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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PRACTICE PROBLEMS 353

Permeable boundarie

Distorted velocity profile

due to mass transfer

Figure 5.P.22-1 Circumferential ﬂow between an inner permeable boundary at R

that is

rotating at a constant angular velocity ω (radians/time) and a stationary outer permeable

boundary at R

. Steady-state mass transfer occurs in the radial direction to maintain the

mass concentration at the inner and outer boundaries at ρ

and ρ

, respectively. Whereas

the circumferential ﬂuid ﬂow does not affect the mass transfer, the radial mass transfer can

distort the velocity proﬁle.

cylindrical coordinates in terms of the mass concentration ρ

and mass

ﬂux n

along with Fick’s law of diffusion expressed in terms of the mass

ﬂux.

(d) Write the requisite boundary conditions on the equations in part (c).

(e) Introduce the following dimensionless variables and determine the unspeci-

ﬁed scale and reference factors:

∗

≡

− ρ

; n

∗

≡

; u

∗

≡

; u

∗

≡

θs

;

∗

≡

; r

∗

≡

r − r

(5.P.22-2)

(f) Determine the criterion for ignoring the curvature effects.

(g) Determine the criterion for ignoring the bulk-ﬂow contribution to Fick’s law.

(h) Determine the criterion for ignoring the concentration-dependence of the

viscosity.

(i) Determine the criterion for ignoring the effect of the radial velocity on

distorting the circumferential velocity proﬁle.

354 APPLICATIONS IN MASS TRANSFER

5.P.23 Bulk-Flow Effects for Free-Convection Mass Transfer from a

Vertical Cylinder

In Section 5.9 we considered solutally driven free convection caused by the tran-

spiration of water vapor through the inner wall of a vertical annular region. We

employed scaling to determine how the describing equations could be simpliﬁed

for large solutal Grashof numbers. Although we allowed for a radial component of

velocity appropriate to a developing free-convection ﬂow, we did not allow for any

effect of the water vapor transpiration on the radial velocity. That is, the transpi-

ration of water vapor will cause a nonzero radial velocity component at the inner

wall. In this problem we use scaling analysis to determine the criterion for ignor-

ing this blowing or transpiration velocity effect. Assume that the water vapor is

transpired into the annular region at a constant radial velocity V

so as to maintain

a constant concentration c

at the wall.

(a) Write the appropriately simpliﬁed form of the continuity equation, equations

of motion, and species-balance equation for this free-convection problem for

which the effects of the transpiration velocity must be considered.

(b) Write the requisite boundary conditions for the equations you derived in

part (a).

velocity at the porous inner wall.

5.P.24 Free-Convection Mass Transfer Adjacent to a Transpiring Vertical

Flat Wall

Consider an inﬁnitely wide vertical ﬂat porous wall of length L that transpires

water vapor into initially quiescent air, as shown in Figure 5.P.24-1. The water

, y)

= 0

Figure 5.P.24-1 Buoyancy-induced free convection next to a vertical porous wall that

transpires water vapor into initially dry air to maintain a concentration c

at its surface;

concentration proﬁles at positions x

and x

along the plate, where x

, and the velocity

proﬁle of the component parallel to the plate are shown.

PRACTICE PROBLEMS 355

concentration at the wall is c

, whereas the quiescent air is assumed to be initially

dry. The air next to the porous wall will become less dense than the air farther

removed from it. Hence, a hydrostatic pressure imbalance will occur that causes

ﬂuid near the wall to rise. We consider this convective ﬂow after the transients

have died out when steady-state free convection prevails. We ignore end effects

at the top and bottom of the plate and viscous dissipation and assume constant

physical properties other than the density in the gravitational body force term in

the equations of motion. Note that this is inherently a developing ﬂow due to

the progressive water transpiration that occurs as the ﬂuid moves up the wall;

therefore, velocity components in both the x-andy-directions must be considered.

Since the density is temperature-dependent, we need an appropriate equation of

state. We consider small density variations and hence represent the density via a

Taylor series expansion about the density ρ

of the dry air given by

ρ = ρ

∂ρ

∂c



= ρ

− ρ

(5.P.24-1)

where β

is the coefﬁcient of solutal volume expansion. It will be convenient in

this problem to split the pressure into dynamic, P , and hydrostatic, P

, contribu-

tions:

P = P(x,y)+P

(x) (5.P.24-2)

(a) Write the appropriately simpliﬁed steady-state continuity equation, equations

of motion, and species-balance equation.

(b) Write the requisite boundary conditions on the equations in part (a).

ﬁed scale and reference factors:

∗

≡

; u

∗

≡

; P

∗

≡

; c

∗

≡

;

∗

≡

; y

∗

≡

; y

∗

≡

(5.P.24-3)

where δ

and δ

are the momentum and solutal boundary-layer thicknesses,

respectively. Indicate why different y-length scales are used for the equations

of motion and species-balance equation.

(d) Determine the scale factors and estimate the thicknesses of both the momen-

tum and solutal boundary layers.

(e) Consider the limit of very large solutal Grashof numbers, where Gr

≡

gβ

c

/μ

,inwhichμ is the shear viscosity, to derive the boundary-

layer equations for solutal free convection.

5.P.25 Correlation for Steady-State Mass Transfer from a Sphere

Correlations for mass transfer to standard geometries are usually characterized in

terms of a dimensionless mass-transfer coefﬁcient known as the Sherwood num-

ber or Nusselt number for mass transfer. The latter is a function of the other

356 APPLICATIONS IN MASS TRANSFER

dimensionless groups characterizing the mass-transfer process, such as the Reynolds

or Peclet number, Schmidt number, and ratios of characteristic lengths that deﬁne

the geometry. Consider steady-state ﬂow over a sphere having radius R that is

transferring a solute A to the surrounding ﬂuid, consisting primarily of a binary

solution of components A and B. Assume that the concentrations of solute A in

the ﬂowing ﬂuid adjacent to and far from the sphere are c

and c

A∞

, respec-

tively. Component B is assumed to be insoluble in the sphere, and the solution is

assumed to be dilute. The physical and transport properties of the surrounding ﬂuid

stream can be assumed to be constant and constitute the density ρ, viscosity μ,

and binary diffusion coefﬁcient D

. The ﬂuid velocity far from the sphere is U

∞

We seek to develop a correlation for the steady-state mass-transfer coefﬁcient de-

ﬁned by

•

≡

−cD

∂x

∂r

− x

A∞

− x

+ N

)

− x

A∞

∼

− x

A∞

(5.P.25-1)

where

is the molar ﬂux of component i in the r-direction averaged over

the surface of the sphere and x

denotes a mole fraction; the simpliﬁcation in

equation (5.P.25-1) follows from applying Fick’s law of diffusion given by equ-

ation (G.3-7) in the Appendices and the assumption of dilute solutions.

(a) Write the equation that relates the average molar ﬂux at the surface of the

sphere to the concentration.

(b) Write the appropriate forms of the continuity equation, equations of motion

and species-balance equation and their boundary conditions for this mass-

transfer problem.

less groups needed to correlate the mass-transfer coefﬁcient deﬁned by

equation (5.P.25-1).

(d) Consider how the correlation that you obtained in part (c) simpliﬁes in the

limit of zero ﬂow, that is, U

∞

= 0.

(e) Compare the result that you obtained in part (c) to the standard correlation

for the Sherwood number given by

Sh = 2 +0.60Re

1/2

1/3

(5.P.25-2)

where the Sherwood and Reynolds numbers are based on the sphere diam-

eter.

(f) In Practice Problem 5.P.9, scaling was applied to the aeration of water by the

sparging of spherical bubbles; each bubble was assumed to be surrounded

by a stagnant ﬁlm having thickness δ

; use the correlation you obtained for

the mass-transfer coefﬁcient to develop a correlation for the thickness of the

region of inﬂuence or solutal boundary layer.

Bird et al., Transport Phenomena, 2nd ed., p. 681.

PRACTICE PROBLEMS 357

5.P.26 Mass-Transfer-Coefﬁcient Correlation for Film Theory

In Section 5.2 we used scaling analysis to develop the classical ﬁlm theory approx-

imation. Consider the following deﬁnition of the mass-transfer coefﬁcient:

•

≡

−ρD

(∂ω

/∂z)

− ω

+ n

)

− ω

(5.P.26-1)

where n

is the mass-transfer ﬂux (mass/area·time) in the z-direction and ω

and

are the mass fractions on each side of the ﬁlm.

(a) Use the scaling approach to dimensional analysis to determine the dimen-

sionless groups required to correlate the dimensionless mass-transfer coefﬁ-

cient (Sherwood number) for ﬁlm theory. Do not assume that the convective

transport is negligible.

(b) Evaluate the Sherwood number using the general solution for the ﬁlm theory

model given by equation (5.2-29), and discuss the physical signiﬁcance of

the result that you obtain; that is, relate the value of the Sherwood number

to the nature of the mass transfer through a stationary ﬁlm.

theory (i.e., when convective transport is included) to the Sherwood number

for the special case of negligible convection.

(d) Discuss how the result that you obtained in part (c) could be used to cor-

rect a mass-transfer coefﬁcient obtained from some literature correlation for

very dilute solutions (low mass transfer rates) to account for the effect of

convective transport.

(e) Discuss when using ﬁlm theory might provide a reliable correction for

the effects of convective transport on a mass-transfer coefﬁcient obtained

from a correlation applicable only to dilute solutions and low mass-transfer

ﬂuxes.

5.P.27 Correlation for Free-Convection Mass Transfer from an Evaporating

Liquid

Consider a horizontal ﬁlm of width W on each side that consists of a pure volatile

liquid A exposed on its upper surface to dry air, referred to a component B,as

shown in Figure 5.P.27-1. The volatile component is less dense than the surround-

ing air, so that steady-state free convection develops. The equilibrium concentration

of the volatile component in the air adjacent to the liquid–gas interface is c

which can be assumed to be dilute. The air is assumed to be insoluble in the liquid

ﬁlm. We seek to develop a correlation for the steady-state mass-transfer coefﬁcient

deﬁned by

•

≡

−cD

∂x

∂y

− x

A∞

− x

+ N

)

− x

A∞

∼

− x

A∞

(5.P.27-1)

358 APPLICATIONS IN MASS TRANSFER

Volatile liquid film

Free-convection

plume

Ambient air Ambient air

Figure 5.P.27-1 Solutally driven buoyancy-induced free convection due to the evaporation

of a pure volatile liquid contained in a trough with sides of width W ; the vapor of the

evaporating liquid is less dense than that of the ambient air.

where N

is the molar ﬂux of component i in the y-direction (perpendicular

to the horizontal plate) averaged over the surface of the ﬁlm and x

denotes a

mole fraction; the simpliﬁcation in equation (5.P.27-1) follows from Fick’s law of

diffusion given by equation (G.1-7) in the Appendices and the assumption of dilute

solutions.

(a) Write the equation that relates the average molar ﬂux at the surface of the

liquid ﬁlm to the concentration.

(b) Write the appropriate forms of the continuity equation, equations of motion,

and species-balance equation and their boundary conditions for this mass-

transfer problem.

groups needed to correlate the mass-transfer coefﬁcient deﬁned by equ-

ation (5.P.27-1).

(d) Compare the result that you obtained in part (c) to the standard correlation

for the Sherwood number given by

Sh = 0.816(Gr

Sc)

0.2

(5.P.27-2)

where the Sherwood and solutal Grashof numbers are based on the half-

width of the ﬁlm.

5.P.28 Correlation for a Tubular Photocatalytic Reactor

Photocatalytic reactors can be used to employ solar energy for promoting hetero-

geneous natural reactions. A typical geometry involves a feed stream containing a

S. N. Singh, R. C. Birkebak, and R. M. Drake, Prog. Heat Mass Transfer, 2:87 (1969).

PRACTICE PROBLEMS 359

solute that diffuses to the transparent walls of the ﬂow channel that are exposed

to sunlight. The solute decomposes due to a photocatalytic reaction at the tube

wall.

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

having constant physical properties in a cylindrical tube of radius R containing

a solute A having an initial concentration c

that undergoes a photocatalytic

reaction along length L, such as that shown in Figure 5.5-1. The heterogeneous

reaction is assumed to be irreversible and ﬁrst-order with a reaction-rate constant

(length/time).

(a) Develop a correlation for the Sherwood number or Nusselt number for mass

transfer as a function of the Reynolds number, Re ≡ Rρ

U/μ, and other rele-

vant dimensionless groups, where the mass-transfer coefﬁcient is deﬁned as

•

≡

−cD

∂x

∂r



r=R

− x

A∞

r=R

− x

+ N

r=R

− x

A∞

(5.P.28-1)

where

denotes the molar ﬂux in the r-direction averaged over the inner

wall of the photocatalytic reactor.

(b) Sketch a plot of n

∗

versus Re based on your dimensional analysis correlation

and the results of Section 5.5 and Example Problem 5.E.8.

Reynolds number beyond which no further increase in the dimensionless

mass-transfer ﬂux to the tube wall is possible for a photocatalytic reactor

system with ﬁxed physical and chemical properties.

(d) Develop an equation for the maximum possible mass-transfer ﬂux to the tube

wall for a photocatalytic reactor system with ﬁxed physical and chemical

properties.

An annular photocatalytic reactor has been considered for removing trace amounts of trichloroethane

(TCE) from air; TCE is a common solvent and chemical intermediate used in industry; owing to its

volatility, it evaporates readily and poses a health hazard if present in air.

6 Applications in Mass Transfer

with Chemical Reaction

Taking all the mutually interfering phenomena which constitute a process of mass

transfer with chemical reaction into account simultaneously is so difﬁcult a task that,

in practice, simplifying hypotheses are always needed. It is more useful in my opinion

to recognize these simplifying hypotheses as justiﬁed on the grounds of physical

intuition, than to perceive a strictly mathematical justiﬁcation.

6.1 INTRODUCTION

Mass transfer with chemical reaction can be very challenging to model since it not

only can be affected by the complex hydrodynamics discussed in Chapter 3 and

heat-transfer effects considered in Chapter 4, but necessarily includes mass transfer,

discussed in Chapter 5. Indeed, reacting systems involve at least one or more ﬂuid

phases and can involve the release or absorption of energy arising from heat-of-

reaction effects. Chemical species are no longer conserved in reacting systems but

can be depleted or generated. Hence, we would certainly agree with the ﬁrst part

of the quotation at the top of the page; that is, simpliﬁcations need to be made to

model mass transfer with chemical reaction. In this chapter we address the second

part of this quotation; that is, scaling analysis provides a mathematical formalism

for justifying these approximations that obviates the need to be gifted with physical

intuition.

In Chapter 5 scaling analysis was applied to several examples that involved

either homogeneous or heterogeneous chemical reactions. However, these examples

involved mass transfer, with chemical reaction occurring on only one length scale.

We considered problems for which mass transfer with chemical reaction occurred

over the length of the reactor, such as in Section 5.4, or over a small element such

as a rising bubble in Practice Problem 5.P.9. It is easy to see how problems can

G. Astarita, Mass Transfer with Chemical Reaction, Elsevier, New York, 1967, pp. v–vi.

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach

to Model Building and the Art of Approximation, By William B. Krantz

360

INTRODUCTION 361

arise involving transport phenomena on multiple scales. Consider, for example, the

bubbling of a gas into a countercurrently ﬂowing liquid in which it is chemisorbed;

that is, the gaseous component dissolves in the liquid while reacting with it. The

solute from the gas phase can enter the liquid only during the short time that it

passes over a gas bubble. If the reaction is relatively slow, none of the transferring

solute will react during the short contact time for passage of the liquid over a gas

bubble. However, once the solute is in the bulk of the liquid, it can both diffuse and

react. Hence, this example involves mass transfer, with chemical reaction occurring

on two scales. There is the microscale, characterizing the ﬂow of the liquid over

a bubble, and the macroscale, characterizing the ﬂow through the chemisorption

device. In this chapter we employ scaling analysis to assess appropriate simpliﬁca-

tions that can be made in both the microscale and macroscale describing equations

for mass transfer with chemical reaction.

To provide a coherent focus in this chapter, we restrict our attention to a partic-

ular form of mass transfer with a chemical reaction: namely, chemisorption. The

latter refers to the use of chemical reaction to enhance the solubility of a solute in

a ﬂuid into which it is being transferred. However, the scaling protocols illustrated

in the chapter can be applied more generally to any type of process involving mass

transfer with chemical reaction.

This chapter has a dual focus in that it applies scaling analysis to mass trans-

fer with chemical reaction, but uses the latter as an example of scaling analysis

in microscale–macroscale modeling. In general, any system involving dispersed

phases will involve phenomena occurring on multiple scales. For example, phase-

transition phenomena such as crystallization, boiling, and condensation will involve

transport occurring on the microscale of a dispersed phase particle and on the

macroscale of the bulk liquid. A ﬂuidized bed reactor will involve heat and or

mass transfer on the microscale of the particles as well as on the macroscale of

the reactor. The scaling protocols illustrated in this chapter can be applied to any

modeling problem involving transport phenomena and chemical reaction occurring

on multiple scales.

The organization of this chapter is somewhat different from that of the preceding

chapters. Rather than considering several different examples to illustrate how scal-

ing is used to arrive at the various approximations made in transport processes, we

focus here on illustrating how scaling is applied to microscale–macroscale modeling

using chemisorption as the example.

We begin with a discussion of the microscale

element in Section 6.2, since it is critical to understand precisely what is meant by

this concept. Since scaling on the microscale involves some concepts that are differ-

ent from those for the macroscale, we treat these two topics in separate sections.

In Section 6.3 we focus on applying systematic scaling analysis to the describ-

ing equations for the microscale element. Scaling of the complementary describing

equations for the macroscale element is considered in Section 6.8. Scaling analysis

Microscale–macroscale modeling of mass transfer with chemical reaction has been treated by other

authors, although there is no general agreement on the terminology used to describe this. Prior treatments

rely on intuition rather than systematic scaling analysis in order to simplify the describing equations at

the two scales. See, for example, Astarita, Mass Transfer with Chemical Reaction.

362 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

of the microscale element leads to the identiﬁcation of various reaction regimes dis-

cussed in Sections 6.4 through 6.7. Scaling analysis of the macroscale element leads

to the identiﬁcation of various reaction domains in the slow reaction regime that

are discussed in Sections 6.9 and 6.10. Scaling analysis will not only develop cri-

teria for determining the relative importance of diffusion and reaction on both the

micro- and macroscale but will also lead to a set of precepts for selecting the type

of reactor that will be most effective for a particular reaction regime or domain,

as shown in Section 6.11. Scaling analysis will be used to explain why the perfor-

mance of a contacting device for mass transfer with chemical reaction depends on the

hydrodynamics for the slow and instantaneous reaction regimes but not for the fast

reaction regime. In Section 6.12, simple models suggested by scaling analysis will

be developed to interrelate the mass-transfer coefﬁcients in the presence and absence

of chemical reaction. The power of scaling analysis to provide considerable insight

into the design of devices involving mass transfer with chemical reaction without

the need to actually solve any model equations analytically or numerically is shown

in Section 6.11 and again in Sections 6.13 and 6.14. In Section 6.13 we illustrate

how microscale–macroscale scaling can be used to design and interpret performance

data for a continuous-stirred tank reactor. Section 6.14 does the same for a packed

gas-absorption column. In Section 6.15 we summarize the implications of scaling

analysis for microscale–macroscale modeling and, in particular, its application to

chemisorption. Unworked practice problems are included at the end of the chapter.

6.2 CONCEPT OF THE MICROSCALE ELEMENT

In developing models for complex mass-transfer processes, in particular those in-

volving dispersed phases, it is convenient to consider the concept of the microscale

element. The latter is associated with mass transfer at the smallest continuum scale.

For example, in a gas-absorption process in which a gas containing a soluble com-

ponent is bubbled through a countercurrently ﬂowing liquid phase, the microscale

element is a gas bubble. In a ﬂuidized bed reactor involving the ﬂow of a gas rel-

ative to suspended recirculating solid particles, the microscale element is a solid

particle. In phase-transition processes involving growth kinetics, the microscale ele-

ment is a dispersed phase particle. In turbulent ﬂows the microscale element can

be an eddy that transports species to and from an interface. The mass-transfer rate

on the microscale is controlled by a region of inﬂuence within the ﬂuid adjacent to

the microscale element. A pivotal consideration in microscale–macroscale model-

ing is that the microscale element is a mathematical point on the macroscale; that is,

the mass-transfer ﬂux from or to a microscale element becomes in the macroscale

balance a homogeneous source or sink term, respectively

Some authors consider the microscale element to be the thin region of ﬂuid adjacent to what we call

the microscale element here. For example, these authors would consider the microscale element to be

a thin element of liquid ﬂowing adjacent to and around a gas bubble rather than the bubble itself.

The deﬁnition of the microscale element used in this chapter is preferred since this is what is reduced

to a mathematical point when one considers the implications of the microscale mass transfer on the