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Unformatted text preview: ‘4 Edited* Excerpts from HEAT
TRANSFER Sixth Edition J. P. HOLMAN Professor of Mechanical Engineering
Southern Methodist University McGrawHill Book Company
New York St. Louis San Francisco Auckland
Bogota Hamburg London
Madrid Mexico Montreal New Delhi
Panama Paris 8510 Pauio Singapore
Sydney Tokyo Toronto *Some of the notation has been modiﬁed to make it consistent
with the boundary layer text. I 11 CONDUCTION HEAT TRANSFER
When a temperature gradient exists in a body, experience has shown that there
is an energy transfer from the hightemperature region to the lowtemperature
region. We say that the energy is transferred by conduction and that the heat
transfer rate per unit area is proportional to the normal temperature gradient: '2' ~ 2"
A ax
When the proportionality constant is inserted,
6T 5T
‘ = — — =—k— 11
q M ea 4' ax ( ) where E is the heattransfer rate and 61%;: is the temperature gradient in the
direction of the heat ﬂow. The positive constant k is called the thermal con
ductivity of the material, and the minus sign is inserted so that the second
principle of thermodynamics will be satisﬁed; i.e., heat must ﬂow downhill on
the temperature scale, as indicated in the coordinate system of Fig. 11. Equa
tion (1—1) is called Fourier's law of heat conduction after the French mathe
matical physicist Joseph Fourier, who made very signiﬁcant contributions to
the analytical treatment of conduction heat transfer. It is important to note that
Eq. (11) is the deﬁning equation for the thermal conductivity and that k has
the units of watts per meter per Celsius degree in a typical system of units in
which the heat ﬂow is expressed in watts. We now set ourselves the problem of determining the basic equation which
governs the transfer of heat in a solid, using Eq. (11) as a starting point. Consider the onedimensional system shown in Fig. 12. If the system is in
a steady state, i.e., if the temperature does not change with time, then the
problem is a simple one, and we need only integrate Eq. (11) and substitute
the appropriate values to solve for the desired quantity. However, if the tem
perature of the solid is changing with time, or if there are heat sources or sinks
within the solid, the situation is more complex. We consider the general case
where the temperature may be changing with time and heat sources may be T Temperature
proﬁle x Fig. 11 Sketch showing direction of heat flow. Fig. 12 Elemental volume for one—dimensional heat
*‘x—’l dx i“ conduction analysis. present within the body. For the element of thickness dx the following energy
balance may be made: Energy conducted in left face + heat generated within element
= change in internal energy + energy conducted out right face These energy quantities are given as follows: . _. 61"
Energy 1n left face = (1; = kA 5
. . 6T
Change In Internal energy = pcA 5 dx 6x —A [kg + 3 (kg) dx]
6x 6x 6x _. 61"
Energy out right face qﬁdx = *kA —]
x+dx where
c = speciﬁc heat of material, J/kg°C
p = density, kg/m3
Combining the relations above gives
6T T .
—kA— =pcAgdxA[k9—+£(ka—7)dx]
6x 61' 6x 6x 6x 6 6T 6T
_._ k _ = _ 2
or ﬁx ( 6):) pc 6r (1 ) energy balance yields .. _. .. ... ._  dE
Qx + qy + q: qx+dx + qv+dy + qz+dz + If:
and the energy quantities are given by
T
qx — —k dy dza—
6):
3T 6 6
‘2 x = — —— — k— d d
g” [kax+ax(6:)dx]yz
... 6T
qy = —kdx dag
6T 3 6T
‘IIV V = _ k— — —' d
9”“ [6y +ay (key) y] ”2
a
"q: = —kdxdy—T
6Z
6T 3
= — k— + — k— d
Ema: [ 62 az( 636140,": y
dE 8T
:2"; — pc 0'): dy dz “1: 3 6T 6 a a 6 6T
— k— + — k — + ~ k — = — 3
6x( 6x) 6y ( 6:) Hz ( 6:) pc 61 (I )
For constant thermal conductivity Eq. (13) is written
aZT 6% 621" 1 6T
—_ + —_ — = — — _
6x2 6y2 + 622 a 61' (1 3a) I 13 CONVECTION HEAT TRANSFER It is well known that a hot plate of metal will cool faster when placed in front
of a fan than when exposed to still air. We say that the heat is convected away,
and we call the process convection heat transfer. The term convection provides
the reader with an intuitive notion concerning the heat—transfer process; how
ever, this intuitive notion must be expanded to enable one to arrive at anything
like an adequate analytical treatment of the problem. For example, we know
that the velocity at which the air blows over the hot plate obviously inﬂuences
the heattransfer rate. But does it inﬂuence the cooling in a linear way; i.e.,
if the velocity is doubled, will the heattransfer rate double? We should
suspect that the heattransfer rate might be different if we cooled the plate
with water instead of air, but, again, how much difference would there be?
These questions may be answered with the aid of some rather basic anal—
yses presented in later chapters. For now, we sketch the physical mechan—
ism of convection heat transfer and show its relation to the conduction
process. Consider the heated plate shown in Fig. 18. The temperature of the plate
is Tw, and the temperature of the ﬂuid is Tm. The velocity of the ﬂow will appear
as shown, being reduced to zero at the plate as a result of viscous action. Since
the velocity of the fluid layer at the wall will be zero, the heat must be transferred
only by conduction at that point. Thus we might compute the heat transfer,
using Eq. (11), with the thermal conductivity of the ﬂuid and the ﬂuid tem
perature gradient at the wall. Why, then, if the heat ﬂows by conduction in
this layer, do we speak of convection heat transfer and need to consider the
velocity of the ﬂuid? The answer is that the temperature gradient is dependent
on the rate at which the ﬂuid carries the heat away; a high velocity produces
a large temperature gradient, and so on Thus the temperature gradient at the
wall depends on the ﬂow ﬁeld, and we must develop in our later analysis ain ex
pression relating the two quantities. Nevertheless, it must be remembered that
the physical mechanism of heat transfer at the wall is a conduction process. To express the overall effect of convection, we use Newton’s law of cooling:
= MU} — m) €1=h(TwTm) (18) Here the heattransfer rate is related to the overall temperature difference
between the wall and ﬂuid and the surface area A. The quantity h is called the
convection heattransfer coefﬁcient, and Eq. (18) is the deﬁning equation. An
analytical calculation of it may be made for some systems. For complex situ
ations it must be determined experimentally. The heattransfer coefﬁcient is Flow Free stream Fig. 1—3 Convection heat trans
fer from a plate. sometimes called the film conductance because of its relation to the conduction
process in the thin stationary layer of ﬂuid at the wall surface. From Eq. (18)
we note that the units of .h are in watts per square meter per Celsius degree
when the heat ﬂow is in watts. In view of the foregoing discussion, one may anticipate that couvection heat
transfer will have a dependence on the viscosity of the ﬂuid in addition to its
dependence on the thermal properties of the ﬂuid (thermal conductivity, speciﬁc
heat, density). This is expected because viscosity inﬂuences the velocity proﬁle
and, correspondingly, the energy—transfer rate in the region near the wall. If a heated plate were exposed to ambient room air without an external Table 12 Approximate Values of Convection HeatTransfer Coefficients I:
Made W/m2  “C Btui'h  ft2  "F
Free convection, AT = 30°C
Vertical plate 0.3 m [1 ft] high 4.5 0.79
in air
Horizontal cylinder, 5cm diameter, 6.5 1.14
in air
Horizontal cylinder, 2cm diameter, 890 157
in water
Forced convection
Airﬂow at 2 mfs over 0.2—m 12 2.1
square plate
Airﬂow at 35 m/s over 0.75m 75 13.2
square plate
Air at 2 atm ﬂowing in 65 11.4
2.5cmdiameter tube at 10 m/s
Water at 0.5 kg/s ﬂowing in 3500 616
2.5—crndiameter tube
Airﬂow across 5cmdiameter 180 32
cylinder with velocity of 50 m/s
Boiling water
In a pool or container 2500—35,000 440—6200
Flowing in a tube  5000—100,000 BSD17,600
Condensation of water vapor, 1 atm
Vertical surfaces 4000—11,300 700—2000 Outside horizontal tubes 9500—25,000 1700—4400 source of motion, a movement of the air would be experienced as a result of
the density gradients near the plate. We call this natural, or free, convection
as opposed to forced convection, which is experienced in the case of the fan
blowing air over a plate. Boiling and condensation phenomena are also grouped
under the general subject of convection heat transfer. The approximate ranges
of convection heattransfer coefﬁcients are indicated in Table 12. 14 RADIATION HEAT TRANSFER transfer through a material medium is involved, heat may also be transferred
through regions where a perfect vacuum exists. The mechanism in this case is
electromagnetic radiation. We shall limit our discussion to electromagnetic
radiation which is propagated as a result of a temperature difference; this is
called thermal radiation. Thermodynamic considerations show that an ideal thermal radiator, or
blackbody, will emit energy at a rate proportional to the fourth power of the
absolute temperature of the body and directly proportional to its surface area. 3...“... = 0.41“ q = 6T4 (19) where or is the proportionality constant and is called the StefanBoltzmann
constant with the value of 5.669 x 10’8 W/m2  K“. Equation (19) is called q = LA; cc am“ — T2“) {110) We have mentioned that a blackbody is a body which radiates energy ac
cording to the T 4 law. We call such a body black because black surfaces, like _
as a piece of metal covered with carbon black, approximate this type of be
havior. Other types of surfaces, like a glossy painted surface or a polished
metal plate, do not radiate as much energy as the blackbody; however, the
total radiation emitted by these bodies still generally follows the T14 propor
tionality. To take account of the “gray” nature of such surfaces we introduce
another factor into Eq. (19), called the emissivity e, which relates the radiation
of the “gray“ surface to that of an ideal black surface. In addition, we must take into account the fact that not all the radiation leaving one surface will
reach the other surface since electromagnetic radiation travels in straight lines
and some will be lost to the surroundings. We therefore introduce two new
factors in Eq. (19) to take into account both situations, so that I; = FEF00A(T14 ._ T24) (111) where F6 is the emissivity function and F6 is the geometric “view factor”
function. The determination of the form of these functions for speciﬁc conﬁg
urations is the subject of a subsequent chapter. It is important to alert the
reader at this time, however, to the fact that these functions usually are not
independent of one another as indicated in Eq. (111). El Radiation in an Enclosure A simple radiation problem is encountered when we have a heat transfer surface
at temperature T1 completely enclosed by a much larger surface maintained at T2. We will show in Chap. 8 that the net radiant exchange in this case can be
calculated with 73 = EtaAI (Tl4 — T24) (112)
Values of e are given in Appendix A. Radiation heattransfer phenomena can be exceedingly complex, and the
calculations are seldom as simple as implied by Eq. (111). For now, we wish
to emphasize the difference in physical mechanism between radiation heat
transfer and conductionconvection systems. ...
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 Fall '08
 Schetz,J

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