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 413

formance data given in Figure 6.P.29-1.

(d) In which ranges of the reaction rate constant would it advantageous

to employ cocurrent (parallel) rather than countercurrent (antiparallel)

ﬂow?

7 Applications in Process Design

The optimist looked at his glass and said to the bartender that it was half full.

The pessimist looked at his glass and said to the bartender that it was half empty.

The engineer looked at both glasses and said to the bartender that he would

◦

(1).

7.1 INTRODUCTION

The focus and organization of this chapter are a dramatic departure from those

of the preceding chapters. In Chapters 3 through 6 we focused primarily on using

scaling to justify classical approximations made in ﬂuid dynamics, heat transfer,

mass transfer, and mass transfer with chemical reaction. As such, the thrust of the

preceding chapters was primarily pedagogical rather than application-oriented. In

contrast, in this chapter we show how scaling analysis has been applied to the

design of four practical engineering processes. Scaling analysis played a pivotal

role in advancing the technologies involved in these four examples. It not only

helped in simplifying the describing equations so that a tractable solution could be

developed but was also used to design the process; that is, to determine process

parameters such as the contact time, oscillation amplitude and frequency, vessel

length, adsorbent particle size, and wall temperature that insured effective and

efﬁcient operation. We also demonstrate the utility of using

◦

(1) scaling analysis to

obtain optimal dimensionless groups for correlating experimental or numerical data.

In Section 7.2 we revisit the design of a membrane–lung oxygenator. In Section

5.10 the scaling approach to dimensional analysis was used to determine the dimen-

sionless groups required to correlate data for the performance enhancement that

could be achieved by applying axial vibrations of the hollow-ﬁber membranes in

this oxygenator. In this chapter,

◦

(1) scaling analysis is used to achieve the mini-

mum parametric representation. Subsequently, we will see that

◦

(1) scaling analysis

provides considerably more information than just the scaling analysis approach to

dimensional analysis or the Pi theorem. Section 7.3 applies scaling analysis to

Anonymous—attributed to a very practical engineer.

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

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

414

DESIGN OF A MEMBRANE–LUNG OXYGENATOR 415

the design of the pulsed pressure swing adsorption (PSA) process. This process

is of particular interest for use in portable oxygenators for people suffering from

severe respiratory problems or chronic obstructive pulmonary disease. This example

involves coupled ﬂow through porous media and mass transfer. In Section 7.4

we consider the thermally induced phase-separation (TIPS) process for fabricating

semipermeable polymeric membranes. The TIPS process is used to make high ﬂux

membranes for water desalination and reclamation as well as for solvent recovery.

This example involves coupled heat and mass transfer. In Section 7.5 we use scal-

ing analysis to design the ﬂuid–wall aerosol ﬂow reactor for converting methane

into hydrogen. The ﬂuid–wall aerosol ﬂow reactor is being developed to use solar

energy to drive the thermal decomposition of methane into hydrogen that can be

used as a clean-burning fuel to address the worldwide concern regarding global

warming. This example involves coupled ﬂuid dynamics, heat transfer, mass trans-

fer, and chemical reactions in a two-phase system. The examples in Sections 7.3,

7.4, and 7.5 use the microscale–macroscale modeling methodology to handle het-

erogeneities such as particles or a dispersed phase. In Section 7.6 we summarize

the scaling principles employed in each of these examples and the role that scal-

ing analysis played in advancing each of these technologies. Unworked practice

problems related to each of the four examples are included at the end of the chapter.

7.2 DESIGN OF A MEMBRANE–LUNG OXYGENATOR

In Section 5.10 we applied the scaling approach to dimensional analysis to develop

a correlation for the performance enhancement achieved by applying axial oscilla-

tions to the hollow ﬁbers in a membrane–lung oxygenator. Recall that the scaling

approach to dimensional analysis outlined in Section 2.4 does not provide as

much information as

◦

(1) scaling analysis for achieving the minimum paramet-

ric representation. In particular, it does not lead to groups whose magnitude can

be used to assess the relative importance of the various terms in the describing

equations. Moreover, it does not provide any insight into the fundamental mech-

anisms involved in the process. It also does not identify regions of inﬂuence or

boundary layers, which in some cases can reduce the number of dimensionless

groups. Here we apply

◦

(1) scaling analysis to design a membrane–lung oxygena-

tor. This example illustrates the value of using

◦

(1) scaling analysis to achieve the

minimum parametric representation in contrast to the simple scaling analysis or Pi

theorem approaches to dimensional analysis.

The membrane–lung oxygenator involves a bundle of permeable cylindrical

hollow-ﬁber membranes encased in a tubular housing. In this application the hollow-

ﬁber membranes do not cause any separation but rather, serve as a gas-permeable

barrier between the blood and the source of oxygen. The mass transfer of oxygen

is controlled on the blood side of the membrane because the hemoglobin “par-

ticles” that scavenge the oxygen are excluded from the wall region due to the

ﬂuid dynamics. Hence, efforts to improve oxygenator performance have focused

on various means to reduce the resistance to mass transfer on the blood side of

the membrane. A very effective way to enhance the mass transfer is to oscillate

416 APPLICATIONS IN PROCESS DESIGN

the hollow ﬁbers relative to the blood ﬂow to increase the concentration gradients

adjacent to the membrane.

In this section we use

◦

(1) scaling analysis to design

an oxygenator that employs periodic oscillations of the hollow ﬁbers to enhance

the mass transfer. It is sufﬁcient to consider the effect of the oscillations on the

oxygen mass transfer to the blood ﬂow in a single hollow-ﬁber membrane having

radius R and length L, as shown in Figure 7.2-1.

The initial steps involved in applying

◦

(1) scaling analysis to this problem are

similar to those used in the scaling analysis approach to dimensional analysis dis-

cussed in Section 5.10. One begins by writing the appropriate describing equations

for the oxygen mass transfer to the blood, which is assumed to be in fully devel-

oped periodically pulsed laminar ﬂow (step 1). The effect of the oscillating wall

on the oxygen mass transfer is assessed in terms of the mass-transfer coefﬁcient

•

deﬁned in terms of N

, the time-average molar ﬂux of oxygen at the inner

membrane wall, as follows:

•

≡

c

(7.2-1)

where c

is the log-mean concentration driving force, deﬁned as

c

≡

− c

) − (c

− c

)

ln[(c

− c

)/(c

− c

)]

(7.2-2)

in which c

is the oxygen concentration at the blood side of the membrane

that is dictated by thermodynamic equilibrium considerations, c

and c

are the

average concentrations in the blood at z = 0andatz = L, respectively, and

is deﬁned as

(

2π

(

2π



2π

dt (7.2-3)

= Aw cos wt

Figure 7.2-1 Single hollow ﬁber of radius R and length L in a membrane–lung oxy-

genator; axial oscillations of amplitude A and angular frequency ω are used to increase

the concentration gradients at the interior wall where the resistance to mass transfer is

concentrated.

R. R. Bilodeau, R. J. Elgas, W. B. Krantz, and M. E. Voorhees, U.S. patent 5,626,759, issued May

6, 1997.

DESIGN OF A MEMBRANE–LUNG OXYGENATOR 417

where ω is the angular frequency of the wall oscillations. The molar ﬂux N

equation (7.2-3) is the local molar ﬂux averaged over the length of the hollow-ﬁber

membrane and is deﬁned as

(

(∂c

/∂r)

r=R

(



∂c

∂r



r=R

dz (7.2-4)

where D

is the binary diffusion coefﬁcient for oxygen in blood, which is assumed

to be constant. The bulk ﬂow contribution to the molar ﬂux has been ignored

because the solutions are dilute. When equations (7.2-3) and (7.2-4) are substituted

into equation (7.2-1), we obtain

•

ωD

2πLc



2π



∂c

∂r



r=R

dz dt (7.2-5)

Both c

appearing in c

and c

in equation (7.2-5) are obtained from a

solution to the axisymmetric form of the species-balance equation in cylindrical

coordinates given by equation (G.2-5) in the Appendices, which when simpliﬁed

appropriately for this unsteady-state convective diffusion problem assumes the form

∂c

∂t

+ u

∂c

∂z

∂

∂r



∂c

∂r



(7.2-6)

When simplifying equation (G.2-5) to arrive at equation (7.2-6), note that each

term has been divided by the molecular weight, which can be assumed to be

constant for the dilute solutions. The axial diffusion term has also been neglected

in equation (7.2-6), owing to the small aspect ratio.

The mass-average velocity

is obtained from a solution to the equations of motion given by equations

(D.2-10) through (D.2-12) in the Appendices. If entrance and exit effects can be

ignored, this will be an unsteady-state fully developed ﬂow for which the equations

of motion reduce to

∂u

∂t

P

+ μ

∂

∂r



∂u

∂r



(7.2-7)

The boundary and periodic solution conditions are given by

∂c

∂r

= 0atr = 0, 0 ≤ z ≤ L (7.2-8)

∂u

∂r

= 0atr = 0, 0 ≤ z ≤ L (7.2-9)

= c

at r = R, 0 ≤ z ≤ L (7.2-10)

= Aω cos ωt at r = R, 0 ≤ z ≤ L (7.2-11)

= c

at z = 0 (7.2-12)

The criterion for ignoring this term is considered in Practice Problem 7.P.1.

The criterion for ignoring transient ﬂow effects is considered in Practice Problem 7.P.2.

418 APPLICATIONS IN PROCESS DESIGN

= c

t+2π/ω

(7.2-13)

= u

t+2π/ω

(7.2-14)

where A is the amplitude of the wall oscillations.

In carrying out an

◦

(1) scaling analysis for this problem, one needs to recog-

nize that even in the absence of any axial oscillations, the mass-transfer problem

would involve a region of inﬂuence or boundary layer; that is, a solutal boundary

layer develops at the wall from the upstream edge of the hollow-ﬁber membrane.

The wall oscillations will inﬂuence the thickness of this solutal boundary layer by

altering the velocity proﬁle near the wall. This suggests that our scaling analysis

should include a reference factor so that the coordinate system can be relocated

on the wall of the hollow-ﬁber membrane. A second consideration is that the axial

oscillations will deﬁne a hydrodynamic region of inﬂuence or momentum bound-

ary layer.

The momentum and solutal boundary-layer thicknesses need not be the

same; indeed, the high Schmidt numbers that characterize liquids imply that the

region of inﬂuence for the mass transfer will be considerably thinner than that for

the ﬂuid ﬂow. This then suggests that the radial coordinate in the equations of

motion should be scaled differently than those for the species-balance equation.

Another consideration is the scaling of the time derivative of the concentration.

Whereas the membrane oscillations clearly establish the amplitude of the velocity

at the wall, it is not clear what the amplitude of the concentration oscillations will

be. In particular, there is no reason to expect that this derivative will scale with

the characteristic concentration change divided by the characteristic time. Hence,

it is appropriate to introduce a separate scale for this time derivative. In view of

these considerations, the following dimensionless variables involving unspeciﬁed

scale and reference factors are deﬁned (steps 2, 3, and 4):

∗

≡

− c

;



∂c

∂t



∗

≡

˙c

∂c

∂t

; u

∗

≡

; y

∗

≡

r − r

;

∗

≡

r − r

; z

∗

≡

; t

∗

≡

(7.2-15)

where δ

and δ

are the solutal and momentum boundary-layer thicknesses, respec-

tively. Substitute these dimensionless variables into the describing equations and

divide through by the dimensional coefﬁcient of one term in each equation (steps

5 and 6):

2πk

•

Lδ

ωD

=−

c

∗



∂c

∗

∂y

∗



∗

=(r

−R)/δ

∗

(7.2-16)

c

∗

≡

−c

∗

+ (c

− c

)/c



− c

−

− c

∗



(7.2-17)

The region of inﬂuence associated with an oscillating boundary was discussed in Section 3.5.

DESIGN OF A MEMBRANE–LUNG OXYGENATOR 419

˙c



∂c

∂t



∗

∂c

∗

∂z

∗

/δ

) − y

∗

∂

∂y

∗



−y

∗



∂c

∗

∂y

∗



(7.2-18)

νt

∂u

∗

∂t

∗

μu

P

/δ

) − y

∗

∂

∂y

∗



− y

∗



∂u

∗

∂y

∗



(7.2-19)

∂c

∗

∂y

∗

= 0aty

∗

, 0 ≤ z

∗

≤

(7.2-20)

∂u

∗

∂y

∗

=0aty

∗

, 0 ≤ z

∗

≤

(7.2-21)

∗

− c

at y

∗

− R

, 0 ≤ z

∗

≤

(7.2-22)

∗

Aω

cos ωt

∗

at y

∗

− R

(7.2-23)

∗

− c

at z

∗

= 0 (7.2-24)

∗

= c

∗

+2π/ωt

(7.2-25)

∗

= u

∗

+2π/ωt

(7.2-26)

The scale and reference factors are determined by bounding the dependent

variables, their derivatives, and the independent variables to be

◦

(1) (step 7). The

reference and scale factors for the concentration are determined by setting the

appropriate dimensionless groups in equations (7.2-22) and (7.2-24) equal to 1 and

zero, respectively, to obtain

− c

= 0 ⇒ c

= c

;

− c

= 1 ⇒ c

= c

− c

(7.2-27)

The radial reference factor and axial length scale are determined by setting the

appropriate dimensionless groups in equation (7.2-22) equal to zero and 1, respec-

tively, to obtain

− R

= 0 ⇒ y

= R;

= 1 ⇒ z

= L (7.2-28)

420 APPLICATIONS IN PROCESS DESIGN

The time scale is obtained by setting the dimensionless group appearing in equations

(7.2-25) and (7.2-26) equal to 1 to obtain

ωt

2π

= 1 ⇒ t

2π

(7.2-29)

The radial length scale for the momentum equation is equal to the penetration

depth of the axial oscillations, which is determined by the viscous diffusion of

the wall motion into the ﬂuid; hence, we set the following dimensionless group in

equation (7.2-19) equal to 1:

νt

= 1 ⇒ δ



2πν

(7.2-30)

The velocity scale is obtained by balancing the pressure term with the viscous term

in equation (7.2-19); hence, the following dimensionless group is set equal to 1:

μu

P

2πν

ωμu

P

= 1 ⇒ u

2π

ωρ

P

16πν

ωR

(7.2-31)

where

U is the average velocity for steady-state fully developed ﬂow through a

circular tube; expressing u

in terms of U is done so that the dimensionless groups

emanating form this

◦

(1) scaling analysis can be recast in terms of those obtained

via the simpler scaling analysis approach to dimensional analysis considered in

Section 5.10. The radial length scale for the species-balance equation is the solutal

boundary-layer thickness, which is determined by balancing the axial convection

and radial diffusion terms; hence, we set the following dimensionless group in

equation (7.2-18) equal to 1:

16πν

Uδ

ωR

= 1 ⇒ δ



ωR

16πνU

(7.2-32)

Determining the proper scale for the time derivative of the concentration is prob-

lematic. An upper bound on this derivative can be obtained by balancing the

unsteady-state term with the radial diffusion term; hence, we set the following

dimensionless group in equation (7.2-18) equal to 1:

˙c

ωR

L ˙c

16πνU(c

− c

)

= 1 ⇒˙c

16πν

U(c

− c

)

ωR

(7.2-33)

Substitute the scale and reference factors deﬁned by equations (7.2-27) through

(7.2-33) into the describing equations given by (7.2-16) through (7.2-26) to obtain

the following:

=−



ωR

(c

∗

)



∂c

∗

∂y

∗



∗

(7.2-34)

(c

∗

)

≡

−c

∗

ln(1 − c

∗

)

(7.2-35)

DESIGN OF A MEMBRANE–LUNG OXYGENATOR 421

64π

∂c

∗

∂t

∗

∂c

∗

∂z

∗



32(ν/ωR

)Gz − y

∗

∂

∂y

∗





ωR

Gz − y

∗



∂c

∗

∂y

∗



(7.2-36)

∂u

∗

∂t

∗

= 1 +



(1/2π)(ωR

/ν) − y

∗

∂

∂y

∗

⎡

⎣

⎛

⎝



2π

ωR

− y

∗

⎞

⎠

∂u

∗

∂y

∗

⎤

⎦

(7.2-37)

∂c

∗

∂y

∗

= 0aty

∗



ωR

Gz, 0 ≤ z

∗

≤ 1

(7.2-38)

∂u

∗

∂y

∗

= 0aty

∗



2π

ωR

, 0 ≤ z

∗

≤ 1

(7.2-39)

∗

= 1aty

∗

= 0, 0 ≤ z

∗

≤ 1 (7.2-40)

∗

16π

Aω

ωR

cos 2πt

∗

at y

∗

= 0 (7.2-41)

∗

= 0atz

∗

= 0 (7.2-42)

∗

= c

∗

(7.2-43)

∗

= u

∗

(7.2-44)

We have recast the describing equations in terms of the dimensionless groups

identiﬁed in Section 5.10, where

Sh ≡

•

is the Sherwood number (7.2-45)

Gz ≡

is the Graetz number (7.2-46)

and Sh

and (c

∗

)

denote the Sherwood number and the dimensionless log-mean

concentration driving force, respectively, in the presence of axial oscillations. Step

8 in the procedure for

◦

(1) scaling analysis would involve assessing whether any

simpliﬁcations can be made in the dimensionless form of the describing equations.

Depending on the oscillation frequency, it might be possible to ignore curvature

effects in equations (7.2-36) and (7.2-37) and to apply the boundary conditions

given by equations (7.2-38) and (7.2-39) at inﬁnity. However, owing to the range

of oscillation frequencies that must be considered, it is not possible in general

422 APPLICATIONS IN PROCESS DESIGN

to make these simpliﬁcations. Moreover, our interest here is not in simplifying

these equations, but rather, in developing a dimensional analysis correlation for the

performance enhancement.

Equations (7.2-34) through (7.2-44) indicate that the Sherwood number for mass

transfer with applied axial oscillations will be a function of four dimensionless

groups; that is,

=−



ωR

(c

∗

)



∂c

∗

∂y

∗



∗



ωR

Aω

, Gz, Sc



(7.2-47)

since the integral and (c

∗

)

introduce only those dimensionless groups appear-

ing in equations (7.2-36) through (7.2-44). The scaling approach to dimensional

analysis developed in Section 5.10 also indicated that the Sherwood number was

a function of four dimensionless groups. However, the

◦

(1) scaling analysis result

carried out here provides additional information in identifying the momentum and

solutal boundary layers; that is, the mass transfer is conﬁned to a region-of-inﬂuence

near the wall that is dictated by the effect of the oscillating boundary on the hydro-

dynamics in the wall region. The oscillating boundary changes the velocity proﬁle

near the wall that in turn increases the concentration proﬁles in this region. Note

that the Schmidt number appeared explicitly only when we recast the temporal

concentration derivative (∂c

/∂t)

∗

in terms of ∂c

∗

/∂t

∗

; that is, to carry out the

time integration of equation (7.2-36) the concentration and time must be nondi-

mensionalized as they are in the other terms in the describing equations.

An effective way to assess the performance enhancement for this novel oxy-

genator design is to consider the ratio of the Sherwood number in the presence and

absence of axial oscillations. An

◦

(1) scaling analysis for a conventional oxygena-

tor that does not employ axial oscillations indicates that

Sh =



c

∗



∂c

∗

∂y

∗



∗

= f(Gz) (7.2-48)

That is, the Sherwood number is a function of only the Graetz number.

Hence, we

conclude that a correlation for the ratio of the Sherwood number in the presence

and absence of axial oscillations is of the form



πν

ωR



(c

∗

)

(

∂c

∗

∂y

∗



∗



with oscillations



c

∗

(

∂c

∗

∂y

∗



∗



without oscillations

= f



ωR

Aω

, Gz, Sc



(7.2-49)

◦

(1) scaling analysis to obtain this result is considered in Practice Problem 7.P.4.