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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EXAMPLE PROBLEMS 323

t (min)

P (mm Hg)

Figure 5.E.7-2 Pressure in the upper chamber versus time from the inception of the per-

meation process through the membrane separating the upper and lower chambers.

= HP at x = L (5.E.7-4)

The boundary condition given by equation (5.E.7-4) is in terms of the unknown

instantaneous pressure in the upper chamber. The auxiliary equation needed to

determine this pressure can be obtained from an integral mass balance on the

upper chamber as follows:

(cV

) =

=−

∂ρ

∂x



x=L

(5.E.7-5)

where c is the molar density of the gas in the upper chamber, R the gas constant,

and T the absolute temperature.

Deﬁne the following dimensionless variables (steps 2, 3, and 4):

∗

≡

; P

∗

≡

;





∗

≡

; x

∗

≡

; t

∗

≡

(5.E.7-6)

Note that we have introduced a separate scale,

, for the time derivative of the

pressure since there is no reason why this should scale with P

and t

Introduce these dimensionless variables into the describing equations and divide

through by the coefﬁcient of one term in each equation (steps 5 and 6):

∂ρ

∗

∂t

∗

∂

∗

∂x

∗2

(5.E.7-7)

∗

= 0,P

∗

at t

∗

= 0 (5.E.7-8)

∗

at x

∗

= 0 (5.E.7-9)

324 APPLICATIONS IN MASS TRANSFER

∗

at x

∗

(5.E.7-10)





∗

=−

RTρ

∂ρ

∗

∂x

∗



∗

=L/x

(5.E.7-11)

The scale factors are determined by means of the following considerations

(step 7). The dimensionless concentration and pressure can be bounded to be

◦

(1) by setting the dimensionless groups in equations (5.E.7-8), (5.E.7-9), and

(5.E.7-10) equal to 1 to obtain ρ

= HP

and P

= P

. Since this is unsteady-state

mass transfer, the time scale is the observation time t

. The length scale is obtained

from the appropriate dimensionless group in equation (5.E.7-10) equal to 1 to obtain

= L. Since the two terms in equation (5.E.7-11) must balance, the dimensionless

group in this equation is set equal to 1 to obtain

= D

RT HP

These choices for the scale factors then result in the following describing equations:

∂ρ

∗

∂t

∗

∂

∗

∂x

∗2

(5.E.7-12)

∗

= 0att

∗

= 0 (5.E.7-13)

∗

= 1atx

∗

= 0 (5.E.7-14)

∗

= P

∗

at x

∗

= 1 (5.E.7-15)





∗

=−

∂ρ

∗

∂x

∗



∗

(5.E.7-16)

where Fo

≡ D

is the solutal Fourier number.

Now let us consider how our scaled describing equations can be used to interpret

the data shown in Figure 5.E.7-2 (step 8). The time required for any pressure

increase to occur in the upper chamber can be estimated from the time required for

the permeating component to penetrate the membrane. This corresponds to setting

the solutal Fourier number equal to 1; that is,

= 1 ⇒ t

(5.E.7-17)

Equation (5.E.7-17) then provides an estimate of the dead time for any pressure

response to occur in the upper chamber for the data shown in Figure 5.E.7-2. Once

the permeating component penetrates through the membrane, a period of unsteady-

state mass transfer will occur during which the pressure will increase nonlinearly

in time. The duration of the latter period can be estimated from the time required

to achieve quasi-steady-state mass-transfer conditions; that is, when

 1orwhent

∼

100L

(5.E.7-18)

EXAMPLE PROBLEMS 325

This then provides an estimate of the time required from the introduction of

the permeating gas to achieve quasi-state mass transfer through the membrane.

We will show that the latter condition corresponds to a linear pressure increase

in time. For observation times greater than that deﬁned by equation (5.E.7-18),

the unsteady-state term in equation (5.E.7-12) can be ignored. If, in addition,

∗

 1, the concentration driving force across the membrane will be constant,

and equation (5.E.7-16) implies that





∗

= 1 ⇒

RT HP

⇒ P =

RT HP

(5.E.7-19)

That is, the pressure will increase linearly in time, as seen in Figure 5.E.7-2 at

longer times. Note that the diffusion coefﬁcient for permeation through the mem-

brane can be obtained from the slope in linear region of the pressure response

curve. However, when P

∗

≥ 0.1 the permeation driving force across the mem-

brane will decrease progressively, causing a less than linear increase in the pres-

sure in the upper chamber. Figure 5.E.7-2 does not show this long-time behavior

since it obviously does not include data taken for sufﬁciently long observation

times.

For quasi-steady-state conditions, equations (5.E.7-12) and (5.E.7-16) can be

solved analytically to obtain the following solution for the pressure in the upper

chamber:

P = P

1 − e

−

(

RT H /M

)

(5.E.7-20)

Note that for small values of the exponent, equation (5.E.7-20) reduces to the

linear response given by equation (5.E.7-19). Hence, in summary, scaling anal-

ysis of the describing equations is able to describe all the principal features of

the pressure-response curve for this standard membrane characterization test proc-

edure.

5.E.8 Large Damk

ohler Number Approximation for Laminar Flow with a

Heterogeneous Reaction

In Section 5.5 we considered steady-state laminar tube ﬂow containing a solute A

that underwent a ﬁrst-order irreversible reaction at the wall as shown in Figure 5.5-

1. We considered a scaling appropriate to a small Damk

ohler number for which

the mass transfer was controlled by the slow rate of heterogeneous reaction. This

implied that the concentration gradient across the tube was negligible and the

concentration was spatially uniform. Here we apply scaling analysis to the comple-

mentary case of a large Damk

ohler number corresponding to a fast heterogeneous

reaction. To supply mass to the tube wall at the same rate that it is consumed

by the heterogeneous reaction, the concentration gradient will be very steep and

occur over a region of inﬂuence or solutal boundary layer whose thickness δ

the appropriate radial length scale.

326 APPLICATIONS IN MASS TRANSFER

The appropriately simpliﬁed species-balance equation and associated boundary

conditions are the same as those in Section 5.5 (step 1):



1 −



∂c

∂z

= D

∂

∂z

+ D

∂

∂r



∂c

∂r



(5.E.8-1)

= c

at z = 0 (5.E.8-2)

= f(r) at z = L (5.E.8-3)

∂c

∂r

= 0atr = 0 (5.E.8-4)

−D

∂c

∂r

= k

•

at r = R (5.E.8-5)

It is convenient to reference the coordinate system to the surface of the tube

where the solutal boundary layer is located. Hence, we deﬁne the following dimen-

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

∗

≡

; r

∗

≡

R − r

; z

∗

≡

(5.E.8-6)

Substitute these dimensionless variables into the describing equations and divide

through by the coefﬁcient of one term in each equation (steps 4 and 5):

Uδ



∗

−

∗2



∂c

∗

∂z

∗

∂

∗

∂z

∗2





− r

∗

∂

∂r

∗



− r

∗



∂c

∗

∂r

∗



(5.E.8-7)

∗

at z

∗

= 0 (5.E.8-8)

∗

= f(r

∗

) at z

∗

(5.E.8-9)

∂c

∗

∂r

∗

= 0atr

∗

(5.E.8-10)

∂c

∗

∂r

∗

•

∗

at r

∗

= 0 (5.E.8-11)

The dimensionless groups in equations (5.E.8-8) and (5.E.8-9), when set equal

to 1 indicate that c

= c

and z

= L (step 7). Since the convection and radial

heat conduction terms must be of the same order, equation (5.E.8-7) provides the

following estimate for δ

Uδ

= 1 ⇒ δ





1/3



4Pe



1/3

(5.E.8-12)

EXAMPLE PROBLEMS 327

where Pe

≡ UR/D

is the solutal Peclet number. Substitution of these scale

factors into equations (5.E.8-7) through (5.E.8-11) then yields the following set of

dimensionless describing equations:

∗

−



4Pe



1/3

∗2

∂c

∗

∂z

∗



4Pe



2/3

∂

∗

∂z

∗2

(

4Pe

R/L

)

1/3

− r

∗

∂

∂r

∗



4Pe



1/3

− r

∗

∂c

∗

∂r

∗

(5.E.8-13)

∗

= 1atz

∗

= 0 (5.E.8-14)

∗

= f(r

∗

) at z

∗

= 1 (5.E.8-15)

∂c

∗

∂r

∗

= 0atr

∗



4Pe



1/3

(5.E.8-16)

∂c

∗

∂r

∗

= Da



4Pe



1/3

∗

at r

∗

= 0 (5.E.8-17)

where Da

≡ k

•

R/D

is the second Damk

ohler number.

Now let us consider how this set of dimensionless describing equations can

be simpliﬁed (step 8). If



/4Pe



2/3

 1, the axial diffusion term can be

ignored in equation (5.E.8-13). If

(

L/4Pe

)

1/3

 1, the curvature effects and

higher-order term in the velocity proﬁle can be ignored in equation (5.E.8-13).

Moreover, the boundary condition given by equation (5.E.8-16) can be applied

at inﬁnity. A zero mass ﬂux far from the tube wall implies no change in the

concentration in the core of the ﬂowing ﬂuid. Hence, equation (5.E.8-16) can be

replaced by the condition that c

∗

= 1asr

∗

→∞. Equation (5.E.8-17) indicates

that as Da

→∞, c

∗

→ 0, to ensure that ∂c

∗

/∂r

∗

remains bounded of

◦

(1).

This implies that the concentration at the tube wall is zero. Hence, for large solu-

tal Peclet numbers and large Damk

ohler numbers, the describing equations sim-

plify to

∗

∂c

∗

∂z

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.E.8-18)

∗

= 1atz

∗

= 0 (5.E.8-19)

∗

= 1atr

∗

→∞ (5.E.8-20)

∗

= 0atr

∗

= 0 (5.E.8-21)

This simpliﬁed system of describing equations can be solved by standard methods,

such as combination of variables.

328 APPLICATIONS IN MASS TRANSFER

Laminar flow of feed

solution through lumen

lumen

Annular membrane wall

Figure 5.E.9-1 Hollow-ﬁber membrane of length L, lumen radius R

, and outer annular

membrane wall radius R

. A feed solution ﬂows axially in fully developed laminar ﬂow.

A reacting component in the feed permeates through the inner wall of the hollow ﬁber and

undergoes a ﬁrst-order homogeneous reaction with an enzyme that is immobilized within

the pores of the annular region.

5.E.9 Small Thiele Modulus Approximation for Mass Transfer in a

Hollow-Fiber Membrane

The development of hollow-ﬁber membranes with diameters of at most a few

hundred micrometers has made it possible to design membrane separation systems

having a very high area-to-volume ratio. This same feature has also permitted

designing efﬁcient catalytic membrane bioreactors by immobilizing an enzyme

within the porous matrix of the tubular membrane. Figure 5.E.9-1 shows both cross-

sectional and proﬁle views of a single hollow-ﬁber membrane of length L and inner

and outer radii R

and R

, respectively. The annular membrane wall is made from

a synthetic polymer that is microporous and serves as host to the immobilized

enzyme. An ultrathin layer at the inner surface of the microporous annular region

is impermeable to the enzyme and thereby conﬁnes it, but is highly permeable

to low-molecular-weight solute(s). A solution of a low-molecular-weight solute is

then pumped through the lumen or hollow core of the hollow ﬁber in steady-state

laminar ﬂow. This solute diffuses through the solution into the annular wall of the

hollow ﬁber, where it reacts with the enzyme via a ﬁrst-order reaction in the solute

concentration. The aspect ratio of hollow ﬁbers is such that axial diffusion can

be ignored in most applications. Moreover, the concentration of the solute in both

the solution and membrane is sufﬁciently dilute so that binary diffusion can be

assumed; that is, the reaction product(s) does(do) not inﬂuence the diffusion of the

solute through either the solution or the membrane. We use scaling to determine

how the describing equations for the membrane enzyme reactor can be simpliﬁed.

The species-balance equation in cylindrical coordinates given by equation (G.2-

10) in the Appendices and the corresponding boundary conditions for both the

lumen and annular wall regions of the membrane, after appropriate simpliﬁcation,

are given by (step 1)

1 −





∂c

∂z

= D

∂

∂r



∂c

∂r



, 0 ≤ r ≤ R

(5.E.9-1)

EXAMPLE PROBLEMS 329

0 = D

∂

∂r



∂c

∂r



− k

≤ r ≤ R

(5.E.9-2)

= c

at z = 0 (5.E.9-3)

∂c

∂r

= 0atr = 0, 0 ≤ z ≤ L (5.E.9-4)

∂c

∂r

= D

∂c

∂r

at r = R

, 0 ≤ z ≤ L (5.E.9-5)

= K

at r = R

, 0 ≤ z ≤ L (5.E.9-6)

∂c

∂r

= 0atr = R

, 0 ≤ z ≤ L (5.E.9-7)

where c

and c

are the molar concentrations of the solute in the lumen solu-

tion and annular membrane wall, respectively, U

the maximum ﬂuid velocity at

the centerline of the lumen, D

and D

the effective binary diffusion coefﬁcients

of the solute in the lumen solution and annular wall, respectively, c

the initial

concentration in the feed solution to the lumen, and K

the distribution coefﬁcient

for the thermodynamic equilibrium of component A between the lumen solution

and the annular membrane wall, respectively.

Deﬁne the following dimensionless variables (steps 2, 3, and 4):

∗

≡

; c

∗

≡

;



∂c

∂r



∗

≡

rls

∂c

∂r

;



∂c

∂z



∗

≡

zls

∂c

∂z

; z

∗

≡

; r

∗

≡

; r

∗

≡

r − r

(5.E.9-8)

Note that we have introduced separate scales for the concentration and radial coor-

dinate within the lumen and the annular wall. We also have allowed separate scales

for both the axial and radial concentration gradients within the lumen since these

do not necessarily scale with the concentration scale divided by the length scale. If

we had naively scaled these derivatives with the concentration scale divided by a

length scale, the forgiving nature of scaling would have indicated a contradiction.

We have also allowed for a reference length factor for the radial coordinate in order

to reference it to zero within the annular membrane wall.

Introduce these dimensionless variables into the describing equations and divide

through by the coefﬁcient of one term in each equation (steps 5 and 6):

1 −





∗2

∂c

∗

∂z

∗

rls

zls

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



, 0 ≤ r

∗

≤

(5.E.9-9)

This is explored further in Practice Problem 5.P.18.

330 APPLICATIONS IN MASS TRANSFER

0 =

∗

+ r

∂

∂r

∗



∗



∂c

∗

∂r

∗



−

∗2

∗

− r

≤ r

∗

≤

− r

(5.E.9-10)

∗

at z

∗

= 0 (5.E.9-11)

∂c

∗

∂r

∗

= 0atr

∗

= 0, 0 ≤ z

∗

≤

(5.E.9-12)

∂c

∗

∂r

∗

rls

∂c

∗

∂r

∗

at r

∗

, 0 ≤ z

∗

≤

(5.E.9-13)

∗

at r

∗

− r

, 0 ≤ z

∗

≤

(5.E.9-14)

∂c

∗

∂r

∗

= 0atr

∗

− r

, 0 ≤ z

∗

≤

(5.E.9-15)

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

concentration in the lumen and annular membrane wall can be bounded to be

◦

(1) by setting the appropriate dimensionless groups in equations (5.E.9-11) and

(5.E.9-14) equal to 1 to obtain c

= c

and c

= K

. The axial length scale is

obtained from the dimensionless group L/z

. The radial length scale in the annular

membrane wall can be referenced to zero by setting the appropriate dimensionless

group equal to zero in equation (5.E.9-14) to obtain r

= R

. Since the radial

diffusion must balance the axial convection in equation (5.E.9-9), the dimensionless

group, which is a measure of this ratio, must be set equal to 1; this establishes a

relationship between the axial and radial concentration gradient scales:

rls

zls

rls

zls

= 1 ⇒ c

zls

rls

(5.E.9-16)

For steady-state to exist, all the diffusing solute must react within the annular

membrane wall. This implies that radial diffusion must balance the homogeneous

reaction within the annular membrane wall. Hence, by setting the dimensionless

group in equation (5.E.9-10) that is a measure of this ratio equal to 1, we obtain

the radial length scale factor:

∗2

= 1 ⇒ r



(5.E.9-17)

Setting the dimensionless group in equation (5.E.9-13) equal to 1 provides the scale

for the radial concentration gradient in the lumen:

rls

√

⇒ c

rls



(5.E.9-18)

EXAMPLE PROBLEMS 331

where Th ≡



is the Thiele modulus, a dimensionless group that is a mea-

sure of the characteristic time for diffusion in the lumen to homogeneous reaction

in the annular membrane wall. It is convenient to express the scale for the radial

concentration gradient within the lumen in terms of the Thiele modulus, since this

provides a means for assessing its magnitude relative to the maximum possible

concentration gradient.

These scale factors then result in the following set of dimensionless describing

equations:



1 − r

∗2



∂c

∗

∂z

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



, 0 ≤ r

∗

≤ 1 (5.E.9-19)

0 =

∗

+ R

√

∂

∂r

∗



∗

√



∂c

∗

∂r

∗



− c

∗

0 ≤ r

∗

≤

− R

√

(5.E.9-20)

∗

= 1atz

∗

= 0 (5.E.9-21)

∂c

∗

∂r

∗

= 0atr

∗

= 0, 0 ≤ z

∗

≤ 1 (5.E.9-22)

∂c

∗

∂r

∗

∂c

∗

∂r

∗

(5.E.9-23)

∗

= c

∗

at r

∗

= 0, 0 ≤ z

∗

≤ 1 (5.E.9-24)

∂c

∗

∂r

∗

= 0atr

∗

− R

√

, 0 ≤ z

∗

≤ 1 (5.E.9-25)

Now let us consider how these dimensionless describing equations can be sim-

pliﬁed (step 8). If the dimensionless group R

√

 1, implying that the

region of inﬂuence wherein the homogeneous reaction converts all of the reacting

solute to product is much thinner than the radius of the lumen, curvature effects on

the mass-transfer process can be ignored. Moreover, if (R

− R

√

 1,

the aforementioned region of inﬂuence is much thinner than the thickness of the

annular membrane wall, the boundary condition given by equation (5.E.9-25) can

be applied at inﬁnity.

The solution to equations (5.E.9-19) and (5.E.9-20) can be simpliﬁed for the

special case of very small Thiele moduli, that is, for Th  1. For this case,

equation (5.E.9-18) indicates that the concentration gradient within the lumen of

the hollow ﬁber is negligibly small, thereby implying that the concentration within

This problem is a mass-transfer analog to the heat-transfer problem considered in Section 4.4; that

is, the low Biot number approximation made in the latter is analogous to the small Thiele modulus

approximation made here.

332 APPLICATIONS IN MASS TRANSFER

the lumen is uniform. Hence, equation (5.E.9-19) can be integrated as follows:





1 − r

∗2



∂c

∗

∂z

∗

2πr

∗



∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



2πr

∗

(5.E.9-26)





1 − r

∗2



∂c

∗

∂z

∗



∂c

∗

/∂r

∗

∂



∗

∂c

∗

∂r

∗



(5.E.9-27)

Use of Leibnitz’s rule given by equation (H.1-2) in the Appendices to integrate

the ﬁrst term and the fact that the concentration is essentially uniform within the

lumen for very small Thiele moduli then yields

∗



∗



1 − r

∗2



∗





1 − r

∗2



∗

∂c

∗

∂r

∗

(5.E.9-28)

Equation (5.E.9-28) now replaces equation (5.E.9-23) as one of the boundary con-

ditions on the differential equation for the mass transfer with the annular membrane

wall. If the conditionsR

√

 1and(R

− R

√

 1 apply, the

solution of the resulting simpliﬁed system of describing equations is straightforward

and given by

∗

= e

−4z

∗

; c

∗

= e

−(4z

∗

)

(5.E.9-29)

5.E.10 Dimensional Analysis for Oxygen Diffusion into a Spherical Red

Blood Cell

Consider a spherical red blood cell of radius R in the blood stream, as shown in

Figure 5.E.10-1. The cell is sufﬁciently large that it moves more slowly than the

blood ﬂow far removed from its surface (where the no-slip condition has to be

satisﬁed). Hence, there is a velocity of the bloodstream U

∞

relative to that of the

red blood cell. Oxygen diffuses through the bloodstream to this red blood cell,

after which it diffuses into the cell and reacts with the hemoglobin. The oxygen

and blood are referred to as components A and B, respectively. We wish to model

the convective oxygen mass transfer through the bloodstream to the cell. However,

since this is a rather complicated problem to solve, we employ the scaling method

for dimensional analysis to obtain a correlation for the dimensionless mass-transfer

coefﬁcient, deﬁned by

≡

− c

A∞

(5.E.10-1)

where N

is the molar ﬂux (moles/area·time) at the outer cell wall, c

the

oxygen concentration (moles/volume) at the outer cell wall, and c

A∞

the oxygen

An analytical solution to this problem accounting for curvature effects is given by W. Lewis and S.

Middleman, A.I.Ch.E.J., 20(5), 1012–1014 (1974).