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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SOLUTALLY DRIVEN FREE CONVECTION DUE TO EVAPOTRANSPIRATION 283

Note that each term in equation (G.2-5) has been divided by the molecular weight,

which is assumed to be constant for the dilute solutions assumed. The corresponding

boundary conditions are given by

= 0,u

= 0,c

= c

at r = R

(5.9-5)

= 0,u

= 0,

∂c

∂r

= 0atr = R

(5.9-6)

= 0,u

= 0,c

= c

A∞

at z = 0 (5.9-7)

= f

(r), u

= f

(r), c

= f

(r) at z = L (5.9-8)

where f

(r), f

(r),andf

(r) are unspeciﬁed functions of r that often are unknown.

Equation (5.9-5) constitutes the no-slip and impermeable boundary conditions at the

inner boundary. The evapotranspiration through the inner boundary will contribute

to the radial component of velocity, which is assumed to be a negligible effect

in this scaling analysis.

Equation (5.9-6) states that the velocities as well as the

molar ﬂux are zero at the impermeable outer boundary. Equation (5.9-7) states that

the ﬂuid is quiescent and at ambient conditions at the bottom of the annular gap.

Equation (5.9-8) is a formal statement that downstream boundary conditions are

required for these coupled elliptic equations.

Since the density is concentration-dependent, we need an appropriate equation

of state. Here we consider small variations and hence will represent the density via

a truncated Taylor series expansion about its value ρ

∞

at the ambient concentra-

tion c

A∞

ρ = ρ|

A∞

∂ρ

∂c



A∞

− c

A∞

) = ρ

∞

− ρ

∞

− c

A∞

) (5.9-9)

where β

is the coefﬁcient of solutal volume expansion. It is convenient to split

the pressure into dynamic, P

, and hydrostatic, P

, contributions

P = P

(r, z) + P

(z) (5.9-10)

When equations (5.9-9) and (5.9-10) are substituted into equation (5.9-1), we obtain

∞

∂u

∂r

+ ρ

∞

∂u

∂z

=−

∂P

∂z

+ μ

∂

∂r



∂u

∂r



+ μ

∂

∂z

+ ρ

∞

g(c

− c

A∞

)

(5.9-11)

Note that the ρ

∞

g term does not appear in equation (5.9-11) because it cancels

with the derivative of the purely hydrostatic contribution to the pressure.

Scaling analysis is used to develop the criterion for making this assumption in Practice Problem

5.P.23.

Splitting the pressure into its dynamic and hydrostatic contributions was discussed in connection with

Example Problem 4.E.7.

284 APPLICATIONS IN MASS TRANSFER

Deﬁne the following dimensionless dependent and independent variables (steps

2, 3, and 4):

∗

≡

; u

∗

≡

; P

∗

≡

; c

∗

≡

− c

;

∗

≡

; r

∗

≡

r − r

; r

∗

≡

r − r

(5.9-12)

Note that we have allowed for different radial length scales for the species-balance

equation and equations of motion since the concentration might experience a char-

acteristic change of

◦

(1) over a different length scale than that of the velocities.

Introduce these dimensionless variables into the describing equations and divide

each equation by the dimensional coefﬁcient of one term that should be retained

in order to maintain physical signiﬁcance (steps 5 and 6):

∞

∗

∂u

∗

∂r

∗

∞

∗

∂u

∗

∂z

∗

=−

μu

∂P

∗

∂z

∗

∂

∂r

∗



∗



∂u

∗

∂r

∗



∂

∗

∂z

∗2

gβ

∞



∗

− c

A∞



(5.9-13)

∞

∗

∂u

∗

∂r

∗

∞

∗

∂u

∗

∂z

∗

=−

μu

∂P

∗

∂r

∗

∂

∂r

∗



∗

+ r

∂

∂r

∗



∗



∗



∂

∗

∂z

∗2

(5.9-14)

∗

+ r

∂

∂r

∗



∗



∗



∂u

∗

∂z

∗

= 0 (5.9-15)

∗

∂c

∗

∂r

∗

∂c

∗

∂z

∗

+ r

∂

∂r

∗



∗



∂c

∗

∂r

∗



∂

∗

∂z

∗2

(5.9-16)

∗

= 0,u

∗

= 0atr

∗

− r

∗

− c

at r

∗

− r

(5.9-17)

∗

= 0,u

∗

= 0atr

∗

− r

∂c

∗

∂r

∗

= 0atr

∗

− r

(5.9-18)

∗

= 0,u

∗

= 0,c

∗

A∞

− c

at z

∗

= 0 (5.9-19)

SOLUTALLY DRIVEN FREE CONVECTION DUE TO EVAPOTRANSPIRATION 285

∗

= f

∗

), u

∗

= f

∗

), c

∗

= f

∗

) at z

∗

(5.9-20)

where ν

∞

≡ μ/ρ

∞

is the kinematic viscosity. Note that the boundary conditions

given by equation (5.9-5) are not applied at the same values of the dimension-

less radial coordinate owing to the different radial scale factors for the equations

of motion and the species-balance equation; the same comment applies to equ-

ation (5.9-6).

The effect of viscosity for this ﬂow will be conﬁned to a thin region near

the vertical cylinder; hence, the radial length scale for the equations of motion

will be the thickness of the momentum boundary layer or region of inﬂuence

;thatis,r

= δ

. The momentum boundary-layer thickness is obtained by

balancing the convection terms with the principal viscous term in equation (5.9-13).

Similarly, the effect of species diffusion will also be conﬁned to a thin region δ

although not necessarily of the same thickness as that of the momentum boundary

layer; hence, the radial length scale for the species-balance equation will be the

thickness of the solutal boundary layer or region of inﬂuence δ

;thatis,r

= δ

The solutal boundary-layer thickness is obtained by balancing the radial species

convection and diffusion terms in equation (5.9-16). To determine the axial velocity

scale u

, we need to balance what causes the ﬂow with the principal resistance

to ﬂow; the former is the gravitationally induced body force, whereas the latter

is the principal viscous term. The transverse velocity scale u

is obtained from

the continuity equation since this is inherently a developing ﬂow. One might be

tempted to obtain P

from the dimensionless group multiplying the pressure term in

equation (5.9-13). However, the pressure term in equation (5.9-13) does not cause

the free-convection ﬂow; the latter is caused by the gravitational body-force term

in this equation. The pressure does cause the ﬂow in the r-direction, which is the

reason why we determine its scale by setting the dimensionless group containing P

in equation (5.9-14) equal to 1. The reference and scale factors for the axial length

and concentration are obtained from the boundary conditions to ensure that these

variables are bounded of

◦

(1). These considerations then result in the following

scale factors (step 7):

= (gβ

c

0.5

; u



∞

gβ

c



0.25

;



gβ

c



0.5

; c

= c

− c

A∞

≡ c

; z

= L;

= δ

0.5

0.25

; r

= δ

0.5

Sc ·Gr

0.25

(5.9-21)

where Re ≡ u

L/ν

∞

the Reynolds number, Pe

≡ u

L/D

= Re · Sc the solu-

tal Peclet number, Sc ≡ ν

∞

the Schmidt number, and Gr

≡ L

gβ

c

/ν

∞

the solutal Grashof number. Note that the Grashof number is a measure of the ratio

286 APPLICATIONS IN MASS TRANSFER

of the free convection to viscous transport of momentum; as such, it is the analog of

the Reynolds number for free convection.

Note that the last of equations (5.9-21)

indicates that δ

<δ

for liquids and δ

∼

for gases.

If we now rewrite our dimensionless describing equations in terms of the scales

deﬁned by equations (5.9-21), we obtain

∗

∂u

∗

∂r

∗

+ u

∗

∂u

∗

∂z

∗

=−

0.5

∂P

∗

∂z

∗

+ R

/δ

∂

∂r

∗



∗



∂u

∗

∂r

∗



0.5

∂

∗

∂z

∗2

+ c

∗

(5.9-22)

∗

∂u

∗

∂r

∗

+ u

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂r

∗

∂

∂r

∗



∗

+ R

/δ

∂

∂r

∗



∗



∗



0.5

∂

∗

∂z

∗2

(5.9-23)

∗

+ R

/δ

∂

∂r

∗



∗



∗



∂u

∗

∂z

∗

= 0 (5.9-24)

∗

∂c

∗

∂r

∗

∂c

∗

∂z

∗

+ (R

/δ

)Sc

∂

∂r

∗



∗



∂c

∗

∂r

∗



0.5

∂

∗

∂z

∗2

(5.9-25)

∗

= 0,u

∗

= 0,c

∗

= 1atr

∗

= r

∗

= 0 (5.9-26)

∗

= 0,u

∗

= 0atr

∗

− R

∂c

∗

∂r

∗

= 0atr

∗

− R

(5.9-27)

∗

= 0,u

∗

= 0,c

∗

= 0atz

∗

= 0 (5.9-28)

∗

= f

∗

), u

∗

= f

∗

), c

∗

= f

∗

) at z

∗

= 1 (5.9-29)

We can now consider how these scaled dimensionless describing equations can

be simpliﬁed (step 8). Note that if the Grashof number is very large such that

0.5

 1, the pressure and axial viscous momentum transfer terms can be dropped

from equation (5.9-22). The former simpliﬁcation implies that the z-component is

decoupled from the solution to the r-component of the equations of motion; hence,

The solutal Rayleigh number, deﬁned as Ra

≡ L

gβ

c

/ν

∞

= Gr

· Pr, is another important

dimensionless group that appears in free-convection problems; it is a measure of the ratio of the free

convection to viscous transport of heat; as such, it is the analog of the solutal Peclet number for free

convection.

DIMENSIONAL ANALYSIS FOR A MEMBRANE–LUNG OXYGENATOR 287

the latter equation can be ignored. Dropping the axial viscous momentum transfer

term from equation (5.9-22) converts it from an elliptic into a parabolic differential

equation; this obviates the need to specify downstream boundary conditions, which

in many cases are unknown. A very large Grashof number also implies that the axial

species-diffusion term can be dropped from equation (5.9-25), which also converts

it from an elliptic into a parabolic differential equation, again avoiding the need to

specify a downstream boundary condition. If in addition, R

 δ

and R

 δ

the curvature effects can be neglected. Finally, if R

− R

 δ

and R

− R

 δ

the boundary conditions given by equation (5.9-27) can be applied at inﬁnity. If the

aforementioned criteria are satisﬁed, the resulting simpliﬁed describing equations

are given by

∗

∂u

∗

∂r

∗

+ u

∗

∂u

∗

∂z

∗

∂

∗

∂r

2∗

+ c

∗

(5.9-30)

∂u

∗

∂r

∗

∂u

∗

∂z

∗

= 0 (5.9-31)

∗

∂c

∗

∂r

∗

∂c

∗

∂z

∗

∂

∗

∂r

∗2

(5.9-32)

∗

= 0,u

∗

= 0,c

∗

= 1atr

∗

= r

∗

= 0 (5.9-33)

∗

= 0,u

∗

= 0,

∂c

∗

∂r

∗

= 0atr

∗

= r

∗

=∞ (5.9-34)

∗

= 0,u

∗

= 0,c

∗

= 0atz

∗

= 0 (5.9-35)

In particular, one sees that the outer cylindrical boundary will have a negligible

effect on the free convection if δ

= L/(R

0.25

)  1. Note that this criterion

will break down for sufﬁciently long cylinders owing to the thickening of the free-

convection boundary layer associated with progressively more evapotranspiration

along the cylinder.

5.10 DIMENSIONAL ANALYSIS FOR A MEMBRANE–LUNG

OXYGENATOR

Section 2.4 we discussed the scaling analysis procedure for dimensional analysis

and its advantages relative to the Pi theorem. Here we apply this procedure to

develop a correlation for the performance of a membrane–lung oxygenator. The

steps referred to here are those outlined in Section 2.4; these differ from those used

in Sections 5.2 through 5.9 since no attempt is made to achieve

◦

(1) scaling. Note

that

◦

(1) scaling analysis, which was illustrated in Sections 5.2 through 5.9, always

leads to the minimum parametric representation for a set of describing equations;

hence, it can always be used to identify the appropriate dimensionless groups.

However, carrying out an

◦

(1) scaling analysis can be somewhat complicated and

288 APPLICATIONS IN MASS TRANSFER

time-consuming. In contrast, the scaling analysis approach to dimensional analysis

illustrated in this section is much easier and quicker to implement. However, it does

not provide as much information as does

◦

(1) scaling analysis for achieving the

minimum parametric representation. In particular, it does not lead to groups whose

magnitude can be used to assess the relative importance of particular terms in the

describing equations. It also does not identify regions of inﬂuence or boundary

layers, whose identiﬁcation in some cases can reduce the number of dimension-

less groups.

The oxygenator of interest here involves a bundle of cylindrical hollow-ﬁber

membranes encased in a tubular housing. Because the hollow-ﬁber membranes

are both microporous and hydrophobic, they provide a means for mass transfer of

oxygen and carbon dioxide to and from the blood, respectively, while preventing

direct contact between the gas and liquid. The mass transfer is controlled on the

blood side because of the inability of the oxygen-absorbing hemoglobin “particles”

to closely approach the inner surface of the membrane. Improvement in oxygenator

performance has focused on various means to reduce the resistance to mass transfer

on the blood side of the membrane. One very effective way to accomplish this is

to oscillate the hollow ﬁbers relative to the blood ﬂow to increase the oxygen

concentration gradients adjacent to the membrane.

We use the scaling analysis

approach to dimensional analysis to determine the dimensionless groups required

to correlate the effects of oscillating the hollow ﬁbers on the performance of the

oxygenator. It is sufﬁcient here to consider the effect of oscillations on the oxygen

mass transfer to the blood ﬂow in a single hollow-ﬁber membrane of radius R and

length L, as shown in Figure 5.10-1.

Step 1 in the scaling procedure for dimensional analysis consists of writing the

appropriate describing equations for the oxygen mass transfer to the blood, which

will be assumed to be in fully developed periodically pulsed laminar ﬂow.

The

= Aw cos wt

Figure 5.10-1 Single hollow ﬁber of radius R and length L in a membrane–lung oxy-

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

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

concentrated.

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

6, 1997.

Note that this is an uncommon example of a fully developed unsteady-state ﬂow; that is, the axial

velocity does not change in the axial direction at any instant of time, yet it is a function of time due to

the oscillating wall.

DIMENSIONAL ANALYSIS FOR A MEMBRANE–LUNG OXYGENATOR 289

mass-transfer coefﬁcient k

•

provides a convenient measure of the effect of the

oscillating wall on the oxygen mass transfer and is deﬁned in terms of

,the

time-average molar ﬂux of oxygen at the inner wall of the hollow-ﬁber mem-

brane, as

•

≡

c

(5.10-1)

where c

is the log-mean concentration driving force, deﬁned as

c

≡

− c

) − (c

− c

)

ln (c

− c

)/(c

− c

)

(5.10-2)

in which c

is the equilibrium oxygen concentration at the blood side of the

membrane, and c

and c

are the average concentrations in the blood at z = 0

and at z = L, respectively, and

≡

(

2π/ω

(

2π/ω

2π



2π/ω

dt (5.10-3)

The molar ﬂux N

appearing in equation (5.10-3) is equal to the local molar ﬂux

averaged over the length of the hollow-ﬁber membrane and is deﬁned

≡

(

(∂c

/∂r)

r=R

(



∂c

∂r



r=R

dz (5.10-4)

where D

is the binary diffusion coefﬁcient for oxygen in blood. Note that the

bulk ﬂow contribution to the molar ﬂux has been ignored because the solutions are

dilute. When equations (5.10-3) and (5.10-4) are substituted into equation (5.10-1),

we obtain

•

ωD

2πLc



2π/ω



∂c

∂r



r=R

dzdt (5.10-5)

Both c

appearing in c

and c

in equation (5.10-5) are obtained from a

solution to the axisymmetric form of the convective diffusion equation in cylindrical

coordinates given by equation (G.2-5) in the Appendices

∂c

∂t

+ u

∂c

∂z

∂

∂r



∂c

∂r



(5.10-6)

in which u

is the local axial ﬂuid velocity. In arriving at equation (5.10-6) each

term in equation (G.2-5) has been divided by the molecular weight, which is

Note that the mass-transfer coefﬁcient is deﬁned in terms of a ﬂux divided by a driving force; as

such, it can be expressed in several different ways, depending on the units chosen for these quantities

as well as whether the total ﬂux or just the diffusive ﬂux is used; the various deﬁnitions of the mass-

transfer coefﬁcient are discussed in standard references such as Bird et al., Transport Phenomena, 2nd

ed., pp. 672–675.

290 APPLICATIONS IN MASS TRANSFER

constant for the assumed dilute solution. An analytical solution for u

for fully

developed laminar ﬂow subject to axial harmonic pulsations of the tube wall has

been developed and is of the form

= f



,ωt,

ωR

Aω



(5.10-7)

where

U is the average ﬂuid velocity in the hollow ﬁber and ν is the kinematic

viscosity of blood. The dimensional analysis correlation given by equation (5.10-7)

is also developed in Practice Problem 3.P.38. The boundary and periodic solution

conditions are given by

∂c

∂r

= 0atr = 0, 0 ≤ z ≤ L (5.10-8)

= c

at r = R, 0 ≤ z ≤ L (5.10-9)

= c

at z = 0 (5.10-10)

= c

t+2π/ω

(5.10-11)

Steps 2 and 3 in the scaling procedure for dimensional analysis involve deﬁning

arbitrary scale factors for all the dependent and independent variables and reference

factors for those not naturally referenced to zero. Hence, we introduce the following

dimensionless variables:

∗

≡

− c

; u

∗

≡

; r

∗

≡

; z

∗

≡

; t

∗

≡

(5.10-12)

Steps 4 and 5 involve introducing these dimensionless variables into the describ-

ing equations and dividing through by the dimensional coefﬁcient of one term in

each equation. In the scaling analysis procedure for dimensional analysis, in con-

trast to

◦

(1) scaling analysis, it makes no difference which term is chosen in this

step. These steps then yield the following dimensionless describing equations:

2πk

•

Lc

∗

ωD



2π/ωt



L/z

∂c

∗

∂r

∗



∗

=R/r

∗

(5.10-13)

c

∗

≡

−c

∗

+ (c

− c

)/c



− c

−

− c

∗



(5.10-14)

∂c

∗

∂t

∗

∂c

∗

∂z

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.10-15)

W.B.Krantz,R.R.Bilodeau,M.E.Voorhees,andR.J.Elgas,J. Membrane Sci., 124, 283–299

(1997).

DIMENSIONAL ANALYSIS FOR A MEMBRANE–LUNG OXYGENATOR 291

∗



∗

,ωt

∗

ωR

Aω



(5.10-16)

∂c

∗

∂r

∗

= 0atr

∗

= 0, 0 ≤ z

∗

≤

(5.10-17)

∗

− c

at r

∗

, 0 ≤ z

∗

≤

(5.10-18)

∗

− c

at z

∗

= 0 (5.10-19)

∗

= c

∗

+2π/ωt

(5.10-20)

Step 6 involves setting the various groups equal to 1 or zero to determine the

scale and reference factors, respectively. Since there is no attempt to achieve

◦

(1) in

the scaling approach for dimensional analysis, it makes no difference what groups

are chosen in this step. Let us set equation (5.10-19) equal to zero to determine

the reference concentration c

= c

and the appropriate group in equation (5.10-

18) equal to 1 to determine the concentration scale c

= c

− c

. Setting the

remaining dimensionless groups in equation (5.10-18) equal to 1 determines the

radial and axial length scales r

= R and z

= L, respectively; note, however, that

there is no assurance that these length factors will scale the corresponding deriva-

tives with respect to these spatial coordinates to be of

◦

(1). The dimensionless

group in equation (5.10-16) provides the velocity scale u

= U. Either the dimen-

sionless group in equation (5.10-15) or in equation (5.10-20) can be set equal to 1

to determine the time scale; let us arbitrarily choose the latter, thereby obtaining

= 2π/ω. These choices then yield the following minimum parametric represen-

tation of the describing equations; that is, in terms of the minimum number of

dimensionless groups:

Sh =

∗

ln(1 − c

∗

)



∂c

∗

∂r

∗



∗

(5.10-21)

(c

∗

)

≡

−c

∗

ln(1 − c

∗

)

(5.10-22)

ωR

2πD

∂c

∗

∂t

∗

2Gz

∗

∂c

∗

∂z

∗

∂

∂r

∗



∗

∂c

∗

∂r

∗



(5.10-23)

∗

= f



∗

ωR

Aω



(5.10-24)

∂c

∗

∂r

∗

= 0atr

∗

= 0, 0 ≤ z

∗

≤ 1 (5.10-25)

∗

= 1atr

∗

= 1, 0 ≤ z

∗

≤ 1 (5.10-26)

∗

= 0atz

∗

= 0 (5.10-27)

∗

= c

∗

(5.10-28)

292 APPLICATIONS IN MASS TRANSFER

where (c

∗

)

denotes c

∗

in the presence of axial oscillations or vibrations and

where

Sh ≡

•

is the Sherwood number (5.10-29)

Gz ≡

πRPe

is the Graetz number (5.10-30)

The Sherwood number provides a measure of the overall mass transfer to that by

diffusion alone; as such, it is analogous to the Nusselt number in heat transfer.

The

Graetz number provides a measure of the relative convection to diffusional transport

of species; as can be seen from equation (5.10-30), it is proportional to the solutal

Peclet number, which is also a measure of this same ratio. However, the Graetz

number takes into account the ﬁnite contact time for mass transfer via the ratio R/L.

Equation (5.10-21) implies that the Sherwood number is a function of the dimen-

sionless groups involved in determining c

∗

and ∂c

∗

/∂r

∗



∗

and hence will be

functions of only the dimensionless groups involved in solving equation (5.10-23).

These include the two dimensionless groups appearing in this equation as well as

the two involved in determining u

∗

as indicated by equation (5.10-24). Hence, we

conclude that membrane–lung oxygenator performance can be correlated in terms

of ﬁve dimensionless groups; that is,

Sh = f



Gz,

ωR

Aω



(5.10-31)

Note that a naive application of the Pi theorem would imply that eight dimensionless

groups would be required (i.e., n = 11 and m = 3).

The ﬁve dimensionless groups in the correlation for membrane–lung oxygenator

performance given by equation (5.10-31) are not unique. Step 7 involves isolating

certain quantities by the procedure indicated formally by equation (2.4-2). Here it

is convenient to isolate the angular frequency into just two groups by the following

operation:

ωR

/ν

= Sc (5.10-32)

where Sc is the Schmidt number. Hence, our modiﬁed correlation for the Sherwood

number is given by

Sh = f



Gz,Sc,

ωR

Aω



(5.10-33)

Recasting our correlation in terms of the Schmidt number is particularly convenient

for designing a membrane blood oxygenator since the Schmidt number is ﬁxed for

The dimensionless group corresponding to the Sherwood number deﬁned here is sometimes referred

to as the Nusselt number for mass transfer.