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 223

4.E.8 Dimensional Analysis Correlation for Electrical Heat Generation

in a Wire

In Example Problem 4.E.2, we applied scaling analysis to unsteady-state radial

heat conduction in a wire due to electrical heat generation. Assume now that we

wish to develop a correlation for the instantaneous average temperature or spatially

averaged temperature of the wire

T(t) shown in Figure 4.E.2-1, which is deﬁned

T(t)≡

πR



T(r,t)2πr dr (4.E.8-1)

Let us recast equation (4.E.8-1) in terms of the dimensionless variables deﬁned in

Example Problem 4.E.2, given by

∗

≡

T − T

; r

∗

≡

; t

∗

≡

αt

(4.E.8-2)

T(t)≡



∗

)2r

∗

+ T

(4.E.8-3)

Employing the results of our scaling analysis for Example Problem 4.E.2 obviates

the need to apply steps 1 through 7 in the scaling analysis procedure for dimensional

analysis outlined in Chapter 2. Equation (4.E.8-3) can be rearranged in form

T(t)− T

]

≡ T

∗

) =



∗

)2r

∗

(4.E.8-4)

Equation (4.E.8-4) implies that

∗

, the dimensionless average temperature, is a

function of t

∗

, the dimensionless time, and any dimensionless groups that enter into

the solution for T

∗

). Equation (4.E.2-15) indicates that once the transients

have died out, only one additional dimensionless group is involved in the solution

for T

∗

): namely, ωR

/α. Hence, the correlation for the instantaneous average

temperature will involve two dimensionless groups and the dimensionless time;

that is,

∗

≡

T − T

)

= f



∗

;

ωR



(4.E.8-5)

Now let us assume that we seek a correlation for

the maximum in the

spatially averaged temperature ﬂuctuation. In principal, this would be obtained by

setting the time derivative of the spatially averaged temperature equal to zero and

determining the corresponding maximum. Hence, the correlation for

∗

, the dimen-

sionless maximum in the average temperature, will involve only two dimensionless

groups; that is,

∗

≡

− T

)

= f



ωR



(4.E.8-6)

224 APPLICATIONS IN HEAT TRANSFER

However, if the dimensionless group ωR

/α  1, corresponding to very low fre-

quency currents, thin wires, or wires having a very high thermal conductivity, the

right-hand side of equation (4.E.8-6) can be expanded in a Taylor series and trun-

cated at the ﬁrst term (step 9). For this special case the dimensionless maximum

spatially averaged temperature is a constant; that is,

∗

≡

− T

)

= a constant (4.E.8-7)

It is instructive to compare the results of scaling for dimensional analysis to those

of the Pi theorem. A naive application of the Pi theorem for correlating the spatially

averaged temperature

T would indicate that four dimensionless groups were neces-

sary; this follows from having nine dimensional quantities (

T,T

,ω,t,R,k,ρ,

and C

) in ﬁve units (mass, length, time, energy, and temperature). However, scal-

ing analysis reveals that the spatially averaged temperature can be correlated with

only three dimensionless groups. The Pi theorem will yield this same result if one

recognizes that the quantities k, ρ,andC

can be combined into a single quantity α

and that

T and T

can be combined into a single quantity T − T

. This reduces the

number of quantities to seven and the number of units to four, thereby indicating

three dimensionless groups. Scaling analysis avoids having to invoke the subtle

arguments required to ensure that the Pi theorem yields the minimum parametric

representation.

4.P PRACTICE PROBLEMS

4.P.1 Steady-State Conduction in a Slab with a Speciﬁed Cooling Flux

Consider steady-state heat conduction in the solid slab considered in Section 4.2

and shown in Figure 4.2-1. The boundary conditions at x = 0,x = W ,andy = H

remain the same; however, the constant-temperature boundary condition at y = 0

is replaced by a constant-heat-ﬂux condition given by q

=−q

where q

> 0.

(a) Use scaling analysis to determine the appropriate temperature scales.

(b) Determine the criterion for ignoring lateral heat conduction.

4.P.2 Steady-State Conduction in a Slab with a Speciﬁed Heat Flux

Consider steady-state heat conduction in the solid slab considered in Section 4.2

and shown in Figure 4.2-1. The boundary conditions at x = 0,x = W ,andy = 0

remain the same; however, the constant-temperature boundary condition at y = H

is replaced by a constant-heat-ﬂux condition given by q

=−q

where q

> 0.

(a) Use scaling analysis to determine the appropriate temperature scales.

(b) Determine the criterion for ignoring lateral heat conduction.

PRACTICE PROBLEMS 225

Figure 4.P.3-1 Steady-state multidimensional heat conduction in a solid rectangular par-

allelepiped of width W ,depthD, and height H and constant physical properties.

4.P.3 Steady-State Heat Conduction in a Rectangular Parallelepiped

Consider the solid rectangular parallelepiped of width W ,depthD, and height H

and constant physical properties shown in Figure 4.P.3-1.

(a) Write the appropriate form of the energy equation along with the boundary

conditions required.

(b) Use scaling analysis to determine the criterion for ignoring conduction in

the x-direction.

the z-direction.

(d) Derive an equation for the thickness of the region of inﬂuence near the

sidewalls at x = 0andx = W wherein lateral heat conduction cannot be

ignored in predicting quantities such as the temperature or heat ﬂux near

these boundaries.

(e) Derive an equation for the thickness of the region of inﬂuence near the

sidewalls at z = 0andz = D wherein lateral heat conduction cannot be

ignored in predicting quantities such as the temperature or heat ﬂux near

these boundaries.

4.P.4 Steady-State Conduction in a Cylinder with Speciﬁed Temperatures

at Its Boundaries

Consider the steady-state heat conduction in a solid cylinder of radius R and con-

stant physical properties due to a high temperature T

applied at z = 0 and a low

temperature T

applied at z = L as well as at the lateral surface as shown in

Figure 4.P.4-1.

(a) Write the appropriate form of the energy equation along with the boundary

conditions required.

(b) Use scaling analysis to determine the criterion for ignoring radial conduc-

tion.

circumferential boundary of the cylinder wherein radial heat conduction

226 APPLICATIONS IN HEAT TRANSFER

Figure 4.P.4-1 Steady-state heat conduction in a solid circular cylinder due to a high

temperature T

applied at z = 0 and a low temperature T

applied at z = L and at the

circumferential boundary at r = R.

cannot be ignored in predicting quantities such as the temperature or heat

ﬂux near this boundary.

4.P.5 Steady-State Conduction in an Annulus with Speciﬁed Temperatures

at Its Boundaries

Consider steady-state heat conduction in a solid with an annular cross-sectional area

and constant physical properties due to a high temperature T

applied at its inner

surface at R

and a low temperature T

applied at its outer surface at R

as well

as at the two ends of the cylinder at z = 0andz = L as shown in Figure 4.P.5-1.

(a) Write the appropriate form of the energy equation and the boundary condi-

tions required.

at r = R

, T = T

at r = R

, T = T

T = T

Figure 4.P.5-1 Steady-state heat conduction in a solid annulus with constant physical prop-

erties whose inner surface at R

is held at a high temperature T

and whose outer surface

at R

and ends at z = 0andz = L are held at a low temperature T

PRACTICE PROBLEMS 227

(b) Use scaling analysis to determine the criterion for ignoring axial conduction;

be certain to include a reference scale for the dimensionless radial coordinate

since it is not naturally referenced to zero for an annulus.

ends of the annulus wherein axial heat conduction cannot be ignored in pre-

dicting quantities such as the temperature or heat ﬂux near this

boundary.

4.P.6 Steady-State Heat Conduction in a Circular Fin

A solid metallic circular ﬂat ﬁn with a constant thermal conductivity k, width H ,

and radius R

is attached to a cylindrical pipe having a radius R

that is maintained

at a constant temperature T

,whereR

− R

 H . The convective heat-transfer

ﬂux q from the surfaces of the ﬁn to the ambient air is described via the lumped-

parameter approach and is given by

q = h(T − T

∞

) (4.P.6-1)

where h is the heat-transfer coefﬁcient and T

∞

is the temperature of the ambi-

ent air far removed from the ﬁn. A schematic of this ﬁn is shown in Figure

4.P.6-1.

(a) Write the appropriate form of the energy equation and the boundary condi-

tions required.

(b) Determine the criterion for assuming that the temperature is uniform across

the thickness of the ﬁn.

T = T

at r = R

T = T

∞

in ambient gas phase

Figure 4.P.6-1 Solid ﬂat circular metallic ﬁn with a constant thermal conductivity k,radius

, and thickness H attached to a cylindrical pipe having a radius R

and temperature T

;

the convective heat-transfer ﬂux from the surfaces of the ﬁn to the ambient air is described

by q = h(T − T

∞

228 APPLICATIONS IN HEAT TRANSFER

imation made in part (b) to obtain an equation for the axial temperature

distribution within the ﬁn.

4.P.7 Unsteady-State Axial Heat Conduction in a Solid Cylinder

A solid cylinder with constant physical properties, radius R, and length L is initially

at a constant temperature T

as shown in Figure 4.P.7-1. At time t = 0thetemper-

ature of the face of this cylinder at z = 0israisedtoT

while the face at z = L is

maintained at T

. The circumferential boundary of the cylinder is perfectly insulated

so that there is no heat transfer in the radial direction.

(a) Write the appropriate form of the energy equation and the initial and bound-

ary conditions required.

(b) Scale the describing equations to determine the criterion for assuming that

steady-state heat transfer is achieved.

which the heat conduction has not penetrated the entire length of the cylin-

der, and determine the criterion for the applicability of this approximation.

(d) Derive an equation for the region of inﬂuence or thermal penetration for the

conditions in part (c).

Figure 4.P.7-1 Unsteady-state axial heat conduction in a solid cylinder with constant phys-

ical properties, a perfectly insulated circumferential boundary, and an initial temperature T

due to a high temperature T

being imposed on the face at z = 0 while the face at z = L is

maintained at T

4.P.8 Unsteady-State Radial Heat Conduction in a Solid Cylinder

A solid cylinder with constant physical properties, radius R, and length L is ini-

tially at a constant temperature T

as shown in Figure 4.P.8-1. At time t = 0the

temperature of the circumferential boundary of this cylinder at r = R is raised to

. The ends of the cylinder at z = 0andz = L are perfectly insulated so that

there is no heat transfer in the axial direction.

(a) Write the appropriate form of the energy equation and the initial and bound-

ary conditions required.

PRACTICE PROBLEMS 229

Solid cylinder initially at T

T = T

, t > 0

Figure 4.P.8-1 Unsteady-state axial heat conduction in a solid cylinder with constant phys-

ical properties, insulated ends, and an initial temperature T

due to a high temperature T

being imposed on the face at r = R.

(b) Scale the describing equations to determine the criterion for assuming that

thermal equilibrium has been achieved.

which the thermal energy has not penetrated entirely through the cylinder

and determine the criterion for the applicability of this approximation.

(d) Derive an equation for the region of inﬂuence or thermal penetration for the

conditions in part (c).

4.P.9 Unsteady-State Radial Heat Conduction in a Spherical Shell

A solid spherical shell with constant physical properties, an inner radius R

,and

an outer radius R

is initially at a constant temperature T

, as shown in Figure

4.P.9-1. At time t = 0 the temperature of the inner boundary at r = R

is raised

to T

and convective heat transfer to a surrounding ﬂowing ﬂuid whose bulk tem-

perature is maintained at T

occurs at the outer boundary at R

; the heat ﬂux

at the outer boundary is described by a constant heat-transfer coefﬁcient and is

given by

= h(T −T

) at r = R

(4.P.9-1)

(a) Write the appropriate form of the energy equation and the initial and bound-

ary conditions required.

(b) Scale the describing equations to determine the criterion for justifying that

steady-state heat transfer can be assumed. Note that it is necessary to intro-

duce a separate scale for the radial temperature gradient since this derivative

does not necessarily scale with the characteristic temperature scale divided

by the length scale; also introduce a reference factor for the spatial variable

since it is not naturally referenced to zero.

230 APPLICATIONS IN HEAT TRANSFER

T = T

, t > 0

= h(T − T

), t > 0

Figure 4.P.9-1 Unsteady-state heat conduction in a solid spherical shell with constant

physical properties, inner radius R

, outer radius R

, and initial temperature T

; at time

t = 0 a high temperature T

is applied at the inner boundary, and convective heat transfer

to the surrounding ﬂuid whose bulk temperature is maintained at T

occurs at the outer

boundary.

ing curvature effects on the heat conduction within the spherical shell.

(d) Solve the simpliﬁed describing equations appropriate to parts (b) and (c) for

the temperature proﬁle in the spherical shell and determine the temperature

at r = R

(e) Scale the describing equations appropriate to very short contact times for

which the thermal energy has not penetrated entirely through the spherical

shell, and determine the criterion for the applicability of this approximation.

(f) Derive an equation for the region of inﬂuence or thermal penetration for the

conditions in part (e).

4.P.10 Steady-State Conduction in a Cylinder with External Phase

Convection

Consider steady-state heat conduction in a solid cylinder of radius R and constant

physical properties due to a high temperature T

applied at z = 0 and a low temper-

ature T

applied at z = L, as shown in Figure 4.P.4-1. However, the circumferential

boundary at r = R is exposed to convective heat transfer in an external ﬂuid phase;

the heat ﬂux in this external phase is described by a lumped-parameter condition

given by

= h(T − T

∞

) (4.P.10-1)

where T

∞

is the temperature in the bulk of the external ﬂuid phase far removed

from the solid cylinder.

(a) Write the appropriate form of the energy equation and the boundary condi-

tions required.

(b) Use scaling analysis to determine the criterion for ignoring radial conduc-

tion. Introduce a separate scale for the radial derivative of the temperature

PRACTICE PROBLEMS 231

since there is no reason to assume that it will scale with the characteristic

temperature scale divided by the characteristic radial length scale.

be ignored everywhere within the solid cylinder, even near the circumfer-

ential boundary; that is, there is no region of inﬂuence in this case near the

circumferential boundary. Explain why there is no region of inﬂuence in this

case, where a lumped-parameter heat-ﬂux boundary condition is prescribed

at the lateral boundaries, whereas there would be a region of inﬂuence if a

constant temperature were prescribed at this boundary.

4.P.11 Unsteady-State Heat Transfer to a Sphere at Small Biot Numbers

In Section 4.4 we considered convective heat transfer to a solid sphere falling at its

terminal velocity through a ﬂuid. Scaling analysis was used to obtain a criterion for

achieving steady-state, which in this case meant thermal equilibrium between the

sphere and the liquid; this criterion is given by equation (4.4-17). However, if the

Biot number is very small, the describing equations can be simpliﬁed and solved

analytically; the resulting solution for the dimensionless temperature is given by

equation (4.4-23).

(a) Use the solution obtained for the small Biot number approximation to deter-

mine the dimensionless time required for the temperature to reach 1% of

its ﬁnal equilibrium temperature and compare this result to the criterion we

derived for achieving steady-state.

(b) The small Biot number approximation implies that the temperature within

the sphere is nearly uniform. Use the solution for the small Biot number

approximation and the results of the scaling analysis in Section 4.4 to esti-

mate the dimensionless time required for the dimensionless temperature at

the center of the sphere to differ from the surface temperature by less than

1.0%.

4.P.12 Unsteady-State Heat Transfer in a Solid Sphere

In Section 4.4 we considered unsteady-state heat transfer to a solid sphere that was

initially at temperature T

due to falling at its terminal velocity through a hot liquid

at temperature T

∞

. We used scaling analysis to arrive at the criterion for assuming

that the temperature of the sphere was essentially uniform across its radius at any

time. This corresponded to the small Biot number approximation, for which the

heat transfer is controlled by the external liquid phase. The exact analytical solution

for this heat-transfer problem is given by

T − T

∞

− T

∞



n=1

−λ

sin

(4.P.12-1)

232 APPLICATIONS IN HEAT TRANSFER

where Fo

≡ αt/R

is the Fourier number for heat transfer and C

is a coefﬁcient

whose value is obtained from

4(sin λ

− λ

cos λ

)

2λ

− sin 2λ

(4.P.12-2)

for which the λ

are the positive roots of

1 − λ

cot λ

= Bi

(4.P.12-3)

where Bi

= hR/k is the Biot number for heat transfer. Compare the approxi-

mate solution that we obtained for the small Biot number approximation given

by equation (4.4-23) to the exact solution given by the above for Bi

= 0.01 and

= 0.1; for the case of the exact solution, compute the temperature at the center

of the sphere.

4.P.13 Unsteady-State Convective Heat Transfer to a Plane Wall

Consider an inﬁnitely long solid plane wall that is initially at a high temperature

with constant physical properties and thickness 2H . This wall is cooled on both

sides by a ﬂowing liquid whose bulk temperature far removed from the wall is T

∞

The heat transfer in the cold liquid is characterized by a heat-transfer coefﬁcient

h. A schematic of this heat-transfer problem is shown in Figure 4.P.13-1.

(a) Write the appropriate form of the energy equation along with the initial and

boundary conditions required.

Solid wall

T = T

, t ≤ 0

T = T

∞

T = T

∞

Figure 4.P.13-1 Inﬁnitely long solid plane wall with constant physical properties, thickness

2H , and initial temperature T

subject to convective cooling at its lateral boundaries by a

liquid whose temperature is T

∞

far from the wall.