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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PRACTICE PROBLEMS 123

= U

cos wt

Figure 3.P.19-1 Unsteady-state laminar ﬂow of an incompressible Newtonian liquid with

constant physical properties in a circular tube of radius R due to an impulsively applied pres-

sure difference P ≡ P

−P

and oscillation of the tube wall described by u

= U

cos ωt;

the axial velocity proﬁles are shown at times t

and t

,wheret

(a) Write the appropriate form of the equations of motion for this unsteady-state

ﬂow.

(b) Write the initial and boundary conditions required to solve the equations of

motion.

is mainly caused primarily by the applied pressure gradient.

(d) Determine the appropriate velocity scale for conditions such that the ﬂow

is caused primarily by the oscillating pipe wall.

(e) Scale the describing equations to determine when the effect of the wall

oscillation can be neglected.

(f) Scale the describing equations to determine when this can be considered

to be a quasi-steady-state creeping ﬂow. Be careful to consider both time

effects: i.e., the transients following startup and the periodic oscillations.

(g) Solve for the velocity proﬁle u

(r, t) for the special case of quasi-steady-state

creeping ﬂow.

3.P.20 Countercurrent Liquid–Gas Flow in a Cylindrical Tube

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

liquid ﬁlm of thickness H with constant physical properties at the inner wall of

a vertical cylindrical tube of radius R and length L due to gravity, pressure, and

interfacial drag arising from the fully developed upward pressure-induced ﬂow of

a gas having constant physical properties in the center of the tube, as shown in

Figure 3.P.20-1.

(a) Write the appropriately simpliﬁed continuity and equations of motion for

both the liquid and gas phases for this ﬂow.

124 APPLICATIONS IN FLUID DYNAMICS

Liquid

Gas

Figure 3.P.20-1 Steady-state fully developed laminar ﬂow of a nonvolatile Newtonian

liquid ﬁlm of thickness H with constant physical properties at the inner wall of a vertical

cylindrical tube of radius R and length L due to gravity, pressure, and interfacial drag arising

from the upward fully developed pressure-induced ﬂow of a gas with constant physical

properties in the center of the tube.

(b) Write the boundary conditions required to solve the differential equations

you obtained in part (a).

is caused primarily by the gravitational body force.

(d) Determine the appropriate velocity scale in the gas phase when this ﬂow is

caused primarily by the applied pressure.

(e) Use scaling to determine the criterion for when the effect of the gas ﬂow

on the liquid ﬁlm ﬂow can be ignored.

(f) Use scaling to determine the criterion for when the effect of the gravitational

body force on the gas ﬂow can be ignored.

(g) For the conditions considered in part (d), determine the criterion for ignoring

curvature effects in the describing equations; that is, for when the describing

equations in cylindrical coordinates reduce to the corresponding equations

in rectangular coordinates.

(h) Determine the appropriate velocity scale in the liquid ﬁlm when this ﬂow

is caused primarily by the drag exerted on it by the upward gas ﬂow.

(i) Use scaling to determine the criterion for when the effect of gravity on the

liquid ﬁlm ﬂow can be ignored.

(j) This ﬂow is an idealization of that within a single packing element such as

a Raschig ring in countercurrent liquid–gas contacting in a packed column.

Flooding in a packed column begins when the upward gas ﬂow causes a

net amount of liquid to be carried upward to the top of the column. Use

the results from your scaling analysis to determine the pressure gradient

required to initiate ﬂooding.

PRACTICE PROBLEMS 125

3.P.21 Stratiﬁed Flow of Two Immiscible Liquid Layers

Consider the steady-state fully developed ﬂow of two immiscible Newtonian liq-

uids with constant physical properties between two stationary parallel impermeable

ﬂat plates, as shown in Figure 3.P.21-1. These ﬂuids have densities ρ

and ρ

,vis-

cosities μ

and μ

, and thicknesses H

and H

, respectively. A high pressure P

is applied at x = 0 and a low pressure P

is applied at x = L. Liquid 1 ﬂows

symmetrically about the center plane between the two ﬂat plates, whereas liquid 2

is conﬁned to a layer having thickness H

− H

adjacent to each of the two ﬂat

plates. The velocities in liquids 1 and 2 are denoted by u

and u

, respectively.

For the scaling analysis in this problem, introduce the following dimensionless

variables in terms of undeﬁned scale and reference factors:

∗

≡

− u

; u

∗

≡

; y

∗

≡

; y

∗

≡

y −y

(3.P.21-1)

(a) Write the appropriate form of the simpliﬁed continuity and equations of

motion along with the necessary boundary conditions for this ﬂow; gravita-

tional body forces may be neglected in your analysis.

(b) Explain why a reference factor is needed for the velocity in liquid 1 and

why a reference factor is needed for the spatial coordinate in liquid 2.

deﬁned above for the case where the ﬂow in both liquids is caused by the

applied pressure P ≡ P

− P

. Indicate why you set various dimensionless

groups equal to zero or 1.

(d) What is the criterion for ignoring the effect of liquid 1 on the ﬂow of liquid

2 for the conditions in part (c)?

(e) Determine the scale and reference factors in the dimensionless variables

deﬁned above for the case where the ﬂow in liquid 1 is caused by the applied

pressure P ≡ P

− P

, whereas the ﬂow in liquid 2 is caused primarily

by the drag force exerted by liquid 1 at the interface. Indicate why you set

various dimensionless groups equal to zero or 1.

Liquid 1

Liquid 2

Figure 3.P.21-1 Steady-state fully developed ﬂow of two immiscible Newtonian liquids

with constant physical properties between two stationary parallel impermeable ﬂat plates.

126 APPLICATIONS IN FLUID DYNAMICS

(f) What is the criterion for determining whether the scaling in part (c) or in

part (e) is appropriate for the describing equations?

(g) Use the results of your scaling analysis in part (e) to estimate the velocity

gradients in both liquids 1 and 2; that is, develop appropriate scales for the

velocity gradients in each liquid.

(h) For the scaling analysis done in part (e), determine the dimensionless groups

that are necessary to correlate the total rate of entropy production

S,which

is given by

S =−







WLdy −







WLdy (3.P.21-2)

where T is the absolute temperature and W and L denote the width and

length, respectively, of the two ﬂat plates.

(i) One means for reducing the viscous drag at the walls experienced in pumping

viscous liquids such as petroleum is to inject a less viscous immiscible liquid

such as water, which will form a layer at the wall. However, for this idea

to work, we have to establish that the less viscous liquid (e.g., water) rather

than the more viscous liquid (e.g., petroleum) will go to the wall region.

This question can be answered by invoking the principle that continuous

steady-state processes seek a state of minimum entropy production. Use this

principle along with the results of your scaling analysis in part (e) to deter-

mine whether the viscous or the less viscous liquid will go to the wall region.

3.P.22 Laminar Cylindrical Jet Flow

In Example Problem 3.E.8 we considered a jet of an incompressible Newtonian

liquid with constant physical properties issuing from a circular oriﬁce with an

initial velocity U

and falling vertically under the inﬂuence of gravity in an inviscid

gas as shown in Figure 3.E.8-1. We scaled the describing equations to explore the

conditions for which quasi-parallel ﬂow can be assumed; that is, when the axial

velocity proﬁle can be assumed to depend only on the axial coordinate. In scaling

the describing equations we introduced a scale for the radial derivative of the axial

velocity since we did not anticipate that this derivative would scale as the ratio

of the characteristic axial velocity scale divided by the characteristic radial length

scale. We anticipated the need to scale this derivative with its own scale since the

axial velocity does not change signiﬁcantly across the jet. It was stated that if we

had scaled this radial derivative with the ratio of the characteristic axial velocity

scale to the characteristic radial length scale, the forgiving nature of scaling would

have indicated a contradiction. To better understand what is meant by this, let us

assume (incorrectly!) that the radial derivative of the axial velocity scales as the

axial velocity scale u

divided by the radial length scale r

. Show that this leads to

an inconsistency in the resulting dimensionless equations in that the dimensionless

group multiplying one term in the describing equations becomes very large, whereas

PRACTICE PROBLEMS 127

the others terms are of

◦

(1); this implies that there is no term to balance this very

large term.

3.P.23 Free Surface Flow Down a Plate with Condensation

Consider the steady-state incompressible laminar ﬁlm ﬂow of a Newtonian liquid

having constant physical properties down an inﬁnitely wide plane surface inclined

at an angle θ to the horizontal as shown in Figure 3.P.23-1. At the interface between

the liquid ﬁlm and the ambient gas phase, a constant amount of liquid

is added

to the ﬁlm ﬂow per unit area of free surface. For example, this might occur due to

condensation occurring at the liquid–gas interface.

(a) Write the appropriate forms of the steady-state equations of motion assuming

that the ambient gas phase exerts negligible drag on the liquid ﬁlm.

(b) Write the boundary conditions required to solve the equations of motion;

note that the continuity of surface normal stresses and tangential stress bal-

ance must account for the surface-curvature effects.

addition at the interface must be taken into consideration.

(d) Scale the describing equations to determine the conditions required to make

the lubrication-ﬂow approximation and to ignore the surface-curvature effects.

h(x)

Δx

mass

area

·time

∼

Liquid

Gas

Figure 3.P.23-1 Steady-state incompressible laminar ﬁlm ﬂow of a Newtonian liquid with

constant physical properties down an inﬁnitely wide plane surface inclined at an angle θ to

the horizontal. Mass addition at the free surface of

units of mass per unit area per unit

time causes the local ﬁlm thickness η to increase as a function of axial distance x.

3.P.24 Free Surface Flow Over a Horizontal Filter

A ﬁlter in a large industrial plant operates by letting a solution ﬂow across a large

sheet of ﬁlter paper as shown in Figure 3.P.24-1. Assume steady-state developing

incompressible laminar ﬂow of a Newtonian liquid with constant physical properties

Scaling was used to determine the criterion for making this assumption in Example Problem 3.E.1.

See the example in Section 3.7 as a guide to scaling this problem.

128 APPLICATIONS IN FLUID DYNAMICS

Filter

Liquid

Gas

h(x)

Δx

mass

area

·time

∼

Figure 3.P.24-1 Gravitationally induced steady-state laminar ﬂow of an incompressible

Newtonian liquid with constant physical properties across a ﬁlter through which the mass

ﬂow rate per unit area is

that is driven by a constant hydrostatic pressure gradient due to a variable ﬁlm

thickness in the x-direction. Assume also that mass is removed from this ﬂow

at the ﬁlter boundary at a constant rate of

units of mass per unit area per

unit time.

(a) Write the appropriate forms of the steady-state equations of motion assuming

that the ambient gas phase exerts negligible drag on the liquid ﬁlm (see

footnote 26).

(b) Write the boundary conditions required to solve the equations of motion;

note that the continuity of surface normal stresses and tangential stress bal-

ance must account for the surface-curvature effects (see footnote 27).

through the ﬁlter must be taken into consideration.

(d) Scale the describing equations to determine the condition(s) required to

make the lubrication-ﬂow approximation and to ignore the surface-curvature

effects.

(e) Solve the simpliﬁed equations of motion that you obtained in part (d) to

obtain the axial velocity proﬁle in terms of the unknown hydrostatic pressure

gradient.

(f) Determine the unknown hydrostatic pressure gradient given that the volu-

metric ﬂow rate Q is speciﬁed.

(g) Determine the component of the velocity normal to the ﬁlter.

3.P.25 Curtain-Coating Flow

The process of curtain coating is shown in Figure 3.P.25-1. This process is used,

for example, to apply protective polymer coatings at high speed to continuous

steel or tin-plate strip. The curtain ﬂow emanates from a slot at x = 0, at which

point it has a velocity U

. The solid guides at z =±W/2 maintain the ﬁlm at a

constant thickness in the z-direction. However, since the ﬁlm accelerates due to the

gravitational body force, it thins in the y-direction, as shown in the side view in

PRACTICE PROBLEMS 129

Solid guides maintain film at

constant width in z-direction

obj

ect to be coated

ecttobecoated

FRONT VIEW SIDE VIEW

Liquid/gas interface

defined by y = h(x)

Liquid

Figure 3.P.25-1 Gravitationally induced steady-state laminar curtain-coating ﬂow of an

incompressible Newtonian liquid with constant physical properties emanating from a rect-

angular slot with an initial velocity of U

; front view shows solid guides that maintain the

ﬁlm at constant width; the axial velocity proﬁle is shown in both views.

Figure 3.P.25-1. The liquid–gas interface is deﬁned by y = η(x),whereη decreases

with increasing x due to the increase in the x-velocity component. The object to

be coated can be ignored entirely in this analysis since we are concerned only with

the falling curtain ﬂow. We assume steady-state laminar ﬂow of an incompressible

Newtonian liquid with constant physical properties and ignore edge effects at the

sidewalls. Note that the applicability of all these assumptions could be determined

using scaling analysis. We use scaling analysis to explore the conditions for which

quasi-parallel ﬂow can be assumed. We then test the applicability of our scaling

analysis results using data from the curtain-coating process.

(a) Write the appropriate form of the three-dimensional equations of motion for

this ﬂow assuming that the ambient gas phase exerts a negligible drag on

the liquid ﬁlm (see footnote 26).

(b) Write the appropriate boundary conditions for this ﬂow assuming that the

surface-tension effects associated with the curvature can be ignored.

How-

ever, do not ignore the effects of curvature in specifying the tangential and

normal stress boundary conditions at the free surface. In deriving the tan-

gential and normal stress boundary conditions at the liquid–gas interface,

use the convection for θ and the normal and tangential unit vectors, n and



t, respectively, shown in Figure 3.P.25-2.

Note that scaling analysis could be used to determine when surface-tension and curvature effects can

be neglected; the latter were considered in Section 3.7.

130 APPLICATIONS IN FLUID DYNAMICS

Gas

Liquid

Slot

h(x)

Figure 3.P.25-2 Enlarged view of the x–y plane for curtain ﬂow, showing normal and

tangential unit vectors, n and



t, respectively, at the liquid–gas interface.

(d) Introduce the following dimensionless variables involving unspeciﬁed scale

and reference factors into the describing equations:

∗

≡

− u

; u

∗

≡

; u

∗

≡

; P

∗

≡

P − P

;

∗

≡

;



∂u

∂y



∗

≡

∂u

∂y

; x

∗

≡

; y

∗

≡

; z

∗

≡

(3.P.25-1)

Note that ∂u

/∂y does not scale with u

since u

does not undergo

a characteristic change of u

over the distance y

. The proper scale for

is obtained from the tangential stress boundary condition. Note that this

scaling for ∂u

/∂y implies that ∂

/∂y

scales with β

.Usex

= L,

where L is some arbitrary but constant downstream distance. Your pressure

scale P

will come from balancing the pressure term with the convection

terms in the y-component of the equations of motion. This follows from the

fact that the pressure gradient arises from the acceleration of the ﬂow.

(e) Simplify the tangential and normal stress boundary conditions appropriate

to the assumption of small curvature; that is, for dη/dx  1.

(f) Assume small curvature and use your scaling analysis in part (d) to deter-

mine the criteria for assuming that this is a quasi-parallel ﬂow; that is, a

ﬂow that can be described by considering only the changes of the x-velocity

component in the x-direction.

(g) Show that the solution to the simpliﬁed set of equations appropriate to the

quasi-parallel-ﬂow approximation is given by

= U

+ 2xg (3.P.25-2)

(h) Table 3.P.25-1 summarizes several data sets for a series of curtain ﬂows for

which the fall velocity u

in cm/s at a position below the slot of x = 5cmis

PRACTICE PROBLEMS 131

TABLE 3.P.25-1 Data Sets for Curtain Flow

Data Viscosity Slot Width Mean Velocity at Measured Velocity at

Set (poise) (mm) Slot Exit (cm/s) x = 5cm(cm/s)

11.2 0.6 18 96

2 1.5 0.6 56 110

3 1.95 0.2 38 100

4 2.0 0.2 47 109

52.6 0.6 18 97

62.7 0.3 33 93

72.8 0.3 13 87

8 3.45 0.6 6 86

9 3.7 1.5 5 86

10 5.3 1.5 12 85

11 9.9 0.6 8 75

reported as a function of the liquid viscosity in poise (g/cm ·s), the slot width

in mm, and the mean velocity U

in cm/s at the exit of the slot. Compare

the fall velocities predicted by the quasi-parallel-ﬂow approximation in part

(g) with the measured fall velocities and also assess the validity of the quasi-

parallel-ﬂow approximation. Discuss any signiﬁcant deviations between the

experimental and model results.

3.P.26 Flow in a Semi-inﬁnite Porous Medium Bounded by a Flat Plate

Consider the steady-state fully developed pressure-driven ﬂow of an incompressible

Newtonian liquid through a liquid-saturated semi-inﬁnite porous medium that is

bounded by a horizontal ﬂat plate as shown in Figure 3.P.26-1.

(a) Write the appropriate form of the equations of motion applicable to porous

media for this ﬂow.

(b) Write the boundary conditions required to solve the equations of motion.

Solid plate

Porous media

Figure 3.P.26-1 Steady-state fully developed ﬂow of an incompressible Newtonian ﬂuid

with constant physical properties through a semi-inﬁnite porous medium bounded by a

horizontal ﬂat plate due to an applied pressure gradient (P

− P

)/L.

132 APPLICATIONS IN FLUID DYNAMICS

(d) Scale the describing equations for this ﬂow to determine the criterion for

ignoring the effect of the ﬂat plate on the velocity proﬁle.

(e) Determine the thickness of the region of inﬂuence δ

within which the effect

of the ﬂat plate on the velocity proﬁle cannot be ignored.

(f) Solve the simpliﬁed form of the equations of motion for the velocity proﬁle

assuming that the effect of the ﬂat plate can be ignored.

(g) Show how the solution for the velocity proﬁle that you derived in part

3.P.27 Porous Media Flow Between Parallel Flat Plates

Consider the steady-state fully developed pressure-driven ﬂow of an incompressible

Newtonian liquid through a liquid-saturated semi-inﬁnite porous medium that is

bounded by horizontal parallel ﬂat plates as shown in Figure 3.P.27-1.

(a) Write the appropriate form of the equations of motion applicable to porous

media for this ﬂow.

(b) Write the boundary conditions required to solve the equations of motion.

(d) Scale the describing equations for this ﬂow to determine the criterion for

ignoring the effect of the solid boundaries on the velocity proﬁle.

(e) Determine the thickness of the region of inﬂuence δ

within which the effect

of the solid boundaries on the velocity proﬁle cannot be ignored.

Solid plate

Porous media

Solid plate

Figure 3.P.27-1 Steady-state fully developed ﬂow of an incompressible Newtonian ﬂuid

with constant physical properties through a porous medium bounded by horizontal parallel

ﬂat plates due to an applied pressure gradient (P

− P

)/L.