Condensation phenomena occur in
many industrial applications. In this section the focus is on the determination
of the condensation heat transfer coefficient and the overall energy balance is
left to the reader. There are two idealized models of condensation (i.e.,
filmwise and dropwise). The former occurs on a cooled surface which is easily
wetted. The vapor condenses in drops which grow by further condensation and
coalesce to form a film over the surface, if the surface-fluid combination is
wettable; if the surface is non-wetting rivulets of liquid flow away and new
drops then begin to form. This review and discussion will mainly deal with
filmwise condensation. The phenomena of dropwise condensation results in local
heat transfer coefficients which are often an order of magnitude greater than
those for filmwise condensation. Even though condensation phenomena can be classified
into these categories of dropwise and film condensation the initial period of
condensation would evolve into a film and probably would not affect the overall
pressure-temperature response unless drop condensation is promoted
(Slaughterbeck, 1970). Rates of heat transfer for film condensation can be
predicted as a function of bulk and surface temperatures, total bulk pressure,
surface and liquid film characteristics, bulk velocity and the presence of
noncondensible gases. Even though film condensation has been investigated
extensively, the majority of these studies were devoted to laminar film
condensation (laminar bulk flow and laminar film). Since the vapor flow in heat
exchange equipment may be turbulent, models and recent data are also reviewed
for the condensation flux with a turbulent mixture flow. A simple engineering
correlation or model is preferred many times for use in engineering design
studies and with existing computer system analyses (Schmitt, et al., 1970;
Tagami, 1965).
Previous theoretical and
experimental investigations are reviewed, in particular, the effects of the presence of noncondensable gases and of
the vapor velocity. These effects
along with the effects of geometry and scale are of major interest at this
time. Because the flow regime for the condensation heat
transfer is well defined (stratified flow), the reader will find a much greater
propensity for detailed mathematical analyses for simple geometries.
Condensation on a vertical or horizontal flat plate, which can be extended to
any arbitrary geometry, is the main focus of this discussion, because of its
generality. The detailed review for film condensation outside a tube can be
found in the work of Lee (1982). The usual modification is to replace the
length scale, L, by tube diameter, D, with a slight change in the
proportionality constant. Table 1 is provided as a summary of the work on
condensation at the present time.
By analogy with the process of
evaporation, liquid may form in one of three ways corresponding to the
existence of an unstable, metastable or stable equilibrium state. Let us
briefly look at each one of these to understand the condensation process. In
practical engineering design of heat exchange equipment the stable condensation
situation needs to exist.
Consider a liquid drop of radius,
r*, in equilibrium with its surrounding vapor at a system temperature,
where R is the vapor gas constant and
one can find the liquid pressure in the
droplet as
One can also use the
Clausius-Clapeyron relation to calculate for this simple situation the
saturation temperature,
Analogous to boiling, the rate of
nucleation of these liquid droplets depends on whether one considers
homogeneous nucleation or heterogeneous nucleation processes. For homogeneous
nucleation the rate expression, dn/dt, is quite similar to that
for boiling,
where r* can be found either from Eq.2
or 3. The term,
where m is the mass of one vapor
molecule. One should note that in a similar fashion to boiling this nucleation
rate is altered if it occurs on solid surfaces since the work required to form
a critical size nuclei (r*) is reduced due to wetting of the solid
surface.
Now in reference to the more
stable situations of a vapor condensing on a planar surface covered by its own
liquid one must consider the local mass transfer situation. Consider a pure
saturated vapor at a pressure,
From kinetic theory it can be
shown that, in a stationary container of molecules, the mass rate of flow (of
molecules) passing in either direction (to right or left) through an imagined
plane is given by
where
In general it can be stated that
the net molecular flux through an interface is the difference between these
fluxes in the directions from gas to liquid and vice-versa,
Since the condition close to the
surface is not one of static thermal equilibrium, for any significant rate of
evaporation or condensation, it is really not meaningful to make use of the thermostatic pressure and temperature on each
side of the interface. Rather there is a
concentration and therefore, a temperature difference,
Only in the
presence of noncondensable gases (continuum) or at low pressure (non-continuum)
is this temperature difference,
Stationary Pure Vapor
For filmwise condensation of a
"stationary" saturated vapor, Nusselt (1916) presented the first
analytical solution for heat transfer on a plane surface (Fig.9.1
) with the following assumptions (Collier, 1981):
1. the flow of
condensate in the film is laminar,
2. the fluid
properties are constant,
3. subcooling of
the condensate may be neglected,
4. momentum
convective changes through the film are negligible,
5. the vapor is
stationary and exerts no drag on the downward motion of the condensate
6. heat transfer
through the film is by conduction only.
The mean
value of the heat transfer coefficient over the whole surface was given by
One
should note here that for a tube, L is replaced by the tube diameter, D,
and 0.943 becomes 0.725. This model had been extended to include the effects of
Nusselt's assumptions. In particular, Bromley (1952) considered the effects of
subcooling within the liquid film and Rohsenow (1956) also allowed for a
non-linear distribution of temperature through the film due to energy
convection. The results indicated that the latent heat of vaporization,
However,
it should be noted that in most engineering applications, the value of
Sparrow
and Gregg (1959) removed assumption (4) and included inertia forces and a
convection term within the condensate film by using a boundary layer treatment
for the condensate film. For common fluids with Prandtl numbers around and
greater than unity, inertia effects are negligible for values of
As a
conclusion for pure steam-water condensation (Pr ;SPMgt;
1), Nusselt's assumptions can be accepted for a stationary vapor without
noncondensable gas in practical engineering situations.
Moving
Pure Vapor
The
effects of vapor velocity and its associated drag on the condensate film have
been found to be significant in many practical problems. For the case of vapor
flow parallel to a horizontal flat plate, Cess (1960) presented uniform
property boundary layer solutions, obtained by means of similarity
transformations by neglecting the inertia and energy convection effects within
the condensate film and assuming that the interfacial velocity was negligible
in comparison with the free stream vapor velocity. Shekriladze and Gomelauri
(1966) simplified the problem and also considered the case of an isothermal
vertical plate with similar assumptions (1973). Mayhew et al. (1966, 1987)
attempted to expand Nusselt's simple approach to take account of vapor friction
as well as momentum drag. South and Denny (1972) proposed an interpolation
formula for the interfacial shear stress in a simplified manner as Mayhew.
However, such an interpolation formula only led to a small difference in the
heat transfer rate.
Jacobs
(1966) used an integral method to solve the boundary layer by matching the mass
flux, shear stress, temperature and velocity at the interface. The inertia and
convection terms in the boundary layer equations of the liquid film were
neglected. The variation of the physical properties and the thermal resistance
at the vapor-liquid interface were also neglected. Since Jacobs used an
incorrect boundary condition for the vapor boundary layer, Fujii and Uehara
(1972) solved the same problem with the correct boundary condition. In addition,
the velocity profile in the vapor layer was taken as a quadratic. They
presented the numerical results and their approximate expressions for the cases
of free convection, forced convection, and mixed convection. The results show
good agreement with numerical calculations and with Cess' approximate solution
(1960).
The
current recommendation in this area is the model developed by this latter work
as a best estimate. One should be cautious as the Nusselt number increases
because this implies a higher vapor and film flow with accompanying film
turbulence, not accounted for in these models. For design purposes the
recommendation is to use Nusselt laminar film model (Equ. 9.9), since it will
predict a slightly lower heat transfer coefficient, with a thicker condensate
film.
Stationary Vapor with a Noncondensable Gas
A noncondensable gas can exist in a condensing
environment and leads to a significant reduction in heat transfer during
condensation. A gas-vapor boundary layer (e.g., air-steam) forms next to the
condensate layer and the partial pressures of gas and vapor vary through the
boundary layer as shown in Fig. 9.2. The buildup of noncondensable gas near the condensate
film inhibits the diffusion of the vapor from the bulk mixture to the liquid
film and reduces the rate of mass and energy transfer. Therefore, it is
necessary to solve simultaneously the conservation equations of mass, momentum
and energy for both the condensate film and the vapor-gas boundary layer
together with the conservation of specified for the vapor-gas layer. At the interface, a
continuity condition of mass, momentum and energy has to be satisfied.
For a stagnant vapor-gas mixture, Sparrow and Eckert (1961) and Sparrow and
Lin (1964) solved the mass, momentum and energy equations for laminar film
condensation on an isothermal vertical plate by using a similarity
transformation. Sparrow and Eckert (1961) considered the notion of the
vapor-gas mixture from the downward motion of the condensate film, whereas
Sparrow and Lin (1964) included free convection arising from density
differences associated with composition differences. These analyses indicated
that the condensing rate is dependent on the bulk gas mass fraction, the
vapor-gas mixture Schmidt number,
To reduce the computation time, Rose (1969) presented an approximate
integral boundary layer solution assuming uniform properties except for density
in the buoyancy term. Plausible velocity and concentration profiles for the
vapor-gas boundary layer were used and it was assumed that these two layers had
equal thickness. The results showed quite good agreement with those of
Minkowycz and Sparrow and this is recommended for use.
Moving
Vapor with a Noncondensable Gas
For a
laminar vapor-gas mixture case, Sparrow, et al. (1967) solved the conservation
equations for the liquid film and the vapor-gas boundary layer neglecting
inertia and convection in the liquid layer and assuming the stream-wise
velocity component at the interface to be zero in the computation of the
velocity field in the vapor-gas boundary layer. Also a reference temperature
was used for the evaluation of properties. The results showed that the effect of noncondensable gas for
the moving vapor-gas mixture case is much less than for the corresponding
stationary vapor-gas mixture. A moving vapor-gas mixture is
considered to have a "sweeping" effect, thereby resulting in a lower
gas concentration at the interface (compared to the corresponding stationary
vapor-gas mixture case). Also, the ratio of the heat flux with a noncondensable gas to
that without a noncondensable gas was calculated to be independent on the bulk
velocity. The computed results reveal that interfacial resistance
has a negligible effect on the heat transfer and that superheating has much
less of an effect than in the corresponding free convection case.
Koh
(1962) and Fujii et al. (1977) solved this problem without the simplifying
assumptions used by Rose (1969) except for uniform properties and showed good
agreement with the approximate analysis. Instead of solving a complete set of
the conservation equations, Rose (1980) used the experimental heat transfer
result for flow over the flat plate with suction (1979). Denny et al. (1971,
1972) also considered the case of downward vapor-gas mixture flow parallel to a
vertical flat plate. They presented a numerical solution of similar mass,
momentum and energy equations for a vapor-gas mixture by means of a forward
marching technique. Interfacial boundary conditions at each step were extracted
from a locally valid Nusselt type analysis of the condensate film. Local
variable properties in the condensate film were evaluated by means of the
reference temperature concept, while those in the vapor-gas layer were treated
exactly. Asano et al. (1978) treated the condensate film as in the Nusselt
analysis but assumed the interfacial shear stress was the same as that for
single-phase flow over an impermeable plate.
The
analytical model described above was solved using only a laminar vapor-gas (or
pure vapor) boundary layer except for Mayhew (1966). Whitley (1976) proposed a
simple model, which uses the analogy between heat and mass transfer for forced
convection condensation of a turbulent mixture boundary layer by neglecting the
interfacial velocity and treating the surface of the condensate film to be
smooth. Kim (1990) improved on Whitley's approach for forced convection and
natural convection applications by extending it to a wavy/turbulent film. By
using well accepted correlations for a flat plate
geometry, the solution procedure is simplified to computing the condensate film
thickness and the local Reynolds and Sherwood numbers in the downstream
direction. This leads to a computationally efficient solution, which can be
easily expanded to include more detailed models of the condensate film. Total
heat flow is controlled by the gas phase heat transfer and the heat flow
through the condensate film. Therefore, the total condensation heat transfer
coefficient can be written as:
Gas
phase heat transfer consists of convection heat transfer and the latent heat
released as a result of mass transfer. Radiation heat transfer can be neglected
in the temperature range of interest (30-
where
The
analogy between momentum-heat-mass transfer is used in
order to find the heat and mass transfer coefficients along the plate. The
friction factor and
For a
smooth surface, the local skin friction factor can be correlated with the local
Reynolds number, and a result like Whitley is obtained
By
substituting equation 15 into equation 14, the local Nusselt number is
obtained,
Utilizing
Reynold's analogy between heat and mass transfer,
equation 16 is modified to obtain the local Sherwood number:
The
turbulent Prandtl and Schmidt numbers in these equations can be replaced with
the equations derived by Jischa and Rieke (Kim, 1990). They derived the
following results from transport equations of turbulent kinetic energy, heat
flux and mass flux, as
where coefficients
Heat and
mass transfer coefficients can now be solved from 20 and 21, respectively. The
condensation heat transfer coefficient hcond can then be obtained by
substituting the following definition into equation 13,
where G is
the mass transfer coefficient and W is the mass fraction.
The
boundary layer thickness is reduced due to the apparent suction effect of the
condensation process. This leads to larger temperature and concentration
gradients close to the interface that, consequently, increase heat and mass
transfer rates. The following correction factors were implemented to account
for the suction effect,
where
where
where
The
droplets or waves that form on the condensate interface can increase the shear
stress and lead to enhanced turbulent mixing at the interface. The effective
surface area of the interface is also increased due to droplets and waves. The
case where waves and droplets are present was modelled as a rough surface (Kim,
1990). Kim integrated the non-dimensional temperature profile and expressed the
resulting
where the
roughness parameter
where
The
Nusselt number can be solved from equation 20,
The Sherwood
number can be obtained utilizing the Reynold's analogy and equations 31 and 32,
In the
aforementioned equations, the effect of the condensate interface structure is
included in the surface roughness parameter
The
current recommendation in this area would be to use this simple engineering
correlation of Kim as an estimate for most situations. If more exact estimates
are necessary then other more detailed fluid mechanics analyses could be used
for the bulk gas flow.
9.3.
Experimental Investigations
Stationary Pure Vapor
A number of
earlier experimental results (before 1950) show some difference with the
predictions of the Nusselt theory (McAdams, 1954). The differences can be
attributed to one or more of the following reasons: 1) significant
forced-convection effects; 2) presence of noncondensable gas; 3) waviness and
turbulence within the condensate film; 4) presence of dropwise condensation.
More recently,
Mills and Seban (1967) condensed steam on a copper vertical flat plate and
Slegers and Seban (1969) conducted some experiments with n-butyl alcohol. These
tests support the Nusselt theory for pure stationary vapor condensation.
Moving Pure Vapor
Mayhew and
Aggarwal (1973) experimented with pure steam condensing on a flat surface. To
avoid air in-leakage, the experiments were carried out at pressures slightly
above atmospheric. Good agreement is obtained between the experimental results
and the calculated values by their own theory. It is very interesting to note
that the results for the counter-current flow cases are always appreciably
higher than those predicted by the author's own model and indeed always higher
than the corresponding co-current velocity vapor values. The reason was
investigated and explained as follows in the original paper;
An obvious explanation was provided by dye-injection tests which showed
that, with counterflow, no laminar film flow could be achieved. The film was
torn off the plate (i.e. flooding occurred at quite moderate values of vapor
velocity. Similar observations with parallel flow confirmed that the film was
always both laminar and smooth. From work with noncondensing films it was
expected that rippled flow would be encountered over part of the surface at the
higher velocities used. In fact remarkable smooth films were observed
suggesting that mass transfer, and possibly also surface tension effects on the
non-isothermal film, must have had a stabilizing effect.
More recently
Asano et al. (1978) reported their data for the condensation of pure saturated
vapors on a vertical flat copper plate and showed good agreement with the
authors' own model.
Stationary Vapor with a Noncondensable
Gas
Perhaps the
earliest definitive experiment of the effect noncondensable gas was done by
Othmer (1929), who introduced air mole fractions of up to 11 The experimental
heat transfer coefficient data of Hampson (1951) and Akers et al. (1960) were
20 Al-Diwany and Rose (1973) reported heat transfer measurements for steam
condensing in the presence at air, argon, neon and helium. The vapor-gas
mixture was passed into the steam chamber via flow straighteners which provided
uniform flow of the mixture towards the condensing surface so as to preclude
forced convection effects. The experimental data for steam-air, steam-argon and
steam-neon showed satisfactory agreement with the predicted theoretical values
of Sparrow but for steam-helium
Recently,
DeVuono and Christensen (1984) reported their experiment of natural convection
of a steam-air mixture at pressures above atmospheric to 0.7 MPa to investigate
the effect of pressure. The experiments were performed on a horizontal copper
tube with 7.94 cm O.D. by 1.22 m of active condensation length. The tube was
mounted in a cylindrical pressure vessel 1.52 m O.D. by 3.35 m long. Saturated
steam was supplied by an external source and allowed to diffuse to the tube
resulting in steady-state, natural convection conditions. An expression, which
is a function of
where
0.0 < Y
< 14.0
Even though
this experiment was done over a large range of pressure for a containment
analysis and showed a significant effect of pressure, the pipe geometry and
length scale make it questionable to apply this correlation to a large scale
system. Unfortunately, the experiment results were not compared with any other
theoretical model.
Moving Vapor with a Noncondensable Gas
Rauscher,
Mills and Denny (1974) performed experiments of filmwise condensation from
steam-air mixtures undergoing forced flow over 0.74 in. O.D. horizontal
tube. The heat transfer coefficient at the stagnation point was reported
for bulk air mass fractions 0 - 7 .
The previous experiments were
separate effect tests in which model development was an integral part of the
research. There have been other data collected in which direct empirical
correlations have resulted or in which analysis is not completed. These
relevant experiments are summarized in Table 9.2.
We consider both separate effects data and large scale tests.
Separate
Effects Experiments
Typically, the heat transfer
coefficients have been observed to decrease significantly with increased
noncondensable gas mass fraction under various conditions and test geometries. The degradation of
heat transfer is caused by the accumulation of a noncondensable gas layer near
the cold wall through which the vapor must diffuse.
Buoyancy Forces in a Stagnant
Gas Mixture
Cho and Stein (1988) investigated
condensation of steam in the presence of air and helium on a small horizontal
plate facing down with stagnant flow conditions. The results with air were
successfully modelled by taking into account the buoyancy forces caused by different
molecular weights of participating gases. Since helium is a lighter gas than
steam, a suppression of natural convection was expected. However, the tests
with a moderate helium content showed higher heat
transfer rates than predicted by a diffusion analysis. A convective heat
transfer mechanism caused by fog and mist formation was hypothesized. Fog that
formed near the cold surface was observed to form localized swirls and
generally move in downward direction. These visual observations seemed to confirm
the presence of hypothesized natural circulation. A similar geometrical
arrangement was also used by Kroger and Rohsenow (1968). Potassium vapor was
condensed in the presence of argon and helium. The diffusion theory
successfully predicted the experimental data with helium. In the case of argon,
experimental results indicated a superimposed natural circulation flow. Vapor
phase instabilities and secondary flow cells were also reported by Spencer,
Chang and Moy (1970). They investigated the condensation of Freon-113 in the
presence of helium, nitrogen and carbon-dioxide on a vertical surface under
stagnant conditions. Both visual observations and heat transfer measurements
were performed. The
results, indicate a modest effect of noncondensable
gas molecular weight. Dehbi (1991) studied the influence of an
air/helium noncondensable mixture on the condensation heat transfer under
stagnant conditions. The condensing surface consisted of a 3500 mm long
vertical copper tube. Helium mass fraction was varied from 1.7 to 8.3 weight
percent. Dehbi reported that the heat transfer rates decreased with a increased helium mass fraction. When the helium mass
fractions were relatively high, sharp stratification patterns were observed as
helium migrated to the top of the test vessel and air/stream mixture stayed at
the bottom. The natural convection patterns in all these data suggest that
scale dependence must be strongly considered.
Forced Flow
As mentioned previously Dallmeyer
(1970) studied condensation of
Barry (1987) performed
condensation experiments with the mixture of steam and air. His apparatus
consisted of a horizontal plate facing upwards. The velocity and mass ratio
range was chosen so that it covered the conditions that are likely to exist in a containment during an accident in a developing parallel
flow situation. Barry's results show expected the effects of velocity and the
mass ratio as mentioned previously. Kutsuna, Inoue and Nakanishi (1987) studied
condensation of steam on a horizontal plate (facing up) in the presence of air. They also reported
increased heat transfer rates due to forced convection. Their results, indicate the expected effects of noncondensable gas
concentration and velocity on the heat transfer coefficients. Tests
were performed with higher steam content than the tests by Barry, and
consequently, the heat transfer coefficient were also significantly higher.
When the tests are performed with a high steam content,
the heat transfer results become very sensitive to the air content. This may be
the reason why the data scatter is markedly higher than in Barry's experiment.
Unfortunately, the experimental uncertainties were not discussed.
Pressure
Several workers have investigated
the effect of the system pressure with stagnant flow conditions (no forced
convection). The heat transfer rates are reported to increase with system
pressure (e.g., Gerstmann, 1964), because the densities of gas components
increase with pressure. Cho and Stein (1988) reported that an increase in the
system pressure (0.31 MPa to 1.24 MPa) also influenced the mode of condensation
on a downward facing surface with helium as the noncondensable gas. Higher
pressures led to mixed mode of condensation (filmwise and dropwise condensation
coexisting) with a downward facing polished surface.
Nuclear reactor safety
evaluations have prompted studies of the effect of pressure on condensation
heat transfer under transient conditions (large concentrations of
noncondensable gases). Robinson and Windebank (1988) studied the effect of
pressure in the range of 0.27-0.62 MPa with an air/stream mixture. The
noncondensable gas mass fraction was varied from 24 to 88 percent. The heat
transfer rates were measured with a cooled disk that was placed inside a
pressure vessel. The results show that heat transfer rates increase with pressure and
decrease with the mass ratio of noncondensable gas. Robinson and
Windebank noted that the velocity field due to the steam injection might have
had an effect on their results. The magnitude of the induced velocities within
the vessel were stated to be below 2
Similar tests were also conducted
by Dehbi. Heat transfer rates were measured at three different pressures (0.15,
0.275 and 0.45 MPa). The noncondensible gas mass fraction in the tests ranged
from 25 to 90 percent. The experimental apparatus consisted of a three meter
and one half long cooled tube (0.038 m Dia) in a pressure vessel. The
motivation behind using a relatively large vertical dimension was to simulate
the length scale of internal containment structures. Surprisingly narrow
pressure vessel was used (L/D = 10). This led to difficulties to establish
homogeneous test conditions in the vessel. In the tests, the mass ratio of air
was 8-33 percent greater in the upper part of the vessel than in the lower
part. Secondly, the flow field created by the natural convection may have been
affected by the sidewalls. Therefore, the results by Dehbi have some
unspecified uncertainty. He confirmed the observations of Robinson that heat
transfer rate increases with system pressure.
Condensate Film Structure
Several studies have been done to
address the effect of condensate film characteristics on the heat transfer
rates. The condensate film characteristics depend on its flow field and the
nature of the condensing surface, e.g. roughness, wetting and orientation.
Forced flow induces interfacial
instabilities that increase the heat transfer rates by reducing the thickness
of gas phase laminar sublayer and enhancing the mixing of both the liquid
(condensate film) and gas phase. Barry (1987) studied the effects of
interfacial structure caused by shear. Since the condensation length was
relatively short, a film injection system was used to produce a condensate film
that was sufficiently thick for measurements. The qualitative results suggested
that enhanced mixing, which is caused by the interfacial film structure,
somewhat compensated for the effect of the noncondensable gas.
The surface finish has a major
effect on the mode of condensation for a downward facing surface and it is the
wetting characteristics of the surface that ultimately determine this. Dropwise
condensation is likely to exist on non-wetting surfaces and filmwise
condensation is likely on wetting surfaces. In dropwise condensation
mode with polished metal surfaces, the heat transfer characteristics are likely
to change due to oxidation of the surface or tarnishing. Thus, one cannot
precisely know the wetting characteristics as surface aging occurs. Gerstmann
and Griffith (1967) studied the condensation of pure, stagnant Freon-113 and
water vapor at atmospheric pressure. Heat transfer measurements and visual
observations of the interfacial behavior were made. Gerstmann and Griffith
observed several distinct flow regimes in the condensate film depending on the
angle of inclination. Unstable condensate film with pendant drops and
lengthwise ridges existed at the horizontal position. The characteristic length
scale of these formations was on the order of the
Integral and
Large Scale Experiments
In addition to the heat transfer
data, integral and large experiments provide valuable background information
about physical conditions such as gas concentrations, temperatures, prevailing
flow fields and system pressures in a particular circumstance. The data from
the experiments can be used as integral benchmarks for models. In the subject
area of condensation these data usually involve the study of condensation in
large containment structures for nuclear reactor safety.
The first integral experiments
were performed by Jubb and Kolflat (1960). The results of these integral tests
were correlated with the experimental parameters. However, some of the
parameters that Jubb and Kolflat used were uniquely dependent on the
experimental apparatus. Therefore, the resulting correlations were not
generally applicable to anything other geometry.
Uchida et al. (1965) and Tagami
(1965) performed experiments using a model of a containment structure (3.4
meters dia and 6.4 meters in height). Three water cooled cylinders inside the
containment structure were used as test surfaces. Uchida measured the post-blowdown
heat transfer rates using a vertical surface with subcooling of
Uchida:
Tagami:
where W is defined as the mass fraction of noncondensable gas. The geometrical aspects and the effect of
velocity field were ignored. Therefore, caution should be used to extrapolate
results from the correlations for the long sections of structural walls.
Unfortunately, it is quoted and used in safety analyses.
The CVTR test series was
conducted using a full scale structure of a decommissioned nuclear power plant
(Schmitt, 1970). The steam was injected through a diffuser (0.25 meters dia and
3 meters in height) located three meters above the operating floor. Three tests
were conducted. The heat transfer rates into the wall were computed from the
measured temperature profiles in the wall using the inverse conduction method.
The velocity field was measured by ultrasonic anemometers. The experimental
data from the CVTR test were used by Kim (1990) as one data set to benchmark
his condensation model. The major conclusion from the analysis of these tests
was that local gas velocities were needed to accurately predict the data.
Historically, integral tests have
been used to find simple correlations that would predict the heat transfer
rates. These correlations have gained wide acceptance and are regularly used in
safety analyses. In this light, it is surprising to find out that until
recently, most of the data from these integral tests have been based on a very
limited number of measurements of the prevailing conditions. Therefore, these
correlations generally have a very limited value in making accurate predictions
of heat transfer rates through different geometries.
Condensation phenomena can be
classified by the presence of noncondensable gas, the gas mixture velocity, the
flow characterization (laminar or turbulent) of the gas mixture and the
condensate film and the interface condition as shown in Table 9.1,
which presented the summary of the theoretical and experimental investigations
discussed.
For all cases with a simple
geometry except the turbulent gas mixture boundary layer and the wavy interface
of both the pure vapor and the vapor-air mixture case, it is seen that
numerical solutions of the conservation equations and more approximate
analytical solutions of the conservation equations agree well with the
corresponding experimental work.
As the geometry of the condensing
surface becomes more complex more prototypic experiments must be performed.
Examples of these cases are provided in the previous section for integral
containment tests. The presence of a turbulent gas mixture (natural or forced
convection) or a wavy/turbulent film interface complicates the analysis even
for simple geometries. Examples of separate effect tests and correlations under
a variety of conditions were also presented in the previous section. In these
situations theoretical analysis of this turbulent condition is still needed as
is consideration of the effect of geometric scale. This may require
multi-dimensional, multi-fluid modelling of the condensation process both near
the wall and gas boundary layers as well as in the bulk gas mixture. If this
approach is taken then one must address the appropriate scaling of these
calculations to produce scaling of these calculations to produce useful
condensation heat transfer design correlations or procedures.
References
2)
Minkowycz, W.J., and Sparrow, E.M., “Condensation
Heat Transfer in the Presence of Noncondensables,
Interfacial resistance, Superheating, Variable Properties, and Diffusion”,
Internal J. of Heat Mass Transfer, Vol. 9, 1966, pp1125-1144
Interfacial
Resistance:
The
interfacial resistance results from the fact that the net condensation of vapor
at the interface is actually the difference between the simultaneous processes
of evaporation and condensation. The kinetic theory of gases shows that an
unbalance between two processes must be accompanied by a temperature jump at
the interface, whence the additional resistance.
Reductions of
more than 50% in heat transfer are observed for bulk mass fraction as small as
0.5%.
3) Sadhal, S.S., and Martin, W.W., “Heat Transfer Through Drop Condensate Using Differential Inequalities,”
Int. J. Heat Mass Transfer, Vol.20, 1977, pp1401-1407
Graham and Griffith(1973) showed that most of the heat is transferred
through droplets of diameter less than 100 micrometer. For such small droplets
the influence of gravity on the droplet shape is negligible and a spherical-segment geometry may be assumed.
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