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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INSTANTANEOUS REACTION REGIME 373

the limiting reactant, component B is not necessarily reduced to zero; moreover,

it diffuses over the full thickness of the microscale liquid layer and reacts with

component A over the thickness δ

. The minimum concentration of component B

can again be estimated from our scaling analysis and is given by equation (6.5-2).

The mass-transfer ﬂux is increased for the fast relative to the slow or intermediate

reaction regimes, as shown by the steeper slope of the concentration proﬁle for

component A at y = 0.

6.7 INSTANTANEOUS REACTION REGIME

As the reaction rate increases in the fast reaction regime, the concentration of

component B at the gas–liquid interface continues to decrease. For sufﬁciently fast

irreversible reaction conditions, it is reduced to zero. An estimate of the reaction

rate required to achieve this condition can be obtained from our reference factor

for component B given by equation (6.5-2):

=ˆc

−

κR

= 0 ⇒ R

ˆc

κδ

(6.7-1)

When the reaction rate is increased from this value, component B becomes the

rate-limiting reactant in the vicinity of the gas–liquid interface; that is, there is

a region near the gas–liquid interface wherein there is no reaction, due to the

depletion of component B. Hence, the reaction will occur only in the region

wherein both components A and B have concentrations greater than zero. Equa-

tion (6.6-2) indicates that the thickness of the region wherein the chemical reaction

occurs decreases to zero as the reaction rate become inﬁnite. This limiting condition

is referred to as the instantaneous reaction regime, for which the criterion is

◦

−ˆc

)

→∞⇒instantaneous reaction regime (6.7-2)

This condition implies that the irreversible reaction is so fast that the two reacting

components cannot coexist anywhere in the microscale element. When instanta-

neous reaction conditions prevail, the resistance to mass transfer in the liquid phase

is greatly reduced. Hence, it is possible for the mass transfer on the microscale to

become gas-phase controlled. If the mass transfer remains liquid-phase controlled,

the interfacial concentration of component A will remain at c

◦

and the reaction

plane will be in the liquid phase; this is referred to as the inner-reaction domain

of the instantaneous reaction regime. However, if the mass transfer becomes gas-

phase controlled, the interfacial concentration of component A will become zero

and the reaction plane will be at the gas–liquid interface; this is referred to as the

surface-reaction domain of the instantaneous reaction regime. Figure 6.3-1 shows

that for the inner-reaction domain of the instantaneous reaction regime the reac-

tion plane separates a region in which component A is diffusing in the absence of

374 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

any B from a region in which component B is diffusing in the absence of any A.

However, for the surface-reaction domain of the instantaneous reaction regime the

concentration of component A remains at zero throughout the microscale element,

owing to the extremely fast reaction that consumes it at the interface between the

two phases. In addition, the concentration of component B becomes zero at the

gas–liquid interface. Figure 6.3-1 shows that the mass-transfer ﬂux is increased for

the instantaneous relative to the slow, intermediate, and fast reaction regimes, as

indicated by the steeper concentration gradient of component A at y = 0. How-

ever, the surface-reaction domain of the instantaneous reaction regime provides

the fastest absorption since the resistance to mass transfer in the liquid becomes

negligible in comparison to that in the gas phase.

The inner-reaction domain of the instantaneous reaction regime is no longer

described by equations (6.3-3) through (6.3-7) since the homogeneous reaction

term does not appear in the describing equations. Moreover, the concentrations of

both components are forced to zero at the reaction plane as shown in Figure 6.3-1.

Since the instantaneous reaction is conﬁned to a reaction plane, the mass transfer

again becomes diffusion-limited; that is, the diffusion of the two components occurs

within the liquid layer thickness deﬁned by δ

. The resulting describing equations

then are given by (step 1)

∂c

∂t

= D

∂

∂y

(6.7-3)

∂c

∂t

= D

∂

∂y

(6.7-4)

= 0,c

=ˆc

at t = 0 (6.7-5)

= c

◦

at y = 0 (6.7-6)

= 0,c

= 0aty = L(t) (6.7-7)

=ˆc

at y = δ

(6.7-8)

The initial condition given by equation (6.7-5) states that for an instantaneous

reaction component A, the limiting reactant cannot be present in the bulk liquid,

whereas component B has a concentration dictated by the local position in the

absorber. Equation (6.7-6) states that the equilibrium concentration of component

A is maintained at the interface. Equation (6.7-7) states that the concentrations

of both reactants are zero at the instantaneous reaction front. Equation (6.7-8)

states that component B has its bulk concentration at the edge of the liquid layer

in the microscale element. An auxiliary condition is needed to locate L(t),the

instantaneous position of the reaction plane; that is, the instantaneous reaction case

involves a moving boundary problem. Consider an integral species balance on

component A:



dy = N

y=0



dy (6.7-9)

INSTANTANEOUS REACTION REGIME 375

Equation (6.7-9) states that the change in the concentration of component A in

the liquid layer of the microscale element is equal to the rate at which it diffuses

in from the gas phase minus the rate at which it is consumed by the irreversible

instantaneous chemical reaction; the latter is equal to the change in the concentra-

tion of component B (a negative number) divided by the stoichiometric coefﬁcient

for the chemical reaction. Equation (6.7-9) can be rearranged using Leibnitz’s rule

for differentiation of an integral given by equation (H.1-2) in the Appendices and

equations (6.7-3) and (6.7-4) to obtain the following auxiliary condition for deter-

mining the instantaneous location of the reaction plane

∂c

∂y

=−

∂c

∂y

at y = L (6.7-10)

Note that velocity of the moving boundary does not appear in equation (6.7-10)

because the concentrations of both A and B are zero at the reaction plane. Our describ-

ing equations for instantaneous reaction conditions then are given by equations (6.7-

3) through (6.7-8) and equation (6.7-10).

Introduce the following dimensionless variables containing unspeciﬁed scale and

reference factors (steps 2, 3, and 4):

∗

≡

; c

∗

≡

; y

∗

≡

; y

∗

≡

y −y

; t

∗

≡

(6.7-11)

We do not need to introduce reference factors for c

and c

since they are naturally

referenced to zero. We have introduced a reference length factor for component B

since its diffusion occurs only on one side of the reaction front. 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):

∂c

∗

∂t

∗

∂

∗

∂y

∗2

(6.7-12)

∂c

∗

∂t

∗

∂

∗

∂y

∗2

(6.7-13)

∗

= 0,c

∗

ˆc

at t

∗

= 0 (6.7-14)

In some formulations of the instantaneous reaction problem, an additional condition that is based

on the fact that the moving boundary is a plane having constant concentration is used to obtain an

equation that relates the velocity of the reaction front dL/dt to the ratio of the partial derivatives of

the concentration ∂c

/∂t and ∂c

/∂y. However, this condition is redundant since it is embodied in

equation (6.7-10). Indeed, the time dependence of L can be obtained by demanding that the solution

to equations (6.7-3) through (6.7-8) satisfy equation (6.7-10). Note, however, that we will see that our

scaling analysis gives us the time dependence of L to within a multiplicative constant of (1):thatis,

L ∝

√

376 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

∗

◦

at y

∗

= 0 (6.7-15)

∗

= 0aty

∗

(6.7-16)

∗

= 0aty

∗

L − y

(6.7-17)

∗

ˆc

at y

∗

− y

(6.7-18)

∂c

∗

∂y

∗

=−

κD

∂c

∗

∂y

∗

at y

∗

(6.7-19)

We again set appropriate dimensionless groups equal to zero or 1 to determine

the reference and scale factors, respectively (step 7). The concentration scales are

obtained by setting c

◦

= 1andˆc

= 1toobtainc

= c

◦

and c

=ˆc

respectively. Since this is inherently an unsteady-state problem, the characteris-

tic time is the observation time t

. Careful consideration must now be given to

determining the proper scales for y

and y

. Note that for an instantaneous

reaction, we expect that L  δ

since the reaction is limited by depletion of

component B, which must diffuse to the reaction front. Hence, we set the dimen-

sionless group in equation (6.7-16) equal to 1 to obtain y

= L. Since the two

terms in equation (6.7-13) must balance, we set y

= 1toobtainy

√

. Since the diffusion of component B is inherently unsteady-state, the two

terms in equation (6.7-19) must balance; thus, we set D

/κD

ˆc

L/κD

◦

√

= 1, thereby obtaining L = (κD

◦

/ ˆc

)

√

When these scale and reference factors are substituted into the describing equations,

we obtain



κc

◦

ˆc



∂c

∗

∂t

∗

∂

∗

∂y

∗2

(6.7-20)

∂c

∗

∂t

∗

∂

∗

∂y

∗2

(6.7-21)

∗

= 0,c

∗

= 1att

∗

= 0 (6.7-22)

∗

= 1aty

∗

= 0 (6.7-23)

∗

= 0aty

∗

= 1 (6.7-24)

∗

= 0aty

∗

= 0 (6.7-25)

∗

= 1aty

∗

− L

√

(6.7-26)

∂c

∗

∂y

∗

=−

∂c

∗

∂y

∗

at y

∗

= 1 (6.7-27)

SCALING THE MACROSCALE ELEMENT 377

Now let us consider how the dimensionless describing equations can be

simpliﬁed (step 8). The boundary condition given by equation (6.7-26) can be

applied at inﬁnity if the following condition is satisﬁed:

− L

√

 1 (6.7-28)

This condition will be satisﬁed for relatively short contact times. The unsteady-

state term in equation (6.7-20) can be dropped if the following condition is

satisﬁed:



κc

◦

ˆc



 1 (6.7-29)

This condition will be satisﬁed if the nonlimiting reactant is present in consider-

able excess relative to the reactant that is being absorbed, since in most cases one

would expect D

∼

. If the condition given by equation (6.7-29) is satisﬁed,

a quasi-steady-state linear concentration proﬁle of component A is obtained. One

can then solve equation (6.7-21) via a penetration theory approach if the condition

given by equation (6.7-28) is satisﬁed.

6.8 SCALING THE MACROSCALE ELEMENT

In Section 6.3 we focused on scaling the describing equations for the microscale

element. The mass-transfer ﬂux determined for the microscale analysis will become

a homogeneous source or sink term in the describing equations for the macroscale

that we consider now. The analysis for the macroscale is greatly simpliﬁed for

the fast and instantaneous reaction regimes, since they reduce the concentration of

the absorbing component to its reaction equilibrium value, which is zero for an

irreversible reaction. Since the latter concentration will prevail on the macroscale

everywhere in the contacting device, no further reaction can occur. However,

the concentration of the nonlimiting reactant, component B, is depleted on the

macroscale, due to its consumption by the chemical reaction on the microscale.

However, as we will see in Section 6.14, the mass-transfer ﬂux of the absorbing

component can be obtained without having to solve for the concentration proﬁle

of the liquid-phase reactant.

Consider now the slow reaction regime for which no reaction occurs on the

microscale and there is no diffusive ﬂux of component B. However, there is a

diffusive ﬂux N

of component A from the gas into the liquid phase. This has

the effect of being a point source of component A and a point sink of compo-

nent B on the macroscale. We assume that conditions are laterally uniform so that

the bulk concentrations of components A and B change only in the axial direc-

tion in Figure 6.2-1. Consider a molar balance on components A and B over a

control volume consisting of a differential slice of the bubble column z that is

378 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

convected at the average axial velocity of the liquid, as shown in Figure 6.2-1

(step 1):

az − R

φa z =

( ˆc

φa z) (6.8-1)

−κR

φa z =

(ˆc

φa z) (6.8-2)

where a is the interfacial area per unit volume of the contacting device, φ the

volume of liquid per unit interfacial area, R

the rate of production (moles/volume

·time) of component A by the homogeneous chemical reaction, κ the stoichio-

metric coefﬁcient for the chemical reaction, and d/dt denotes the total deriva-

tive.

In the slow reaction regime N

can be obtained directly from k

•

,the

mass-transfer coefﬁcient for purely diffusive transfer in the liquid phase via the

equation

= k

•

◦

−ˆc

) (6.8-3)

Hence, equations (6.8-1) and (6.8-2) can be written as

•

◦

−ˆc

) − R

φ = φ

∂ ˆc

∂t

(6.8-4)

−κR

∂ ˆc

∂t

(6.8-5)

The initial conditions required to solve equations (6.8-4) and (6.8-5) are given

ˆc

= 0, ˆc

=ˆc

at t = 0 (6.8-6)

that is, we are assuming that the nonreacting solvent that enters the bubble column

contains only liquid component B at an initial concentration ˆc

Introduce the following dimensionless variables and scale factors (steps 2, 3,

and 4):

ˆc

∗

≡

ˆc

;ˆc

∗

≡

ˆc

−ˆc

ˆc

; R

∗

≡

; t

∗

≡

(6.8-7)

Introduce these variables into equations (6.8-4) through (6.8-6) and divide through

by the coefﬁcient of one term in each equation to obtain the following dimensionless

describing equations (steps 5 and 6):



◦

ˆc

−ˆc

∗



−

•

ˆc

∗

•

∂ ˆc

∗

∂t

∗

(6.8-8)

−

κR

ˆc

∗

∂ ˆc

∗

∂t

∗

(6.8-9)

In a convected coordinate system, one takes the total derivative of an extensive quantity such as the

total number of moles, but the partial derivative of an intensive quantity such as the molar concentration;

the partial derivative with respect to the temporal coordinate is evaluated at a ﬁxed value of the spatial

coordinate in the convected coordinate system.

SCALING THE MACROSCALE ELEMENT 379

ˆc

∗

= 0, ˆc

∗

ˆc

−ˆc

ˆc

at t

∗

= 0 (6.8-10)

Note that we have divided through by the coefﬁcient of the ﬁrst term in equa-

tion (6.8-4) since this term must be retained if component A is transferred from the

gas to the liquid phase. Equation (6.8-8) indicates that the dimensionless concen-

tration can be bounded of

◦

(1)ifwesetc

◦

/ ˆc

= 1 in order to obtain ˆc

= c

◦

The appropriate time scale is the observation or contact time t

that is required for

the liquid to ﬂow through the bubble column. The dimensionless reaction rate is

bounded to be

◦

(1) by again choosing its scale factor to be some characteristic

maximum value R

, which could be determined if the reaction rate were speci-

ﬁed. Setting the dimensionless group in equation (6.8-10) equal to 1 indicates that

ˆc

=ˆc

−ˆc

. When this scale is substituted into the dimensionless group in

equation (6.8-9) and the latter is set equal to −1 (because R

< 0), we obtain

ˆc

=ˆc

− κR

. Substituting these scale and reference factors into the describ-

ing equations then yields the following set of dimensionless describing equations

for the macroscale element (step 7):

(1 −ˆc

∗

)



 

mass transfer

from microscale

to macroscale

−

•

◦

∗

  

chemical reaction

•

∂ ˆc

∗

∂t

∗

  

accumulation in

convected system

(6.8-11)

∗

∂ ˆc

∗

∂t

∗

(6.8-12)

ˆc

∗

= 0, ˆc

∗

= 1att

∗

= 0 (6.8-13)

Now let us consider how the describing equations for the macroscale element can

be simpliﬁed (step 8). These equations again involve three characteristic time scales:

∼ contact or residence time in the bubble column (6.8-14)

•

∼ characteristic time for interphase mass transfer on the macroscale

(6.8-15)

◦

∼ characteristic time for reaction on the macroscale (6.8-16)

Consider ﬁrst the case where the characteristic time for reaction is extremely long

compared to that for interphase mass transfer. This corresponds to purely physical

absorption in the absence of any chemical reaction for which the criterion is

•

◦

∼

0 ⇒ purely physical absorption (6.8-17)

Now let us consider the more interesting case for which chemical reaction

occurs. Note that the accumulation or unsteady-state term in equation (6.8-11) is

380 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

equal to the difference between the rate of mass transfer from the microscale to the

macroscale and the rate at which the transferring component is consumed by the

chemical reaction. Although both the mass transfer and reaction terms can be large,

their difference can be quite small under some conditions that we will now explore.

The accumulation or unsteady-state term will be very small when the following

criterion is satisﬁed:

•

 1 ⇒ species accumulation is very small (6.8-18)

Equation (6.8-18) implies that the time scale for interphase mass transfer is much

shorter than the residence time in the contacting device. When the criterion given by

equation (6.8-18) is satisﬁed, it means that the mass transfer and reaction terms are

nearly equal in magnitude such that their difference is very small. This approxima-

tionisreferredtoasthequasistationary hypothesis. The quasistationary hypothesis

is rigorous for a continuous stirred tank reactor (CSTR), within which there is

a uniform concentration. However, it is an approximation for contactors such as

packed columns, for which the concentration changes with axial position.

6.9 KINETIC DOMAIN OF THE SLOW REACTION REGIME

Now let us assume that the quasistationary hypothesis applies, in which case we

can write equation (6.8-11) as follows:

(1 −ˆc

∗

)



 

mass transfer

from microscale

to macroscale

−

•

◦

∗

  

chemical reaction

•

∂ ˆc

∗

∂t

∗

  

accumulation in

convected system

 1 (6.9-1)

Let us ﬁrst consider the case for which the characteristic time for mass transfer

from the microscale to macroscale is much shorter than the characteristic time for

reaction. This means that component A is transferred from the gas to the liquid

phase much faster than it can be consumed by the chemical reaction. This is

referred to as the kinetic domain of the slow reaction regime.

The criterion for its

applicability is

•

◦

 1 ⇒ kinetic domain of the slow reaction regime (6.9-2)

Since the mass transfer and reaction terms in equation (6.9-1) must nearly balance,

the kinetic domain of the slow reaction regime implies that ˆc

∗

∼

1orthatˆc

∼

◦

;

that is, the bulk liquid is maintained nearly at the thermodynamic equilibrium

concentration for component A. The driving force for the chemical reaction then

is proportional to c

◦

for an irreversible reaction.

The kinetic domain of the slow reaction regime is often referred to as the kinetic regime of the slow

reaction regime; the terminology used here is preferred because it more clearly indicates that the domain

is a subcategory of the regime.

IMPLICATIONS OF SCALING ANALYSIS FOR REACTOR DESIGN 381

6.10 DIFFUSIONAL DOMAIN OF THE SLOW REACTION REGIME

Let us again assume that the quasi-stationary hypothesis applies so that equa-

tion (6.9-1) applies. Now consider the case for which the characteristic time for

mass transfer from the microscale to macroscale is much longer than the charac-

teristic time for reaction. This means that component A is transferred from the gas

to the liquid phase much slower than it is consumed by the chemical reaction. This

is referred to as the diffusional domain of the slow reaction regime.

The criterion

for its applicability is the following:

•

◦

 1 ⇒ diffusional domain of the slow reaction regime (6.10-1)

For the diffusional domain of the slow reaction regime, it would appear that the

reaction term is much larger than the mass-transfer term in equation (6.9-1). How-

ever, since these two terms must nearly balance, we conclude that R

∗

 1. This in

turn implies that the driving force for the chemical reaction must be very small (i.e.,

ˆc

∗

∼

0); that is, the chemical reaction is sufﬁciently fast so that it maintains the

bulk concentration at the reaction equilibrium concentration. This in turn implies

that the driving force for mass transfer from the microscale to the macroscale is c

◦

for an irreversible reaction. It might seem contradictory that the chemical reaction

is fast enough in the diffusional domain of the slow reaction regime to reduce the

bulk concentration to zero, whereas in the intermediate reaction regime for which

the reaction is faster, the bulk concentration is not necessarily reduced to zero. In

the intermediate reaction regime the reaction occurs on the microscale and therefore

steepens the concentration gradient of component A at the gas–liquid interface and

increases the mass-transfer coefﬁcient from that for purely physical absorption.

In contrast, the mass-transfer coefﬁcient for the diffusional domain of the slow

reaction regime is the same as that for purely physical absorption. The faster mass-

transfer rate in the intermediate regime relative to the diffusional domain of the

slow reaction regime corresponds to a steeper concentration gradient of component

A at y = 0inFigure6.3-1.

6.11 IMPLICATIONS OF SCALING ANALYSIS FOR REACTOR

DESIGN

The scaling analysis that we have applied to the microscale–macroscale modeling

of mass transfer with chemical reaction can be used to develop a set of design

precepts for selecting the optimal reactor. These will be determined by considering

how the process parameters affect the total absorption rate per unit volume N

We will proceed from the slowest to the fastest reaction conditions. Hence, let

us ﬁrst consider purely diffusional mass transfer in the absence of any chemical

The diffusional domain of the slow reaction regime is often referred to as the diffusional regime of

the slow reaction regime.

382 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

reaction for which the mass-transfer rate per unit volume of reactor is given by

a = k

•

a(c

◦

−ˆc

) for purely physical absorption (6.11-1)

Correlations for k

•

, the mass-transfer coefﬁcient for purely physical absorption, in

terms of the Sherwood number or Nusselt number for mass transfer are available

in the literature for many contacting geometries. If the contact time is long in

comparison to the diffusion time, a ﬁlm theory model applies and the mass-transfer

coefﬁcient for purely physical absorption is given by equation (6.3-25); that is,

•

steady-state physical absorption (6.11-2)

If the contact time is short in comparison to the diffusion time, a penetration theory

model applies and the mass-transfer coefﬁcient for purely physical absorption is

given by equation (6.3-29); that is,

•



unsteady-state physical absorption (6.11-3)

Equation (6.11-1) implies that the total absorption rate per unit volume for purely

physical absorption is:

•

Proportional to the interfacial area

•

Independent of the liquid-phase holdup φa

•

Proportional to the mass-transfer coefﬁcient k

•

that characterizes the transport

between the microscale and macroscale elements

•

Proportional to the overall driving force for the absorbing component

These considerations indicate that appropriate contactors for purely physical absorp-

tion are packed columns that provide large interfacial area (i.e., large values of a),

good hydrodynamics (i.e., high Reynolds numbers) to promote large mass-transfer

coefﬁcients (i.e., large values of k

•

), and maintain large overall driving forces

(i.e., large values of c

◦

−ˆc

are maintained by avoiding global mixing). On the

other hand, contactors such as stirred tanks are not appropriate since they have

low interfacial area and promote large scale mixing that reduces the overall driving

force for mass transfer.

Now let us consider the kinetic domain of the slow reaction regime. For this

domain the chemical reaction is very slow on both the micro- and macroscales.

This means that the mass-transfer coefﬁcient for transfer from the microscale to

the macroscale is the same as for mass transfer in the absence of chemical reac-

tion. On the macroscale the kinetic domain of the slow reaction regime means

that the reaction is so slow that the local bulk ﬂuid is nearly at the thermody-

namic equilibrium concentration c

◦

. Since the mass-transfer and chemical reaction

terms in equation (6.9-1) must nearly balance if the quasi-stationary hypothesis is