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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SMALL PECLET NUMBER APPROXIMATION 163

This equation can be integrated to give the following solution for the temperature

of the sphere as a function of time:

∗

= 1 − e

−3Bi

·t

∗

(4.4-22)

For very small Biot numbers, equation (4.4-22) simpliﬁes to

∗

∼

3Bi

· t

∗

(4.4-23)

The error in the solution given by equation (4.4-23) will be in the range 10 to

100% if Bi

◦

(0.1) and 1 to 10% if Bi

◦

(0.01).

Scaling this problem has illustrated an important simpliﬁcation: the small Biot

number approximation that can be made when considering convective heat transfer

to or from a solid object having ﬁnite dimensions.

The Biot number is a measure

of the resistance to heat conduction in the solid object relative to the resistance to

convective heat transfer in the surrounding ﬂuid. For sufﬁciently small Biot num-

bers the heat transfer will be controlled totally by convection into the surrounding

ﬂuid. Under these conditions there will be a uniform temperature in the solid object,

which permits a straightforward analytical solution. Note that most heat-transfer

textbooks do not provide any rigorous justiﬁcation for the equations appropriate to

the small Biot number approximation.

4.5 SMALL PECLET NUMBER APPROXIMATION

The problem considered in Section 4.4 involved convective heat transfer in the

ﬂuid phase adjacent to a solid sphere. However, a lumped-parameter approach was

used to account for this convection. In the present example problem we consider

the convective heat transfer explicitly. Consider the steady-state fully developed

laminar ﬂow of a viscous Newtonian ﬂuid with constant physical properties between

two inﬁnitely wide parallel plates separated by a distance 2H and of length L,as

shown in Figure 4.5-1. The upstream (entering) temperature of the ﬂuid is T

.The

temperature of the upper and lower plates is also maintained at T

. Since the ﬂow

is assumed to be laminar and fully developed, the velocity proﬁle is given by

= U



1 −







(4.5-1)

where U

is the maximum velocity. As a result of this shear ﬂow, there will be

heat generation via viscous dissipation. The latter will cause both radial and axial

conduction as well as axial convection of heat.

Note that the small Biot number approximation is sometimes referred to as the lumped capacitance

approximation.

164 APPLICATIONS IN HEAT TRANSFER

Figure 4.5-1 Steady-state fully developed laminar ﬂow of a viscous Newtonian ﬂuid with

constant physical properties between two inﬁnitely wide parallel plates separated by a dis-

tance 2H and of length L; the temperature of the entering ﬂuid as well as that of the upper

and lower plates is T

; the shear ﬂow causes viscous heat generation; the fully developed

velocity proﬁle is shown as well as a representative developing temperature proﬁle.

The thermal energy equation given by equation (F.1-2) in the Appendices, appro-

priately simpliﬁed using equation (4.5-1) and for the conditions deﬁned in the

problem statement and the associated boundary conditions are given by (step 1)

ρC



1 −







∂T

∂x

= k

∂

∂x

+ k

∂

∂y

4μU

(4.5-2)

T = T

at x = 0 (4.5-3)

T = f(y) at x = L (4.5-4)

T = T

at y =±H (4.5-5)

∂T

∂y

= 0aty = 0 (4.5-6)

where f(y) is some function of y, which might be unknown. This is a nontrivial

problem to solve, due to the elliptic nature of the describing equations. The presence

of the second-order axial derivative requires that a downstream boundary condition

be speciﬁed. In many problems such as this, these downstream conditions are not

known, which precludes solving the describing equations numerically. Clearly, one

would like to know when and how these describing equations might be simpliﬁed

to permit a tractable solution. In particular, one would like to know when the

axial conduction and convection terms might be neglected. We use

◦

(1) scaling to

determine the criteria for neglecting these terms.

SMALL PECLET NUMBER APPROXIMATION 165

Deﬁne the following dimensionless variables involving unspeciﬁed scale factors

(steps 2, 3, and 4):

∗

≡

T − T

; x

∗

≡

; y

∗

≡

(4.5-7)

We have introduced a reference factor for the temperature since it is not natu-

rally referenced to zero. Introduce these dimensionless variables into the describing

equations and divide through by the coefﬁcient of one term in each equation that

should be retained (steps 5 and 6):

ρC



1 −



∗





∂T

∗

∂x

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

4μU

∗2

(4.5-8)

∗

− T

at x

∗

= 0 (4.5-9)

∗

= f(y

∗

) at x

∗

(4.5-10)

∗

− T

at y

∗

=±

(4.5-11)

∂T

∗

∂y

∗

= 0aty

∗

= 0 (4.5-12)

Step 7 involves bounding the independent and dependent dimensionless variables

to be

◦

(1). This can be done for the dimensionless spatial coordinates by setting

the dimensionless groups containing x

and y

in equations (4.5-10) and (4.5-11)

equalto1;thatis,

= 1 ⇒ x

= L;

= 1 ⇒ y

= H (4.5-13)

To determine the reference and scale factors for the temperature, we need to con-

sider the conditions for which we are scaling. We are seeking to determine when

axial conduction and convection can be ignored relative to heat generation by vis-

cous dissipation and transverse conduction. If the former are negligible, the heat

generation by viscous dissipation must be balanced by the transverse heat conduc-

tion. Hence, we can bound our dimensionless temperature to be

◦

(1) by setting the

dimensionless group in equation (4.5-9) or (4.5-10) equal to zero, thereby determin-

ing the reference temperature, and by setting the dimensionless group multiplying

the heat generation term in equation (4.4-8) equal to 1, to determine the temperature

scale; that is,

− T

= 0 ⇒ T

= T

;

μU

= 1 ⇒ T

μU

(4.5-14)

166 APPLICATIONS IN HEAT TRANSFER

Substitution of the scale and reference factors deﬁned by equations (4.5-13) and

(4.5-14) into the dimensionless describing equations given by equations (4.5-8)

through (4.5-12) yields

ρC

(1 − y

∗2

)

∂T

∗

∂x

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

+ 4y

∗2

(4.5-15)

∗

= 0atx

∗

= 0 (4.5-16)

∗

= f(y

∗

) at x

∗

= 1 (4.5-17)

∗

= 0aty

∗

=±1 (4.5-18)

∂T

∗

∂y

∗

= 0aty

∗

= 0 (4.5-19)

Now let us explore possible simpliﬁcations of the describing equations (step 8).

The criterion for ignoring axial conduction is

 1 ⇒ axial conduction can be ignored (4.5-20)

that is, the aspect ratio cannot be too large. Note, however, that the length L was

arbitrary in that L could denote any value of the axial coordinate in the principal

direction of ﬂow. This is the principle of local scaling whereby we scale the

problem for some ﬁxed but arbitrary value of some coordinate, usually that in the

principal direction of ﬂow.

To ignore axial convection of heat, the dimensionless group multiplying the ﬁrst

term in equation (4.5-15) must be very small; that is,

ρC

= Re ·Pr

= Pe

 1 (4.5-21)

where α ≡ k/ρC

is the thermal diffusivity, ν ≡ μ/ρ is the kinematic viscosity,

Re ≡ ρU

H/μ is the Reynolds number, Pr ≡ ν/α is the Prandtl number, and

≡ U

H/α is the Peclet number for heat transfer. The Reynolds number is a

measure of the ratio of the convective to viscous transport of momentum. The

Prandtl number is a measure of the ratio of the viscous transport of momentum

to heat conduction. Hence, the Peclet number is a measure of the ratio of heat

convection to heat conduction. We see that the criterion for ignoring axial heat

convection is that the product of the Peclet number and the aspect ratio must

be very small. The Peclet number in heat transfer has a role analogous to that

of the Reynolds number in ﬂuid dynamics; that is, when it is small, it justiﬁes

ignoring axial convective transport. We will see in the next example problem that

when it is large, it justiﬁes a boundary-layer approximation. Note that ignoring

convective transport in the energy equation in this example is analogous to ignoring

BOUNDARY-LAYER OR LARGE PECLET NUMBER APPROXIMATION 167

convective transport in the equations of motion, which is the basis for the creeping-

ﬂow approximation.

If the conditions in equations (4.5-20) and (4.5-21) are satisﬁed, equations (4.5-

15) through (4.5-19) reduce to

0 =

∗

∗2

+ 4y

∗2

(4.5-22)

∗

= 0aty

∗

=±1 (4.5-23)

∂T

∗

∂y

∗

= 0aty

∗

= 0 (4.5-24)

The solution to this simpliﬁed set of describing equations is straightforward and

given by

∗

(1 − y

∗4

) (4.5-25)

We see from this solution that our dimensionless temperature is bounded of

◦

(1),

which conﬁrms that our scaling analysis is correct.

4.6 BOUNDARY-LAYER OR LARGE PECLET NUMBER

APPROXIMATION

In the preceding example we saw that a low Peclet number justiﬁed ignoring the

convective transport of thermal energy, which is analogous to a low Reynolds

number justifying neglecting convective transport of momentum in the creeping-

ﬂow approximation considered in Chapter 3. In this example we consider the other

end of the Peclet number spectrum by exploring the implications of a high Peclet

number on heat transfer. Consider the steady-state laminar uniform (plug) ﬂow of a

Newtonian ﬂuid with constant physical properties and temperature T

∞

intercepting

a stationary, semi-inﬁnitely long inﬁnitely wide horizontal ﬂat plate maintained at a

constant temperature T

such that T

∞

, as shown in Figure 4.6-1. Gravitational

and viscous heating effects can be assumed to be negligible. To fully understand

this example, it would be helpful to review Section 3.4, in which the boundary-

layer approximation in ﬂuid dynamics was considered. As stated in Section 3.4,

boundary-layer ﬂows are examples of region-of-inﬂuence scaling,forwhichwe

determine the thickness of a region wherein some effect is conﬁned, in this case

the inﬂuence of the heated ﬂat plate that is propagated by conduction. This example

also illustrates the principle of local scaling, in which we carry out the scaling at

some arbitrary but ﬁxed value of the axial coordinate.

The creeping-ﬂow approximation was considered in Section 3.3 and Example Problem 3.E.2.

168 APPLICATIONS IN HEAT TRANSFER

∞

T = T

∞

Figure 4.6-1 Steady-state laminar uniform ﬂow of a Newtonian ﬂuid with constant phys-

ical properties and temperature T

∞

intercepting a stationary, semi-inﬁnitely long inﬁnitely

wide horizontal ﬂat plate maintained at a constant temperature T

such that T

∞

;the

solid line shows the hypothetical momentum boundary-layer thickness δ

, whereas the dash-

dotted line shows the hypothetical thermal boundary-layer thickness δ

for Pr > 1, where Pr

is the Prandtl number.

The continuity equation given by equation (C.1-1), equations of motion given

by equations (D.1-10) and (D.1-11), and thermal energy equation given by equa-

tion (F.1-2) in the Appendices simplify to the following for the assumed ﬂow

conditions (step 1):

ρu

∂u

∂x

+ ρu

∂u

∂y

=−

∂P

∂x

+ μ



∂

∂x

∂

∂y



(4.6-1)

ρu

∂u

∂x

+ ρu

∂u

∂y

=−

∂P

∂y

+ μ



∂

∂x

∂

∂y



(4.6-2)

∂u

∂x

∂u

∂y

= 0 (4.6-3)

ρC

∂T

∂x

+ ρC

∂T

∂y

= k



∂

∂x

∂

∂y



(4.6-4)

The corresponding boundary conditions for this ﬂow are given by

= U

∞

= 0,T= T

∞

at x = 0 (4.6-5)

= f

(y), u

= f

(y), T = f

(y) at x = L (4.6-6)

= 0,u

= 0,T= T

at y = 0 (4.6-7)

= U

∞

= 0,T= T

∞

at y =∞ (4.6-8)

where f

(y), f

(y),andf

(y) are unspeciﬁed functions. The boundary conditions

for the equations of motion were discussed in Section 3.4. The boundary conditions

BOUNDARY-LAYER OR LARGE PECLET NUMBER APPROXIMATION 169

on the energy equation given by equations (4.6-5) and (4.6-7) are the prescribed

temperature of the entering ﬂuid and at the plate, respectively. Equation (4.6-6)

merely states that to solve this elliptic energy equation, a downstream bound-

ary condition must be speciﬁed; this condition might not be known in practice.

Equation (4.6-8) states that the temperature becomes equal to that of the entering

ﬂuid inﬁnitely far above the ﬂat plate.

Deﬁne the following dimensionless dependent and independent variables (steps

2, 3, and 4):

∗

≡

; u

∗

≡

; P

∗

≡

;

∗

≡

T − T

; x

∗

≡

; y

∗

≡

; y

∗

≡

(4.6-9)

Note that we have allowed for different y-length scales for the energy equation

and equations of motion; that is, the temperature might experience a characteristic

change of

◦

(1) over a different length scale than the velocities. Introduce these

dimensionless variables into the describing equations and divide each equation

through by the dimensional coefﬁcient of one term that should be retained to

maintain physical signiﬁcance (steps 5 and 6):

∗

∂u

∗

∂x

∗

∂u

∗

∂y

∗

=−

ρu

∂P

∗

∂x

∗

ρu

∂

∗

∂x

∗2

μx

ρu

∂

∗

∂y

∗2

(4.6-10)

∗

∂u

∗

∂x

∗

∂u

∗

∂y

∗

=−

ρu

∂P

∗

∂y

∗

ρu

∂

∗

∂x

∗2

μx

ρu

∂

∗

∂y

∗2

(4.6-11)

∂u

∗

∂x

∗

∂u

∗

∂y

∗

= 0 (4.6-12)

∗

∂T

∗

∂x

∗

+ u

∗

∂T

∗

∂y

∗

ρC

∂

∗

∂x

∗2

ρC

∂

∗

∂y

∗2

(4.6-13)

∗

∞

∗

= 0,T

∗

∞

− T

at x

∗

= 0

(4.6-14)

∗

= f

∗

), u

∗

), T

∗

) at x

∗

(4.6-15)

170 APPLICATIONS IN HEAT TRANSFER

∗

= 0,u

∗

= 0,T

∗

− T

at y

∗

= y

∗

= 0 (4.6-16)

∗

∞

∗

= 0,T

∗

∞

− T

at y

∗

= y

∗

=∞ (4.6-17)

Note that we have divided equations (4.6-10) and (4.6-11) by the dimensional

coefﬁcient of the axial convection term since we are considering a high Reynolds

number ﬂow and high Peclet number heat transfer for which the convection terms

must be retained. However, we have divided equation (4.6-13) by the dimensional

coefﬁcient of the transverse convection term. The reason for doing this is not

obvious at this point. However, we will see that this is the principal convective

term in the energy equation for Pr > 1.

For a high Reynolds number ﬂow for which the action of viscosity is conﬁned

to the vicinity of the boundaries, the y-length scale for the velocities will be the

thickness of the momentum boundary layer or region of inﬂuence δ

;thatis,we

say that y

= δ

. The scale factors for the velocity components, pressure, and

axial coordinate are determined in exactly the same manner as was described in

detail in Section 3.4 and are given by (step 7)

= U

∞

; x

= L; u

∞

; P

= ρU

∞

(4.6-18)

In view of the fact that the principal viscous term in equation (4.6-10) has to be

important at least within some small region in the vicinity of the ﬂat plate, we set

the dimensionless group in front of this term equal to 1 to ensure that this term

is of the same size as the convection terms that are being retained. This yields

the following equation for the thickness of the region of inﬂuence or momentum

boundary layer:

μL

ρU

∞

⇒ δ

√

(4.6-19)

where Re is the local Reynolds number based on using L as the characteristic

length. Note that L is arbitrary in that it can be any ﬁxed value of the axial length

coordinate; that is, our scaling was done for an arbitrary length L of a semi-inﬁnitely

long ﬂat plate; this is what is meant by the concept of local scaling. The reference

and scale factors for the temperature are determined by setting the appropriate group

in equations (4.6-14) or (4.6-17) equal to zero and in equation (4.6-16) equal to 1

to obtain

= T

∞

; T

= T

− T

∞

(4.6-20)

In view of the fact that the principal conduction term in equation (4.6-13) has to

be important at least within some small region in the vicinity of the ﬂat plate, we

set the dimensionless group in front of this term equal to 1 to ensure that this term

is of the same size as the convection terms that are being retained. This yields the

BOUNDARY-LAYER OR LARGE PECLET NUMBER APPROXIMATION 171

following equation for the thickness of the region of inﬂuence or thermal boundary

layer:

ρC

∞

√

(4.6-21)

where Pe

≡ U

∞

L/α is the local Peclet number for heat transfer based on using L

as the characteristic length. A comparison of equations (4.6-19) and (4.6-21) again

indicates that in heat transfer the Peclet number plays a role analogous to that of

the Reynolds number in ﬂuid dynamics. Note that for liquids Pr > 1; hence, the

thermal boundary-layer thickness will be less than the momentum boundary-layer

thickness. For gases Pr

∼

1; hence, the thermal and momentum boundary layers

have nearly the same thickness. However, for liquid metals, Pr < 1; hence, the

thermal boundary-layer thickness will be greater than the momentum boundary-

layer thickness. The general behavior of δ

(x) and δ

(x) for the case when Pr > 1

is shown in Figure 4.6-1.

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

deﬁned by equations (4.6-18) through (4.6-21), we obtain

∗

∂u

∗

∂x

∗

+ u

∗

∂u

∗

∂y

∗

=−

∂P

∗

∂x

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.6-22)

∗

∂u

∗

∂x

∗

+ u

∗

∂u

∗

∂y

∗

=−

∂P

∗

∂y

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.6-23)

∂u

∗

∂x

∗

∂u

∗

∂y

∗

= 0 (4.6-24)

∗

∂T

∗

∂x

∗

+ u

∗

∂T

∗

∂y

∗

Pr ·Pe

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.6-25)

∗

= 1,u

∗

= 0,T

∗

= 0atx

∗

= 0 (4.6-26)

∗

= f

∗

), u

∗

= f

∗

), T

∗

= f

∗

) at x

∗

= 1 (4.6-27)

∗

= 0,u

∗

= 0,T

∗

= 1aty

∗

= y

∗

= 0 (4.6-28)

∗

= 1,u

∗

= 0,T

∗

= 0aty

∗

= y

∗

=∞ (4.6-29)

The system of equations above is difﬁcult to solve for several reasons. First, elliptic

differential equations are involved that require specifying some downstream bound-

ary conditions that in practice are generally not known. Second, as discussed in

Section 3.4, equations (4.6-22) and (4.6-23) are coupled due to the pressure. Third,

the energy equation is coupled to the solution for the equations of motion through

the two velocity components. However, if the ﬂuid properties are not temperature-

dependent, this coupling is unidirectional in that the equations of motion can be

172 APPLICATIONS IN HEAT TRANSFER

solved independent of the energy equation. In view of these complications, we seek

to explore the conditions required to eliminate these two complications.

Note that in the limit of Re  1andPe

 1, the system of equations above

reduces to (step 8)

∗

∂u

∗

∂x

∗

+ u

∗

∂u

∗

∂y

∗

∂

∗

∂y

∗2

(4.6-30)

∂u

∗

∂x

∗

∂u

∗

∂y

∗

= 0 (4.6-31)

∗

∂T

∗

∂x

∗

+ u

∗

∂T

∗

∂y

∗

∂

∗

∂y

∗2

(4.6-32)

∗

= 1,u

∗

= 0,T

∗

= 0atx

∗

= 0 (4.6-33)

∗

= 0,u

∗

= 0,T

∗

= 1aty

∗

= y

∗

= 0 (4.6-34)

∗

= 1,T

∗

= 0aty

∗

= y

∗

=∞ (4.6-35)

Equations (4.6-30) through (4.6-35) are the classical boundary-layer equations for

ﬂow over a heated ﬂat plate. Note that by showing that the pressure term in

equation (4.6-22) is negligible in the limit of high Reynolds number, we have

eliminated the coupling between x-andy-components of the equations of motion.

Moreover, we have shown that the axial viscous term in equation (4.6-22) and

the axial heat-conduction term in equation (4.6-25) are negligible in the limit of

very high Reynolds and Peclet numbers, respectively, and thereby have converted

the elliptic into parabolic differential equations that do not require a downstream

boundary condition. Note that the dimensionless y-coordinate in the equations of

motion is deﬁned differently from that in the energy equation. If these equations

are recast in terms of dimensional variables and a stream function and similarity

variable are introduced, they can be transformed into a set of nonlinear ordinary dif-

ferential equations solved that can be solved via approximate analytical techniques

or numerically.

Note that the criterion for the applicability of the hydrodynamic boundary-layer

approximation is

Re ≡

ρU

∞

 1 ⇒ hydrodynamic boundary layer (4.6-36)

whereas the criterion for the applicability of the thermal boundary-layer approxi-

mation is

ρC

∞

 1 ⇒ thermal boundary layer (4.6-37)

See, for example, Bird et al., Transport Phenomena, 2nd ed., pp. 388–390.