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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SUMMARY 193

,α

,andk

; the Pi theorem would indicate zero groups for these four

quantities (i.e., n − m = 4 −4 = 0) when in fact they lead to two dimensionless

groups. This example clearly indicates the pitfalls of using the Pi theorem for

dimensional analysis. Scaling analysis provides a systematic method for avoiding

these problems associated with the Pi theorem.

4.11 SUMMARY

The example in Section 4.2 provided an introduction to the step-by-step procedure

for scaling analysis in heat transfer. Scaling was used in this problem to assess the

criterion for ignoring edge effects so that the heat transfer could be considered to

be one-dimensional when predicting the temperature sufﬁciently far removed from

the sidewalls or integral quantities such as the total heat-transfer rate. This problem

involved the introduction of both reference and scale factors since the temperature

was not naturally referenced to zero. It also introduced region-of-inﬂuence scaling

to determine the thickness of the zone near the sidewalls, wherein the effects of

lateral heat transfer could never be ignored when predicting quantities such as

the temperature or heat ﬂux at or near the sidewalls. This example provided a

means for estimating the error incurred when the assumptions suggested by scaling

analysis are invoked since an analytical solution was available for this heat-transfer

problem. Demanding that a quantity be

◦

(0.1) in order to ignore some term in

the describing equations typically results in an error of 40 to 50%; demanding

that it be

◦

(0.01) reduces the error to less than 10%. However, the error that

is encountered also depends on the quantity being considered; for example, point

quantities within a region of inﬂuence might incur very large errors even when the

relevant dimensionless group is very small.

In Section 4.3 we applied scaling analysis to unsteady-state one-dimensional

heat conduction in a ﬂat solid slab. This example led to two time scales, the

observation time and the characteristic conduction time, whose ratio is the Fourier

number for heat transfer. If the Fourier number is very large, the process can be

assumed to be steady-state, whereas if it is very small, the unsteady-state heat

transfer is conﬁned to a region of inﬂuence or thermal boundary layer. We referred

to these as the ﬁlm theory and penetration theory approximations, respectively,

although this terminology is generally used only for the analogous approximations

in mass-transfer modeling.

Unsteady-state convective heat transfer from a solid sphere was considered in

Section 4.4. A lumped-parameter boundary condition involving an appropriate heat-

transfer coefﬁcient was used to describe the heat transfer in the ﬂuid phase. It

was necessary to introduce a separate scale for the temperature gradient in this

problem since the temperature did not go through a characteristic change over the

characteristic length. Scaling this problem introduced the Biot number, which is

the dimensionless ratio of the heat-conduction resistance in the solid sphere to

the convective heat-transfer resistance in the surrounding ﬂuid. The criterion for

assuming steady-state in this problem involved an interrelationship between the

194 APPLICATIONS IN HEAT TRANSFER

Fourier and Biot numbers; the observation time required to achieve steady-state

was found to decrease with increasing Biot number. Scaling analysis provided a

systematic method for arriving at the simpliﬁed equations appropriate to the small

Biot number approximation whereby the temperature can be assumed to be uniform

within the conducting object. Most heat-transfer textbooks do not provide any

rigorous justiﬁcation for the low Biot number approximation.

In Section 4.5 we considered fully developed laminar ﬂow between two ﬂat

plates with heat generation due to viscous dissipation. The presence of both the

transverse and axial conduction terms made the describing equations elliptic. This

complicated the solution since the required downstream boundary condition often

is unknown. The concept of local scaling in heat transfer was introduced in this

problem, whereby one considers the describing equations within a domain deﬁned

by some arbitrary distance in the principal direction of ﬂow that is assumed to be

constant during the scaling analysis. In contrast to the preceding three examples,

there was no explicit temperature scale in this problem; rather, the temperature scale

was determined by balancing the viscous dissipation and transverse heat-conduction

terms. Scaling analysis led to two important dimensionless groups in heat transfer:

the Peclet and Prandtl numbers. The former is a measure of the ratio of the convec-

tion to conduction of heat, whereas the latter is a measure of the viscous transport

of momentum to the conductive transfer of heat. The Peclet number has a role in

heat transfer that is analogous to that of the Reynolds number in ﬂuid dynamics. For

example, we found that the convective heat transfer could be ignored if the Peclet

number was very small; this is analogous to the low Reynolds number or creeping-

ﬂow approximation in ﬂuid dynamics. We also found that the complicating effects

of axial heat conduction could be ignored if the width-to-length aspect ratio was

very small. The combination of small Peclet number and small aspect ratio in heat

transfer is analogous to the lubrication-ﬂow approximation in ﬂuid dynamics.

Scaling analysis was applied to the complementary problem of high Peclet num-

ber, coupled heat and momentum transfer in Section 4.6. The problem considered

here was heat transfer from a hot ﬂat plate to the developing ﬂow over this surface.

In this problem the transverse derivative of the temperature and axial velocity were

scaled with different characteristic lengths: the thermal and momentum boundary-

layer thicknesses, respectively. The relative thickness of the two boundary layers

depended on the Prandtl number. For liquids whose Prandtl number is much greater

than 1, the thermal was thinner than the momentum boundary layer. For liq-

uid metals whose Prandtl number is less than 1, the thermal is thicker than the

momentum boundary layer. For gases whose Prandtl number is nearly 1, the two

boundary layers have essentially the same thickness. We found that the boundary-

layer approximation is reasonable when the Peclet and Reynolds numbers based

on the local axial length scale become large (i.e., Pe

≡ U

∞

L/α = Re ·Pr 1

and Re ≡ U

∞

ρL/μ 1). Note that for ordinary liquids, thermal boundary-layer

analysis might apply, whereas the momentum boundary-layer analysis might not.

Note also that the boundary-layer approximation must break down in the vicinity

of the leading edge, where L becomes small. Hence, this problem involved both

a transverse and an axial region of inﬂuence; boundary-layer theory is applicable

SUMMARY 195

within the former and beyond the latter. This upstream limitation of boundary-layer

theory, which emerges from scaling analysis, is not mentioned in some transport

and ﬂuid mechanics textbooks. This problem also illustrated how scaling analysis

can suggest a strategy for solving the describing equations by using the solution

to the parabolic boundary-layer equations to specify the downstream boundary

conditions on the elliptic equations that must be solved near the leading edge.

Scaling analysis was applied to heat transfer associated with phase change in

Section 4.7. Conductive heat transfer causes a melting front to penetrate progres-

sively into frozen porous media. This provided an introduction to moving boundary

problems, those for which boundary conditions must be applied at surfaces whose

location in turn had to be determined by solving the describing equations. Since

the location of the moving boundary constituted an additional unknown, an auxil-

iary condition was needed. This was obtained by an integral energy balance over

the domain of interest. An analogy was drawn between this auxiliary condition in

heat transfer and the kinematic condition in ﬂuid dynamics in that both are based on

integral balances done to derive an additional equation to determine the location of

an unspeciﬁed boundary. The forgiving nature of scaling analysis was illustrated by

intentionally scaling one of the dependent variables incorrectly and thereby arriving

at a contradiction: that the unsteady-state term was not multiplied by the reciprocal of

the Fourier number. A proper analysis required that the time derivative of the moving

boundary location be scaled with a velocity scale rather than with the ratio of the char-

acteristic length divided by the time scale. We then found that for sufﬁciently large

Fourier numbers, quasi-steady-state thermal penetration of the melting front could be

assumed. This problem also introduced the pseudo-convection term that arises when

the scaling introduces either a scale or a reference factor that is time-dependent.

In Section 4.8 we illustrated how scaling analysis could be used to determine

when it is reasonable to ignore the temperature dependence of the viscosity for a

problem involving laminar ﬂow between two ﬂat plates with transverse heat con-

duction and viscous dissipation. Since we sought to determine only if any signiﬁcant

variation in the viscosity occurred, it was sufﬁcient to expand the equation describ-

ing the temperature dependence of the viscosity in a Taylor series and retain only

the ﬁrst-order correction. Scaling then identiﬁed the condition required to ignore

this ﬁrst-order correction. The scaling procedure used here for assessing when the

temperature dependence of the viscosity can be ignored can be applied to assess

the dependence of any physical or transport property on state variables such as

temperature, pressure, or concentration.

In Section 4.9 we applied scaling to a free-convection heat-transfer problem;

that is, to a problem wherein the driving force for ﬂow was internal to the system,

in this case due to density variations created by temperature gradients. Scaling was

employed to determine when the temperature dependence of the density could be

represented by the ﬁrst two terms in a Taylor series expansion about the density at

the average temperature. This is the basis of the classical Boussinesq approximation

in the analysis of free convection.

Scaling was applied to dimensional analysis in Section 4.10. In contrast to

◦

(1) scaling analysis, the scaling approach to dimensional analysis merely seeks to

196 APPLICATIONS IN HEAT TRANSFER

arrive at the minimum parametric representation of the problem; that is, to obtain

a set of dimensionless describing equations in terms of the minimum number of

dimensionless parameters. The goal of this problem was to determine the relation-

ship between the required cooking time and turkey mass based on discrete data

taken from a cookbook. This example illustrated the advantages of the scaling

analysis methodology for dimensional analysis relative to using the conventional

Pi theorem approach in that the latter did not lead to the minimum parametric

representation. Scaling analysis led to a single dimensionless group that involved

the two quantities that we sought to interrelate. By using this dimensionless group

in conjunction with the discrete cookbook data it was possible to develop a fully

predictive equation that related the required cooking time to the mass of the turkey.

4.E EXAMPLE PROBLEMS

4.E.1 Steady-State Heat Conduction in a Rectangular Fin

A solid metallic ﬂat ﬁn with a constant thermal conductivity k, length L, width

W , and height H such that H  L<W is attached at one end to a surface that is

maintained at a constant temperature T

. The convective heat-transfer ﬂux q

from

the surfaces of the ﬁn to the ambient air is describing via the lumped-parameter

approach and given by

= h(T − T

∞

) (4.E.1-1)

where h is the heat-transfer coefﬁcient and T

∞

is the temperature of the ambient

air far removed from the ﬁn. A schematic of the ﬁn is shown in Figure 4.E.1-1.

We use scaling analysis to explore what simpliﬁcations are possible in describing

heat transfer in this ﬁn.

The describing equations obtained by considering only the conduction terms

in equation (F.1-2) in the Appendices and the necessary boundary conditions are

given by (step 1)

Solid metallic fin

Ambient gas phase at T

∞

Figure 4.E.1-1 Flat solid metallic ﬁn that has a constant thermal conductivity and length

L, width W , and height H such that H  L<W; the convective heat-transfer ﬂux q

from

the surfaces of the ﬁn to the ambient air is described by h(T − T

∞

EXAMPLE PROBLEMS 197

0 =

∂

∂x

∂

∂y

∂

∂z

(4.E.1-2)

T = T

at x = 0, 0 ≤ y ≤ H, 0 ≤ z ≤ W (4.E.1-3)

−k

∂T

∂x

= h(T −T

∞

) at x = L, 0 ≤ y ≤

, 0 ≤ z ≤

(4.E.1-4)

∂T

∂y

= 0aty = 0, 0 ≤ x ≤ L, 0 ≤ z ≤

(4.E.1-5)

−k

∂T

∂y

= h(T −T

∞

) at y =

, 0 ≤ x ≤ L, 0 ≤ z ≤

(4.E.1-6)

∂T

∂z

= 0atz = 0, 0 ≤ x ≤ L, 0 ≤ y ≤

(4.E.1-7)

−k

∂T

∂z

= h(T −T

∞

) at z =

, 0 ≤ x ≤ L, 0 ≤ y ≤

(4.E.1-8)

Because of the planar symmetry, we have considered the heat transfer in only the

upper half-width of the ﬁn.

Introduce the following scale and reference factors (steps 2, 3, and 4):

∗

≡

T − T

; x

∗

≡

; y

∗

≡

; z

∗

≡

;



∂T

∂x



∗

≡

∂T

∂x

;



∂T

∂y



∗

≡

∂T

∂y

;



∂T

∂z



∗

≡

∂T

∂z

(4.E.1-9)

Note that we have allowed for separate scales for the temperature gradients in the

x-, y-, and z-directions since there is no reason to assume that the temperature

will change from its maximum to its minimum value over the length, thickness,

or width of the ﬁn, respectively. Substitute these dimensionless variables into the

describing equations to obtain (steps 5 and 6)

0 =

∂

∂x

∗



∂T

∂x



∗

∂

∂y

∗



∂T

∂y



∗

∂

∂z

∗



∂T

∂z



∗

(4.E.1-10)

∗

− T

at x

∗

= 0, 0 ≤ y

∗

≤

, 0 ≤ z

∗

≤

(4.E.1-11)



∂T

∂x



∗

=−



∗

− T

∞



at x

∗

, 0 ≤ y

∗

≤

0 ≤ z

∗

≤

(4.E.1-12)



∂T

∂y



∗

= 0aty

∗

= 0, 0 ≤ x

∗

≤

, 0 ≤ z

∗

≤

(4.E.1-13)

198 APPLICATIONS IN HEAT TRANSFER



∂T

∂y



∗

=−



∗

− T

∞



at y

∗

, 0 ≤ x

∗

≤

0 ≤ z

∗

≤

(4.E.1-14)



∂T

∂z



∗

= 0atz

∗

= 0, 0 ≤ x

∗

≤

, 0 ≤ y

∗

≤

(4.E.1-15)



∂T

∂z



∗

=−



∗

− T

∞



at z

∗

, 0 ≤ x

∗

≤

0 ≤ y

∗

≤

(4.E.1-16)

Inspection of the dimensionless describing equations indicates that the tempera-

ture can be bounded to be

◦

(1) if we set T

= T

∞

and T

= T

− T

∞

. The dimen-

sionless spatial coordinates can be bounded to be

◦

(1) if we set x

= L, y

= H ,

and z

= W. The dimensionless groups in equations (4.E.1-14) and (4.E.1-16) pro-

vide the following scales for the temperature gradients in the y-andz-directions,

respectively (step 7):

= T

h(T

− T

∞

)

− T

∞

)

= Bi

− T

∞

(4.E.1-17)

where Bi

≡ hH/k is the Biot number for heat transfer based on the smallest

length dimension H . However, it would not be appropriate to determine T

from

the dimensionless group in equation (4.E.1-12) since the small amount of heat

leaving the tip of the ﬁn does not cause the axial temperature gradient; rather, we

expect that the axial temperature gradient is caused by the heat being transferred

from the side of the ﬁn.

When the scale and reference factors are substituted into equations (4.E.1-10)

through (4.E.1-16), we obtain the following dimensionless describing equations:

0 =

∂

∂x

∗



∂T

∂x



∗

+ Bi

L(T

− T

∞

)

∂

∂y

∗



∂T

∂y



∗

+ Bi

L(T

− T

∞

)

WHT

∂

∂z

∗



∂T

∂z



∗

(4.E.1-18)

∗

= 1atx

∗

= 0, 0 ≤ y

∗

≤ 1, 0 ≤ z

∗

≤ 1 (4.E.1-19)



∂T

∂x



∗

=−

h(T

− T

∞

)

∗

at x

∗

= 1, 0 ≤ y

∗

≤ 1, 0 ≤ z

∗

≤ 1

(4.E.1-20)



∂T

∂y



∗

= 0aty

∗

= 0, 0 ≤ x

∗

≤ 1, 0 ≤ z

∗

≤ 1 (4.E.1-21)

EXAMPLE PROBLEMS 199



∂T

∂y



∗

=−T

∗

at y

∗

= 1, 0 ≤ x

∗

≤ 1, 0 ≤ z

∗

≤ 1 (4.E.1-22)



∂T

∂z



∗

= 0atz

∗

= 0, 0 ≤ x

∗

≤ 1, 0 ≤ y

∗

≤ 1 (4.E.1-23)



∂T

∂z



∗

=−T

∗

at z

∗

= 1, 0 ≤ x

∗

≤ 1, 0 ≤ y

∗

≤ 1 (4.E.1-24)

The axial conduction must be balanced by at least one of the other two terms

in equation (4.E.1-18). However, since H  W , it is clear that conduction in the

y-direction is far more important than that in the z-direction. Hence, we set the

dimensionless group multiplying the conduction term in the y-direction equal to 1

to obtain T

L(T

− T

∞

)

= 1 ⇒ T

= Bi

L(T

− T

∞

)

(4.E.1-25)

Substituting this value for T

into equation (4.E.1-18) then yields

0 =

∂

∂x

∗



∂T

∂x



∗

∂

∂y

∗



∂T

∂y



∗

∂

∂z

∗



∂T

∂z



∗

(4.E.1-26)

Inspection of equation (4.E.1-26) indicates that conduction in the z-direction can

be ignored if H/W  1, which is the case for a thin wide ﬁn such as that being

considered here (step 8). Furthermore, if we invoke the low Biot number approxi-

mation developed in Section 4.4 (i.e., Bi

 1), equation (4.E.1-17) indicates that

the temperature gradient in the y-direction will be essentially zero, which implies

that the temperature will be constant across the thickness of the ﬁn. Analogously

to the procedure used in Section 4.4, let us integrate equation (4.E.1-26) across the

cross-sectional area of the ﬁn:



−1

∂

∂x

∗



∂T

∂x



∗

Wdy

∗



−1

∂

∂y

∗



∂T

∂y



∗

Wdy

∗

= 0 (4.E.1-27)

∂

∂x

∗



∂T

∂x



∗



−1

∗



−T

∗

∂



∂T

∂y



∗

= 0 (4.E.1-28)

∂

∂x

∗



∂T

∂x



∗

− T

∗

= 0 (4.E.1-29)

Equation (4.E.1-29) is subject to the boundary conditions given by equations

(4.E.1-19) and (4.E.1-20), given by

∗

= 1atx

∗

= 0 (4.E.1-30)



∂T

∂x



∗

=−

∗

∼

0atx

∗

= 1 (4.E.1-31)

200 APPLICATIONS IN HEAT TRANSFER

The solution to this simpliﬁed system of equations is straightforward and given by

∗

x

∗

1 + e

−2



−2x

∗

+ e

−2



, where  ≡



(4.E.1-32)

The solution given by equation (4.E.1-32) is applicable for small Biot numbers

for which the temperature across the thickness of the ﬁn can be assumed to be

constant and for which heat transfer along the sides and tip of the ﬁn can be

neglected. These are reasonable assumptions for thin ﬁns made from a highly

conducting metal.

4.E.2 Unsteady-State Resistance Heating in a Wire

Consider a long solid wire having radius R as shown in Figure 4.E.2-1, whose ini-

tial temperature is T

. At time t = 0 an alternating current begins to ﬂow through the

wire that causes electrical resistance heating whose volumetric energy generation

rate G

is given by

= G

cos ωt (energy generation rate per unit volume) (4.E.2-1)

where ω is the angular frequency in radians per second. The temperature at the

surface of the wire is held constant at the initial temperature T

. We use scaling to

explore how the describing equations might be simpliﬁed.

The appropriately simpliﬁed form of the thermal energy equation given by

equation (F.2-2) in the Appendices and the required initial and boundary conditions

are given by (step 1)

ρC

∂T

∂t

= k





+ G

cos ωt (4.E.2-2)

T = T

at t = 0 (4.E.2-3)

T is bounded or

∂T

∂r

= 0atr = 0 (4.E.2-4)

T = T

at r = R (4.E.2-5)

T = T

G = G

cos wt

Figure 4.E.2-1 Unsteady-state heat conduction in a solid cylinder of radius R due to

electrical heat generation at a volumetric rate given by G

= G

cos ωt.

EXAMPLE PROBLEMS 201

The boundary condition given by equation (4.E.2-4) merely states that the temper-

ature cannot be inﬁnite or is an extremum (a maximum in this case) at r = 0; this

type of boundary condition is often invoked in problems involving cylindrical or

spherical coordinates. Introduce the following scale and reference factors (steps 2,

3, and 4):

∗

≡

T − T

; r

∗

≡

; t

∗

≡

(4.E.2-6)

Substitute these dimensionless variables into the describing equations and divide

through by the dimensional coefﬁcient of one term in each of the equations to

obtain (steps 5 and 6)

αt

∂T

∗

∂t

∗



∗



cos ωt

∗

(4.E.2-7)

∗

− T

at t

∗

= 0 (4.E.2-8)

∗

is bounded or

∂T

∗

∂r

∗

= 0atr

∗

= 0 (4.E.2-9)

∗

− T

at r

∗

(4.E.2-10)

Setting the appropriate dimensionless groups in equation (4.E.2-8) and (4.E.2-

10) equal to zero and 1, respectively, gives T

= T

and r

= R (step 7). Since

the heat generation must be balanced by the radial heat conduction, we set the

dimensionless group in equation (4.E.2-7) equal to 1 to obtain T

= G

/k.There

are three relevant time scales whose relative values determine the simpliﬁcations

that are possible for this problem (step 8):

= t

time scale corresponding to the observation time

= t

time scale characterizing the heat conduction (4.E.2-11)

= t

2π

time scale characterizing the periodic rate of heat generation

Note that we use the reciprocal of the cyclic rather than the angular frequency to

properly characterize the time scale for the periodic rate of heat generation.

Consider ﬁrst the case where t

= t

, for which equation (4.E.2-7) simpliﬁes to

∂T

∗

∂t

∗



∗



+ cos ωt

∗

(4.E.2-12)

where Fo

≡ R

/αt

is the Fourier number for heat transfer. It is not possible for the

Fourier number to be much less than 1 since the conduction term has to be retained

202 APPLICATIONS IN HEAT TRANSFER

at all times. If Fo

∼

1,t

= R

/α and equation (4.E.2-12) can be written as

∂T

∗

∂t

∗



∗



+ cos

ωR

∗

(4.E.2-13)

Equation (4.E.2-13) can be simpliﬁed further if ωR

/α  1. This implies that

the heat conduction occurs much faster than the periodic heat generation; that is,

the heat conduction smoothes out the variations in temperature due to the time-

dependence of the heat generation, and equation (4.E.2-13) simpliﬁes to

∂T

∗

∂t

∗



∗



+ 1 (4.E.2-14)

This equation describes the transient heat conduction period for the special case of

the conduction time scale being much smaller than the characteristic time scale for

the periodic heat generation.

If Fo

 1, the transient heat conduction effects have died out; however, the

problem is still possibly unsteady state, due to the time-dependent heat-generation

term. For this case the characteristic time scale is t

= t

= 2π/ω and equa-

tion (4.E.2-7) assumes the form

ωR

2πα

∂T

∗

∂t

∗



∗



+ cos

ωR

2πα

∗

(4.E.2-15)

Equation (4.E.2-15), which describes unsteady-state heat conduction due to time-

dependent heat generation, can be simpliﬁed further if ωR

/2πα  1, whereby it

reduces to

0 =

∗



∗



+ 1 (4.E.2-16)

which describes steady-state heat conduction due to constant heat generation.

4.E.3 Convective Heat Transfer with Injection Through Permeable Walls

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

with constant physical properties between two inﬁnitely wide permeable parallel

plates separated by a distance H as shown in Figure 4.E.3-1. There is constant

injection of the same ﬂuid at velocity V through the permeable plate at y = H

and constant withdrawal of the same incompressible ﬂuid through the permeable

plate at y = 0. The continuity equation can be used to prove that the injection and

withdrawal velocities at the upper and lower plates must be equal for the ﬂow to

be fully developed. The upstream (entering) temperature of the ﬂuid is T

.The

temperature of the upper plate at y = H is also T

, whereas the temperature of

the lower plate at y = 0isT

, such that T

. Since the ﬂow is assumed to be