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 203

Injection of fluid at a constant velocity V

Withdrawal of fluid at a constant velocity V

T = T

at x = 0

T (x

, y)

Figure 4.E.3-1 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separated by a distance H ; the same ﬂuid is injected and withdrawn at a constant velocity

V at the upper and lower plates, respectively; the ﬂuid enters at T

; the upper and lower

plates are maintained at temperatures T

and T

, respectively, where T

laminar and fully developed, the velocity proﬁle is given by

= U

1 −



1 −



(4.E.3-1)

where U

is the maximum velocity. We use scaling to explore simplifying approx-

imations that might be invoked for this heat-transfer problem.

We begin by appropriately simplifying the energy equation given by

equation (F.1-2) in the Appendices (step 1). Note that we must include axial con-

vection, transverse convection, axial conduction, transverse conduction, and viscous

dissipation.

ρC

∂T

∂x

− ρC

∂T

∂y

= k

∂

∂x

+ k

∂

∂y

+ μ





(4.E.3-2)

When equation (4.E.3-1) is substituted into the above, we obtain the following set

of describing equations:

ρC

1 −



1 −



∂T

∂x

− ρC

∂T

∂y

= k

∂

∂x

+ k

∂

∂y

16μU



1 −



(4.E.3-3)

T = T

at x = 0 (4.E.3-4)

T = f(y) at x = L (4.E.3-5)

T = T

at y = H (4.E.3-6)

T = T

at y = 0 (4.E.3-7)

Equation (4.E.3-5) prescribes the downstream boundary condition in terms of the

function f(y), which might be unknown. We use

◦

(1) scaling to explore when

204 APPLICATIONS IN HEAT TRANSFER

the axial conduction, viscous heat generation, and axial convection terms can be

neglected. Introduce the following dimensionless variables involving unspeciﬁed

scale and reference factors (2, 3, and 4):

∗

≡

T − T

; x

∗

≡

; y

∗

≡

(4.E.3-8)

Substitute these dimensionless variables into the describing equations and divide

through by the coefﬁcient of one term in each equation (steps 5 and 6):

ρC



1 −



1 − 2

∗





∂T

∗

∂x

∗

−

ρC

∂T

∗

∂y

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

16μU



1 − 2

∗



(4.E.3-9)

∗

− T

at x

∗

= 0 (4.E.3-10)

∗

= f(y

∗

) at x

∗

(4.E.3-11)

∗

− T

at y

∗

(4.E.3-12)

∗

− T

at y

∗

= 0 (4.E.3-13)

We have divided by the coefﬁcient of the conduction term in the y-direction since

this term must be retained to satisfy the boundary conditions at the upper and lower

plates.

We can bound the dimensionless temperature to be

◦

(1) by setting the dimen-

sionless group in equation (4.E.3-13) equal to zero, thereby determining the ref-

erence temperature, and by setting the dimensionless group in equation (4.E.3-10)

equal to 1 to determine the temperature scale (step 7). We can bound the dimen-

sionless axial and transverse coordinates to be

◦

(1) by setting the dimensionless

groups in equations (4.E.3-11) and (4.E.3-12) equal to 1. Hence, we obtain the

following scale and reference factors:

= T

; T

= T

− T

; x

= L; y

= H (4.E.3-14)

Substitution of these scale and reference factors into equations (4.E.3-9) through

(4.E.3-13) yields

1 − (1 − 2y

∗

)

∂T

∗

∂x

∗

−

ρC

∂T

∗

∂y

∗

(4.E.3-15)

∂

∗

∂x

∗2

∂

∗

∂y

∗2

16μU

k(T

− T

)

(1 − 2y

∗

)

EXAMPLE PROBLEMS 205

∗

= 1atx

∗

= 0 (4.E.3-16)

∗

= f(y

∗

) at x

∗

= 1 (4.E.3-17)

∗

= 1aty

∗

= 1 (4.E.3-18)

∗

= 0aty

∗

= 0 (4.E.3-19)

where Pe

≡ HU

/α is the Peclet number for heat transfer.

The dimensionless describing equations given by equations (4.E.3-15) through

(4.E.3-19) can be simpliﬁed if any of the following conditions are satisﬁed

(step 8):

 1 ⇒ axial heat convection can be ignored (4.E.3-20)

 1 ⇒ axial heat conduction can be ignored (4.E.3-21)

16μU

k(T

− T

)

 1 ⇒ viscous heat generation can be ignored (4.E.3-22)

Note in particular that if the condition deﬁned by equation (4.E.3-21) is satisﬁed,

the elliptic thermal energy equation is reduced to a parabolic differential equation,

thereby obviating the need to satisfy any downstream boundary condition on the

temperature.

Let us now assume that the conditions deﬁned by equations (4.E.3-20) through

(4.E.3-22) are satisﬁed; that is, axial heat conduction and convection as well as

viscous dissipation can be ignored. Hence, only the effects of transverse heat

conduction and convection remain in our describing equations. If the transverse

injection and withdrawal of ﬂuid is sufﬁciently large, one might anticipate that

there will be a region of inﬂuence near the lower plate where essentially all the

heat transfer is occurring. We use scaling analysis to determine the thickness of this

region. In this case the transverse length scale will not be H since the dimension-

less temperature will experience a change of

◦

(1) over a much shorter distance.

Hence, we must rescale the describing equations to determine the thickness δ

of the

region of inﬂuence or thermal boundary layer near the lower plate. Our simpliﬁed

dimensional describing equations now are given by

−ρC

∂T

∂y

= k

∂

∂y

(4.E.3-23)

T = T

at y = H (4.E.3-24)

T = T

at y = 0 (4.E.3-25)

206 APPLICATIONS IN HEAT TRANSFER

If we again introduce the dimensionless variables deﬁned by equations (4.E.3-8),

we obtain the following set of dimensionless describing equations:

−

ρC

∂T

∗

∂y

∗

∂

∗

∂y

∗2

(4.E.3-26)

∗

− T

at y

∗

(4.E.3-27)

∗

− T

at y

∗

= 0 (4.E.3-28)

Our reference and scale factors for the temperature will be the same as before.

Since the two terms in equation (4.E.3-26) must be

◦

(1) to balance each other,

the dimensionless group multiplying the ﬁrst term must be equal to 1; this then

determines the transverse length scale as follows:

ρC

= 1 ⇒ y

≡ δ

ρC

(4.E.3-29)

where we have identiﬁed the transverse length scale with the thickness of the

region of inﬂuence δ

. We see that δ

decreases with increasing injection velocity

and decreasing thermal diffusivity α.

If the injection velocity is sufﬁciently large, δ

will be very small so that

H/δ

→∞. Hence, our scaled dimensionless describing equations reduce to

−

∂T

∗

∂y

∗

∂

∗

∂y

∗2

(4.E.3-30)

∗

= 1aty

∗

→∞ (4.E.3-31)

∗

= 0aty

∗

= 0 (4.E.3-32)

These equations admit an analytical solution given by

∗

= 1 − e

−y

∗

(4.E.3-33)

Note that this solution indicates that T

∗

is bounded of

◦

(1), as it should be if our

scaling analysis was carried out properly.

4.E.4 Steady-State Heat Transfer to Falling Film Flow

Consider the fully developed laminar ﬂow of a Newtonian liquid ﬁlm of thickness

H and constant physical properties down a solid vertical wall maintained at a

temperature T

, as shown in Figure 4.E.4-1. The temperature of the liquid at x = 0

is T

. Assume that negligible heat is transferred to the adjacent gas phase along the

length of the liquid ﬁlm and that viscous heat generation can be ignored. We seek

EXAMPLE PROBLEMS 207

Liquid film

Inviscid gas phase

T (x, y)

(y)

T = T

at x = 0

Figure 4.E.4-1 Steady-state heat transfer from a vertical plate at temperature T

to a liquid

ﬁlm in fully developed laminar ﬂow of initial temperature T

, thickness H , and constant

physical properties.

to determine how the describing equations can be simpliﬁed. The velocity proﬁle

for fully developed laminar ﬁlm ﬂow is given by

= U



−







(4.E.4-1)

where U

is the maximum ﬂuid velocity, that is, the velocity at the liquid–gas

interface.

The thermal energy equation given by the appropriately simpliﬁed form of

equation (F.1-2) in the Appendices, along with the required boundary conditions,

are given by (step 1)



−







∂T

∂x

∂

∂x

∂

∂y

(4.E.4-2)

T = T

at x = 0, 0 ≤ y ≤ H (4.E.4-3)

T = f(y) at x = L, 0 ≤ y ≤ H (4.E.4-4)

T = T

at y = 0, 0 ≤ x ≤ L (4.E.4-5)

∂T

∂y

= 0aty = H, 0 ≤ x ≤ L (4.E.4-6)

where α = k/ρC

is the thermal diffusivity. Equation (4.E.4-2) is an elliptic dif-

ferential equation that requires a downstream boundary condition, which we have

indicated formally by equation (4.E.4-4); in practice, the inability to specify this

208 APPLICATIONS IN HEAT TRANSFER

downstream boundary condition precludes solving problems of this type even

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

∗

≡

T − T

; x

∗

≡

; y

∗

≡

(4.E.4-7)

The resulting dimensionless describing equations are given by (steps 5 and 6)

ρC

kHx



∗

−

∗2



∂T

∗

∂x

∗

∂

∗

∂x

∗2

∂

∗

∂y

∗2

(4.E.4-8)

∗

− T

at x

∗

= 0, 0 ≤ y

∗

≤

(4.E.4-9)

∗

f(y

∗

) − T

at x

∗

, 0 ≤ y

∗

≤

(4.E.4-10)

∗

− T

at y

∗

= 0, 0 ≤ x

∗

≤

(4.E.4-11)

∂T

∗

∂y

∗

= 0aty

∗

, 0 ≤ x

∗

≤

(4.E.4-12)

The dimensionless temperature can be bounded to be

◦

(1) by setting the

dimensionless groups in equations (4.E.4-9) and (4.E.4-11) equal to zero and 1,

respectively, to obtain T = T

and T

= T

− T

(step 7). The streamwise spatial

coordinate can be bounded to be

◦

(1) by setting the appropriate group appear-

ing in equations (4.E.4-10) through (4.E.4-12) to obtain x

= L.Therearetwo

choices for bounding the cross-stream spatial coordinate to be

◦

(1): by setting

the appropriate group in any one of equations (4.E.4-9), (4.E.4-10), and (4.E.4-12)

equal to 1, or by setting the dimensionless group multiplying the convection term

in equation (4.E.4-8) equal to 1. However, the convection term has to balance the

principal conduction term in equation (4.E.4-8); this yields the following equation

for y

ρC

kHx

ρC

kHL

⇒

≡



ρC



1/3





1/3

(4.E.4-13)

We see that y

= δ

is a region of inﬂuence near the hot wall whose thickness

increases slowly with axial length and is inversely proportional to the thermal

Peclet number, Pe

≡ U

H/α. Sufﬁciently far downstream, δ

will become equal

to the ﬁlm thickness H .

For large Peclet numbers the ratio δ

/H will be quite small. This permits signiﬁ-

cant simpliﬁcation of the describing equations (step 8). In particular, the aspect ratio

will be quite small, which permits ignoring the axial conduction term, thereby

avoiding the complication of having to specify a downstream boundary condition.

In addition, for sufﬁciently large Peclet numbers, the quadratic term in the equation

EXAMPLE PROBLEMS 209

for the velocity proﬁle can be ignored since it is only the velocity in the vicinity

of the solid wall that is important for a thin region of inﬂuence. Finally, since

H/δ

 1 for large Peclet numbers, the boundary condition by equation (4.E.4-

12) can be applied at y

∗

→∞. The resulting simpliﬁed dimensionless describing

equations are given by

∗

∂T

∗

∂x

∗

∂

∗

∂y

∗2

(4.E.4-14)

∗

= 0atx

∗

= 0, 0 ≤ y

∗

≤∞ (4.E.4-15)

∗

= 1aty

∗

= 0, 0 ≤ x

∗

≤ 1 (4.E.4-16)

∂T

∗

∂y

∗

= 0 ⇒ T

∗

= 0asy

∗

→∞, 0 ≤ x

∗

≤ 1 (4.E.4-17)

This system of equations admits a solution via a similarity solution or combination

of variables, as is shown in standard references.

4.E.5 Unsteady-State Heat Transfer from a Sphere at Large Biot Numbers

In Section 4.4 we considered heat transfer from a solid sphere initially at temper-

ature T

of radius R falling at its terminal velocity through a liquid at temperature

∞

for which the convective heat transfer was characterized via a heat-transfer

coefﬁcient, as shown in Figure 4.4-1. We considered a scaling appropriate to a

small Biot number for which all the resistance to heat transfer was in the exter-

nal liquid. This implied that the temperature gradient in the sphere was negligible

and the temperature was spatially uniform. Here we apply scaling analysis to the

complementary case of a large Biot number corresponding to rapid heat transfer in

the external liquid. To have continuity of heat ﬂux at the surface of the sphere, the

temperature gradient in the sphere will occur over a region of inﬂuence or thermal

boundary layer whose thickness δ

is the appropriate radial length scale.

The dimensional describing equations will be the same as those in Section 4.4

(step 1):

∂T

∂t

= α

∂

∂r



∂T

∂r



(4.E.5-1)

T = T

at t ≤ 0, 0 ≤ r ≤ R (4.E.5-2)

∂T

∂r

= 0atr = 0,t>0 (4.E.5-3)

∂T

∂r

= h(T

∞

− T) at r = R, t > 0 (4.E.5-4)

R. B. Bird, W. E. Stewart, and E. L. Lightfoot, Transport Phenomena, Wiley, New York, 1960,

pp. 349–350.

210 APPLICATIONS IN HEAT TRANSFER

It is convenient to reference the coordinate system to the surface of the sphere

where the thermal boundary layer is located. Hence, we deﬁne the following dimen-

sionless variables (steps 2, 3, and 4):

∗

≡

T − T

; r

∗

≡

R − r

; t

∗

≡

(4.E.5-5)

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):

αt

∂T

∗

∂t

∗

R/δ

− r

∗

∂

∂r

∗



− r

∗



∂T

∗

∂r

∗

(4.E.5-6)

∗

− T

at t

∗

≤ 0, 0 ≤ r

∗

≤

(4.E.5-7)

∂T

∗

∂r

∗

= 0atr

∗

> 0 (4.E.5-8)

∂T

∗

∂r

∗

=−

hδ



∞

− T

∗



at r

∗

= 0,t

∗

> 0 (4.E.5-9)

When set equal to zero and 1, respectively, the dimensionless groups in equa-

tions (4.E.5-7) and (4.E.5-9), indicate that T

= T

and T

= T

∞

− T

(step 7).

Since this is inherently unsteady state, the appropriate time scale is the observation

time t

. The fact that the unsteady-state and radial heat conduction terms in equa-

tion (4.E.5-6) must be of the same order provides an estimate for δ

αt

= 1 ⇒ δ

√

αt

= R



(4.E.5-10)

where Fo

≡ αt

is the thermal Fourier number. Note that the thermal boundary

layer penetrates progressively farther into the sphere in time. Eventually, it will

penetrate to the center of the sphere when Fo

∼

1, corresponding to an observation

time t

∼

/α.

When these scale and reference factors are substituted into equations (4.E.5-6)

through (4.E.5-9), we obtain the following dimensionless describing equations:

∂T

∗

∂t

∗



√



− r

∗



∂

∂r

∗



√

− r

∗



∂T

∗

∂r

∗

(4.E.5-11)

∗

= 0att

∗

≤ 0, 0 ≤ r

∗

≤

√

(4.E.5-12)

EXAMPLE PROBLEMS 211

∂T

∗

∂r

∗

= 0atr

∗

√

∗

> 0 (4.E.5-13)

∂T

∗

∂r

∗

=−Bi



(1 − T

∗

) at r

∗

= 0,t

∗

> 0 (4.E.5-14)

where Bi

≡ hR/k is the thermal Biot number.

Now let us consider how this system of describing equations can be simpliﬁed

(step 8). We note that the curvature effects can be ignored when

√

 1, cor-

responding to short contact times for which the thermal boundary layer will be

thin in comparison to the radius of the sphere. If

√

 1, the boundary con-

dition given by equation (4.E.5-13) can be applied at inﬁnity. A zero conductive

heat ﬂux far from the surface of the sphere implies no change in the temperature

outside the thermal boundary layer. Hence, equation (4.E.5-13) can be replaced

by the condition that T

∗

= 0asr

∗

→∞. Equation (4.E.5-14) indicates that as

→∞,T

∗

→ 1 to ensure that ∂T

∗

/∂r

∗

remains bounded of

◦

(1). This implies

that the surface temperature of the sphere is at T

∞

and that there is essentially no

temperature gradient in the liquid. Hence, for

√

 1 and large Biot numbers,

the describing equations simplify to

∂T

∗

∂t

∗

∂

∗

∂r

∗2

(4.E.5-15)

∗

= 0att

∗

≤ 0, 0 ≤ r

∗

< ∞ (4.E.5-16)

∗

= 0atr

∗

→∞,t

∗

> 0 (4.E.5-17)

∗

= 1atr

∗

= 0,t

∗

> 0 (4.E.5-18)

This simpliﬁed set of describing equations can be solved via standard methods such

as combination of variables.

4.E.6 Evaporative Cooling of a Liquid Film

An inﬁnitely wide ﬁlm of an incompressible volatile pure (i.e., single compo-

nent) liquid has an initial thickness of L

and is resting on a solid boundary that

is maintained at a constant temperature T

. Initially, the entire liquid ﬁlm is at

this temperature. At time t = 0, evaporation from the ﬁlm begins that causes the

ﬁlm thickness to decrease, thus implying that this is a moving boundary prob-

lem. The surrounding gas phase is assumed to transfer negligible heat to the

liquid ﬁlm. Hence, the latent heat of vaporization (evaporation) must be supplied

entirely by heat conduction from the liquid ﬁlm. This, of course, causes heat trans-

fer within the liquid ﬁlm. The evaporative mass ﬂux at the free surface n

given by

= k

•

◦

(units of mass/area · time) (4.E.6-1)

212 APPLICATIONS IN HEAT TRANSFER

Volatile liquid

Gas

= k

•

p°

L(t)

T = T

Figure 4.E.6-1 Evaporative cooling of an inﬁnitely wide planar ﬁlm of an incompressible

volatile liquid of vapor pressure p

◦

and initial temperature T

, into an adjacent insoluble gas

phase whose mass-transfer coefﬁcient is k

•

; the gas phase is assumed to transfer negligible

heat to the liquid.

in which k

•

is the constant mass-transfer coefﬁcient in the ambient gas phase and

◦

is the temperature-dependent vapor pressure of the liquid at the instantaneous

temperature of the liquid–gas interface. Note that equation (4.E.6-1) assumes that

the bulk of the ambient gas phase does not contain any of the evaporating com-

ponent. For moderate departures of the temperature from T

, the temperature

dependence of the vapor pressure can be approximated by

◦

= p

◦

+ β(T − T

) (4.E.6-2)

where p

◦

is the vapor pressure at T

and β is a constant. We seek to develop a

model for this evaporative cooling, ﬁlm-thinning process, whose essential features

are shown in Figure 4.E.6-1. In particular, we explore conditions for which the

describing equations can be simpliﬁed.

The appropriately simpliﬁed form of equation (F.1-2) in the Appendices along

with the initial condition and boundary condition at x = 0 are given by (step 1)

∂T

∂t

= k

∂

∂x

(4.E.6-3)

T = T

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

T = T

at x = 0 (4.E.6-5)

where k

,ρ

,andC

are the thermal conductivity, density, and heat capacity of

the liquid, respectively. The boundary condition at the moving upper interface is

obtained from an integral energy balance as follows:



(T − T

◦

)dx +



∞

(T − T

◦

)dx = q

(4.E.6-6)

where T

◦

is an arbitrary reference temperature for the enthalpy or heat content

and q

is the heat transferred into the liquid ﬁlm at x = 0. Applying Leibnitz’s