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Unformatted text preview: cen58933_ch10.qxd 9/4/2002 12:37 PM Page 515 CHAPTER BOILING AND
C O N D E N S AT I O N
e know from thermodynamics that when the temperature of a liquid
at a specified pressure is raised to the saturation temperature Tsat at
that pressure, boiling occurs. Likewise, when the temperature of a
vapor is lowered to Tsat, condensation occurs. In this chapter we study the
rates of heat transfer during such liquid-to-vapor and vapor-to-liquid phase
transformations.
Although boiling and condensation exhibit some unique features, they are
considered to be forms of convection heat transfer since they involve fluid
motion (such as the rise of the bubbles to the top and the flow of condensate
to the bottom). Boiling and condensation differ from other forms of convection in that they depend on the latent heat of vaporization hfg of the fluid and
the surface tension at the liquid–vapor interface, in addition to the properties of the fluid in each phase. Noting that under equilibrium conditions the
temperature remains constant during a phase-change process at a fixed pressure, large amounts of heat (due to the large latent heat of vaporization released or absorbed) can be transferred during boiling and condensation
essentially at constant temperature. In practice, however, it is necessary to
maintain some difference between the surface temperature Ts and Tsat for effective heat transfer. Heat transfer coefficients h associated with boiling and
condensation are typically much higher than those encountered in other forms
of convection processes that involve a single phase.
We start this chapter with a discussion of the boiling curve and the modes of
pool boiling such as free convection boiling, nucleate boiling, and film boiling. We then discuss boiling in the presence of forced convection. In the
second part of this chapter, we describe the physical mechanism of film condensation and discuss condensation heat transfer in several geometrical
arrangements and orientations. Finally, we introduce dropwise condensation
and discuss ways of maintaining it. W 10
CONTENTS
10–1 Boiling Heat Transfer 516
10–2 Pool Boiling 518
10–3 Flow Boiling 530
10–4 Condensation
Heat Transfer 532
10–5 Film Condensation 532
10–6 Film Condensation Inside Horizontal Tubes 545
10–7 Dropwise Condensation 545
Topic of Special Interest:
Heat Pipes 546 515 cen58933_ch10.qxd 9/4/2002 12:37 PM Page 516 516
HEAT TRANSFER
Evaporation
Air 10–1 Water
20°C Boiling Water
100°C Heating FIGURE 10–1
A liquid-to-vapor phase change
process is called evaporation if it
occurs at a liquid–vapor interface
and boiling if it occurs at a
solid–liquid interface.
P = 1 atm Water
Tsat = 100°C Bubbles 110°C Heating element FIGURE 10–2
Boiling occurs when a liquid is
brought into contact with a surface
at a temperature above the saturation
temperature of the liquid. I BOILING HEAT TRANSFER Many familiar engineering applications involve condensation and boiling heat
transfer. In a household refrigerator, for example, the refrigerant absorbs heat
from the refrigerated space by boiling in the evaporator section and rejects
heat to the kitchen air by condensing in the condenser section (the long coils
behind the refrigerator). Also, in steam power plants, heat is transferred to the
steam in the boiler where water is vaporized, and the waste heat is rejected
from the steam in the condenser where the steam is condensed. Some electronic components are cooled by boiling by immersing them in a fluid with an
appropriate boiling temperature.
Boiling is a liquid-to-vapor phase change process just like evaporation, but
there are significant differences between the two. Evaporation occurs at the
liquid–vapor interface when the vapor pressure is less than the saturation
pressure of the liquid at a given temperature. Water in a lake at 20°C, for
example, will evaporate to air at 20°C and 60 percent relative humidity since
the saturation pressure of water at 20°C is 2.3 kPa and the vapor pressure of
air at 20°C and 60 percent relative humidity is 1.4 kPa (evaporation rates are
determined in Chapter 14). Other examples of evaporation are the drying of
clothes, fruits, and vegetables; the evaporation of sweat to cool the human
body; and the rejection of waste heat in wet cooling towers. Note that evaporation involves no bubble formation or bubble motion (Fig. 10–1).
Boiling, on the other hand, occurs at the solid–liquid interface when a liquid is brought into contact with a surface maintained at a temperature Ts sufficiently above the saturation temperature Tsat of the liquid (Fig. 10–2). At 1
atm, for example, liquid water in contact with a solid surface at 110°C will
boil since the saturation temperature of water at 1 atm is 100°C. The boiling
process is characterized by the rapid formation of vapor bubbles at the
solid–liquid interface that detach from the surface when they reach a certain
size and attempt to rise to the free surface of the liquid. When cooking, we do
not say water is boiling until we see the bubbles rising to the top. Boiling is a
complicated phenomenon because of the large number of variables involved
in the process and the complex fluid motion patterns caused by the bubble formation and growth.
As a form of convection heat transfer, the boiling heat flux from a solid
surface to the fluid is expressed from Newton’s law of cooling as
q·boiling h(Ts Tsat) h Texcess (W/m2) (10-1) where Texcess Ts Tsat is called the excess temperature, which represents
the excess of the surface above the saturation temperature of the fluid.
In the preceding chapters we considered forced and free convection heat
transfer involving a single phase of a fluid. The analysis of such convection
processes involves the thermophysical properties , , k, and Cp of the fluid.
The analysis of boiling heat transfer involves these properties of the liquid
(indicated by the subscript l) or vapor (indicated by the subscript v) as well as
the properties hfg (the latent heat of vaporization) and (the surface tension).
The hfg represents the energy absorbed as a unit mass of liquid vaporizes
at a specified temperature or pressure and is the primary quantity of energy cen58933_ch10.qxd 9/4/2002 12:38 PM Page 517 517
CHAPTER 10 transferred during boiling heat transfer. The hfg values of water at various temperatures are given in Table A-9.
Bubbles owe their existence to the surface-tension at the liquid–vapor interface due to the attraction force on molecules at the interface toward the liquid phase. The surface tension decreases with increasing temperature and
becomes zero at the critical temperature. This explains why no bubbles are
formed during boiling at supercritical pressures and temperatures. Surface
tension has the unit N/m.
The boiling processes in practice do not occur under equilibrium conditions,
and normally the bubbles are not in thermodynamic equilibrium with the surrounding liquid. That is, the temperature and pressure of the vapor in a bubble
are usually different than those of the liquid. The pressure difference between
the liquid and the vapor is balanced by the surface tension at the interface. The
temperature difference between the vapor in a bubble and the surrounding liquid is the driving force for heat transfer between the two phases. When the liquid is at a lower temperature than the bubble, heat will be transferred from the
bubble into the liquid, causing some of the vapor inside the bubble to condense and the bubble to collapse eventually. When the liquid is at a higher
temperature than the bubble, heat will be transferred from the liquid to the
bubble, causing the bubble to grow and rise to the top under the influence of
buoyancy.
Boiling is classified as pool boiling or flow boiling, depending on the presence of bulk fluid motion (Fig. 10–3). Boiling is called pool boiling in the absence of bulk fluid flow and flow boiling (or forced convection boiling) in the
presence of it. In pool boiling, the fluid is stationary, and any motion of the
fluid is due to natural convection currents and the motion of the bubbles under the influence of buoyancy. The boiling of water in a pan on top of a stove
is an example of pool boiling. Pool boiling of a fluid can also be achieved by
placing a heating coil in the fluid. In flow boiling, the fluid is forced to move
in a heated pipe or over a surface by external means such as a pump. Therefore, flow boiling is always accompanied by other convection effects.
Pool and flow boiling are further classified as subcooled boiling or saturated boiling, depending on the bulk liquid temperature (Fig. 10–4). Boiling
is said to be subcooled (or local) when the temperature of the main body of
the liquid is below the saturation temperature Tsat (i.e., the bulk of the liquid is
subcooled) and saturated (or bulk) when the temperature of the liquid is
equal to Tsat (i.e., the bulk of the liquid is saturated). At the early stages of boiling, the bubbles are confined to a narrow region near the hot surface. This is
because the liquid adjacent to the hot surface vaporizes as a result of being
heated above its saturation temperature. But these bubbles disappear soon after they move away from the hot surface as a result of heat transfer from the
bubbles to the cooler liquid surrounding them. This happens when the bulk of
the liquid is at a lower temperature than the saturation temperature. The bubbles serve as “energy movers” from the hot surface into the liquid body by absorbing heat from the hot surface and releasing it into the liquid as they
condense and collapse. Boiling in this case is confined to a region in the locality of the hot surface and is appropriately called local or subcooled boiling.
When the entire liquid body reaches the saturation temperature, the bubbles
start rising to the top. We can see bubbles throughout the bulk of the liquid, Heating Heating (a) Pool boiling (b) Flow boiling FIGURE 10–3
Classification of boiling on the basis
of the presence of bulk fluid motion.
P = 1 atm P = 1 atm Subcooled 80°C
water
107°C Saturated 100°C
water
107°C
Bubble Heating Heating (a) Subcooled boiling (b) Saturated boiling FIGURE 10–4
Classification of boiling
on the basis of the presence of
bulk liquid temperature. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 518 518
HEAT TRANSFER and boiling in this case is called the bulk or saturated boiling. Next, we consider different boiling regimes in detail. 10–2 I POOL BOILING So far we presented some general discussions on boiling. Now we turn our
attention to the physical mechanisms involved in pool boiling, that is, the
boiling of stationary fluids. In pool boiling, the fluid is not forced to flow
by a mover such as a pump, and any motion of the fluid is due to natural
convection currents and the motion of the bubbles under the influence of
buoyancy.
As a familiar example of pool boiling, consider the boiling of tap water in a
pan on top of a stove. The water will initially be at about 15°C, far below the
saturation temperature of 100°C at standard atmospheric pressure. At the early
stages of boiling, you will not notice anything significant except some bubbles
that stick to the surface of the pan. These bubbles are caused by the release of
air molecules dissolved in liquid water and should not be confused with vapor
bubbles. As the water temperature rises, you will notice chunks of liquid water rolling up and down as a result of natural convection currents, followed by
the first vapor bubbles forming at the bottom surface of the pan. These bubbles get smaller as they detach from the surface and start rising, and eventually collapse in the cooler water above. This is subcooled boiling since the
bulk of the liquid water has not reached saturation temperature yet. The intensity of bubble formation increases as the water temperature rises further, and
you will notice waves of vapor bubbles coming from the bottom and rising to
the top when the water temperature reaches the saturation temperature (100°C
at standard atmospheric conditions). This full scale boiling is the saturated
boiling.
100°C 100°C 103°C 110°C Heating
(a) Natural convection
boiling Heating
(b) Nucleate boiling Vapor film Vapor pockets
100°C 100°C 180°C 400°C Heating
(c) Transition boiling Heating
(d) Film boiling FIGURE 10–5
Different boiling regimes
in pool boiling. Boiling Regimes and the Boiling Curve
Boiling is probably the most familiar form of heat transfer, yet it remains to be
the least understood form. After hundreds of papers written on the subject, we
still do not fully understand the process of bubble formation and we must still
rely on empirical or semi-empirical relations to predict the rate of boiling heat
transfer.
The pioneering work on boiling was done in 1934 by S. Nukiyama, who
used electrically heated nichrome and platinum wires immersed in liquids in
his experiments. Nukiyama noticed that boiling takes different forms, depending on the value of the excess temperature Texcess. Four different boiling
regimes are observed: natural convection boiling, nucleate boiling, transition
boiling, and film boiling (Fig. 10–5). These regimes are illustrated on the boiling curve in Figure 10–6, which is a plot of boiling heat flux versus the excess temperature. Although the boiling curve given in this figure is for water,
the general shape of the boiling curve remains the same for different fluids.
The specific shape of the curve depends on the fluid–heating surface material combination and the fluid pressure, but it is practically independent of
the geometry of the heating surface. We will describe each boiling regime
in detail. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 519 519
CHAPTER 10
Natural convection
boiling Bubbles
collapse
in the
liquid 106
·
qboiling, W/m2 Nucleate
boiling Transition
boiling C Film
boiling Maximum
(critical)
·
heat flux, qmax
E 105
B 104 103 Bubbles
rise to the
free surface A 1 ~5 10 D ·
Leidenfrost point, qmin ~30 100 ~120
∆Texcess = Ts – Tsat , °C 1000 Natural Convection Boiling (to Point A on the Boiling Curve)
We learned in thermodynamics that a pure substance at a specified pressure
starts boiling when it reaches the saturation temperature at that pressure. But
in practice we do not see any bubbles forming on the heating surface until the
liquid is heated a few degrees above the saturation temperature (about 2 to
6°C for water). Therefore, the liquid is slightly superheated in this case
(a metastable condition) and evaporates when it rises to the free surface.
The fluid motion in this mode of boiling is governed by natural convection
currents, and heat transfer from the heating surface to the fluid is by natural
convection. Nucleate Boiling (between Points A and C )
The first bubbles start forming at point A of the boiling curve at various preferential sites on the heating surface. The bubbles form at an increasing rate at
an increasing number of nucleation sites as we move along the boiling curve
toward point C.
The nucleate boiling regime can be separated into two distinct regions. In
region A–B, isolated bubbles are formed at various preferential nucleation
sites on the heated surface. But these bubbles are dissipated in the liquid
shortly after they separate from the surface. The space vacated by the rising
bubbles is filled by the liquid in the vicinity of the heater surface, and the
process is repeated. The stirring and agitation caused by the entrainment of the
liquid to the heater surface is primarily responsible for the increased heat
transfer coefficient and heat flux in this region of nucleate boiling.
In region B–C, the heater temperature is further increased, and bubbles form
at such great rates at such a large number of nucleation sites that they form
numerous continuous columns of vapor in the liquid. These bubbles move all
the way up to the free surface, where they break up and release their vapor
content. The large heat fluxes obtainable in this region are caused by the combined effect of liquid entrainment and evaporation. FIGURE 10–6
Typical boiling curve for water
at 1 atm pressure. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 520 520
HEAT TRANSFER At large values of Texcess, the rate of evaporation at the heater surface
reaches such high values that a large fraction of the heater surface is covered
by bubbles, making it difficult for the liquid to reach the heater surface and
wet it. Consequently, the heat flux increases at a lower rate with increasing
Texcess, and reaches a maximum at point C. The heat flux at this point is
·
called the critical (or maximum) heat flux, qmax. For water, the critical heat
2
flux exceeds 1 MW/m .
Nucleate boiling is the most desirable boiling regime in practice because
high heat transfer rates can be achieved in this regime with relatively small
values of Texcess, typically under 30°C for water. The photographs in Figure
10–7 show the nature of bubble formation and bubble motion associated with
nucleate, transition, and film boiling. Transition Boiling (between Points C and D on the
Boiling Curve)
As the heater temperature and thus the Texcess is increased past point C, the
heat flux decreases, as shown in Figure 10–6. This is because a large fraction
of the heater surface is covered by a vapor film, which acts as an insulation
due to the low thermal conductivity of the vapor relative to that of the liquid.
In the transition boiling regime, both nucleate and film boiling partially occur.
Nucleate boiling at point C is completely replaced by film boiling at point D.
Operation in the transition boiling regime, which is also called the unstable
film boiling regime, is avoided in practice. For water, transition boiling occurs
over the excess temperature range from about 30°C to about 120°C. Film Boiling (beyond Point D ) ·
q
W
—2
–
m
·
qmax Sudden jump
in temperature
Bypassed
part of the
boiling
curve 106 Sudden drop
in temperature
1 ·
qmin 1000
10
100
∆Texcess = Ts – Tsat, °C FIGURE 10–8
The actual boiling curve obtained
with heated platinum wire in water
as the heat flux is increased and
then decreased. In this region the heater surface is completely covered by a continuous stable
vapor film. Point D, where the heat flux reaches a minimum, is called the
Leidenfrost point, in honor of J. C. Leidenfrost, who observed in 1756 that
liquid droplets on a very hot surface jump around and slowly boil away. The
presence of a vapor film between the heater surface and the liquid is responsible for the low heat transfer rates in the film boiling region. The heat transfer
rate increases with increasing excess temperature as a result of heat transfer
from the heated surface to the liquid through the vapor film by radiation,
which becomes significant at high temperatures.
A typical boiling process will not follow the boiling curve beyond point C,
as Nukiyama has observed during his experiments. Nukiyama noticed, with
surprise, that when the power applied to the nichrome wire immersed in wa·
ter exceeded qmax even slightly, the wire temperature increased suddenly to the
melting point of the wire and burnout occurred beyond his control. When he
repeated the experiments with platinum wire, which has a much higher melting point, he was able to avoid burnout and maintain heat fluxes higher than
·
qmax. When he gradually reduced power, he obtained the cooling curve shown
·
in Figure 10–8 with a sudden drop in excess temperature when qmin is reached.
Note that the boiling process cannot follow the transition boiling part of the
boiling curve past point C unless the power applied is reduced suddenly.
The burnout phenomenon in boiling can be explained as follows: In order to
·
move beyond point C where qmax occurs, we must increase the heater surface
temperature Ts. To increase Ts, however, we must increase the heat flux. But cen58933_ch10.qxd 9/4/2002 12:38 PM Page 521 521
CHAPTER 10 (a) (b) (c) the fluid cannot receive this increased energy at an excess temperature just beyond point C. Therefore, the heater surface ends up absorbing the increased
energy, causing the heater surface temperature Ts to rise. But the fluid can receive even less energy at this increased excess temperature, causing the heater
surface temperature Ts to rise even further. This continues until the surface FIGURE 10–7
Various boiling regimes during
boiling of methanol on a horizontal
1-cm-diameter steam-heated
copper tube: (a) nucleate boiling,
(b) transition boiling, and (c) film
boiling (from J. W. Westwater and
J. G. Santangelo, University of
Illinois at Champaign-Urbana). cen58933_ch10.qxd 9/4/2002 12:38 PM Page 522 522
HEAT TRANSFER
·
q
W
—2
–
m
·
·
qmax qmax = constant
C E Sudden jump
in temperature
Ts, °C
Tmelting FIGURE 10–9
An attempt to increase the boiling heat
flux beyond the critical value often
causes the temperature of the heating
element to jump suddenly to a value
that is above the melting point,
resulting in burnout. temperature reaches a point at which it no longer rises and the heat supplied
can be transferred to the fluid steadily. This is point E on the boiling curve,
which corresponds to very high surface temperatures. Therefore, any attempt
·
to increase the heat flux beyond qmax will cause the operation point on the boiling curve to jump suddenly from point C to point E. However, surface temperature that corresponds to point E is beyond the melting point of most heater
materials, and burnout occurs. Therefore, point C on the boiling curve is also
called the burnout point, and the heat flux at this point the burnout heat flux
(Fig. 10–9).
·
Most boiling heat transfer equipment in practice operate slightly below qmax
to avoid any disastrous burnout. However, in cryogenic applications involving
fluids with very low boiling points such as oxygen and nitrogen, point E usually falls below the melting point of the heater materials, and steady film boiling can be used in those cases without any danger of burnout. Heat Transfer Correlations in Pool Boiling
Boiling regimes discussed above differ considerably in their character, and
thus different heat transfer relations need to be used for different boiling
regimes. In the natural convection boiling regime, boiling is governed by natural convection currents, and heat transfer rates in this case can be determined
accurately using natural convection relations presented in Chapter 9. Nucleate Boiling
In the nucleate boiling regime, the rate of heat transfer strongly depends on
the nature of nucleation (the number of active nucleation sites on the surface,
the rate of bubble formation at each site, etc.), which is difficult to predict.
The type and the condition of the heated surface also affect the heat transfer.
These complications made it difficult to develop theoretical relations for heat
transfer in the nucleate boiling regime, and people had to rely on relations
based on experimental data. The most widely used correlation for the rate of
heat transfer in the nucleate boiling regime was proposed in 1952 by
Rohsenow, and expressed as
q·nucleate l hfg g( l v) 1/2 Cp(Ts Tsat)
Csf hfg Prn
l 3 where
·
q nucleate
l hfg
g
l
v Cpl
Ts
Tsat
Csf
Prl
n nucleate boiling heat flux, W/m2
viscosity of the liquid, kg/m · s
enthalpy of vaporization, J/kg
gravitational acceleration, m/s2
density of the liquid, kg/m3
density of the vapor, kg/m3
surface tension of liquid–vapor interface, N/m
specific heat of the liquid, J/kg · °C
surface temperature of the heater, °C
saturation temperature of the fluid, °C
experimental constant that depends on surface–fluid combination
Prandtl number of the liquid
experimental constant that depends on the fluid (10-2) cen58933_ch10.qxd 9/4/2002 12:38 PM Page 523 523
CHAPTER 10 It can be shown easily that using property values in the specified units in
the Rohsenow equation produces the desired unit W/m2 for the boiling heat
flux, thus saving one from having to go through tedious unit manipulations
(Fig. 10–10).
The surface tension at the vapor–liquid interface is given in Table 10–1 for
water, and Table 10–2 for some other fluids. Experimentally determined values of the constant Csf are given in Table 10–3 for various fluid–surface combinations. These values can be used for any geometry since it is found that the
rate of heat transfer during nucleate boiling is essentially independent of the
geometry and orientation of the heated surface. The fluid properties in Eq.
10–2 are to be evaluated at the saturation temperature Tsat.
The condition of the heater surface greatly affects heat transfer, and the
Rohsenow equation given above is applicable to clean and relatively smooth
surfaces. The results obtained using the Rohsenow equation can be in error by
100% for the heat transfer rate for a given excess temperature and by 30%
for the excess temperature for a given heat transfer rate. Therefore, care
should be exercised in the interpretation of the results.
Recall from thermodynamics that the enthalpy of vaporization hfg of a pure
substance decreases with increasing pressure (or temperature) and reaches
zero at the critical point. Noting that hfg appears in the denominator of the
Rohsenow equation, we should see a significant rise in the rate of heat transfer at high pressures during nucleate boiling. Peak Heat Flux
In the design of boiling heat transfer equipment, it is extremely important for
the designer to have a knowledge of the maximum heat flux in order to avoid
the danger of burnout. The maximum (or critical) heat flux in nucleate pool
boiling was determined theoretically by S. S. Kutateladze in Russia in 1948
and N. Zuber in the United States in 1958 using quite different approaches,
and is expressed as (Fig. 10–11)
·
q max Ccr hfg[ g 2 ( l )]1/4 (10-3) where Ccr is a constant whose value depends on the heater geometry. Exhaustive experimental studies by Lienhard and his coworkers indicated that the
value of Ccr is about 0.15. Specific values of Ccr for different heater geometries are listed in Table 10–4. Note that the heaters are classified as being large
or small based on the value of the parameter L*.
Equation 10–3 will give the maximum heat flux in W/m2 if the properties
are used in the units specified earlier in their descriptions following Eq. 10–2.
The maximum heat flux is independent of the fluid–heating surface combination, as well as the viscosity, thermal conductivity, and the specific heat of the
liquid.
Note that v increases but and hfg decrease with increasing pressure, and
·
thus the change in qmax with pressure depends on which effect dominates. The
·
experimental studies of Cichelli and Bonilla indicate that qmax increases with
pressure up to about one-third of the critical pressure, and then starts to de·
crease and becomes zero at the critical pressure. Also note that qmax is proportional to hfg, and large maximum heat fluxes can be obtained using fluids with
a large enthalpy of vaporization, such as water. ·
q kg
J
m · s kg
m kg
s2 m3
N
m
W1
m m2
W/m2 1/23 J
°C
kg · °C
J
kg 3 1/2 (1)3 FIGURE 10–10
Equation 10–2 gives the
boiling heat flux in W/m2 when
the quantities are expressed in the
units specified in their descriptions.
TABLE 10–1
Surface tension of liquid–vapor
interface for water
T, °C , N/m* 0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
374 0.0757
0.0727
0.0696
0.0662
0.0627
0.0589
0.0550
0.0509
0.0466
0.0422
0,0377
0.0331
0.0284
0.0237
0.0190
0.0144
0.0099
0.0056
0.0019
0.0 *Multiply by 0.06852 to convert to lbf/ft or by
2.2046 to convert to lbm/s2. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 524 524
HEAT TRANSFER TABLE 10–2 TABLE 10–3 Surface tension of some fluids (from
Suryanarayana, Ref. 26; originally
based on data from Jasper, Ref. 14) Values of the coefficient Csf and n for various fluid–surface combinations Substance
and Temp.
Range Water–copper (polished)
Water–copper (scored)
Water–stainless steel (mechanically polished)
Water–stainless steel (ground and polished)
Water–stainless steel (teflon pitted)
Water–stainless steel (chemically etched)
Water–brass
Water–nickel
Water–platinum
n-Pentane–copper (polished)
n-Pentane–chromium
Benzene–chromium
Ethyl alcohol–chromium
Carbon tetrachloride–copper
Isopropanol–copper Fluid-Heating Surface Combination Surface Tension,
, N/m* (T in °C) Ammonia, 75 to 40°C:
0.0264 0.000223T
Benzene, 10 to 80°C:
0.0315 0.000129T
Butane, 70 to 20°C:
0.0149 0.000121T
Carbon dioxide, 30 to 20°C:
0.0043 0.000160T
Ethyl alcohol, 10 to 70°C:
0.0241 0.000083T
Mercury, 5 to 200°C:
0.4906 0.000205T
Methyl alcohol, 10 to 60°C:
0.0240 0.000077T
Pentane, 10 to 30°C:
0.0183 0.000110T
Propane, 90 to 10°C:
0.0092 0.000087T *Multiply by 0.06852 to convert to lbf/ft or by
2.2046 to convert to lbm/s2. Csf n 0.0130
0.0068
0.0130
0.0060
0.0058
0.0130
0.0060
0.0060
0.0130
0.0154
0.0150
0.1010
0.0027
0.0130
0.0025 1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.7
1.7
1.7
1.7
1.7
1.7 Minimum Heat Flux
Minimum heat flux, which occurs at the Leidenfrost point, is of practical interest since it represents the lower limit for the heat flux in the film boiling
regime. Using the stability theory, Zuber derived the following expression for
the minimum heat flux for a large horizontal plate,
q·min 0.09 hfg g(
(l ) l 1/4 (10-4) 2 ) where the constant 0.09 was determined by Berenson in 1961. He replaced the
theoretically determined value of 24 by 0.09 to match the experimental data
better. Still, the relation above can be in error by 50 percent or more. TABLE 10–4
Values of the coefficient Ccr for use in Eq. 10–3 for maximum heat flux
(dimensionless parameter L* L[g( l
)/ ]1/2) Heater Geometry
Large horizontal flat heater
Small horizontal flat heater1
Large horizontal cylinder
Small horizontal cylinder
Large sphere
Small sphere
1 K1 /[g( l v)Aheater] Ccr Charac.
Dimension
of Heater, L Range of L* 0.149
Width or diameter
L* 27
18.9K1
Width or diameter
9 L* 20
0.12
Radius
L* 1.2
0.12L* 0.25
Radius
0.15 L* 1.2
0.11
Radius
L* 4.26
0.227L* 0.5
Radius
0.15 L* 4.26 cen58933_ch10.qxd 9/4/2002 12:38 PM Page 525 525
CHAPTER 10
·
q Film Boiling
Using an analysis similar to Nusselt’s theory on filmwise condensation presented in the next section, Bromley developed a theory for the prediction of
heat flux for stable film boiling on the outside of a horizontal cylinder. The
heat flux for film boiling on a horizontal cylinder or sphere of diameter D is
given by
·
q film Cfilm gk3 ( )[hfg 0.4Cp (Ts
D(Ts Tsat) l Tsat) (Ts Tsat) (10-5) 0.62 for horizontal cylinders
0.67 for spheres (Ts4 4
Tsat) (10-6) where
is the emissivity of the heating surface and
5.67
10 8 W/m2 · K4 is the Stefan–Boltzman constant. Note that the temperature in
this case must be expressed in K, not °C, and that surface tension and the
Stefan–Boltzman constant share the same symbol.
You may be tempted to simply add the convection and radiation heat transfers to determine the total heat transfer during film boiling. However, these
two mechanisms of heat transfer adversely affect each other, causing the total
heat transfer to be less than their sum. For example, the radiation heat transfer
from the surface to the liquid enhances the rate of evaporation, and thus
the thickness of the vapor film, which impedes convection heat transfer. For
·
·
qrad qfilm, Bromley determined that the relation
·
q total ·
q film 3·
q
4 rad Natural
convection
relations Minimum
heat flux
relation
Ts – Tsat Other properties are as listed before in connection with Eq. 10–2. We used a
modified latent heat of vaporization in Eq. 10–5 to account for the heat transfer associated with the superheating of the vapor.
The vapor properties are to be evaluated at the film temperature, given as
Tf (Ts Tsat)/2, which is the average temperature of the vapor film. The
liquid properties and hfg are to be evaluated at the saturation temperature at the
specified pressure. Again, this relation will give the film boiling heat flux in
W/m2 if the properties are used in the units specified earlier in their descriptions following Eq. 10–2.
At high surface temperatures (typically above 300°C), heat transfer across
the vapor film by radiation becomes significant and needs to be considered
(Fig. 10–12). Treating the vapor film as a transparent medium sandwiched between two large parallel plates and approximating the liquid as a blackbody,
radiation heat transfer can be determined from
·
q rad Nucleate
boiling
relations 1/4 where kv is the thermal conductivity of the vapor in W/m · °C and
Cfilm Film
boiling
relations Critical heat
flux relation (10-7) correlates experimental data well.
Operation in the transition boiling regime is normally avoided in the design
of heat transfer equipment, and thus no major attempt has been made to develop general correlations for boiling heat transfer in this regime. FIGURE 10–11
Different relations are
used to determine the heat
flux in different boiling regimes.
P = 1 atm 100°C
400°C
Vapor
·
qfilm boiling ·
qrad
Heating FIGURE 10–12
At high heater surface temperatures,
radiation heat transfer becomes
significant during film boiling. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 526 526
HEAT TRANSFER Note that the gravitational acceleration g, whose value is approximately
9.81 m/s2 at sea level, appears in all of the relations above for boiling heat
transfer. The effects of low and high gravity (as encountered in aerospace
applications and turbomachinery) are studied experimentally. The studies
confirm that the critical heat flux and heat flux in film boiling are proportional
to g1/4. However, they indicate that heat flux in nucleate boiling is practically
independent of gravity g, instead of being proportional to g1/2, as dictated by
Eq. 10–2. Enhancement of Heat Transfer in Pool Boiling Liquid
Vapor Nucleation sites for vapor FIGURE 10–13
The cavities on a rough surface act as
nucleation sites and enhance
boiling heat transfer. P = 1 atm 100°C The pool boiling heat transfer relations given above apply to smooth surfaces.
Below we will discuss some methods to enhance heat transfer in pool boiling.
We pointed out earlier that the rate of heat transfer in the nucleate boiling
regime strongly depends on the number of active nucleation sites on the surface, and the rate of bubble formation at each site. Therefore, any modification that will enhance nucleation on the heating surface will also enhance heat
transfer in nucleate boiling. It is observed that irregularities on the heating
surface, including roughness and dirt, serve as additional nucleation sites during boiling, as shown in Figure 10–13. For example, the first bubbles in a pan
filled with water are most likely to form at the scratches at the bottom surface.
These scratches act like “nests” for the bubbles to form and thus increase the
rate of bubble formation. Berensen has shown that heat flux in the nucleate
boiling regime can be increased by a factor of 10 by roughening the heating
surface. However, these high heat transfer rates cannot be sustained for long
since the effect of surface roughness is observed to decay with time, and the
heat flux to drop eventually to values encountered on smooth surfaces. The effect of surface roughness is negligible on the critical heat flux and the heat
flux in film boiling.
Surfaces that provide enhanced heat transfer in nucleate boiling permanently are being manufactured and are available in the market. Enhancement
in nucleation and thus heat transfer in such special surfaces is achieved either
by coating the surface with a thin layer (much less than 1 mm) of very porous
material or by forming cavities on the surface mechanically to facilitate continuous vapor formation. Such surfaces are reported to enhance heat transfer
in the nucleate boiling regime by a factor of up to 10, and the critical heat flux
by a factor of 3. The enhancement provided by one such material prepared by
machine roughening, the thermoexcel-E, is shown in Figure 10–14. The use
of finned surfaces is also known to enhance nucleate boiling heat transfer and
the critical heat flux.
Boiling heat transfer can also be enhanced by other techniques such as mechanical agitation and surface vibration. These techniques are not practical,
however, because of the complications involved. Water
108°C Heating FIGURE 10–15
Schematic for Example 10–1. EXAMPLE 10–1 Nucleate Boiling of Water in a Pan Water is to be boiled at atmospheric pressure in a mechanically polished stainless steel pan placed on top of a heating unit, as shown in Figure 10–15. The
inner surface of the bottom of the pan is maintained at 108°C. If the diameter
of the bottom of the pan is 30 cm, determine (a) the rate of heat transfer to the
water and (b) the rate of evaporation of water. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 527 527
CHAPTER 10
Tsat = 0°C
105 0.5 be n tu Finn 104 Pore Tunnel Plai Th ed t
ube erm qc (kcal/(m 2 h)
′′ oe
xc
elE Vapor
Liquid 1
2
5
(Ts – Tsat ) (°C) FIGURE 10–14
The enhancement of boiling heat
transfer in Freon-12 by a mechanically
roughened surface, thermoexcel-E. 10 SOLUTION Water is boiled at 1 atm pressure on a stainless steel surface. The
rate of heat transfer to the water and the rate of evaporation of water are to be
determined.
Assumptions 1 Steady operating conditions exist. 2 Heat losses from the heater
and the pan are negligible.
Properties The properties of water at the saturation temperature of 100°C are
0.0589 N/m (Table 10–1) and, from Table A-9,
l
v Prl 957.9 kg/m3
0.6 kg/m3
1.75 hfg
l Cpl 2257.0 103 J/kg
0.282 10 3 kg · m/s
4217 J/kg · °C Also, Csf 0.0130 and n 1.0 for the boiling of water on a mechanically polished stainless steel surface (Table 10–3). Note that we expressed the properties in units specified under Eq. 10–2 in connection with their definitions in
order to avoid unit manipulations.
Analysis
(a) The excess temperature in this case is T
Ts
Tsat
108 100 8°C which is relatively low (less than 30°C). Therefore, nucleate
boiling will occur. The heat flux in this case can be determined from the
Rohsenow relation to be ·
q nucleate Cpl (Ts Tsat) 3
Csf hfg Prn
l
9.81 (957.9
(0.282 10 3)(2257 103)
0.0589
3
4217(108 100)
0.0130(2257 103)1.75
l hfg 7.20 g( l 1/2 ) 104 W/m2 The surface area of the bottom of the pan is A D2/4 (0.3 m)2/4 0.07069 m2 0.6) 1/2 cen58933_ch10.qxd 9/4/2002 12:38 PM Page 528 528
HEAT TRANSFER Then the rate of heat transfer during nucleate boiling becomes ·
Q boiling Aq·nucleate (0.07069 m2)(7.20 104 W/m2) 5093 W (b) The rate of evaporation of water is determined from ·
m evaporation ·
Q boiling
hfg 5093 J/s
2257 103 J/kg 2.26 10 3 kg/s That is, water in the pan will boil at a rate of more than 2 grams per second. EXAMPLE 10–2 Water in a tank is to be boiled at sea level by a 1-cm-diameter nickel plated
steel heating element equipped with electrical resistance wires inside, as shown
in Figure 10–16. Determine the maximum heat flux that can be attained in the
nucleate boiling regime and the surface temperature of the heater surface in
that case. P = 1 atm
Water, 100°C
d Ts = ?
Heating element FIGURE 10–16
Schematic for Example 10–2. Peak Heat Flux in Nucleate Boiling ·
qmax SOLUTION Water is boiled at 1 atm pressure on a nickel plated steel surface. The maximum (critical) heat flux and the surface temperature are to be
determined.
Assumptions 1 Steady operating conditions exist. 2 Heat losses from the boiler
are negligible.
Properties The properties of water at the saturation temperature of 100°C are
0.0589 N/m (Table 10–1) and, from Table A-9,
l
v Prl 957.9 kg/m3
0.6 kg/m3
1.75 hfg
l Cpl 2257 103 J/kg
0.282 10 3 kg · m/s
4217 J/kg · °C Also, Csf 0.0060 and n 1.0 for the boiling of water on a nickel plated surface (Table 10–3). Note that we expressed the properties in units specified
under Eqs. 10–2 and 10–3 in connection with their definitions in order to avoid
unit manipulations.
Analysis The heating element in this case can be considered to be a short
cylinder whose characteristic dimension is its radius. That is, L r 0.005 m.
The dimensionless parameter L* and the constant Ccr are determined from
Table 10–4 to be L* L g( l ) 1/2 (0.005) (9.81)(957.8 0.6)
0.0589 1/2 2.00 1.2 which corresponds to Ccr 0.12.
Then the maximum or critical heat flux is determined from Eq. 10–3 to be q·max Ccr hfg [ g 2 ( l
)]1/4
0.12(2257 103)[0.0589
1.02 106 W/m2 9.8 (0.6)2(957.9 0.6)]1/4 cen58933_ch10.qxd 9/4/2002 12:38 PM Page 529 529
CHAPTER 10 The Rohsenow relation, which gives the nucleate boiling heat flux for a specified surface temperature, can also be used to determine the surface temperature when the heat flux is given. Substituting the maximum heat flux into Eq.
10–2 together with other properties gives ·
q nucleate
1,017,200 Ts l hfg (0.282 g( l ) 1/2 10 3)(2257 4217(Ts
0.0130(2257
119°C Cpl (Ts Tsat)
Csf hfg Prn
l
103) 3 9.81(957.9 0.6)
0.0589 1/2 100)
103) 1.75 Discussion Note that heat fluxes on the order of 1 MW/m2 can be obtained in
nucleate boiling with a temperature difference of less than 20°C. P = 1 atm EXAMPLE 10–3 Film Boiling of Water on a Heating Element Water is boiled at atmospheric pressure by a horizontal polished copper heating
element of diameter D 5 mm and emissivity
0.05 immersed in water, as
shown in Figure 10–17. If the surface temperature of the heating wire is
350°C, determine the rate of heat transfer from the wire to the water per unit
length of the wire. SOLUTION Water is boiled at 1 atm by a horizontal polished copper heating
element. The rate of heat transfer to the water per unit length of the heater is to
be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat losses from the boiler
are negligible.
Properties The properties of water at the saturation temperature of 100°C are
hfg 2257 103 J/kg and l 957.9 kg/m3 (Table A-9). The properties of vapor at the film temperature of Tf (Tsat Ts)/2 (100 350)/2 225°C
498 K (which is sufficiently close to 500 K) are, from Table A-16, 0.441 kg/m3
1.73 10 5 kg/m · s Cp
k 1977 J/kg · °C
0.0357 W/m · °C Note that we expressed the properties in units that will cancel each other in
boiling heat transfer relations. Also note that we used vapor properties at 1 atm
pressure from Table A-16 instead of the properties of saturated vapor from Table
A-9 at 250°C since the latter are at the saturation pressure of 4.0 MPa.
Analysis The excess temperature in this case is T
Ts
Tsat
350 100 250°C, which is much larger than 30°C for water. Therefore, film
boiling will occur. The film boiling heat flux in this case can be determined from
Eq. 10–5 to be 100°C Heating
element
Vapor
film FIGURE 10–17
Schematic for Example 10–3. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 530 530
HEAT TRANSFER q·film 0.62 gk3 ( )[hfg 0.4Cp (Ts
D(Ts Tsat) l Tsat)] 9.81(0.0357)3 (0.441)(957.9 0.441)
[(2257 103 0.4 1977(250)]
0.62
(1.73 10 5)(5 10 3)(250)
5.93 104 W/m2 1/4 (Ts Tsat) 1/4 250 The radiation heat flux is determined from Eq. 10–6 to be q·rad 4
(Ts4 Tsat)
(0.05)(5.67 10
157 W/m2 8 W/m2 · K4)[(250 273 K)4 (100 273 K)4] Note that heat transfer by radiation is negligible in this case because of the low
emissivity of the surface and the relatively low surface temperature of the heating element. Then the total heat flux becomes (Eq. 10–7) q·total q·film 3·
q
4 rad 5.93 104 3
4 157 5.94 104 W/m2 Finally, the rate of heat transfer from the heating element to the water is determined by multiplying the heat flux by the heat transfer surface area, ·
Q total ·
q ·
qmax High
velocity
Low
velocity it y Nucleate pool
boiling regime io n oc
vel
ow
L ity Fre e
onv
ec ct ∆Texcess FIGURE 10–18
The effect of forced convection on
external flow boiling for different
flow velocities. 104 W/m2) Discussion Note that the 5-mm-diameter copper heating element will consume
about 1 kW of electric power per unit length in steady operation in the film boiling regime. This energy is transferred to the water through the vapor film that
forms around the wire. 10–3
lo c
h ve
Hig ·
Aq·total ( DL)q total
(
0.005 m 1 m)(5.94
933 W I FLOW BOILING The pool boiling we considered so far involves a pool of seemingly motionless liquid, with vapor bubbles rising to the top as a result of buoyancy effects.
In flow boiling, the fluid is forced to move by an external source such as a
pump as it undergoes a phase-change process. The boiling in this case exhibits
the combined effects of convection and pool boiling. The flow boiling is also
classified as either external and internal flow boiling depending on whether
the fluid is forced to flow over a heated surface or inside a heated tube.
External flow boiling over a plate or cylinder is similar to pool boiling, but
the added motion increases both the nucleate boiling heat flux and the critical
heat flux considerably, as shown in Figure 10–18. Note that the higher the velocity, the higher the nucleate boiling heat flux and the critical heat flux. In experiments with water, critical heat flux values as high as 35 MW/m2 have been
obtained (compare this to the pool boiling value of 1.3 MW/m2 at 1 atm pressure) by increasing the fluid velocity. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 531 531
CHAPTER 10 Internal flow boiling is much more complicated in nature because there is
no free surface for the vapor to escape, and thus both the liquid and the vapor
are forced to flow together. The two-phase flow in a tube exhibits different
flow boiling regimes, depending on the relative amounts of the liquid and the
vapor phases. This complicates the analysis even further.
The different stages encountered in flow boiling in a heated tube are illustrated in Figure 10–19 together with the variation of the heat transfer coefficient along the tube. Initially, the liquid is subcooled and heat transfer to the
liquid is by forced convection. Then bubbles start forming on the inner surfaces of the tube, and the detached bubbles are drafted into the mainstream.
This gives the fluid flow a bubbly appearance, and thus the name bubbly flow
regime. As the fluid is heated further, the bubbles grow in size and eventually
coalesce into slugs of vapor. Up to half of the volume in the tube in this slugflow regime is occupied by vapor. After a while the core of the flow consists
of vapor only, and the liquid is confined only in the annular space between the
vapor core and the tube walls. This is the annular-flow regime, and very high
heat transfer coefficients are realized in this regime. As the heating continues,
the annular liquid layer gets thinner and thinner, and eventually dry spots start
to appear on the inner surfaces of the tube. The appearance of dry spots is accompanied by a sharp decrease in the heat transfer coefficient. This transition
regime continues until the inner surface of the tube is completely dry. Any liquid at this moment is in the form of droplets suspended in the vapor core,
which resembles a mist, and we have a mist-flow regime until all the liquid
droplets are vaporized. At the end of the mist-flow regime we have saturated
vapor, which becomes superheated with any further heat transfer.
Note that the tube contains a liquid before the bubbly flow regime and a
vapor after the mist-flow regime. Heat transfer in those two cases can be
determined using the appropriate relations for single-phase convection heat
transfer. Many correlations are proposed for the determination of heat transfer
High
x=1 Forced convection
Liquid
droplets Low Mist flow Vapor core
Bubbles
in liquid
Liquid
core Quality Transition flow Annular flow Slug flow Bubbly flow x=0
Forced convection
Coefficient of heat transfer FIGURE 10–19
Different flow regimes
encountered in flow boiling
in a tube under forced convection. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 532 532
HEAT TRANSFER in the two-phase flow (bubbly flow, slug-flow, annular-flow, and mist-flow)
cases, but they are beyond the scope of this introductory text. A crude estimate
for heat flux in flow boiling can be obtained by simply adding the forced convection and pool boiling heat fluxes. 10–4 80°C 80°C Droplets
Liquid film
(a) Film
condensation (b) Dropwise
condensation FIGURE 10–20
When a vapor is exposed to a
surface at a temperature below Tsat,
condensation in the form of a liquid
film or individual droplets occurs
on the surface.
Cold 0
plate y x
g
·
m(x) Vapor,
Liquid-vapor
interface T(y) Ts
Tsat Temperature
profile Tv, Velocity
(y) profile
Liquid, l FIGURE 10–21
Film condensation on a vertical plate. I CONDENSATION HEAT TRANSFER Condensation occurs when the temperature of a vapor is reduced below its
saturation temperature Tsat. This is usually done by bringing the vapor into
contact with a solid surface whose temperature Ts is below the saturation temperature Tsat of the vapor. But condensation can also occur on the free surface
of a liquid or even in a gas when the temperature of the liquid or the gas to
which the vapor is exposed is below Tsat. In the latter case, the liquid droplets
suspended in the gas form a fog. In this chapter, we will consider condensation on solid surfaces only.
Two distinct forms of condensation are observed: film condensation and
dropwise condensation. In film condensation, the condensate wets the surface and forms a liquid film on the surface that slides down under the influence of gravity. The thickness of the liquid film increases in the flow direction
as more vapor condenses on the film. This is how condensation normally occurs in practice. In dropwise condensation, the condensed vapor forms
droplets on the surface instead of a continuous film, and the surface is covered
by countless droplets of varying diameters (Fig. 10–20).
In film condensation, the surface is blanketed by a liquid film of increasing
thickness, and this “liquid wall” between solid surface and the vapor serves as
a resistance to heat transfer. The heat of vaporization hfg released as the vapor
condenses must pass through this resistance before it can reach the solid surface and be transferred to the medium on the other side. In dropwise condensation, however, the droplets slide down when they reach a certain size,
clearing the surface and exposing it to vapor. There is no liquid film in this
case to resist heat transfer. As a result, heat transfer rates that are more than
10 times larger than those associated with film condensation can be achieved
with dropwise condensation. Therefore, dropwise condensation is the preferred mode of condensation in heat transfer applications, and people have
long tried to achieve sustained dropwise condensation by using various vapor
additives and surface coatings. These attempts have not been very successful,
however, since the dropwise condensation achieved did not last long and converted to film condensation after some time. Therefore, it is common practice
to be conservative and assume film condensation in the design of heat transfer equipment. 10–5 I FILM CONDENSATION We now consider film condensation on a vertical plate, as shown in Figure
10–21. The liquid film starts forming at the top of the plate and flows downward under the influence of gravity. The thickness of the film increases in
the flow direction x because of continued condensation at the liquid–vapor interface. Heat in the amount hfg (the latent heat of vaporization) is released during condensation and is transferred through the film to the plate surface at
temperature Ts. Note that Ts must be below the saturation temperature Tsat of
the vapor for condensation to occur. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 533 533
CHAPTER 10 Typical velocity and temperature profiles of the condensate are also given
in Figure 10–21. Note that the velocity of the condensate at the wall is zero because of the “no-slip” condition and reaches a maximum at the liquid–vapor
interface. The temperature of the condensate is Tsat at the interface and decreases gradually to Ts at the wall.
As was the case in forced convection involving a single phase, heat transfer
in condensation also depends on whether the condensate flow is laminar or
turbulent. Again the criterion for the flow regime is provided by the Reynolds
number, which is defined as
Re Dh l
l l 4 Ac
p l 4 l l ·
4m
pl l
l l (10-8) where
Dh
p
Ac
l
l ·
m 4Ac /p 4
hydraulic diameter of the condensate flow, m
wetted perimeter of the condensate, m
p
wetted perimeter film thickness, m2, cross-sectional area of the
condensate flow at the lowest part of the flow
density of the liquid, kg/m3
viscosity of the liquid, kg/m · s
average velocity of the condensate at the lowest part of the flow, m/s
mass flow rate of the condensate at the lowest part, kg/s
l l Ac The evaluation of the hydraulic diameter Dh for some common geometries is
illustrated in Figure 10–22. Note that the hydraulic diameter is again defined
such that it reduces to the ordinary diameter for flow in a circular tube, as was
done in Chapter 8 for internal flow, and it is equivalent to 4 times the thickness of the condensate film at the location where the hydraulic diameter is
evaluated. That is, Dh 4 .
The latent heat of vaporization hfg is the heat released as a unit mass of
vapor condenses, and it normally represents the heat transfer per unit mass of
condensate formed during condensation. However, the condensate in an actual
L
D
L D δ δ
p=L
Ac = L δ
4A
—
Dh = — c = 4δ
p (a) Vertical plate p = πD
Ac = π Dδ
4A
—
Dh = — c = 4δ
p
(b) Vertical cylinder δ
p = 2L
Ac = 2Lδ
4A
—
Dh = — c = 4δ
p
(c) Horizontal cylinder FIGURE 10–22
The wetted perimeter p, the
condensate cross-sectional area Ac,
and the hydraulic diameter Dh for
some common geometries. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 534 534
HEAT TRANSFER condensation process is cooled further to some average temperature between
Tsat and Ts, releasing more heat in the process. Therefore, the actual heat transfer will be larger. Rohsenow showed in 1956 that the cooling of the liquid below the saturation temperature can be accounted for by replacing hfg by the
modified latent heat of vaporization h* , defined as
fg
h*
fg hfg 0.68Cpl (Tsat Ts) (10-9a) where Cpl is the specific heat of the liquid at the average film temperature.
We can have a similar argument for vapor that enters the condenser as
superheated vapor at a temperature T instead of as saturated vapor. In this
case the vapor must be cooled first to Tsat before it can condense, and this heat
must be transferred to the wall as well. The amount of heat released as a unit
mass of superheated vapor at a temperature T is cooled to Tsat is simply
Tsat), where Cp is the specific heat of the vapor at the average temCp (T
Tsat)/2. The modified latent heat of vaporization in this case
perature of (T
becomes
h*
fg hfg 0.68Cpl (Tsat Ts) Cp (T Tsat) (10-9b) With these considerations, the rate of heat transfer can be expressed as
·
Q conden hAs(Tsat Ts) mh*
fg (10-10) where As is the heat transfer area (the surface area on which condensation oc·
curs). Solving for m from the equation above and substituting it into Eq. 10–8
gives yet another relation for the Reynolds number,
Re Re = 0
Laminar
(wave-free)
Re ≅ 30 ·
4Q conden
p l h*
fg 4As h(Tsat Ts)
p l h*
fg (10-11) This relation is convenient to use to determine the Reynolds number when the
condensation heat transfer coefficient or the rate of heat transfer is known.
The temperature of the liquid film varies from Tsat on the liquid–vapor interface to Ts at the wall surface. Therefore, the properties of the liquid should
be evaluated at the film temperature Tf (Tsat Ts)/2, which is approximately
the average temperature of the liquid. The hfg, however, should be evaluated
at Tsat since it is not affected by the subcooling of the liquid. Flow Regimes
Laminar
(wavy)
Re ≅ 1800
Turbulent FIGURE 10–23
Flow regimes during film
condensation on a vertical plate. The Reynolds number for condensation on the outer surfaces of vertical tubes
or plates increases in the flow direction due to the increase of the liquid film
thickness . The flow of liquid film exhibits different regimes, depending on
the value of the Reynolds number. It is observed that the outer surface of the
liquid film remains smooth and wave-free for about Re 30, as shown in Figure 10–23, and thus the flow is clearly laminar. Ripples or waves appear on
the free surface of the condensate flow as the Reynolds number increases, and
the condensate flow becomes fully turbulent at about Re 1800. The condensate flow is called wavy-laminar in the range of 450 Re 1800 and
turbulent for Re 1800. However, some disagreement exists about the value
of Re at which the flow becomes wavy-laminar or turbulent. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 535 535
CHAPTER 10 δ–y Heat Transfer Correlations for Film Condensation
Below we discuss relations for the average heat transfer coefficient h for the
case of laminar film condensation for various geometries. dx
Shear force
du
µl — (bdx)
dy 1 Vertical Plates
Consider a vertical plate of height L and width b maintained at a constant temperature Ts that is exposed to vapor at the saturation temperature Tsat. The
downward direction is taken as the positive x-direction with the origin placed
at the top of the plate where condensation initiates, as shown in Figure 10–24.
The surface temperature is below the saturation temperature (Ts Tsat) and
thus the vapor condenses on the surface. The liquid film flows downward under the influence of gravity. The film thickness and thus the mass flow rate
of the condensate increases with x as a result of continued condensation on the
existing film. Then heat transfer from the vapor to the plate must occur
through the film, which offers resistance to heat transfer. Obviously the
thicker the film, the larger its thermal resistance and thus the lower the rate of
heat transfer.
The analytical relation for the heat transfer coefficient in film condensation
on a vertical plate described above was first developed by Nusselt in 1916
under the following simplifying assumptions:
1. Both the plate and the vapor are maintained at constant temperatures of
Ts and Tsat, respectively, and the temperature across the liquid film varies
linearly.
2. Heat transfer across the liquid film is by pure conduction (no convection
currents in the liquid film).
3. The velocity of the vapor is low (or zero) so that it exerts no drag on the
condensate (no viscous shear on the liquid–vapor interface).
4. The flow of the condensate is laminar and the properties of the liquid
are constant.
5. The acceleration of the condensate layer is negligible.
Then Newton’s second law of motion for the volume element shown in Figure
10–24 in the vertical x-direction can be written as
Fx max 0 since the acceleration of the fluid is zero. Noting that the only force acting
downward is the weight of the liquid element, and the forces acting upward
are the viscous shear (or fluid friction) force at the left and the buoyancy
force, the force balance on the volume element becomes
Weight
l g( Fdownward ↓ Fupward ↑
Viscous shear force Buoyancy force
du
y)(bdx)
(bdx)
g(
y)(bdx)
l
dy Canceling the plate width b and solving for du/dy gives
du
dy g( l )g(
l y) Weight
ρl g(δ – y) (bdx) Buoyancy force
ρv g(δ – y) (bdx) y 0
x δ
dx
=0
at y = 0 y Idealized
velocity
profile
No vapor drag
Idealized
temperature
profile Ts
g
Liquid, l Tsat
Linear FIGURE 10–24
The volume element of condensate
on a vertical plate considered
in Nusselt’s analysis. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 536 536
HEAT TRANSFER Integrating from y 0 where u 0 (because of the no-slip boundary condition) to y y where u u(y) gives
g( u(y) )g l y2
2 y l (10-12) The mass flow rate of the condensate at a location x, where the boundary layer
thickness is , is determined from
·
m (x) l u(y)dA A l u(y)bdy y0 (10-13) Substituting the u(y) relation from Equation 10–12 into Eq. 10–13 gives
gb l( ·
m (x) ) l 3 3 2 d
dx (10-14) l whose derivative with respect to x is
·
dm
dx gb l( ) l
l (10-15) which represents the rate of condensation of vapor over a vertical distance dx.
The rate of heat transfer from the vapor to the plate through the liquid film is
simply equal to the heat released as the vapor is condensed and is expressed as
·
dQ ·
hfg dm kl (bdx) Tsat Ts ·
dm
dx → kl b Tsat
hfg Ts (10-16) ·
Equating Eqs. 10–15 and 10–16 for dm /dx to each other and separating the
variables give
3 l kl (Tsat
g l( l d Ts)
dx
)hfg (10-17) Integrating from x
0 where
0 (the top of the plate) to x
x where
(x), the liquid film thickness at any location x is determined to be
4 (x) kl (Tsat
g l( l
l Ts)x
)hfg 1/4 (10-18) The heat transfer rate from the vapor to the plate at a location x can be
expressed as
q·x hx(Tsat Ts) kl Tsat Ts → hx kl
(x) (10-19) Substituting the (x) expression from Eq. 10–18, the local heat transfer coefficient hx is determined to be
hx g l( l
)hfg k3
l
4 l (Tsat Ts)x 1/4 (10-20) cen58933_ch10.qxd 9/4/2002 12:38 PM Page 537 537
CHAPTER 10 The average heat transfer coefficient over the entire plate is determined from
its definition by substituting the hx relation and performing the integration.
It gives
h 1
L have L
0 4
h
3x hx dx 0.943 L g l( l
l (Tsat )hfg k3
l
Ts)L 1/4 (10-21) Equation 10–21, which is obtained with the simplifying assumptions stated
earlier, provides good insight on the functional dependence of the condensation heat transfer coefficient. However, it is observed to underpredict heat
transfer because it does not take into account the effects of the nonlinear temperature profile in the liquid film and the cooling of the liquid below the saturation temperature. Both of these effects can be accounted for by replacing hfg
by h* given by Eq. 10–9. With this modification, the average heat transfer
fg
coefficient for laminar film condensation over a vertical flat plate of height L
is determined to be
hvert 0.943 g l(
l l (Tsat )h* k3
fg l
Ts)L 1/4 (W/m2 · °C), 0 Re 30 (10-22) where
g
l,
l h*
fg
kl
L
Ts
Tsat gravitational acceleration, m/s2
densities of the liquid and vapor, respectively, kg/m3
viscosity of the liquid, kg/m · s
hfg 0.68Cpl (Tsat Ts) modified latent heat of vaporization, J/kg
thermal conductivity of the liquid, W/m · °C
height of the vertical plate, m
surface temperature of the plate, °C
saturation temperature of the condensing fluid, °C At a given temperature,
l and thus l
l except near the critical
point of the substance. Using this approximation and substituting Eqs. 10–14
kl
and
and 10–18 at x
L into Eq. 10–8 by noting that x L
hx L
4
h
(Eqs. 10–19 and 10–21) give
hvert
3xL
Re 4g l ( ) l 3 3 2
l 4g
3 2
l
2
l 3 kl
hx kl
4g
2 3h
3l
vert/4 L 3 (10-23) Then the heat transfer coefficient hvert in terms of Re becomes
hvert 1.47kl Re 1/3 g
2
l 1/3 , 0 Re 30 (10-24) l The results obtained from the theoretical relations above are in excellent
agreement with the experimental results. It can be shown easily that using
property values in Eqs. 10–22 and 10–24 in the specified units gives the condensation heat transfer coefficient in W/m2 · °C, thus saving one from having cen58933_ch10.qxd 9/4/2002 12:38 PM Page 538 538
HEAT TRANSFER hvert W
m kg kg J
s2 m3 m3 kg m · °C
kg
m · s · °C · m 3 1/4 J
W3
m1
s m6 m3 · °C3 °C
1/4
W4
m8 · °C4
W/m2 · °C FIGURE 10–25
Equation 10–22 gives the
condensation heat transfer coefficient
in W/m2 · °C when the quantities are
expressed in the units specified
in their descriptions. to go through tedious unit manipulations each time (Fig. 10–25). This is also
true for the equations below. All properties of the liquid are to be evaluated at
the film temperature Tf (Tsat Ts)/2. The hfg and are to be evaluated at
the saturation temperature Tsat.
Wavy Laminar Flow on Vertical Plates At Reynolds numbers greater than about 30, it is observed that waves form at
the liquid–vapor interface although the flow in liquid film remains laminar.
The flow in this case is said to be wavy laminar. The waves at the liquid–
vapor interface tend to increase heat transfer. But the waves also complicate
the analysis and make it very difficult to obtain analytical solutions. Therefore, we have to rely on experimental studies. The increase in heat transfer due
to the wave effect is, on average, about 20 percent, but it can exceed 50 percent. The exact amount of enhancement depends on the Reynolds number.
Based on his experimental studies, Kutateladze (1963, Ref. 15) recommended
the following relation for the average heat transfer coefficient in wavy lamiRe 1800,
nar condensate flow for
l and 30
hvert, wavy Re kl
1.08 Re1.22 g
5.2 2
l 1/3 30 , Re 1800 (10-25) l A simpler alternative to the relation above proposed by Kutateladze (1963,
Ref. 15) is
hvert, wavy 0.8 Re0.11 hvert (smooth) (10-26) which relates the heat transfer coefficient in wavy laminar flow to that in
wave-free laminar flow. McAdams (1954, Ref. 2) went even further and
suggested accounting for the increase in heat transfer in the wavy region by
simply increasing the heat transfer coefficient determined from Eq. 10–22 for
the laminar case by 20 percent. Holman (1990) suggested using Eq. 10–22
for the wavy region also, with the understanding that this is a conservative
approach that provides a safety margin in thermal design. In this book we will
use Eq. 10–25.
A relation for the Reynolds number in the wavy laminar region can be
determined by substituting the h relation in Eq. 10–25 into the Re relation in
Eq. 10–11 and simplifying. It yields
Revert, wavy 4.81 3.70 Lkl (Tsat
l h*
fg Ts) g 1/3 0.820 , 2
l v l (10-27) Turbulent Flow on Vertical Plates At a Reynolds number of about 1800, the condensate flow becomes turbulent.
Several empirical relations of varying degrees of complexity are proposed for
the heat transfer coefficient for turbulent flow. Again assuming
l for
simplicity, Labuntsov (1957, Ref. 17) proposed the following relation for the
turbulent flow of condensate on vertical plates:
hvert, turbulent 8750 Re kl
58 Pr 0.5 (Re0.75 g
253) 2
l 1/3 , Re 1800
l (10-28) cen58933_ch10.qxd 9/4/2002 12:38 PM Page 539 539
CHAPTER 10
1.0
Pr = 10 Eq. 10-24 5
h (νl2/g)1/ 3
————
—
kl 3
Eq. 10-25 2
1
Eq. 10-28 Wave-free
laminar
0.1
10 Wavy laminar
30 100 Turbulent
1000 1800
Re 10,000 FIGURE 10–26
Nondimensionalized heat transfer
coefficients for the wave-free laminar,
wavy laminar, and turbulent flow
of condensate on vertical plates. The physical properties of the condensate are again to be evaluated at the film
temperature Tf (Tsat Ts)/2. The Re relation in this case is obtained by substituting the h relation above into the Re relation in Eq. 10–11, which gives
Revert, turbulent 0.0690 Lkl Pr0.5 (Tsat
l h*
fg Ts) g
2
l 1/3 4/3 151 Pr0.5 253 (10-29) Nondimensionalized heat transfer coefficients for the wave-free laminar,
wavy laminar, and turbulent flow of condensate on vertical plates are plotted
in Figure 10–26. 2 Inclined Plates
Equation 10–12 was developed for vertical plates, but it can also be used for
laminar film condensation on the upper surfaces of plates that are inclined by
an angle from the vertical, by replacing g in that equation by g cos (Fig.
10–27). This approximation gives satisfactory results especially for
60°.
Note that the condensation heat transfer coefficients on vertical and inclined
plates are related to each other by
hinclined hvert (cos )1/4 (laminar) Vapor
θ (10-30) Equation 10–30 is developed for laminar flow of condensate, but it can also
be used for wavy laminar flows as an approximation. Inclined
plate Condensate 3 Vertical Tubes
Equation 10–22 for vertical plates can also be used to calculate the average
heat transfer coefficient for laminar film condensation on the outer surfaces of
vertical tubes provided that the tube diameter is large relative to the thickness
of the liquid film. 4 Horizontal Tubes and Spheres
Nusselt’s analysis of film condensation on vertical plates can also be extended
to horizontal tubes and spheres. The average heat transfer coefficient for film
condensation on the outer surfaces of a horizontal tube is determined to be FIGURE 10–27
Film condensation on
an inclined plate. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 540 540
HEAT TRANSFER hhoriz 0.729 ) h* k3
fg l
Ts)D g l( l
l(Tsat 1/4 (W/m2 · °C) (10-31) where D is the diameter of the horizontal tube. Equation 10–31 can easily be
modified for a sphere by replacing the constant 0.729 by 0.815.
A comparison of the heat transfer coefficient relations for a vertical tube of
height L and a horizontal tube of diameter D yields
hvert
hhoriz 1.29 D
L 1/4 (10-32) Setting hvertical hhorizontal gives L 1.294 D 2.77D, which implies that for
a tube whose length is 2.77 times its diameter, the average heat transfer coefficient for laminar film condensation will be the same whether the tube is positioned horizontally or vertically. For L 2.77D, the heat transfer coefficient
will be higher in the horizontal position. Considering that the length of a tube
in any practical application is several times its diameter, it is common practice
to place the tubes in a condenser horizontally to maximize the condensation
heat transfer coefficient on the outer surfaces of the tubes. 5 Horizontal Tube Banks
Horizontal tubes stacked on top of each other as shown in Figure 10–28 are
commonly used in condenser design. The average thickness of the liquid film
at the lower tubes is much larger as a result of condensate falling on top of
them from the tubes directly above. Therefore, the average heat transfer coefficient at the lower tubes in such arrangements is smaller. Assuming the condensate from the tubes above to the ones below drain smoothly, the average
film condensation heat transfer coefficient for all tubes in a vertical tier can be
expressed as
hhoriz, N tubes FIGURE 10–28
Film condensation on a
vertical tier of horizontal tubes. 0.729 g l( l
l (Tsat ) h* k3
fg l
Ts) ND 1/4 1
h
N1/4 horiz, 1 tube (10-33) Note that Eq. 10–33 can be obtained from the heat transfer coefficient relation
for a horizontal tube by replacing D in that relation by ND. This relation does
not account for the increase in heat transfer due to the ripple formation and
turbulence caused during drainage, and thus generally yields conservative
results. Effect of Vapor Velocity
In the analysis above we assumed the vapor velocity to be small and thus the
vapor drag exerted on the liquid film to be negligible, which is usually the
case. However, when the vapor velocity is high, the vapor will “pull” the liquid at the interface along since the vapor velocity at the interface must drop to
the value of the liquid velocity. If the vapor flows downward (i.e., in the same
direction as the liquid), this additional force will increase the average velocity
of the liquid and thus decrease the film thickness. This, in turn, will decrease
the thermal resistance of the liquid film and thus increase heat transfer.
Upward vapor flow has the opposite effects: the vapor exerts a force on the cen58933_ch10.qxd 9/4/2002 12:38 PM Page 541 541
CHAPTER 10 liquid in the opposite direction to flow, thickens the liquid film, and thus
decreases heat transfer. Condensation in the presence of high vapor flow is
studied [e.g., Shekriladze and Gomelauri (1966), Ref. 23] and heat transfer relations are obtained, but a detailed analysis of this topic is beyond the scope of
this introductory text. The Presence of Noncondensable Gases in Condensers
Most condensers used in steam power plants operate at pressures well below
the atmospheric pressure (usually under 0.1 atm) to maximize cycle thermal
efficiency, and operation at such low pressures raises the possibility of air (a
noncondensable gas) leaking into the condensers. Experimental studies show
that the presence of noncondensable gases in the vapor has a detrimental effect on condensation heat transfer. Even small amounts of a noncondensable
gas in the vapor cause significant drops in heat transfer coefficient during condensation. For example, the presence of less than 1 percent (by mass) of air in
steam can reduce the condensation heat transfer coefficient by more than half.
Therefore, it is common practice to periodically vent out the noncondensable
gases that accumulate in the condensers to ensure proper operation.
The drastic reduction in the condensation heat transfer coefficient in the
presence of a noncondensable gas can be explained as follows: When the vapor mixed with a noncondensable gas condenses, only the noncondensable
gas remains in the vicinity of the surface (Fig. 10–29). This gas layer acts as a
barrier between the vapor and the surface, and makes it difficult for the vapor
to reach the surface. The vapor now must diffuse through the noncondensable
gas first before reaching the surface, and this reduces the effectiveness of the
condensation process.
Experimental studies show that heat transfer in the presence of a noncondensable gas strongly depends on the nature of the vapor flow and the flow
velocity. As you would expect, a high flow velocity is more likely to remove
the stagnant noncondensable gas from the vicinity of the surface, and thus improve heat transfer. EXAMPLE 10–4 Vapor + Noncondensable gas Cold
surface Condensate
Noncondensable gas
Vapor FIGURE 10–29
The presence of a noncondensable
gas in a vapor prevents the vapor
molecules from reaching the cold
surface easily, and thus impedes
condensation heat transfer. Condensation of Steam on a Vertical Plate Saturated steam at atmospheric pressure condenses on a 2-m-high and 3-mwide vertical plate that is maintained at 80°C by circulating cooling water
through the other side (Fig. 10–30). Determine (a) the rate of heat transfer by
condensation to the plate and (b) the rate at which the condensate drips off the
plate at the bottom. SOLUTION Saturated steam at 1 atm condenses on a vertical plate. The rates
of heat transfer and condensation are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The plate is isothermal.
3 The condensate flow is wavy-laminar over the entire plate (will be verified).
4 The density of vapor is much smaller than the density of liquid,
l.
Properties The properties of water at the saturation temperature of 100°C are
hfg
2257
103 J/kg and
0.60 kg/m3. The properties of liquid water
at the film temperature of Tf
(Tsat
Ts)/2
(100
80)/2
90°C are
(Table A-9) 1 atm
3m
Ts = 80°C 2m Condensate FIGURE 10–30
Schematic for Example 10–4. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 542 542
HEAT TRANSFER l
l
l 965.3 kg/m3
0.315 10 3 kg/m · s
0.326 10 6 m2/s
l/ l Cpl
kl 4206 J/kg · °C
0.675 W/m · °C Analysis (a) The modified latent heat of vaporization is h*
fg hfg 0.68Cpl (Tsat Ts)
2257 103 J/kg 0.68
2314 103 J/kg (4206 J/kg · °C)(100 80)°C For wavy-laminar flow, the Reynolds number is determined from Eq. 10–27
to be Re Revertical, wavy 4.81 3.70 Lkl (Tsat
l h*
fg Ts) g 1/3 0.820 2
l 3.70(3 m)(0.675 W/m · °C)(100 90)°C
(0.315 10 3 kg/m · s)(2314 103 J/kg)
1/3 0.82
9.81 m/s2
6
2
2
(0.326 10 m /s)
1287
4.81 which is between 30 and 1800, and thus our assumption of wavy laminar flow
is verified. Then the condensation heat transfer coefficient is determined from
Eq. 10–25 to be h Re kl
g 1/3
1.08 Re1.22 5.2 2
l
1287 (0.675 W/m · °C)
9.81 m/s2
1.08(1287)1.22 5.2
(0.326 10 6 m2/s)2
hvertical, wavy 1/3 5848 W/m2 · °C The heat transfer surface area of the plate is As W L (3 m)(2 m) 6 m2.
Then the rate of heat transfer during this condensation process becomes ·
Q 30° hAs(Tsat Ts) (5848 W/m2 · °C)(6 m2)(100 80)°C 7.02 105 W (b) The rate of condensation of steam is determined from Steam
100°C ·
m condensation ·
Q
h*
fg 7.02
2314 105 J/s
103 J/kg 0.303 kg/s That is, steam will condense on the surface at a rate of 303 grams per second.
80°C EXAMPLE 10–5
FIGURE 10–31
Schematic for Example 10–5. Condensation of Steam on a Tilted Plate What would your answer be to the preceding example problem if the plate were
tilted 30° from the vertical, as shown in Figure 10–31? cen58933_ch10.qxd 9/4/2002 12:38 PM Page 543 543
CHAPTER 10 SOLUTION (a) The heat transfer coefficient in this case can be determined
from the vertical plate relation by replacing g by g cos . But we will use
Eq. 10–30 instead since we already know the value for the vertical plate from
the preceding example:
h hvert (cos )1/4 hinclined (5848 W/m2 · °C)(cos 30°)1/4 5641 W/m2 · °C The heat transfer surface area of the plate is still 6 m2. Then the rate of condensation heat transfer in the tilted plate case becomes ·
Q hAs(Tsat Ts) (5641 W/m2 · °C)(6 m2)(100 80)°C 6.77 105 W (b) The rate of condensation of steam is again determined from ·
m condensation ·
Q
h*
fg 6.77
2314 105 J/s
103 J/kg 0.293 kg/s Discussion Note that the rate of condensation decreased by about 3.6 percent
when the plate is tilted. EXAMPLE 10–6 Condensation of Steam on Horizontal Tubes The condenser of a steam power plant operates at a pressure of 7.38 kPa.
Steam at this pressure condenses on the outer surfaces of horizontal pipes
through which cooling water circulates. The outer diameter of the pipes is 3 cm,
and the outer surfaces of the pipes are maintained at 30°C (Fig. 10–32). Determine (a) the rate of heat transfer to the cooling water circulating in the pipes
and (b) the rate of condensation of steam per unit length of a horizontal pipe. SOLUTION Saturated steam at a pressure of 7.38 kPa (Table A-9) condenses
on a horizontal tube at 30°C. The rates of heat transfer and condensation are to
be determined.
Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal.
Properties The properties of water at the saturation temperature of 40°C
corresponding to 7.38 kPa are hfg
2407 103 J/kg and
0.05 kg/m3.
The properties of liquid water at the film temperature of Tf
(Tsat
Ts)/2
(40 30)/2 35°C are (Table A-9)
l
l 994 kg/m3
0.720 10 3 kg/m · s Cpl
kl 4178 J/kg · °C
0.623 W/m · °C Analysis (a) The modified latent heat of vaporization is h*
fg hfg 0.68Cpl (Tsat Ts)
2407 103 J/kg 0.68
2435 103 J/kg (4178 J/kg · °C)(40 30)°C Noting that
994), the heat transfer coefficient for conl (since 0.05
densation on a single horizontal tube is determined from Eq. 10–31 to be Steam, 40°C 30°C Cooling water FIGURE 10–32
Schematic for Example 10–6. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 544 544
HEAT TRANSFER h 1/4
g l( l
) h* k3 1/4
g 2 h* k3
fg l
l fg l
0.729
(Tsat Ts) D
Ts) D
1 (Tsat
(9.81 m/s2)(994 kg/m3)2 (2435 103 J/kg)(0.623 W/m · °C)3
0.729
(0.720 10 3 kg/m · s)(40 30)°C(0.03 m)
2
9292 W/m · °C hhorizontal 0.729 1/4 The heat transfer surface area of the pipe per unit of its length is As
DL
(0.03 m)(1 m) 0.09425 m2. Then the rate of heat transfer during this condensation process becomes ·
Q hAs(Tsat Ts) (9292 W/m2 · °C)(0.09425 m2)(40 30)°C 8758 W (b) The rate of condensation of steam is ·
m condensation ·
Q
h*
fg 8578 J/s
2435 103 J/kg 0.00360 kg/s Therefore, steam will condense on the horizontal tube at a rate of 3.6 g/s or
12.9 kg/h per meter of its length. EXAMPLE 10–7 Condensation of Steam on Horizontal Tube Banks Repeat the proceeding example problem for the case of 12 horizontal tubes
arranged in a rectangular array of 3 tubes high and 4 tubes wide, as shown in
Figure 10–33. SOLUTION (a) Condensation heat transfer on a tube is not influenced by the
presence of other tubes in its neighborhood unless the condensate from other
tubes drips on it. In our case, the horizontal tubes are arranged in four vertical
tiers, each tier consisting of 3 tubes. The average heat transfer coefficient for a
vertical tier of N horizontal tubes is related to the one for a single horizontal
tube by Eq. 10–33 and is determined to be
Condensate
flow 1
h
N1/4 horiz, 1 tube hhoriz, N tubes FIGURE 10–33
Schematic for Example 10–7. 1
(9292 W/m2 · °C)
31/4 7060 W/m2 · °C Each vertical tier consists of 3 tubes, and thus the heat transfer coefficient determined above is valid for each of the four tiers. In other words, this value can
be taken to be the average heat transfer coefficient for all 12 tubes.
The surface area for all 12 tubes per unit length of the tubes is As Ntotal DL 12 (0.03 m)(1 m) 1.1310 m2 Then the rate of heat transfer during this condensation process becomes ·
Q hAs(Tsat Ts) (7060 W/m2 · °C)(1.131 m2)(40 30)°C (b) The rate of condensation of steam is again determined from 79,850 W cen58933_ch10.qxd 9/4/2002 12:38 PM Page 545 545
CHAPTER 10 ·
m condensation ·
Q
h*
fg 79,850 J/s
2435 103 J/kg 0.0328 kg/s Therefore, steam will condense on the horizontal pipes at a rate of 32.8 g/s per
meter length of the tubes. 10–6 I FILM CONDENSATION
INSIDE HORIZONTAL TUBES So far we have discussed film condensation on the outer surfaces of tubes and
other geometries, which is characterized by negligible vapor velocity and the
unrestricted flow of the condensate. Most condensation processes encountered
in refrigeration and air-conditioning applications, however, involve condensation on the inner surfaces of horizontal or vertical tubes. Heat transfer analysis of condensation inside tubes is complicated by the fact that it is strongly
influenced by the vapor velocity and the rate of liquid accumulation on the
walls of the tubes (Fig. 10–34).
For low vapor velocities, Chato recommends this expression for
condensation Liquid Vapor hinternal 0.555 g l( l
l (Tsat ) k3
l
Ts) hfg 3
C (T
8 pl sat 1/4 Ts) (10-34) for Tube Revapor D
inlet 35,000 (10-35) FIGURE 10–34
Condensate flow in a horizontal tube
with large vapor velocities. where the Reynolds number of the vapor is to be evaluated at the tube inlet
conditions using the internal tube diameter as the characteristic length. Heat
transfer coefficient correlations for higher vapor velocities are given by
Rohsenow. 10–7 I DROPWISE CONDENSATION Dropwise condensation, characterized by countless droplets of varying diameters on the condensing surface instead of a continuous liquid film, is one of
the most effective mechanisms of heat transfer, and extremely large heat transfer coefficients can be achieved with this mechanism (Fig. 10–35).
In dropwise condensation, the small droplets that form at the nucleation
sites on the surface grow as a result of continued condensation, coalesce into
large droplets, and slide down when they reach a certain size, clearing the surface and exposing it to vapor. There is no liquid film in this case to resist heat
transfer. As a result, with dropwise condensation, heat transfer coefficients can
be achieved that are more than 10 times larger than those associated with film
condensation. Large heat transfer coefficients enable designers to achieve a
specified heat transfer rate with a smaller surface area, and thus a smaller (and FIGURE 10–35
Dropwise condensation of steam on a
vertical surface (from Hampson
and Özisik, Ref. 11).
¸ cen58933_ch10.qxd 9/4/2002 12:38 PM Page 546 546
HEAT TRANSFER less expensive) condenser. Therefore, dropwise condensation is the preferred
mode of condensation in heat transfer applications.
The challenge in dropwise condensation is not to achieve it, but rather, to
sustain it for prolonged periods of time. Dropwise condensation is achieved
by adding a promoting chemical into the vapor, treating the surface with a
promoter chemical, or coating the surface with a polymer such as teflon or a
noble metal such as gold, silver, rhodium, palladium, or platinum. The promoters used include various waxes and fatty acids such as oleic, stearic, and
linoic acids. They lose their effectiveness after a while, however, because of
fouling, oxidation, and the removal of the promoter from the surface. It is possible to sustain dropwise condensation for over a year by the combined effects
of surface coating and periodic injection of the promoter into the vapor. However, any gain in heat transfer must be weighed against the cost associated
with sustaining dropwise condensation.
Dropwise condensation has been studied experimentally for a number of
surface–fluid combinations. Of these, the studies on the condensation of steam
on copper surfaces has attracted the most attention because of their widespread use in steam power plants. P. Griffith (1983) recommends these simple
correlations for dropwise condensation of steam on copper surfaces:
hdropwise 51,104 2044Tsat
,
255,310 22°C
Tsat Tsat 100°C
100°C (10-36)
(10-37) where Tsat is in °C and the heat transfer coefficient hdropwise is in W/m2 · °C.
The very high heat transfer coefficients achievable with dropwise condensation are of little significance if the material of the condensing surface is not
a good conductor like copper or if the thermal resistance on the other side of
the surface is too large. In steady operation, heat transfer from one medium to
another depends on the sum of the thermal resistances on the path of heat
flow, and a large thermal resistance may overshadow all others and dominate
the heat transfer process. In such cases, improving the accuracy of a small resistance (such as one due to condensation or boiling) makes hardly any difference in overall heat transfer calculations. TOPIC OF SPECIAL INTEREST* Heat Pipes
A heat pipe is a simple device with no moving parts that can transfer large
quantities of heat over fairly large distances essentially at a constant temperature without requiring any power input. A heat pipe is basically a
sealed slender tube containing a wick structure lined on the inner surface
and a small amount of fluid such as water at the saturated state, as shown
in Figure 10–36. It is composed of three sections: the evaporator section at
one end, where heat is absorbed and the fluid is vaporized; a condenser
section at the other end, where the vapor is condensed and heat is rejected;
and the adiabatic section in between, where the vapor and the liquid phases
of the fluid flow in opposite directions through the core and the wick,
*This section can be skipped without a loss in continuity. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 547 547
CHAPTER 10
Wick Tube wall Liquid
flow Copper
tube Heat
in Heat
out Vapor
core Vapor flow Wick
(liquid flow
passage)
Cross-section
of a heat pipe Evaporation
section Adiabatic
section Condenser
section FIGURE 10–36
Schematic and operation of a heat pipe. respectively, to complete the cycle with no significant heat transfer between the fluid and the surrounding medium.
The type of fluid and the operating pressure inside the heat pipe depend
on the operating temperature of the heat pipe. For example, the criticaland triple-point temperatures of water are 0.01°C and 374.1°C, respectively. Therefore, water can undergo a liquid-to-vapor or vapor-to-liquid
phase change process in this temperature range only, and thus it will not be
a suitable fluid for applications involving temperatures beyond this range.
Furthermore, water will undergo a phase-change process at a specified temperature only if its pressure equals the saturation pressure at that temperature. For example, if a heat pipe with water as the working fluid is designed
to remove heat at 70°C, the pressure inside the heat pipe must be maintained at 31.2 kPa, which is the boiling pressure of water at this temperature. Note that this value is well below the atmospheric pressure of
101 kPa, and thus the heat pipe will operate in a vacuum environment in
this case. If the pressure inside is maintained at the local atmospheric pressure instead, heat transfer would result in an increase in the temperature of
the water instead of evaporation.
Although water is a suitable fluid to use in the moderate temperature
range encountered in electronic equipment, several other fluids can be used
in the construction of heat pipes to enable them to be used in cryogenic as
well as high-temperature applications. The suitable temperature ranges for
some common heat pipe fluids are given in Table 10–5. Note that the overall temperature range extends from almost absolute zero for cryogenic fluids such as helium to over 1600°C for liquid metals such as lithium. The
ultimate temperature limits for a fluid are the triple- and critical-point temperatures. However, a narrower temperature range is used in practice to
avoid the extreme pressures and low heats of vaporization that occur near
the critical point. Other desirable characteristics of the candidate fluids are
having a high surface tension to enhance the capillary effect and being
compatible with the wick material, as well as being readily available,
chemically stable, nontoxic, and inexpensive.
The concept of heat pipe was originally conceived by R. S. Gaugler of
the General Motors Corporation, who filed a patent application for it in TABLE 10–5
Suitable temperature ranges for
some fluids used in heat pipes
Fluid Temperature
Range, °C Helium
Hydrogen
Neon
Nitrogen
Methane
Ammonia
Water
Mercury
Cesium
Sodium
Lithium 271
259
248
210
182
78
5
200
400
500
850 to
to
to
to
to
to
to
to
to
to
to 268
240
230
150
82
130
230
500
1000
1200
1600 cen58933_ch10.qxd 9/4/2002 12:38 PM Page 548 548
HEAT TRANSFER 1942. However, it did not receive much attention until 1962, when it was
suggested for use in space applications. Since then, heat pipes have found
a wide range of applications, including the cooling of electronic equipment. The Operation of a Heat Pipe
The operation of a heat pipe is based on the following physical principles:
• At a specified pressure, a liquid will vaporize or a vapor will condense
at a certain temperature, called the saturation temperature. Thus,
fixing the pressure inside a heat pipe fixes the temperature at which
phase change will occur.
• At a specified pressure or temperature, the amount of heat absorbed as
a unit mass of liquid vaporizes is equal to the amount of heat rejected
as that vapor condenses.
• The capillary pressure developed in a wick will move a liquid
in the wick even against the gravitational field as a result of the
capillary effect.
• A fluid in a channel flows in the direction of decreasing pressure.
Initially, the wick of the heat pipe is saturated with liquid and the core
section is filled with vapor. When the evaporator end of the heat pipe is
brought into contact with a hot surface or is placed into a hot environment,
heat will transfer into the heat pipe. Being at a saturated state, the liquid in
the evaporator end of the heat pipe will vaporize as a result of this heat
transfer, causing the vapor pressure there to rise. This resulting pressure
difference drives the vapor through the core of the heat pipe from the evaporator toward the condenser section. The condenser end of the heat pipe is
in a cooler environment, and thus its surface is slightly cooler. The vapor
that comes into contact with this cooler surface condenses, releasing the
heat a vaporization, which is rejected to the surrounding medium. The liquid then returns to the evaporator end of the heat pipe through the wick as
a result of capillary action in the wick, completing the cycle. As a result,
heat is absorbed at one end of the heat pipe and is rejected at the other end,
with the fluid inside serving as a transport medium for heat.
The boiling and condensation processes are associated with extremely
high heat transfer coefficients, and thus it is natural to expect the heat pipe
to be a very effective heat transfer device, since its operation is based on alternative boiling and condensation of the working fluid. Indeed, heat pipes
have effective conductivities several hundred times that of copper or silver.
That is, replacing a copper bar between two mediums at different temperatures by a heat pipe of equal size can increase the rate of heat transfer between those two mediums by several hundred times. A simple heat pipe
with water as the working fluid has an effective thermal conductivity of the
order of 100,000 W/m · °C compared with about 400 W/m · °C for copper.
For a heat pipe, it is not unusual to have an effective conductivity
of 400,000 W/m · °C, which is 1000 times that of copper. A 15-cm-long,
0.6-cm-diameter horizontal cylindrical heat pipe with water inside, for
example, can transfer heat at a rate of 300 W. Therefore, heat pipes are
preferred in some critical applications, despite their high initial cost. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 549 549
CHAPTER 10 There is a small pressure difference between the evaporator and
condenser ends, and thus a small temperature difference between the two
ends of the heat pipe. This temperature difference is usually between 1°C
and 5°C. The Construction of a Heat Pipe
The wick of a heat pipe provides the means for the return of the liquid to
the evaporator. Therefore, the structure of the wick has a strong effect on
the performance of a heat pipe, and the design and construction of the wick
are the most critical aspects of the manufacturing process.
The wicks are often made of porous ceramic or woven stainless wire
mesh. They can also be made together with the tube by extruding axial
grooves along its inner surface, but this approach presents manufacturing
difficulties.
The performance of a wick depends on its structure. The characteristics
of a wick can be changed by changing the size and the number of the pores
per unit volume and the continuity of the passageway. Liquid motion in the
wick depends on the dynamic balance between two opposing effects: the
capillary pressure, which creates the suction effect to draw the liquid, and
the internal resistance to flow as a result of friction between the mesh surfaces and the liquid. A small pore size increases the capillary action, since
the capillary pressure is inversely proportional to the effective capillary radius of the mesh. But decreasing the pore size and thus the capillary radius
also increases the friction force opposing the motion. Therefore, the core
size of the mesh should be reduced so long as the increase in capillary force
is greater than the increase in the friction force.
Note that the optimum pore size will be different for different fluids and
different orientations of the heat pipe. An improperly designed wick will
result in an inadequate liquid supply and eventual failure of the heat pipe.
Capillary action permits the heat pipe to operate in any orientation in a
gravity field. However, the performance of a heat pipe will be best when
the capillary and gravity forces act in the same direction (evaporator end
down) and will be worst when these two forces act in opposite directions
(evaporator end up). Gravity does not affect the capillary force when the
heat pipe is in the horizontal position. The heat removal capacity of a horizontal heat pipe can be doubled by installing it vertically with evaporator
end down so that gravity helps the capillary action. In the opposite case,
vertical orientation with evaporator end up, the performance declines considerably relative the horizontal case since the capillary force in this case
must work against the gravity force.
Most heat pipes are cylindrical in shape. However, they can be manufactured in a variety of shapes involving 90° bends, S-turns, or spirals. They
can also be made as a flat layer with a thickness of about 0.3 cm. Flat heat
pipes are very suitable for cooling high-power-output (say, 50 W or
greater) PCBs. In this case, flat heat pipes are attached directly to the back
surface of the PCB, and they absorb and transfer the heat to the edges.
Cooling fins are usually attached to the condenser end of the heat pipe to
improve its effectiveness and to eliminate a bottleneck in the path of heat
flow from the components to the environment when the ultimate heat sink
is the ambient air. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 550 550
HEAT TRANSFER
Evaporator end FIGURE 10–37
Variation of the heat removal
capacity of a heat pipe with tilt angle
from the horizontal when the liquid
flows in the wick against gravity
(from Steinberg, Ref. 18). TABLE 10–6
Typical heat removal capacity of
various heat pipes
Outside
Diameter,
cm (in.)
0.64(1)
4
0.95(3)
8
1.27(1)
2 Length,
cm (in.) Heat
Removal
Rate, W 15.2(6)
30.5(12)
45.7(18)
15.2(6)
30.5(12)
45.7(18)
15.2(6)
30.5(12)
45.7(18) 300
175
150
500
375
350
700
575
550 ∆T = 3°C 180 W
Heat
pipe 0.6 cm
L = 30 cm FIGURE 10–38
Schematic for Example 10–8. Power handling capability, % 100 Coarse wick
Medium wick θ 80
Fine wick
60 Condenser end
40
20
0
0° 10° 20° 30° 40° 50° 60° 70° 80° 90° Angle θ The decline in the performance of a 122–cm-long water heat pipe with
the tilt angle from the horizontal is shown in Figure 10–37 for heat pipes
with coarse, medium, and fine wicks. Note that for the horizontal case, the
heat pipe with a coarse wick performs best, but the performance drops off
sharply as the evaporator end is raised from the horizontal. The heat pipe
with a fine wick does not perform as well in the horizontal position but
maintains its level of performance greatly at tilted positions. It is clear from
this figure that heat pipes that work against gravity must be equipped with
fine wicks. The heat removal capacities of various heat pipes are given in
Table 10–6.
A major concern about the performance of a heat pipe is degradation
with time. Some heat pipes have failed within just a few months after they
are put into operation. The major cause of degradation appears to be contamination that occurs during the sealing of the ends of the heat pipe tube
and affects the vapor pressure. This form of contamination has been minimized by electron beam welding in clean rooms. Contamination of the
wick prior to installation in the tube is another cause of degradation. Cleanliness of the wick is essential for its reliable operation for a long time. Heat
pipes usually undergo extensive testing and quality control process before
they are put into actual use.
An important consideration in the design of heat pipes is the compatibility of the materials used for the tube, wick, and fluid. Otherwise, reaction
between the incompatible materials produces noncondensable gases, which
degrade the performance of the heat pipe. For example, the reaction between stainless steel and water in some early heat pipes generated hydrogen gas, which destroyed the heat pipe. 180 W EXAMPLE 10–8 Replacing a Heat Pipe by a Copper Rod A 30-cm-long cylindrical heat pipe having a diameter of 0.6 cm is dissipating
heat at a rate of 180 W, with a temperature difference of 3°C across the heat
pipe, as shown in Figure 10–38. If we were to use a 30-cm-long copper rod in- cen58933_ch10.qxd 9/4/2002 12:38 PM Page 551 551
CHAPTER 10 stead to remove heat at the same rate, determine the diameter and the mass of
the copper rod that needs to be installed. SOLUTION A cylindrical heat pipe dissipates heat at a specified rate. The
diameter and mass of a copper rod that can conduct heat at the same rate are
to be determined.
Assumptions Steady operating conditions exist.
Properties The properties of copper at room temperature are
8950 kg/m3
and k 386 W/m · °C.
·
Analysis The rate of heat transfer Q through the copper rod can be expressed as ·
Q kA T
L where k is the thermal conductivity, L is the length, and T is the temperature
difference across the copper bar. Solving for the cross-sectional area A and
substituting the specified values gives
L·
Q
kT A 0.3 m
(180 W)
(386 W/m · °C)(3°C) 0.04663 m2 466.3 cm2 Then the diameter and the mass of the copper rod become A 1
D2
4 m V →
AL D 4 A/ 4(466.3 cm2)/ (8590 kg/m3)(0.04663 m2)(0.3 m) 24.4 cm
125.2 kg Therefore, the diameter of the copper rod needs to be almost 25 times that of
the heat pipe to transfer heat at the same rate. Also, the rod would have a mass
of 125.2 kg, which is impossible for an average person to lift. SUMMARY
Boiling occurs when a liquid is in contact with a surface maintained at a temperature Ts sufficiently above the saturation temperature Tsat of the liquid. Boiling is classified as pool boiling
or flow boiling depending on the presence of bulk fluid motion.
Boiling is called pool boiling in the absence of bulk fluid flow
and flow boiling (or forced convection boiling) in its presence.
Pool and flow boiling are further classified as subcooled boiling and saturated boiling depending on the bulk liquid temperature. Boiling is said to be subcooled (or local) when the
temperature of the main body of the liquid is below the saturation temperature Tsat and saturated (or bulk) when the temperature of the liquid is equal to Tsat. Boiling exhibits different
regimes depending on the value of the excess temperature
Texcess. Four different boiling regimes are observed: natural
convection boiling, nucleate boiling, transition boiling, and
film boiling. These regimes are illustrated on the boiling curve. The rate of evaporation and the rate of heat transfer in nucleate
boiling increase with increasing Texcess and reach a maximum
at some point. The heat flux at this point is called the critical
(or maximum) heat flux, q·max. The rate of heat transfer in nucleate pool boiling is determined from
·
q nucleate l hfg g( l ) 1/2 Cpl (Ts Tsat)
Csf hfg Prln 3 The maximum (or critical) heat flux in nucleate pool boiling is
determined from
q·max Ccr hfg[ g 2 ( l )]1/4 where the value of the constant Ccr is about 0.15. The minimum
heat flux is given by cen58933_ch10.qxd 9/4/2002 12:38 PM Page 552 552
HEAT TRANSFER ·
q min 0.09 hfg g(
( ) l 1/4 and fully turbulent for Re 1800. Heat transfer coefficients in
the wavy-laminar and turbulent flow regions are determined
from 2 ) l The heat flux for stable film boiling on the outside of a horizontal cylinder or sphere of diameter D is given by
q·film gk3 Cfilm
(Ts ( )[hfg 0.4Cp (Ts
D(Ts Tsat) l Tsat) hvert, wavy Re kl
1.08 Re1.22 1/4 hvert, turbulent 8750 g 1/3 ,
5.2 2
l
Re kl
58 Pr 0.5 (Re0.75 Tsat) 30 Re 1800 l g 1/3 ,
253) 2
l
Re 1800
l where the constant Cfilm
0.62 for horizontal cylinders and
0.67 for spheres, and k is the thermal conductivity of the
vapor. The vapor properties are to be evaluated at the film temperature Tf (Tsat Ts)/2, which is the average temperature
of the vapor film. The liquid properties and hfg are to be evaluated at the saturation temperature at the specified pressure.
Two distinct forms of condensation are observed in nature:
film condensation and dropwise condensation. In film condensation, the condensate wets the surface and forms a liquid film
on the surface that slides down under the influence of gravity.
In dropwise condensation, the condensed vapor forms countless droplets of varying diameters on the surface instead of a
continuous film.
The Reynolds number for the condensate flow is defined as
Dh Re l l l 4A
p l l
l ·
4m
pl and
·
4Q conden
p l h*
fg Re 4 As h(Tsat Ts)
p l h*
fg where h* is the modified latent heat of vaporization, defined as
fg
h*
fg hfg 0.68Cpl (Tsat 0.943 g l(
l l (Ts ) h* k3
fg l
Tsat)L hhoriz 1/4 (W/m2 · °C) All properties of the liquid are to be evaluated at the film temperature Tf (Tsat Ts)/2. The hfg and are to be evaluated at
Tsat. Condensate flow is smooth and wave-free laminar for
about Re 30, wavy-laminar in the range of 30 Re 1800, 0.729 g l( l
l (Ts ) h* k3
fg l
Tsat)D 1/4 (W/m2 · °C) where D is the diameter of the horizontal tube. This relation
can easily be modified for a sphere by replacing the constant
0.729 by 0.815. It can also be used for N horizontal tubes
stacked on top of each other by replacing D in the denominator
by ND.
For low vapor velocities, film condensation heat transfer
inside horizontal tubes can be determined from
hinternal Ts) and represents heat transfer during condensation per unit
mass of condensate. Here Cpl is the specific heat of the liquid in
J/kg · °C.
Using some simplifying assumptions, the average heat
transfer coefficient for film condensation on a vertical plate of
height L is determined to be
hvert Equations for vertical plates can also be used for laminar
film condensation on the upper surfaces of the plates that are
inclined by an angle from the vertical, by replacing g in that
equation by g cos . Vertical plate equations can also be used to
calculate the average heat transfer coefficient for laminar film
condensation on the outer surfaces of vertical tubes provided
that the tube diameter is large relative to the thickness of the
liquid film.
The average heat transfer coefficient for film condensation
on the outer surfaces of a horizontal tube is determined to be 0.555 g l( l
l (Tsat ) k3
l
h
Ts) fg 3
C (T
8 pl sat 1/4 Ts) and
Revapor D
inlet 35,000 where the Reynolds number of the vapor is to be evaluated at
the tube inlet conditions using the internal tube diameter as the
characteristic length. Finally, the heat transfer coefficient for
dropwise condensation of steam on copper surfaces is given by
hdropwise 51,104 2044Tsat
,
255,310 22°C
Tsat Tsat 100°C
100°C where Tsat is in °C and the heat transfer coefficient hdropwise is in
W/m2 · °C. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 553 553
CHAPTER 10 REFERENCES AND SUGGESTED READING
1. N. Arai, T. Fukushima, A. Arai, T. Nakajima, K. Fujie,
and Y. Nakayama. “Heat Transfer Tubes Enhancing
Boiling and Condensation in Heat Exchangers of a
Refrigeration Machine.” ASHRAE Journal 83 (1977),
p. 58.
2. P. J. Berensen. “Film Boiling Heat Transfer for a
Horizontal Surface.” Journal of Heat Transfer 83 (1961),
pp. 351–358.
3. P. J. Berensen. “Experiments in Pool Boiling Heat
Transfer.” International Journal of Heat Mass Transfer 5
(1962), pp. 985–999.
4. L. A. Bromley. “Heat Transfer in Stable Film Boiling.”
Chemical Engineering Prog. 46 (1950), pp. 221–227.
5. J. C. Chato. “Laminar Condensation inside Horizontal
and Inclined Tubes.” ASHRAE Journal 4 (1962), p. 52.
6. S. W. Chi. Heat Theory and Practice. Washington, D.C.:
Hemisphere, 1976.
7. M. T. Cichelli and C. F. Bonilla. “Heat Transfer to Liquids
Boiling under Pressure.” Transactions of AIChE 41
(1945), pp. 755–787.
8. R. A. Colclaser, D. A. Neaman, and C. F. Hawkins.
Electronic Circuit Analysis. New York: John Wiley &
Sons, 1984.
9. J. W. Dally. Packaging of Electronic Systems. New York:
McGraw-Hill, 1960.
10. P. Griffith. “Dropwise Condensation.” In Heat Exchanger
Design Handbook, ed. E. U. Schlunder, Vol 2, Ch. 2.6.5.
New York: Hemisphere, 1983.
11. H. Hampson and N. Özisik. “An Investigation into the
¸
Condensation of Steam.” Proceedings of the Institute of
Mechanical Engineers, London 1B (1952), pp. 282–294.
12. J. P. Holman. Heat Transfer. 8th ed. New York:
McGraw-Hill, 1997.
13. F. P. Incropera and D. P. DeWitt. lntroduction to Heat
Transfer. 4th ed. New York: John Wiley & Sons, 2002.
14. J. J. Jasper. “The Surface Tension of Pure Liquid
Compounds.” Journal of Physical and Chemical
Reference Data 1, No. 4 (1972), pp. 841–1009. 15. R. Kemp. “The Heat Pipe—A New Tune on an Old Pipe.”
Electronics and Power (August 9, 1973), p. 326.
16. S. S. Kutateladze. Fundamentals of Heat Transfer. New
York: Academic Press, 1963.
17. S. S. Kutateladze. “On the Transition to Film Boiling
under Natural Convection.” Kotloturbostroenie 3 (1948),
p. 48.
18. D. A. Labuntsov. “Heat Transfer in Film Condensation of
Pure Steam on Vertical Surfaces and Horizontal Tubes.”
Teploenergetika 4 (l957), pp. 72–80.
19. J. H. Lienhard and V. K. Dhir. “Extended Hydrodynamic
Theory of the Peak and Minimum Pool Boiling Heat
Fluxes.” NASA Report, NASA-CR-2270, July 1973.
20. J. H. Lienhard and V. K. Dhir. “Hydrodynamic Prediction
of Peak Pool Boiling Heat Fluxes from Finite Bodies.”
Journal of Heat Transfer 95 (1973), pp. 152–158.
21. W. H. McAdams. Heat Transmission. 3rd ed. New York:
McGraw-Hill, 1954.
22. W. M. Rohsenow. “A Method of Correlating Heat
Transfer Data for Surface Boiling of Liquids.” ASME
Transactions 74 (1952), pp. 969–975.
23. D. S. Steinberg. Cooling Techniques for Electronic
Equipment. New York: John Wiley & Sons, 1980.
24. W. M. Rohsenow. “Film Condensation.” In Handbook of
Heat Transfer, ed. W. M. Rohsenow and J. P. Hartnett, Ch.
12A. New York: McGraw-Hill, 1973.
25. I. G. Shekriladze, I. G. Gomelauri, and V. I. Gomelauri.
“Theoretical Study of Laminar Film Condensation of
Flowing Vapor.” International Journal of Heat Mass
Transfer 9 (1966), pp. 591–592.
26. N. V. Suryanarayana. Engineering Heat Transfer. St. Paul,
MN: West Publishing, 1995.
27. J. W. Westwater and J. G. Santangelo. Industrial
Engineering Chemistry 47 (1955), p. 1605.
28. N. Zuber. “On the Stability of Boiling Heat Transfer.”
ASME Transactions 80 (1958), pp. 711–720. PROBLEMS*
Boiling Heat Transfer
10–1C What is boiling? What mechanisms are responsible
for the very high heat transfer coefficients in nucleate boiling?
10–2C Does the amount of heat absorbed as 1 kg of saturated
liquid water boils at 100°C have to be equal to the amount of
heat released as 1 kg of saturated water vapor condenses at
100°C? *Problems designated by a “C” are concept questions, and
students are encouraged to answer them all. Problems designated
by an “E” are in English units, and the SI users can ignore them.
Problems with an EES-CD icon
are solved using EES, and
complete solutions together with parametric studies are included
on the enclosed CD. Problems with a computer-EES icon
are
comprehensive in nature, and are intended to be solved with a
computer, preferably using the EES software that accompanies
this text. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 554 554
HEAT TRANSFER 10–3C What is the difference between evaporation and
boiling?
10–4C What is the difference between pool boiling and flow
boiling?
10–5C What is the difference between subcooled and saturated boiling?
10–6C Draw the boiling curve and identify the different boiling regimes. Also, explain the characteristics of each regime.
10–7C How does film boiling differ from nucleate boiling?
Is the boiling heat flux necessarily higher in the stable film
boiling regime than it is in the nucleate boiling regime?
10–8C Draw the boiling curve and identify the burnout point
on the curve. Explain how burnout is caused. Why is the
burnout point avoided in the design of boilers?
10–9C Discuss some methods of enhancing pool boiling heat
transfer permanently.
10–10C Name the different boiling regimes in the order they
occur in a vertical tube during flow boiling. from 70 kPa and 101.3 kPa. Plot the maximum heat flux and
the temperature difference as a function of the atmospheric
pressure, and discuss the results.
10–14E Water is boiled at atmospheric pressure by a horizontal polished copper heating element of diameter D 0.5 in.
and emissivity
0.08 immersed in water. If the surface temperature of the heating element is 788°F, determine the rate of
heat transfer to the water per unit length of the heating element.
Answer: 2465 Btu/h 10–15E Repeat Problem 10–14E for a heating element temperature of 988°F.
10–16 Water is to be boiled at sea level in a 30-cm-diameter
mechanically polished AISI 304 stainless steel pan placed on
top of a 3-kW electric burner. If 60 percent of the heat generated by the burner is transferred to the water during boiling,
determine the temperature of the inner surface of the bottom
of the pan. Also, determine the temperature difference between
the inner and outer surfaces of the bottom of the pan if it is
6 mm thick. 10–11 Water is to be boiled at atmospheric pressure in a mechanically polished steel pan placed on top of a heating unit.
The inner surface of the bottom of the pan is maintained at
110°C. If the diameter of the bottom of the pan is 25 cm, determine (a) the rate of heat transfer to the water and (b) the rate of
evaporation. 1 atm 3 kW 1 atm FIGURE P10–16
110°C 10–17 Repeat Problem 10–16 for a location at an elevation of
1500 m where the atmospheric pressure is 84.5 kPa and thus
the boiling temperature of water is 95°C.
Answers: 100.9°C, 10.3°C 10–18 Water is boiled at sea level in a coffee maker equipped
with a 20-cm long 0.4-cm-diameter immersion-type electric
1 atm FIGURE P10–11
10–12 Water is to be boiled at atmospheric pressure on a
3-cm-diameter mechanically polished steel heater. Determine
the maximum heat flux that can be attained in the nucleate
boiling regime and the surface temperature of the heater surface in that case.
Reconsider Problem 10–12. Using EES (or
other) software, investigate the effect of local
atmospheric pressure on the maximum heat flux and the temperature difference Ts Tsat. Let the atmospheric pressure vary Coffee
maker
1L 10–13 FIGURE P10–18 cen58933_ch10.qxd 9/4/2002 12:38 PM Page 555 555
CHAPTER 10 heating element made of mechanically polished stainless steel.
The coffee maker initially contains 1 L of water at 18°C. Once
boiling starts, it is observed that half of the water in the coffee
maker evaporates in 25 min. Determine the power rating of the
electric heating element immersed in water and the surface
temperature of the heating element. Also determine how long it
will take for this heater to raise the temperature of 1 L of cold
water from 18°C to the boiling temperature.
10–19 Vent
Boiler 150°C Repeat Problem 10–18 for a copper heating element. 10–20 A 65-cm-long, 2-cm-diameter brass heating element is
to be used to boil water at 120°C. If the surface temperature of
the heating element is not to exceed 125°C, determine the
highest rate of steam production in the boiler, in kg/h. Water 165°C Answer: 19.4 kg/h 10–21 To understand the burnout phenomenon, boiling experiments are conducted in water at atmospheric pressure using
an electrically heated 30-cm-long, 3-mm-diameter nickelplated horizontal wire. Determine (a) the critical heat flux and
(b) the increase in the temperature of the wire as the operating
point jumps from the nucleate boiling to the film boiling
regime at the critical heat flux. Take the emissivity of the wire
to be 0.5.
10–22 Reconsider Problem 10–21. Using EES (or
other) software, investigate the effects of the
local atmospheric pressure and the emissivity of the wire on
the critical heat flux and the temperature rise of wire. Let the
atmospheric pressure vary from 70 kPa and 101.3 kPa and
the emissivity from 0.1 to 1.0. Plot the critical heat flux and the
temperature rise as functions of the atmospheric pressure and
the emissivity, and discuss the results.
10–23 Water is boiled at 1 atm pressure in a 20-cm-internaldiameter teflon-pitted stainless steel pan on an electric range. If
it is observed that the water level in the pan drops by 10 cm in
30 min, determine the inner surface temperature of the pan.
Answer: 111.5°C 10–24 Repeat Problem 10–23 for a polished copper pan. 10–25 In a gas-fired boiler, water is boiled at 150°C by hot
gases flowing through 50-m-long, 5-cm-outer-diameter mechanically polished stainless steel pipes submerged in water. If
the outer surface temperature of the pipes is 165°C, determine
(a) the rate of heat transfer from the hot gases to water, (b) the
rate of evaporation, (c) the ratio of the critical heat flux to the
present heat flux, and (d) the surface temperature of the pipe at
which critical heat flux occurs.
Answers: (a) 10,865 kW, (b) 5.139 kg/s, (c) 1.34, (d) 166.5°C 10–26 Repeat Problem 10–25 for a boiling temperature of
160°C.
10–27E Water is boiled at 250°F by a 2-ft-long and 0.5-in.diameter nickel-plated electric heating element maintained
at 280°F. Determine (a) the boiling heat transfer coefficient, Hot gases FIGURE P10–25
(b) the electric power consumed by the heating element, and
(c) the rate of evaporation of water.
10–28E Repeat Problem 10–27E for a platinum-plated heating element.
10–29E Reconsider Problem 10–27E. Using EES (or
other) software, investigate the effect of surface temperature of the heating element on the boiling heat
transfer coefficient, the electric power, and the rate of evaporation of water. Let the surface temperature vary from 260°F to
300°F. Plot the boiling heat transfer coefficient, the electric
power consumption, and the rate of evaporation of water as a
function of the surface temperature, and discuss the results.
10–30 Cold water enters a steam generator at 15°C and
leaves as saturated steam at 100°C. Determine the fraction of
heat used to preheat the liquid water from 15°C to the saturation temperature of 100°C in the steam generator.
Answer: 13.6 percent 10–31 Cold water enters a steam generator at 20°C and
leaves as saturated steam at the boiler pressure. At what pressure will the amount of heat needed to preheat the water to saturation temperature be equal to the heat needed to vaporize the
liquid at the boiler pressure?
10–32 Reconsider Problem 10–31. Using EES (or
other) software, plot the boiler pressure as a
function of the cold water temperature as the
temperature varies from 0°C to 30°C, and discuss the results.
10–33 A 50-cm-long, 2-mm-diameter electric resistance
wire submerged in water is used to determine the boiling
heat transfer coefficient in water at 1 atm experimentally. The
wire temperature is measured to be 130°C when a wattmeter cen58933_ch10.qxd 9/4/2002 12:38 PM Page 556 556
HEAT TRANSFER will the average heat transfer coefficient for the entire stack of
tubes be equal to half of what it is for a single horizontal tube? 1 atm
3.8 kW 130°C FIGURE P10–33
indicates the electric power consumed to be 3.8 kW. Using
Newton’s law of cooling, determine the boiling heat transfer
coefficient. Answer: 16 10–44 Saturated steam at 1 atm condenses on a 3-m-high and
5-m-wide vertical plate that is maintained at 90°C by circulating cooling water through the other side. Determine (a) the rate
of heat transfer by condensation to the plate, and (b) the rate at
which the condensate drips off the plate at the bottom.
Answers: (a) 942 kW, (b) 0.412 kg/s
1 atm
Steam 5m Condensation Heat Transfer
10–34C What is condensation? How does it occur?
10–35C What is the difference between film and dropwise
condensation? Which is a more effective mechanism of heat
transfer? 90°C
3m 10–36C In condensate flow, how is the wetted perimeter
defined? How does wetted perimeter differ from ordinary
perimeter?
10–37C What is the modified latent heat of vaporization?
For what is it used? How does it differ from the ordinary latent
heat of vaporization? ·
m FIGURE P10–44 10–38C Consider film condensation on a vertical plate. Will
the heat flux be higher at the top or at the bottom of the plate?
Why? 10–45 Repeat Problem 10–44 for the case of the plate being
tilted 60° from the vertical. 10–39C Consider film condensation on the outer surfaces of
a tube whose length is 10 times its diameter. For which orientation of the tube will the heat transfer rate be the highest:
horizontal or vertical? Explain. Disregard the base and top surfaces of the tube. 10–46 Saturated steam at 30°C condenses on the outside of a
4-cm-outer-diameter, 2-m-long vertical tube. The temperature
of the tube is maintained at 20°C by the cooling water. Determine (a) the rate of heat transfer from the steam to the cooling
water, (b) the rate of condensation of steam, and (c) the approximate thickness of the liquid film at the bottom of the tube. 10–40C Consider film condensation on the outer surfaces of
four long tubes. For which orientation of the tubes will the condensation heat transfer coefficient be the highest: (a) vertical,
(b) horizontal side by side, (c) horizontal but in a vertical tier
(directly on top of each other), or (d) a horizontal stack of two
tubes high and two tubes wide?
10–41C How does the presence of a noncondensable gas in a
vapor influence the condensation heat transfer?
10–42 The Reynolds number for condensate flow is defined
·
as Re 4m /p l, where p is the wetted perimeter. Obtain simplified relations for the Reynolds number by expressing p and
·
m by their equivalence for the following geometries: (a) a vertical plate of height L and width w, (b) a tilted plate of height L
and width w inclined at an angle from the vertical, (c) a vertical cylinder of length L and diameter D, (d) a horizontal
cylinder of length L and diameter D, and (e) a sphere of diameter D.
10–43 Consider film condensation on the outer surfaces of N
horizontal tubes arranged in a vertical tier. For what value of N 4 cm
Steam
30°C
Condensate L=2m 20°C FIGURE P10–46
10–47E Saturated steam at 95°F is condensed on the outer
surfaces of an array of horizontal pipes through which cooling
water circulates. The outer diameter of the pipes is 1 in. and the
outer surfaces of the pipes are maintained at 85°F. Determine
(a) the rate of heat transfer to the cooling water circulating in
the pipes and (b) the rate of condensation of steam per unit
length of a single horizontal pipe. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 557 557
CHAPTER 10 10–48E Repeat Problem 10–47E for the case of 32 horizontal pipes arranged in a rectangular array of 4 pipes high and
8 pipes wide.
10–49 Saturated steam at 55°C is to be condensed at a rate of
10 kg/h on the outside of a 3-cm-outer-diameter vertical tube
whose surface is maintained at 45°C by the cooling water. Determine the tube length required.
10–50 Repeat Problem 10–49 for a horizontal tube. Answer: 0.70 m 10–51 Saturated steam at 100°C condenses on a 2-m 2-m
plate that is tilted 40° from the vertical. The plate is maintained
at 80°C by cooling it from the other side. Determine (a) the average heat transfer coefficient over the entire plate and (b) the
rate at which the condensate drips off the plate at the bottom.
10–52 Reconsider Problem 10–51. Using EES (or
other) software, investigate the effects of plate
temperature and the angle of the plate from the vertical on the
average heat transfer coefficient and the rate at which the condensate drips off. Let the plate temperature vary from 40°C to
90°C and the plate angle from 0° to 60°. Plot the heat transfer
coefficient and the rate at which the condensate drips off as the
functions of the plate temperature and the tilt angle, and discuss the results.
10–53 Saturated ammonia vapor at 10°C condenses on the
outside of a 2-cm-outer-diameter, 8-m-long horizontal tube
whose outer surface is maintained at 10°C. Determine (a) the
rate of heat transfer from the ammonia and (b) the rate of condensation of ammonia.
10–54 The condenser of a steam power plant operates at a
pressure of 4.25 kPa. The condenser consists of 100 horizontal
tubes arranged in a 10 10 square array. The tubes are 8 m Cooling
water
20°C
L=8m Reconsider Problem 10–54. Using EES (or
other) software, investigate the effect of the
condenser pressure on the rate of heat transfer and the rate of
condensation of the steam. Let the condenser pressure vary
from 3 kPa to 15 kPa. Plot the rate of heat transfer and the rate
of condensation of the steam as a function of the condenser
pressure, and discuss the results. 10–56 A large heat exchanger has several columns of tubes,
with 20 tubes in each column. The outer diameter of the tubes
is 1.5 cm. Saturated steam at 50°C condenses on the outer surfaces of the tubes, which are maintained at 20°C. Determine
(a) the average heat transfer coefficient and (b) the rate of condensation of steam per m length of a column.
10–57 Saturated refrigerant-134a vapor at 30°C is to be condensed in a 5-m-long, 1-cm-diameter horizontal tube that is
maintained at a temperature of 20°C. If the refrigerant enters
the tube at a rate of 2.5 kg/min, determine the fraction of the
refrigerant that will have condensed at the end of the tube.
10–58 Repeat Problem 10–57 for a tube length of 8 m. Answer: 17.2 percent 10–59 Reconsider Problem 10–57. Using EES (or
other) software, plot the fraction of the refrigerant condensed at the end of the tube as a function of the temperature of the saturated R-134a vapor as the temperature
varies from 25°C to 50°C, and discuss the results. Special Topic: Heat Pipes 10–61C A heat pipe with water as the working fluid is said to
have an effective thermal conductivity of 100,000 W/m · °C,
which is more than 100,000 times the conductivity of water.
How can this happen? P = 4.25 kPa n = 100
tubes 10–55 10–60C What is a heat pipe? How does it operate? Does it
have any moving parts? Saturated
steam N = 10 long and have an outer diameter of 3 cm. If the tube surfaces
are at 20°C, determine (a) the rate of heat transfer from the
steam to the cooling water and (b) the rate of condensation of
steam in the condenser.
Answers: (a) 3678 kW, (b) 1.496 kg/s 10–62C What is the effect of a small amount of noncondensable gas such as air on the performance of a heat pipe?
10–63C Why do water-based heat pipes used in the cooling
of electronic equipment operate below atmospheric pressure?
10–64C What happens when the wick of a heat pipe is too
coarse or too fine?
10–65C Does the orientation of a heat pipe affect its performance? Does it matter if the evaporator end of the heat pipe is
up or down? Explain. FIGURE P10–54 10–66C How can the liquid in a heat pipe move up against
gravity without a pump? For heat pipes that work against gravity, is it better to have coarse or fine wicks? Why? cen58933_ch10.qxd 9/4/2002 12:38 PM Page 558 558
HEAT TRANSFER 10–67C What are the important considerations in the design
and manufacture of heat pipes? coefficient, (b) the rate of heat transfer, and (c) the rate of condensation of ammonia. 10–68C What is the major cause for the premature degradation of the performance of some heat pipes? 10–74 Saturated isobutane vapor in a binary geothermal
power plant is to be condensed outside an array of eight horizontal tubes. Determine the ratio of the condensation rate for
the cases of the tubes being arranged in a horizontal tier versus
Answer: 1.68
in a vertical tier of horizontal tubes. 10–69 A 40-cm-long cylindrical heat pipe having a diameter
of 0.5 cm is dissipating heat at a rate of 150 W, with a temperature difference of 4°C across the heat pipe. If we were
to use a 40-cm-long copper rod (k 386 W/m · °C and
8950 kg/m3) instead to remove heat at the same rate, determine
the diameter and the mass of the copper rod that needs to be installed.
10–70 Repeat Problem 10–69 for an aluminum rod instead of
copper.
10–71E A plate that supports 10 power transistors, each dissipating 35 W, is to be cooled with 1-ft-long heat pipes having
a diameter of 1 in. Using Table 10–6, determine how many
4
pipes need to be attached to this plate.
Answer: 2
Heat sink
Heat
pipe 10–75E The condenser of a steam power plant operates at a
pressure of 0.95 psia. The condenser consists of 144 horizontal
tubes arranged in a 12 12 square array. The tubes are 15 ft
long and have an outer diameter of 1.2 in. If the outer surfaces
of the tubes are maintained at 80°F, determine (a) the rate of
heat transfer from the steam to the cooling water and (b) the
rate of condensation of steam in the condenser.
10–76E Repeat Problem 10–75E for a tube diameter of 2 in. 10–77 Water is boiled at 100°C electrically by a 80-cm-long,
2-mm-diameter horizontal resistance wire made of chemically
etched stainless steel. Determine (a) the rate of heat transfer to
the water and the rate of evaporation of water if the temperature of the wire is 115°C and (b) the maximum rate of evaporation in the nucleate boiling regime.
Answers: (a) 2387 W, 3.81 kg/h, (b) 1280 kW/m2 Transistor Steam Water
100°C FIGURE P10–71E
115°C Review Problems
10–72 Steam at 40°C condenses on the outside of a 3-cm diameter thin horizontal copper tube by cooling water that enters
the tube at 25°C at an average velocity of 2 m/s and leaves at
35°C. Determine the rate of condensation of steam, the average
overall heat transfer coefficient between the steam and the
cooling water, and the tube length.
Steam
40°C
Cooling
water 35°C 25°C FIGURE P10–72
10–73 Saturated ammonia vapor at 25°C condenses on the
outside of a 2-m-long, 3.2-cm-outer-diameter vertical tube
maintained at 15°C. Determine (a) the average heat transfer FIGURE P10–77
10–78E Saturated steam at 100°F is condensed on a 6-ft-high
vertical plate that is maintained at 80°F. Determine the rate of
heat transfer from the steam to the plate and the rate of condensation per foot width of the plate.
10–79 Saturated refrigerant-134a vapor at 35°C is to be condensed on the outer surface of a 7-m-long, 1.5-cm-diameter
horizontal tube that is maintained at a temperature of 25°C.
Determine the rate at which the refrigerant will condense, in
kg/min.
10–80 Repeat Problem 10–79 for a tube diameter of 3 cm. 10–81 Saturated steam at 270.1 kPa condenses inside a horizontal, 6-m-long, 3-cm-internal-diameter pipe whose surface is
maintained at 110°C. Assuming low vapor velocity, determine cen58933_ch10.qxd 9/4/2002 12:38 PM Page 559 559
CHAPTER 10 the average heat transfer coefficient and the rate of condensation of the steam inside the pipe.
Answers: 3345 W/m2 · °C, 0.0174 kg/s 10–82 A 1.5-cm-diameter silver sphere initially at
30°C is suspended in a room filled with saturated steam at 100°C. Using the lumped system analysis, determine how long it will take for the temperature of the ball to
rise to 50°C. Also, determine the amount of steam that condenses during this process and verify that the lumped system
analysis is applicable.
10–83 Repeat Problem 10–82 for a 3-cm-diameter copper ball.
10–84 You have probably noticed that water vapor that condenses on a canned drink slides down, clearing the surface for
further condensation. Therefore, condensation in this case can
be considered to be dropwise. Determine the condensation heat
transfer coefficient on a cold canned drink at 5°C that is placed
in a large container filled with saturated steam at 95°C.
Steam
95°C 5°C FIGURE P10–84 behind the refrigerator. Heat transfer from the outer surface of
the coil to the surroundings is by natural convection and radiation. Obtaining information about the operating conditions of
the refrigerator, including the pressures and temperatures of the
refrigerant at the inlet and the exit of the coil, show that the coil
is selected properly, and determine the safety margin in the
selection.
10–88 Water-cooled steam condensers are commonly used in
steam power plants. Obtain information about water-cooled
steam condensers by doing a literature search on the topic and
also by contacting some condenser manufacturers. In a report,
describe the various types, the way they are designed, the limitation on each type, and the selection criteria.
10–89 Steam boilers have long been used to provide process
heat as well as to generate power. Write an essay on the history
of steam boilers and the evolution of modern supercritical
steam power plants. What was the role of the American Society
of Mechanical Engineers in this development?
10–90 The technology for power generation using geothermal energy is well established, and numerous geothermal
power plants throughout the world are currently generating
electricity economically. Binary geothermal plants utilize a
volatile secondary fluid such as isobutane, n-pentane, and
R-114 in a closed loop. Consider a binary geothermal plant
with R-114 as the working fluid that is flowing at a rate of
600 kg/s. The R-114 is vaporized in a boiler at 115°C by the
geothermal fluid that enters at 165°C and is condensed at 30°C
outside the tubes by cooling water that enters the tubes at 18°C.
Design the condenser of this binary plant.
Specify (a) the length, diameter, and number of tubes and
their arrangement in the condenser, (b) the mass flow rate of
cooling water, and (c) the flow rate of make-up water needed if
a cooling tower is used to reject the waste heat from the cooling water. The liquid velocity is to remain under 6 m/s and the
length of the tubes is limited to 8 m. 10–85 A resistance heater made of 2-mm-diameter nickel
wire is used to heat water at 1 atm pressure. Determine the
highest temperature at which this heater can operate safely
without the danger of burning out.
Answer: 109.6°C 10–91 A manufacturing facility requires saturated steam at
120°C at a rate of 1.2 kg/min. Design an electric steam boiler
for this purpose under these constraints: Computer, Design, and Essay Problems • The boiler will be in cylindrical shape with a height-todiameter ratio of 1.5. The boiler can be horizontal or
vertical. 10–86 Design the condenser of a steam power plant that has
a thermal efficiency of 40 percent and generates 10 MW of net
electric power. Steam enters the condenser as saturated vapor
at 10 kPa, and it is to be condensed outside horizontal tubes
through which cooling water from a nearby river flows. The
temperature rise of the cooling water is limited to 8°C, and the
velocity of the cooling water in the pipes is limited to 6 m/s to
keep the pressure drop at an acceptable level. Specify the pipe
diameter, total pipe length, and the arrangement of the pipes to
minimize the condenser volume.
10–87 The refrigerant in a household refrigerator is condensed as it flows through the coil that is typically placed Boiler FIGURE P10–91 cen58933_ch10.qxd 9/4/2002 12:38 PM Page 560 560
HEAT TRANSFER • The boiler will operate in the nucleate boiling regime,
and the design heat flux will not exceed 60 percent of the
critical heat flux to provide an adequate safety margin.
• A commercially available plug-in type electrical heating
element made of mechanically polished stainless steel will
be used. The diameter of the heater cannot be between
0.5 cm and 3 cm.
• Half of the volume of the boiler should be occupied by
steam, and the boiler should be large enough to hold
enough water for 2 h supply of steam. Also, the boiler
will be well insulated. late the surface area of the heater. First, boil water in a pan using the heating element and measure the temperature of the
boiling water away from the heating element. Based on your
reading, estimate the elevation of your location, and compare it
to the actual value. Then glue the tip of the thermocouple wire
of the thermometer to the midsection of the heater surface. The
temperature reading in this case will give the surface temperature of the heater. Assuming the rated power of the heater to be
the actual power consumption during heating (you can check
this by measuring the electric current and voltage), calculate
the heat transfer coefficients from Newton’s law of cooling. You are to specify the following: (1) The height and inner diameter of the tank, (2) the length, diameter, power rating, and
surface temperature of the electric heating element, (3) the
maximum rate of steam production during short periods of
overload conditions, and how it can be accomplished.
10–92 Repeat Problem 10–91 for a boiler that produces
steam at 150°C at a rate of 2.5 kg/min.
10–93 Conduct this experiment to determine the boiling heat
transfer coefficient. You will need a portable immersion-type
electric heating element, an indoor-outdoor thermometer, and
metal glue (all can be purchased for about $15 in a hardware
store). You will also need a piece of string and a ruler to calcu- FIGURE P10–93 ...
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