% of the desired isotope to a

product of 90% purity and a waste of 10% with another that processes 50 moles/hr of

mole% material to product and waste of 95% and 2%, respectively. Which fractionation is

more effective? Assume the Dirac separative capacity to be an accurate measure of effectiveness.

Irreversible second-order reaction in an agitated tank. Consider a system similar to that dis-

cussed in Problem 23B.4, except that the solute disappears according to a second-order reac-

tion; that is,

RA,t,,

-k;'Vci.

Develop an expression for

function of time by the

following method:

(a) Use a macroscopic mass balance for the tank to obtain a differential equation describing

the evolution of

with time.

(b) Rewrite the differential equation and the accompanying initial condition in terms of the

variable

The nonlinear differential equation obtained in this way is a

Bernoulli difj'erential equation.

(c)

Now put

l/u and perform the integration. Then rewrite the result in terms of the

original variable

c,.

23C.2.

Protein purification (Fig. 23C.2). It is desired to purify a binary protein mixture using an

ideal cascade of individual ultrafiltration stages of the type shown in the figure. The larger of

the two membrane units is the source of separation and each protein flux across the mem-

brane is expressed by

where

is the transmembrane protein flux of species

is its concentration in the upstream

solution (assumed to be well mixed),

is the transmembrane superficial velocity, and

is a

protein-specific

sieving factor.

The smaller membrane unit is used solely to maintain a solvent

balance and can be ignored for the purposes of this problem.

(a) Show that the enrichment of protein 1 relative to

is given by

where

and

are the mole ratios of protein 1 to protein

in the product and waste

streams, respectively, and

a,,

S,/S2.

(b)

Determine the number of stages required in an ideal cascade to produce 99% pure protein

1 from a 90% feed in 95% yield as a function of

a,,.

It is suggested that

a,,

be varied from

200.

(c)

Calculate the output concentrations, yield, and stream flow rates for a three-stage cascade,

with

a,,

40, and with a feed of 90% purity to the middle stage.

(d)

Compare the Dirac separative capacity of this three-stage cascade with that of a single

unit with the same molar ratio of product to feed.

23C.3.

Physical significance of the zeroth and first moments. Consider some simple flow systems,

such as plug flow and well-stirred vessels, individually and in series or parallel arrange-

ments. Show that flow rates and volumes can be obtained from the moments defined in

Ex-

ample 23.6-3.

F=P+W

zF=yP+xW

return

Fig.

23C.2.

membrane-based binary splitter.

762

Chapter 23 Macroscopic Balances for Multicomponent Systems

Fig.

23C.4.

schematic

representation of a

"sandwich-type'' cross-

flow heat exchanger.

23C.4.

Analogy between the unsteady operation of an adsorption column and a cross-flow heat

exchanger1 (Fig. 23C.4). In the heat exchanger shown in the figure, the two fluid streams flow

at right angles to one another, and the heat flux parallel to the wall is neglected. Here ex-

change of heat is clearly less than for a countercurrent exchanger of the same surface and

overall heat transfer coefficient under otherwise identical conditions. The heat flow in these

exchangers may be expressed for constant Ul,, as

Here

is the total rate of heat transfer,

is the heat transfer surface area, and ATl, is the loga-

rithmic mean of (T,ll

T,,) and (T,,

Tcl), as defined in the figure. Note that Th2 and

T,,

are

the flow-averaged temperatures of the two exit streams. We may then regard

as the ratio

heat transferred in cross flow to that which would be transferred in counterflow.

Use Eq. 23.6-27 to write an expression for

as a function of the stream rates, physical

properties, heat transfer area, and localpve~all heat transfer coefficient. Express the result in

terms of definite integrals, and assume Cphr Cpcr and

U,,,

to be constant.

23D.1.

Unsteady-state operation of a packed column. Show that Eq. 23.6-25 is a valid solution

Eqs. 23.6-23 and 24. The following approach is recommended:

(a) Take the Laplace transform of Eqs. 23.6-23 and 24 with respect to

Eliminate the trans-

form of

from the resulting expressions. Show that the transform of

may be written for the

given boundary conditions as

[s/(s+l)lt

(23D.1-1)

in which

is the Laplace transform of

Nusselt,

Tech. Math. Therm.,

1,417 (1930);

Smith,

Engineering,

138,479 (1934).

Problems

763

(b) Rewrite this expression in the form

Invert this expression to obtain Eq. 23.6-25 by making use of the identity

23D.2.

Additivity of the lower moments. Consider a pair of flow systems meeting the requirements

of Example 23.6-3 arranged in series. Show that (i) the zeroth moment is the same at the sys-

tem inlets and outlets of the first and second systems, and (ii) the first absolute moment and

the second and third central moments, but not the fourth central moment, are additive.

Sug-

gestion:

For the second and higher moments it is helpful to recognize that the output from the

second unit, following a pulse input to the first may be obtained by the use of the convolution

integral

where

is a system response to a pulse input. A simple way of proceeding is to recognize that

the Laplace transform of

c(t)

may be written as

It then follows that

and similarly for the higher derivatives. Now it may also be shown that

23D.3.

Start-up of a chemical reactor. Rework Example 23.6-1 by use of Laplace transforms of Eqs.

23.6-2 and 3.

23D.4.

Transient behavior of

reactors in ~eries.~ There are

identical chemical reactors of vol-

ume

connected in series, each equipped with a perfect stirrer. Initially, each tank is filled

with pure solvent

At zero time, a solution of

is introduced to the first tank at

con-

stant volumetric flow rate

and a constant concentration c,(O). This solution also contains a

small amount of a dissolved catalyst, introduced just prior to discharge into the first tank,

which causes the following first-order reactions to occur:

k;Ls

kYhc

ASBSC

(23D.4-1)

ki'b

The rate constants in these reactions are assumed constant throughout the system. Let

Q/V,

the inverse of the "effective residence time" in each tank. Obtain an expression for

c,(n),

the

concentration of chemical species

in the nth tank at any time

Acrivos

and

Amundson,

Ind.

Eng.

Chem., 47,1533-1541 (1955).

Chapter

Other Mechanisms

for

524.1.

924.2.

524.3'

524.4'

924.5'

524.6'

Mass Transport

The equation of change for entropy

The

flux expressions for heat and mass

Concentration diffusion and driving forces

Applications of the generalized Maxwell-Stefan equations

Mass transfer across selectively permeable membranes

Diffusion in porous media

In Chapter

we stated that the molecular transport of momentum is related to the veloc-

ity gradient by Newton's law of viscosity. In Chapter

we gave Fourier's law, which

says that molecular heat transport occurs because of

gradient in temperature. How-

ever, when we discussed mixtures in Chapter

19,

we pointed out an extra contribution

the molecular heat flux that accounts for the amount of enthalpy transported by the in-

terdiffusion of the various species. In Chapter

we gave Fick's (first) law of diffusion,

which says that molecular mass transport occurs as the result of a concentration gradi-

ent. We indicated there that other driving forces may contribute to the mass flux. The

purposes of this chapter are to describe the most important of these additional driving

forces and to illustrate some applications.

Important among these forces are the gradients of electrical potential and pressure,

which govern the behavior of ionic systems and permselective membranes as well as

ul-

tracentrifuges. Electrokinetic phenomena in particular are rapidly gaining in importance.

Induced dipoles can produce separations, such as dielectrophoresis and magnetophoresis,

which are useful in specialized applications. In addition, we shall find that temperature

gradients can cause mass fluxes by a process known as thermal diffusion1 or the Soret ef-

fect, and that concentration gradients can produce heat transfer by the diffusion-therrn~,~

or the Dufour, effect. Finally, it is important to realize that in systems containing three or

more components, the behavior of any one species is influenced by the concentration

gradients of all other species present.

Fortunately the wide range of behavior resulting from these various driving forces

can be described compactly via the framework provided by nonequilibrium thermody-

The

effect was first observed

liquids by

Ludwig,

Sitzber. Akad. Wiss. Wien

20,539 (1856),

but

is named after

Ch.

Soret,

Arch. Sci. Phys.

Nat.,

Genhe,

2,4841 (1879); 4,209-213 (1880);

Comptes Rendus

Acad. Sci., Paris,

91,289-291 (1880).

The first observations

gases were made by

Chapman and

F. W.

Dootson,

Phil. Mag.,

33,248-253 (1917).

Dufour,

Arch. Sci. Pkys. Nut. Gentve,

45,9-12 (1872);

Ann.

Phys.

(5) 28,490492 (1873).

s24.1

The Equation of Change for Entropy

765

narni~s;~ this topic is summarized in

gg24.1

and

This discussion concludes with the

generalized Maxwell-Stefan equations. In the remaining sections we show how various

specializations of these equations can be used to provide convenient descriptions of se-

lected diffusional processes.

Those who do not wish to read the first two sections can go directly to the later sec-

tions, where the essential results of nonequilibrium thermodynamics are summarized.

524.1

THE EQUATION

CHANGE FOR ENTROPY

Nonequilibrium thermodynamics makes use of four postulates above and beyond those

of equilibrium thermodynamics:'

The equilibrium thermodynamic relations apply to systems that are not in equi-

librium, provided that the gradients are not too large (quasi-equilibrium postulate).

All fluxes in the system may be written as linear relations involving all the forces

(linearity postulate).

No coupling of fluxes and forces occurs if the difference in tensorial order of the

flux and force is an odd number (Curie's postulate).'

In the absence of magnetic fields the matrix of the coefficients in the flux-force re-

lations is symmetric (Onsager's reciprocal relationsh3

In this and the following section we will use these postulates, which arose from

need to

describe various observed phenomena and also from kinetic theory developments. Note

that the nonequilibrium theory we

are

using excludes consideration of non-Newtonian

fluid^.^

In Problem

11D.1

we saw how to derive Jaumann's entropy balance equation,

in which

is the entropy per unit mass of a multicomponent fluid,

is the entropy-flux

vector, and

is the rate of entropy production per unit volume. At this point we do not

know what

and

are, and hence our first task is to find expressions for these quantities

The discussion here is for multicomponent systems.

discussion for binary systems can be found

in L. Landau and E.

Lifshitz,

Fluid Mechanics,

2nd edition, Pergamon Press (1987), Chapter

VI.