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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THERMALLY DRIVEN FREE CONVECTION: BOUSSINESQ APPROXIMATION 183

When the reference and scale factors deﬁned by equations (4.8-14), (4.8-15),

and (4.8-16) are substituted into our dimensionless describing equations, we

obtain

0 = 1 +

∗

1 −

kμ



P



∗

(4.8-17)

0 =

∗

∗2

1 −

kμ



P



∗



∗



(4.8-18)

∗

= 0,

∗

= 0aty

∗

= 0 (4.8-19)

∗

= 0,T

∗

= 0aty

∗

=±1 (4.8-20)

We see that the criterion for ignoring the temperature dependence of the viscosity

is given by (step 8)

kμ



P



 1 (4.8-21)

If the criterion given by equation (4.8-21) is satisﬁed, equations (4.8-17) through

(4.8-20) reduce to exactly the same equations that were considered in Section 4.5;

that is, equations (4.5-22) through (4.5-24) for which the solution for the velocity

proﬁle is given by equation (4.5-1) and for which the temperature proﬁle is given

by equation (4.5-25).

4.9 THERMALLY DRIVEN FREE CONVECTION: BOUSSINESQ

APPROXIMATION

Thus far we have considered problems that have involved either pure conduc-

tion or heat transfer with forced convection, that is, ﬂow that is caused by some

external driving force, such as a pump, fan, moving boundary, gravitational ﬁeld,

or other body force. Here we consider an example of thermal free convection,

convection caused internal to the system, owing to a temperature gradient that cre-

ates unstable density variations. Note that unstable density variations can also be

caused by concentration gradients, in which case it is referred to as solutal free

convection. Free convection can also arise due to surface-tension gradients at an

interface.

Consider a ﬂuid with density ρ and viscosity μ that is conﬁned between two

vertical parallel plates of vertical height L separated by a distance 2H , as shown

in Figure 4.9-1. We assume that the space between the plates is capped at the

top and the bottom but that L  H . The vertical plate at y =−H is main-

tained at a constant temperature T

, whereas the vertical plate at y =+H is

maintained at a constant temperature T

,whereT

. Because the density of

184 APPLICATIONS IN HEAT TRANSFER

Temperatur

profile

Velocity

profile

Figure 4.9-1 Steady-state fully developed buoyancy-induced free convection of a ﬂuid

conﬁned between two vertical plates maintained at temperatures T

and T

,whereT

showing representative temperature and velocity proﬁles.

a ﬂuid decreases with increasing temperature, there will be a tendency for the

ﬂuid near the hot wall to ﬂow upward and for the ﬂuid near the cold wall to

ﬂow downward. As long as the plates are not too close together or the ﬂuid is

not too viscous, a steady-state free-convection ﬂow can be generated for which

the mass ﬂow upward is equal to the mass ﬂow downward.

We use scaling

analysis to determine the conditions that permit the equations describing free-

convection heat transfer to be simpliﬁed. We ignore end effects at the top and

bottom of the vertical parallel plates as well as heat generation due to viscous

dissipation.

Appropriate simpliﬁcation of the equations of motion given by equations (D.1-

10) and (D.1-11) and the thermal energy equation given by equation (F.1-2) in

To determine if free-convection ﬂow will be generated, it is necessary to carry out a stability analysis;

the latter leads to the conditions required for free convection to be initiated expressed in terms of

a critical value of the thermal Grashof number, Gr

≡ H

gβ

ρ /ν

, or thermal Rayleigh number,

≡ Gr

· Pr, where ρ in the present case is the difference in density between the ﬂuid at the cold

and hot plates; note that if the plates are too closely spaced or if the viscosity is sufﬁciently high, the

Grashof number can be below the critical value for the inception of free convection; the reader interested

in more information on stability theory applied to free convection is referred to standard references such

as P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge,

England, 1981, or J. S. Turner, Buoyancy Effects in Fluids, Cambridge University Press, Cambridge,

England, 1973.

THERMALLY DRIVEN FREE CONVECTION: BOUSSINESQ APPROXIMATION 185

the Appendices and speciﬁcation of the required boundary conditions yield the

following set of describing equations (step 1):

0 =−

∂P

∂x

+ μ

− ρg (4.9-1)

0 =−

∂P

∂y

(4.9-2)

0 =

(4.9-3)

= 0,T= T

at y =−H (4.9-4)

= 0,T= T

at y = H (4.9-5)

Note that axial derivatives of both velocity and temperature are assumed to be zero

under the assumptions of fully developed ﬂow and no end effects. Since the density

is temperature dependent, we need an appropriate equation of state. Here we con-

sider small density variations and hence represent the density via a Taylor series

expansion about the density

ρ at the average temperature between the two plates,

T ≡ (T

+ T

)/2, given by

ρ = ρ|

∂ρ

∂T



(T − T)+

∂

∂T



(T − T)

+··· (4.9-6)

ρ =

ρ −ρβ

(T − T)+ ργ

(T − T)

+··· (4.9-7)

where β

is the coefﬁcient of volume expansion and γ

is a positive constant. When

equation (4.9-7) is substituted into equation (4.9-1) and truncated at the third term

in the expansion, we obtain

0 =−

∂P

∂x

+ μ

− ρg + ρgβ

(T − T)− ρgγ

(T − T)

(4.9-8)

Equation (4.9-3) can be integrated directly subject to the boundary conditions given

by equations (4.9-4) and (4.9-5) to obtain the following temperature proﬁle:

T =

+ T

−

− T

T −

T

(4.9-9)

Moreover, equation (4.9-2) in combination with equation (4.9-1) implies that

∂P/∂x = dP/dx = a constant. Hence, we can evaluate dP/dx at any lateral

position between the two vertical plates. It is convenient to evaluate the pres-

sure gradient at the centerline between the two plates where T =

T ; therefore,

dP/dx =−

ρg. With this simpliﬁcation, equation (4.9-8) takes the form

0 = μ

+ ρgβ

(T − T)− ρgγ

(T − T)

(4.9-10)

186 APPLICATIONS IN HEAT TRANSFER

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

∗

≡

; T

∗

≡

T − T

; y

∗

≡

(4.9-11)

Substitute 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):

∗

∗2

ρgβ

μu

∗

− T

−

ρgγ

μu

∗

− T

(4.9-12)

0 =

∗

∗2

(4.9-13)

∗

= 0,T

∗

− T

at y

∗

=−

(4.9-14)

∗

= 0,T

∗

− T

at y

∗

(4.9-15)

The dimensionless groups in equations (4.9-14) and (4.9-15) when set equal to

1 and zero, respectively, provide the following reference and scale factors (step 7):

− T

= 0 ⇒ T

= T

− T

= 1 ⇒ T

= T

− T

(4.9-16)

In addition, the dimensionless groups in these equations provide the characteristic

length scale

= 1 ⇒ y

= H (4.9-17)

Since what causes this ﬂow (i.e., the leading-order gravitational body force term)

must balance the viscous resisting force, the corresponding dimensionless group

that provides a measure of this ratio in equation (4.9-12) must be equal to 1, which

provides the characteristic scale for the velocity:

ρgβ

μu

= 1 ⇒ u

ρgβ

− T

)

(4.9-18)

If the reference and scale factors deﬁned by equations (4.9-16) through (4.9-18)

are substituted into our dimensionless describing equations, we obtain

0 =

∗

∗2



∗

−



−

− T

)



∗

−



(4.9-19)

0 =

∗

∗2

(4.9-20)

DIMENSIONAL ANALYSIS CORRELATION FOR COOKING A TURKEY 187

∗

= 0,T

∗

= 1aty

∗

=−1 (4.9-21)

∗

= 0,T

∗

= 0aty

∗

= 1 (4.9-22)

We see that the criterion for ignoring the higher-order temperature dependence of

the density is given by (step 8)

− T

)

 1 (4.9-23)

This simpliﬁcation, which considers only the leading-order effects of the tempera-

ture on the density, is referred to as the Boussinesq approximation.

4.10 DIMENSIONAL ANALYSIS CORRELATION FOR COOKING

A TURKEY

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 using

◦

(1) scaling analysis since this procedure leads

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

ever, the preceding sections indicated that carrying out an

◦

(1) scaling analysis

can be somewhat complicated and time consuming. In contrast, the scaling analy-

sis 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 can in some

cases reduce the number of dimensionless groups. This ﬁrst example of the use

of scaling for dimensional analysis in heat-transfer applications will provide more

details on the steps involved. We will also compare the results of scaling analysis

to the results obtained from using the Pi theorem, to 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 4.2 through 4.9, since no

attempt is made to achieve

◦

(1) scaling.

This ﬁrst example of the use of scaling analysis for dimensional analysis in heat

transfer will consider developing a correlation for determining the cooking time

of a turkey. In particular, we seek to determine how long it will take to cook the

28-lb (12.7-kg) turkey shown in Figure 4.10-1. Cookbooks do not provide equations

to determine the cooking time. Rather, they usually provide discrete data for the

required cooking time t

as a function of the mass of the turkey M, such as shown

in the table that accompanies Figure 4.10-1. A problem arises in that this table does

not indicate in any precise way how much time is required to cook a 28-lb turkey.

A crude way to estimate this time might be to do some type of extrapolation from

188 APPLICATIONS IN HEAT TRANSFER

Ready to cook weight

M (lb)

5 to 8

8 to 12

12 to 16

16 to 20

20 to 24

Total cooking tim

(hr)

3 to 3½

3½ to 4½

4½ to 5½

5½ to 6½

6½ to 7

Figure 4.10-1 Schematic of a very large turkey of a characteristic length L along with

cookbook data for the cooking time t

as a function of the mass of the turkey M.(Data

from General Mills, Betty Crocker’s Cookbook, Golden Press, New York, 1972.)

the data given in this table. However, a better way is to use these data and scaling

analysis to determine the underlying correlation between the weight of the turkey

and the cooking time.

The time it takes to fully cook the turkey is that required to bring the center of the

bird from its initial temperature T

up to the temperature T

speciﬁed by the cook-

book (typically, 165

◦

For73.9

◦

C) by placing it in a preheated oven maintained at a

temperature T

(typically 325

◦

F or 163

◦

C). It is reasonable to assume that the heat

transfer is controlled by conduction in the turkey. To determine the time required

to reach this temperature, in principle we would have to solve the unsteady-state

heat-conduction equation. This is complicated by the fact that the center of the

turkey is usually ﬁlled with dressing that has physical properties that differ from

those of the turkey itself. A further complication is that appropriate boundary con-

ditions need to be speciﬁed on the surface of the turkey and at the interface between

the turkey and the dressing along with appropriate conditions at the center of the

stuffed turkey. Indeed, this would be a complicated problem to solve numerically!

However, we will see that with the aid of the data given in the cookbook, we can

determine the cooking time without having to solve the describing equations.

We begin by writing the equations we would solve for the cooking time, if indeed

we could solve these equations (step 1 in the scaling procedure for dimensional

analysis); namely, the unsteady-state heat-conduction equations in both the turkey

proper and in the dressing in its interior, which are given in generalized vector

notation by equation (B.4-2) in the Appendices:

∂T

∂t

= α

∇

T (applicable in the turkey) (4.10-1)

∂T

∂t

= α

∇

T (applicable in the dressing) (4.10-2)

DIMENSIONAL ANALYSIS CORRELATION FOR COOKING A TURKEY 189

where ∇

denotes the Laplacian operator and α

≡ k

/ρ

and α

≡ k

are the thermal diffusivities of the turkey and dressing, respectively, in

which k

,ρ

,andC

are the thermal conductivity, density, and heat capacity of

medium i, respectively. Note that we have chosen to write the thermal energy

equation in generalized notation that is not speciﬁc to any particular coordinate

system; this is convenient to do in dimensional analysis since there is no need to

scale speciﬁc spatial derivatives to be

◦

(1). The initial and boundary conditions

are given by

T = T

at t = 0 (4.10-3)

T = T

at the surface of the turkey (4.10-4)

T |

= T |

−

at the interface between the turkey and dressing (4.10-5)

∇T |

= k

∇T |

−

at the interface between the turkey and dressing

(4.10-6)

n ·∇T = 0 along the plane of symmetry in the turkey (4.10-7)

where + and − denote the turkey and dressing side of the interface, respectively,

and n is a unit vector normal to the plane of symmetry in the turkey.

To solve equations (4.10-1) through (4.10-7), we would need to describe math-

ematically the surface of the turkey and its interface with the dressing; this would

be prohibitively difﬁcult to do in practice. However, in dimensional analysis this

challenging task can be avoided by recognizing that it is reasonable to assume

that all turkeys are geometrically similar. By this we mean that although their

mass might differ, their geometry is more or less the same. Assume that it takes

p geometric parameters to characterize the shape of a turkey. Recall that one can

form p − 1 dimensionless ratios from p quantities having one dimension (length in

this case). For geometrically similar turkeys, these p − 1 dimensionless geometric

ratios will be the same. Hence, one needs to include in the dimensional analysis

only one geometric quantity such as some characteristic body dimension along with

the quantities that characterize the heat transfer. This arbitrary characteristic length

is chosen to be the maximum body width L as shown in Figure 4.10-1.

Steps 2 and 3 in the scaling procedure for dimensional analysis involve deﬁning

arbitrary scale factors for all the dependent and independent variables and reference

factors for those not naturally referenced to zero. Hence, we introduce the following

dimensionless variables:

∗

≡

T − T

;∇

∗

≡ L∇; ∇

2∗

≡ L

∇

; t

∗

≡

(4.10-8)

Note that we have chosen L for our length scale since we arbitrarily chose it to form

the p − 1 geometric ratios that deﬁne the surface of the turkey and its interface

with the dressing.

Steps 4 and 5 involve introducing these dimensionless variables into the describ-

ing equations and dividing through by the dimensional coefﬁcient of one term in

190 APPLICATIONS IN HEAT TRANSFER

each equation. In dimensional analysis it makes no difference which term you

choose. These steps yield the following dimensionless describing equations:

∂T

∗

∂t

∗

∇

∗2

∗

(applicable in the turkey) (4.10-9)

∂T

∗

∂t

∗

∇

∗2

∗

(applicable in the dressing) (4.10-10)

∗

− T

at t

∗

= 0 (4.10-11)

∗

− T

at the surface of the turkey (4.10-12)

∗

= T

∗

−

at the interface between the turkey and dressing (4.10-13)

∇

∗

∇

∗

−

at the interface between the turkey and dressing

(4.10-14)

n ·∇

∗

= 0 along the plane of symmetry in the turkey (4.10-15)

Step 6 involves setting the various groups equal to 1 or zero to determine the

scale and reference factors, respectively. It makes no difference in dimensional anal-

ysis which groups we set equal to 1 since there is no attempt to achieve

◦

(1) scaling.

We need to do the latter only if we are seeking to simplify the equations by dropping

one or more of the terms. In dimensional analysis we are seeking to determine the

minimum parametric representation. Let us set equation (4.10-11) equal to zero

to determine the reference temperature and set equation (4.10-12) equal to 1 to

determine the temperature scale. Finally, let us set the dimensionless group in

equation (4.10-9) equal to 1 to obtain the time scale; we could equally well have

set the dimensionless group in equation (4.10-10) equal to 1 to determine this time

scale. These choices then yield the following minimum parametric representation of

the describing equations; that is, in terms of the minimum number of dimensionless

groups:

∂T

∗

∂t

∗

=∇

∗2

∗

(applicable in the turkey) (4.10-16)

∂T

∗

∂t

∗

∇

∗2

∗

(applicable in the dressing) (4.10-17)

∗

= 0att

∗

= 0 (4.10-18)

∗

= 1 at the surface of the turkey (4.10-19)

∗

= T

∗

−

at the interface between the turkey and dressing (4.10-20)

∇

∗

∇

∗

−

at the interface between the turkey and dressing

(4.10-21)

DIMENSIONAL ANALYSIS CORRELATION FOR COOKING A TURKEY 191

n ·∇

∗

= 0 along the plane of symmetry in the turkey (4.10-22)

The solution to equations (4.10-16) through (4.10-22) for the dimensionless tem-

perature as a function of the dimensionless time then will be of the form

∗

= f



∗



(4.10-23)

However, we seek the particular value of the dimensionless time t

∗

at which the

center of the turkey reaches the temperature T

. The center of the turkey is located

at some speciﬁc values of the dimensionless coordinates x

∗

, y

∗

,andz

∗

. Hence, our

correlation for the dimensionless cooking time is given by

∗

− T

= f



∗



(4.10-24)

An equivalent statement is

∗

= f



− T



(4.10-25)

Hence, for speciﬁed cooking conditions and geometrically similar turkeys and

dressing with speciﬁed physical and transport properties, we conclude that t

∝ L

It is reasonable to assume for geometrically similar turkeys that the characteristic

length L will be proportional to the mass of the turkey; that is,

L = AM

(4.10-26)

When equation (4.10-26) is substituted into equation (4.10-25), we obtain the fol-

lowing equation that can be used to correlate the data given in Table 4.10-1 for the

cooking time as a function of turkey weight:





− T



= A





− T



= A



(4.10-27)

where A



and B



are empirical constants that will be determined by ﬁtting the

cooking-time data given in the cookbook as a function of the mass of the turkey.

Figure 4.10-2 shows a plot of the cooking time t

as a function of the mass M

of the turkey. The data points represent average values of the cooking time and

turkey mass for each table entry in Figure 4.10-1; the corresponding error bars

then represent the maximum deviation of the values in Figure 4.10-1 about these

average values for each of the ﬁve data points. The trend line in Figure 4.10-2 ﬁts

the data with a regression coefﬁcient R

= 0.994 and is given by the equation

= M

0.61

(4.10-28)

192 APPLICATIONS IN HEAT TRANSFER

0 2 4 6 8 1012141618202224

Turkey mass, M (lb)

Cooking time, t

(hrs)

= M

0.61

= 0.994

Figure 4.10-2 Cooking time in hours as a function of turkey mass in pounds. The solid

line shows the correlation suggested by using the scaling approach to dimensional analysis.

(Data from General Mills, Betty Crocker’s Cookbook, Golden Press, New York, 1972.)

Note that if the turkey body were perfectly spherical with a diameter L, the exponent

in equation (4.10-28) would be 0.67. Equation (4.10-28) now permits determining

the time required to cook a 28-lb turkey, which is found to be 7.6 hours.

Equation (4.10-25) indicates that the cooking time can be correlated in terms

of four dimensionless groups and p − 1 geometric ratios required to specify the

shape of a turkey. It is of interest to explore the possible consequences of using

the Pi theorem to develop a correlation for the cooking time. The Pi theorem

approach would require ﬁrst somehow identifying the quantities that enter into the

correlation; these would include t

,α

,L,T

,andT

. Note that the

quantity L can be expressed in terms of the mass M of the turkey via a relation of

the form of equation (4.10-26). These nine quantities are expressed in terms of four

units: length, time, energy, and temperature (note that energy is a fundamental unit

in the system of thermodynamics when no exchange is involved between internal

and mechanical energy). Hence, we might conclude that ﬁve dimensionless groups

are required to correlate the cooking time (i.e., n − m = 9 −4 = 5) in addition

to p − 1 geometric aspect ratios when, in fact, our scaling analysis indicated that

only four dimensional groups are needed in addition to the p − 1 aspect ratios. A

more enlightened use of the Pi theorem cleverly might recognize that the quantities

,andT

can be combined into the single variable (T

− T

)/(T

− T

),which

then leads to the conclusion that only three dimensionless groups (n − m = 7 −4 =

3) plus the p − 1 geometric aspect ratios are required. However, this conclusion

drawn from the Pi theorem is error since our scaling analysis indicates that the

minimum parametric representation involves four dimensionless groups. This error

arises owing to a breakdown of the Pi theorem associated with the quantities