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 213

rule for differentiating an integral given by equation (H.1-2) in the Appendices and

substituting equation (4.E.6-3) yields

−T

◦

)



∂T

∂t

dx−ρ

−T

◦

)



∞

∂T

∂t

dx=q

(4.E.6-7)

(ρ

−ρ

)(T

−T

◦

)



∂

∂x

dx+



∞

∂

∂x

dx=q

(4.E.6-8)

(ρ

− ρ

)(T

− T

◦

)

+ k

∂T

∂x



x=L

− k

∂T

∂x



x=0

∂T

∂x



x=∞

− k

∂T

∂x



x=L

(4.E.6-9)

The ﬁrst term in equation (4.E.6-9) is the difference in heat content between the

gas and the liquid, which is proportional to H

, the latent heat of vaporization

(energy/mass). The ﬁfth term in this equation is assumed to be zero, whereas the

third term is equal to the last term. Hence, our boundary condition at the moving

free interface simpliﬁes to

∂T

∂x

= H

at x = L (4.E.6-10)

Equation (4.E.6-10) merely states that the heat conducted to the moving interface

is equal to that required to vaporize the liquid.

An auxiliary condition is still needed to determine the location of the moving

interface. Since mass is lost at this moving boundary, this auxiliary condition is

obtained from an integral mass balance given by



dx +



∞

dx = 0 (4.E.6-11)

Applying Leibnitz’s rule for differentiating an integral given by equation (H.1-

2) in the Appendices and substituting the one-dimensional form of the continuity

equation given by equation (C.1-1) in the Appendices yields

−



∂ρ

∂x

dx − ρ

−



∞

∂ρ

∂x

dx = 0

(4.E.6-12)

(

− ρ

)

− ρ

x=L

+ ρ

x=0

− ρ

x=∞

+ ρ

x=L

= 0

(4.E.6-13)

214 APPLICATIONS IN HEAT TRANSFER

where u

denotes the velocity in the x-direction. Due to the incompressibility of

the liquid, the second and third terms in equation (4.E.6-13) cancel. The fourth

term is identically zero, and the last term is equal to the evaporative mass-transfer

ﬂux n

= k

•

◦

+ β(T − T

)]. Hence, the auxiliary condition for determining

the location of the moving boundary simpliﬁes to

(

− ρ

)

∼

= n

= k

•

◦

+ β

(

T − T

)

at x = L

(4.E.6-14)

Equation (4.E.6-14) merely states that the rate of mass loss in the liquid ﬁlm is

equal to the rate of evaporation at the free surface. This equation requires an initial

condition given by

L = L

at t = 0 (4.E.6-15)

We anticipate that for very short times, the heat transfer will be conﬁned to

a thin region of inﬂuence near the moving interface. Hence, to explore the full

spectrum of possible simpliﬁcations of the describing equations, it is convenient to

carry out a coordinate transformation whereby we locate the origin of our spatial

coordinate at the liquid–gas interface. Deﬁne a new spatial coordinate as follows:

˜x ≡ L(t) − x (4.E.6-16)

The describing equations in this new coordinate system are given by

∂T

∂t

∂T

∂ ˜x

= α

∂

∂ ˜x

(4.E.6-17)

T = T

at t = 0 (4.E.6-18)

T = T

at ˜x = L (4.E.6-19)

−k

∂T

∂ ˜x

= H

at ˜x = 0 (4.E.6-20)

= k

•

◦

− β(T − T

)]at˜x = 0 (4.E.6-21)

L = L

at t = 0 (4.E.6-22)

Note that the coordinate transformation introduces a pseudo-convection term in the

energy equation.

We ﬁrst seek to determine the criteria for ignoring the temperature dependence

of the vapor pressure and assuming that this evaporative cooling process is at

quasi-steady-state. Hence, introduce the following scale and reference factors

EXAMPLE PROBLEMS 215

(steps 2, 3, and 4):

∗

T − T

;˜x

∗

≡

˜x

; t

∗

≡

; L

∗

≡

;





∗

≡

(4.E.6-23)

Note that we have given dL/dt its own scale since there is no reason for it to

scale as ˜x

. Substitute these dimensionless variables into equations (4.E.6-17)

through (4.E.6-22) to obtain the following set of dimensionless describing equations

(steps 5 and 6):

˜x

∂T

∗

∂t

∗

˜x





∗

∂T

∗

∂ ˜x

∗

∂

∗

∂ ˜x

∗2

(4.E.6-24)

∗

− T

at t

∗

= 0 (4.E.6-25)

∗

− T

at ˜x

∗

˜x

(4.E.6-26)

−

∂T

∗

∂ ˜x

∗

H

˜x





∗

at ˜x

∗

= 0 (4.E.6-27)





∗

•

◦

− β

(

∗

+ T

− T

)

at ˜x

∗

= 0 (4.E.6-28)

∗

at t

∗

= 0 (4.E.6-29)

The dimensionless temperature can be referenced to zero by setting the group in

either equation (4.E.6-25) or (4.E.6-26) equal to zero. For sufﬁciently long times

such that the heat conduction penetrates through nearly the entire liquid ﬁlm, the

dimensionless spatial coordinate can be bounded between zero and 1 by setting the

appropriate group in equation (4.E.6-26) equal to 1. The instantaneous liquid ﬁlm

thickness can bounded between zero and 1 by setting the dimensionless group in

equation (4.E.6-29) equal to 1. Since this is an unsteady-state problem, the time

scale is the observation time t

. We can ensure that the dimensionless temperature

derivative is

◦

(1) by setting the dimensionless group in equation (4.E.6-27) equal

to 1; this also determines the temperature scale factor. We can ensure that the

dimensionless velocity of the free surface is

◦

(1) by setting the dimensionless

group corresponding to the leading-order term in equation (4.E.6-28) equal to 1;

this also determines the scale for the free surface velocity. This results in the

following dimensionless variables:

∗

(T − T

)

H

•

◦

;˜x

∗

˜x

; t

∗

; L

∗

;





∗

≡

•

◦

(4.E.6-30)

216 APPLICATIONS IN HEAT TRANSFER

These variables result in the following set of dimensionless describing equations:

∂T

∗

∂t

∗

•

◦





∗

∂T

∗

∂ ˜x

∗

∂

∗

∂ ˜x

∗2

(4.E.6-31)

∗

= 0att

∗

= 0 (4.E.6-32)

∗

= 0at˜x

∗

= 1 (4.E.6-33)

−

∂T

∗

∂ ˜x

∗





∗

at ˜x

∗

= 0 (4.E.6-34)





∗

= 1 −

•

βH

∗

at ˜x

∗

= 0 (4.E.6-35)

∗

= 1att

∗

= 0 (4.E.6-36)

where Fo

≡ α

is the Fourier number for heat transfer.

Equation (4.E.6-35) indicates that the temperature dependence of the vapor pres-

sure can be ignored if the following criterion is satisﬁed (step 8):

•

βH

 1 (4.E.6-37)

Note that any factors that reduce the evaporative cooling favor satisfy this crite-

rion; these include a reduced gas-phase mass-transfer coefﬁcient, a smaller heat

of vaporization, and a higher thermal conductivity. Note also that this criterion

becomes progressively easier to satisfy as time increases, owing to the presence

of L in the numerator. Equation (4.E.6-31) indicates that quasi-steady-state will

prevail if the following criterion is satisﬁed:

 1 (4.E.6-38)

If quasi-steady-state applies, equations (4.E.6-31) through (4.E.6-35) can be solved

analytically for both constant as well as variable vapor pressure. However, the initial

condition given by equation (4.E.6-36) cannot be applied, due to the long-time

constraint implied by equation (4.E.6-38).

The quasi-steady-state approximation applies at very long observation times. It

would be of value to explore whether a complementary short-time approximation

can be developed as well. We anticipate that a short-time solution would apply when

the heat transfer is conﬁned to a thin region of inﬂuence δ

near the upper moving

interface. The dimensionless temperature goes through a change of

◦

(1) over the

thickness δ

; therefore, to bound our spatial coordinate to be

◦

(1), we choose our

length scale factor ˜x

= δ

. Hence, our dimensionless describing equations assume

EXAMPLE PROBLEMS 217

the following form:

∂T

∗

∂t

∗

•

◦





∗

∂T

∗

∂ ˜x

∗

∂

∗

∂ ˜x

∗2

(4.E.6-39)

∗

= 0att

∗

= 0 (4.E.6-40)

∗

= 0at˜x

∗

(4.E.6-41)

−

∂T

∗

∂ ˜x

∗





∗

at ˜x

∗

= 0 (4.E.6-42)





∗

= 1 −

•

βH

∗

at ˜x

∗

= 0 (4.E.6-43)

∗

= 1att

∗

= 0 (4.E.6-44)

Since we are considering simpliﬁcations appropriate to very short observation times,

this is inherently an unsteady-state problem. This implies that the ﬁrst and third

terms in equation (4.E.6-39) must be retained; this provides a measure of the thick-

ness of the region of inﬂuence given by

= 1 ⇒ δ

√

(4.E.6-45)

Since for very short times δ

 L, the boundary condition given by equation

(4.E.6-41) can be applied at inﬁnity. This permits obtaining an analytical solution

using the method of combination of variables if the temperature dependence of the

vapor pressure can be ignored. For this short observation time scaling, the criterion

for ignoring the temperature dependence of the vapor pressure is given by

•

βH

 1 (4.E.6-46)

Since for short times δ

 L, the criterion given by equation (4.E.6-46) is much

easier to satisfy than that given by equation (4.E.6-37), which is applicable at

longer times. Note also that the pseudo-convection term in equation (4.E.6-39) can

be dropped if the following criterion is satisﬁed:

•

◦

•

◦



 1 (4.E.6-47)

This example illustrates one of the principal values of scaling in arriving at

appropriately simpliﬁed forms of the describing equations that admit either analyt-

ical or much simpler numerical solutions. In this case we were able to simplify the

describing equations so that analytical solutions could be obtained for both very

short and very long observation times. These analytical solutions could be used to

check the validity of a numerical solution to the complete describing equations.

218 APPLICATIONS IN HEAT TRANSFER

T (x

, y)

∞

< T

Figure 4.E.7-1 Buoyancy-induced free-convection ﬂow next to a vertical heated plate of

temperature T

immersed in an inﬁnite ﬂuid whose temperature far from the plate is T

∞

;

sketch shows the temperature proﬁles at two positions along the plate, x

and x

,where

, and the velocity proﬁle of the component u

parallel to the plate.

4.E.7 Free-Convection Heat Transfer Adjacent to a Vertical Heated

Flat Plate

Consider a vertical ﬂat plate of length L and temperature T

that is immersed in

an initially quiescent ﬂuid of temperature T

∞

that can be assumed to have

inﬁnite extent in all directions, as shown in Figure 4.E.7-1. The ﬂuid next to the

heated plate will become less dense than the ﬂuid farther removed from it. Hence,

a hydrostatic pressure imbalance will occur that causes ﬂuid near the plate to

rise; in contrast to the free-convection problem considered in Section 4.9, no ﬂuid

will descend in this case, due to the assumption of inﬁnite extent. We consider this

convective ﬂow after the transients have died out when steady-state free convection

prevails. We ignore end effects at the top and bottom ends of the plate and viscous

dissipation and assume constant physical properties other than the density in the

gravitational body-force term in the equations of motion.

Note that this is inherently a developing ﬂow, due to the progressive heating

that occurs as the ﬂuid moves up the plate; therefore, velocity components in both

the x-andy-directions must be considered. Hence, equations (C.1-1), (D.1-10),

(D.1-11), and (F.1-2) in the Appendices simplify to (step 1)

ρu

∂u

∂x

+ ρu

∂u

∂y

=−

∂P

∂x

+ μ



∂

∂x

∂

∂y



− ρg (4.E.7-1)

ρu

∂u

∂x

+ ρu

∂u

∂y

=−

∂P

∂y

+ μ



∂

∂x

∂

∂y



(4.E.7-2)

∂u

∂x

∂u

∂y

= 0 (4.E.7-3)

ρC

∂T

∂x

+ ρC

∂T

∂y

= k



∂

∂x

∂

∂y



(4.E.7-4)

EXAMPLE PROBLEMS 219

The corresponding boundary conditions are given by

= 0,u

= 0,T= T

∞

at x = 0 (4.E.7-5)

= f

(y), u

= f

(y), T = f

(y) at x = L (4.E.7-6)

= 0,u

= 0,T= T

at y = 0 (4.E.7-7)

= 0,u

= 0,T= T

∞

at y →∞ (4.E.7-8)

where f

(y), f

(y),andf

(y) are functions of y that often are unknown. Since the

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

we consider small density variations and hence represent the density by means of a

Taylor series expansion about the density ρ

∞

at the cold temperature T

∞

given by

ρ = ρ|

∞

∂ρ

∂T



∞

(T − T

∞

) = ρ

∞

− ρ

∞

(T − T

∞

) (4.E.7-9)

where β

is the coefﬁcient of thermal volume expansion. In addition, it is conve-

nient to split the pressure into dynamic, P

, and hydrostatic, P

, contributions as

follows

P = P

(x, y) + P

(x) (4.E.7-10)

When equations (4.E.7-9) and (4.E.7-10) are substituted into equation (4.E.7-1),

we obtain

∞

∂u

∂x

+ ρ

∞

∂u

∂y

=−

∂P

∂x

+ μ



∂

∂x

∂

∂y



+ ρ

∞

g(T − T

∞

)

(4.E.7-11)

Note that the ρ

∞

g term does not appear in equation (4.E.7.11) since it cancels

with the derivative of the purely hydrostatic contribution to the pressure. Note that

higher-order effects of the temperature on the density are ignored in the convection

terms; that is, the density appearing in the convection terms is evaluated at T

∞

and

hence is denoted by ρ

∞

Deﬁne the following dimensionless dependent and independent variables (steps

2, 3, and 4):

∗

≡

; u

∗

≡

; P

∗

≡

; T

∗

≡

T − T

;

∗

≡

; y

∗

≡

; y

∗

≡

(4.E.7-12)

Splitting the pressure into its dynamic and hydrostatic contributions is standard practice when the

latter does not cause the ﬂow; this permits eliminating the contribution of the gravitational body force

from the describing equations as seen in the present example.

220 APPLICATIONS IN HEAT TRANSFER

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

and equations of motion since 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 by

the dimensional coefﬁcient of one term that should be retained to maintain physical

signiﬁcance (steps 5 and 6):

∞

∗

∂u

∗

∂x

∗

∞

∗

∂u

∗

∂y

∗

=−

μu

∂P

∗

∂x

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

gβ

∞



∗

− T

∞



(4.E.7-13)

∞

∗

∂u

∗

∂x

∗

∞

∗

∂u

∗

∂y

∗

=−

μu

∂P

∗

∂y

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.E.7-14)

∂u

∗

∂x

∗

∂u

∗

∂y

∗

= 0 (4.E.7-15)

∗

∂T

∗

∂x

∗

+ u

∗

∂T

∗

∂y

∗

∞

∂

∗

∂x

∗2

∞

∂

∗

∂y

∗2

(4.E.7-16)

∗

= 0,u

∗

= 0,T

∗

∞

− T

at x

∗

= 0 (4.E.7-17)

∗

= f

∗

), u

∗

= f

∗

), T

∗

= f

∗

) at x

∗

(4.E.7-18)

∗

= 0,u

∗

= 0,T

∗

− T

at y

∗

= y

∗

= 0 (4.E.7-19)

∗

= 0,u

∗

= 0,T

∗

∞

− T

at y

∗

= y

∗

→∞ (4.E.7-20)

where ν

∞

≡ μ/ρ

∞

is the kinematic viscosity and α

∞

≡ k/ρ

∞

is the thermal

diffusivity.

For this ﬂow the effect of viscosity will be conﬁned to a thin region near the

vertical plate; hence, the y-length scale for the equations of motion will be the thick-

ness of the momentum boundary layer or region of inﬂuence δ

;thatis,y

= δ

(step 7). The momentum boundary-layer thickness is obtained by balancing the

convection terms with the principal viscous term in equation (4.E.7-13). Similarly,

the effect of heat conduction will be conﬁned to a thin region δ

, although not nec-

essarily of the same thickness as that of the momentum boundary layer; hence, the

y-length scale for the energy equation will be the thickness of the thermal boundary

layer or region of inﬂuence δ

;thatis,y

= δ

. The thermal boundary-layer thick-

ness is obtained by balancing the transverse heat-convection and heat-conduction

terms in equation (4.E.7-16). To determine the axial velocity scale u

, we need

to balance what causes the ﬂow with the principal resistance to ﬂow; the former

is the gravitationally induced body force; the latter is the principal viscous term.

EXAMPLE PROBLEMS 221

The transverse velocity scale u

is obtained from the continuity equation since

this is inherently a developing ﬂow. One might be tempted to obtain P

from the

dimensionless group multiplying the pressure term in equation (4.E.7-13). How-

ever, the pressure term in equation (4.E.7-13) does not cause the free convection

ﬂow; the latter is caused by the gravitational body-force term in this equation.

However, the pressure does cause the ﬂow in the y-direction, which is the rea-

son why we determine its scale by setting the dimensionless group containing P

in equation (4.E.7-14) equal to 1. The axial length scale and temperature refer-

ence and scale factors are obtained from the boundary conditions as described

in Sections 3.4 and 4.6. These considerations then result in the following scale

factors:

= (gβ

T L)

0.5

; u



∞

gβ

T



0.25

; P



gβ

T



0.5

;

= T

− T

∞

≡ T ; x

= L; y

= δ

0.5

0.25

;

(4.E.7-21)

= δ

0.5

Pr · Gr

0.25

where Re ≡ u

L/ν

∞

is the Reynolds number, Pe

≡ u

L/α

∞

= Re · Pr is the

Peclet number for heat transfer, Pr ≡ ν

∞

/α

∞

is the Prandtl number, and Gr

≡

gβ

T /ν

∞

is the Grashof number for heat transfer. Note that the Grashof

number is a measure of the ratio of the free convection to viscous transport of

momentum; as such, it is the analog of the Reynolds number for free convection.

Note that the last of equations (4.E.7-21) indicates that δ

<δ

for normal liquids,

∼

for gases, and δ

>δ

for liquid metals.

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

deﬁned by equations (4.E.7-21), we obtain

∗

∂u

∗

∂x

∗

+ u

∗

∂u

∗

∂y

∗

=−

0.5

∂P

∗

∂x

∗

0.5

∂

∗

∂x

∗2

∂

∗

∂y

∗2

+ T

∗

(4.E.7-22)

∗

∂u

∗

∂x

∗

+ u

∗

∂u

∗

∂y

∗

=−

∂P

∗

∂y

∗

0.5

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.E.7-23)

∂u

∗

∂x

∗

∂u

∗

∂y

∗

= 0 (4.E.7-24)

∗

∂T

∗

∂x

∗

+ u

∗

∂T

∗

∂y

∗

0.5

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.E.7-25)

The thermal Rayleigh number, deﬁned as Ra

≡ L

∞

gβ

T /ν

∞

= Gr

· Pr, is another important

dimensionless group that appears in free-convection problems; it is a measure of the ratio of the free

convection to viscous transport of heat; as such, it is the analog of the thermal Peclet number for free

convection.

222 APPLICATIONS IN HEAT TRANSFER

∗

= 0,u

∗

= 0,T

∗

= 0atx

∗

= 0 (4.E.7-26)

∗

= f

∗

), u

∗

= f

∗

), T

∗

= f

∗

) at x

∗

= 1 (4.E.7-27)

∗

= 0,u

∗

= 0,T

∗

= 1aty

∗

= y

∗

= 0 (4.E.7-28)

∗

= 0,u

∗

= 0,T

∗

= 0aty

∗

= y

∗

=∞ (4.E.7-29)

We now can consider how these scaled dimensionless describing equations can

be simpliﬁed (step 8). Note that if the Grashof number is very large, such that

0.5

 1, the pressure and axial viscous momentum transfer terms can be dropped

from equation (4.E.7-22). The former simpliﬁcation implies that the x-component

is decoupled from the solution to the y-component of the equations of motion;

hence, the latter equation can be ignored. Dropping the axial viscous momen-

tum transfer term from equation (4.E.7-22) converts it from an elliptic into a

parabolic differential equation; this obviates the need to specify downstream bound-

ary conditions that in many cases are unknown. A very large Grashof number also

implies that the axial heat-conduction term can be dropped from equation (4.E.7-

23), which also converts it from an elliptic into a parabolic differential equation,

again avoiding the need to specify a downstream boundary condition. The resulting

describing equations that are applicable in the limit of very large Grashof number

are given by

∗

∂u

∗

∂x

∗

+ u

∗

∂u

∗

∂y

∗

∂

∗

∂y

∗2

+ T

∗

(4.E.7-30)

∂u

∗

∂x

∗

∂u

∗

∂y

∗

= 0 (4.E.7-31)

∗

∂T

∗

∂x

∗

+ u

∗

∂T

∗

∂y

∗

∂

∗

∂y

∗2

(4.E.7-32)

∗

= 0,u

∗

= 0,T

∗

= 0atx

∗

= 0 (4.E.7-33)

∗

= 0,u

∗

= 0,T

∗

= 1aty

∗

= y

∗

= 0 (4.E.7-34)

∗

= 0,u

∗

= 0,T

∗

= 0aty

∗

= y

∗

→∞

(4.E.7-35)

These simpliﬁed describing equations are often given in standard references

with little or no justiﬁcation.

Scaling analysis clearly provides a systematic

method for developing these simpliﬁed equations and for understanding their lim-

itations. The latter are explored further in the practice problems at the end of the

chapter.

See, for example, Bird et al., Transport Phenomena, 2nd ed., pp. 346–349; note that equations (4.E.7-

30) and (4.E.7-33) differ from those in Transport Phenomena due to allowing for different radial length

scales in the equations of motion and energy equation.