A HEAT TRANSFER TEXTBOOK - THIRD EDITION Episode 2 Part 10 pptx

The bubble will be in mechanical equilibrium when the pressure dif-ference between the inside and the outside of the bubble is balanced by the forces of surface tension, σ , as indicate

Figure 9.4 Enlarged sketch of a typical metal surface.

9.2 Nucleate boilingInception of boiling

Figure9.4shows a highly enlarged sketch of a heater surface Most finishing operations score tiny grooves on the surface, but they also typ-

metal-ically involve some chattering or bouncing action, which hammers small

holes into the surface When a surface is wetted, liquid is prevented bysurface tension from entering these holes, so small gas or vapor pocketsare formed These little pockets are the sites at which bubble nucleationoccurs

To see why vapor pockets serve as nucleation sites, consider Fig.9.5.Here we see the problem in highly idealized form Suppose that a spher-ical bubble of pure saturated steam is at equilibrium with an infinitesuperheated liquid To determine the size of such a bubble, we imposethe conditions of mechanical and thermal equilibrium

The bubble will be in mechanical equilibrium when the pressure

dif-ference between the inside and the outside of the bubble is balanced by

the forces of surface tension, σ , as indicated in the cutaway sketch in

Fig.9.5 Since thermal equilibrium requires that the temperature must

be the same inside and outside the bubble, and since the vapor inside

must be saturated at Tsup because it is in contact with its liquid, theforce balance takes the form

psat at Tsup − pambient (9.1)

The p–v diagram in Fig. 9.5 shows the state points of the internalvapor and external liquid for a bubble at equilibrium Notice that the

external liquid is superheated to (Tsup− Tsat) K above its boiling point at

the ambient pressure; but the vapor inside, being held at just the rightelevated pressure by surface tension, is just saturated

§9.2 Nucleate boiling 465

Figure 9.5 The conditions required for simultaneous

mechan-ical and thermal equilibrium of a vapor bubble

1.256

1− 0.625



1− Tsat T c

mN

where both Tsat and the thermodynamical critical temperature, T c =

647.096 K, are expressed in K The units of σ are millinewtons (mN)

per meter Table9.1gives additional values of σ for several substances.

Equation 9.2a is a specialized refinement of a simple, but quite

ac-curate and widely-used, semi-empirical equation for correlating surface

466

We include correlating equations of this form for CO2, propane, and some

refrigerants at the bottom of Table 9.1 Equations of this general form

are discussed in Reference [9.8]

It is easy to see that the equilibrium bubble, whose radius is described

by eqn (9.1), is unstable If its radius is less than this value, surface

tension will overbalance [psat(Tsup)− pambient] Thus, vapor inside will

condense at this higher pressure and the bubble will collapse If the

bubble radius is slightly larger than the equation specifies, liquid at the

interface will evaporate and the bubble will begin to grow

Thus, as the heater surface temperature is increased, higher and higher

values of [psat(Tsup)−pambient] will result and the equilibrium radius, R b,

will decrease in accordance with eqn (9.1) It follows that smaller and

smaller vapor pockets will be triggered into active bubble growth as the

temperature is increased As an approximation, we can use eqn (9.1)

to specify the radius of those vapor pockets that become active

nucle-ation sites More accurate estimates can be made using Hsu’s [9.9]

bub-ble inception theory, the subsequent work by Rohsenow and others (see,

e.g., [9.10]), or the still more recent technical literature

Example 9.1

Estimate the approximate size of active nucleation sites in water at

1 atm on a wall superheated by 8 K and by 16 K This is roughly in

the regime of isolated bubbles indicated in Fig.9.2

Solution. psat = 1.203 × 105 N/m2at 108◦ C and 1.769 × 105 N/m2

at 116◦ C, and σ is given as 57.36 mN/m at Tsat = 108 ◦C and as

55.78 mN/m at Tsat= 116 ◦C by eqn (9.2a) Then, at 108◦ C, R b from

This means that active nucleation sites would be holes with diametersvery roughly on the order of magnitude of 0.005 mm or 5µm—at least

on the heater represented by Fig 9.2 That is within the range ofroughness of commercially finished surfaces

Region of isolated bubbles

The mechanism of heat transfer enhancement in the isolated bubbleregime was hotly argued in the years following World War II A few con-clusions have emerged from that debate, and we shall attempt to identifythem There is little doubt that bubbles act in some way as small pumpsthat keep replacing liquid heated at the wall with cool liquid The ques-tion is that of specifying the correct mechanism Figure 9.6shows theway bubbles probably act to remove hot liquid from the wall and intro-duce cold liquid to be heated

It is apparent that the number of active nucleation sites generating

bubbles will strongly influence q On the basis of his experiments,

Yam-agata showed in 1955 (see, e.g., [9.11]) that

where∆T ≡ T w − Tsat and n is the site density or number of active sites

per square meter A great deal of subsequent work has been done to

fix the constant of proportionality and the constant exponents, a and b The exponents turn out to be approximately a = 1.2 and b = 1

3.The problem with eqn (9.3) is that it introduces what engineers call

a nuisance variable A nuisance variable is one that varies from system

to system and cannot easily be evaluated—the site density, n, in this case Normally, n increases with ∆T in some way, but how? If all sites were identical in size, all sites would be activated simultaneously, and q

would be a discontinuous function of∆T When the sites have a typical distribution of sizes, n (and hence q) can increase very strongly with ∆T

It is a lucky fact that for a large class of factory-finished materials, n

varies approximately as ∆T5 or 6, so q varies roughly as ∆T3 This has

made it possible for various authors to correlate q approximately for a

large variety of materials One of the first and most useful correlationsfor nucleate boiling was that of Rohsenow [9.12] in 1952 It is

§9.2 Nucleate boiling 469

A bubble growing and departing in saturated liquid.

The bubble grows, absorbing heat from the

superheated liquid on its periphery As it leaves, it

entrains cold liquid onto the plate which then warms

up until nucleation occurs and the cycle repeats

A bubble growing in subcooled liquid.

When the bubble protrudes into coldliquid, steam can condense on the topwhile evaporation continues on thebottom This provides a short-circuit forcooling the wall Then, when the bubblecaves in, cold liquid is brought to the wall

Figure 9.6 Heat removal by bubble action during boiling Dark

regions denote locally superheated liquid

where all properties, unless otherwise noted, are for liquid at Tsat The

constant Csf is an empirical correction for typical surface conditions

Table 9.2 includes a set of values of Csf for common surfaces (taken

from [9.12]) as well as the Prandtl number exponent, s A more extensive

compilation of these constants was published by Pioro in 1999 [9.13]

We noted, initially, that there are two nucleate boiling regimes, and

the Yamagata equation (9.3) applies only to the first of them Rohsenow’s

equation is frankly empirical and does not depend on the rational

anal-ysis of either nucleate boiling process It turns out that it represents

q( ∆T ) in both regimes, but it is not terribly accurate in either one

Fig-ure9.7shows Rohsenow’s original comparison of eqn (9.4) with data for

water over a large range of conditions It shows typical errors in heat

flux of 100% and typical errors in∆T of about 25%.

Thus, our ability to predict the nucleate pool boiling heat flux is poor

Our ability to predict ∆T is better because, with q ∝ ∆T3, a large error

in q gives a much smaller error in ∆T It appears that any substantial

improvement in this situation will have to wait until someone has

man-aged to deal realistically with the nuisance variable, n Current research

efforts are dealing with this matter, and we can simply hope that such

work will eventually produce a method for achieving reliable heat

trans-fer design relationships for nucleate boiling

Table 9.2 Selected values of the surface correction factor for

use with eqn (9.4) [9.12]

It is indeed fortunate that we do not often have to calculate q, given

∆T , in the nucleate boiling regime More often, the major problem is

to avoid exceeding qmax We turn our attention in the next section topredicting this limit

Example 9.2

What is Csffor the heater surface in Fig.9.2?

Solution. From eqn (9.4) we obtain

where, since the liquid is water, we take s to be 1.0 Then, for water at

Tsat = 100 ◦ C: c p = 4.22 kJ/kg·K, Pr = 1.75, (ρ f − ρ g ) = 958 kg/m3,

σ = 0.0589 N/m or kg/s2, h fg = 2257 kJ/kg, µ = 0.000282 kg/m·s.

This value compares favorably with Csffor a platinum or copper

sur-face under water

9.3 Peak pool boiling heat fluxTransitional boiling regime and Taylor instability

It will help us to understand the peak heat flux if we first consider theprocess that connects the peak and the minimum heat fluxes Duringhigh heat flux transitional boiling, a large amount of vapor is gluttedabout the heater It wants to buoy upward, but it has no clearly definedescape route The jets that carry vapor away from the heater in the re-gion of slugs and columns are unstable and cannot serve that function inthis regime Therefore, vapor buoys up in big slugs—then liquid falls in,touches the surface briefly, and a new slug begins to form Figure9.3cshows part of this process

The high and low heat flux transitional boiling regimes are different

in character The low heat flux region does not look like Fig.9.2c but is most indistinguishable from the film boiling shown in Fig.9.2d However,both processes display a common conceptual key: In both, the heater isalmost completely blanketed with vapor In both, we must contend withthe unstable configuration of a liquid on top of a vapor

al-Figure 9.8 shows two commonplace examples of such behavior Ineither an inverted honey jar or the water condensing from a cold waterpipe, we have seen how a heavy fluid falls into a light one (water or honey,

in this case, collapses into air) The heavy phase falls down at one node

of a wave and the light fluid rises into the other node

The collapse process is called Taylor instability after G I Taylor, who first predicted it The so-called Taylor wavelength, λ d, is the length ofthe wave that grows fastest and therefore predominates during the col-lapse of an infinite plane horizontal interface It can be predicted using

dimensional analysis The dimensional functional equation for λ dis

λ d = fnσ , g

ρ f − ρ g



(9.5)since the wave is formed as a result of the balancing forces of surfacetension against inertia and gravity There are three variables involving m

and kg/s2, so we look for just one dimensionless group:

§9.3 Peak pool boiling heat flux 473

a Taylor instability in the surface of the honey

in an inverted honey jar

b Taylor instability in the interface of the water condensing on

the underside of a small cold water pipe

Figure 9.8 Two examples of Taylor instabilities that one might

Experiment 9.3

Hang a metal rod in the horizontal position by threads at both ends.The rod should be about 30 cm in length and perhaps 1 to 2 cm in diam-eter Pour motor oil or glycerin in a narrow cake pan and lift the pan upunder the rod until it is submerged Then lower the pan and watch theliquid drain into it Take note of the wave action on the underside of therod The same thing can be done in an even more satisfactory way byrunning cold water through a horizontal copper tube above a beaker ofboiling water The condensing liquid will also come off in a Taylor wavesuch as is shown in Fig.9.8 In either case, the waves will approximate

λ d1 (the length of a one-dimensional wave, since they are arrayed on aline), but the wavelength will be influenced by the curvature of the rod

Throughout the transitional boiling regime, vapor rises into liquid on

the nodes of Taylor waves, and at qmaxthis rising vapor forms into jets.These jets arrange themselves on a staggered square grid, as shown inFig.9.9 The basic spacing of the grid is λ d2(the two-dimensional Taylorwavelength) Since

[recall eqn (9.6)], the spacing of the most basic module of jets is actually

λ d1, as shown in Fig.9.9.Next we must consider how the jets become unstable at the peak, tobring about burnout

Helmholtz instability of vapor jets

Figure9.10shows a commonplace example of what is called Helmholtz

instability This is the phenomenon that causes the vapor jets to cave in

when the vapor velocity in them reaches a critical value Any flag in abreeze will constantly be in a state of collapse as the result of relativelyhigh pressures where the velocity is low and relatively low pressureswhere the velocity is high, as is indicated in the top view

This same instability is shown as it occurs in a vapor jet wall inFig 9.11 This situation differs from the flag in one important partic-ular There is surface tension in the jet walls, which tends to balance theflow-induced pressure forces that bring about collapse Thus, while the

flag is unstable in any breeze, the vapor velocity in the jet must reach a limiting value, u g, before the jet becomes unstable

a Plan view of bubbles rising from surface

b Waveform underneath the bubbles shown in a.

Figure 9.9 The array of vapor jets as seen on an infinite

hori-zontal heater surface

475

Figure 9.10 The flapping of a flag due to Helmholtz instability.

Lamb [9.16] gives the following relation between the vapor flow u g,shown in Fig.9.11, and the wavelength of a disturbance in the jet wall,

§9.3 Peak pool boiling heat flux 477

Figure 9.11 Helmholtz instability of vapor jets.

will be better developed than the others and therefore more liable to

collapse

Example 9.3

Saturated water at 1 atm flows down the periphery of the inside of a

10 cm I.D vertical tube Steam flows upward in the center The wall of

the pipe has circumferential corrugations in it, with a 4 cm wavelength

in the axial direction Neglect problems raised by curvature and the

finite thickness of the liquid, and estimate the steam velocity required

to destabilize the liquid flow over these corrugations, assuming that

the liquid moves slowly

Solution.The flow will be Helmholtz-stable until the steam velocity

reaches the value given by eqn (9.8):

u g =

2

2π (0.0589) 0.577(0.04 m) Thus, the maximum stable steam velocity would be u g = 4 m/s.

Beyond that, the liquid will form whitecaps and be blown back

upward

Example 9.4

Capillary forces hold mercury in place between two parallel steel plateswith a lid across the top The plates are slowly pulled apart until themercury interface collapses Approximately what is the maximumspacing?

Solution. The mercury is most susceptible to Taylor instabilitywhen the spacing reaches the wavelength given by eqn (9.6):

λ d1= 2π √3

2

σ g(ρ f − ρ g ) = 2π √3

2

0.487 9.8(13600) = 0.021 m = 2.1 cm

(Actually, this spacing would give the maximum rate of collapse It

can be shown that collapse would begin at 1 3 times this value, or

For any heater configuration, two things must be determined One

is the length of the particular disturbance in the jet wall, λ H, which will

trigger Helmholtz instability and fix u gin eqn (9.8) for use in eqn (9.9)

The other is the ratio A j A h The prediction of qmaxin any pool boilingconfiguration always comes down to these two problems

qmax on an infinite horizontal plate The original analysis of this type

was done by Zuber in his doctoral dissertation at UCLA in 1958 (see [9.17])

He first guessed that the jet radius was λ d1 4 This guess has receivedcorroboration by subsequent investigators, and (with reference to Fig.9.9)