Computational Fluid Mechanics and Heat Transfer Third Edition_12 docx

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

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:

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 Nuturb 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:

or Ra∗ are given in [8.13])

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-

con-§8.4 Natural convection in other situations 427

fer for it Review articles on the subject have been written by Chen and

Armaly [8.30] and by Aung [8.31]

Example 8.5

A horizontal circular disk heater of diameter 0.17 m faces downward

in air at 27◦C If it delivers 15 W, estimate its average surface

temper-ature

Solution. We have no formula for this situation, so the problem

calls for some judicious guesswork Following the lead of Churchill

and Chu, we replace RaD with Ra∗ D /Nu D in eqn (8.39):

∆T = 1.18 q w D k

gβq w D4kνα

= 140 K

In the preceding computation, all properties were evaluated at T ∞

Now we must return the calculation, reevaluating all properties except

= 142 K

so the surface temperature is 27+ 142 = 169 ◦C.

That is rather hot Obviously, the cooling process is quite

ineffec-tive in this case

8.5 Film condensation

Dimensional analysis and experimental data

The dimensional functional equation for h (or h) during film

as a product, because gravity only enters the problem as it acts upon thedensity difference [cf eqn (8.4)]

The problem is therefore expressed nine variables in J, kg, m, s, and

◦C (where we once more avoid resolving J into N· m since heat is not

being converted into work in this situation) It follows that we look for

9− 5 = 4 pi-groups The ones we choose are

Two of these groups are new to us The group Π3 is called the Jakob

number, Ja, to honor Max Jakob’s important pioneering work during the

1930s on problems of phase change It compares the maximum sensibleheat absorbed by the liquid to the latent heat absorbed The group Π4

does not normally bear anyone’s name, but, if it is multiplied by Ja, itmay be regarded as a Rayleigh number for the condensate film

Notice that if we condensed water at 1 atm on a wall 10◦C below

Tsat, then Ja would equal 4.174(10/2257) = 0.0185 Although 10 ◦C is a

fairly large temperature difference in a condensation process, it gives amaximum sensible heat that is less than 2% of the latent heat The Jakobnumber is accordingly small in most cases of practical interest, and itturns out that sensible heat can often be neglected (There are important

7Note that, throughout this section, k, µ, c p, and Pr refer to properties of the liquid, rather than the vapor.

§8.5 Film condensation 429

exceptions to this.) The same is true of the role of the Prandtl number

Therefore, during film condensation

NuL = fn



 ρ f (ρ f − ρ g )gh fg L3µk(Tsat− T w )



Equation (8.46) is not restricted to any geometrical configuration,

since the same variables govern h during film condensation on any body.

Figure 8.10, for example, shows laminar film condensation data given

for spheres by Dhir8 [8.32] They have been correlated according to

eqn (8.12) The data are for only one value of Pr but for a range of

Π4 and Ja They generally correlate well within ±10%, despite a broad

variation of the not-very-influential variable, Ja A predictive curve [8.32]

is included in Fig.8.10for future reference

Laminar film condensation on a vertical plate

Consider the following feature of film condensation The latent heat of

a liquid is normally a very large number Therefore, even a high rate of

heat transfer will typically result in only very thin films These films move

relatively slowly, so it is safe to ignore the inertia terms in the momentum

This result will give u = u(y, δ) (where δ is the local b.l thickness)

when it is integrated We recognize that δ = δ(x), so that u is not strictly

dependent on y alone However, the y-dependence is predominant, and

it is reasonable to use the approximate momentum equation

8 Professor Dhir very kindly recalculated his data into the form shown in Fig 8.10

for use here.

Figure 8.10 Correlation of the data of Dhir [8.32] for laminarfilm condensation on spheres at one value of Pr and for a range

ofΠ4and Ja, with properties evaluated at (Tsat+ T w )/2

Ana-lytical prediction from [8.33]

This simplification was made by Nusselt in 1916 when he set down theoriginal analysis of film condensation [8.34] He also eliminated the con-vective terms from the energy equation (6.40):

u = (ρ f − ρ g )gδ

2

2µ

2

To complete the analysis, we must calculate δ This can be done in

two steps First, we express the mass flow rate per unit width of film, ˙m,

in terms of δ, with the help of eqn (8.48):

Second, we neglect the sensible heat absorbed by that part of the film

cooled below Tsat and express the local heat flux in terms of the rate of

change of ˙m (see Fig.8.11):

Substituting eqn (8.50) in eqn (8.51), we obtain a first-order

differen-tial equation for δ:

Figure 8.11 Heat and mass flow in an element of a condensing film.

Both Nusselt and, subsequently, Rohsenow [8.35] showed how to rect the film thickness calculation for the sensible heat that is needed to

cor-cool the inner parts of the film below Tsat Rohsenow’s calculation was, inpart, an assessment of Nusselt’s linear-temperature-profile assumption,

and it led to a corrected latent heat—designated h  fg—which accountedfor subcooling in the liquid film when Pr is large Rohsenow’s result,which we show below to be strictly true only for large Pr, was

in the energy equation

The physical properties inΠ4, Ja, and Pr (with the exception of h fg)

are to be evaluated at the mean film temperature However, if Tsat− T w

is small—and it often is—one might approximate them at Tsat

At this point we should ask just how great the missing influence of

Pr is and what degree of approximation is involved in representing the

influence of Ja with the use of h  fg Sparrow and Gregg [8.36] answered

these questions with a complete b.l analysis of film condensation They

did not introduce Ja in a corrected latent heat but instead showed its

influence directly

Figure8.12displays two figures from the Sparrow and Gregg paper

The first shows heat transfer results plotted in the form

Nux

43

Π4 = fn (Ja, Pr)
→ constant as Ja
→ 0 (8.58)Notice that the calculation approaches Nusselt’s simple result for all

Pr as Ja→ 0 It also approaches Nusselt’s result, even for fairly large

values of Ja, if Pr is not small The second figure shows how the

tem-perature deviates from the linear profile that we assumed to exist in the

film in developing eqn (8.49) If we remember that a Jakob number of

0.02 is about as large as we normally find in laminar condensation, it is

clear that the linear temperature profile is a very sound assumption for

nonmetallic liquids

Figure 8.12 Results of the exact b.l analysis of laminar film

condensation on a vertical plate [8.36]

Sadasivan and Lienhard [8.37] have shown that the Sparrow-Gregg

for-mulation can be expressed with high accuracy, for Pr  0.6, by including

Pr in the latent heat correction Thus they wrote

h  fg = h fg

!

1+0.683 − 0.228 Pr Ja"

(8.59)which includes eqn (8.54) for Pr→ ∞ as we anticipated.

§8.5 Film condensation 435

The Sparrow and Gregg analysis proves that Nusselt’s analysis is quite

accurate for all Prandtl numbers above the liquid-metal range The very

high Ja flows, for which Nusselt’s theory requires some correction,

usu-ally result in thicker films, which become turbulent so the exact analysis

no longer applies

The average heat transfer coefficient is calculated in the usual way for

Twall= constant:

h = 1L

Water at atmospheric pressure condenses on a strip 30 cm in height

that is held at 90◦C Calculate the overall heat transfer per meter, the

film thickness at the bottom, and the mass rate of condensation per

1/4

= 0.000138 x 1/4

Then

δ(L) = 0.000102 m = 0.102 mm

Notice how thin the film is Finally, we use eqns (8.56) and (8.59) tocompute

Condensation on other bodies

Nusselt himself extended his prediction to certain other bodies but wasrestricted by the lack of a digital computer from evaluating as many cases

as he might have In 1971 Dhir and Lienhard [8.33] showed how Nusselt’smethod could be readily extended to a large class of problems They

showed that one need only to replace the gravity, g, with an effective gravity, geff:

• g = g(x), the component of gravity (or other body force) along x;

g can vary from point to point as it does in Fig.8.13b and c

influence directly

Figure8.12displays two figures from the Sparrow and Gregg paper

The first shows heat transfer results plotted in the form

Nux

43...

that is held at 90◦C Calculate the overall heat transfer per meter, the

film thickness at the bottom, and the mass rate of condensation per

1/4...

is small? ?and it often is—one might approximate them at Tsat

At this point we should ask just how great the missing influence of

Pr is and what degree of