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923.1

The Macroscopic Mass Balances

727

In this chapter we summarize the macroscopic balances for the more general situa-

tion described above. Each of these balances will now contain one extra term, to account

for mass, momentum, or energy transport across the bounding surfaces. The balances

thus obtained are capable of describing industrial mass transfer processes, such as ab-

sorption, extraction, ion exchange, and selective adsorption. Inasmuch as entire treatises

have been devoted to these topics, all we try to do here is to show how the material dis-

cussed in the preceding chapters paves the way for the study of mass transfer opera-

tions. The reader interested in pursuing these topics further should consult the available

textbooks and treatises.'-"

The main emphasis on this chapter is on the mass balances for mixtures. For that

reason,

523.1

is accompanied by five examples, which illustrate problems arising in envi-

ronmental science, isotope separation, economic evaluation, and biomedical science. In

ss23.2

23.4

the other macroscopic balances are given. In Table

23.5-1

they are summa-

rized for systems with multiple inlets and outlets. The last two sections of the chapter il-

lustrate applications of the macroscopic balances to more complex systems.

523.1

THE

MACROSCOPIC MASS BALANCES

The statement of the law of conservation of mass of chemical species

in a multicompo-

nent macroscopic flow system is

This is a generalization of

Eq.

7.1-2.

Here

ma,,,

is the instantaneous total mass of

in the

system, and

-Awn

w,,

p,l(v,)S,

pa2(v2)S2

is the difference between the mass

rates of flow of species

across planes

and

The quantity

w,,,

is the mass rate of addi-

tion of species

to the system by mass transfer across the bounding surface. Note that

w,,,

is positive when mass is

added

to the system, just as

and

are taken to be posi-

tive

the total energy balance when heat is added to the system and work is done

the system by moving parts. Finally, the symbol

r,,t,,

stands for the net rate of produc-

tion of species

by homogeneous and heterogeneous reactions within the system.'

Recall that in Table

15.5-1

the molecular and eddy transport of momentum and

en-

ergy across surfaces

and

in the direction of flow were neglected with respect to the

convective transport. The same is done everywhere in this chapter-in

Eq.

23.1-1

and in

the other macroscopic balances presented here.

McCabe,

Smith, and

Harriot,

Unit Operations of Chemical Engineering,

McGraw-Hill,

New York, 6th edition

(2000).

Sherwood,

Pigford, and

Wilke,

Mass Transfer,

McGraw-Hill, New York (1975).

Treybal,

Mass Transfer Operations,

3rd edition, McGraw-Hill, New York (1980).

King,

Separation Processes,

McGraw-Hill, New York (1971).

Holland,

Multicomponent Distillation,

McGraw-Hill, New York (1963).

Lo, M. H.

Baird, and

Hanson, eds.,

Handbook of Solvent Extracfion,

Wiley-Interscience,

New York (1983).

Yang,

Gas Separations

Adsorption Processes,

Butterworth, Boston (1987).

Seader and

Henley,

Separation Process Principles,

Wiley, New York (1998).

The quantities

ma,,,,

w,,,

and

r,,,,

may be expressed as integrals:

in which

is the outwardly directed unit normal vector, and

is that portion of the bounding surface

on which mass transfer occurs. The integrands in

r,,,,

are

the net rates of production of species

homogeneous and heterogeneous reactions, respectively.

728

Chapter

Macroscopic Balances for Multicomponent Systems

If all

equations in Eq. 23.1-1 are summed, we get

in which w,

Saw,,,,

and use has been made of the law of conservation of mass in the

form Z,r,,,,

is often convenient to write Eq. 23.1-1 in molar units:

Here the capital letters represent the molar counterparts of the lowercase symbols in

Eq.

23.1-1. When Eq. 23.1-3 is summed over all species, the result is

Note that the last term is not in general zero, because moles are produced or consumed

in many reaction systems.

In some applications, such as spatially continuous mass transfer operations, it is cus-

tomary to rewrite Eq. 23.1-1 or

for a differential element of the system (that is, in the

"d-form" discussed in

515.4).

Then

the differentials

dw,,

W,,

can

be expressed in

terms of local mass transfer coefficients.

fluid stream emerges from a chemical plant with a constant mass flow rate

and dis-

charges into a river

(Fig.

23.1-la). It contains a waste material

at mass fraction

w,,,

which is

Disposal

unstable and decomposes at a rate proportional to its concentration according to the expres-

Unstable Waste

sion

-k';bA-that is, by a first-order reaction.

Product

To reduce pollution it is decided to allow the effluent stream to pass through a holding

tank of volume

before discharging into the river (Fig. 23.1-lb). The tank is equipped

with

Volume flow rate

w/p

Concentration of

in effluent

PAO

(a)

Volume

flow

rate

w/p

Concentration,

Well-stirred

tank

with volume

(b)

Fig.

23.1-1.

(a)

Waste stream with unstable pollutant emptying directly into a river.

(b)

Waste stream with

holding tank that allows the unstable pollutant to decay prior to going into the river.

(c)

Sketch showing the

concentration of pollutant being discharged into the river after the holding tank has been filled (the dimen-

sionless quantity

k;"V/Q).

523.1 The Macroscopic Mass Balances

729

SOLUTION

an efficient stirrer that keeps the fluid in the tank at very nearly uniform composition. At time

the fluid begins to flow into the empty tank. No liquid flows out until the tank has been

filled up to the volume V.

Develop an expression for the concentration of the fluid in the tank as a function of time,

both during the tank-filling process and after the tank has been completely filled.

(a)

We begin by considering the period during which the tank is being filled-that is the pe-

riod t

pV/w, where

is the density of the fluid mixture. We apply the macroscopic mass

balance of Eq. 23.1-1 to the holding tank. The quantity mA,tot on the left side is wto, at time t.

The mass rate of flow entering the tank is wwAo, and there is no outflow during the tank-filling

stage. No

is entering or leaving through a mass transfer interface. The mass rate of produc-

tion of species

Y,,~,,

(wt/p)(-k;"pA)

-k~m,,t,t. Therefore the macroscopic mass balance

for species

during the filling period is

This first-order differential equation can be solved with the initial condition that mA,tot

to give

This may be written

terms of the instantaneous mass fraction of

in the tank by using the

relation mA,tot

wtw,:

The mass fraction of

at the instant when the tank is full,

o,,

is then given by

in which

kYpv/w

k;"V/Q.

(b)

The mass balance on the tank after it has been filled is

or, in dimensionless form, with

(w/pV)t,

This first-order differential equation can be solved with the initial condition that

w,,

to give

This shows that as time progresses the mass fraction of the pollutant being discharged into

the river decreases exponentially, with a limiting value of

The curve for the mass concentration as a function of time after the filling of the tank is shown

in Fig. 23.1-l(c). This curve can be used to determine conditions such that the effluent concen-

tration will be in the permitted range. Equation 23.1-12 can be used to decide on the size of

holding tank that is required.

730

Chapter 23 Macroscopic Balances for Multicomponent Systems

product

Fig.

23.1-2.

Binary splitter, in which a feed stream is split into

a product stream and a waste stream.

EXAMPLE

23.1-2

Bind

Splitters

Describe the operation of a binary splitter, one of the commonest and simplest separation de-

vices (see Fig. 23.1-2). Here a binary mixture of

and

enters the apparatus in a feed stream

at a molar rate F, and by some separation mechanism it is split into a product stream with a

molar rate P and a waste stream with molar rate

The mole fraction of

(the desired com-

ponent) in the feed stream is

and the mole fractions in the product and waste streams are

and x, respectively.

SOL

UTION

We start by writing the steady-state macroscopic mass balances for component

and for the

entire fluid as

ZF =yP

(23.1-13)

F=P+W (23.1-14)

It is customary to define the ratio

P/F

of the molar rates of the product and feed streams

as the cut. Equation 23.1-13 then becomes, after eliminating

by use of Eq. 23.1-14,

Normally the cut

and the feed composition z are taken to be known.

We now need

relation between the feed and waste compositions, and it is conventional

to write an equation relating the compositions of the two outgoing streams:

Here

is known as the separation facfor, also usually taken as known, and which characterizes

the separation capability of the splitter. Here

and

are the mole ratios defined by

x=- and

1-x

(23.1-17,18)

1-Y

In terms of the mole fractions, Eq. 23.1-16 may be written as

Equations 23.1-15 and 19 (or 20) describe completely the splitter operation.

For vapor-liquid splitting-that is, equilibrium distillation-it is typical to define the

ideal splitter in terms of an operation in which the product and waste streams are in equilib-

rium. For this situation,

is the relative volatility, and for thermodynamically ideal systems, it

just the ratio of the component vapor pressures. Even for nonideal systems,

changes rela-

tively slowly with composition.

For real splitters one can then define

in terms of an empirical correction factor-for ex-

ample, the eficiency-defined by

Ea*

(23.1-21)

where

is the separation factor for the ideal model, and

is a correction factor that accounts

for the failure of the actual system to meet the ideal behavior.

We thus find that, for a given feed composition, the enrichment (y

z)/z produced by the

splitter is a function of the cut

and the separation factor

The enrichment can be calculated

from the following equation, which is obtained by combining Eqs. 23.1-15 and 20:

s23.1 The Macroscopic Mass Balances

731

Fig.

23.1-3.

Behavior of a binary

splitter.

cut

This is a quadratic equation for

that can be solved when

is given, and then the enrichment

z)/z is obtained. An example is given in Fig. 23.1-3 where both

z)/z

and

58(y

z)/z

are plotted as functions of

for

and

1.25 (a reasonable value for many processes). It

may be seen that, whereas the maximum enrichment

z)/z is obtained for vanishingly

small cuts, the product of enrichment and product rate is greatest at an intermediate

value.

Finding an optimum

value is a problem that must be addressed on economic grounds.

Simple splitters of the general type pictured in Fig. 23.1-2 are very widely used as build-

ing blocks in multistage separation processes. These include evaporators and crystallizers,

which typically have a very high separation factor

per stage, and systems for distillation,

gas absorption, and liquid extraction, where

can vary widely. All of these applications are

well covered in standard texts on unit operations.

Membrane processes are rapidly increasing

importance, and many of the design prin-

ciples were developed for the isotope fractionation industry.' Discussions of modern applica-

tions are also a~ailable.~

EXAMPLE

23.1-3

The Macroscopic

Balances and Dirac's

"Separative Capacity"

and

"Value Function"

During the Manhattan Project of World War

11,

the British physicist Dira~~,~,~ used the macro-

scopic mass balances for a binary splitter to develop a criterion for comparing the effective-

ness of different separation processes-for example, thermal diffusion and centrifugation.

The same criterion has also proven useful in the evaluation of bioseparations.

We imagine the simple separation system shown in Fig. 23.1-2 in which

is the molar

rate of flow of the feed stream, which contains

binary mixture of

and

and Ware

the molar rates of flow of the product and waste streams. The mole fractions of species

the three streams are z,

and x, respectively.

In the system there is some mechanism (for example, a membrane) for increasing the

concentration of

in the product stream and decreasing it in the waste stream. We then may

define a

separation

factor

as in Eqs. 23.1-16 to 18

E. Von Halle and

Schacter,

Diffusion Separation Methods,

in Volume

Kirk-Othmer Encyclopedia

Chemical Technology

(M.

Howe-Grant, ed.), 4th edition, Wiley, New York (19931, pp. 149-203.

S. W. Ho and

Sirkar,

Membrane Handbook,

Van Nostrand Reinhold, New York (1992),

p. 954;

D. Noble and S.

Stern,

Membrane Separations Technology,

Elsevier, Amsterdam (19951, p.

718.

Dirac, British Ministry of Supply (1941); this is reprinted in

The Collected Works of

Dirac

(1924-19481,

(R.

Dalitz, ed.) Cambridge University Press (1995). Nobel Laureate

Paul

Adrien

Maurice

Dirac (1902-1984), one of the leaders in the development of quantum mechanics, developed the

relativistic wave equation and predicted the existence of the positron.

Cohen,

Theory of Isotope Separation,

McGraw-Hill, New York (1951).

732

Chapter 23 Macroscopic Balances for Multicomponent Systems

SOLUTION

We have written this in

second form, because we will consider only systems in which there

is only a slight enrichment of species

so that

is a very small quantity. When Eq. 23.1-

23 is solved for

as a function of x we then get

Next we define the Dirac separative capacity

of the system as the net increase in "value"

(this could, for example, be the monetary value) of the streams that are participating in the

system:

in which v(x) is the Dirac value function. (In the separation science literature, the separative ca-

pacity is often given the symbol

6U.)

Show how the separative capacity and value function can be obtained by using the defin-

ition in Eq. 23.1-25 along with the mass balances for the system.

The total mass balance and the mass balance for species

are:

We now divide Eq. 23.1-27 by

and then use Eq. 23.1-26 to eliminate

Then introducing

the quantity

P/F

(called the "cuV), we get

Next we divide Eq. 23.1-25 by

and introduce

to get

Inasmuch as the differences between the concentrations of the streams are quite small, we can

expand v(y) and v(x) about

and get

where the primes indicate differentiation with respect to

When these expressions are put

into Eq. 23.1-29 and we use Eq. 23.1-28, we get

When we use

Eq.

23.1-24, this last equation becomes

We now assume that the separative capacity of the system is virtually independent of concen-

tration. Therefore we set the concentration-dependent factor in Eq. 23.1-33 equal to unity, so

that

is the final expression for the separative capacity. According to this expression, the separative

capacity has

maximum when the system is operated at

It remains to obtain the Dirac value function, which must satisfy the differential equation

g23.1 The Macroscopic Mass Balances

733

When this equation is integrated, we get

The two integration constants may be assigned arbitrarily, and several different choices have

been used. However, the most common choice is v($)

and v1(;)

This leads to

which is the symmetrical solution, in the sense that v(l

v(z) and vl(l

-vl(z).

The value function v(z) and the separative capacity

have proven useful in comparing

separations made in different kinds of equipment as well as different concentration ranges.

From an economic standpoint v(z) as given by Eq. 23.1-37 has been found useful for determin-

ing price differences for isotope mixtures of differing purity.

EXAMPLE

23.1-4

Compartmental

One of the simplest and most useful applications of the species macroscopic mass balance is

compartmental analysis, in which a complex system is treated as a network of perfect mixers,

each of constant volume, connected by ducts of negligible volume, with no dispersion occur-

Analysis

ring in the connecting ducts. Imagine mixing units, labeled

2,3,

. . .

containing

various species (labeled with indices

.).

Then the mass concentration

p,,

of species

in unit

changes with time according to the equation

Here

is the volume of unit n,

Q,,

is the volumetric flow rate of solvent flow from unit

unit

and r,, is the rate of formation of species

per unit volume in unit n.

Show how such a model can be specialized to describe the removal of toxic metabolic

products (that is, the toxic materials resulting from the human metabolism) from a patient by

hemodialysis. Hemodialysis is the periodic removal of toxic metabolites achieved by contacting

the blood and a dialysis fluid in countercurrent flow, separated by a cellophane membrane

that

permeable to the metabolite.

SOLUTION

The simple two-compartment model of Fig. 23.1-4 has been found to be adequate for repre-

senting the hemodialysis system. Here the large block, or compartment

(labeled "body") rep-

resents the combined body fluids, except for those in the blood, which are represented by

compartment

The blood circulates via a branching system of vessels through compartment

at a volumetric rate

and in the process extracts solute across the vessel walls. This process is

highly efficient, and a single solute is assumed to leave compartment

at concentration

p,,

equal to the concentration throughout that compartment. At the same time, the solute is being

formed within the body fluids at a constant rate

and during dialysis it is being extracted

from the blood by the dialyzer at a rate

Dp,.

The proportionality constant

is known as the

"dialyzer clearance" and is fixed by the dialyzer design and operating conditions.

Fig.

23.1-4.

Two-compartment model

used to analyze the functioning of a

dialyzer.

734

Chapter 23 Macroscopic Balances for Multicomponent Systems

The very complex process actually taking place is modeled by the two equations

with

0 between the dialysis periods. Because we are considering a single solute, the con-

centrations have only one subscript to indicate the compartment. We measure the time

from

the start of a dialysis procedure, when the blood and body fluids are very nearly in equilib-

rium with each other, so that we may write the initial conditions as

C.: at

(23.1-41)

where

is a constant. We now want to get an explicit expression for the toxic metabolite con-

centration in the blood as a function of time.

We start by adding Eqs. 23.1-39 and 40 and solving for

dp,/dt.

The latter is then substi-

tuted into the time derivative of Eq. 23.1-40 to obtain a differential equation for the metabolite

concentration in the blood:

with

C.:

The second initial condition is obtained by use of Eqs. 23.1-40 and 41.

This equation is now to be solved with the following specific parameter values, which

are typical for the removal of creatinine from

70-kg adult human:

Quantity (liters) (liters) (liters per min) (liters per min) (g/min)

per liter)

Magnitude 43 4.5 5.4 0.3 0.0024 0.140

The differential equation and initial conditions now take the form:

C.:

at time

and

dp2

-0.00933 (23.1-45)

in which concentration is in grams per liter and time is in minutes. The complementary func-

tion that satisfies the associated homogeneous equation is

P,,~,

exp(0.006043t)

exp(1.386t) (23.1-46)

and the particular integral is

p2,pi

0.0080

The complete solution to the nonhomogeneous equation

given by the sum of the comple-

mentary function and the particular integral. When the constants of integration are deter-

mined from the initial conditions, we get

during the dialysis period.

s23.1 The Macroscopic Mass Balances

735

For the recovery period following dialysis, we assume here that the patient has no

kid-

ney function, so the clearance

is zero. Equation 23.1-42 takes the simpler form

where

is the concentration during the recovery period. The complementary function and

particular integral are

Gt'

in which

is the time measured from the start of the recovery period. Inserting the numer-

ical values, we then get for the concentration during the recovery period and its time

derivative

The integration constants are to be determined from the matching conditions at

We need a second initial condition for determining the integration constants in Eq. 23.1-53.

This can be obtained from Eq. 23.1-40 and the corresponding equation for

(i.e., with

O),

combined with the two relations in Eqs. 23.1-55 and 56. This relation is

For illustrative purposes, we shall end the dialysis at 50 min, for which

We now have enough information to determine the constants of integration, and therefore we

get for the concentration in the blood during the recovery period

Equations 23.1-48 and 59 are plotted in Fig. 23.1-5.

perhaps more interest is Fig. 23.1-6, which shows the application of Eqs. 23.1-39 and

40 to an actual patient. Here the points represent data and the lines are the model predictions.

Here only the dialyzer clearance and the creatinine concentrations are known, and the data of

0.14

0.13

.r(

0.12

0.11

0.10

0.09

0 50 100

200

Fig.

23.1-5.

Pharmacokinetics of dialysis: model

Time,

minutes

prediction.

736

Chapter 23 Macroscopic Balances for Multicomponent Systems

Fig.

23.1-6. Experimental (dots) and

simulated creatinine data (solid curve)

for a dialysis patient

[R.

Bell,

Cur-

tiss, and

Babb,

Trans. Amer. Soc.

Artificial Internal Organs,

11, 183

(1965)l.

Time (days)

the first cycle are used to estimate the remaining parameters. The resulting model is then

used to predict the next three cycles. We see that this approach does an excellent job of corre-

lating the data and has predictive value. Note that the sudden rise in creatinine concentration

min results from the fact that the dialyzer is no longer removing it from the blood. As

result, the disequilibrium between the blood and the rest of the body then becomes smaller.

Similar compartmental models have wide application in medicine, where they are re-

ferred to as

pharmacokinetic model~.~

priori pharmacokinetic modeling, where model para-

meters are determined separately from the process being modeled, was pioneered by Bischoff

and Dedri~k.~

the foregoing example it is clear, even on casual inspection, that neither the body fluids nor

the circulating blood have much in common with ideal mixing tanks, and it is therefore of

Time

and

some interest to examine the success of the simple compartmental model critically. To make

Ahdel Insensitivity

start in that direction, compare the response (that is, the output concentration) of two quite

different systems in Fig. 23.1-7 to an exponentially decaying solute input: one in which the en-

PFR

Dimensionless time

t/tO

Fig.

23.1-7. Responses of the PFR and the CSTR to a pulse input.

Welling,

Pharmacokinetics,

American Chemical Society (1997).

Bischoff

and

Dedrick,

Pkarm. Sci.,

87,

1347-1357 (1968);

AIChE

Symposium Series,

64,

32-44 (1968).