Heat Transfer Handbook part 66 potx

For example, bubble growth and departure are influenced by the orientation of the surface, the thickness and temperature profile in the thermal boundary layer, the proximity of neighboring

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TABLE 9.1 Advancing Contact Angles

Source: Shakirand Thome (1986).

9.3.3 Size Range of Active Nucleation Sites

Above, a single uniform temperature was assumed for the wall and liquid A more practical case is when there is a temperature gradient in the form of a thermal bound-ary layer in the liquid adjacent to the wall, such as illustrated in Fig 9.5 for a conical nucleation site, where a vapor nucleus of radiusrnucsits at the cavity mouth The bulk liquid temperature isT∞, the wall temperature isT w(whereT w ≥ T∞) and a linear temperature gradient is assumed in the thermal boundary layer of thicknessδ If αnc

is the natural convection heat transfer coefficient andλLis the thermal conductivity

of the liquid, the boundary layer thickness is approximately

δ = αλL

Hsu (1962) postulated that a nucleus sitting in such a temperature gradient activates

if the superheat at the top of the vapor nucleus is greater than that required for its equilibrium [i.e., eq (9.10)], including the distortion of the temperature profile by the bubble nucleus itself Nucleation occurs if the local liquid temperature profile intersects the equilibrium nucleation curve The first site to activate is at the tangency between the nucleation superheat curve and the liquid temperature profile line Hsu assumed that distortion put the location of this temperature at a distance 2rnucfrom the surface, while Han and Griffith (1965) put the distance at 1.5rnucbased on potential flow theory If the liquid pool is at the saturation temperature (i.e., ifT∞= Tsatand

1.5rnucis assumed fordisplacement of the isotherms), the cavity size satisfying the condition of tangency isrnuc = δ/2, which is approximately 50 µm forwaterat its

normal boiling point

Much larger superheats or heat fluxes are typically necessary, however, to initiate boiling on a heated surface This discrepancy results from the fact that a range of cav-ity sizes exists on a real surface, and large cavities that respect the criterion above may

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Figure 9.5 Nucleation in a temperature gradient

not be available In this case,Twmust be increased until the liquid temperature pro-file intersects the equilibrium bubble curve corresponding to a smaller cavity Thus, there is a maximum nucleation radiusrmaxand a minimum nucleation radiusrminthat will activate among those available, wherermaxis the largest cavity size available that meets the nucleation criterion, and similarly,rminis the smallest cavity size that meets the criterion nearer the surface In reality, no such cavity may be available, or only one at the tangency point, or finally, numerous cavities in the size range fromrminto

rmax, and hence this introduces the concept of a size range of active nucleation sites

Actual prediction of the nucleation superheat for a surface is complicated by our lack

of knowledge of the size and shape of the cavities actually available Consequently,

it is more typical to observe the superheat at which nucleation occurs experimentally and then to back-calculate the effective nucleation radius using eq (9.10)

9.3.4 Nucleation Site Density

As the heat flux at the surface is increased, more and more nucleation sites activate

and the question arises as to how many sites are active per unit area The nucleation

site density can be determined using several approaches First, active boiling sites can be counted with heat flux or wall superheat as the independent variable, typically using photographs or videos of the boiling process Second, the nucleation site den-sity can be inferred from the measured heat flux by postulating some heat transfer mechanisms There is rarely agreement between these two approaches Third, the

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size and distribution of cavities on the surface can be determined using a scanning electron microscope However, this is tedious and the resulting distribution of cavi-ties only describes the actual surface investigated As a consequence, prediction of the nucleation site density is only an approximate estimate

9.4 BUBBLE DYNAMICS

The growth of vapor bubbles can be a particularly complex physical phenomenon For example, bubble growth and departure are influenced by the orientation of the surface, the thickness and temperature profile in the thermal boundary layer, the proximity of neighboring bubbles, transient thermal diffusion within the wall to the adjacent liquid, wake effects of the previous bubble, bubble shape during growth, and so on Bubble dynamics play a key role in the development of any analytical model purporting to predict nucleate pool boiling heat transfer coefficients The simplest case to analyze

is that of a single spherical bubble growing within an infinite, uniformly superheated liquid remote from a wall This is presented below A comprehensive treatment of bubble growth theory can be found in van Stralen and Cole (1979)

9.4.1 Bubble Growth

A spherical bubble growing in a uniformly superheated liquid is the simplest geom-etry to analyze The pressure and temperature inside the bubble arepGandTG; the bubble radius isR and is a function of time t from the initiation of growth (growth rate

isdR/dt) The pressure and temperature in the liquid are p∞andT∞ Othereffects influencing the bubble are ignored, such as the static head of the liquid, and the center point of the bubble is assumed to be immobile At inception, the superheat is sufficient for nucleation Then, as the bubble grows, the pressure inside the bubble decreases and with it,Tsatat the bubble interface Enthalpy stored in the superheated liquid adja-cent to the interface is converted into latent heat at the bubble interface and hence the interfacial temperature falls, creating a thermal diffusion shell around the bubble Mo-mentum is imparted to the surrounding liquid as the bubble grows and heat diffuses from the superheated bulk to the interface at a rate equal to the rate at which latent heat is liberated at the interface In addition, the equilibrium vapor pressure curve is assumed to describe this dynamic process, and it is assumed that the vapor pressure

in the bubble corresponds to the saturation pressure at the vapor temperature [i.e., that

pG = psat(TG)] Bubble growth under these conditions is controlled by two factors:

1 Inertia The initial growth of a bubble is very fast, limited only by the

momen-tum available to displace the surrounding liquid from its path That is, inertia must be imparted to the liquid to accelerate it away in front of the growing bubble

2 Heat diffusion As the bubble grows in size, the effect of inertia becomes

negli-gible, and growth continues by virtue of diffusion of heat from the superheated

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liquid to the interface, although at a much slower growth rate than during the inertia-controlled stage of growth

For inertia-controlled bubble growth during the initial stage of bubble growth, Rayleigh (1917) modeled incompressible radially symmetric flow of the liquid sur-rounding a bubble for a spherical bubble with a differential element of radiusr and

thicknessdr fora bubble of radius R The Rayleigh equation is

R d2R

dt2 +3 2

dR dt

2

ρL

p G − p∞−2σ

R



(9.16)

where the vapor pressure in the bubble isp G and that at the interface isp L Since

2 G − p∞, the term 2σ/R can be ignored Utilizing a linearized version

of the Clapeyron equation for the small pressure differences involved, the Rayleigh equation reduces to

R d2R

dt2 +3 2

dR dt

2

= ρG

ρL

T∞− Tsat(p∞)

Finally, integrating from the initial condition ofR = 0 at t = 0, the Rayleigh bubble

growth equation for inertia-controlled growth is obtained:

R(t) =

 2 3



T∞− Tsat(p∞)

Tsat(p∞)



hLGρ G

ρL

1/2

For heat diffusion–controlled growth, Plesset and Zwick (1954) derived the following bubble growth equation for relatively large superheats:

R(t) = Ja



12aLt

whereaLis the thermal diffusivity of the liquid:

a L= λL

and the Jakob numberis

Ja=ρL cpL(T∞− Tsat)

Thus, for heat diffusion–controlled bubble growth, the radiusR increases with time

ast1/2, while it grows linearly with time during the initial inertia-controlled stage

of growth Mikic et al (1970) combined the Rayleigh and Plesset–Zwick equations

to arrive at an asymptotic bubble growth equation valid for the entire bubble growth period:

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R+= 2 3



(t++ 1)3/2 − (t+)3/2− 1 (9.22) where

t+= tA2

A =



2 [T∞− Tsat(p∞)] hLGρ G

3ρL Tsat(p∞)

1/2

(9.25)

B =

12a L

π

1/2

This expression reduces to eqs (9.18) and (9.19) at the two extremes oft+, respec-tively

Bubble growth at heated walls differs significantly from these ideal conditions since growth occurs in a thermal boundary layer that may be thicker or thinner than the bubble itself The velocity field in the liquid created by the growing bubble is affected

by the wall and with it the inertia force imposed on the liquid, which may change the bubble shape from spherical to hemispherical or to some other more complex shape The hydrodynamic wake of a departing bubble may disturb the velocity field

of the next bubble or that of adjacent bubbles Furthermore, rapidly growing bubbles trap a thin evaporating liquid microlayer on the heated surface For example, Fig

9.6 illustrates microlayer evaporation underneath a growing bubble and macrolayer evaporation from the thermal boundary layer to the bubble as proposed in a model by van Stralen (1966)

Figure 9.6 Bubble growth model of van Stralen (1966)

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9.4.2 Bubble Departure

Bubble departure is another fundamental process of importance in nucleate boiling

The diameter at which a bubble departs from the surface during its growth is con-trolled by buoyancy and inertia forces (each attempting to detach the bubble from the surface) and surface tension and hydrodynamic drag forces (both resisting its departure) Furthermore, the shape of the bubble may deviate significantly from the idealized spherical shape While slowly growing bubbles tend to remain spherical, rapidly growing bubbles tend to be hemispherical Numerous other shapes are ob-served using high-speed movie cameras or videos

The simplest case to analyze is that of a large, slowly growing bubble on a flat surface facing upward, for which the hydrodynamic and inertia forces are negligible

Departure occurs when the buoyancy force trying to lift the bubble off overcomes the surface tension force trying to hold it on The surface tension force also depends on the contact angleβ (i.e., a contact angle approaching 90° increases the surface tension force and hence the bubble departure diameter) Fritz (1935) proposed the first bubble departure equation that equated these two forces The Fritz equation, utilizing the contact angleβ (i.e., β = π/2 = 90° for a right angle) and surface tension σ, gives

the bubble departure diameter as

doF = 0.0208β

g(ρ L− ρG )

1/2

(9.27)

The contact angle relative to the surface (through the liquid) is input in degrees The Fritz equation has been extended empirically to pressures ranging from 0.1 to 19.8 MPa with the following correction todoF :

do = 0.0012

ρL− ρG

ρG

0.9

More complex bubble departure models include the inertia, buoyancy, drag, and surface tension forces The surface tension may also act to pinch off the bubble as

it begins to depart from the surface As an illustration of one of these theories, that

of Keshock and Siegel (1964) is presented here Assuming a spherical bubble, their force balance of the static and dynamic forces acting on a departing bubble is

whereFbandFpare the buoyancy and the excess pressure forces, respectively, acting

to lift the bubble off the surface andFi , Fσ, and F D are the inertia, surface tension,

and liquid drag forces resisting bubble departure The buoyancy force is

Fb =πd o3

The excess pressure on the dry area where the bubble is attached to the wall is

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∆p = 2σ sin β

db +

σ

rb 

2σ sin β

The first term comes from the Laplace equation (9.1) applied to the bubble base diameter, including the contact angleβ, while the second term accounts for the effect

of curvature at the base of the bubble Keshock and Siegel assumed that the second term is negligible compared to the first, such that the excess pressure force acting on the base area of diameterd bis

F p =πdb

The inertia force imparted on the surrounding liquid by the growing bubble is

wherem is the mass of the liquid displaced by the bubble and u is the interfacial

velocity Assuming that the liquid affected is 11/16 of the bubble volume, the inertia force is

F i = d

dt

11

16ρL4π [R(t)]3

3

dR(t)

dt d=d o

(9.34)

The interfacial velocity dR(t)/dt may be determined with the Plesset and Zwick

bubble growth model presented earlier The surface tension force acting on the dry perimeter at the base of the bubble of diameterdbis

Assuming a spherical bubble rising freely in a liquid at a velocity equal to that of bubble growth at the moment of departure, the hydrodynamic drag force resisting bubble departure is

F D= 1

2ρL C D π [R(t)]2



dR(t) dt

2

(9.36a) or

F D= π

4C DRebubµL[R(t)] dR(t)

where the drag coefficient isCDand the bubble Reynolds numberis

Rebub =ρL[2R(t)] [dR(t)/dt]µ

L

(9.37)

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The drag coefficient is related to the bubble Reynolds number as

C D = a

Rebub

(9.38) (where Keshock and Siegel usea = 45), so the drag force is

F D= π

4aµ L[R(t)] dR(t)

In the model above, the drag force is typically negligible, while the inertia force becomes significant only at large∆T The excess pressure and surface tension forces

are difficult to calculate for a real case sincedb is unknown (db would typically be largerthan the cavity mouth since the liquid film trapped undera growing bubble partially dries out)

9.4.3 Bubble Departure Frequency

The bubble departure frequency is

f = 1

wheretg is the bubble growth time andtwis the waiting time between the departure

of one bubble and the initiation of growth of the next Bubble departure frequencies range from as low as 1 Hz at very small superheats to over 100 Hz at high superheats

The bubble growth timetgcan be obtained by calculating the bubble departure diam-eter[e.g., using eq (9.28)] and solving fortimet in the Plesset–Zwick bubble growth

equation presented earlier After a bubble departs, the length of pause before the next bubble begins to grow depends on the rate at which the surface and adjacent liquid are reheated by transient heat conduction from the wall

Once the bubble departure diameter and frequency are known, the volumetric vapor flow rate from a single boiling site may be determined Combining this with the bubble nucleation site density, the volumetric vapor flow rate per unit area from the heated surface can be estimated, and hence latent heat transport from the surface can also be calculated Thus, the latent heat flux can be determined, and subtracting this from the total heat flux measured, the sensible heat flux leaving the surface in the form

of superheated liquid is obtained The effects of sequential bubbles and neighboring bubbles on one anotherand theircompetition forthe heat stored within the thermal boundary layer, however, make an accurate prediction of the vapor flow rate from the surface very difficult to obtain

9.5 POOL BOILING HEAT TRANSFER

Methods for predicting heat transfer coefficients in the various pool boiling regimes are described in this section Most of the attention is placed on nucleate boiling, but

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heat transfer in the film boiling and transition boiling regimes are also addressed

Methods are also provided to predict departure from nucleate boiling (referred to as DNB orcritical heat flux) and the minimum heat flux of film boiling

9.5.1 Nucleate Boiling Heat Transfer Mechanisms

Before presenting a selection of nucleate pool boiling correlations, the heat transfer mechanisms playing a role in nucleate pool boiling, illustrated in Fig 9.7, are identi-fied as follows:

1 Bubble agitation The systematic pumping motion of the growing and departing

bubbles agitates the liquid, pushing it back and forth across the heater surface, which

in effect transforms the otherwise natural convection process into a localized forced convection process Sensible heat is transported away in the form of superheated liquid and depends on the intensity of the boiling process

( )a

( )b

( )c

Bubble Liquid motion

Hot liquid

Superheated liquid Heated surface

Microlayer Evaporation

Figure 9.7 Heat transfer mechanisms in nucleate pool boiling: (a) bubble agitation; (b) vapor–liquid exchange; (c) evaporation.

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2 Vapor–liquid exchange The wakes of departing bubbles remove the thermal

boundary layer from the heated surface, and this creates a cyclic thermal boundary layer stripping process Sensible heat is transported away in the form of superheated liquid, whose rate of removal is proportional to the thickness of the layer, its mean temperature, the area of the boundary layer removed by a departing bubble, the bubble departure frequency, and the density of active boiling sites

3 Evaporation Heat is conducted into the thermal boundary layer and then to the

bubble interface, where it is converted to latent heat Macroevaporation occurs over the top of the bubble while microevaporation occurs underneath the bubble across the thin liquid layer trapped between the bubble and the surface, the latter often referred to

as microlayer evaporation The rate of latent heat transport depends on the volumetric

flow of vapor away from the surface per unit area

The mechanisms above compete forthe same heat in the liquid and hence overlap with one another, thermally speaking At low heat fluxes characteristic of the isolated bubble region, natural convection also occurs on inactive areas of the surface, where

no bubbles are growing

9.5.2 Nucleate Pool Boiling Correlations

The complexity of the nucleate pool boiling process is such that accurate, reliable analytically based design theories are yet to be available Unresolved problems are related primarily to predicting such things as boiling nucleation superheats, boiling site densities for a given surface, and thermal interaction between neighboring boil-ing sites As a consequence, completely empirical methods are used for predictboil-ing nucleate pool boiling heat transfer coefficients In a pool boiling experiment, the wall superheat∆T is measured versus the heat flux q, and the nucleate boiling heat

trans-fercoefficient is obtained from its definition (αnb ≡ q/∆T ) These data may be fit

with expressions such asq ∝ ∆T n , αnb ∝ ∆T n, orαnb ∝ q n, where the exponent

n is on the order of 3, 2, or 0.7, respectively Pool boiling correlations are typically

formulated in similar fashion where expressions in the formαnb∝ q nare the easiest

to apply since heat flux is an imposed design variable while the wall temperature in

∆T is unknown and part of the solution Literally hundreds of pool boiling

correla-tions have been proposed; below a representative selection of recommended methods

is presented plus the classic Rohsenow method

Bubble Agitation Correlation of Rohsenow Rohsenow (1962) proposed the first widely quoted correlation He assumed the boiling process is dominated by the bubble agitation mechanism depicted in Fig 9.7, whose bubble-induced forced-convective heat transfer process could be correlated with the standard single-phase forced-convection correlation relation:

where the Nusselt numberforboiling is defined as

9.5.1 Nucleate Boiling Heat Transfer Mechanisms

Before presenting a selection of nucleate pool boiling correlations, the heat transfer mechanisms... predictboil-ing nucleate pool boiling heat transfer coefficients In a pool boiling experiment, the wall superheat∆T is measured versus the heat flux q, and the nucleate boiling heat

trans-fercoefficient... rate from the surface very difficult to obtain

9.5 POOL BOILING HEAT TRANSFER< /b>

Methods for predicting heat transfer coefficients in the various pool boiling regimes are described