Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation

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EXAMPLE PROBLEMS 303

5.E.2 Taylor Dispersion

Sir Geoffrey Taylor developed a widely used approximation for the manner in which

a pulse or step change in concentration disperses as it is convected downstream; this

phenomenon, which is called Taylor dispersion, causes the sharp boundary at the

leading edge of the pulse or step change to become diffuse, due to the combined

action of species diffusion and differential convection arising from the velocity

proﬁle.

Sir Geoffrey Taylor developed this approximation via very clever intuitive

arguments. In this example we develop the Taylor dispersion approximation using

systematic scaling analysis.

Consider steady-state fully developed laminar ﬂow of a Newtonian ﬂuid com-

posed of pure component B having constant physical properties in a cylindrical tube

having radius R, as shown in Figure 5.E.2-1. At time t = 0, the ﬂuid is changed

instantaneously to pure component A while maintaining all other aspects of the

ﬂow. Although the interface between ﬂuid A and B is initially planar, it will begin

to disperse as a result of convection and molecular diffusion. Convective transport

will cause component A near the center of the tube to move farther downstream

than that near the wall, due to the parabolic velocity proﬁle. Superimposed on this

convection of component A are both radial and axial diffusion; the former will

cause diffusion of component A between the center of the tube and the wall. This

combined species diffusion and convection, which is due to a nonuniform velocity

proﬁle, is referred to as Taylor dispersion.

We begin by writing the appropriately simpliﬁed species-balance equation given

by equation (G.2-5) in the Appendices (step 1):

∂c

∂t

+ u

∂c

∂z

= D

∂

∂z

+ D

∂

∂r



∂c

∂r



(5.E.2-1)

where the axial velocity is given by

= 2U



1 −



(5.E.2-2)

Pure component APure component B

Increasing time

Figure 5.E.2-1 Fully developed laminar ﬂow in a cylindrical tube of radius R showing

the displacement of liquid B by the continuous injection of immiscible liquid A due to

Taylor dispersion that arises from the combined effects of diffusion and convection via a

nonuniform velocity proﬁle.

G. I. Taylor, Proc. R. Soc. London, 225A, 473–477 (1954).

304 APPLICATIONS IN MASS TRANSFER

in which U is the average velocity. Note that each term in equation (G.2-5) has been

divided by the molecular weight of component A in arriving at equation (5.E.2-1).

The center of the dispersion zone containing both components A and B will be

convected at the average velocity. Hence, we deﬁne a new axial coordinate ˜z ≡

z −

Ut and transform equations (5.E.2-1) and (5.E.2-2) to a convected coordinate

system as follows:

∂c

∂t



1 − 2



∂c

∂ ˜z

= D

∂

∂ ˜z

+ D

∂

∂r



∂c

∂r



(5.E.2-3)

where the time derivative is now evaluated at constant ˜z rather than at constant z;

that is, the time derivative of the concentration is evaluated at a position relative

to the convected dispersion zone. Note that this coordinate transformation modiﬁes

the convection term in the species-balance equation. The corresponding initial and

boundary conditions are given by

= 0att = 0 (5.E.2-4)

is bounded at r = 0 (5.E.2-5)

∂c

∂r

= 0atr = R (5.E.2-6)

= c

at ˜z =−Ut (5.E.2-7)

= 0at˜z →∞ (5.E.2-8)

The boundary condition given by equation (5.E.2-5) merely states that the concen-

tration remains ﬁnite at the centerline; this is a common condition applied at a

point or axis of symmetry. Equation (5.E.2-6) states that the walls of the tube are

impermeable.

Deﬁne the following dimensionless variables (steps 2, 3, and 4):

∗

≡

; r

∗

≡

;˜z

∗

≡

˜z

; t

∗

≡

(5.E.2-9)

Introduce these dimensionless variables into the describing equations and divide

through by the coefﬁcient of one term in each equation (steps 5 and 6):

∂c

∗

∂t

∗

˜z



1 − 2

∗2



∂c

∗

∂ ˜z

∗

∂

∗

∂ ˜z

∗2

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.E.2-10)

∗

= 0att

∗

= 0 (5.E.2-11)

∗

is bounded at r

∗

= 0 (5.E.2-12)

∂c

∗

∂r

∗

= 0atr

∗

(5.E.2-13)

EXAMPLE PROBLEMS 305

∗

at ˜z

∗

=−

˜z

∗

(5.E.2-14)

∗

= 0at˜z

∗

→∞ (5.E.2-15)

Now let us determine the scale factors (step 7). The dimensionless concentra-

tion and radial coordinate can be bounded to be

◦

(1) by the following choice of

scale factors that emanate from the dimensionless groups in equations (5.E.2-13)

and (5.E.2-14): c

= c

and r

= R. Since this is inherently unsteady-state mass

transfer, we choose the observation time as our time scale; that is, t

= t

. Since the

principal terms in the dispersion process are axial convection and radial diffusion,

the dimensionless group multiplying the convection term in equation (5.E.2-10) is

set equal to 1, which provides the axial length scale; that is, ˜z

= UR

.Our

dimensionless describing equations then assume the form

∂c

∗

∂t

∗



1 − 2r

∗2



∂c

∗

∂ ˜z

∗

∂

∗

∂ ˜z

∗2

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.E.2-16)

∗

= 0att

∗

= 0 (5.E.2-17)

∗

is bounded at r

∗

= 0 (5.E.2-18)

∂c

∗

∂r

∗

= 0atr

∗

= 1 (5.E.2-19)

∗

= 1at˜z

∗

=−Fo

∗

(5.E.2-20)

∗

= 0at˜z

∗

→∞ (5.E.2-21)

where

≡

(5.E.2-22)

is the solutal Fourier number or Fourier number for mass transfer, which is a

measure of the ratio of the observation time to the characteristic diffusion time, and

≡

= Re · Sc (5.E.2-23)

is the solutal Peclet number or Peclet number for mass transfer, which is a measure

of the convection to diffusion of species, Re is the Reynolds number, which is a

measure of the convection to viscous transport of momentum, and Sc is the Schmidt

number, which is a measure of the viscous transport of momentum to diffusive

transport of species.

If the following conditions apply, these describing equations can be reduced

to those considered by Sir Geoffrey Taylor in his classical development of Taylor

dispersion (step 8; see footnote 27):

 1 ⇒ quasi-steady-state can be assumed (5.E.2-24)

306 APPLICATIONS IN MASS TRANSFER

 1 ⇒ axial diffusion can be ignored (5.E.2-25)

Note that ignoring this unsteady-state term in equation (5.E.2-16) term does not

mean that the dispersion is not time-dependent; it merely means that it is not time-

dependent in a convected coordinate system; the time dependence enters through

the axial coordinate in the convected coordinate system. Sir Geoffrey developed

an approximate solution to the resulting simpliﬁed system of describing equations

by assuming that ∂c

∗

/ − ∂ ˜z

∗

was constant (see footnote 27). He used intuitive

arguments to conclude that his solution was a reasonable approximation to the

solution for the full set of describing equations if Fo

 0.25 and Pe

 6.9.

These conditions agree well with those given equations (5.E.2-24) and (5.E.2-25).

The full set of describing equations were solved numerically by Gill et al.; their

solution conﬁrms the criteria given by equations (5.E.2-24) and (5.E.2-25) for the

applicability of Sir Geoffrey’s approximate solution.

5.E.3 Convective Diffusion in a Tapered Pore

Consider steady-state binary gas-phase diffusion at constant pressure and tempera-

ture through a pore having a nonconstant circular cross-sectional area whose radius

R is given by R = R

− β

√

z,whereβ is a constant. Figure 5.E.3-1 shows a

cross-sectional view of this model pore along a plane that cuts through its axis of

symmetry. The binary gas mixture at the mouth of the pore is assumed to be dilute

in the diffusing component, whose concentration is maintained at a constant value

(moles/volume). The concentration of the diffusing component is maintained

at zero at the other end of the pore, where z = L. The diffusion coefﬁcient may be

R = R

− b z

= c

= 0

→

Figure 5.E.3-1 Binary diffusion of a dilute gas at constant temperature and pressure

through a tapered pore having length L and a circular cross-section with a radius given

by R = R

− β

√

V. Ananthakrishnan, W. N. Gill, and A. J. Barduhn, A.I.Ch.E. J., 11(6), 1063–1072 (1965).

EXAMPLE PROBLEMS 307

assumed to be constant. Scaling analysis is used to determine when radial diffusion

can be ignored relative to axial diffusion.

The appropriately simpliﬁed species-balance equation given by equation (G.2-

10) in the Appendices and corresponding boundary conditions are given by

0 = D

∂

∂z

+ D

∂

∂r



∂c

∂r



(5.E.3-1)

= c

at z = 0 (5.E.3-2)

= 0atz = L (5.E.3-3)

∂c

∂r

= 0atr = 0for0≤ z ≤ L (5.E.3-4)



·n = 0atr = R(z) = R

− β

√

z, 0 ≤ z ≤ L (5.E.3-5)

where n is the unit normal vector to the pore wall as shown in Figure 5.E.3-1 (step

1).

Deﬁne the following dimensionless variables (steps 2, 3, and 4):

∗

≡

; N

∗

≡

; z

∗

≡

; r

∗

≡

(5.E.3-6)

A scale factor is introduced for the mass-transfer ﬂux N

, although this is a for-

mality since it will not be necessary to determine this scale factor to assess when

radial diffusion can be ignored.

Introduce these dimensionless variables into the describing equations and divide

through by the coefﬁcient of one term in each equation (steps 5 and 6):

0 =

∂

∗

∂z

∗2

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.E.3-7)

∗

at z

∗

= 0 (5.E.3-8)

∗

= 0atz

∗

(5.E.3-9)

∂c

∗

∂r

∗

= 0atr

∗

= 0for0≤ z

∗

≤

(5.E.3-10)



∗

·n = 0atr

∗

−

√

∗

, 0 ≤ z

∗

≤

(5.E.3-11)

The following considerations determine the scale factors (step 7). The dimen-

sionless concentration and axial coordinate can be bounded to be

◦

(1) by the

following choice of scale factors, which emanate from the dimensionless groups

in equations (5.E.3-8) and (5.E.3-9): c

= c

and z

= L. Note that to ensure that

308 APPLICATIONS IN MASS TRANSFER

∗

◦

(1), we want the largest possible value for r

; this corresponds to setting

the appropriate dimensionless group in equation (5.E.3-11) equal to 1 to obtain

= R

Our dimensionless describing equations then assume the form

0 =

∂

∗

∂z

∗2





∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.E.3-12)

∗

= 1atz

∗

= 0 (5.E.3-13)

∗

= 0atz

∗

= 1 (5.E.3-14)

∂c

∗

∂r

∗

= 0atr

∗

= 0for0≤ z

∗

≤ 1 (5.E.3-15)



∗

·n = 0atr

∗

= 1 −

√

∗

, 0 ≤ z

∗

≤ 1 (5.E.3-16)

Equation (5.E.3-12) then indicates that to ignore radial relative to axial diffusion

the following criterion must be satisﬁed (step 8):





 1 (5.E.3-17)

Note that this criterion will always break down for sufﬁciently long pores. This

limitation is explored further in Practice Problem 5.P.3.

5.E.4 Dissolution of a Spherical Capsule

Consider a solid spherical capsule of a pure material A having an initial radius

. Assume that this capsule is ingested into the stomach, where it progressively

dissolves while undergoing a ﬁrst-order reaction with the stomach ﬂuid B given by

= k

(moles/volume·time), in which k

is the reaction-rate constant. Equilib-

rium is assumed at the interface between the capsule and stomach ﬂuid, at which

the concentration of A is c

(moles/volume).

The solution will be assumed

to be sufﬁciently dilute so that any bulk ﬂow arising from the mass transfer is

negligible; that is, the ﬂuid phase is assumed to be quiescent so that the mass

transfer is purely diffusive. A schematic of this dissolution process is shown in

Figure 5.E.4-1. We use scaling to explore how the describing equations can be

simpliﬁed.

Note that in practice it would be difﬁcult to measure this equilibrium concentration if component A

reacts in the liquid phase with component B. However, it would be possible to infer the equilibrium

concentration from measurements of the dissolution process if the kinetic constant for the homogeneous

reaction were known.

EXAMPLE PROBLEMS 309

R(t)

Solid spherical

capsule of pure

component A

Stomach liquid B in

which component A

dissolves and reacts

Moving interfacial

boundary due to

dissolution of capsule

Figure 5.E.4-1 Solid spherical capsule of pure component A dissolving in stomach liquid

B, accompanied by a ﬁrst-order homogeneous chemical reaction.

The appropriately simpliﬁed form of the species-balance equation in spherical

coordinates given by equation (G.3-10) in the Appendices and the corresponding

initial and boundary conditions are (step 1)

∂c

∂t

= D



∂

∂r



∂c

∂r



− k

(5.E.4-1)

= 0,R= R

at t = 0 (5.E.4-2)

= c

at r = R(t) (5.E.4-3)

= 0asr →∞ (5.E.4-4)

where c

is the molar concentration (moles/volume) and R

is the initial radius

of the capsule. Since the dissolution of the spherical capsule implies that this

is a moving boundary problem, an auxiliary condition is needed to locate the

instantaneous position of the interface, denoted here by R(t). This comes from an

integral species balance on the spherical capsule and the stomach ﬂuid, shown in

detail here to illustrate how the homogeneous reaction term is handled:



4πr

dr +



∞

4πr

dr =−



∞

4πr

dr (5.E.4-5)

where M

and ρ

are the molecular weight and solid mass density of pure com-

ponent A, respectively. The last term in equation (5.E.4-5) accounts for loss of

species A due to the homogeneous chemical reaction. Application of Leibnitz’s

rule for differentiating an integral given by equation (H.1-2) in the Appendices

and substitution of equation (5.E.4-1) into equation (5.E.4-5) then yields



− c



− D

∂c

∂r

= 0atr = R (5.E.4-6)

Note that for dilute solutions, c

 ρ

. The initial condition needed to solve

equation (5.E.4-6) is given in equation (5.E.4-2).

310 APPLICATIONS IN MASS TRANSFER

Deﬁne the following dimensionless variables (steps 2, 3, and 4):

∗

≡

; R

∗

≡

;





∗

≡

; r

∗

≡

r − r

; t

∗

≡

(5.E.4-7)

Note that we have introduced a reference factor since r is not naturally referenced

to zero. We also have introduced a separate scale factor,

, for the dissolution

velocity of the spherical capsule dR/dt since there is no reason to assume that this

will scale as r

Introduce these dimensionless variables into the describing equations and divide

through by the dimensional coefﬁcient of one term in each equation (steps 5 and 6):

∂c

∗

∂t

∗

+ r

)

∂

∂r

∗



∗

+ r

)

∂c

∗

∂r

∗



−

∗

(5.E.4-8)

∗

= 0,R

∗

at t

∗

= 0 (5.E.4-9)

∗

at r

∗

R − r

(5.E.4-10)

∗

= 0asr

∗

→∞ (5.E.4-11)





∗

−

∂c

∗

∂r

∗

= 0atr

∗

R − r

(5.E.4-12)

The boundary condition given by equation (5.E.4-10) indicates that we can bound

∗

to be

◦

(1) by setting c

= c

and can reference r

∗

to zero by choosing r

= R.

The proper scale factor for the spatial coordinate depends on the conditions for

which we are scaling this problem. Let us ﬁrst consider the case where the homo-

geneous chemical reaction is sufﬁciently slow such that quasi-steady-state is never

possible. Hence, the proper time scale is the observation time t

. For this case the

diffusion term must always balance the unsteady-state term in equation (5.E.4-8);

the proper scale for the spatial coordinate is r

√

. The dimensionless

sphere size can be bounded to be

◦

(1) by setting the dimensionless group in

equation (5.E.4-9) to 1 to obtain R

= R

. Since the two terms in equation (5.E.4-

12) must balance, the scale for the dissolution velocity of the capsule is found to

= M

√

/ρ

√

These scale and reference factors then result in the following dimensionless

describing equations (step 7):

∂c

∗

∂t

∗

−





∗

∂c

∗

∂r

∗



∗

√



∂

∂r

∗



∗

√



∂c

∗

∂r

∗

− k

∗

(5.E.4-13)

EXAMPLE PROBLEMS 311

∗

= 0,R

∗

= 1att

∗

= 0 (5.E.4-14)

∗

= 1atr

∗

= 0 (5.E.4-15)

∗

= 0asr

∗

→∞ (5.E.4-16)





∗

−

∂c

∗

∂r

∗

= 0atr

∗

= 0 (5.E.4-17)

Note that an additional term now appears in equation (5.E.4-13) because of the

transformation from a ﬁxed coordinate system to one that is referenced to the

moving interface of the spherical capsule. This is another example of the fact that

one has to be careful when applying the chain rule of differentiation in transforming

to the dimensionless variables.

Now let us consider how equations (5.E.4-13) through (5.E.4-17) can be sim-

pliﬁed (step 8). Note that the characteristic length r

√

deﬁnes a region

of inﬂuence wherein the diffusive mass transfer is essentially conﬁned. For sufﬁ-

ciently short times such that R/

√

 1, one can ignore the curvature effects

in equation (5.E.4-13). Note, however, that this approximation will break down

for sufﬁciently long times or when the capsule size becomes very small. One can

estimate when the curvature effects become important by using the scale factor

to obtain an approximate solution for the instantaneous location of the capsule

interface applicable for short contact times:

∼

=−

√

⇒ R

∼

−

√

(5.E.4-18)

Hence, curvature effects associated with the spherical geometry will need to be

considered when

√

− 2M

√

/ρ

◦

(1) (5.E.4-19)

Note that equation (5.E.4-18) also provides an estimate of the time required for

the capsule to dissolve. A further simpliﬁcation of equation (5.E.4-13) is possible

if k

 1, in which case the effect of the homogeneous chemical reaction can

be ignored. Note, however, that this approximation will always break down for

sufﬁciently long contact times such that k

◦

(1). In arriving at this set of

describing equations we already have made the dilute solution assumption; that is,

/ρ

 1, which implies that the pseudo-convection term arising from the

coordinate transformation can also be ignored.

Now let us consider conditions for which the homogeneous reaction term is

always important. In this case the characteristic length scale is obtained by bal-

ancing the diffusion term with the reaction term in equation (5.E.4-8), thereby

obtaining r

√

. This in turn implies that

= M

√

/ρ

.All

the other scale and reference factors remain the same. Hence, our dimensionless

312 APPLICATIONS IN MASS TRANSFER

describing equations assume the form

∂c

∗

∂t

∗

−





∗

∂c

∗

∂r

∗



∗

+ R

√



∂

∂r

∗

⎡

⎣

∗

+ R



∂c

∗

∂r

∗

⎤

⎦

− c

∗

(5.E.4-20)

∗

= 0,R

∗

= 1att

∗

= 0 (5.E.4-21)

∗

= 1atr

∗

= 0 (5.E.4-22)

∗

= 0asr

∗

→∞ (5.E.4-23)





∗

−

∂c

∗

∂r

∗

= 0atr

∗

= 0 (5.E.4-24)

The condition for ignoring the pseudo-convection term arising from the coordi-

nate transformation again is M

/ρ

 1. In this case the curvature effects can

be ignored when R

√

 1. However, this condition will break down for

sufﬁciently long times when the capsule radius becomes very small. Again, one

can estimate when the curvature effects become important by using the scale factor

to obtain an approximate solution for the instantaneous location of the capsule

interface:

∼

=−

√

⇒ R

∼

−

√

(5.E.4-25)

Hence, curvature effects associated with the spherical geometry will need to be

considered when



−

√





= O(1) (5.E.4-26)

For fast reaction conditions it is also possible to achieve quasi-steady-state mass

transfer since the unsteady-state term in equation (5.E.4-20) becomes insigniﬁcant

when k

 1. In the latter case all the mass transfer and accompanying chemical

reaction occur within a region of inﬂuence whose constant thickness is given by

√

; for very fast reaction conditions this boundary layer can become

very thin. However, unless the reaction is instantaneous, the unsteady-state term

will become signiﬁcant for sufﬁciently short observation times.

5.E.5 Mass Transfer to a Rotating Disk: Uniformly Accessible Surface

In Example Problem 3.E.4 we scaled the hydrodynamics for a circular disk rotating

about its axis at a constant angular frequency ω (radians/second) in an inﬁnite

Newtonian ﬂuid having constant physical properties. We used scaling analysis in

the aforementioned example to provide an estimate for the region of inﬂuence