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 103

∂u

∂r

= 0,u

= 0atr = 0 (3.E.10-4)

= 0,u

=−U

at r = R (3.E.10-5)

= 0,u

= 0atz = 0 (3.E.10-6)

= f

(r), u

= f

(r), P = P

atm

at z = L (3.E.10-7)

2πRzU



2πrdr (3.E.10-8)

where f

(r) and f

(r) are undetermined functions that are required for complete-

ness in specifying the boundary conditions. The fact that these functions are gener-

ally unknown in practice signiﬁcantly complicates solving the full set of describing

equations. Equation (3.E.10-5) accounts for the fact that permeation through the

walls of the hollow-ﬁber membrane results in a nonzero radial velocity component.

This permeation also causes this to be a developing ﬂow. Equation (3.E.10-8) is a

statement that for an incompressible liquid the total permeation over a length z of

the ﬁber wall is equal to the volumetric ﬂow rate at that axial position. This serves

as an auxiliary condition to determine the unspeciﬁed axial pressure gradient.

Deﬁne the following scale factors, reference factor, and dimensionless variables

(steps 2, 3, and 4):

∗

≡

; u

∗

≡

; P

∗

≡

P − P

; r

∗

≡

; z

∗

≡

(3.E.10-9)

Introduce these dimensionless variables into equations (3.E.10-1) through

(3.E.10-8) and divide each equation through by the dimensional coefﬁcient of a

term that must be retained to obtain (steps 5 and 6)

ρu

∗

∂u

∗

∂r

∗

ρu

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂r

∗

μu

∂

∂r

∗



∗

∂

∂r

∗

)



μu

∂

∗

∂z

∗2

(3.E.10-10)

ρu

∗

∂u

∗

∂r

∗

ρu

μz

∗

∂u

∗

∂z

∗

=−

μu

∂P

∗

∂z

∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



∂

∗

∂z

∗2

(3.E.10-11)

∗

∂

∂r

∗

) +

∂u

∗

∂z

∗

= 0 (3.E.10-12)

∂u

∗

∂r

∗

= 0,u

∗

= 0atr

∗

= 0 (3.E.10-13)

∗

= 0,u

∗

=−

at r

∗

(3.E.10-14)

104 APPLICATIONS IN FLUID DYNAMICS

∗

= 0,u

∗

= 0atz

∗

= 0 (3.E.10-15)

∗

= f

∗

), u

∗

= f

∗

), P

∗

atm

− P

at z

∗

(3.E.10-16)

∗



R/r

∗

(3.E.10-17)

The dimensionless geometric ratios in equations (3.E.10-14) and (3.E.10-16)

can be set equal to 1, thereby determining the length scales r

= R and z

= L

(step 7). Since the permeation through the hollow-ﬁber membrane wall causes the

radial ﬂow, we set the dimensionless group containing U

in equation (3.E.10-14)

equal to 1, thereby obtaining u

= U

. Since this is a developing ﬂow, setting the

dimensionless group equal to 1 in equation (3.E.10-12) gives u

= LU

/R.Since

the axial ﬂow is caused by the axial pressure gradient, we set the dimensionless

group containing P

in equation (3.E.10-11) equal to 1, thereby obtaining P

μU

. Finally, our dimensionless pressure will be bounded between zero

and 1 if we set the dimensionless group in equation (3.E.10-16) equal to zero,

which gives P

= P

atm

. When these values for the scale and reference factors are

substituted into equations (3.E.10-10) through (3.E.10-17), we obtain the following

minimum parametric representation of the describing equations:

∗

∂u

∗

∂r

∗

+ Re

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂r

∗

∂

∂r

∗



∗

∂

∂r

∗

)



∂

∗

∂z

∗2

(3.E.10-18)

Reu

∗

∂u

∗

∂r

∗

+ Reu

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂z

∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



∂

∗

∂z

∗2

(3.E.10-19)

∗

∂

∂r

∗

) +

∂u

∗

∂z

∗

= 0 (3.E.10-20)

∂u

∗

∂r

∗

= 0,u

∗

= 0atr

∗

= 0 (3.E.10-21)

∗

= 0,u

∗

=−1atr

∗

= 1 (3.E.10-22)

∗

= 0,u

∗

= 0atz

∗

= 0 (3.E.10-23)

∗

= f

∗

), u

∗

= f

∗

), P

∗

= 0atz

∗

= 1 (3.E.10-24)

∗



∗

(3.E.10-25)

The fact that equation (3.E.10-25) does not contain any dimensionless groups is

an indication that we have scaled the describing equations properly. That is, the

scaling has normalized the integral mass balance in this equation to be

◦

(1), as it

should be, since the two terms in this equation must balance each other.

EXAMPLE PROBLEMS 105

Inspection of equations (3.E.9-18) through (3.E.10-25) indicates that consider-

able simpliﬁcation is possible if the following conditions apply (step 8):

Re ≡

ρU

 1and

 1 (3.E.10-26)

We recognize these conditions to be those for assuming lubrication ﬂow that was

considered in Section 3.3. However, for this ﬂow the Reynolds number charac-

terizes the ratio of the radial convection to viscous force. If the conditions above

apply, our describing equations simplify to

0 =

∂P

∗

∂r

∗

(3.E.10-27)

0 =−

∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



(3.E.10-28)

∗

∂

∂r

∗

) +

∂u

∗

∂z

∗

= 0 (3.E.10-29)

∂u

∗

∂r

∗

= 0atr

∗

= 0 (3.E.10-30)

∗

= 0,u

∗

=−1atr

∗

= 1 (3.E.10-31)

∗

= 0atz

∗

= 0 (3.E.10-32)

∗

= 0atz

∗

= 1 (3.E.10-33)

∗



∗

(3.E.10-34)

Equation (3.E.10-27) implies that P

∗

= P

∗

), which permits integrating equa-

tion (3.E.10-28) directly to obtain the axial velocity proﬁle in terms of the unspec-

iﬁed axial pressure gradient. The latter can be obtained by substituting the axial

velocity proﬁle into equation (3.E.10-34). The corresponding radial velocity pro-

ﬁle can be obtained from equation (3.E.10-29). The resulting solutions for the

dimensionless velocity and pressure proﬁles are given by

∗

= 4z

∗

(1 − r

∗2

); u

∗

= r

∗

∗2

− 2); P

∗

= 8(1 − z

∗2

) (3.E.10-35)

3.E.11 Falling Head Method for Determining Soil Permeability

The falling head method is used to determine the permeability of soils. This test,

shown in Figure 3.E.11-1, involves driving a pipe of radius R into the soil until

it penetrates the water table, assumed here to be at the end of the tube. The pipe

is ﬁlled with water to a height L

and the time t

required to drain it to a height

is measured. We use the scaling method for dimensional analysis to develop a

correlation for the draining time.

Step 1, the scaling procedure for dimensional analysis, involves writing the

appropriate describing equations to determine the quantity of interest. The draining

106 APPLICATIONS IN FLUID DYNAMICS

Wet porous soil

Water table

Cylindrical

tube

Ground surface

Dry porous soil

Figure 3.E.11-1 Falling head apparatus for measuring the permeability of soils; a cylin-

drical tube of radius R is pushed to the depth of the water table, which is deﬁned to be at

z = 0, and ﬁlled to an initial depth of L

with water; the instantaneous water depth in the

tube L(t) decreases due to permeation into the soil.

time t

is related to the axial velocity u

and Darcy’s law via an instantaneous

mass balance on the water in the tube:

− L

=−



u



z=0

dt =





+ ρg





z=0

dt, 0 ≤ r ≤ R

(3.E.11-1)

where k

is the permeability, μ the viscosity, ρ the density, and g the gravitational

acceleration. Note that we have ignored the effect of the Brinkman term for ﬂow

through the porous medium based on the assumption that k

 1, as discussed

in Section 3.8. To evaluate the integral above it would be necessary to solve for the

pressure distribution in the porous medium. This is obtained in turn by solving the

Darcy ﬂow equations in the porous medium subject to the appropriate boundary

conditions. The incompressible continuity equation given by equation (C.2-1) in the

Appendices in combination with Darcy’s law for ﬂow through porous media given

by equation (E.2-3) in the Appendices implies that the pressure P is obtained from a

solution to the axisymmetric form of Laplace’s equation in cylindrical coordinates:

∂

∂r



∂P

∂r



∂

∂z

= 0 (3.E.11-2)

EXAMPLE PROBLEMS 107

The pressure is subject to the following boundary conditions:

P = P

atm

+ ρgL(t) at z = 0, 0 ≤ r ≤ R, L

≤ L(t) ≤ L

(3.E.11-3)

P = P

atm

at z = 0,R≤ r<∞ (3.E.11-4)

u

=−

∂P

∂z

= 0asz →−∞, 0 ≤ r<∞ (3.E.11-5)

P = P

atm

− ρgz as r →∞, −∞ <z≤ 0 (3.E.11-6)

Note that in specifying the boundary conditions for this partial differential equation,

one must also specify the domain of any other variables that are not speciﬁed in

the particular boundary condition; in some cases, such as for the domain of the

boundary condition given by equation (3.E.11-3), this introduces additional param-

eters into the dimensional analysis. The boundary conditions given by equation

(3.E.11-3) involve the instantaneous liquid depth L(t) in the tube. This can be

obtained from a mass balance for the amount of liquid ﬂowing from the tube up

to the instantaneous time t and is given by

− L =−



u



z=0

dt =





+ ρg





z=0

dt 0 ≤ r ≤ R

(3.E.11-7)

Deﬁne the following scale and reference factors and corresponding dimension-

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

∗

≡

P − P

; L

∗

≡

L − L

; r

∗

≡

; z

∗

≡

; t

∗

≡

(3.E.11-8)

Substitute these dimensionless variables into equations (3.E.11-1) through (3.E.11-7)

and divide each equation by the dimensional coefﬁcient of one of its terms (steps 5 and

6). Since this is dimensional analysis rather than

◦

(1) scaling, we can divide through

by the dimensional coefﬁcient of any arbitrary term in each equation:

μz

− L

)





∗

ρgz





∗

, 0 ≤ r

∗

≤

(3.E.11-9)

∗

∂

∂r

∗



∗

∂P

∗

∂r

∗



∂

∗

∂z

∗2

= 0 (3.E.11-10)

∗

atm

+ ρgL



∗



− P

at z = 0

⎧

⎪

⎨

⎪

⎩

0 ≤ r

∗

≤

≤ L

∗



1 +



≤

(3.E.11-11)

∗

atm

− P

at z

∗

= 0,

≤ r

∗

< ∞ (3.E.11-12)

108 APPLICATIONS IN FLUID DYNAMICS

∂P

∗

∂z

∗

= 0asz

∗

→−∞, 0 ≤ r

∗

< ∞ (3.E.11-13)

∗

atm

− ρgz

∗

− P

as r

∗

→∞, −∞ <z

∗

≤ 0 (3.E.11-14)

μz



1−

∗





∗



∗

ρgz





z=0

∗

, 0 ≤ r

∗

≤

(3.E.11-15)

We are free to choose which groups we set equal to zero or 1 in order to determine

our scale and reference factors since we are not concerned about scaling our variables

to be of order one. Hence, let us arbitrarily make the following choices (step 7):

= 1 ⇒ r

= R;

= 1 ⇒ L

= L

;

⇒ L

= L

ρgL

= 1 ⇒ P

= ρgL

;

atm

− P

= 0 ⇒ P

= P

atm

;

ρgz

= 1 ⇒ z

= L

;

ρgt

μ(L

− L

)

= 1 ⇒ t

μ(L

− L

)

ρg

(3.E.11-16)

When these scale and reference factors are substituted into the equations (3.E.11-9)

through (3.E.10.15), we obtain the following minimum parametric representation

of the dimensionless describing equations:

1 =







∗

+ 1





∗

, 0 ≤ r

∗

≤ 1 (3.E.11-17)

∗

∂

∂r

∗



∗

∂P

∗

∂r

∗



+ 

∂

∗

∂z

∗2

= 0 (3.E.11-18)

∗

= 1atz

∗

= 0, 0 ≤ r

∗

≤ 1, 0 ≤ L

∗

≤ 1 − 

(3.E.11-19)

∗

= 0atz

∗

= 0, 1 ≤ r

∗

< ∞ (3.E.11-20)

∂P

∗

∂z

∗

= 0asz

∗

→−∞, 0 ≤ r

∗

< ∞ (3.E.11-21)

∗

= 1asr

∗

→∞, −∞ <z

∗

≤ 0 (3.E.11-22)

1 − L

∗



∗



∗

+ 1





z=0

∗

, 0 ≤ r

∗

≤ 1 (3.E.11-23)

where the following dimensionless groups have been deﬁned:



≡

ρgt

μ(L

− L

)

; 

≡

; 

≡

(3.E.11-24)

EXAMPLE PROBLEMS 109

A solution to equation (3.E.11-18) subject to the boundary conditions given

by equations (3.E.11-19) through (3.E.11-23) will yield the dimensionless pressure

∗

as a function of r

∗

,andt

∗

and the dimensionless groups 

and 

When the axial pressure gradient is evaluated at exit of the tube, where it is not a

function of r

∗

, and substituted into equation (3.E.11-17), the resulting solution for

the dimensionless draining time 

will be a function only of the dimensionless

groups 

and 

; that is, the minimum parametric representation is given by a

general correlation of the form



≡

ρgt

μ(L

− L

)

= f

(

,

) = f





(3.E.11-25)

where f

(

,

) denotes some unspeciﬁed function of the dimensionless groups



and 

that would have to be determined empirically. Note that a naive appli-

cation of the Pi theorem for eight quantities in three units (i.e., n = 8andm = 3)

would have suggested that ﬁve (m − n = 5) rather than three dimensionless groups

were necessary to correlate the draining time. For the Pi theorem to yield the min-

imum parametric representation, it is necessary to recognize that the grouping

ρg/μ can be considered as a single quantity, in which case only two units need

to be considered.

It would be necessary to take a considerable number of data to establish the

correlation indicated in equation (3.E.11-25). We show here how a general cor-

relation for the draining time can be established from very limited data for a

speciﬁc tube, using water in a particular soil. The following empirical correla-

tion has been obtained using water and a 5-cm-radius pipe for a soil having a

permeability k

= 5.9×10

−6

223

= 4.94 ln

(3.E.11-26)

This empirical result is a special case of equation (3.E.11-25) and can be used

to determine the functional form of a more general correlation for the draining

time that can be used for different tubes, ﬂuids, and soils. This can be seen more

easily by using step 8 of the scaling procedure for dimensional analysis outlined

in Section 2.4 in order to eliminate both L

and L

from the dimensionless group

that contains the draining time t

. This involves generating a new dimensionless

group formed from the original set of groups by means of the operation



≡



ρg

μR

(

1 − L

)

= f

(

,

) (3.E.11-27)

where f

(

,

) denotes an unspeciﬁed function of the dimensionless groups 

and 

that would have to be determined empirically. Equation (3.E.11-27) can

R. J. Ray, A Rayleigh free convection compliant ice front model for sorted patterned ground, M.S.

thesis, University of Colorado, Boulder, Co, 1981.

110 APPLICATIONS IN FLUID DYNAMICS

be recast into the following form using step 11:

ρg

μR

≡ 

= f

(

,

) (3.E.11-28)

where f

(

,

) denotes an unspeciﬁed function of the dimensionless groups 

and 

that would have to be determined empirically. In general, the radius R

of the tubes used to determine soil permeability is much less than the initial ﬁll

height L

.IfR  L

,then

 1 and we can use the procedure in step 9 of the

scaling analysis procedure for dimensional analysis to eliminate group 

from the

correlation; hence,

ρg

μR

≡ 

= f

(

) (3.E.11-29)

where f

(

) is an unspeciﬁed function of 

only. A comparison of equa-

tion (3.E.11-29) with equation (3.E.11-26) implies the following:

= 

μR

ρg

=−4.94 ln 

(3.E.11-30)

A generalized correlation relating soil permeability k

and draining time t

can be

obtained by substituting values for the quantities in equation (3.E.11-30) to obtain

the following correlation:



=−

4.94(5.9×10

−6

)



1g/cm



980 cm/s



(

0.01 g/cm · s

)

(5cm)

ln 

⇒ 

=−0.572 ln 

⇒

ρg

μR

=−0.572 ln

(3.E.11-31)

In this case, using the scaling analysis approach for dimensional analysis in com-

bination with data for a speciﬁc falling head test gives the functional form of a

generalized correlation that relates the measured draining time t

to the soil perme-

ability k

and relevant physical properties and process parameters. The generalized

correlation given by equation (3.E.11-31) applies for any falling head test, irrespec-

tive of the ﬂuid, pipe size, and soil, provided that the dimensionless group 

 1.

3.P PRACTICE PROBLEMS

3.P.1 Alternative Scales for Laminar Flow Between Stationary and Moving

Parallel Plates

Consider the steady-state fully developed laminar ﬂow of an incompressible vis-

cous Newtonian ﬂuid with constant physical properties between two inﬁnitely wide

parallel ﬂat plates due to both an applied axial pressure gradient and to the upper

plate moving at a constant velocity U

as shown in Figure 3.2-1. In Section 3.2

we introduce scales for the velocity and y-coordinate to determine the criterion

necessary to ignore the effect of the motion of the upper plate.

PRACTICE PROBLEMS 111

(a) Rescale this problem by introducing an additional scale for the second

derivative as well as the velocity and y-coordinate; you will ﬁnd that there

is no dimensionless group to determine the velocity scale; however, this can

be determined by integrating the scale for the second derivative.

(b) Rescale this problem by introducing an additional scale for the ﬁrst deriva-

tive as well as the velocity and y-coordinate. Again you will ﬁnd that there

is no dimensionless group to determine the velocity scale; however, this can

be determined by integrating the scale for the ﬁrst derivative.

3.P.2 Laminar Flow Between Stationary and Moving Parallel Plates

Consider the steady-state fully developed laminar ﬂow of an incompressible vis-

cous Newtonian ﬂuid with constant physical properties between two inﬁnitely wide

parallel ﬂat plates due to both an applied axial pressure gradient and to the upper

plate moving at a constant velocity U

, as shown in Figure 3.2-1. In Section 3.2

we scaled this ﬂow to determine the criterion necessary to ignore the effect of the

motion of the upper plate. We found that the motion of the upper plate would not

affect quantities such as the average velocity, volumetric ﬂow rate, or drag at the

stationary plate if equation (3.2-19) were satisﬁed. However, there was a region of

inﬂuence next to the upper plate within which the motion of the plate could never

be ignored. In this problem we explore complementary ﬂow conditions for which

the ﬂow is caused primarily by the motion of the upper plate.

(a) Determine the criterion necessary to neglect the effect of the applied pressure

on quantities such as the average velocity or volumetric ﬂow rate.

(b) Determine if there is a region of inﬂuence within which the effect of the

pressure on the ﬂow can never be ignored in determining point quantities

such as the local velocity or drag at the wall.

tions such that the criterion you derived in part (a) is satisﬁed.

3.P.3 Gravity and Pressure-Driven Laminar Flow in a Vertical Tube

Consider the steady-state fully developed laminar ﬂow of a Newtonian liquid with

constant physical properties in a vertical tube of radius R that is subject to both

gravity and a constant axial pressure gradient, as shown in Figure 3.P.3-1.

(a) Write the appropriate form of the simpliﬁed equations of motion for this

ﬂow.

(b) Write the boundary conditions required for the differential equations above.

effect of the applied pressure gradient on the velocity proﬁle.

(d) Solve the resulting simpliﬁed describing equations to obtain the velocity

proﬁle.

112 APPLICATIONS IN FLUID DYNAMICS

Figure 3.P.3-1 Steady-state fully developed laminar ﬂow of a Newtonian liquid with con-

stant physical properties in a vertical tube of radius R and length L subject to both gravity

and a constant axial pressure gradient (P

− P

)/L.

3.P.4 Axial Flow in a Rotating Tube

Consider the steady-state fully developed laminar ﬂow of an incompressible

Newtonian liquid with constant physical properties in a vertical cylindrical tube

of radius R and length L subject to a constant axial pressure gradient (P

− P

)/L

and a constant angular rotation about its axis of symmetry at ω radians per second,

as shown in Figure 3.P.4-1.

Figure 3.P.4-1 Steady-state fully developed laminar ﬂow of an incompressible Newtonian

liquid with constant physical properties in a vertical cylindrical tube of radius R and length

L subject to a constant axial pressure gradient (P

− P

)/L and a constant angular rotation

at ω radians per second.