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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SUMMARY 293

this system. Hence, one needs to consider only three variable groups in correlating

performance data for a membrane blood oxygenator. Note that the angular fre-

quency could be isolated into just one dimensionless group. However, the resulting

group would still contain the variable amplitude and hence would not be ﬁxed by

specifying the blood–oxygen system.

Step 8 in the scaling procedure for dimensional analysis involves exploring the

correlation for various limiting values of the dimensionless groups. For large Sc

(i.e., for blood) we can use the expansion in Sc

−1

suggested by equation (2.4-3)

truncated to one term to conclude that the oxygenator performance can be correlated

in terms of only four dimensionless groups; that is,

Sh = f



Gz,

ωR

Aω



for Sc

−1

→ 0 (5.10-34)

The dimensionless group ωR

/ν is the ratio of the characteristic time for viscous

penetration to that for the periodic wall oscillations. One might anticipate that the

wall oscillations will not have much effect on the Sherwood number for very small

values of ωR

/ν since this implies negligible wall motion. We would also not

expect much effect on the Sherwood number for very large values of ωR

/ν,since

this implies negligible time for the wall oscillations to penetrate the ﬂuid. The

dimensionless group Aω/

U is the ratio of the wall velocity to the average velocity

of the ﬂuid. If Aω/

U is very small, the wall oscillations are insigniﬁcant relative to

the ﬂuid velocity, in which case they will have a negligible effect on the Sherwood

number. If Aω/

U is very large, it implies a very slow ﬂow for which the oxygen

transport is not limited by contact time; hence, the oscillations will also have a neg-

ligible effect for large values of this group. These arguments suggest that the effects

of the oscillations on the Sherwood number will exhibit a maximum with respect to

the values of ωR

/ν and Aω/U ; that is, the effect of wall oscillations on the mass

transfer involves a “tuned” response whereby the maximum effect will be achieved

only over a relatively narrow range of oscillation amplitudes and frequencies.

The design of a membrane–lung oxygenator will be revisited in Chapter 7.

This same problem will be analyzed in Section 7.2 using

◦

(1) scaling analysis

to achieve the minimum parametric representation rather than using the scaling

approach to dimensional analysis. We will see that

◦

(1) scaling analysis yields far

more information about the design of a membrane–lung oxygenator. In particular,

it will show that the oxygenator performance can be correlated in terms of four

dimensionless groups rather than ﬁve. Moreover,

◦

(1) scaling analysis permits pre-

dicting the optimum frequency, within a multiplicative constant of

◦

(1), required

to achieve maximum enhancement of the oxygen mass transfer to the blood.

5.11 SUMMARY

In Section 5.2, we provided an introduction to the step-by-step procedure for apply-

ing scaling analysis in mass transfer. We considered binary diffusion in a stagnant

294 APPLICATIONS IN MASS TRANSFER

ﬁlm of liquid subject to an instantaneous change in the concentration at one of

its boundaries. Scaling was used to determine the criterion for applying the ﬁlm

theory model. This criterion requires that the Fourier number or ratio of the contact

to diffusion time be very large. Scaling was also used to assess when the bulk-ﬂow

contribution that arises from a net mass-transfer ﬂux can be neglected. This crite-

rion requires that either the mass ﬂuxes be nearly equal in magnitude or that the

relative concentration change across the ﬁlm be small. When this bulk-ﬂow con-

tribution is potentially important, it is necessary either to know some relationship

between the mass-transfer ﬂuxes or to have an equation of state for the mass den-

sity. In this problem the ratio of the mass ﬂuxes was speciﬁed.

Since this problem

could be solved analytically, it was possible to estimate the error incurred when

the bulk-ﬂow effect was ignored. This was seen to be less than 0.1% when the

criterion given by equation (5.2-28) was

◦

(0.01). Note that the ﬁlm theory model

developed in this section is frequently used to correct mass-transfer coefﬁcients

obtained from literature correlations for the effects of bulk ﬂow.

Interestingly,

the ratio of the mass-transfer coefﬁcient in the presence to that in the absence of

bulk-ﬂow effects is not a strong function of the actual mass-transfer conﬁguration.

For this reason ﬁlm theory can be used to correct for the bulk-ﬂow effects even

when the model is not applicable to the particular mass-transfer problem.

In Section 5.3 we considered the same physical situation as in Section 5.2. How-

ever, short-contact-time or small Fourier number scaling was done whereby the

diffusion and unsteady-state terms were balanced in the species-balance equation.

This led to a length scale that was identiﬁed with a region of inﬂuence or solu-

tal boundary layer within which all the mass transfer is conﬁned. The resulting

simpliﬁed describing equations provide the basis for penetration theory. The latter

is the short-contact-time complement to ﬁlm theory. Penetration theory provides a

better model for correcting mass-transfer coefﬁcients obtained from the literature

for bulk-ﬂow effects when the contact time or Fourier number is small.

Mass transfer in fully developed laminar ﬂow between permeable parallel mem-

brane walls in the presence of a homogeneous chemical reaction was considered

in Section 5.4. The presence of both the transverse and axial diffusion terms made

the describing equations elliptic. This complicated the solution since the required

downstream boundary condition is often unknown. The concept of local scaling in

mass transfer was introduced in this problem, whereby one considers the describ-

ing equations within a domain deﬁned by some arbitrary distance in the principal

direction of ﬂow that is assumed to be constant during the scaling analysis. In

contrast to the preceding two examples, there was no explicit concentration scale

in this problem. Hence, the concentration scale was determined by balancing the

mass ﬂux through the permeable membrane walls and transverse diffusion terms.

Scaling analysis led to three important dimensionless groups: the Peclet number

for mass transfer, Schmidt number, and Thiele modulus. The Peclet number for

mass transfer is a measure of the convection to diffusion of mass. The Schmidt

Note that Example Problem 5.E.1 considers scaling to determine when the bulk-ﬂow effect can be

neglected when an equation of state for the mass density is known.

The interested reader is referred to Bird et al., Transport Phenomena, pp 704–706.

SUMMARY 295

number is a measure of the viscous transport of momentum to the diffusive transfer

of mass. The Thiele modulus is a measure of the characteristic time for diffusion

relative to that for homogeneous reaction. The Peclet number has a role in mass

transfer analogous to that of the Reynolds number in ﬂuid dynamics. For example,

we found that the convective mass transfer could be ignored if the Peclet number

was very small; this is analogous to the low Reynolds number or creeping-ﬂow

approximation in ﬂuid dynamics. We also found that the complicating effects of

axial diffusion could be ignored if the aspect ratio was very small. The combina-

tion of small Peclet number and small aspect ratio in mass transfer is analogous to

the lubrication-ﬂow approximation in ﬂuid dynamics. We found that for very large

Thiele moduli, the homogeneous chemical reaction was conﬁned to a thin region

of inﬂuence or boundary layer near the permeable membrane walls.

In Section 5.5 we considered a complementary problem to that discussed in

Section 5.4: mass transfer with a heterogeneous chemical reaction at the wall for

fully developed laminar tube ﬂow. In this case, scaling was used to determine

when the classical plug-ﬂow reactor approximation can be made; that is, when

one can ignore any radial concentration gradients and thereby represent the axial

convection using the average rather than the local velocity. It was necessary to

introduce separate scales for both the axial and radial concentration gradients since

neither of these scaled with the characteristic length divided by the characteristic

time. This problem introduced the second Damk

ohler number, which is a measure

of the time scale for radial diffusion to that for the heterogeneous reaction. When

this dimensionless group was small, the plug-ﬂow reactor approximation can be

made. The small Damk

ohler number approximation in mass transfer is analogous to

the small Biot number approximation in heat transfer in that both imply negligible

resistance to transfer within the control volume relative to that at some boundary.

The problem considered in Section 5.6 involved mass transfer to falling ﬁlm

ﬂow. Scaling was used to determine when the mass transfer could be assumed

to be conﬁned to a thin boundary layer or region of inﬂuence near the interface.

For large Peclet numbers the mass transfer will be conﬁned to a boundary layer

that is sufﬁciently thin to assume that the ﬁlm is inﬁnitely thick. Moreover, if

the product of the Peclet number and the length-to-thickness aspect ratio is large,

axial diffusion can be neglected. These two approximations greatly simplify the

describing equations; in particular, they obviate the need to apply a downstream

boundary condition, which in many cases is not known.

In Section 5.7 we used scaling to assess when the quasi-steady-state (QSS)

approximation can be made for the evaporation of a pure volatile liquid into an

insoluble gas in a cylindrical tube. Quasi-steady-state implies that time does not

enter explicitly in the describing differential equations, but implicitly through one or

more boundary conditions. This is a moving boundary problem for which an aux-

iliary condition is required to locate the interface. The velocity that arises because

of the diffusive mass transfer was determined in this problem from the additional

condition that the gas was insoluble in the liquid. A proper scaling analysis required

introducing a reference factor for the independent variable since it was not naturally

referenced to zero. Moreover, it was necessary to introduce a separate scale factor

296 APPLICATIONS IN MASS TRANSFER

for the interface velocity since it does not necessarily scale with the ratio of

the characteristic length divided by the characteristic time. Since this problem

involved a transformation from a stationary to a moving coordinate system, a

pseudo-convection term was generated. Scaling analysis indicated that QSS can

be assumed if the Fourier number is very large. Moreover, it provided criteria for

when the convective mass transfer and pseudoconvection term could be neglected.

In Section 5.8 we applied scaling analysis to permeation through a polymeric

membrane whose swelling caused the diffusivity to be concentration-dependent. For

this reason it was necessary to scale the diffusivity as well. Two length scales were

possible depending on how markedly the diffusivity changed with concentration.

If the resistance to mass transfer was distributed through the entire cross-section

of the membrane, the appropriate length scale was its thickness. However, if the

diffusivity decreased markedly with concentration near one of the boundaries, the

resistance to mass transfer was conﬁned to a thin region of inﬂuence whose thick-

ness was the appropriate length scale. To determine whether the diffusivity could be

assumed to be constant, the equation describing its concentration-dependence was

expanded in a Taylor series in which only the ﬁrst-order correction was retained

in the scaling analysis. Scaling then identiﬁed the condition required to ignore this

ﬁrst-order correction. For this problem it was possible to compare the solution to

the complete describing equations with the simpliﬁed form of these equations for

both negligible swelling and signiﬁcant swelling; this conﬁrmed that the simpliﬁed

equation emanating from scaling provided accurate solutions when the appropriate

criteria for the validity of these approximation were satisﬁed.

In Section 5.9 we applied scaling to a free-convection mass-transfer problem,

that is, to a problem wherein the driving force for ﬂow was internal to the system,

in this case due to density variations created by concentration gradients. Scaling

was employed to arrive at the free-convection boundary-layer equations and to

determine when curvature effects could be neglected. This problem introduced the

solutal Grashof and Rayleigh numbers, which are the free convection analogues

of the Reynolds and Peclet numbers; that is, the former is a measure of the ratio

of the free convection to viscous transport of momentum, whereas the latter is a

measure of the free convection to diffusive transport of species.

Scaling was applied to dimensional analysis in Section 5.10. In contrast to

◦

(1) scaling analysis, the scaling approach to dimensional analysis merely seeks to

arrive at the minimum parametric representation of the problem; that is, to obtain

a set of describing equations in terms of the minimum number of dimensionless

parameters. The scaling approach to dimensional analysis was applied here to a

novel membrane–lung oxygenator that employed axial oscillations to enhance the

mass transfer. Scaling analysis was used to determine the dimensionless groups

required to correlate the effects of the oscillations on the performance of the oxy-

genator. This problem introduced the Sherwood number, a dimensionless group that

is a measure of the ratio of the overall mass transfer to that by diffusion alone, and

the Graetz number. The latter is closely related to the Peclet number since it is a

measure of the ratio of the convection to diffusion of species. However, the Graetz

number includes an aspect ratio that accounts for the effect of a limited contact

EXAMPLE PROBLEMS 297

time. This example illustrated the advantages of the scaling analysis methodology

for dimensional analysis relative to using the conventional Pi theorem approach in

that the latter did not lead to the minimum parametric representation.

5.E EXAMPLE PROBLEMS

5.E.1 Evaporative Casting of a Polymer Film

Consider a binary mixture of volatile solvent A and nonvolatile polymer B that has

initial mass fractions ω

and ω

, respectively, cast as a thin planar ﬁlm having

an initial thickness L

on an impermeable plate as shown in Figure 5.E.1-1. At

time t = 0, the volatile solvent begins to evaporate into the gas phase, thereby

causing the ﬁlm to thin so that its instantaneous thickness is L(t); evaporative

cooling effects will be assumed to be negligible in this analysis. The evaporative

mass transfer is described via a lumped-parameter approach with an appropriate

mass-transfer coefﬁcient. The overall mass density of the polymer ﬁlm is assumed

to be given by

ρ = ω

+ ω

⇒ ρ = ρ

+ ρ

(5.E.1-1)

where ρ

is the pure component mass density of component i and ρ

≡ρ

− ρ

;

note that an alternative equation of state for the density could be used in this

scaling analysis. We use scaling to determine criteria for when the following

approximations can be made: Convective mass transfer arising from densiﬁca-

tion can be neglected; convective mass-transfer effects on the ﬁlm thinning can

Impermeable support plate

L(t)

Casting solution

Gas phase

Polymer

concentration

profile

Solvent

concentration

profile

Direction of

increasing

concentration

Figure 5.E.1-1 Representative concentration proﬁles during the evaporative casting of a

dense ﬁlm from a solution of a volatile solvent and nonvolatile polymer on an impermeable

plate; the instantaneous thickness of the planar ﬁlm is L(t).

298 APPLICATIONS IN MASS TRANSFER

be ignored; the density can be assumed to be constant; quasi-steady-state applies;

and the mass transfer can be assumed to be gas-phase controlled. This example

illustrates how to handle a mass-transfer problem involving a nonconstant den-

sity and to scale properties such as the mass density that are functions of the

dependent variable; it also provides another example of a moving boundary

problem.

The appropriately simpliﬁed forms of the continuity and species-balance equ-

ations given by (C.1-1) and (G.1-1) in the Appendices are (step 1)

∂ρ

∂t

=−

∂

∂z

(ρu) ⇒ ρ

∂ω

∂t

=−

∂

∂z

(ρu) (5.E.1-2)

∂ρ

∂t

∂(ω

ρ)

∂t

=−

∂n

∂z

(5.E.1-3)

where

=−ρD

∂ω

∂z

+ ω

ρu (5.E.1-4)

in which u is the mass-average velocity. Equations (5.E.1-2) and (5.E.1-3) can be

combined to obtain

∂

∂z

(ρu) =

ρ

/ρ

1 + ω

ρ

/ρ

∂n

∂z

(5.E.1-5)

Equation (5.E.1-5) can, in turn, be integrated to obtain the following explicit

equation for the mass-average velocity u:

u =



ρ

/ρ

1 + ω

ρ

/ρ

∂n

∂z

dz (5.E.1-6)

u =



ρ

/ρ

1 + ω

ρ

/ρ

(5.E.1-7)

in which the following boundary conditions corresponding to an impermeable lower

solid boundary have been used:

u = 0,n

= 0atz = 0 (5.E.1-8)

A boundary condition is also required at the liquid–gas interface; this is obtained

by an integral species balance over the entire ﬁlm thickness:



L(t)

(ω

ρ)dz +



∞

L(t)

( ˜ω

˜ρ)dz = 0atz = L(t) (5.E.1-9)

where ˜ω

and ˜ρ denote the mass fraction of component A and mass density in

the gas phase. Applying Leibnitz’s rule for differentiating an integral given by

EXAMPLE PROBLEMS 299

equation (H.1-2) in the Appendices and substituting equations (5.E.1-3) and (5.E.1-

4) for the casting solution and similar equations for the gas phase yields

ρD

∂ω

∂z

= ω

ρu −k

•

− ω

at z = L(t), 0 ≤ t ≤∞

(5.E.1-10)

In arriving at equation (5.E.1-10), the mass ﬂux of component A in the gas phase

at the liquid–gas interface was represented via a lumped-parameter approach in

terms of the mass-transfer coefﬁcient k

•

. This introduces the distribution coefﬁ-

cient K

, which incorporates the effects of nonideal solution behavior and other

concentration-dependent factors needed to interrelate the liquid and gas concentra-

tions at the interface. Equation (5.E.1-10) also assumes that the bulk of the gas

phase contains none of the evaporating solvent and that the mass density of the

gas phase is much less than that of the casting solution.

Since this is a moving boundary problem, owing to the mass loss and densiﬁca-

tion, an auxiliary condition is required to locate the instantaneous position of the

interface. This is determined by an integral mass balance as follows:



L(t)

ρdz=−k

•

at z = L(t) (5.E.1-11)

Applying Leibnitz’s rule for differentiating an integral given by equation (H.1-2) in

the Appendices and substituting equation (5.E.1-2) yields the following auxiliary

condition to determine L(t):

= ρu − k

•

at z = L(t) (5.E.1-12)

In arriving at equation (5.E.1-12) we have used the boundary condition that u = 0at

z = 0, corresponding to an impermeable boundary. The initial conditions required

to solve Equations (5.E.1-3), (5.E.1-10), and (5.E-12) are given by

= ω

,L= L

at t = 0, 0 ≤ z ≤ L(t) (5.E.1-13)

Equation (5.E.1-12) can be used to simplify equation (5.E.1-10) to yield the fol-

lowing form of the boundary condition at the moving interface:

ρD

∂ω

∂z

=−k

•

(1 − ω

) at z = L(t), 0 ≤ t ≤∞ (5.E.1-14)

Introduce the following dimensionless dependent and independent variables

(steps 2, 3, and 4):

∗

≡

; ρ

∗

≡

; u

∗

≡

; n

∗

≡

;

∗

≡

; z

∗

≡

; t

∗

≡

; L

∗

≡

;





∗

≡

(5.E.1-15)

300 APPLICATIONS IN MASS TRANSFER

Note that we have introduced a scale factor for ω

even though it is dimension-

less with a maximum possible value of 1 because we seek to bound it of

◦

(1).

Indeed, the maximum concentration ω

could be much less than 1. However, it

is not necessary to introduce a reference factor for ω

since its minimum value is

zero, corresponding to complete evaporation of the volatile component. We have

introduced a separate scale for dL/dt, the velocity of the interface since there

is no reason why this should scale as z

However, we have not introduced

a separate scale for the derivative of the concentration since we are considering

a longer time scaling for which the concentration will undergo its characteristic

change over the instantaneous thickness of the ﬁlm, not over some boundary layer

near the upper interface.

Scale factors are also introduced for the mass density ρ

and distribution coefﬁcient K

since they depend on the concentration.

Introduce these dimensionless variables into equations ((5.E.1-1)), (5.E.1-3),

(5.E.1-4), (5.E.1-5), (5.E.1-7), (5.E.1-8), (5.E.1-12), and (5.E.1-14), and divide

each equation through by the dimensional coefﬁcient of one term that should be

retained in order to maintain physical signiﬁcance to obtain the following set of

dimensionless describing equations (steps 5 and 6):

∗

ρ

∗

(5.E.1-16)

∂(ω

∗

)

∂t

∗

=−

∂n

∗

∂z

∗

(5.E.1-17)

∗

−

∗

∂ω

∗

∂z

∗

(5.E.1-18)

∗

ρ

∗



∗

1/ρ

∗

1 + (ρ

/ρ

)(ω

∗

/ρ

∗

)

∂n

∗

∂z

∗

(5.E.1-19)

∗

at t

∗

= 0, 0 ≤ z

∗

≤

(5.E.1-20)

∗

= 0,u

∗

= 0atz

∗

= 0, 0 ≤ t

∗

≤∞ (5.E.1-21)

∗

∂ω

∗

∂z

∗

=−

•

∗

(1 − ω

∗

) at z

∗

, 0 ≤ t

∗

≤∞

(5.E.1-22)

∗





∗

−

•

∗

at z

∗

(5.E.1-23)

The consequences of scaling the interface velocity with z

were discussed and illustrated in

Section 4.7.

However, for a short-time scaling where the concentration change is conﬁned to a region of inﬂuence

near the upper surface, one must introduce a separate scale for the spatial derivative of the concentration,

which is determined from the mass-ﬂux boundary condition given by equation (5.E.1–14); this short-

time scaling is considered in Practice Problem 5.P.20.

EXAMPLE PROBLEMS 301

The following considerations dictate determining our scale factors (step 7).

Assessing when quasi-steady-state can be assumed necessarily implies a scaling

analysis for longer contact times, for which the effect of the evaporation will have

penetrated through the entire liquid ﬁlm. This consideration is important since it

implies that the characteristic length is the entire thickness of the liquid ﬁlm rather

than some region of inﬂuence near the liquid–gas interface. Hence, to bound the

dimensionless spatial coordinate and ﬁlm thickness to be

◦

(1), we set the appropri-

ate dimensionless groups in equation (5.E.1-20) equal to 1 to obtain z

= L(t) and

= L

. Note that z and L scale differently since for longer contact times, z ranges

between 0 and L(t), whereas L(t) ranges between L

initially to some smaller value

at the end of the evaporation process. The scale for the mass fraction comes from

equation (5.E.1-20) and is given by ω

= ω

. The density scale comes from

equation (5.E.1-16) and is given by ρ

= ρ

. Since we seek to determine when

the quasi-steady-state assumption is applicable, our time scale is the observation

time; that is, t

= t

. The scale for the species mass ﬂux comes from balancing the

principal terms in equation (5.E.1-18) to obtain n

= D

/L. The scale for

the front velocity comes from balancing the principal terms in equation (5.E.1-23),

which yields

= k

•

/ρ

. A proper scale for K

that bounds K

∗

to be

◦

(1) is the initial value of K

denoted here by K

for which the concentration is

known. The scale for the mass-average velocity comes from equation (5.E.1-19),

which yields u

= D

ρ

/ρ

These scale factors then result in the following set of dimensionless describ-

ing equations, which will permit us to assess when quasi-steady and negligible

convection can be assumed (step 8):

∗

= 1 +

ρ

∗

(5.E.1-24)

∂(ω

∗

)

∂t

∗

− Bi

∗





∗

∂(ω

∗

)

∂z

∗

=−

∂n

∗

∂z

∗

(5.E.1-25)

∗

ρ

∗

− ρ

∗

∂ω

∗

∂z

∗

(5.E.1-26)

∗



∗

1/ρ

∗

1 + ω

(ρ

/ρ

)(ω

∗

/ρ

∗

)

∂n

∗

∂z

∗

(5.E.1-27)

= 1,L

∗

= 1att

∗

= 0, 0 ≤ z

∗

≤ 1 (5.E.1-28)

∗

= 0,u

∗

= 0, at z

∗

= 0, 0 ≤ t

∗

≤∞ (5.E.1-29)

∗

∂ω

∗

∂z

∗

=−Bi

∗

(1 − ω

∗

) at z

∗

= 1, 0 ≤ t

∗

≤∞

(5.E.1-30)

∗





∗

ρ

∗

− K

∗

at z

∗

= 1 (5.E.1-31)

302 APPLICATIONS IN MASS TRANSFER

where Fo

≡ D

and Bi

≡ k

•

L/D

are the solutal Fourier and

solutal Biot numbers, respectively, for mass transfer. Note that an additional

pseudo-convection term now appears in equation (5.E.1-25) because of the trans-

formation from a ﬁxed coordinate system to one that is referenced to the moving

interface between the liquid and gas phases. Pseudo-convection terms will always

arise when one transforms from a stationary coordinate system to a moving sys-

tem. Equation (5.E.1-26) indicates that the convection arising from densiﬁcation

will have a negligible effect on the mass-transfer ﬂux if the following criterion is

satisﬁed:

ρ

 1 ⇒

ρ

◦

(0.01) (5.E.1-32)

This criterion indicates that convective transport can be ignored when the densities

of the two components are nearly the same (i.e., ρ

∼

0) or for very dilute

solutions. Equation (5.E.1-31) indicates that the convection will have a negligible

effect on the ﬁlm thinning if the following criterion is satisﬁed:

ρ

 1 ⇒

ρ

◦

(0.01) (5.E.1-33)

This criterion is also satisﬁed when the density difference between the two com-

ponents is small or when the Biot number is very large, which implies a negligible

resistance to mass transfer in the gas phase relative to that in the liquid. Quasi-

steady-state mass transfer can be assumed if the following criterion is satisﬁed:

 1 (5.E.1-34)

corresponding to sufﬁciently long contact times. Note that quasi-steady-state implies

that the time dependence enters implicitly through the boundary condition at the

upper surface rather than explicitly via the unsteady-state term in equation (5.E.1-

25). The concentration dependence of the density given by equation (5.E.1-24) can

be ignored if the following criterion is satisﬁed:

ρ

 1 ⇒

ρ

◦

(1) (5.E.1-35)

Equation (5.E.1-30) indicates that an additional simpliﬁcation is possible when the

Biot number is very small: namely, that

∂ω

∗

∂z

∗

 1ifBi

 1 (5.E.1-36)

This simpliﬁed problem, which is the small Biot number approximation for mass

transfer, is solved in a manner similar to that shown in detail in Section 4.4 for an

analogous problem in heat transfer.