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 333

Red blood cell

Blood flow relative to blood cell

Oxygen mass

transfer from

bloodstream to

red blood cell

Figure 5.E.10-1 Steady-state oxygen mass transfer from a ﬂowing bloodstream to a spher-

ical red blood cell.

concentration in the bulk of the bloodstream far from the red blood cell. We assume

that c

A∞

∞

, and the physical properties of the blood are known. Moreover,

we assume that the oxygen concentration in the blood is dilute.

Equation (5.E.10-1) requires determination of N

, which is given by

=−

1 − x

A∞

∂c

∂r



∼

−D

∂c

∂r



(5.E.10-2)

where D

is the effective binary diffusion coefﬁcient of oxygen in blood and

is the oxygen mole fraction at the outer wall of the red blood cell. The

molar concentration in equation (5.E.10-2) would have to be determined from a

solution to the coupled species-balance equation and equations of motion given

by equations (B.5-3) and (B.2-3) in the Appendices, respectively, which when

appropriately simpliﬁed in a spherical coordinate system referenced to the center

of the spherical red blood cell are given by

u ·∇c

= D

∇

(5.E.10-3)

ρ u ·∇u = μ∇

u −∇P (5.E.10-4)

where u is the vector velocity in the bloodstream, ρ and μ are the mass density

and shear viscosity of the blood, respectively, and P is the pressure. Note that each

term in equation (B.5-3) has been divided by the molecular weight of component

A in arriving at equation (5.E.10-3). The boundary conditions required to solve

equations (5.E.10-3) and (5.E.10-4) are given formally as follows:

= c

at r = R (5.E.10-5)

n ·u = 0atr = R (5.E.10-6)

334 APPLICATIONS IN MASS TRANSFER

= c

A∞

as r →∞ (5.E.10-7)

u ·



= U

∞

cos θ, u ·



= U

∞

sin θ as r →∞ (5.E.10-8)

where



and



are unit vectors in the radial and circumferential directions,

respectively. It is not necessary to specify any boundary conditions on the pres-

sure if the velocity of the bloodstream far from the red blood cell is speciﬁed.

Equations (5.E.10-1) through (5.E.10-8) constitute step 1 in the scaling analysis

procedure for dimensional analysis.

Deﬁne arbitrary scale factors for all the dependent and independent variables and

reference factors for those variables not naturally referenced to zero (steps 2 and 3):

∗

≡

c − c

; P

∗

≡

;u

∗

≡

u

; r

∗

≡

(5.E.10-9)

Note that we do not need to scale the angular coordinates θ and φ, since they are

dimensionless and bounded of

◦

(1).

Introduce these dimensionless variables into the describing equations and divide

through by the dimensional coefﬁcient of one term in each equation (steps 4 and 5):

− c

A∞

=−

∂c

∗

∂r

∗

at r

∗

(5.E.10-10)

u

∗

·∇

∗

=∇

∗2

∗

(5.E.10-11)

ρv

u

∗

·∇

∗

u

∗

=∇

∗2

u

∗

−

μu

∇

∗

(5.E.10-12)

∗

− c

at r

∗

(5.E.10-13)

n ·u

∗

= 0atr

∗

(5.E.10-14)

∗

A∞

− c

as r

∗

→∞ (5.E.10-15)

u

∗



∞

cos θ, u

∗



∞

sin θ as r

∗

→∞ (5.E.10-16)

Step 6 involves setting the various groups equal to 1 or zero to determine

the scale and reference factors, respectively. Since there is no attempt to achieve

◦

(1) in dimensional analysis scaling, it makes no difference which groups we

choose. Hence, the two dimensionless groups in equation (5.E.10-13), when set

equal to zero and 1, respectively, indicate that c

= c

and r

= R.The

dimensionless group in equation (5.E.10-15), when set equal to 1, indicates that

= c

− c

. Setting the dimensionless group in equation (5.E.10-16) equal

to 1 indicates that u

= U

∞

. The dimensionless group in equation (5.E.10-12)

multiplying the dimensionless pressure gradient, when set equal to 1, indicates

EXAMPLE PROBLEMS 335

that P

= μU

∞

/R. These choices then yield the following minimum parametric

representation of the describing equations:

Sh =

∂c

∗

∂r

∗

at r

∗

= 1 (5.E.10-17)

·u

∗

·∇

∗

=∇

∗2

∗

(5.E.10-18)

Re ·u

∗

·∇

∗

u

∗

=∇

∗2

u

∗

−∇

∗

(5.E.10-19)

∗

= 0atr

∗

= 1 (5.E.10-20)

n ·u

∗

= 0atr

∗

= 1 (5.E.10-21)

∗

= 1asr

∗

→∞ (5.E.10-22)

u

∗



= cos θ, u

∗



= sin θ as r

∗

→∞ (5.E.10-23)

where

Sh ≡

is the Sherwood number (5.E.10-24)

≡

∞

is the solutal Peclet number (5.E.10-25)

Re ≡

ρU

∞

is the Reynolds number (5.E.10-26)

Note that Sherwood number provides a measure of the overall mass transfer to that

by diffusion alone; as such, it is analogous to the Nusselt number in heat transfer.

The solutal Peclet number is a measure of the convective to diffusional transport

of species. Equation (5.E.10-17) implies that the Sherwood number is a function of

the dimensionless groups involved in determining ∂c

∗

/∂r

∗

and hence will be

a function of only the dimensionless groups involved in solving equations (5.E.10-

18) and (5.E.10-19), which introduce the Peclet and Reynolds numbers. Hence, we

conclude that oxygen transfer to a red blood cell can be correlated in terms of three

dimensionless groups, that is,

Sh = f(Re, Pe

) (5.E.10-27)

Note that a naive application of the Pi theorem would imply that ﬁve dimensionless

groups would be required (i.e., n = 8andn = 3).

The three dimensionless groups in the correlation given by equation (5.E.10-

27) are not unique. It is convenient to isolate the velocity into just one group by

applying the formalism indicated in equation (2.4-2); that is,

Sc = Pe

−1

ρD

≡ the Schmidt number (5.E.10-28)

Hence, our modiﬁed correlation for the Sherwood number is given by

Sh = f(Re, Sc) (5.E.10-29)

336 APPLICATIONS IN MASS TRANSFER

A frequently used equation for the Sherwood number for mass transfer to a

single sphere that has a constant velocity relative to a ﬂuid far removed from

it is

Sh = 1 +0.4536

√

Re · Sc = 1 + 0.4536



(5.E.10-30)

Equation (5.E.10-30) indicates that the Sherwood number can be correlated in terms

of just one dimensionless group, the solutal Peclet number. Note that this correlation

is applicable only in the limit of very small Reynolds numbers. This more restrictive

correlation can be obtained from the general correlation given by equation (5.E.10-

29) by invoking the formalism suggested by equation (2.4-3); that is, by expanding

equation (5.E.10-29) in a Taylor series for small Reynolds number (step 9):

Sh = f(Re, Pe

) = f |

∂f

∂Re



Re + O(Re

) (5.E.10-31)

Hence, in the limit of very small Reynolds number, we conclude that

Sh = f(Pe

) = f(Re · Sc) (5.E.10-32)

which is consistent with the form of equation (5.E.10-30).

5.P PRACTICE PROBLEMS

5.P.1 Penetration Theory Approximation for a Speciﬁed Equation of State

Consider unsteady-state mass transfer in the liquid ﬁlm considered in Section 5.3

and shown in Figure 5.2.1. The boundary conditions at z = 0andz = H remain

the same. However, rather than specifying the ratio of the mass ﬂuxes, an equation

of state is given for the mass density of the form

ρ = ω

+ ω

(5.P.1-1)

where ω

and ρ

are the mass fraction and pure component mass density of com-

ponent i. Use scaling analysis to determine when the convective transport arising

from the diffusive mass transfer can be neglected.

5.P.2 Error Estimate for Penetration Theory Approximation

Consider unsteady-state mass transfer through the liquid ﬁlm considered in Sec-

tion5.3andshowninFigure5.2-1.

Bird et al., Transport Phenomena, 2nd ed., p. 678. Note that this reference deﬁnes the Sherwood and

Reynolds numbers in terms of the sphere diameter rather than the radius.

PRACTICE PROBLEMS 337

(a) Estimate the time required for the mass transfer to penetrate the entire thick-

ness of the ﬁlm.

(b) Consider the special case of unimolecular diffusion of component A in

stationary component B for which equation (5.2-6) implies that κ = 0. Solve

equations (5.3-2) through (5.3-6) for the concentration proﬁle ρ

∗

),in-

voking the penetration theory approximation given by equation (5.3-7) while

retaining the convective mass-transfer contribution to the mass ﬂux in equ-

ation (5.3-3).

∗

) as a function of the dimensionless group 

deﬁned by

equation (5.2-26) to show the error encountered when the convective contri-

bution to the total mass-transfer ﬂux deﬁned by equation (5.3-3) is ignored.

5.P.3 Diffusion in a Tapered Pore

In Section 5.E.3 we considered steady-state binary gas-phase diffusion at constant

pressure and temperature through the pore shown in Figure 5.E.3-1, which has a

nonconstant circular cross-sectional area with a radius R = R

− β

√

z,whereβ

is a constant. The concentration at the mouth of the pore was held constant at

(moles/volume), whereas it was maintained at zero at z = L. We scaled this

problem to determine a criterion to assess when radial diffusion could be neglected.

Determine when this assumption is no longer valid.

5.P.4 Liquid Evaporation for Short Contact Times

In Section 5.7 we considered the unsteady-state evaporation of a pure liquid con-

tained in a cylindrical tube that initially contained none of the evaporating compo-

nent in the gas phase and for which the evaporating component concentration was

maintained at zero, as shown in Figure 5.7-1. We considered a long-time scaling

to assess when quasi-steady-state could be assumed. Here we consider this same

problem for very short contact times.

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

and boundary conditions.

(b) Consider an integral mass balance in order 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 entrance

of the tube 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 the entrance of the tube.

338 APPLICATIONS IN MASS TRANSFER

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

transfer ﬂux.

5.P.5 Mass Transfer to Film Flow Down a Vertical Plane

In Section 5.6 we applied scaling analysis to the absorption of a soluble gas into

falling ﬁlm ﬂow. We found that if the solutal Peclet number were sufﬁciently

large, the mass transfer was conﬁned to a region of inﬂuence in the vicinity of the

liquid–gas interface. Moreover, if the solutal Peclet number were sufﬁciently large,

we found that the convective mass transfer was inﬂuenced only by the velocity at

the liquid–gas interface. However, both of these approximations have limitations

that we explore in this problem.

(a) Based on the scaling analysis in Section 5.6, determine when the region of

inﬂuence or mass-transfer boundary layer penetrates the full distance from

the liquid–gas interface to the solid boundary.

(b) Determine when the assumption that the convective mass transfer is inﬂu-

enced only by the surface velocity breaks down.

ﬁlm wherein axial diffusion cannot be neglected.

(d) Discuss how the scaling analysis done in Section 5.6 can be used to solve

the full set of elliptic differential equations that are required to describe

mass transfer at the top of the liquid ﬁlm.

5.P.6 Mass Transfer to Film Flow Down a Vertical Cylinder

Consider the absorption of a sparingly soluble component from an inviscid gas into

a liquid ﬁlm in fully developed laminar ﬂow down a cylindrical wire of radius R

The liquid–gas interface is located at R

, as shown in Figure 5.P.6-1. The liquid

ﬁlm has an initial concentration c

at z = 0 and an interfacial concentration c

established through equilibrium with the adjacent gas phase. The velocity proﬁle

in the liquid ﬁlm is given by

= U

− R

) − 2R

ln(r/R

)

− R

) − 2R

ln(R

)

(5.P.6-1)

where U

is the maximum liquid velocity: namely, at the liquid–gas interface.

(a) Use scaling analysis to develop a criterion for assuming that the mass transfer

is conﬁned to a region of inﬂuence or boundary layer near the liquid–gas

interface.

(b) Develop a criterion for assuming that the convective mass transfer is inﬂu-

enced only by the surface velocity of the liquid ﬁlm.

(d) Develop a criterion for assuming that the curvature effects can be ignored.

PRACTICE PROBLEMS 339

Cylindrical rod

Liquid film

Velocity profile

Figure 5.P.6-1 Mass transfer of a sparingly soluble solute from an inviscid gas to a liquid

ﬁlm ﬂowing down a vertical cylindrical rod.

5.P.7 Mass Transfer with Chemical Reaction for Flow Between

Semipermeable Membranes

In Section 5.4 we considered fully developed laminar ﬂow between two parallel

semipermeable membrane boundaries through which a component was injected that

reacted with the liquid feed stream. We considered a low solutal Peclet number

scaling for which the convective transport of species could be neglected. However,

we considered the implications of both a low and a high Thiele modulus, corre-

sponding to very slow and very fast homogeneous reaction conditions. Consider

this same ﬂow geometry, as shown in Figure 5.4-1 for the special case of a high

Thiele modulus when convective mass transfer is important. For this scaling use

the following dimensionless variables:

∗

≡

;



∂c

∂x



∗

≡

∂c

∂x

; x

∗

≡

; y

∗

≡

− y

(5.P.7-1)

(a) Indicate why a separate scale is needed for the axial derivative of the con-

centration and why a reference factor is needed for the transverse coordinate.

(b) Determine the scale and reference factors for this convective mass-transfer

problem.

aries within which all the permeating reactant is consumed.

(d) Determine the criterion for assuming that the velocity proﬁle can be lin-

earized.

(e) Determine the criterion for ignoring the axial diffusion of species.

(f) Indicate when the criterion in part (e) breaks down.

(g) When the criterion in part (e) is not satisﬁed, the full elliptic problem must

be considered. Often an appropriate downstream boundary condition to solve

340 APPLICATIONS IN MASS TRANSFER

this elliptic problem is not known. Indicate how the results of the scaling

analysis in parts (a) through (e) can be used to solve the full elliptic problem

in the entry region.

5.P.8 Entrance Effect Limitations for Laminar Tube Flow with a Fast

Heterogeneous Reaction

In Example Problem 5.E.8 we considered fully developed laminar ﬂow in a cylin-

drical tube for which a solute underwent an irreversible heterogeneous reaction at

the wall. We considered the special case of a very large Damk

ohler number for

which our scaling analysis indicated that there was a region of inﬂuence or solutal

boundary layer near the tube wall across which the concentration dropped from its

initial value to essentially zero.

(a) Scaling analysis for large Damk

ohler numbers led to a simpliﬁed set of

describing equations given by (5.E.8-18) through (5.E.8-21). Indicate when

the criteria leading to these simpliﬁed equations break down.

(b) Determine the thickness of the region of inﬂuence near the entrance of the

tube wherein the axial diffusion term cannot be neglected.

an elliptic system of equations for which a downstream boundary condition

is required. Indicate how the results of the scaling analysis in Example Prob-

lem 5.E.8 can be used to solve the full elliptic problem in the entry region.

(d) In Section 5.4 we considered steady-state convective mass transfer for the

case of a homogeneous chemical reaction and found that for sufﬁciently small

Peclet numbers, the convective transport of species could be neglected. Is a

small Peclet number approximation ever justiﬁed for steady-state mass trans-

fer in the convective mass-transfer problem being considered in Example

Problem 5.E.8?

5.P.9 Aeration of Water Containing Aerobic Bacteria

Consider a spherical bubble consisting of pure oxygen with a radius R rising at

its terminal velocity U

through at stationary tank of water of depth L.Thewater

contains aerobic bacteria that consume dissolved oxygen via a zeroth-order reaction

whose rate constant is k

(moles/volume·time). The water is assumed to have no

oxygen at the bottom of the tank where the bubbles enter. The bubbles can be

assumed to become saturated with water vapor very quickly relative to the oxygen

transfer to the liquid. The corresponding equilibrium concentration of oxygen in

water, denoted by c

(moles/volume), will be assumed to be unaffected by the

small increase in the bubble pressure associated with the decrease in bubble size due

to oxygen absorption. We assume that the bubbles rise such that the hydrodynamics

cause the mass transfer to be conﬁned to a ﬁlm of liquid having a constant thickness

that surrounds each bubble as shown in Figure 5.P.9-1. However, δ

is not

necessarily thin in comparison to the radius of the bubble. We ignore any effects

PRACTICE PROBLEMS 341

Oxygen

bubble

Stagnant film determined

by hydrodynamics

= 0

Liquid water containing

aerobic bacteria

Figure 5.P.9-1 Water-saturated oxygen bubble of radius R rising at its terminal velocity

in liquid water containing aerobic bacteria that consume the oxygen via a zeroth-order

reaction.

of the change in bubble size and also assume that the molar density of the liquid,

c, remains constant. Use the following dimensionless variables containing arbitrary

scale and reference factors for scaling this problem:

∗

≡

; N

∗

≡

; r

∗

≡

r − r

; t

∗

≡

(5.P.9-1)

(a) Consider a spherical coordinate system located at the center of a single

rising bubble as shown in Figure 5.P.9-1; write the appropriate form of the

species-balance equation in spherical coordinates along with the requisite

initial and boundary conditions; do not ignore the bulk-ﬂow contribution

to the molar ﬂux; note that if the oxygen bubbles are saturated with water

vapor, this is unimolecular diffusion.

(b) Introduce the dimensionless variables deﬁned above into your describing

equations and determine the relevant scale and reference factors; note that

the observation time for this problem is well-deﬁned since it will be the

time required for a bubble to rise at its terminal velocity through the entire

depth of liquid.

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

(e) Determine the criterion for ignoring the effect of the bacterial consumption

of oxygen.

(f) Consider the case of a very fast bacterial consumption of oxygen such that

the oxygen concentration is essentially reduced to zero within a distance

much smaller than the ﬁlm thickness δ

; determine the region of inﬂu-

ence or boundary-layer thickness wherein all the diffusive mass transfer

occurs.

342 APPLICATIONS IN MASS TRANSFER

(g) Discuss how the scaling of this problem would change if we allowed

for the effects of the decrease in bubble size due to the oxygen absorp-

tion.

5.P.10 Dissolution of a Spherical Capsule for a Concentrated Solution

In Section 5.E.4 we considered the dissolution of a spherical capsule of pure solid

component A as shown in Figure 5.E.4-1 and used scaling analysis to assess how

the describing equations could be simpliﬁed. One assumption that we made was

that the solution was dilute, which permitted us to ignore the bulk-ﬂow contribution

to Fick’s law of diffusion. In this problem we no longer assume dilute solutions

but assume constant physical properties and that the effect of any reaction products

resulting from the dissolution can be ignored.

(a) Rescale this problem appropriate to a slow chemical reaction to assess when

the reaction and the bulk-ﬂow contributions can be ignored.

(b) Rescale this problem appropriate to a fast chemical reaction to assess when

quasi-steady-state can be assumed and when the bulk-ﬂow contribution can

be ignored.

5.P.11 Dissolution of a Spherical Capsule for a Bimolecular Reaction

In Section 5.E.4 we considered the mass transfer from a spherical capsule of pure

solid component A as shown in Figure 5.E.4-1 for the case of a ﬁrst-order dissolu-

tion chemical reaction. Assume now that the dissolution kinetics are governed by

a bimolecular reaction of component A with the stomach liquid B, for which the

reaction rate (moles/volume·time) is given by R

= k

,wherek

is the second-

order reaction-rate constant. We do not assume dilute solutions in this problem but

assume constant physical properties and that the effect of any reaction products

resulting from the dissolution can be ignored.

(a) Write the appropriately simpliﬁed species-balance equations and their initial

and boundary conditions in spherical coordinates; express the concentrations

in terms of mole fractions.

(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

spherical capsule and the stomach liquid.

neous chemical reaction.

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

(e) Determine the criterion for ignoring the bulk-ﬂow contribution to the mass-

transfer ﬂux.

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

transfer.