A HEAT TRANSFER TEXTBOOK - THIRD EDITION Episode 2 Part 8 docx

§8.4 Natural convection in other situations 417Figure 8.6 The data of many investigators for heat transfer from isothermal horizontal cylinders during natural convec-tion, as correlated

Variable-properties problem

Sparrow and Gregg [8.7] provide an extended discussion of the influence

of physical property variations on predicted values of Nu They found

that while β for gases should be evaluated at T ∞, all other properties

should be evaluated at T r, where

T r = T w − C (T w − T ∞ ) (8.28)and where C = 0.38 for gases Most books recommend that a simple

mean between T w and T ∞ (or C = 0.50) be used A simple mean seldom

differs much from the more precise result above, of course

It has also been shown by Barrow and Sitharamarao [8.8] that when

β ∆T is no longer  1, the Squire-Eckert formula should be corrected as

This same correction can be applied to the Churchill-Chu correlation or

to other expressions for Nu Since β = 1 T ∞for an ideal gas, eqn (8.29)gives only about a 1.5% correction for a 330 K plate heating 300 K air

Note on the validity of the boundary layer approximations

The boundary layer approximations are sometimes put to a rather vere test in natural convection problems Thermal b.l thicknesses areoften fairly large, and the usual analyses that take the b.l to be thin can

se-be significantly in error This is particularly true as Gr se-becomes small.Figure8.5includes three pictures that illustrate this These pictures areinterferograms (or in the case of Fig.8.5c, data deduced from interfer-ograms) An interferogram is a photograph made in a kind of lightingthat causes regions of uniform density to appear as alternating light anddark bands

Figure8.5a was made at the University of Kentucky by G.S Wang and

R Eichhorn The Grashof number based on the radius of the leadingedge is 2250 in this case This is low enough to result in a b.l that islarger than the radius near the leading edge Figure8.5b and c are fromKraus’s classic study of natural convection visualization methods [8.9].Figure8.5c shows that, at Gr= 585, the b.l assumptions are quite unrea-

sonable since the cylinder is small in comparison with the large region

of thermal disturbance

a A 1.34 cm wide flat plate with a

rounded leading edge in air T w =

46.5 ◦C,∆T = 17.0 ◦C, Grradius 2250

b A square cylinder with a fairly low

value of Gr (Rendering of an ogram shown in [8.9].)

interfer-c Measured isotherms around a cylinder

in airwhen GrD ≈ 585 (from [8.9])

Figure 8.5 The thickening of the b.l during natural

con-vection at low Gr, as illustrated by interferograms made on

two-dimensional bodies (The dark lines in the pictures are

isotherms.)

415

The analysis of free convection becomes a far more complicated lem at low Gr’s, since the b.l equations can no longer be used We shallnot discuss any of the numerical solutions of the full Navier-Stokes equa-tions that have been carried out in this regime We shall instead note thatcorrelations of data using functional equations of the form

prob-Nu= fn(Ra, Pr)

will be the first thing that we resort to in such cases Indeed, Fig.8.3veals that Churchill and Chu’s equation (8.27) already serves this purpose

re-in the case of the vertical isothermal plate, at low values of Ra≡ Gr Pr.

8.4 Natural convection in other situationsNatural convection from horizontal isothermal cylinders

Churchill and Chu [8.10] provide yet another comprehensive correlation

of existing data For horizontal isothermal cylinders, they find that anequation with the same form as eqn (8.27) correlates the data for hor-izontal cylinders as well Horizontal cylinder data from a variety ofsources, over about 24 orders of magnitude of the Rayleigh number based

on the diameter, RaD, are shown in Fig.8.6 The equation that correlatesthem is

NuD = 0.36 + 0.518 Ra

1/4 D

!

1+ (0.559/Pr) 9/16"4/9 (8.30)

They recommend that eqn (8.30) be used in the range 10−6  Ra D  109.When RaD is greater than 109, the flow becomes turbulent The fol-lowing equation is a little more complex, but it gives comparable accuracyover a larger range:

§8.4 Natural convection in other situations 417

Figure 8.6 The data of many investigators for heat transfer

from isothermal horizontal cylinders during natural

convec-tion, as correlated by Churchill and Chu [8.10]

Example 8.4

Space vehicles are subject to a “g-jitter,” or background variation of

acceleration, on the order of 10−6 or 10−5 earth gravities Brief

pe-riods of gravity up to 10−4 or 10−2 earth gravities can be exerted

by accelerating the whole vehicle A certain line carrying hot oil is

½ cm in diameter and it is at 127◦ C How does Q vary with g-level if

T ∞ = 27 ◦C in the air around the tube?

Solution. The average b.l temperature is 350 K We evaluate

prop-erties at this temperature and write g as g e × (g-level), where g e is g

at the earth’s surface and the g-level is the fraction of g e in the space

g-level NuD h = Nu D



0.0297 0.005

Natural convection from vertical cylinders

The heat transfer from the wall of a cylinder with its axis running cally is the same as that from a vertical plate, so long as the thermal b.l isthin However, if the b.l is thick, as is indicated in Fig.8.7, heat transferwill be enhanced by the curvature of the thermal b.l This correction wasfirst considered some years ago by Sparrow and Gregg, and the analysiswas subsequently extended with the help of more powerful numericalmethods by Cebeci [8.11]

verti-Figure 8.7includes the corrections to the vertical plate results thatwere calculated for many Pr’s by Cebeci The left-hand graph gives acorrection that must be multiplied by the local flat-plate Nusselt number

to get the vertical cylinder result Notice that the correction increaseswhen the Grashof number decreases The right-hand curve gives a similar

correction for the overall Nusselt number on a cylinder of height L Notice

that in either situation, the correction for all but liquid metals is less than

1% if D/(x or L) < 0.02 Gr 1/4 x or L

Heat transfer from general submerged bodies Spheres The sphere is an interesting case because it has a clearly speci-

fiable value of NuD as RaD → 0 We look first at this limit When the

buoyancy forces approach zero by virtue of:

• low gravity, • very high viscosity,

• small diameter, • a very small value of β,

then heated fluid will no longer be buoyed away convectively In that case,only conduction will serve to remove heat Using shape factor number 4

§8.4 Natural convection in other situations 419

Figure 8.7 Corrections for h and h on vertical

isother-mal plates to make them apply to vertical isotherisother-mal

fore has the lead constant, 2, in it.5 A typical example is that of Yuge [8.12]

for spheres immersed in gases:

NuD = 2 + 0.43 Ra 1/4 D , RaD < 105 (8.33)

A more complex expression [8.13] encompasses other Prandtl numbers:

NuD = 2 + 0.589 Ra

1/4 D

!

1+ (0.492/Pr) 9/16"4/9 RaD < 1012 (8.34)This result has an estimated uncertainty of 5% in air and an rms error of

about 10% at higher Prandtl numbers

5 It is important to note that while NuD for spheres approaches a limiting value at

small RaD, no such limit exists for cylinders or vertical surfaces The constants in

eqns ( 8.27 ) and ( 8.30 ) are not valid at extremely low values of Ra

Rough estimate of Nu for other bodies In 1973 Lienhard [8.14] notedthat, for laminar convection in which the b.l does not separate, the ex-pression

Nuτ  0.52 Ra 1/4

would predict heat transfer from any submerged body within about 10%

if Pr is not 1 The characteristic dimension in eqn (8.35) is the length

of travel, τ, of fluid in the unseparated b.l.

In the case of spheres without separation, for example, τ = πD/2, the

distance from the bottom to the top around the circumference Thus, forspheres, eqn (8.35) becomes

This is within 8% of Yuge’s correlation if RaDremains fairly large

Laminar heat transfer from inclined and horizontal plates

In 1953, Rich [8.15] showed that heat transfer from inclined plates could

be predicted by vertical plate formulas if the component of the gravityvector along the surface of the plate was used in the calculation of the

Grashof number Thus, the heat transfer rate decreases as (cos θ) 1/4,

where θ is the angle of inclination measured from the vertical, as shown

in Fig.8.8.Subsequent studies have shown that Rich’s result is substantially cor-rect for the lower surface of a heated plate or the upper surface of acooled plate For the upper surface of a heated plate or the lower surface

of a cooled plate, the boundary layer becomes unstable and separates at

a relatively low value of Gr Experimental observations of such ity have been reported by Fujii and Imura [8.16], Vliet [8.17], Pera andGebhart [8.18], and Al-Arabi and El-Riedy [8.19], among others

instabil-§8.4 Natural convection in other situations 421

Figure 8.8 Natural convection b.l.’s on some inclined and

hor-izontal surfaces The b.l separation, shown here for the

unsta-ble cases in (a) and (b), occurs only at sufficiently large values

of Gr

In the limit θ = 90 ◦— a horizontal plate — the fluid flow above a hot

plate or below a cold plate must form one or more plumes, as shown in

Fig.8.8c and d In such cases, the b.l is unstable for all but small Rayleigh

numbers, and even then a plume must leave the center of the plate The

unstable cases can only be represented with empirical correlations

Theoretical considerations, and experiments, show that the Nusselt

number for laminar b.l.s on horizontal and slightly inclined plates varies

as Ra1/5[8.20, 8.21] For the unstable cases, when the Rayleigh number

exceeds 104 or so, the experimental variation is as Ra1/4, and once the

flow is fully turbulent, for Rayleigh numbers above about 107,

experi-ments show a Ra1/3 variation of the Nusselt number [8.22,8.23] In thelatter case, both NuL and Ra1/3 L are proportional to L, so that the heat transfer coefficient is independent of L Moreover, the flow field in these

situations is driven mainly by the component of gravity normal to theplate

Unstable Cases: For the lower side of cold plates and the upper side

of hot plates, the boundary layer becomes increasingly unstable as Ra isincreased

• For inclinations θ  45 ◦and 105 Ra L  109, replace g with g cos θ

in eqn (8.27)

• For horizontal plates with Rayleigh numbers above 107, nearly tical results have been obtained by many investigators From theseresults, Raithby and Hollands propose [8.13]:

scale L is immaterial Fujii and Imura’s results support using the

above for 60◦  θ  90 ◦ with g in the Rayleigh number.

For high Ra in gases, temperature differences and variable ties effects can be large From experiments on upward facing plates,Clausing and Berton [8.23] suggest evaluating all gas properties at

proper-a reference temperproper-ature, in kelvin, of

Tref= T w − 0.83 (T w − T ∞ ) for 1  T w /T ∞  3.

• For horizontal plates of area A and perimeter P at lower Rayleigh

numbers, Raithby and Hollands suggest [8.13]

scale L ∗ = A/P, is used in the Rayleigh and Nusselt numbers If

§8.4 Natural convection in other situations 423

NuL ∗  10, the b.l.s will be thick, and they suggest correcting the

result to

Nucorrected= 1.4

ln

1+ 1.4 NuL ∗ (8.37b)These equations are recommended6for 1 < Ra L ∗ < 107

• In general, for inclined plates in the unstable cases, Raithby and

Hollands [8.13] recommend that the heat flow be computed first

using the formula for a vertical plate with g cos θ and then using

the formula for a horizontal plate with g sin θ (i.e., the component

of gravity normal to the plate) and that the larger value of the heat

flow be taken

Stable Cases: For the upper side of cold plates and the lower side of hot

plates, the flow is generally stable The following results assume that the

flow is not obstructed at the edges of the plate; a surrounding adiabatic

surface, for example, will lowerh [8.24,8.25]

• For θ < 88 ◦and 105  Ra L  1011, eqn (8.27) is still valid for the

upper side of cold plates and the lower side of hot plates when g

is replaced with g cos θ in the Rayleigh number [8.16]

• For downward-facing hot plates and upward-facing cold plates of

width L with very slight inclinations, Fujii and Imura give:

NuL = 0.58 Ra 1/5

This is valid for 106 < Ra L < 109 if 87◦  θ  90 ◦and for 109 

RaL < 1011 if 89◦  θ  90 ◦ RaL is based on g (not g cos θ).

Fujii and Imura’s results are for two-dimensional plates—ones in

which infinite breadth has been approximated by suppression of

end effects

For circular plates of diameter D in the stable horizontal

configu-rations, the data of Kadambi and Drake [8.26] suggest that

in which Nu turb is calculated from eqn ( 8.36) using L ∗ The formula is useful for

numerical progamming, but its effect on h is usually small.

Natural convection with uniform heat flux

When q w is specified instead of ∆T ≡ (T w − T ∞ ), ∆T becomes the known dependent variable Because h ≡ q w / ∆T , the dependent variable

un-appears in the Nusselt number; however, for natural convection, it alsoappears in the Rayleigh number Thus, the situation is more complicatedthan in forced convection

Since Nu often varies as Ra1/4, we may write

values of ∆T , we can use ∆T evaluated at the midpoint of the plate in

both the Rayleigh number, RaL, and the average Nusselt number, NuL =

q w L/k ∆T Churchill and Chu, for example, show that their vertical plate

correlation, eqn (8.27), represents q w = constant data exceptionally well

in the range RaL > 1 when Ra Lis based on∆T at the middle of the plate.

This approach eliminates the variation of∆T with x from the calculation,

but the temperature difference at the middle of the plate must still befound by iteration

To avoid iterating, we need to eliminate∆T from the Rayleigh number.

We can do this by introducing a modified Rayleigh number, Ra∗ x, definedas

9/164/9

§8.4 Natural convection in other situations 425

Figure 8.9 The mean value of ∆T ≡ T w − T ∞ during natural

lations for laminar natural convection from vertical plates with a uniform

wall heat flux:

Some other natural convection problems

There are many natural convection situations that are beyond the scope

of this book but which arise in practice

Natural convection in enclosures When a natural convection process

occurs within a confined space, the heated fluid buoys up and then lows the contours of the container, releasing heat and in some way re-turning to the heater This recirculation process normally enhances heattransfer beyond that which would occur by conduction through the sta-tionary fluid These processes are of importance to energy conserva-tion processes in buildings (as in multiply glazed windows, uninsulatedwalls, and attics), to crystal growth and solidification processes, to hot

fol-or cold liquid stfol-orage systems, and to countless other configurations.Survey articles on natural convection in enclosures have been written byYang [8.27], Raithby and Hollands [8.13], and Catton [8.28]

Combined natural and forced convection When forced convection along,

say, a vertical wall occurs at a relatively low velocity but at a relativelyhigh heating rate, the resulting density changes can give rise to a super-imposed natural convection process We saw in footnote2on page402

that Gr1/2 L plays the role of of a natural convection Reynolds number, itfollows that we can estimate of the relative importance of natural andforced convection can be gained by considering the ratio

GrL

Re2L = strength of natural convection flow

strength of forced convection flow (8.45)

where ReLis for the forced convection along the wall If this ratio is smallcompared to one, the flow is essentially that due to forced convection,whereas if it is large compared to one, we have natural convection When

GrL Re2L is on the order of one, we have a mixed convection process.

It should be clear that the relative orientation of the forced flow andthe natural convection flow matters For example, compare cool air flow-ing downward past a hot wall to cool air flowing upward along a hot wall

The former situation is called opposing flow and the latter is called

as-sisting flow Opposing flow may lead to boundary layer separation and

degraded heat transfer

Churchill [8.29] has provided an extensive discussion of both the ditions that give rise to mixed convection and the prediction of heat trans-