Bird R.B., Stewart W.E., Lightfoot E.N. Transport Phenomena

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s10.7

Heat Conduction in a Cooling Fin

307

Region 23:

At the two fluid-solid interfaces we can write Newton's law of cooling:

Surface

Surface 3:

Addition of the preceding five equations gives an equation for

T,.

Then the equation is

solved for

to give

We now define an "overall heat transfer coefficient based on the inner surface"

Combination of the last two equations gives, on generalizing to

system with

annular

layers,

The

subscript

"0"

indicates that the overall heat transfer coefficient is referred to the

radius

ro.

510.7

HEAT

CONDUCTION IN A COOLING FIN'

Another simple, but practical application of heat conduction is the calculation of the effi-

ciency of a cooling fin. Fins are used to increase the area available for heat transfer be-

tween metal walls and poorly conducting fluids such as gases. A simple rectangular fin

is shown

Fig.

10.7-1.

The wall temperature is

and the ambient air temperature is

T,.

wall temperature

known to be

Fig.

10.7-1.

A simple cooling fin with

B<<LandB<<

For further information

fins,

see

Jakob,

Heat Transfer,

Vol.

Wiley, New York

(19491,

Chapter

11;

and

Baehr and

Stephan,

Heat and Mass Transfeu,

Springer, Berlin

(1998),

52.2.3.

308

Chapter

Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

reasonably good description of the system may be obtained by approximating the

true physical situation by a simplified model:

True situation Model

is a function of

and z, but the

dependence on

is most important.

small quantity of heat is lost from the

fin at the end (area 2BW) and at the

edges (area (2BL

2BL).

The heat transfer coefficient is a function

of position.

T is a function of z alone.

heat is lost from the end or from the

edges.

The heat flux at the surface is given

h(T

T,), where

is constant and

depends on z.

The energy balance is made over a segment

of the bar. Since the bar is stationary, the

terms containing

in the combined energy flux vector

may be discarded, and the only

contribution to the energy flux is

Therefore the energy balance is

Division by

2BW

and taking the limit as

approaches zero gives

We now insert Fourier's law

(q,

-kdT/dz), in which

is the thermal conductivity

the metal. If we assume that

is constant, we then get

This equation is to be solved with the boundary conditions

B.C.

1: at

0, T

B.C.

2: atz=L,

dTO

We now introduce the following dimensionless quantities:

@=--

dimensionless temperature

T',

dimensionless distance

hL2

dimensionless heat transfer coefficient2

(10.7-8)

The problem then takes the form

d2@

N20

with

@Ii=.

and

(10.7-9,10,11)

dl2

The quantity may be rewritten as

(hL/k)(L/B)

Bi(L/B), where Bi is called the Biot

number,

named after Jean Baptiste Biot

(1774-1862)

(pronounced "Bee-oh"). Professor of physics at

the

CollPge de France, he received the Rumford Medal for his development of

simple, nondestructive test

to determine sugar concentration.

s10.7 Heat Conduction in a Cooling Fin

309

Equation 10.7-9 may be integrated to give hyperbolic functions (see Eq. C.l-4 and

9C.5).

When the two integration constants have been determined, we get

cosh

(tanh

sinh

(10.7-12)

This may be rearranged to give

cosh N(1

cosh

This result is reasonable only if the heat lost at the end and at the edges is negligible.

The "effectiveness" of the fin surface is defined3 by

actual rate of heat loss from the

fin

rate of heat loss from an isothermal fin at

T,.

(10.7-14)

For the problem being considered here

is then

in which

is the dimensionless quantity defined in Eq. 10.7-8.

Fig.

10.7-2 a thermocouple is shown in a cylindrical well inserted into a gas stream. Esti-

mate the true temperature of the gas stream if

Error

Thermocouple

Measurement

500°F

temperature indicated by thermocouple

350°F

wall temperature

120 Btu/hr. ft2

heat transfer coefficient

60 Btu/hr ft3

thermal conductivity of well wall

0.08 in.

thickness of well wall

0.2 ft

length of well

Thermocouvle wires

Pipe wall

T,.

to potentlometer

Well wall of

thickness

Thermocouple

Fig.

10.7-2.

thermocouple in a cylindrical

junction at

well.

Jakob,

Heat Transfer,

Vol.

Wiley, New York

(19491,

235.

310

Chapter 10

Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

SOLUTION

The thermocouple well wall of thickness

is in contact with the gas stream on one side

only,

and the tube thickness is small compared with the diameter. Hence the temperature distribu-

tion along this wall will be about the same as that along a bar of thickness 2B, in contact

with

the gas stream on both sides. According to

Eq.

10.7-13, the temperature at the end of the well

(that registered by the thermocouple) satisfies

Hence the actual ambient gas temperature is obtained by solving this equation for

T,:

and the result is

Therefore, the reading is 10

I?'

too low.

This example has focused on one kind of error that can occur in thermometry.

Fre-

quently a simple analysis, such as the foregoing, can be used to estimate the measurement

errors4

510.8

FORCED CONVECTION

In the preceding sections the emphasis has been placed on heat conduction in solids.

In this and the following section we study two limiting types of heat transport in

flu-

ids:

forced convection

and

free convection

(also called

natural

convection).

The main

dif-

ferences between these two modes of convection are shown in Fig.

10.8-1.

Most

industrial heat transfer problems are usually put into either one or the other

these

two limiting categories. In some problems, however, both effects must be taken

into

account, and then we speak of

mixed convection

(see 514.6 for some empiricisms

for

handling this situation).

In this section we consider forced convection in a circular tube, a limiting case

which is simple enough to be solved analyti~all~.',~

viscous fluid with physical prop-

erties

(p,

Cp)

assumed constant is in laminar flow in a circular tube of radius

For

the fluid temperature is uniform at the inlet temperature

TI.

For

there

constant radial heat flux

-qo

at the wall. Such a situation exists, for example, when

pipe is wrapped uniformly with an electrical heating coil, in which case

is positive.

the pipe is being chilled, then

q,,

has to be taken as negative.

As indicated in Fig. 10.8-1, the first step in solving a forced convection heat transfer

problem is the calculation of the velocity profiles in the system. We have seen in

52.3

For further discussion, see

Jakob,

Heat Transfer,

Vol.

11,

Wiley, New York (1949), Chapter

33,

pp. 147-201.

Eagle and

Ferguson,

Proc. Roy. Soc. (London),

A127,540-566 (1930).

Goldstein,

Modern Developments in

Fluid

Dynamics,

Oxford

University

Press (1938), Dover

Edition (1965),

Vol.

11,

622.

510.8

Forced Convection

311

Forced Convection

Heat Transfer

Heat swept to right by forced

stream of air

The flow patterns are

determined primarily by

some external force

First, the velocity profiles are

found; then they are used to

find the temperature profiles

(usual procedure for fluids

with constant physical

properties)

The Nusselt number depends

on the Reynolds and Prandtl

numbers (see Chapter

14)

Free Convection

Heat Transfer

Heat transported upward by

heated air that rises

The flow patterns are

determined by the buoyant

force on the heated fluid

The velocity profiles and

temperature profiles are

interdependent

The Nusselt number depends

on the Grashof and Prandtl

numbers (see Chapter

14)

Fig.

10.8-1.

comparison of forced and free convection

non-

isothermal systems.

how this may be done for tube flow by using the shell balance method. We know that

the velocity distribution so obtained is v,

0, and

This parabolic distribution is valid sufficiently far downstream from the inlet that the en-

trance length has been exceeded.

In this problem, heat is being transported in both the r and the z directions. There-

fore, for the energy balance we use

"washer-shaped" system, which is formed by the

intersection of an annular region of thickness Ar with a slab of thickness Az (see Fig. 10.8-

2). In this problem, we are dealing with a flowing fluid, and therefore all terms in the

vector will be retained. The various contributions to

Eq.

10.1-1 are

Total energy in at r

e,l,

2mAz

(2me,)Jr Az (10.8-2)

Total energy out at

erlr+hr 27dr

Ar)Az

(2~re,)),+~~

(10.8-3)

Total energy in at z ezlZ -2mAr (10.8-4)

Total energy out at z

eZlz+~,

2m.A~

(10.8-5)

Work done on fluid by gravity pv,g,

2.rruArA.z (10.8-6)

The last contribution is the rate at which work is done on the fluid within the ring by

gravity-that is, the force per unit volume

pg,

times the volume 2m Ar Az multiplied by

the downward velocity of the fluid.

312

Chapter

Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Fluid

inlet

temperature

Fig.

10.8-2.

Heating of a fluid in laminar flow through

cir-

cular tube, showing the annular ring over which the energy

balance is made.

The energy balance is obtained by summing these contributions and equating

the

sum to zero. Then we divide by

Ar Az

to get

the limit as

and

go to zero, we find

The subscript z in

has been omitted, since the gravity vector is acting in the

direc-

tion.

Next we use Eqs. 9.8-6 and 9.8-8 to write out the expressions for the

and z-compo-

nents of the combined energy flux vector, using the fact that the only nonzero compo-

nent of

is the axial component

v,:

Substituting these flux expressions into Eq. 10.8-8 and using the fact that

depends

only

gives, after some rearrangement,

The second bracket is exactly zero, as can be seen from Eq. 3.6-4, which is the z-component

s10.8

Forced Convection

313

term containing the viscosity is the viscous heating, which we shall neglect in this dis-

cussion. The last term in the first bracket, corresponding to heat conduction in the axial

direction, will be omitted, since we know from experience that it is usually small in com-

parison with the heat convection in the axial direction. Therefore, the equation that we

want to solve here is

This partial differential equation, when solved, describes the temperature in the fluid as

a function of r and

The boundary conditions are

B.C. 1: at

finite (10.8-13)

B.C. 2:

at r

(constant)

(10.8-14)

B.C. 3: at

(10.8-15)

We now put the problem statement into dimensionless form. The choice of the dimen-

sionless quantities is arbitrary. We choose

Generally one tries to select dimensionless quantities so as to minimize the number of

parameters in the final problem formulation. In this problem the choice

r/R is a

natural one, because of the appearance of r/R in the differential equation. The choice for

the dimensionless temperature is suggested by the second and third boundary condi-

tions. Having specified these two dimensionless variables, the choice of dimensionless

axial coordinate follows naturally.

The resulting problem statement, in dimensionless form, is now

with boundary conditions

B.C. 1: at

finite (10.8-20)

B.C. 2:

at(= I,

---

(10.8-21)

B.C. 3: at

(10.8-22)

The partial differential equation in Eq. 10.8-19 has been solved for these boundary condi-

tions: but in this section we do not give the complete solution.

It is, however, instructive to obtain the asymptotic solution to Eq. 10.8-19 for large

After the fluid is sufficiently far downstream from the beginning of the heated section,

one expects that the constant heat flux through the wall will result in a rise of the fluid

temperature that is linear in

One further expects that the shape of the temperature pro-

files as a function of

will ultimately not undergo further change with increasing (see

Fig. 10.8-3). Hence a solution of the following form seems reasonable for large 6:

in which

is a constant to be determined presently.

Siegel, E.

Sparrow, and

T. M.

Hallman,

Appl.

Sci.

Research,

A7,386-392

(1958).

See

Example

12.2-1

for

the

complete solution and Example

12.2-2

for

the

asymptotic solution for small

314

Chapter 10

Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Tube

wall

z=o

Region of

small

Region

large

slope at

same for all

Shape of profiles

is same-they

are

displaced

upward with

increasing

Fig.

10.8-3.

Sketch showing how one expects the temperature

T(r,

to look for the system shown in Fig. 10.8-2 when the

fluid is heated by means of a heating coil wrapped uniformly

around the tube (corresponding to

positive).

The function in Eq. 10.8-23 is clearly not the complete solution to the problem;

does allow the partial differential equation and boundary conditions

and

to be satis-

fied, but clearly does not satisfy boundary condition 3. Hence we replace the latter by

integral condition (see Fig. 10.8-41,

Condition

21~Xzq~

/021

loR

T,)u,r dr

0.8-24)

or, in dimensionless form,

This condition states that the energy entering through the walls over a distance

the

same as the difference between the energy leaving through the cross section at

and that

entering at

Substitution of the postulated function of Eq. 10.8-23 into Eq. 10.8-19 leads to the

fol-

lowing ordinary differential equation for

(see Eq. C.l-11):

s10.8

Forced Convection

315

Uniform

Plane at arbitrary

temperature

downstream position

No energy enters here, ~eat'in by Energy leaving here is

since datum temperature heating coil

was chosen to be

2rRzqO

Fig.

10.8-4.

Energy balance used for boundary condition 4

given in

Eq.

10.8-24.

This equation may be integrated twice with respect to

and the result substituted into

Eq.

10.8-23 to give

The three constants are determined from the conditions 1,2, and

above:

B.C. 1:

B.C. 2:

Condition

Substitution of these values into

Eq.

10.8-27 gives finally

Both averages are functions of

The quantity

(T)

is the arithmetic average of the temper-

atures over the cross section at

The "bulk temperature"

is the temperature one

would obtain if the tube were chopped off at

and if the fluid issuing forth were col-

lected in a container and thoroughly mixed. This average temperature is sometimes re-

ferred to as the "cup-mixing temperature" or the "flow-average temperature."

O(&[)

at4

(10.8-31)

This result gives the dimensionless temperature as a function of the dimensionless radial

and axial coordinates. It is exact in the limit as

for

0.1, it predicts the local

value of O to within about

2%.

Once the temperature distribution is known, one can get various derived quantities.

There are two kinds of average t$mperatures commonly used in connection with the

flow of fluids with constant

and

C,:

316

Chapter 10

Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Now let us evaluate the local heat transfer driving force,

Tb, which is the differ-

ence between the wall and bulk temperatures at a distance

down the tube:

where

is the tube diameter. We may now rearrange this result in the form of a dimen-

sionless wall heat flux

which, in Chapter 14, will be identified as a

Nusselt

number.

Before leaving this section, we point out that the dimensionless axial coordinate

in-

troduced above may be rewritten in the following way:

Here

is the tube diameter, Re is the Reynolds number used in Part I, and Pr and

Pi.

are

the Prandtl and Pkclet numbers introduced in Chapter

We shall find in Chapter 11 that

the Reynolds and Prandtl numbers can be expected to appear in forced convection

prob-

lems. This point will be reinforced in Chapter

in connection with correlations for heat

transfer coefficients.

510.9

FREE

CONVECTION

In 510.8 we gave an example of forced convection. In this section we turn our attention

to an elementary free convection problem-namely, the flow between two parallel walls

maintained at different temperatures (see Fig. 10.9-1).

fluid with density

and viscosity

is located between two vertical walls

dis-

tance

apart. The heated wall at

-B

is maintained at temperature

T,,

and the

cooled wall at

is maintained at temperature

TI.

It is assumed that the tempera-

ture difference is sufficiently small that terms containing

(An2

can be neglected.

Because of the temperature gradient in the system, the fluid near the hot wall rises

and that near the cold wall descends. The system is closed at the top and bottom,

that

the fluid is continuously circulating between the plates. The mass rate of flow of

the

Fig.

10.9-1.

Laminar free convection flow between

two vertical plates at two different temperatures.

The

velocity is a cubic function of the coordinate