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 343

(g) Use your scaling analysis with the simpliﬁcation considered in part (e) to

estimate the time required for the capsule to dissolve.

5.P.12 Slow Dissolution of a Cylindrical Capsule

Consider the dissolution of a medication in the form of a cylindrical capsule of pure

solid component A with initial radius and length, R

and L

, respectively, as shown

in Figure 5.P.12-1. We assume that this cylindrical solid capsule is swallowed and

dissolves in the acidic aqueous liquid environment of the stomach via binary diffu-

sion accompanied by a relatively slow zeroth-order homogeneous chemical reaction

for which the rate constant is k

(moles/volume·time). The equilibrium solubility

mole fraction of component A in the stomach liquid is x

. The binary solution can

be assumed to be dilute and to have constant physical and transport properties. How-

ever, mass transfer must be considered from both the circumferential area and the

end of the cylindrical capsule. We scale this problem to assess when the mass trans-

fer from the circular ends of the cylindrical capsule can be neglected; this has sig-

niﬁcant consequences for how you determine your length scales. We also make the

assumption that the capsule remains a cylinder throughout the dissolution process.

In fact, variations in the mass-transfer ﬂuxes along the length and ends of the cylin-

der will gradually cause it to become ellipsoidal and eventually spherical. However,

during the early stage of the dissolution process, the cylindrical shape will be main-

tained at least approximately. Note that because of the plane of symmetry in the cap-

sule, one can consider the mass-transfer problem only for one-half of the capsule.

(a) Write the appropriately simpliﬁed species-balance equation and its initial and

boundary conditions in cylindrical coordinates; express the concentration in

terms of mole fraction.

Cylindrical capsule of

pure component A

Stomach liquid

Figure 5.P.12-1 Unsteady-state dissolution of a medication in the form of a cylindrical

capsule of pure component A into the stomach liquid, where it is consumed via a zeroth-order

homogeneous chemical reaction.

344 APPLICATIONS IN MASS TRANSFER

(b) Carry out an integral mass balance to obtain the auxiliary equation required

to determine the location of boundary that deﬁnes the interface between

the cylindrical capsule and the stomach liquid; account for both a changing

cylindrical radius and length.

homogeneous chemical reaction.

(d) Based on your scaling analysis, discuss whether quasi-steady-state mass

transfer is ever possible.

(e) Determine the criterion for ignoring mass transfer from the ends of the

cylindrical capsule.

(f) Determine the criterion for ignoring the curvature effects on the mass

transfer.

(g) Use your scaling analysis with the simpliﬁcation considered in part (e) to

estimate the time required for the cylindrical capsule to dissolve.

5.P.13 Dissolution of a Cylindrical Capsule in the Fast Reaction Limit

Let us consider the mass-transfer problem deﬁned in Practice Problem 5.P.12 for

the special case of a fast reaction.

(a) Write the appropriately simpliﬁed species-balance equation and its initial and

boundary conditions in cylindrical coordinates; express the concentration in

terms of mole fraction.

(b) Carry out an integral mass balance to obtain the auxiliary equation required

to determine the location of boundary that deﬁnes the interface between

the cylindrical capsule and the stomach liquid; account for both a changing

cylindrical radius and length.

neous chemical reaction.

(d) Determine the criterion for assuming quasi-steady-state mass transfer.

(e) Determine the criterion for ignoring the curvature effects on the mass

transfer.

(f) Use your scaling analysis with the simpliﬁcation considered in part (e) to

estimate the time required for the cylindrical capsule to dissolve completely;

do not ignore the dissolution from the ends of the capsule in determining

your estimate.

5.P.14 Diffusional Growth of a Nucleated Water Droplet

Nucleation of a liquid droplet from a gas phase requires that the latter be suf-

ﬁciently supersaturated with the nucleating component. This is required because

a submicroscopic nucleus has a large surface area relative to its volume, which

increases its Gibbs free energy. Supersaturation in turn increases the Gibbs free

energy of this component in the gas phase, thereby permitting a decrease in the

PRACTICE PROBLEMS 345

R(t)

A∞

r → ∞

Growing

water droplet

Air supersaturated

with water vapor

Figure 5.P.14-1 Water droplet of instantaneous radius R(t) growing in supersaturated air

at concentration c

A∞

due to diffusion and instantaneous heterogeneous nucleation on its

surface, at which the saturated gas-phase concentration is c

system Gibbs free energy via nucleation. Once nucleation occurs, the dispersed

phase droplets grow via coalescence with neighboring droplets and diffusion of the

nucleating component from the supersaturated gas accompanied by heterogeneous

nucleation on their surface. Here we explore the diffusive growth of a water droplet

in supersaturated air as shown schematically in Figure 5.P.14-1. We assume that

the heterogeneous nucleation occurs instantaneously. The droplet has a negligible

initial radius and an instantaneous radius R(t). The supersaturation concentration is

A∞

, whereas the thermodynamic equilibrium concentration of water in the atmo-

sphere at the prevailing temperature is c

. Since nuclei are quite small, we assume

that this water droplet initiates from a zero radius. We use scaling to explore how

this mass-transfer problem can be simpliﬁed.

(a) Write the appropriately simpliﬁed species-balance equation and its initial

and boundary conditions in spherical coordinates; express the concentration

in terms of mole fraction and assume dilute solutions with constant physical

properties.

(b) Carry out an integral mass balance to obtain the auxiliary equation required

to determine the location of boundary that deﬁnes the interface between the

water droplet and the atmosphere.

solutal boundary layer wherein essentially all the mass transfer is occurring.

(d) Determine the criterion for ignoring the curvature effects on the mass

transfer.

(e) Can quasi-steady-state conditions ever be achieved in this mass-transfer

problem?

5.P.15 Crystallization from a Supersaturated Liquid

Consider the one-dimensional diffusional growth of a planar crystal of pure com-

ponent A from its binary solution with component B as shown in Figure 5.P.15-1.

The binary solution is assumed to be supersaturated with a concentration ρ

A∞

far from the growing crystal face, whereas the equilibrium concentration at the

boundary between the solid crystal and liquid solution is ρ

346 APPLICATIONS IN MASS TRANSFER

L(t)

Planar crystal of pure component A

Solution of component A in B

Concentration profile

of component A

A∞

Figure 5.P.12-1 One-dimensional growth of a planar crystal of pure component A from

its supersaturated solution with component B.

(a) Write the appropriately simpliﬁed species-balance equation and its initial

and boundary conditions in rectangular coordinates; since this involves an

incompressible liquid phase, express the concentration in terms of mass

density and mass fraction; do not ignore the bulk ﬂow contribution to the

mass-transfer ﬂux.

(b) Carry out an integral mass balance to obtain the auxiliary equation required

to determine the location of boundary that deﬁnes the interface between the

crystal and the liquid solution.

wherein all the mass transfer is effectively occurring.

(d) Is quasi-steady-state ever possible for this mass-transfer problem?

(e) Use scaling analysis to determine the criterion for ignoring the bulk-ﬂow

contribution to the mass-transfer ﬂux.

5.P.16 Growth of a Liquid Droplet by means of Diffusion and

Heterogeneous Nucleation

In Practice Problem 5.P.14 we considered the diffusional growth of a liquid droplet

assuming that the heterogeneous nucleation was instantaneous. The latter is a rea-

sonable assumption for heterogeneous nucleation from the gas phase that involves

relatively low molecular weight “simple” molecules. However, for heterogeneous

nucleation of a liquid phase in another immiscible liquid phase involving more

complex molecules (e.g., an amorphous polymer component from its solution in an

organic solvent), steric effects can result in noninstantaneous heterogeneous nucle-

ation. In this problem we assume that the heterogeneous nucleation of the pure

PRACTICE PROBLEMS 347

dispersed phase component from its solution, which constitutes the continuous

phase, is characterized by a zeroth-order rate constant

(moles/area·time). The

supersaturation concentration far from the surface of the growing droplet is denoted

A∞

; the thermodynamic equilibrium concentration of the nucleating component

in the continuous phase is denoted by ρ

. Mass rather than molar concentrations

are used here since we are considering a liquid phase. The molecular weight of the

crystallizing component is M

(a) Write the appropriately simpliﬁed species-balance equation and its initial

and boundary conditions in spherical coordinates; express the concentration

in terms of mass fraction.

(b) Carry out an integral species balance to obtain the auxiliary equation required

to determine the location of boundary that deﬁnes the interface between the

liquid droplet and the continuous liquid phase; consider carefully how the

heterogeneous nucleation affects this integral balance.

wherein all the mass transfer effectively is occurring.

(d) Determine the criterion for ignoring the curvature effects on the mass

transfer.

(e) Determine the criterion for ignoring the heterogeneous nucleation.

(f) Determine the criterion for assuming quasi-steady-state.

5.P.17 Rusting of a Planar Surface

Consider a ﬂat piece of initially pure iron that is immersed so that its upper surface

is exposed continuously to liquid water saturated with dissolved oxygen (O

whose concentration is denoted by c

A∞

. The oxygen in the water will diffuse to

the surface of the iron and promote rusting via the formation of iron oxide, Fe

based on the reaction

Fe +O

→

(5.P.17-1)

Note that reaction (5.P.17-1) implies a relationship between the molar ﬂuxes of

oxygen and iron as well as the molar rate of growth of the rust layer. The rate

of conversion of oxygen to rust is assumed to occur by means of a ﬁrst-order

heterogeneous reaction whose reaction rate is given by

(moles/area · time) (5.P.17-2)

in which

(length/time) is the heterogeneous reaction-rate constant and A denotes

the molecular oxygen. The rate of growth of the rust layer decreases progressively

in time since the oxygen must diffuse through both the water and the increasing

thickness of the rust layer in order to reach the surface of the iron. Since the oxy-

gen diffuses through the water-saturated microporous structure of the rust layer,

348 APPLICATIONS IN MASS TRANSFER

its local concentration in the latter is equal to εc

,inwhichε is the porosity

and c

is the local concentration of oxygen in the pores. Owing to the microp-

orous structure, the effective oxygen diffusivity in the rust, which is denoted by

, will be less than that in the water, which is denoted by D

. The densi-

ties of the water, water-saturated rust, and pure iron are denoted by ρ

,ρ

,and

, respectively, in which B, C,andD denote the water, rust, and iron, respec-

tively. The corresponding molecular weights are denoted by M

,andM

respectively. Note that this problem involves two moving boundaries: the interface

between the rust and the water and that between the iron and the rust, as shown

in Figure 5.P.17-1. One might anticipate that initially, oxygen diffusion through

the water controls the rate of rusting, whereas at longer times, oxygen diffusion

through the rust becomes controlling. We use scaling analysis to explore how the

describing equations for this mass-transfer process can be simpliﬁed. In scaling this

problem, use the following dimensionless variables involving unspeciﬁed scale and

reference factors:

∗

≡

− c

; c

∗

≡

− c

; z

∗

≡

z − z

;

∗

≡

z − z

; t

∗

≡

; (5.P.17-3)

∗

≡

L − L

; L

∗

≡

;





∗

≡





;





∗

≡





where the subscripts W, R,andI refer to the water, rust, and iron layers, respec-

tively.

(a) Explain why it is necessary to deﬁne separate reference and scale factors

for the concentrations and spatial coordinates in the water and rust layers.

(b) Explain why it is necessary to deﬁne separate scale factors for the thick-

nesses of the rust and iron layers as well as the velocities of each layer.

and boundary conditions for both the water and the rust layer; use molar

concentrations and assume dilute solutions so that the bulk-ﬂow contribution

to the mass-transfer ﬂux can be ignored.

(d) Carry out an integral mass balance on the iron and rust layers to obtain the

auxiliary equations required to determine the location of the two moving

boundaries. Note that the growth rate of the rust layer must satisfy the

molar exchange dictated by reaction (5.P.17-1).

(e) Use scaling analysis to estimate the thickness of the region of inﬂuence in

the water layer.

(f) Determine the criteria for quasi-steady-state to apply.

PRACTICE PROBLEMS 349

Un-oxidized iron

Rust

Liquid water saturated

with dissolved oxygen

Concentration profile

Figure 5.P.17-1 Rusting of a ﬂat piece of pure iron immersed in liquid water that is

saturated with dissolved oxygen; this involves two moving boundaries: the interface between

the rust layer and the water and that between the iron and the rust layer.

(g) Use scaling analysis to determine when the resistance to mass transfer in

the rust layer can be ignored.

(h) Use scaling analysis to determine when the resistance to mass transfer in

the water bath can be ignored.

5.P.18 Mass Transfer in a Hollow-Fiber Membrane

In Example Problem 5.E.9 we applied scaling analysis to a hollow-ﬁber membrane

reactor as shown in Figure 5.E.9-1. An enzyme was immobilized in the microporous

annular wall of the hollow ﬁber. A low-molecular-weight solute permeated through

the ultrathin inner wall of this annular region and reacted with the conﬁned enzyme.

We used scaling analysis to explore how the describing equations in the annular

region could be simpliﬁed. In doing this we allowed for separate scale factors for the

axial and radial concentration gradients. Rather than allowing for separate scales for

the axial and radial concentration gradients, assume that these two derivatives scale

with the characteristic concentration scale divided by the characteristic length scales

in the axial and radial directions, respectively. This will lead to a contradiction in

that one or more terms will not be bounded of

◦

(1). This exercise provides another

example of the forgiving nature of scaling; that is, scaling tells you if the ordering

analysis has been done correctly.

5.P.19 Permeation Accompanied by Membrane Swelling

Consider a mass-transfer process that involves maintaining concentrations ρ

(mass/unit volume) and zero of component A on opposing sides of a polymeric

membrane having thickness L as shown in Figure 5.P.19-1. The permeating

350 APPLICATIONS IN MASS TRANSFER

= 0

Polymeric

membrane

Concentration gradien

Figure 5.P.19-1 Steady-state diffusion of a solute A that causes membrane swelling, sug-

gested by the progressive shading, which in turn increases the effective diffusion coefﬁ-

cient.

component A swells the membrane, which in turn causes an increase in the effective

binary diffusion coefﬁcient that is given by

= (1 − ω

βρ

(5.P.19-1)

where D

is the binary diffusion coefﬁcient in the membrane at inﬁnite dilution of

component A, ω

the mass fraction of component A,andβ a positive constant.

(a) Write the appropriately simpliﬁed species-balance equation and its boundary

conditions; assume dilute solutions so that the bulk-ﬂow contribution to the

mass-transfer ﬂux can be ignored.

(b) Scale the describing equations to determine when the concentration depen-

dence of the diffusion coefﬁcient can be neglected.

layer near one side of the membrane within which the resistance to diffusion

will be concentrated; use scaling analysis to determine the thickness of this

region of inﬂuence.

(d) Use the results of your scaling analyses in parts (b) and (c) to develop a

criterion for determining when signiﬁcant swelling is occurring in the mem-

brane. Hint: Use scaling to estimate the mass-transfer ﬂux in the presence

and absence of swelling.

5.P.20 Evaporative Polymer Film Casting for Short Contact Times

In Example Problem 5.E.1 we considered the evaporative casting of a dense ﬁlm

from a solution of a polymer in a volatile solvent as shown in Figure 5.E.1-1.

PRACTICE PROBLEMS 351

We developed a long-contact-time scaling to determine the criterion for assuming

quasi-steady-state. In this problem we consider a short-contact-time scaling.

(a) Write the appropriately simpliﬁed species-balance equation and its initial

and boundary conditions.

(b) Consider an integral mass balance to derive an auxiliary condition needed

to determine the location of the moving liquid–gas interface.

determine the thickness of the region of inﬂuence wherein all the mass

transfer is effectively concentrated.

(d) Develop a criterion for assuming that the boundary condition at the solid sup-

port surface can be applied at inﬁnity; note that if this criterion is satisﬁed,

a pseudo-convection term is not generated when converting the describing

equations to a translated coordinate system.

(e) Use your scaling analysis result in part (c) to determine when the region of

inﬂuence penetrates to the solid support surface.

(f) Determine the criterion for ignoring the convective contribution to the mass-

transfer ﬂux.

5.P.21 Mass Transfer with a Homogeneous Chemical Reaction and a

Concentration-Dependent Diffusivity

Consider a liquid ﬁlm consisting of components A and B of thickness L for which

the concentration of a diffusing component is maintained at c

(moles/volume)

at z = 0andzeroatz = L. Component A undergoes a ﬁrst-order irreversible

homogeneous chemical reaction with a rate constant k

(time

−1

) as shown in

Figure 5.P.21-1. In addition, the diffusion coefﬁcient D

decreases linearly with

concentration as follows:

= D

− βc

(5.P.21-1)

where D

is the diffusion coefﬁcient at inﬁnite dilution and β is a positive constant.

The effects of the reaction product(s) and any change in the ﬁlm thickness can be

neglected.

(a) Write the appropriately simpliﬁed steady-state species-balance equation and

boundary conditions.

(b) Scale these describing equations to determine a criterion for ignoring the

homogeneous reaction term.

concentration dependence of the binary diffusion coefﬁcient.

(d) Scale the describing equations for the special case of a very fast reaction

that consumes all the diffusing reactant before it can reach the boundary at

z = L. Use your scaling to determine the thickness of the region of inﬂuence

wherein all the diffusion takes place.

352 APPLICATIONS IN MASS TRANSFER

Liquid film of component

B in

which a chemical reaction is

consuming component

Concentration gradient

of component

Figure 5.P.21-1 Diffusion of component A in a liquid ﬁlm of component B within which

A is being consumed by a homogeneous ﬁrst-order chemical reaction and for which the

diffusivity is concentration dependent.

5.P.22 Mass Transfer in the Annular Region Between Fixed and Rotating

Cylinders

Consider a liquid conﬁned to the annular gap between two concentric cylindrical

shells of radii R

and R

, as shown in Figure 5.P.22-1. The inner cylinder rotates

at a constant angular velocity of ω (radians/time), while the outer cylinder remains

stationary. The cylindrical boundaries at R

and R

are permeable, so they can be

maintained at constant compositions ρ

and ρ

(mass/volume), respectively. In this

case the mass transfer is not inﬂuenced by the rotating ﬂuid ﬂow. However, the

mass transfer can inﬂuence the ﬂuid ﬂow in two ways. First, the shear viscosity μ

can be concentration-dependent. Second, the bulk ﬂow velocity established by the

radial mass transfer can distort the velocity proﬁle in the circumferential direction.

In this problem we use scaling analysis to explore when these effects need to be

considered. Assume that the concentration dependence of the viscosity is given by

μ = μ

βρ

(5.P.22-1)

where β is a constant that can be either positive or negative.

(a) Write the appropriately simpliﬁed steady-state continuity and equations of

motion in cylindrical coordinates.

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

that the radial component of velocity will be equal to the mass ﬂux divided

by the mass density at the inner and outer cylindrical boundaries.