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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FLUID-WALL AEROSOL FLOW REACTOR FOR HYDROGEN 463

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 3 456789 10

, fractional conversion

−1

, dimensionless wall temperature

----

Scaling analysis estimate

Numerical solution, Π

= 1.2 × 10

Numerical solution, Π

= 1.2 × 10

Figure 7.5-2 Fractional conversion as a function of the dimensionless wall temperature

estimated by scaling analysis and predicted by a numerical solution to the full set of describ-

ing equations for two values of the dimensionless heat-transfer coefﬁcient, 

corresponding to higher reactor wall temperatures result in increased conversion,

whereas smaller values of 

corresponding to larger carbon particles result in

decreased conversion.

The redeﬁned dimensionless groups given by equations (7.5-55) through

(7.5-61) are useful for isolating various dimensional parameters so that their effect

on the fractional conversion and other quantities of interest can be studied. How-

ever, to understand the reasons for the trends observed, the original dimensionless

groups determined by scaling analysis are required since these groups are propor-

tional to the relative magnitudes of the various terms in the describing equations.

The increase in X

with increasing T

is a direct effect of increasing the reaction-

rate parameter via 

in equation (7.5-34). The decrease in X

with an increase in

particle size can be attributed to a decrease in the heat-transfer coefﬁcient only if the

gas and carbon particles are not essentially in thermal equilibrium; that is, if 

◦

(1). Note that 

= 

− T

)

. Hence, we ﬁnd that 3.5 ≤ 

≤ 43

for the temperature range 2400K ≥ T

 1200 K and 

= 1.2 × 10

, whereas

35 ≤ 

≤ 430 for 

= 1.2 × 10

;thatis,

is large at all but the highest tem-

peratures for the larger particle size. Hence, the inverse relationship between R

and h

cannot explain the decrease in X

with decreasing values of 

. However,

the particle size also affects conversion through the dimensionless group 

,which

is directly proportional to R

through the parameter a

= 3/R

; this group is a

measure of the sensible heat required for the carbon particles. Hence, decreasing



by an order of magnitude also increases 

proportionately. The larger effect

of a decrease in 

on X

observed at lower values of 

−1

is because 

is pro-

portional to (T

− T

)

−3

; that is, as the wall temperature increases, 

decreases

and the sensible heat requirement for the carbon particles becomes less important.

464 APPLICATIONS IN PROCESS DESIGN

In summary, one sees in this example the utility of ﬁrst scaling the describing

equations to obtain the minimum parametric representation. The magnitude of

the resulting dimensionless groups provides a direct assessment of the relative

importance of the various terms in the describing equations. In particular, these

dimensionless groups allow one to assess what approximations, if any, might be

applicable. However, the dimensionless groups that result from scaling analysis

often are not optimal for correlating experimental or numerical data. Hence, it is

convenient to redeﬁne the dimensionless groups using the formalisms introduced

in Chapter 2 in order to isolate particular dimensional quantities of interest into

just one dimensionless group. However, to interpret trends in the observed perfor-

mance of the particular device or process of interest, one must revert back to the

original dimensionless groups obtained from scaling analysis, since the magnitude

of the latter is a direct measure of the physicochemical phenomena involved in the

describing equations.

7.6 SUMMARY

In contrast to Chapters 3 through 6, in which scaling analysis was used essentially

as a pedagogical tool, in this chapter we illustrated how it can be used in the

design of improved and new processes. This was demonstrated via four examples

that focused on emerging technologies, all of which involved coupled transport.

Three of these examples also used the microscale–macroscale modeling concept to

address the heterogeneous nature of the systems. Scaling the describing equations

for complex technologies is a trial-and-error process; that is, one generally is not

certain which terms need to be balanced to determine the proper scale factors.

However, the forgiving nature of scaling will always indicate when terms have

not been balanced properly. This is indicated when one or more terms are not

bounded of

◦

(1) when the relevant physical properties and process parameters are

substituted into the dimensionless describing equations. The scaling analysis in

each of these examples was the ﬁnal result of this trial-and-error procedure for

speciﬁed sets of properties and parameters. However, in the Practice Problems at

the end of the chapter we explore other possible scalings.

In Section 7.2 we applied

◦

(1) scaling analysis to the design of a membrane–

lung oxygenator. The same problem was analyzed in Section 5.10 using the simple

scaling analysis approach to dimensional analysis. Whereas both methods resulted

in a correlation for the Sherwood number in terms of the same four dimen-

sionless groups, the

◦

(1) scaling analysis approach provided considerably more

information on the performance of this device. Additional insight into the design

of the membrane-lung oxygenator was possible because the

◦

(1) scaling analy-

sis identiﬁed both a momentum boundary layer within which the effect of the

wall oscillations on the velocity proﬁle was conﬁned and a solutal boundary layer

across within which the mass transfer occurred. The

◦

(1) scaling analysis pro-

vided an explanation for why the enhancement in mass transfer involved a “tuned”

response with respect to the oscillation frequency. It also led to a design equation

SUMMARY 465

for predicting the effect of the process parameters on the frequency required to

achieve optimal enhancement in the mass transfer. This design problem effectively

illustrated the advantages of using

◦

(1) scaling analysis to achieve the minimum

parametric representation relative to the simple scaling analysis or Pi theorem

approaches to dimensional analysis.

In Section 7.3 we considered pulsed single-bed pressure swing adsorption (PSA)

for the production of oxygen-enriched air. This example demonstrated the value of

◦

(1) scaling analysis to design a new process. Whereas the design of conven-

tional pulsed packed single-bed PSA is well established, no performance data are

available for the novel pulsed monolithic single-bed PSA process. This example

involved coupled ﬂuid dynamics and mass transfer with adsorption and employed

microscale–macroscale modeling to address the adsorption from the bulk gas ﬂow

onto the discrete adsorbent particles. We found that proper scaling was indicated

by all the terms in the describing equations being bounded of

◦

(1). This usually

means that all the dimensionless groups are bounded of

◦

(1), although strictly

speaking, it means that the product of any dimensionless group and the variables

it multiplies must be bounded of

◦

(1). Scaling analysis identiﬁed several marked

differences between packed single-bed PSA technology and the monolithic bed

process. Scaling analysis indicated that local adsorption equilibrium was achieved

nearly instantaneously in the monolithic bed PSA process; in contrast, the adsorp-

tion in the packed bed PSA process was mass-transfer controlled. This difference

between monolithic and packed bed adsorption is analogous to that between the fast

and slow reaction regimes in the microscale–macroscale analysis of mass transfer

with chemical reaction that was considered in Chapter 6. Scaling analysis indicated

that for the monolithic bed process the optimal pressurization time was the contact

time required for the adsorption wave to pass through the bed; in contrast, for the

packed bed process it was the time required to achieve equilibrium in the adsorption

bed. This example also illustrated the advantages of using

◦

(1) scaling analysis for

dimensional analysis. Whereas a na

ıve application of the Pi theorem indicated that

11 dimensionless groups were required,

◦

(1) scaling analysis reduced the number

of dimensionless groups required to correlate the product purity and product recov-

ery to just two for the production of oxygen-enriched air using a speciﬁed adsorbent.

In Section 7.4 we applied scaling analysis to the thermally induced phase-

separation (TIPS) process for the fabrication of polymeric membranes. The TIPS

process involves coupled heat and mass transfer as well as nucleation and growth

of the dispersed solid phase in the casting solution. As such, this example employed

the microscale–macroscale concept to address the growth of the nucleated parti-

cles on the microscale that were considered to be a homogeneous source of diluent

mass on the macroscale of the casting solution. The dispersed phase particles were

assumed to be in both thermal and solutal equilibrium with the bulk of the sur-

rounding casting solution. This assumption is a thermal–solutal analog of the fast

reaction regime approximation considered in Chapter 6. This moving boundary

problem was atypical because the location of the interface between the phase-

separated region and homogeneous solution was not dictated by a classical Stefan

condition involving latent heat release but by a dynamic condition that depended

466 APPLICATIONS IN PROCESS DESIGN

on the nucleation process. Scaling analysis was used in this example to determine

when the mass transport, latent heat effects, and heat loss to the ambient gas phase

could be ignored in the describing equations. Although it was not necessary to scale

the liquid volume fraction in this example, since it was dimensionless and bounded

◦

(1), a separate scale was introduced for its time rate of change. Since we were

interested in scaling the TIPS process at short times during which the pore-size

distribution of the resulting membrane is created, we determined the length scale

by balancing the accumulation and conduction terms. This scaling introduced the

Lewis number, which is the ratio of heat conduction to species diffusion. A large

Lewis number, which is typical for polymer solutions, implied that mass-transfer

effects could be ignored in the describing equations. When the dimensionless group

that characterized the ratio of the characteristic time for heat conduction to that

for latent heat generation was small, the latter effect could be ignored. For suf-

ﬁciently short penetration depths relative to the casting solution thickness, heat

loss to the ambient gas phase could be ignored. The predictions for the polymer

spherulite diameter of a numerical solution to the simpliﬁed describing equations

compared well with scanning electron microscopy measurements for a typical TIPS

membrane-casting system.

In Section 7.5 we applied scaling analysis to the design of the ﬂuid-wall aerosol

reactor process for the direct conversion of methane to hydrogen that produces a

clean-burning fuel without the production of any greenhouse gases. This example

involved coupled heat and mass transfer with chemical reaction. It also involved the

concept of microscale–macroscale modeling. In this case the microscale element, a

carbon particle, was not necessarily assumed to be in local thermal equilibrium with

the gas phase on the macroscale of the reactor. As such, this example was a thermal

analog of the intermediate reaction regime considered in Chapter 6; that is, some

heat transfer from the carbon particles to the gas phase was assumed to be necessary

to promote the decomposition reaction (i.e., the thermal analog of the slow reaction

regime would not be practical); however, the gas phase did not need be in thermal

equilibrium with the carbon particles (i.e., the fast reaction regime analog). These

considerations required including separate energy conservation equations for the

dispersed carbon particle and continuous gas phases. Scaling analysis was employed

to determine when thermal equilibrium between the carbon particles and gas phase

could be assumed and to estimate the local temperature and degree of conversion

in the gas phase. Independent scales were introduced for the spatial derivatives of

the conversion and temperatures of both the gas phase and carbon particles. One

challenge in this scaling analysis was to determine the proper terms to balance in

the energy equation for the gas phase; that is, it was not clear whether the principal

heat source was sensible heat introduced via the injected hydrogen, transfer from

the carbon particles, transfer from the carbon particles, or convective transfer from

the heated wall. However, the forgiving nature of scaling ensured that the proper

terms were balanced for speciﬁc design conditions. This example illustrated the

interesting situation where a dimensionless group was very large in a term that had

to be bounded of

◦

(1). In particular, a large dimensionless group multiplied a term

containing the difference between the dimensionless carbon particle and gas-phase

PRACTICE PROBLEMS 467

temperatures. This condition then implied that these two temperatures were nearly

equal; that is, the carbon particles and gas phase were in local thermal equilibrium

(i.e., the thermal analog of the fast reaction regime). The temperature dependence

of the nth-order decomposition reaction rate kinetics was pivotal in determining

the conversion achieved in this reaction. Hence, in this example scaling analysis

was used to obtain estimates of the local gas-phase temperature and degree of

conversion. The scaling approach to dimensional analysis was then used to isolate

into separate dimensionless groups key design parameters such as the reactor wall

temperature, carbon particle radius, and ﬂow rate of carbon particles introduced in

the feed. Predictions of a numerical solution to the full set of describing equations

were then used to illustrate that scaling analysis provides reliable estimates for

the degree of conversion. Systematic deviations from the scaling analysis estimates

based on the local thermal equilibrium assumption between the carbon particles and

the gas phase were then explained in terms of the relevant dimensionless groups.

In summary, this example provides a very effective illustration of how scaling

analysis can be used to design and estimate the performance of an entirely new

process for which no prior operating data are available.

7.P PRACTICE PROBLEMS

7.P.1 Axial Diffusion Effects in an Oscillated Membrane–Lung Oxygenator

In Section 7.2 we scaled the coupled ﬂuid dynamics and mass-transfer problem

associated with applying axial oscillations to improve the performance of a hollow-

ﬁber membrane oxygenator. We assumed that axial diffusion effects were negligible

in our scaling analysis. Retain the axial diffusion term in the describing equations

and use scaling analysis to develop a criterion for assessing when axial diffusion

can be ignored.

7.P.2 Transient Flow Effects in an Oscillated Membrane–Lung Oxygenator

In Section 7.2 we scaled the coupled ﬂuid dynamics and mass-transfer problem

associated with applying axial oscillations in order to improve the performance of

a hollow-ﬁber membrane oxygenator. We assumed that transient ﬂow effects could

be neglected: that is, the unsteady-state ﬂow effects associated with the initiation

of the axial oscillations.

(a) Indicate how the describing equations for this problem need to be modiﬁed to

apply scaling analysis to assess when the transients can be

neglected.

(b) Scale the modiﬁed describing equations to develop appropriate criteria for

neglecting the transient effects.

transient effects can be ignored.

468 APPLICATIONS IN PROCESS DESIGN

7.P.3 Effect of Process Parameters on the Performance of an Oscillated

Membrane–Lung Oxygenator

In Section 7.2 we scaled the coupled ﬂuid dynamics and mass-transfer problem

associated with applying axial oscillations to improve the performance of a hollow-

ﬁber membrane oxygenator. Our scaling analysis indicated that the dimensionless

mass-transfer coefﬁcient or Sherwood number was a function of three dimensionless

groups, as indicated in equation (7.2-47). Discuss how changes in the relevant

physical properties and process parameters affect the performance of an oscillated

membrane–lung oxygenator.

7.P.4 Correlation for the Sherwood Number for a Membrane–Lung

Oxygenator Without Axial Oscillations

In Section 7.2 we stated without proof that the Sherwood number for a hollow-ﬁber

membrane oxygenator was correlated in terms of just one dimensionless group, as

indicated by equation (7.2-48). In this problem we prove this result.

(a) Consider the mass-transfer geometry shown in Figure 7.2-1; however,

assume that no oscillations are applied and that the liquid within the hollow

ﬁber is in fully developed laminar ﬂow. Write the appropriate describing

equations and associated boundary conditions.

(b) Scale the describing equations to determine the relevant scale and reference

factors. Ignore axial diffusion. In carrying out this scaling analysis, recall

that in the absence of axial oscillations the velocity proﬁle is that for fully

developed laminar ﬂow in a circular tube. Hence, it is necessary to scale

only the species-balance equation and its associated boundary conditions.

group(s) required to correlate the Sherwood number.

7.P.5 Wall Effects in Pulsed PSA

In scaling the pulsed PSA process, we assumed plug ﬂow through the packed

adsorbent bed. However, due to the no-slip condition at the wall of the vessel that

holds the adsorbent, there is a region of inﬂuence wherein a steep radial velocity

gradient exists.

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

wherein the velocity proﬁle is not uniform; that is, the gas is not in plug ﬂow.

(b) Develop a criterion for ignoring the velocity gradient near the wall for

pulsed PSA.

7.P.6 Alternative Boundary Conditions for Pulsed PSA: Speciﬁed

Superﬁcial Velocity and Downstream Pressure

The pulsed single-bed PSA process can be operated by specifying the feed com-

position and any two process parameters, consisting of the upstream total pressure,

PRACTICE PROBLEMS 469

downstream total pressure, and upstream superﬁcial velocity U

. In Section 7.3 we

considered operating the PSA process by controlling the upstream and downstream

pressures. In this problem we consider an alternative mode of operation.

(a) Consider operation of the PSA process whereby the feed composition and

upstream superﬁcial velocity as well as the downstream total pressure are

speciﬁed. Scale the modiﬁed describing equations and determine the relevant

scale factors. Note that in this case it is necessary to introduce a scale factor

for the axial pressure gradient.

(b) Determine the characteristic times in terms of the relevant physical proper-

ties and design parameters for these modiﬁed boundary conditions.

(d) Determine the criterion for assuming a linear adsorption isotherm for both

components.

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

(f) Determine the criterion for assuming local adsorption equilibrium.

7.P.7 Alternative Boundary Conditions for Pulsed PSA: Speciﬁed

Superﬁcial Velocity and Upstream Pressure

The pulsed single-bed PSA process can be operated by specifying the feed com-

position and any two process parameters, consisting of the upstream total pressure,

downstream total pressure, and upstream superﬁcial velocity U

. In Section 7.3 we

considered operating the PSA process by controlling the upstream and downstream

pressures. In this problem we consider an alternative mode of operation.

(a) Consider operation of the PSA process whereby the feed composition and

upstream superﬁcial velocity pressure are speciﬁed. Scale the modiﬁed

describing equations and determine the relevant scale factors. Note that in

this case it is necessary to introduce a scale factor for the axial pressure

gradient.

(b) Determine the characteristic times in terms of the relevant physical proper-

ties and design parameters for these modiﬁed boundary conditions.

(d) Determine the criterion for assuming a linear adsorption isotherm for both

components.

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

(f) Determine the criterion for assuming local adsorption equilibrium.

7.P.8 Pressure and Velocity Dependence of the Axial Dispersion Coefﬁcient

for Pulsed PSA

In the scaling analysis for the pulsed PSA process that was carried out in Section

7.3, the axial dispersion coefﬁcient was assumed to be constant. In fact, the axial

470 APPLICATIONS IN PROCESS DESIGN

dispersion coefﬁcient depends on both the total pressure and the superﬁcial velocity

through the porous media via the equation

= 0.7D

(7.P.8-1)

where R

is the radius of the adsorbent particles, and D

, the binary diffusion

for components A (nitrogen) and B (oxygen), is given by the following

0.0018583



(

1/M

+ 1/M

)

Pσ



(7.P.8-2)

in which M

is the molecular weight of component i, σ

the collision diameter

determined from the Lennard-Jones intermolecular potential function, and  a

dimensionless function of the temperature and intermolecular potential.

(a) Use scaling analysis to develop a criterion for determining when the effect

of the superﬁcial velocity on the axial dispersion coefﬁcient can be ignored.

(b) Use scaling analysis to develop a criterion for determining when the effect

of the total pressure on the axial dispersion coefﬁcient can be ignored.

7.P.9 Scaling the Depressurization Step for Pulsed PSA

In Section 7.3 we scaled the pressurization step for a pulsed PSA oxygenator, during

which product enrichment in oxygen is effected by the selective adsorption of nitro-

gen from an air feed stream. The second step in the pulsed PSA process involves

depressurization of the packed adsorption bed to cause desorption and regeneration

of the adsorbent. Assume that depressurization involves instantaneously reducing the

pressure at the feed end of the adsorbent bed to atmospheric, while simultaneously

closing a valve at the product end of the bed so that no outward ﬂow can occur.

Hence, all the desorbed gas exits through the feed end of the adsorbent bed. The

initial condition for this assumed depressurization step constitutes the axial distri-

butions of the pressure and concentration that were established at the end of the

pressurization step; these can be indicated formally as P(0,z) and

(0,z).The

spatial distribution of the pressure and concentration necessarily change in time as

the adsorbent is regenerated. In particular, the pressure and concentration will display

maxima that progressively move through the adsorbent bed. Ideally, one chooses a

depressurization time that is sufﬁciently long to essentially restore the adsorbent bed

to its loading in equilibrium with air at atmospheric pressure. In this problem we

consider the scaling of this depressurization step.

(a) Write the appropriate describing equations along with the required initial

and boundary conditions for the depressurization step in pulsed PSA.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ,

2002, p. 526.

PRACTICE PROBLEMS 471

(b) Determine the scale and reference factors as well as the relevant dimension-

less groups.

and design parameters.

(d) Estimate the time required to essentially regenerate the adsorbent bed to a

loading corresponding to equilibrium with air at atmospheric pressure for

the physical properties and design parameters given in Table 7.3-1.

(e) In contrast to the pressurization step, the depressurization step will involve

total and partial pressure distributions that display maxima along the adsor-

bent bed length. Use scaling analysis to estimate the instantaneous location

of the maxima in the pressure. Note that this involves estimating the instan-

taneous thickness of the region of inﬂuence wherein the pressure gradient is

conﬁned.

(f) Use scaling analysis to estimate the time required for the maxima in pressure

to propagate through the entire thickness of the adsorbent bed for the physical

properties and design parameters given in Table 7.3-1.

7.P.10 Dimensional Analysis Correlation for a Bed-Size Factor

for Pulsed PSA

In Section 7.3 we used the scaling analysis approach for dimensional analysis to

develop a correlation for product recovery. In particular, we isolated the applied

pressure and pressurization time into separate dimensionless groups. Use the scaling

analysis approach for dimensional analysis to develop a correlation for the bed-size

factor, ψ

, which is deﬁned by

≡

(7.P.10-1)

where m

is the total mass of adsorbent in the bed and m

is the total mass of

oxygen produced per day. Isolate the applied pressure and pressurization time into

separate groups.

7.P.11 Estimation of the Time Required for TIPS Membrane Casting

The TIPS membrane-casting process was scaled in Section 7.4 to determine the

conditions for which the describing equations could be simpliﬁed. However, more

can be done with the results of this scaling analysis.

(a) Consider conditions for which heat loss to the ambient gas phase can be

ignored and estimate the time required for the boundary between the phase-

separated and single-phase regions to penetrate through the entire casting

solution.

(b) Why would this estimate be inaccurate when signiﬁcant heat transfer occurs

at the upper boundary to the ambient gas phase?

472 APPLICATIONS IN PROCESS DESIGN

7.P.12 Scaling of the TIPS Process for Concentrated Casting Solutions

The scaling in Section 7.4 assumed that the diluent mass fraction ω was

◦

(1). How-

ever, it is possible to have casting solutions concentrated in polymer for which the

diluent mass fraction is considerably less than 1. Rescale the describing equations

for the TIPS process for a casting solution that is very concentrated in polymer. In

this case it will be necessary to scale the diluent mass fraction even though it is

dimensionless.

7.P.13 TIPS Membrane Casting with Both Convective and Radiative Heat

Transfer to the Ambient Gas Phase

In Section 7.4 we allowed for heat loss from the casting solution to the ambient

gas phase only by convective heat transfer, which was described by a lumped-

parameter boundary condition given by equation (7.4-9). Consider now heat loss

by both radiation and convection, for which the radiative heat ﬂux is given by

= εσ (T

− T

∞

) (7.P.13-1)

where ε and T are the emissivity and absolute temperature of the upper surface of

the casting solution, σ is the Stefan–Boltzmann constant, and T

∞

is the temperature

in the bulk of the ambient gas phase. In working this problem, assume that the

energy and species-balance equations can be decoupled and that latent heat effects

can be ignored.

(a) Rescale the TIPS-casting process considered in Section 7.4 for this modiﬁed

boundary condition.

(b) Determine the criterion for ignoring heat loss to the ambient gas phase at

the upper boundary.

transfer at the upper boundary.

7.P.14 TIPS Casting on a Cold Boundary with a Constant Heat Flux

The TIPS membrane-casting process considered in Section 7.4 assumed that the

cold boundary was maintained at a ﬁxed temperature. Here we consider maintaining

a constant heat ﬂux at this boundary, which implies that its temperature must

decrease in time as the phase-separation front penetrates farther into the casting

solution.

(a) Rescale the describing equations assuming that the heat ﬂux is speciﬁed at

this cold boundary; that is, assume a boundary condition at the cold surface

given by

=−k

∂T

∂z

=−q

, where q

is a positive constant (7.P.14-1)