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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POROUS MEDIA FLOW 53

solid. The local Reynolds number that characterizes ﬂow through the pores is

well into the creeping-ﬂow regime, due to the small pore size; hence, one can

safely ignore the inertia terms in the equations of motion. Moreover, the resistance

to ﬂow offered by the pore walls within the porous medium is generally much

greater than that of the solid walls that bound the entire porous medium (the

wall of the cylindrical tube in the present problem); the former can be described

by Darcy’s law, whereas the latter is usually neglected. However, the effect of

the boundaries of the porous medium cannot be ignored within some region of

inﬂuence adjacent to the boundaries. We seek to use scaling analysis to determine

the criterion for when the effect of the lateral boundaries on the ﬂow through the

porous medium can be neglected and to determine the thickness of the region of

inﬂuence within which the effect of drag on the lateral boundaries on the ﬂow must

be considered.

The appropriately simpliﬁed form of equations (E.2-1) and (E.2-3) in the Appen-

dices for steady-state creeping ﬂow through a porous medium are given by (step 1)

0 =−

∂P

∂z

−

u

+ μ



du



(3.8-1)

0 =−

∂P

∂r

(3.8-2)

u

= 0atr = R (3.8-3)

du

= 0atr = 0 (3.8-4)

P = P

at z = 0 (3.8-5)

where u

is the superﬁcial ﬂow velocity through the porous medium (i.e., the

ﬂow velocity averaged over a differential cross-sectional area of the heterogeneous

medium in contrast to the ﬂow velocity through a pore) and k

is the Darcy

permeability of the microporous solid. Note that we have ignored any effect of

hydrostatic pressure in equation (3.8-2). Equations (3.8-1) and (3.8-2) imply that

the axial pressure gradient is a constant. Hence, our describing equations sim-

plify to

0 =

P

−

u

+ μ



du



(3.8-6)

u

= 0atr = R (3.8-7)

du

= 0atr = 0 (3.8-8)

where P ≡ P

− P

,inwhichP

and P

are the pressures at z = 0andz = L,

respectively. The equations above differ from the conventional equations of motion

by the inclusion of the Darcy term, which accounts for the resistance to ﬂow offered

by the porous medium.

54 APPLICATIONS IN FLUID DYNAMICS

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

and 4):

u

∗

≡

u

; r

∗

≡

(3.8-9)

Substitute these variables into the describing equations and divide through by the

dimensional coefﬁcient of a term that must be retained to obtain the following set

of dimensionless describing equations (steps 5 and 6):

0 = 1 −

μLu

P

u

∗

μLu

P r

∗



∗

du

∗



(3.8-10)

u

∗

= 0atr

∗

(3.8-11)

du

∗

= 0atr

∗

= 0 (3.8-12)

The radial length scale r

is determined by setting the dimensionless group in

equation (3.8-11) equal to 1. Since pressure causes the ﬂow and the porous medium

is assumed to offer the primary resistance to ﬂow, the velocity scale u

is deter-

mined by setting the dimensionless group in the Darcy term in equation (3.8-10)

equal to 1. These considerations then determine the following scale and reference

factors (step 7):

= R;u

P

μL

(3.8-13)

Note that the velocity scale in equation (3.8-13) is the axial velocity that would

be predicted if just the pressure and Darcy ﬂow terms were retained in equation

(3.8-10).

When the scales deﬁned by equation (3.8-13) are substituted into equations

(3.8-10) through (3.8-12), we obtain

0 = 1 −u

∗



∗

du

∗



(3.8-14)

u

∗

= 0atr

∗

= 1 (3.8-15)

du

∗

= 0atr

∗

= 0 (3.8-16)

Hence, we see that the effects of viscous drag at the inner wall of the cylindrical

tube can be neglected if the following condition is satisﬁed (step 8):

POROUS MEDIA FLOW 55

 1 ⇒ viscous drag at boundary of porous medium can be neglected

(3.8-17)

The velocity obtained by solving equation (3.8-14) while dropping the last

term will be accurate provided that the condition indicated in equation (3.8-17)

is satisﬁed; that is, if k

◦

(0.01) and provided that one is not interested

in predicting the velocity very close to the tube wall or the viscous drag at the

wall. It is of interest to determine the region of inﬂuence δ

, wherein the effect

of the tube wall cannot be ignored. To do this it is necessary to translate the

coordinate system to the tube wall via the transformation ˜r ≡ R − r.Sincewe

are considering a region of inﬂuence, ˜r

= δ

; the velocity scale still is given by

equation (3.8-13). The resulting transformed dimensionless describing equations

are given by

0 = 1 −u

∗

R/δ

−˜r

∗



−˜r

∗



du

∗

d ˜r

∗



(3.8-18)

u

∗

= 0at˜r

∗

= 0 (3.8-19)

du

∗

= 0at˜r

∗

(3.8-20)

Within the region of inﬂuence the last term in equation (3.8-18) cannot be ignored;

hence, we set the dimensionless group multiplying this term equal to 1 to determine

the thickness of the region of inﬂuence:

= 1 ⇒ δ



(3.8-21)

For typical Darcy permeabilities, δ

is on the order of 10 to 100 pore diame-

ters. Hence, in most cases one can ignore the effect of the system boundaries

on the relationship between the average superﬁcial velocity through the porous

medium and the pressure gradient used. The curvature effects can be ignored

within the region of inﬂuence near the wall if the following criterion is satis-

ﬁed:

 1 ⇒ curvature effects can be ignored (3.8-22)

in which case the describing equations simplify to

0 = 1 −u

∗

u

∗

∗2

(3.8-23)

56 APPLICATIONS IN FLUID DYNAMICS

u

∗

= 0at˜r

∗

= 0 (3.8-24)

du

∗

= 0as˜r

∗

→∞ (3.8-25)

3.9 COMPRESSIBLE FLUID FLOW

In the examples we have considered thus far, we have assumed that the ﬂowing ﬂuid

was incompressible. This is certainly a tenuous assumption for gas ﬂows and can

be questioned even for complex liquid ﬂows.

In this example we use systematic

scaling analysis to assess when a ﬂow can be considered to be incompressible.

This example will again involve introducing separate scale factors for a spatial

derivative rather than assuming that it scales with the ratio of the characteristic

value for the dependent variable divided by the characteristic length.

Consider the steady-state pressure-driven ﬂow in a cylindrical tube of a com-

pressible gas whose other physical properties will be assumed to be constant as

shown in Figure 3.9-1. Due to the compressibility, the density of the gas will

decrease due to the pressure drop in the axial direction; it also changes in the

radial direction, however, this effect is usually quite small (although it will be

included in this scaling analysis). Correspondingly, there will be an increase in the

axial velocity, thereby implying that this is a developing ﬂow with both nonzero

axial and radial velocity components. Hence, we must use the appropriate form of

the steady-state equations of motion that allow for a compressible ﬂow. We use

scaling analysis to determine the conditions for which this ﬂow can be considered

to be incompressible and fully developed. Although we allow only for a variable

density in this problem, the manner in which we use scaling to assess when this

Figure 3.9-1 Steady-state pressure-driven ﬂow in a cylindrical tube of radius R and length

L of a compressible gas for which the other physical properties are assumed to be constant.

Examples of complex ﬂuids include the ﬂow of microemulsions, proteins, micellar solutions, and

suspensions, as well as other nonhomogeneous liquids.

COMPRESSIBLE FLUID FLOW 57

property can be assumed to be constant can be used to handle other variable

properties, such as the viscosity.

The describing equations in terms of the components of the viscous stress tensor

are obtained by appropriately simplifying the continuity equations and equations

of motion given by equations (C.2-1), (D.2-1), and (D.2-3) in the Appendices:

ρu

∂u

∂r

+ ρu

∂u

∂z

=−

∂P

∂z

−

∂

∂r

(rτ

) −

∂τ

∂z

(3.9-1)

ρu

∂u

∂r

+ ρu

∂u

∂z

=−

∂P

∂r

−

∂

∂r

(rτ

) −

θθ

−

∂τ

∂z

(3.9-2)

∂

∂r

(ρru

) +

∂

∂z

(ρu

) = 0 (3.9-3)

where τ

is the viscous shear stress associated with the transfer of j -momentum in

the i-direction. These equations have been given in terms of the components of the

viscous stress tensor to emphasize that the term τ

θθ

is not zero even though this is

an axisymmetric ﬂow. To complete the speciﬁcation of the describing equations,

we need to provide an equation of state that relates the density to the pressure (the

temperature is assumed to be constant) and need to specify appropriate boundary

conditions; this will be done after we have rearranged the equations above into a

more convenient form.

When the appropriate components of the viscous stress tensor given by equations

(D.2-4) through (D.2-9) in the Appendices are substituted into equations (3.9-1) and

(3.9-2) and the resulting equations simpliﬁed using the continuity equation given

by equation (3.9-3), we obtain

ρu

∂u

∂r

+ ρu

∂u

∂z

=−

∂P

∂z

∂

∂r



∂u

∂r



−

∂

∂z



∂ρ

∂r

∂ρ

∂z



+ μ

∂

∂z

(3.9-4)

ρu

∂u

∂r

+ρu

∂u

∂z

=−

∂P

∂r

+μ

∂

∂r



∂

∂r

(ru

)



−

∂

∂r



∂ρ

∂r

∂ρ

∂z



+μ

∂

∂z

(3.9-5)

The derivatives of the density can be expressed in terms of derivatives of the

pressure as follows:

∂ρ

∂r

∂ρ

∂P



∂P

∂r

∂P

∂r

;

∂ρ

∂z

∂ρ

∂P



∂P

∂z

∂P

∂z

(3.9-6)

in which c is the speed of sound in the gas and γ is the ratio of the heat capacity at

constant pressure to that at constant volume. To assess the effect of pressure on the

density, we represent the density in terms of a Taylor series expansion about some

58 APPLICATIONS IN FLUID DYNAMICS

reference density ρ

, which is chosen to be at the downstream low pressure P

ρ = ρ

∂ρ

∂P



(P − P

) +

◦

(P − P

)

= ρ

(P − P

) +

◦

(P − P

)

(3.9-7)

Note that it is sufﬁcient to represent the density as a Taylor series truncated at

the second term since we need consider only small variations in the density. That

is, because we seek to determine the conditions for which incompressible ﬂow

can be assumed, we need explore only small variations in the density since large

variations in the density most certainly will be associated with compressible ﬂow.

Hence, our describing equations and associated boundary conditions are given by

(step 1)

ρu

∂u

∂r

+ ρu

∂u

∂z

=−

∂P

∂z

∂

∂r



∂u

∂r



−

∂

∂z



γu

ρc

∂P

∂r

γu

ρc

∂P

∂z



+ μ

∂

∂z

(3.9-8)

ρu

∂u

∂r

+ ρu

∂u

∂z

=−

∂P

∂r

+ μ

∂

∂r



∂

∂r

(ru

)



−

∂

∂r



γu

ρc

∂P

∂r

γu

ρc

∂P

∂z



+ μ

∂

∂z

(3.9-9)

∂

∂r

(ρru

) +

∂

∂z

(ρu

) = 0 (3.9-10)

ρ = ρ

(P − P

) (3.9-11)

∂u

∂r

= 0,u

= 0,P= p(z) at r = 0 (3.9-12)

= 0,u

= 0atr = R (3.9-13)

= f

(r), u

= f

(r), P = P

at z = 0 (3.9-14)

= g

(r), u

= g

(r) at z = L (3.9-15)

in which p(z), f

(r), g

(r), f

(r),andg

(r) are unspeciﬁed functions that in prin-

ciple would need to be known in order to integrate the full set of describing

equations. If the conditions that we seek to determine for assuming fully devel-

oped incompressible ﬂow are satisﬁed, it will not be necessary to know these

unspeciﬁed functions.

COMPRESSIBLE FLUID FLOW 59

Introduce the following scale factors, reference factors, and dimensionless vari-

ables (steps 2, 3, and 4):

∗

≡

; u

∗

≡

; P

∗

≡

P − P

;



∂P

∂r



∗

≡

∂P

∂r

;

∗

≡

; r

∗

≡

; z

∗

≡

(3.9-16)

We again allow for a reference pressure P

in our deﬁnition of the dimensionless

pressure since the latter is not naturally referenced to zero. Note that we must also

scale the density in this problem since it is one of the dependent variables. How-

ever, we do not need to introduce a reference factor for the density even though it

is not referenced to zero. The reason for this is that we are considering only small

variations in density; that is, the density does not vary signiﬁcantly in either coordi-

nate direction. We have introduced a scale for the radial derivative of the pressure

denoted by P

since we do not anticipate that this will scale in the same way as

the axial pressure gradient. The question might arise as to how one knows whether

to scale a derivative as the ratio of some dependent variable scale divided by some

independent variable scale or to introduce a separate scale for the entire derivative.

The answer is contained simply in the forgiving nature of scaling. That is, if we

were to assume that the radial pressure derivative scales as the pressure scale P

divided by the radial length scale r

, we would ﬁnd that the dimensionless group in

front of the dimensionless radial pressure derivative was much larger than that of

any other term in the r-component of the equations of motion. This clearly would

indicate that we scaled incorrectly. Hence, determining whether a derivative needs

its own scale is often a matter of trial and error. If any term is scaled incorrectly,

the forgiving nature of scaling will indicate a contradiction in the dimensionless

equations. One then rescales until a self-consistent set of dimensionless equations

is obtained; that is, a system of equations for which balancing terms are of

◦

(1)

and all other terms, including those multiplied by dimensionless groups, are of

◦

(1). The consequence of not introducing a separate scale for the radial pressure

derivative is explored in Practice Problem 3.P.31.

Substitute the variables deﬁned in equation (3.9-16) into the describing equations

and divide through by the dimensional coefﬁcient of a term that must be retained

to obtain the following set of dimensionless describing equations (steps 5 and 6):

∗

∂u

∗

∂r

∗

μz

∗

∂u

∗

∂z

∗

=−

μu

∂P

∗

∂z

∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



−

γu

∂

∂z

∗



∗



∂P

∂r



∗



−

γP

∂

∂z

∗



∗

∂P

∗

∂z

∗



∂

∗

∂z

∗2

(3.9-17)

60 APPLICATIONS IN FLUID DYNAMICS

∗

∂u

∗

∂r

∗

μz

∗

∂u

∗

∂z

∗

=−

μu



∂P

∂r



∗

∂

∂r

∗



∗

∂

∂r

∗

)



−

γP

∂

∂r

∗



∗



∂P

∂r



∗



−

γu

∂

∂r

∗



∗

∂P

∗

∂z

∗



∂

∗

∂z

∗2

(3.9-18)

∗

∂

∂r

∗

(ρ

∗

) +

∂

∂z

∗

(ρ

∗

) = 0 (3.9-19)

∗

γP



∗

− P



(3.9-20)

∂u

∗

∂r

∗

= 0,u

∗

= 0,P

∗

p(z

∗

) − P

at r

∗

= 0 (3.9-21)

∗

= 0,u

∗

= 0atr

∗

(3.9-22)

∗

= f

∗

), u

∗

= f

∗

), P

∗

− P

at z

∗

= 0 (3.9-23)

∗

= g

∗

), u

∗

= g

∗

) at z

∗

(3.9-24)

The radial and axial length scales, r

and z

, can be bounded between zero and 1

by setting the dimensionless groups in equations (3.9-22) and (3.9-24) equal to 1.

The dimensionless pressure can be bounded between zero and 1 by setting the

dimensionless group containing the reference pressure in equation (3.9-20) equal

to zero and the dimensionless group containing the pressure scale in equation (3.9-

23) equal to 1. Since pressure causes the axial ﬂow and the principal viscous term

(i.e., involving the second-order radial derivative) must be retained, the velocity

scale u

is determined by setting the dimensionless group multiplying the pressure

term in equation (3.9-17) equal to 1. Since compressibility implies a developing

ﬂow, the dimensionless group in the continuity equation given by (3.9-19) must be

equal to 1. The density scale is obtained by setting the dimensionless group in the

principal term in equation (3.9-20) equal to 1. Since the radial pressure gradient

causes the ﬂow in the radial direction, the radial pressure gradient scale P

determined by setting the dimensionless group multiplying the pressure term in

equation (3.9-18) equal to 1. These considerations then determine the following

scale and reference factors (step 7):

= R; z

= L; P

= P

; P

= P

− P

≡ P ; u

P

μL

;

P

μL

; ρ

= ρ

; P

RP

(3.9-25)

COMPRESSIBLE FLUID FLOW 61

Note that the axial velocity scale u

is within a multiplicative constant of

◦

(1)

of the value for the maximum velocity in fully developed incompressible ﬂow in

a cylindrical tube.

When the scales deﬁned by equation (3.9-25) are substituted into equations

(3.9-17) through (3.9-24), we obtain

∗

∂u

∗

∂r

∗

+ Re

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂z

∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



−

∂

∂z

∗



∗



∂P

∂r



∗



−

∂

∂z

∗



∗

∂P

∗

∂z

∗



∂

∗

∂z

∗2

(3.9-26)

∗

∂u

∗

∂r

∗

+ Re

∗

∂u

∗

∂z

∗

=−



∂P

∂r



∗

∂

∂r

∗



∗

∂

∂r

∗

)



−

∂

∂r

∗



∗



∂P

∂r



∗



−

∂

∂r

∗



∗

∂P

∗

∂z

∗



∂

∗

∂z

∗

(3.9-27)

∗

∂

∂r

∗

(ρ

∗

) +

∂

∂z

∗

(ρ

∗

) = 0 (3.9-28)

∗

= 1 + γ

∗

(3.9-29)

∂u

∗

∂r

∗

= 0,u

∗

= 0,P

∗

) − P

P

at r

∗

= 0 (3.9-30)

∗

= 0,u

∗

= 0atr

∗

= 1 (3.9-31)

∗

= f

∗

), u

∗

= f

∗

(r), P

∗

= 1atz

∗

= 0 (3.9-32)

∗

= g

∗

), u

∗

= g

∗

) at z

∗

= 1 (3.9-33)

where Re ≡ Rρ

/μ is the Reynolds number and Ma ≡ u

/c is the Mach num-

ber; the latter is the ratio of the characteristic velocity of the ﬂuid divided by

the speed of sound in the medium. We see that equations (3.9-26), (3.9-27), and

(3.9-29) can be simpliﬁed signiﬁcantly if we can make the incompressible ﬂow

approximation, which requires that the Mach number be very small; that is, Ma 

1. Note that the size of the Mach number required to ensure incompressible ﬂow

depends on both the Reynolds number and the aspect ratio. These equations can

be further simpliﬁed if we can assume fully developed ﬂow, which requires that

Re(R/L)  1; this condition is ensured if Re(R/L) =

◦

(0.01). In summary, the

62 APPLICATIONS IN FLUID DYNAMICS

following conditions justify simplifying the describing equations for this ﬂow:

Ma =

P

μLc

 1 ⇒ incompressible ﬂow approximation (3.9-34)

 1 ⇒ L = 100

ρu

⇒ fully developed ﬂow approximation (3.9-35)

If the conditions in equations (3.9-34) and (3.9-35) apply, equations (3.9-26)

through (3.9-33) simplify to (step 8)

0 = 1 +

∗



∗



(3.9-36)

∗

= 0atr

∗

= 0 (3.9-37)

∗

= 0atr

∗

= 1 (3.9-38)

Of course, these are just the describing equations for fully developed ﬂow of a

Newtonian ﬂuid with constant physical properties in a cylindrical tube.

3.10 DIMENSIONAL ANALYSIS CORRELATION FOR THE

TERMINAL VELOCITY

In dimensional analysis we seek to determine the dimensionless groups required

to correlate data or to scale a process up or down. These dimensionless groups

can always be determined by means of

◦

(1) scaling analysis since this procedure

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

However, in the preceding sections we indicated that carrying out an

◦

(1) scaling

analysis can be somewhat complicated and time consuming. In contrast, the scal-

ing analysis approach to dimensional analysis illustrated in this section is much

easier and quicker to implement. Note, however, that 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 dimensionless groups. This ﬁrst example

of the use of scaling for dimensional analysis in ﬂuid-dynamics applications will

provide more details on the steps involved. We also compare the results of scaling

analysis to those obtained from using the Pi theorem, underscore the advantages

of using the former to achieve the minimum parametric representation. The steps

referred to here are those outlined in Section 2.4 for the scaling approach to dimen-

sional analysis; these differ from those used in Sections 3.2 through 3.9 since no

attempt is made to achieve

◦

(1) scaling.