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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PULSED SINGLE-BED PRESSURE-SWING ADSORPTION 433

TABLE 7.3-1 System Parameters for the Pulsed Single-Bed PSA Process

Parameter Packed Adsorption Column Monolithic Adsorbent

(m) 5 ×10

−4

1 ×10

−6

−1

) 6.0 ×10

−3

1.50 × 10

) 6.76 × 10

−10

2.71 × 10

−15

ε 0.35 0.35

s) 1.0 ×10

−5

1.0 ×10

−6

∞

(mol

) 2.8 ×10

2.8 ×10

∞

(mol

) 2.2 ×10

2.2 ×10

(Pa

−1

) 5.0 ×10

−7

5.0 ×10

−7

(Pa

−1

) 1.8 ×10

−7

1.8 ×10

−7

μ(Pa ·s) 1.8 ×10

−5

1.8 ×10

−5

(Pa) 1.3 ×10

1.5 ×10

(Pa) 1.0 ×10

1.0 ×10

T(K) 293 293

L(m) 1.0 2.0 ×10

−3

(s) ——

Source: E. M. Kopaygorodsky, V. V. Guliants, and W. B. Krantz, A.I.Ch.E. J., 50(5) 953 (2004).

TABLE 7.3-2 Dimensionless Groups Characterizing Pulsed Packed and Monolithic

Single-Bed PSA for the Process Parameters Given in Table 7.3-1 for Producing an

Oxygen-Enriched Product from Air

Dimensionless

Group Packed Bed Monolithic Bed



0.311

0.186



3.11 × 10

−6

0.0465



0.0118 1768



2.63 1.58



0.283 0.283



3.33 2.00



0.015 0.025



0.00540 0.00900

If the pressurization time is too short, breakthrough of the adsorption front will

not have occurred, and no enrichment will be achieved. On the other hand, if the

pressurization time is too long, the bed will become completely saturated at the

prevailing local pressure and no enrichment will be achieved in the product gas.

The optimal pressurization time achieves the maximum possible enrichment of the

product gas. The process parameters in Table 7.3-1 lead to the values of the dimen-

sionless groups that characterize the pulsed packed and monolithic single-bed PSA

processes summarized in Table 7.3-2.

To have a workable oxygenator, it is essential that the dimensionless group



< 1; that is, the adsorbent must preferentially adsorb nitrogen to produce an

oxygen-enriched product. We see from Table 7.3-2 that 

= 0.283, which indi-

cates that this adsorbent is satisfactory. The value of 

 1 for pulsed packed

434 APPLICATIONS IN PROCESS DESIGN

single-bed PSA, which implies that axial dispersion can be ignored. Although 

is much larger for pulsed monolithic single-bed PSA, axial dispersion can still be

neglected in modeling this process without incurring signiﬁcant error. The small

values of 

and 

imply that the adsorption isotherms for both nitrogen and

oxygen can be linearized in the partial pressure without incurring signiﬁcant error.

A dramatic difference between pulsed packed and monolithic single-bed PSA

is apparent from the values of 

. Whereas 

 1 for pulsed packed single-bed

PSA, 

 1 for pulsed monolithic single-bed PSA. Since a proper scaling implies

that all the terms in the describing equations be bounded of

◦

(1), the product of



with q

e∗

− q

∗

must be of

◦

(1) in equation (7.3-47); this implies that q

∗

∼

e∗

for monolithic bed PSA. A similar argument for equation (7.3-46) implies that

∗

∼

e∗

. Hence, the adsorption time is so fast in comparison to the contact time

(i.e., t

 t

) that we conclude that local thermodynamic adsorption equilibrium

prevails for pulsed monolithic single-bed PSA. In contrast, the contact time is

considerably faster than the adsorption time for pulsed packed single-bed PSA (i.e.,

 t

), thus implying that adsorption in this process is mass-transfer controlled.

This marked difference between the adsorption behavior for pulsed packed and

monolithic single-bed PSA arises because of the very small particle size associated

with the latter. The small particle size implies a much smaller permeability and

therefore a much slower contact time relative to the adsorption time; moreover, in

contrast to packed bed PSA, the interparticle mass-transfer resistance is negligible

for the monolithic bed process.

The marked difference between the characteristic adsorption time and the contact

time for packed relative to monolithic bed PSA implies that a different design

strategy must be used for each process. Since the adsorption time is much longer

than the contact time for pulsed packed single-bed PSA, the pressurization time

must be approximately equal to the adsorption time. Scaling analysis then suggests

the following criterion for determining the pressurization time for conventional

pulsed packed single-bed PSA:

= t

(1 − ε)RT k

∞

(7.3-65)

For the design parameters given in Table 7.3-1, this criterion suggests a

pressurization time of 75.2 seconds, which agrees well with typical values used

in practice for packed pulsed single-bed PSA. On the other hand, since the pulsed

monolithic single-bed PSA process occurs at local adsorption equilibrium, the pres-

surization time must be approximately equal to the contact time required for the

adsorption “wave” to pass through the column. Scaling analysis then suggests the

following criterion for determining the pressurization time for monolithic single-bed

PSA:

= t

μL

P

(7.3-66)

For the design parameters given in Table 7.3-1, this criterion suggests a pres-

surization time of 0.532 seconds. These estimates for the optimal pressurization

PULSED SINGLE-BED PRESSURE-SWING ADSORPTION 435

times permit determining the values of 

, the dimensionless group that pro-

vides a measure of the importance of the unsteady-state term in equations (7.3-46)

and (7.3-47). For packed single-bed PSA, 

= 0.00414, which implies that the

pressurization step can be considered to be a steady-state process. However, for

monolithic bed PSA, 

= 0.350, which implies that the pressurization step is

inherently unsteady-state.

The fact that 

 1 for monolithic single-bed PSA implies that the length

scale for the pressure and partial pressure gradients for this process is not equal

to the length of the adsorption column as was assumed in our scaling analysis.

Indeed, this large value of 

suggests that monolithic bed PSA involves a region

of inﬂuence or boundary layer across which the pressure drop occurs. Hence, the

describing equations must be rescaled for conditions such that the appropriate length

scale for the pressure gradients is the boundary-layer thickness δ

. The thickness

of the region of inﬂuence or boundary layer wherein the total and partial pressures

undergo their characteristic change is obtained by balancing the convection term

with the accumulation term in equation (7.3-22) or (7.3-23) since the monolithic

bed PSA process is inherently unsteady state; this results in the following equation

for the boundary-layer thickness:

≡ δ



P t

εμ

(7.3-67)

where t

is the observation time. For the design parameters given in Table 7.3-1,

the boundary-layer thickness is given by

= 4.64 × 10

−3

√

(7.3-68)

Clearly, 0 ≤ δ

≤ L; that is, once the boundary layer has penetrated to the end of

the monolithic bed, breakthrough of the adsorption wave occurs. One implication

of this region of inﬂuence is that any numerical technique employed to solve the

describing equations must allow for adequate resolution of the steep total and partial

pressure gradients at very short contact times.

Whereas the process parameters in Table 7.3-1 imply that neglecting axial dis-

persion is a very good approximation for pulsed packed single-bed PSA, they

indicate that it could be important for pulsed monolithic single-bed PSA at least

under some operating conditions. When axial dispersion cannot be neglected, it

is necessary to apply a downstream boundary condition on the partial pressure.

Due to the very rapid adsorption time scale for the monolithic bed PSA process,

the total and partial pressure proﬁles propagate as sharp fronts through the adsor-

bent bed. Hence, until breakthrough of the adsorption wave occurs, a downstream

boundary condition demanding that the spatial derivative of the partial pressure be

zero is appropriate. Note that the downstream boundary condition given in dimen-

sional form by equation (7.3-18) can be applied for the total pressure throughout

the entire adsorption step. However, once breakthrough has occured, demanding

that the derivative of the partial pressure be zero is not justiﬁed. This assumption

436 APPLICATIONS IN PROCESS DESIGN

can introduce some error even when equilibrium adsorption applies since the latter

changes with the local total pressure. That is, if there is a pressure drop owing to

the resistance offered by the adsorption bed to gas ﬂow, the partial pressure must

change with axial distance as well. Our scaling analysis suggests an alternative

downstream boundary condition for the partial pressure that is far less restrictive

than demanding that the partial pressure gradient be zero. Since both 

and 

are

small, it is a reasonable assumption after breakthrough of the adsorption wave has

occurred to ignore the accumulation and axial dispersion terms in equation (7.3-47)

at the downstream end of the adsorption bed. Hence, a more realistic downstream

boundary condition for the partial pressure is given by

∂(U

∗

)

∂z

∗

= 

e∗

− q

∗

) at z

∗

= 1 (7.3-69)

This boundary condition states that the change in the net convection of the adsorbed

component is due to its adsorption. Although this condition neglects any accumu-

lation or axial dispersion of the adsorbed component at the downstream end of the

bed, it does allow for a nonzero partial pressure gradient.

The pulsed single-bed PSA process provides a good example to illustrate the use

of advanced dimensional analysis concepts for correlating experimental or numer-

ical data. The quantities of particular interest in characterizing PSA performance

are the product purity y

and recovery ψ

. The latter is equal to the ratio of the

volumetric ﬂow of the product (oxygen) to that of the feed and is deﬁned by the

following equation (step 1 in the scaling procedure for dimensional analysis):

x=L

x=0

(∂P /∂x)

x=L

(∂P /∂x)

x=0

(7.3-70)

The values of y

and ∂P

∂x at the feed and product ends of the adsorption bed

would have to be obtained by solving equations (7.3-46) through (7.3-56). Steps

2 through 7 in the scaling approach to dimensional analysis have already been

done when we scaled the pulsed PSA process that resulted in the eight dimension-

less groups deﬁned by equations (7.3-57) through (7.3-64). Moreover, Table 7.3-2

indicates that for both the packed and monolithic pulsed single-bed PSA pro-

cess, the groups 

,

,and

are sufﬁciently small so that axial dispersion

and nonlinear adsorption effects can be ignored without incurring signiﬁcant error;

that is, these dimensionless groups can be ignored in developing our dimensional

analysis correlations for the product purity and recovery. However, the remaining

ﬁve dimensionless groups are not optimal for correlating numerical or experimental

data for y

and ψ

. For an air feed delivering an enriched oxygen product at a

ﬁxed low pressure P

(usually atmospheric) and a speciﬁed adsorbent, bed length,

and temperature, the performance of pulsed single-bed PSA depends only on the

pressurization time t

and the applied pressure P

. Ideally, one would like to iso-

late these two quantities to simplify correlating either experimental or numerical

data. This can be done by ﬁrst applying step 10 in the procedure for dimensional

analysis that is indicated formally by equation (2.4-4); that is, since P

and P

PULSED SINGLE-BED PRESSURE-SWING ADSORPTION 437

appear independently in the dimensionless groups 

and 

, respectively, the

quantity P can be replaced by P

in the dimensionless groups 

and 

.This

same principle permits replacing P by P

in the dimensionless groups 

and



. These operations then result in the following new dimensionless groups that

replace 

,

,and



≡

εμL

(7.3-71)



≡

(1 − ε)RT k

∞

μL

(7.3-72)



≡ y

(7.3-73)



≡

(7.3-74)

where 

and 

have been deﬁned so that they are directly proportional to t

and P

, respectively. Note that we have isolated the pressurization time t

in 

and the applied pressure P

in 

. Hence, dimensional analysis correlations for

the product purity and recovery will be of the form

= f

(

,

)

= f

(

,

)

(7.3-75)

However, for an air feed delivering an enriched oxygen product at a ﬁxed low

pressure P

and a speciﬁed adsorbent, bed length, and temperature, the dimen-

sionless groups 

,

,and

are constant. Hence, experimental or numerical

data for the product purity and recovery can be correlated in terms of just two

dimensionless groups:

= f

(

,

) = f



εμL



= f

(

,

) = f



εμL



(7.3-76)

One would anticipate that both the product purity and recovery would be monoton-

ically increasing functions of 

, due to the increased adsorption of nitrogen at

higher operating pressures. However, both the product purity and recovery display

a maximum as a function of 

, due to the fact that longer pressurization times

imply progressively more loading of the adsorbent.

This design problem again illustrates the advantages of employing

◦

(1) scaling

to determine the dimensionless groups required to correlate experimental or numer-

ical data. A naive application of the Pi theorem indicates that 11 dimensionless

groups would be required to correlate the product purity and product recovery; that

is, n = 16 (the 15 quantities in Table 7.3-1 and the gas constant R) and m = 5(M,

mol, L, t,andT). Scaling analysis is able to reduce the number of dimensionless

438 APPLICATIONS IN PROCESS DESIGN

groups to ﬁve by systematically identifying combinations of the quantities (i.e.,

RT, k

∞

,andk

∞

) and assessing reasonable simplifying assumptions (that the

groups 

,

,and

can be neglected); that is, the number of quantities is

reduced to 13 in terms of ﬁve units, and three of the dimensionless groups are

sufﬁciently small to permit neglecting them. The latter approximations involve

ignoring axial dispersion and nonlinear adsorption effects. However, by recogniz-

ing that several process parameters are ﬁxed once the adsorbent is speciﬁed for

producing an oxygen-enriched product from air, it was possible to correlate either

the product purity or recovery in terms of just two variable dimensionless groups.

7.4 THERMALLY INDUCED PHASE-SEPARATION PROCESS

FOR POLYMERIC MEMBRANE FABRICATION

This application of scaling analysis focuses on the formation of microporous mem-

branes through a process known as thermally induced phase separation (TIPS).

The goal of scaling in this example is to explore the conditions for which the

complex describing equations can be simpliﬁed to permit a tractable numerical

solution. Before considering the describing equations, a brief background on the

TIPS process for polymeric membrane formation will be given.

A membrane is a semipermeable medium that permits the passage of some

molecules, colloidal aggregates, or particles relative to others. Most microporous

membranes are made from polymeric materials through a technique known as phase

inversion because it involves converting a single-phase solution of polymer in a

solvent, referred to as the casting solution, into a two-phase dispersion in which

the polymer ultimately becomes a continuous microporous solid-phase matrix. The

TIPS process involves forming a single-phase solution of the polymer in a solvent,

referred to as the diluent, at an elevated temperature. However, the diluent becomes

a nonsolvent below some lower temperature. Hence, phase separation in the TIPS

process occurs by casting a thin ﬁlm of the hot single-phase polymer solution onto

a cold surface. The resulting front that separates the phase-separated region from

the single-phase solution then propagates in time away from the cold boundary.

For solutions of amorphous (i.e., having no crystalline regions) or nonconcentrated

semicrystalline (i.e., having both amorphous and crystalline regions) polymers,

TIPS involves the formation of two liquid phases. However, for a more concentrated

semicrystalline polymer solution, which is the focus of this example, TIPS involves

the formation of solid polymer and a polymer-lean liquid phase. This solid polymer

phase will evolve through a nucleation and growth process that can be described by

Avrami theory.

The dispersed-phase polymer particles eventually fuse together

to create a continuous polymer network that provides the structural matrix of the

microporous membrane. The dispersed polymer-lean phase then is extracted with

an appropriate solvent to create a microporous network. Subsequent post-casting

treatment can involve annealing the membrane at a temperature slightly above that

M. Avrami, J. Chem. Phys., 7, 1103–1112 (1939); 8, 212–224 (1940); 9, 177–184 (1941).

THERMALLY INDUCED PHASE-SEPARATION PROCESS 439

of the glass-transition point of the polymer (i.e., the temperature above which

signiﬁcant segmental and polymer chain motion becomes possible) to decrease the

pore size and increase the selectively of the membrane. In this example, which is

based on the work of Li et al.,

we are concerned with developing a model to

describe the evolution of the microstructure during the phase-separation process.

Prior to the work described here, there was no model available to describe structural

evolution during the solid–liquid TIPS process.

The physical considerations that need to be incorporated into a model for

structural evolution during the TIPS process are summarized here. Commercial

TIPS casting is usually done continuously; hence, it is reasonable to assume one-

dimensional transport. The TIPS process will involve heat transfer from the hot

polymer casting solution to the cold boundary, and possibly to the ambient gas

phase as well. This cooling will eventually cause phase separation of pure polymer

dispersed in a continuous polymer-lean phase. This phase separation in turn can

cause diffusive mass transfer due to the concentration gradients that are created by

the nucleation and growth of the solid phase. This constitutes a moving boundary

problem since the front separating the single- and two-phase regions will propagate

away from the cold boundary. This moving boundary is deﬁned by the nonequilib-

rium thermodynamic condition, which relates the temperature at which nucleation

of the polymer phase begins to the instantaneous cooling rate; this condition can be

determined experimentally via differential scanning calorimetry. The temperature

for the inception of nucleation increases in time since the cooling rate decreases as

the phase-separation front moves away from the cold surface. One complication is

that the physical and transport properties of the polymer-lean and solid polymer in

the phase-separated region can be different. A model must somehow account for the

presence of the two phases in this phase-separated region. Note also that the volume

of the dispersed phase will increase in time at any plane within the phase-separated

region, owing to the progressive precipitation of the polymer. In the following, an

appropriate set of describing equations is developed that accounts for the principal

features of the TIPS casting process. This problem involves a microscale element,

(i.e., a dispersed polymer phase particle) and a macroscale element, which is a

differential thickness of the casting solution.

A schematic of TIPS casting is shown in Figure 7.4-1. The origin of the coor-

dinate system is located at the cold boundary. Allowance is made for possible heat

loss to the ambient gas phase at the upper boundary of the casting solution. The

moving boundary between the single- and two-phase regions is located at z = L(t),

where z is the spatial coordinate, L is the location of the moving boundary, and

t is time. Representative temperature and diluent-concentration proﬁles are shown

in the ﬁgure. Note that both the temperature and diluent-concentration proﬁles as

well as their slopes are continuous at the moving boundary, owing to the fact that

no phase separation has occurred at this plane. That is, this boundary is deﬁned

to be the plane at which nucleation becomes possible at the instantaneous cooling

rate. However, no nuclei have formed at this boundary, because there has not been

D.Li,A.R.Greenberg,W.B.Krantz,andR.L.Sani,J. Membrane Sci., 279, 50 (2006).

440 APPLICATIONS IN PROCESS DESIGN

= L(t)

T (z, t) w

(z, t)

z = H

Single-phase region

Cold boundary at T

Heat loss to ambient gas phase at flux q = h(T − T

∞

)

Moving boundary

defined by T

= f(T )

Phase-separated region

Figure 7.4-1 Thermally induced phase-separation (TIPS) process showing representative

temperature and diluent-concentration proﬁles; phase separation in the initially hot polymer

solution propagates from the lower cold boundary; the moving boundary separating the

homogeneous and phase-separated regions is denoted by L(t); the solid volume fraction in

the phase-separated region increases in time, as shown schematically by the progressively

darker shading toward the cold boundary; heat loss at the interface between the casting

solution and ambient gas causes a temperature decrease near this upper boundary.

enough time for them to initiate and grow. Nuclei will form and grow at this ﬁxed

plane when the moving boundary moves upward due to the progressive cooling.

Hence, this moving boundary problem is not a classical Stefan problem deﬁned

by a latent heat effect associated with phase change at the moving boundary for

which the temperature gradients (and both the concentration and its gradient for a

multicomponent phase-change process) are discontinuous.

The appropriately simpliﬁed energy equation given by equation (F.1-2) in the

Appendices is given by (step 1)

ρC

∂T

∂t

∂

∂z



∂T

∂z



− ρ

H

∂ψ

∂t

(7.4-1)

in which T is the temperature and ψ is the liquid volume fraction in the phase-

separated region. The properties ρ, C

,andk are the mass density, heat capacity,

and thermal conductivity, respectively. The heat-generation term introduces H

the latent heat of fusion of the polymer, and ρ

, the density of the pure polymer. The

species-balance given by equation (G.1-5) in the Appendices needs to be modiﬁed

to account for the fact that diffusion can occur only in the continuous liquid phase.

For this reason the liquid volume fraction ψ appears in the species-balance, which

is given by

∂(ω

ψ)

∂t

= ω

∂ψ

∂t

+ ψ

∂ω

∂t

∂

∂z



∂ω

∂z



(7.4-2)

THERMALLY INDUCED PHASE-SEPARATION PROCESS 441

where ω

is the mass fraction of the diluent and D

is the binary diffusion

coefﬁcient. Equation (7.4-2) also applies to the single-phase region in which ψ = 1.

The physical properties of the polymer solution depend on the concentration.

The physical properties in the phase-separated region depend on the liquid volume

fraction and the concentration. It is reasonable to assume that the relevant solution

properties are weight-fraction-weighted sums of the pure component properties, and

that the relevant properties in the two-phase region are volume-fraction-weighted

sums of the properties of the solution and pure polymer phases. Hence, the follow-

ing equations are used for the relevant physical properties:

ρC

= (1 − ψ)(ρC

)

+ ψ [ω

(ρC

)

+ (1 −ω

)(ρC

)

] (7.4-3)

k = (1 −ψ)k

+ ψ [ω

+ (1 − ω

] (7.4-4)

where the subscripts P and D denote polymer and diluent properties, respectively.

Equations (7.4-3) and (7.4-4) apply in both the single- and two-phase regions since

in the single-phase region the liquid volume fraction ψ = 1.

Equations (7.4-1) and (7.4-2) constitute two equations in three dependent vari-

ables, T,ω,andψ. An additional equation to determine the liquid volume fraction

is provided by Avrami theory for the nucleation and growth process

ψ = ψ

∞

+ (1 − ψ

∞

) exp(−Kt

) (7.4-5)

where ψ

∞

is the equilibrium liquid volume fraction at the prevailing temperature,

K the crystallization rate constant, and n is referred to as the Avrami exponent.

Note that t

is the time for growth of the solid phase; t

is initialized at each

location of the plane that separates the phase-separated and single-phase regions;

hence, t

will become progressively shorter at planes farther removed from the

cold boundary, at which phase separation begins.

The initial and boundary conditions required to solve equations (7.4-1) and

(7.4-2) are given by

T = T

,ω

= ω

,ψ= 1att = 0 (7.4-6)

T = T

∂ω

∂z

= 0atz = 0 (7.4-7)

ψ = 1forz ≥ L (7.4-8)

−k

∂T

∂z

= h(T − T

∞

∂ω

∂z

= 0atz = H (7.4-9)

Equation (7.4-7) speciﬁes the temperature at the impermeable cold boundary.

Equation (7.4-8) speciﬁes that there is no phase separation at or above the mov-

ing boundary at L. Equation (7.4-9) speciﬁes that the heat conducted to the upper

Avrami, J. Chem. Phys., 7, 1103–1112 (1939); 8, 212–224 (1940); 9, 177–184 (1941).

442 APPLICATIONS IN PROCESS DESIGN

interface is equal to the convective heat transfer into the ambient gas phase, whose

temperature is T

∞

, and that the diluent is nonvolatile.

An auxiliary equation is needed to locate the position of the moving boundary

that separates the single- and two-phase regions. This is obtained from measure-

ments of the nonequilibrium crystallization temperature T

as a function of cooling

rate

T ≡ ∂T

∂t using differential scanning calorimetry; that is,

= f



∂T

∂t



= f(

T) at z = L (7.4-10)

When

T → 0,T

→ a constant determined only by the composition of the solution.

To use this model to predict the size of the dispersed-phase particles, it is nec-

essary to have a relationship between the number of nuclei generated as a function

of the degree of subcooling at the moving boundary between the phase-separated

and single-phase regions. However, this relationship is not needed to scale the

describing equations and hence is not given here. This relationship implies that the

number of nuclei decreases with a reduced degree of subcooling, which in turn is

proportional to the cooling rate. Hence, for a constant cold boundary temperature,

the dispersed phase particle size will progressively increase away from this bound-

ary owing to a gradual decrease in the cooling rate. This is the underlying reason

why the TIPS process can create an asymmetric membrane structure whereby the

permselective layer created near the cold boundary is supported by a more open

highly permeable substructure.

The describing equations given (7.4-1) through (7.4-10) are nontrivial to solve

numerically, for several reasons. First, this involves a set of coupled parabolic and

hyperbolic partial differential equations. These equations are stiff; that is, the time

derivatives can be quite large initially. Moreover, this is a moving boundary prob-

lem for which it is necessary to obtain the instantaneous cooling rate to determine

the extent of the phase-separated region. Another complication is that the origin of

the time scale for the time dependence of ψ is different for each plane in the phase-

separated region since nucleation begins progressively later as the moving boundary

progresses into the single-phase region. Hence, it is necessary to determine both

the rate of cooling

T and the time at which equation (7.4-10) is satisﬁed for each

plane in the phase-separated region. Coping with these complexities certainly con-

tributed to the delay in developing a model for TIPS membrane formation via a

nucleation-and-growth process. Hence, we seek to scale these describing equations

to determine the conditions for which a tractable numerical solution can be obtained.

We begin by introducing appropriate reference and scale factors via the following

dimensionless variables (steps 2, 3, and 4):

∗

≡

T − T

; z

∗

≡

; t

∗

≡

;

∗

≡

∂ψ

∂t

; k

∗

≡

;

(ρC

)

∗

≡

(ρC

)

(ρC

)

; D

∗

≡

(7.4-11)