Heat Transfer with Boiling and Condensation

10 Heat Transfer with Boiling and Condensation

In this chapter we intend to study heat transfer during boiling and condensation. These two phenomena are rather different in nature, and are usually dealt with in separate chapters in textbooks; however, there are two reasons for bringing them together in a single chapter. The first is that boiling and condensation most often occur together in real processes. For example, in Fig. 10.la we have shown a refrigeration cycle where boiling is caused by thermal energy supplied by air coming from the freezer compartment of a refrigerator. The vapor from the boiler is compressed and passed through a condenser. This is usually constructed from a bank of finned tubes placed at the back of the refrigerator where heat is transferred to the surrounding air by free convection. On some models the condenser is placed on the bottom of the refrigerator and a fan is necessary to provide a sufficient flow of cooling air. In Fig. 10.1 b we have illustrated a simple distillation column which is used to separate a feed stream into a low boiling point top product and a high boiling point bottom product. The vapor leaving the top of the column is condensed and some of it is returned to flow down the column. At the bottom of the column a boiler provides a source of vapor which rises through the column contacting the liquid and thus enhancing the mass transfer which acts to separate the constituents in the feed. In Fig. 10.lc a typical steam power plant has been illustrated. An economical condenser is always desirable, and it is this aspect of the power plant operation which is usually responsible for the location of these plants along our coastlines where a large supply of cold ocean water is available at the expense of scenic blight and possible marine thermal pollution. In each of these three examples the boiling and condensation processes are intimately related, for in general what is boiled in one part of the process is condensed in another. If one is concerned with boiling heat transfer in some process, it is likely that a condensation heat transfer process is also of importance. There is a second reason for covering boiling and condensation in a single chapter, and that is that both are two-phase flow processes, thus the analysis presented in Sec. 10.3 of mass, momentum, and energy transfer at a vapor-liquid interface is applicable to both processes. Before going on to the study of boiling we need to briefly review some concepts previously encountered in thermodynamics and fluid mechanics courses. Vapor pressure If we enclose a liquid and its vapor in a system, such as that illustrated in Fig. 10.2, and allow equilibrium conditions to be reached we find that for each temperature, Tsah at which we maintain the system there corresponds a pressure, pvap, at which the two phases can co-exist. We will refer to this temperature as the saturation temperature and the pressure as the vapor pressure. Boiling, or formation of vapor bubbles within the liquid phase, usually occurs when the vapor pressure is greater than the surrounding ambient pressure. This can be caused by either raising the temperature of the liquid above the saturation temperature or lowering the ambient pressure below the vapor pressure. The former occurs when a coffee 473

474

Heat Transfer with Boiling and Condensation cool liquid at high pressure Condenser, usually finned and located on the back . bottom of the refri gerator

pressure drop

cool vapor at low pressure

I

warm air

cool air to frost-free freezer compartment (a) Refrigeration

vapor 1-----'-'--"-'---"'-----'-----__

r:

top product

liquid

feed stream

vapor

liquid

'-------h';,

' - - - - - - - bottom product (b) Distillation

Heat Transfer with Boiling and Condensation

475

mechanical power out

highpressure steam

steam

f------''----,

thermal energy in

hinh-nr,,,",,",,,ul

water

WATER

(e) Steam power plant

Fig. 10.1 Boiling and condensation processes.

Temperature = T. o' Pressure = p.o.

Fig. 10.2 Vapor pressure.

pot is placed on a hot stove, and the latter can occur if a deep-sea diver is forced to surface quickly, thus causing the nitrogen in his blood to "boil" giving rise to the bends. If we measure Lat and p yap we can construct a vapor-pressure curve such as the one shown in Fig. 10.3 for water. On the basis of simplifying assumptions which are generally valid away from the critical point one can derive the Clausius-Clapeyron equation which relates the vapor pressure to the temperature by the expression

.2)_

I (Pvap -I1Hvap ( 1 1) n pvap,! R T sat ,2 - T sat ,!

Eq. 10-1 illustrates the exponential dependence of vapor pressure on temperature.

(10-1)

476

Heat Transfer with Boiling and Condensation critical point

pvap

liquid ice

triple point

Fig. 10.3 Vapor-pressure curve for water.

Surface tension

The tangential stress associated with an interface is referred to as the surface tension or interfacial tension. It can give rise to enormously important forces in two-phase systems when the ratio of surface area to bulk volume becomes large. If we write a force balance on the hemisphere illustrated in Fig. 10.4 we obtain Pi (7Tr2)

= Po( 7Tr2) + u(27Tr)

I forces = P, (7Tr2) - Pot 7Tr2) -

r

r

(10-2)

0"(27Tr)

r

A:ea Aiea petri meter Interior Ambient Surface pressure pressure tension

Fig. 10.4 Surface tension.

where Pi is the interior pressure and po is the pressure on the outside of the sphere. The pressure difference across the interface under equilibrium conditions is given by flp

2u

= -, r

pressure difference across a spherical interface

(10-3)

Pool Boiling

477

For an arbitrary interface this relation takes the form

ap

=

u

(1r! + 1), r2

pressure difference across an arbitrary interface

(10-4)

where r! and r2 are the two principal radii of curvature. From Eqs. 10-1 and 10-3 we can calculate the temperature T sat,2 at which a vapor bubble of radius r will be in equilibrium with the surrounding liquid. T sat ,!

T sat,2

1- (RTsat,!) In (1 +~) aH rp

=

vap

(10-5)

vap.!

If the temperature is larger than T sat .2 the bubble will grow because of vaporization at the vapor-liquid interface, while condensation occurs and the bubble collapses if the temperature is less than T sat,2. If T sat,!

is the boiling point, i.e., Pvap.! is equal to the ambient pressure, then T sat ,2- T sat,! is referred to as the superheat and will often be denoted simply by a T. For small bubbles Eq. 10-5 indicates that the superheat can become quite large. When 2u / rp vap.! ~ 1 the superheat is approximated by T

- T sat.2

= sat,1

(RTsat'2Tsat,!)(~) LiH yap

rpvap.l'

(10-6)

indicating that large values of the superheat are to be expected for small values of the radius.

P ART I

10.1

BOILING

Pool Boiling

Pool boiling is a fairly descriptive term which refers to the type of boiling that occurs when a pool of liquid is brought into contact with a heated surface. A common example is a pan of water being heated on the kitchen stove, but a more practical example would be the situation illustrated in Fig. 10.1.1 where we have shown a heated tube immersed in a pool of liquid. In order to explore the different regimes of pool boiling we wish to consider the case where the tube shown in Fig. 10.1.1 is being heated by condensing steam. The temperature of the tube can be controlled by fixing the pressure of the steam. Imagine now that we have a pool of water at some temperature T liq which is less than T sat , i.e., think of a pan of water at 7rF. Into this pool of water we immerse our tube heated by condensing steam and adjust the tube temperature To so that it is greater than T liq but less than T sat . Under these conditions free convection occurs as illustrated in Fig. 1O.1.1a and the temperature of the water begins to rise. If the vapor pressure at the surface of the pool is greater than the partial pressure of the water in the air above the pool, evaporation will take place at the surface. The heat transfer rate for this process can be treated by means of the theory presented in Secs. 5.11 and 7.9. As the temperature of the liquid increases we can raise the tube temperature by increasing the steam pressure until To = T liq = T sat , i.e., for water at one atmosphere the temperature of the tube and the liquid is 212°F. Let us imagine now that we further increase the tube temperature and measure the heat flux q = q . olr-ro by determining the rate at which condensate is formed. We can now plot q versus To- T sat = aT to obtain a curve similar to that shown in Fig. 10.1.2. For small values of aT no boiling takes place and the heat transfer mechanism is simply free convection. This condition occurs in Region I of Fig. 10.1.2. When a T becomes large enough (usually a few degrees Fahrenheit) vapor bubbles are formed at a few selected sites on the surface of the tube and a stream of bubbles will rise from these sites to the surface of the pool as illustrated in Fig. 10.1.1b.t The dependence of q on a T for this condition is shown as Region II in Fig. 10.1.2. As we continue to increase tIf the liquid temperature is lower than the saturation temperature, T hq < T,." the bubbles formed will rise from the surface and collapse as heat is transferred from the bubble to the subcooled liquid and condensation takes place. This phenomenon is usually referred to as subcooled boiling.

Evaporation may take place if the vapor pressure is greater than the partial pressure. _..A4 . . . . . . . .

~~ ..

•

.......... 4

............................................................

GAS ~:..!.!.~

To < T.... free convection Fig. 10.1.la

Regimes of pool boiling.

Fig.10.1.1b Regimes of pool boiling; (To - T ..t ) - 4-lOoF, nucleate boiling. (Repetitive formation of bubbles at a nucleation site. A natural nucleation site exists at the left end of a heated !-in. diameter copper rod. The column of 8 bubbles on the left has formed from one site. The liquid is isopropyl alcohol. From K. W. Haley and J. W. Westwater, "Heat Transfer from a Fin to a Boiling Liquid," Chern. Eng. Science 20, 711 (1%4).) 478

Pool Boiling

Fig. lO.l.le Regimes of pool boiling; (To - T,.t) - 10-65°F, nucleate boiling. (Nucleate boiling of methanol on a ~-in. steam-heated copper tube. The overall t:..T is 67°F, and the heat flux is 76800 Btu/hrsqft. Note that part of the tube is bare. This part will produce no bubbles until the temperature of the metal is increased. From J. W. Westwater, "The Boiling of Liquids," Scientific American 190, No.6, 64-67 (1954).)

Fig.lO.l.ld Regimes of pool boiling; (To - T a• t )

65-200°F, partial nucleate boiling and unstable film boiling. (Transition boiling of methanol on a ~-in. steam-heated copper tube. The overall t:.. T is 124°F, and the heat flux is 27 200 Btu/hrsqft. From J. W. Westwater and J. O. Santangelo, "Photographic Study of Boiling," Ind. Eng. Chern. 47, 1605-1610 (1955).) -

479

480


Fig. 10.l.le Regimes of pool boiling; (To - L .. ) > 200°F, stable film boiling, radiation becoming increasingly important. (Film boiling of methanol on a Hn. steam-heated copper tube. The overall 8 Tis 148°F and the heat flux is 12970 Btu/hrsqft or about 8 per cent of the peak flux. From 1. W. Westwater and J. G. Santangelo, "Photographic Study of Boiling," Ind. Eng. Chon. 47, 1605-16\0 (\955).)

II

10·

IV -... ~

III

":-......

10'

10'

':::....

-

I

II

.....

I

:::l

0-

/

L

.r:

iii

1

"-

/

/

10'

II

102

,lL 10 2

3 4

6 810

100 8T

Fig. 10.1.2

=

(To- T••• ). of

Boiling curve.

1000

4000

Pool BOiling

481

the tube temperature more nucleation sites become active and the concentration of bubbles around the tube increases as shown in Fig. 10.1.1c. The bubbles grow and break away from the surface giving rise to considerable fluid motion which tends to bring the cooler liquid into contact with the hot surface. This is illustrated in Fig. 10.1.3. As more nucleating sites become active the overall fluid motion increases and the liquid at the saturation temperature

(c)

Fig. 10.1.3 Fluid motion during bubble growth and departure.

heat flux q increases rapidly with increasing a T. Eventually the high concentration of bubbles around the tube begins to hinder the fluid motion and the plot of q versus a T begins to level off and the critical superheat, a Te , is reached. Further increases in a T lead to severe vapor blanketing of the tube and the heat flux begins to decrease in this region. This represents a curious phenomenon for which the rate of heat transfer decreases with increasing temperature difference. It is a region which one wishes to avoid in the design of boilers and knowledge of a Te for a given process is crucial to the efficient operation of the process. If the tube at which the boiling takes place is electrically heated the transition region cannot be reached for a slight increase in the heat flux will cause the system to quickly pass the transition region and progress toward the new stable condition. For most metals this gives rise to temperatures greater than the melting point of the tube and the tube is destroyed. Because of this the critical superheat is often referred to as the burnout point. In the transition region portions of the tube become completely surrounded by a vapor film as

482


illustrated in Fig. 10.1.1d. For still larger values of d T the vapor-blanketing mechanism comes into full play and a stable film surrounds the tube. This is known as stable film boiling and is designated by Region IV in Fig. 10.1.2. Everyone has observed film boiling in the Leidenfrost phenomenon of water droplets dancing on a very hot surface. The vapor film formed between the droplet and the surface provides a nearly frictionless support and the drops move rapidly about the surface. During film boiling the vapor film undulates in a rather regular manner and comparatively large bubbles break off from the film as illustrated in Fig. 10.1.1e. For values of d T larger than about 1000°F the heat flux begins to increase as radiation becomes the dominant heat transfer mechanism. Nucleation From the discussion on vapor pressure and surface tension given in the previous section we know that the superheat d T tends toward infinity as the bubble size tends toward zero. This relation, which can be inferred from Eq. 10-5, does not actually hold true for r ~O for molecular effects come into play and vapor bubbles can be formed at finite values of the superheat. Nevertheless, for practical purposes, boiling will not occur unless nucleation sites exist which provide for bubble formation at low values of the superheat. From numerous experimental studies it is clear that nucleation sites consist of cavities in the solid surface such as the one shown in Fig. 10.1.3 and reproduced in greater detail in Fig. 10.1.4. For most liquid-solid

noncondensable gas

vapor (a)

(b) Fig. 10.1.4 Nucleation sites.

systems the contact angle f3 is less than 'IT /2 and the surface curvature which occurs at a nucleating site such as the one shown in Fig. 10.1.4a is such that the pressure is increased because of surface tension. Under these circumstances the nucleating site is active as long as it contains some noncondensable gas such as air; however, as the cavity becomes filled with vapor it tends to condense and the site may become filled with liquid and therefore inactive. The demise of active nucleating sites does not seem to occur often presumably because the noncondensable gas is difficult to remove completely and because of the existence of re-entrant sections such as illustrated in Fig. 10.1.4b. There surface tension lowers the pressure in the cavity and a stable vapor filled cavity can exist. In any natural surface there will be a spectrum of cavity sizes such as we have illustrated in Fig. 10.1.5. Under these conditions cavity I will begin boiling at relatively low values of d T while cavity II will be the last nucleating site as d T is increased. This distribution of cavity sizes is of course responsible for the nucleate boiling phenomena illustrated in Figs. 10.1.1b,c and for the shape of the curve in Region II of Fig. 10.1.2. If all the surface cavities were nearly the same size the plot of q versus dT would be very steep, whereas a broad distribution of cavity sizes would yield a slower increase of q with d T.

Dimensional Analysis

483

Fig. 10.1.5 Distribution of cavity sizes on a real surface.

When the heat flux at a surface is sufficiently high so that film boiling takes place the nucleating sites are of no importance; however, most boiling processes operate in the nucleate boiling regime and knowledge of the q versus ~ T curve and the critical superheat ~ Tc are of utmost importance to the designer. Clearly the boiling curve can differ considerably depending on the nature of the surface and the contact angle at the liquid-solid-vapor interface, and we can expect that correlating boiling heat transfer data will be a difficult task.

10.2

Dimensional Analysis for a Two-Phase System with Phase Changes

Having briefly explored the phenomena of pool boiling and nucleation, we now need to consider the dimensional analysis of a two-phase system so that we can understand the correlations for the nucleate boiling Nusselt number and the critical heat flux Nusselt number. We can neglect the temperature dependence of the physical properties in the governing equations and write them as: p

(~; + v . VV) = V. pCp

-

VP + P g + I.l V2 v,

V =

0,

(~; + V' VT) =

equations of motion for both the vapor and liquid phases

continuity equation for both the vapor and liquid phases

kV 2 T,

thermal energy equation for both the vapor and liquid phases

(10.2-1) (10.2-2) (10.2-3)

The governing differential equations are easily expressed and put into dimensionless form; however, as we shall see in subsequent paragraphs it is the boundary conditions that require considerable thought. Although the development of the boundary conditions is tedious, the effort is necessary if we are to understand the form of the correlations for boiling and the analysis of film condensation. In constructing these boundary conditions we will denote the solid surface as 51 and the vapor-liquid interface as d(t) as illustrated in Fig. 10.2.1. At the solid-vapor and solid-liquid interface we express the velocity and temperature boundary conditions as B.C. 1

v=o,

T

=

To,

at

[f

(10.2-4)

Here we have designated the temperature at 51 as To and assumed that the thermal conductivity of the solid is sufficiently high so that the temperature in the solid is essentially uniform. Developing the boundary


484

solid-liquid and solid-vapor interface denoted by 9'

Fig. 10.2.1 Interfaces for a boiling process.

conditions at the vapor-liquid interface will require considerable analysis, t and we will accomplish this by applying the macroscopic mass, momentum, and total energy balances to the small section of interface illustrated in Fig. 10.2.1 and shown in more detail in Fig. 10.2.2:

~ dt ~ dt

J

pv dV

+

'Ya(t)

f

'Ya(t)

J

p dV + J

p(v - w) • n dA

= 0,

macroscopic mass balance

(10.2-5)

od,(t)

pv(v - w) . n dA = J

Ae(t)

t(n)

dA +

oda(t)

J

pg dV,

macroscopic momentum balance

'Ya(t)

(10.2-6)

df

dt

'Ya(t)

pedV+

J

A,(t)

pe(v-w)'ndA=-J

q.ndA+J

oda(t)

t(n)'vdA

oda(t)

+

f

p g • v d V,

macroscopic total energy balance

(10.2-7)

'Va (t)

In applying the macroscopic balances we plan to let the control volume shrink to an arbitrarily small volume containing the interface, thus all the volume integrals in Eqs. 10.2-5 through 10.2-7 will tend toward zero.

tThe question might naturally arise as to why we do not simply impose the normal constraints of continuity of v, T, q . n, and ~.) at the interface. The answer is that the vapor-liquid interface is treated as a singular surface which means that we will assign certain intrinsic properties to the surface itself, i.e., surface tension for example, and for singular surfaces the normal continuity requirements are no longer valid.


control volume 'V.(t) moving with interface

485

u

Fig. 10.2.2 Element of vapor-liquid interface.

Mass

The mass balance is particularly simple to apply and it yields p~(v~

- w) . n~ + pg(vg - w) . Dg

=

0

(10.2-8a)

Here the subscript e is used to denote the liquid phase and the sUbscript g is used to denote the vapor or gas phase. Note that n~ and ng are not the outwardly directed unit normal vectors for the liquid and vapor phases; rather they are the inwardly directed normal vectors. In addition to the relation given by Eq. 1O.2-8a for the normal components of the velocity we will continue to impose the traditional constraint on the tangential components of the velocity at the interface, i.e., V~ •

A=

Vg

•

A,

(1O.2-8b)

at the interface

where A is any tangent vector to the surface.

Momentum

A rigorous application of the momentum equation to the control volume shown in Fig. 10.2.2 has been presented by Slattery [1]; and we will draw upon that result to provide us with the proper expression for the contribution of surface tension. Application of Eq. 10.2-6 leads to (10.2-9) Here r, and r2 represent the two principal radii of curvature and we have included the effect of surface tension on a purely intuitive basis. We will now neglect viscous effects so that the stress vectors are given by (10.2-lOa) (10.2-lOb) Substituting Eqs. 10.2-10 into Eq. 10.2-9 and forming the scalar product with De = -Dg yields (10.2-11)


486

We can make use of the mass balance to simplify this result to (PR - p,)

=

u(l_+l) + p,(v, r, r,

w) - D,(V, - Vg

)

-

(10.2-12)

0,

Note that if there is no change of phase, Vr •

n

=v

R

•

n

=W

•

n,

for no change of phase

(10.2-13)

and Eq. 10.2-12, which neglects viscous effects, reduces to the result given by Eq. 10-4 at the beginning of this chapter. Energy Application of the macroscopic total energy balance to the element of interface presents some difficult steps if we do a thorough analysis of the rate of work term, and in order to simplify our development we will neglect the work done by viscous stresses and include the work done by surface tension forces in an intuitive manner. Subject to these restrictions application of Eq. 10.2-7 leads to p,e, (VI

-

w) . nf

+PRe.

(V. -

w) . n R = V, . (- D,p, )

+

V• . ( -

D.P.) - u

(lr, +l) W . De - (q, . n, +q •. D.) r2 (10.2-14)

Adding and subtracting PeW· (Peer

De

and pg W

+P, )(v, -w)· +(PRe. +P.)(V R Of

• Dg

allows us to rearrange the energy equation to obtain

-W)· Dg =

(PR - p,)w·

Of -

u(l+l) r, r,

W·

n, - (q, -qg).

0,

(10.2-15) Here we have used Of = - o. in some of the terms. Noting that the enthalpy of the gas and liquid phases can be expressed as

leads to

Expressing the latent heat of vaporization as !1H vap simplify Eq. 10.2-16 to obtain - ll.Hvapp,(V, -

w)·

Of

+(q, - q.).

=

h. - hi and using the mass balance allows us to

0, =

[CPR - p,) -

u

(~+-fJ ] W· 0,

( 10.2-17)

We can eliminate the pressure and surface tension terms by use of Eq. 10.2-12 so that Eq. 10.2-17 takes the form (10.2-18) At this point it becomes apparent that V!1H vap has the units of velocity. Furthermore an order-ofmagnitude estimate indicates that !1H vap ~ (v I - v.) . Of (w . 0, ) which allows us to further simplify Eq. 10.2-18. A summary of our boundary conditions at the vapor-liquid interface can now be expressed as B.C.2

p,(Ve-W)·Ot=P.(V.-W)·OR

mass

(10.2-19)


B.C.3 B.CA

487

(10.2-20) (- keVTe

+ kg VT

g )·

ne = I1Hvap pe(Ve - w)· ne,

energy

(10.2-21)

In addition we will require that the temperature at the vapor-liquid interface is the saturation temperature T = T sat ,

B.C.5

(10.2-22)

at d(t)

and we will assume that the contact angle f3 is all that is needed to describe the state of the solid-liquid-vapor interface. We now have available the governing differential equations, Eqs. 10.2-1 through 10.2-3, and the boundary conditions for the vapor-liquid interface, Eqs. 10.2-19 through 10.2-22. The "no slip" condition and the continuity of temperature condition at the solid-vapor and solid-liquid interface have previously been given as B.C.l. We are now confronted with the problem of choosing a characteristic length and velocity, L * and u *. Aside from perhaps some average cavity diameter the characteristic length and velocity for nucleate boiling are not obvious quantities. Many investigators like to think of the bubble diameter and velocity as it leaves the surface as being representative of L * and u *; however, these quantities are unknown except by experimental observation or by solution of the very complex governing equations and boundary conditions. Because of this the bubble diameter and velocity are not suitable characteristic quantities, and any dimensional analysis which begins with these quantities must always replace them with other parameters which are known a priori. An examination of the governing equations and boundary conditions indicates that a wide variety of characteristic values can be constructed in terms of known parameters. For example, we could use the following characteristic lengths

L *-

v

L*-

V I1Hvap' L * and u * one

1 -

2 -

ex

V cp(To- Tsat)'

L * _ I1HYap 3 -

g

In choosing the values of usually tries to pick quantities which strongly influence the process under consideration. Certainly the bubble diameter and velocity strongly influence the nucleate boiling heat transfer rate, but these are unknown except to the extent that their order-of-magnitude can be predicted by approximate theories. With essentially no real justification other than the knowledge of the form of current correlations we will choose L * and u * as follows: (10.2-23)

u* = exeg/I1Hvap

(10.2-24)

Our thought here is that the bubble diameter should become large as either g ~ 0 or (pe - pg) ~ 0, and that the bubble growth rate will increase with increasing thermal diffusivity, ex, since large values of ex mean that energy is rapidly supplied to the interface where vaporization occurs. Similarly, large values of I1Hvap should mean a slow bubble growth rate since more energy must be supplied to the interface for each cubic centimeter of vapor formed. Having chosen L * and u * we can put Eqs. 10.2-1 through 10.2-2 in dimensionless form to obtain (10.2-25a) (10.2-25b) Here N pr is the Prandtl number and N p and N v represent density and kinematic viscosity ratios: N p = pg/pe

(10.2-26a)

Nv =

(1O.2"26b)

Vg/Ve

488


The two continuity equations take the form (l0.2-27a) (l0.2-27b) and the two thermal energy equations become (l0.2-28a)

(l0.2-28b) where 0

= (To - T)/(To -

T sat ) and the dimensionless number No is given by

( 10.2-29)

No = a.la,

Going on to the boundary conditions we obtain U = 0 = 0,

B.C.1'

(Ut - W) . De = Np(U. - W) . D.,

B.C.2'

(10.2-30)

on Y

mass at d(t)

B.C.3' Here N

momentum at d (t ) We

(10.2-31) (10.2-32)

represents a nucleate boiling Weber number given by (10.2-33)

B.C.4'

(10.2-34)

The dimensionless number Nk is just the ratio of thermal conductivities, Nk = k.lk,

(10.2-35)

and N 80 will be referred to as the boiling number and is defined by N

- ke(To - T sat )

peae flH Yap

80 -

B.C.5'

0=1,

atd(t)

(10.2-36) (10.2-37)

The gravitational potential function can be expressed as

Ac , Bromley's theory begins to break down and bubble formation at the top of the vapor film plays an increasingly important role in the vapor removal process. For the full range of tube diameters Jordan suggests the following equations: - 0 60 [k g3g tlHvap(pe - pg )Jl/4 -1/4 hco -· vgD(To-Tsa,) Ac , 3

g - pg h c =o0· 62 [k g tlHvap(pe D('T' -Tsat ) Vg .1 0

)J

1I4

D- 1/ 4

(l0.4-6a)

forAc