Computational Fluid Mechanics and Heat Transfer Third Edition_14 docx

Flowing fluids undergo boiling or condensation in many of the cases inwhich we transfer heat to fluids moving through tubes.. 498 Heat transfer in boiling and other phase-change configurati

496 Heat transfer in boiling and other phase-change configurations §9.7

range of the data

The influence of fluid flow on film boiling Bromley et al [9.40] showedthat the film boiling heat flux during forced flow normal to a cylindershould take the form

q = constant

k

g ρ g h  fg ∆T u ∞ D

1/2

(9.42)

for u2∞ /(gD) ≥ 4 with h  fg from eqn (9.29) Their data fixed the constant

at 2.70 Witte [9.41] obtained the same relationship for flow over a sphereand recommended a value of 2.98 for the constant

Additional work in the literature deals with forced film boiling onplane surfaces and combined forced and subcooled film boiling in a vari-ety of geometries [9.42] Although these studies are beyond our presentscope, it is worth noting that one may attain very high cooling rates usingfilm boiling with both forced convection and subcooling

Flowing fluids undergo boiling or condensation in many of the cases inwhich we transfer heat to fluids moving through tubes For example,such phase change occurs in all vapor-compression power cycles and

refrigerators When we use the terms boiler, condenser, steam generator,

or evaporator we usually refer to equipment that involves heat transfer

within tubes The prediction of heat transfer coefficients in these systems

is often essential to determining U and sizing the equipment So let us

consider the problem of predicting boiling heat transfer to liquids flowingthrough tubes

Figure 9.18 The development of a two-phase flow in a vertical

tube with a uniform wall heat flux (not to scale)

497

498 Heat transfer in boiling and other phase-change configurations §9.7

Relationship between heat transfer and temperature difference

Forced convection boiling in a tube or duct is a process that becomes veryhard to delineate because it takes so many forms In addition to the usualsystem variables that must be considered in pool boiling, the formation

of many regimes of boiling requires that we understand several boilingmechanisms and the transitions between them, as well

Collier and Thome’s excellent book, Convective Boiling and

Condensa-tion [9.43], provides a comprehensive discussion of the issues involved

in forced convection boiling Figure9.18 is their representation of the

fairly simple case of flow of liquid in a uniform wall heat flux tube in

which body forces can be neglected This situation is representative of a

fairly low heat flux at the wall The vapor fraction, or quality, of the flow

increases steadily until the wall “dries out.” Then the wall temperaturerises rapidly With a very high wall heat flux, the pipe could burn outbefore dryout occurs

Figure9.19, also provided by Collier, shows how the regimes shown inFig.9.18are distributed in heat flux and in position along the tube Noticethat, at high enough heat fluxes, burnout can be made to occur at any sta-

tion in the pipe In the subcooled nucleate boiling regime (B in Fig.9.18)

and the low quality saturated regime (C), the heat transfer can be

pre-dicted using eqn (9.37) in Section 9.6 But in the subsequent regimes

of slug flow and annular flow (D, E, and F ) the heat transfer mechanism

changes substantially Nucleation is increasingly suppressed, and ization takes place mainly at the free surface of the liquid film on thetube wall

vapor-Most efforts to model flow boiling differentiate between

nucleate-boiling-controlled heat transfer and convective boiling heat transfer In

those regimes where fully developed nucleate boiling occurs (the later

parts of C), the heat transfer coefficient is essentially unaffected by the

mass flow rate and the flow quality Locally, conditions are similar to poolboiling In convective boiling, on the other hand, vaporization occursaway from the wall, with a liquid-phase convection process dominating

at the wall For example, in the annular regions E and F , heat is convected

from the wall by the liquid film, and vaporization occurs at the interface

of the film with the vapor in the core of the tube Convective boilingcan also dominate at low heat fluxes or high mass flow rates, where wallnucleate is again suppressed Vaporization then occurs mainly on en-trained bubbles in the core of the tube In convective boiling, the heattransfer coefficient is essentially independent of the heat flux, but it is

§9.7 Forced convection boiling in tubes 499

Figure 9.19 The influence of heat flux on two-phase flow behavior.

strongly affected by the mass flow rate and quality

Building a model to capture these complicated and competing trends

has presented a challenge to researchers for several decades One early

effort by Chen [9.44] used a weighted sum of a nucleate boiling heat

trans-fer coefficient and a convective boiling coefficient, where the weighting

depended on local flow conditions This model represents water data to

an accuracy of about ±30% [9.45], but it does not work well with most

other fluids Chen’s mechanistic approach was substantially improved

in a more complex version due to Steiner and Taborek [9.46] Many other

investigators have instead pursued correlations built from dimensional

analysis and physical reasoning

To proceed with a dimensional analysis, we first note that the liquid

and vapor phases may have different velocities Thus, we avoid

intro-500 Heat transfer in boiling and other phase-change configurations §9.7

ducing a flow speed and instead rely on the the superficial mass flux, G,

through the pipe:

tion for the flow boiling heat transfer coefficient, hfb, should take the

following form for saturated flow in vertical tubes:

hfb= fnhlo, G, x, h fg , q w , ρ f , ρ g , D

(9.45)

It should be noted that other liquid properties, such as viscosity and

con-ductivity, are represented indirectly through hlo This functional

equa-tion has eight dimensional variables (and one dimensionless variable, x)

in five dimensions (m, kg, s, J, K) We thus obtain three more

dimension-less groups to go with x, specifically

Bo≡ qw

§9.7 Forced convection boiling in tubes 501

Table 9.4 Fluid-dependent parameter F in the Kandlikar

cor-relation for copper tubing Additional values are given in [9.47]

When the convection number is large (Co  1), as for low quality,

nucleate boiling dominates In this range, hfb/hlorises with increasing Bo

and is approximately independent of Co When the convection number

is smaller, as at higher quality, the effect of the boiling number declines

and hfb/hloincreases with decreasing Co

Correlations having the general form of eqn (9.49) were developed

by Schrock and Grossman [9.48], Shah [9.49], and Gungor and

Winter-ton [9.50] Kandlikar [9.45, 9.47, 9.51] refined this approach further,

obtaining good accuracy and better capturing the parametric trends His

method is to calculate hfb/hlofrom each of the following two correlations

and to choose the larger value:

where “nbd” means “nucleate boiling dominant” and “cbd” means

“con-vective boiling dominant”

In these equations, the orientation factor, f o, is set to unity for

ver-tical tubes4 and F is a fluid-dependent parameter whose value is given

4 The value for horizontal tubes is given in eqn ( 9.52 ).

502 Heat transfer in boiling and other phase-change configurations §9.7

in Table9.4 The parameter F arises here for the same reason that

fluid-dependent parameters appear in nucleate boiling correlations: surfacetension, contact angles, and other fluid-dependent variables influencenucleation and bubble growth The values in Table9.4are for commer-cial grades of copper tubing For stainless steel tubing, Kandlikar recom-

mends F = 1 for all fluids Equations (9.50) are applicable for the

satu-rated boiling regimes (C through F ) with quality in the range 0 < x ≤ 0.8.

For subcooled conditions, see Problem9.21

Example 9.9

0.6 kg/s of saturated H2O at T b = 207 ◦C flows in a 5 cm diameter

ver-tical tube heated at a rate of 184,000 W/m2 Find the wall temperature

at a point where the quality x is 20%.

Solution. Data for water are taken from Tables A.3–A.5 We first

compute hlo

G = A m˙

pipe = 0.001964 0.6 = 305.6 kg/m2sand

Relo= GD

µ f = (305.6)(0.05)

1.297 × 10 −4 = 1.178 × 105From eqns (7.42) and (7.43):

for water, and for a vertical tube, f o = 1 Also,

§9.7 Forced convection boiling in tubes 503

Substituting into eqns (9.50):

hfb

nbd= (3, 115)(1 − 0.2) 0.8

0.6683 (0.3110) −0.2 (1) + 1058 (3.147 × 10 −4 ) 0.7 (1)

= 11, 950 W/m2K

hfb

cbd= (3, 115)(1 − 0.2) 0.8

1.136 (0.3110) −0.9 (1) + 667.2 (3.147 × 10 −4 ) 0.7 (1)

The Kandlikar correlation leads to mean deviations of 16% for water

and 19% for the various refrigerants The Gungor and Winterton

corre-lation [9.50], which is popular for its simplicity, does not contain

fluid-specific coefficients, but it is somewhat less accurate than either the

Kan-dlikar equations or the more complex Steiner and Taborek method [9.45,

9.46] These three approaches, however, are among the best available

Two-phase flow and heat transfer in horizontal tubes

The preceding discussion of flow boiling in tubes is largely restricted to

vertical tubes Several of the flow regimes in Fig 9.18 will be altered

as shown in Fig 9.20if the tube is oriented horizontally The reason is

that, especially at low quality, liquid will tend to flow along the bottom of

the pipe and vapor along the top The patterns shown in Fig.9.20, by the

way, will also be observed during the reverse process—condensation—or

during adiabatic two-phase flow

Which flow pattern actually occurs depends on several parameters

in a fairly complex way While many methods have been suggested to

predict what flow pattern will result for a given set of conditions in the

pipe, one of the best is that developed by Dukler, Taitel, and their

co-workers Their two-phase flow-regime maps are summarized in [9.52]

and [9.53]

For the prediction of heat transfer, the most important additional

parameter is the Froude number, Frlo, which characterizes the strength

of the flow’s inertia (or momentum) relative to the gravitational forces

504 Heat transfer in boiling and other phase-change configurations §9.7

Figure 9.20 The discernible flow

regimes during boiling, condensation, or

adiabatic flow from left to right in

horizontal tubes

that drive the separation of the liquid and vapor phases:

Frlo≡ G2

When Frlo< 0.04, the top of the tube becomes relatively dry and hfb/hlo

begins to decline as the Froude number decreases further

Kandlikar found that he could modify his correlation to account for

gravitational effects in horizontal tubes by changing the value of f o ineqns (9.50):

Peak heat flux

We have seen that there are two limiting heat fluxes in flow boiling in atube: dryout and burnout The latter is the more dangerous of the twosince it occurs at higher heat fluxes and gives rise to more catastrophictemperature rises Collier and Thome provide an extensive discussion ofthe subject [9.43], as does Hewitt [9.54]

§9.8 Forced convective condensation heat transfer 505

One effective set of empirical formulas was developed by Katto [9.55]

He used dimensional analysis to show that



where L is the length of the tube and D its diameter Since G2L σ ρ f

is a Weber number, we can see that this equation is of the same form

as eqn (9.39) Katto identifies several regimes of flow boiling with both

saturated and subcooled liquid entering the pipe For each of these

re-gions, he and Ohne [9.56] later fit a successful correlation of this form to

existing data

Pressure gradients in flow boiling

Pressure gradients in flow boiling interact with the flow pattern and the

void fraction, and they can change the local saturation temperature of the

fluid Gravity, flow acceleration, and friction all contribute to pressure

change, and friction can be particularly hard to predict In particular, the

frictional pressure gradient can increase greatly as the flow quality rises

from the pure liquid state to the pure vapor state; the change can amount

to more than two orders of magnitude at low pressures Data correlations

are usually used to estimate the frictional pressure loss, but they are,

at best, accurate to within about ±30% Whalley [9.57] provides a nice

introduction such methods Certain complex models, designed for use

in computer codes, can be used to make more accurate predictions [9.58]

When vapor is blown or forced past a cool wall, it exerts a shear stress

on the condensate film If the direction of forced flow is downward, it

will drag the condensate film along, thinning it out and enhancing heat

transfer It is not hard to show (see Problem9.22) that



(9.53)

where τ δis the shear stress exerted by the vapor flow on the condensate

film

Equation (9.53) is the starting point for any analysis of forced

convec-tion condensaconvec-tion on an external surface Notice that if τ δis negative—if

506 Heat transfer in boiling and other phase-change configurations §9.9

the shear opposes the direction of gravity—then it will have the effect of

thickening δ and reducing heat transfer Indeed, if for any value of δ,

τ δ = − 3g(ρ f − ρg)

the shear stress will have the effect of halting the flow of condensate

completely for a moment until δ grows to a larger value.

Heat transfer solutions based on eqn (9.53) are complex because theyrequire that one solve the boundary layer problem in the vapor in order

to evaluate τ δ; and this solution must be matched with the velocity atthe outside surface of the condensate film Collier and Thome [9.43,

§10.5] discuss such solutions in some detail One explicit result has beenobtained in this way for condensation on the outside of a horizontalcylinder by Shekriladze and Gomelauri [9.59]:

An automobile windshield normally is covered with droplets during alight rainfall They are hard to see through, and one must keep the wind-shield wiper moving constantly to achieve any kind of visibility A glasswindshield is normally quite clean and is free of any natural oxides, sothe water forms a contact angle on it and any film will be unstable Thewater tends to pull into droplets, which intersect the surface at the con-tact angle Visibility can be improved by mixing a surfactant chemicalinto the window-washing water to reduce surface tension It can also be

§9.9 Dropwise condensation 507

improved by preparing the surface with a “wetting agent” to reduce the

contact angle.5

Such behavior can also occur on a metallic condensing surface, but

there is an important difference: Such surfaces are generally wetting

Wetting can be temporarily suppressed, and dropwise condensation can

be encouraged, by treating an otherwise clean surface (or the vapor) with

oil, kerosene, or a fatty acid But these contaminants wash away fairly

quickly More permanent solutions have proven very elusive, with the

result the liquid condensed in heat exchangers almost always forms a

film

It is regrettable that this is the case, because what is called

drop-wise condensation is an extremely effective heat removal mechanism.

Figure 9.21 shows how it works Droplets grow from active nucleation

sites on the surface, and in this sense there is a great similarity between

nucleate boiling and dropwise condensation The similarity persists as

the droplets grow, touch, and merge with one another until one is large

enough to be pulled away from its position by gravity It then slides off,

wiping away the smaller droplets in its path and leaving a dry swathe in

its wake New droplets immediately begin to grow at the nucleation sites

in the path

The repeated re-creation of the early droplet growth cycle creates a

very efficient heat removal mechanism It is typically ten times more

effective than film condensation under the same temperature difference

Indeed, condensing heat transfer coefficients as high as 200,000 W/m2K

can be obtained with water at 1 atm Were it possible to sustain dropwise

condensation, we would certainly design equipment in such a way as to

make use of it

Unfortunately, laboratory experiments on dropwise condensation are

almost always done on surfaces that have been prepared with oleic, stearic,

or other fatty acids, or, more recently, with dioctadecyl disulphide These

nonwetting agents, or promoters as they are called, are discussed in

[9.60,9.61] While promoters are normally impractical for industrial use,

since they either wash away or oxidize, experienced plant engineers have

sometimes added rancid butter through the cup valves of commercial

condensers to get at least temporary dropwise condensation

Finally, we note that the obvious tactic of coating the surface with a

5 A way in which one can accomplish these ends is by wiping the wet window with

a cigarette It is hard to tell which of the two effects the many nasty chemicals in the

cigarette achieve.

a The process of liquid removal during dropwise

con-densation

b Typical photograph of dropwise condensation

pro-vided by Professor Borivoje B Miki´c Notice the dry paths

on the left and in the wake of the middle droplet

Figure 9.21 Dropwise condensation.

508

§9.10 The heat pipe 509

thin, nonwetting, polymer film (such as PTFE, or Teflon) adds just enough

conduction resistance to reduce the overall heat transfer coefficient to a

value similar to film condensation, fully defeating its purpose!

(Suffi-ciently thin polymer layers have not been found to be durable.) Noble

metals, such as gold, platinum, and palladium, can also be used as

non-wetting coating, and they have sufficiently high thermal conductivity to

avoid the problem encountered with polymeric coatings For gold,

how-ever, the minimum effective coating thickness is about 0.2 µm, or about

1/8 Troy ounce per square meter [9.62] Such coatings are far too

expen-sive for the vast majority of technical applications

A heat pipe is a device that combines the high efficiencies of boiling and

condensation It is aptly named because it literally pipes heat from a hot

region to a cold one

The operation of a heat pipe is shown in Fig.9.22 The pipe is a tube

that can be bent or turned in any way that is convenient The inside of

the tube is lined with a layer of wicking material The wick is wetted with

an appropriate liquid One end of the tube is exposed to a heat source

that evaporates the liquid from the wick The vapor then flows from the

hot end of the tube to the cold end, where it is condensed Capillary

action moves the condensed liquid axially along the wick, back to the

evaporator where it is again vaporized

Placing a heat pipe between a hot region and a cold one is thus

sim-ilar to connecting the regions with a material of extremely high thermal

conductivity—potentially orders of magnitude higher than any solid

ma-terial Such devices are used not only for achieving high heat transfer

rates between a source and a sink but for a variety of less obvious

pur-poses They are used, for example, to level out temperatures in systems,

since they function almost isothermally and offer very little thermal

re-sistance

Design considerations in matching a heat pipe to a given application

center on the following issues

• Selection of the right liquid The intended operating temperature of

the heat pipe can be met only with a fluid whose saturation

tem-peratures cover the design temperature range Depending on the

temperature range needed, the liquid can be a cryogen, an organic

510 Heat transfer in boiling and other phase-change configurations §9.10

Figure 9.22 A typical heat pipe configuration.

liquid, water, a liquid metal, or, in principle, almost any fluid ever, the following characteristics will serve to limit the vapor massflow per watt, provide good capillary action in the wick, and controlthe temperature rise between the wall and the wick:

How-i) High latent heatii) High surface tensioniii) Low liquid viscositiesiv) High thermal conductivityTwo liquids that meet these four criteria admirably are water andmercury, although toxicity and wetting problems discourage theuse of the latter Ammonia is useful at temperatures that are abit too low for water At high temperatures, sodium and lithiumhave good characteristics, while nitrogen is good for cryogenic tem-

peratures Fluids can be compared using the merit number, M =

h fg σ /ν f (see Problem9.36)

• Selection of the tube material The tube material must be compatible

with the working fluid Gas generation and corrosion are particularconsiderations Copper tubes are widely used with water, methanol,and acetone, but they cannot be used with ammonia Stainless steel

§9.10 The heat pipe 511

tubes can be used with ammonia and many liquid metals, but are

not suitable for long term service with water In some aerospace

applications, aluminum is used for its low weight; however, it is

compatible with working fluids other than ammonia

• Selection and installation of the wick Like the tube material, the

wick material must be compatible with the working fluid In

ad-dition, the working fluid must be able to wet the wick Wicks can

be fabricated from a metallic mesh, from a layer of sintered beads,

or simply by scoring grooves along the inside surface of the tube

Many ingenious schemes have been created for bonding the wick to

the inside of the pipe and keeping it at optimum porosity

• Operating limits of the heat pipe The heat transfer through a heat

pipe is restricted by

i) Viscous drag in the wick at low temperature

ii) The sonic, or choking, speed of the vapor

iii) Drag of the vapor on the counterflowing liquid in the wick

iv) Ability of capillary forces in the wick to pump the liquid through

the pressure rise between evaporator and condenser

v) The boiling burnout heat flux in the evaporator section

These items much each be dealt with in detail during the design of

a new heat pipe [9.63]

• Control of the pipe performance Often a given heat pipe will be

called upon to function over a range of conditions—under varying

evaporator heat loads, for example One way to vary its

perfor-mance is through the introduction of a non-condensible gas in the

pipe This gas will collect at the condenser, limiting the area of

the condenser that vapor can reach By varying the amount of gas,

the thermal resistance of the heat pipe can be controlled In the

absence of active control of the gas, an increase in the heat load

at the evaporator will raise the pressure in the pipe, compressing

the noncondensible gas and lowering the thermal resistance of the

pipe The result is that the temperature at the evaporator remains

essentially constant even as the heat load rises as falls

Heat pipes have proven useful in cooling high power-density

elec-tronic devices The evaporator is located on a small elecelec-tronic component

512 Chapter 9: Heat transfer in boiling and other phase-change configurations

Figure 9.23 A heat sink for cooling a microprocessor tesy of Dr A B Patel, Aavid Thermalloy LLC

Cour-to be cooled, perhaps a microprocessor, and the condenser is finned andcooled by a forced air flow (in a desktop or mainframe computer) or isunfinned and cooled by conduction into the exterior casing or structuralframe (in a laptop computer) These applications rely on having a heatpipe with much larger condenser area than evaporator area Thus, theheat fluxes on the condenser are kept relatively low This facilitates suchuncomplicated means for the ultimate heat disposal as using a small fan

to blow air over the condenser

One heat-pipe-based electronics heat sink is shown in Fig.9.23 Thecopper block at center is attached to a microprocessor, and the evapora-tor sections of two heat pipes are embedded in the block The condensersections of the pipes have copper fins pressed along their length A pair

of spring clips holds the unit in place These particular heat pipes havecopper tubes with water as the working fluid

The reader interested in designing or selecting a heat pipe will find abroad discussion of such devices in the book by Dunn and Reay [9.63]

Problems 513

Problems

9.1 A large square tank with insulated sides has a copper base

1.27 cm thick The base is heated to 650◦C and saturated water

is suddenly poured in the tank Plot the temperature of the

base as a function of time on the basis of Fig.9.2if the bottom

of the base is insulated In your graph, indicate the regimes

of boiling and note the temperature at which cooling is most

rapid

9.2 Predict qmaxfor the two heaters in Fig.9.3b At what

percent-age of qmax is each one operating?

9.3 A very clean glass container of water at 70◦C is depressurized

until it is subcooled 30◦C Then it suddenly and explosively

“flashes” (or boils) What is the pressure at which this

hap-pens? Approximately what diameter of gas bubble, or other

disturbance in the liquid, caused it to flash?

9.4 Plot the unstable bubble radius as a function of liquid

super-heat for water at 1 atm Comment on the significance of your

curve

9.5 In chemistry class you have probably witnessed the phenomenon

of “bumping” in a test tube (the explosive boiling that blows

the contents of the tube all over the ceiling) Yet you have

never seen this happen in a kitchen pot Explain why not

9.6 Use van der Waal’s equation of state to approximate the

high-est reduced temperature to which water can be superheated at

low pressure How many degrees of superheat does this

sug-gest that water can sustain at the low pressure of 1 atm? (It

turns out that this calculation is accurate within about 10%.)

What would R bbe at this superheat?

9.7 Use Yamagata’s equation, (9.3), to determine how nucleation

site density increases with∆T for Berenson’s curves in Fig.9.14

(That is, find c in the relation n = constant ∆T c.)

9.8 Suppose that Csffor a given surface is high by 50% What will

be the percentage error in q calculated for a given value of ∆T ?

[Low by 70%.]

514 Chapter 9: Heat transfer in boiling and other phase-change configurations

9.9 Water at 100 atm boils on a nickel heater whose temperature

is 6◦ C above Tsat Find h and q.

9.10 Water boils on a large flat plate at 1 atm Calculate qmaxif the

plate is operated on the surface of the moon (at16of gearth−normal).

What would qmax be in a space vehicle experiencing 10−4 of

gearth−normal?

9.11 Water boils on a 0.002 m diameter horizontal copper wire Plot,

to scale, as much of the boiling curve on log q vs log ∆T

coor-dinates as you can The system is at 1 atm

9.12 Redo Problem 9.11 for a 0.03 m diameter sphere in water at

10 atm

9.13 Verify eqn (9.17)

9.14 Make a sketch of the q vs (T w −Tsat) relation for a pool boiling

process, and invent a graphical method for locating the points

where h is maximum and minimum.

9.15 A 2 mm diameter jet of methanol is directed normal to the

center of a 1.5 cm diameter disk heater at 1 m/s How many

watts can safely be supplied by the heater?

9.16 Saturated water at 1 atm boils on a ½ cm diameter platinum

rod Estimate the temperature of the rod at burnout

9.17 Plot (T w − Tsat) and the quality x as a function of position x

for the conditions in Example9.9 Set x = 0 where x = 0 and

end the plot where the quality reaches 80%

9.18 Plot (T w − Tsat) and the quality x as a function of position in

an 8 cm I.D pipe if 0.3 kg/s of water at 100 ◦C passes through

it and q w = 200, 000 W/m2

9.19 Use dimensional analysis to verify the form of eqn (9.8)

9.20 Compare the peak heat flux calculated from the data given in

Problem 5.6with the appropriate prediction [The prediction

is within 11%.]

by Schrock and Grossman [9.48], Shah [9.49], and Gungor and

Winter-ton [9.50] Kandlikar [9.45, 9.47, 9.51] refined this approach further,

obtaining good accuracy and better capturing... the wick to

the inside of the pipe and keeping it at optimum porosity

• Operating limits of the heat pipe The heat transfer through a heat< /i>

pipe is restricted by

i)... developed by Dukler, Taitel, and their

co-workers Their two-phase flow-regime maps are summarized in [9.52]

and [9.53]

For the prediction of heat transfer, the most important