International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

www.elsevier.com/locate/ijhmt

Condensation in horizontal tubes, part 2: new heat

transfer model based on flow regimes
a,*
J.R. Thome , J. El Hajal a, A. Cavallini b

a
Laboratory of Heat and Mass Transfer, Faculty of Engineering Science, Swiss Federal Institute of Technology,
Lausanne, CH-1015 Lausanne, Switzerland
b
Dipartimento di Fisca Tecnica, University of Padova, Padova, I-35131, Italy
Received 24 June 2002; received in revised form 4 March 2003

Abstract
A new general flow pattern/flow structure based heat transfer model for condensation inside horizontal, plain tubes
is proposed based on simplified flow structures of the flow regimes, and also includes the effect of liquid–vapor in-
terfacial roughness on heat transfer. The model predicts local condensation heat transfer coefficients for the following
flow regimes: annular, intermittent, stratified-wavy, fully stratified and mist flow. The new model has been compared to
test data for 15 fluids (R-11, R-12, R-22, R-32, R-113, R-125, R-134a, R-236ea, a R-32/R-125 near-azeotrope, R-404A,
R-410A, propane, n-butane, iso-butane and propylene) obtained in nine independent research laboratories. The new
model has been tested over the following range of conditions: mass velocities from 24 to 1022 kg/(m2 s), vapor qualities
from 0.03 to 0.97, reduced pressures from 0.02 to 0.80 and tube internal diameters from 3.1 to 21.4 mm. Overall, the
model predicts 85% of the heat transfer coefficients in the non-hydrocarbon database (1850 points) to within
20% with
nearly uniform accuracy for each flow regime and predicts 75% of the entire database to within
20% when including
the hydrocarbons (2771 points), the latter all from a single laboratory whose data had some unusual experimental
trends over part of their test range.
Ó 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Intube condensation; Heat transfer model; Refrigerants; Hydrocarbons

1. Introduction between the stratified flows and non-stratified flows, and

so forth. In Part I of this two-part paper, a new flow
In the second half of the past century, numerous pattern map and void fraction equation were proposed
models have been proposed in the literature to predict for condensation. In this part, that map and void frac-
local heat transfer coefficients for condensation of pure tion equation are used to proposed a new flow patterned
saturated vapors under forced convection conditions based condensation heat transfer model for condensa-
inside plain horizontal tubes. These methods have only tion inside horizontal, plain tubes. The objective is to
been partially successful because of the limited databases obtain a method with a minimum of empirical constants
used, use of poor or overly simplistic transition criteria and exponents that not only gives a good statistical
representation of the data, but also correctly captures
the trends in the data. This latter point is particularly
important when such a method is used in an optimiza-
tion program for the design of air-cooled condensers.
*
Corresponding author. Tel.: +41-21-693-5981; fax: +41-21- The basic idea here is to adapt the principles used in the
693-5960. flow pattern-based intube flow boiling heat transfer
E-mail addresses: john.thome@epfl.ch (J.R. Thome), jean. model of Kattan et al. [1–3] for predicting intube con-
elhajal@epfl.ch (J. El Hajal), alcav@unipd.it (A. Cavallini). densation heat transfer.
0017-9310/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0017-9310(03)00140-6
3366 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Nomenclature

A total cross-sectional area of tube (m2 ) uV mean vapor velocity (m/s)

AL cross-sectional area of tube occupied by x vapor quality (–)
liquid (m2 ) xIA vapor quality at transition from intermittent
AV cross-sectional area of tube occupied by to annular flow (–)
vapor (m2 ) Dx change in vapor quality from inlet to outlet
c convective film constant (–) (–)
cpL liquid specific heat (J/(kg K)) Xtt Martinelli parameter with both phases tur-
d internal tube diameter (m) bulent [–]
fi interfacial roughness factor (–)
Greek symbols
g acceleration of gravity (m/s2 )
ac convective condensation heat transfer co-
G total mass velocity of liquid and vapor (kg/
efficient (W/(m2 K))
(m2 s))
acb convective boiling heat transfer coefficient
Gbubbly bubbly flow transition mass velocity (kg/
(W/(m2 K))
(m2 s))
af Nusselt film condensing coefficient on top
Gmist mist flow transition mass velocity (kg/(m2 s))
perimeter of tube (W/(m2 K))
Gstrat stratified flow transition mass velocity (kg/
atp local perimeter averaged condensing heat
(m2 s))
transfer coefficient (W/(m2 K))
Gwavy wavy flow transition mass velocity (kg/
d liquid film thickness of annular ring (m)
(m2 s))
din initial liquid film thickness after desuper-
hLG latent heat of vaporization (J/kg)
heating zone (m)
j exponent (j ¼ 1=2) (–)
Ddi interfacial roughness (m)
k exponent (k ¼ 1=4) (–)
e void fraction of vapor (–)
m exponent on PrL (–)
kL liquid thermal conductivity (W/(m K))
n exponent on ReL (–)
kT Taylor instability wavelength (m)
pcrit critical pressure (N/m2 )
lL liquid dynamic viscosity (N s/m2 )
pr reduced pressure (pr ¼ psat =pcrit ) (–)
h upper angle of the tube not wetted by
psat saturation pressure (N/m2 )
stratified liquid (rad)
PrL liquid Prandtl number (–)
hstrat stratified angle around upper perimeter of
q heat flux from fluid to tube (W/m2 )
the tube (rad)
r internal radius of tube (m)
qG vapor density (kg/m3 )
ReL Reynolds number of liquid film (–)
qL liquid density (kg/m3 )
Tsat saturation temperature (K)
r surface tension (N/m)
Tw wall temperature (K)
si interfacial shear of vapor on liquid film
DTwater fall in water temperature from inlet to outlet
(J/m3 )
(K)
uL mean liquid velocity in film (m/s)

2. Literature review of condensation models used for the calculation of heat transfer coefficients: an
interfacial shear model or a two-phase multiplier cor-
In the early models, the flow patterns were classified relation. The latter is the most common approach
under only two categories, as either stratified (or strati- consisting in calculating the Nusselt number during
fying, or wavy) flow or as annular flow. In the first case, condensation by multiplying the Nusselt number for
the gravity dominated flow has been modelled consid- turbulent single-phase flow by a suitable two-phase
ering a thick condensate layer flowing along the bottom multiplier. The two-phase multiplier is usually given as a
of the tube, while a thin liquid film forms on the wall in function of some of the following parameters: vapor
the upper portion of the tube. Heat transfer through quality, viscosity and density ratios between the liquid
the thin film is treated by a classical Nusselt type ana- and vapor phases, reduced pressure, liquid Froude
lysis, while heat transfer through the thick condensate number, Martinelli Xtt parameter, etc. For the single-
layer can either be neglected as in Jaster and Kosky phase Nusselt number upon which these multipliers
[4], or treated as a convective process. In the shear act, the equation by Dittus and Boelter [5] is often the
dominated annular flow, two different approaches were basis.
 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3367

Worth mentioning, although very old, are the equa- mental data, that is generally within
20%. This applies
tions by Akers et al. [6] and Akers and Rosson [7], still in particular, with somewhat different approximations,
recommended as a design tool in the ASHRAE Hand- to the methods by Shah, Cavallini-Zecchin, Dobson-
book [8]. Three straightforward equations are given, Chato, Tang, Traviss et al. and Haraguchi et al. Never-
each of them applicable within definite ranges of suitable theless, this did not hold true when the same methods
dimensionless parameters; they cover both annular and were applied to condensation of some of the new gen-
stratified flow conditions. The one for annular flow is of eration HFC refrigerants, those that are commonly used
the two-phase multiplier type. Cavallini and Zecchin [9] at higher pressure (also higher reduced pressure) than
proposed a simple dimensionless semi-empirical equa- the traditional refrigerants. In this case the available
tion to be applied when annular flow is present during predicting methods, developed in the past in connection
condensation. The same authors showed later [10] that with availability of experimental data only concerning
their equation represented, within narrow limits, the traditional refrigerants, either cannot be applied because
results of a flow-dynamic analysis of the condensation of the cited limits in their application ranges, or unac-
phenomenon. Similar analyses, based on the assumption ceptably overpredict the experimental data, typically by
that the Von Karman velocity profile for pipe flow holds 20–40%. To overcome this situation, Cavallini et al. [20]
true in the condensate annulus, had previously been presented a new heat transfer flow pattern-based
used by Kosky and Staub [11] and Traviss et al. [12], to method, that was able to give satisfactory predictions
develop their own calculation procedures, valid of also with the new generation high-pressure refrigerants
course only with an annular flow pattern. The method while their flow pattern map was discussed in Part I of
by Kosky and Staub is for instance recommended by this two-part paper. In the annular flow regime, the
Butterworth [13] in the Heat Transfer Design Handbook. method employs an interfacial shear model similar to
Analyses based on a similar type of approach are also that developed by Kosky and Staub [11], with a modified
suggested for design purposes in Germany in their VDI Friedel [21] correlation for the calculation of the fric-
W€ armeatlas, i.e. VDI Heat Atlas [14] translated in tional pressure drop and related wall shear stress. For
English, and still in the most recent version in German, their comparison, Cavallini et al. [20] used an extensive
VDI W€armeatlas [15]. A two-phase multiplier correla- experimental data bank formed from quite a few data
tion extensively used in North America is the one by sets taken in independent research laboratories. Details
Shah [16], which should coherently be applied only in of this data bank are reported in Table 1, where they
the presence of an annular flow pattern, even if the au- represent the first nine listings. While their new model
thor did not establish this limitation, presenting his ex- was an improvement upon past methods, it also in-
pression as a ‘‘generalised’’ one. cluded a large number of empirical constants and also
Tang [17], based on his condensation measurements, predicts an unlikely jump in the heat transfer coefficient
also developed a new heat transfer two-phase multiplier across one flow transition boundary. Hence, the objec-
correlation valid in the annular flow regime, when mass tive here is to present a new condensation heat transfer
velocities are larger than 300 kg/(m2 s) and the reduced model that avoids these pitfalls and also will predict
pressure is between 0.2 and 0.53. The computational hydrocarbon data.
method suggested by Haraguchi et al. [18] is extensively
used in Japan. It consists of two dimensionless equa-
tions, one each for both annular and stratified flow 3. Heat transfer database
conditions, combined together in an asymptotic way.
Appropriate applicability ranges are specified. Dobson The database of condensation heat transfer coeffi-
and Chato [19] presented a set of equations able to cients available for the current study is described in
predict the heat transfer coefficient both in the stratified Table 1. These studies are contributions from nine in-
flow regime and in the annular one; the annular flow dependent research laboratories and cover 15 fluids and
correlation was derived using the two-phase multiplier a wide range of test conditions. Only studies from the
approach. In some circumstances quite discontinuous 1990s to the present have been considered as they are
results are calculated when passing from one flow regime more accurate than the older databases, i.e. newer
to the other, contrary to experimental evidence. studies use Coriolis mass flow meters for both the
Recently, Cavallini et al. [20] compared most of the coolant and the refrigerant, more accurate data acqui-
above prediction methods to heat transfer coefficients of sition systems and so on. The newest study, that of
halogenated refrigerants condensing inside horizontal Liebenberg [29], only includes his R-22 data available at
plain tubes, for experimental data obtained by inde- this time.
pendent research workers. The outcome of this com- This list of test fluids includes single component
parison showed that a few of the above methods, when refrigerants (R-11, R-12, R-22, R-32, R-113, R-125,
employed with condensation of the old generation re- R-134a, R-236ea), binary azeotropic or very near azeo-
frigerants, were able to predict satisfactorily the experi- tropic refrigerant mixtures (60% R-32/40% R-125,
3368 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Table 1
Condensation heat transfer data bank (pure fluids and quasi-azeotropic mixtures)
Author(s) Data points Refrigerants d (mm) Tsat (°C) Tsat  Tw (°C) G (kg/(m2 s))
Cavallini et al. [22,23] 425 R-22, R-134a, R-410A, 8.0 27–60 2.4–15.4 63–773
R-125, R-32, R-236ea
Dobson–Chato [19] 644 R-22, R-134a, R-410A, 3.1–7.0 33.5–46.4 1.1–8.8 24–812
R-32/R-125 (60/40% by mass)
Zhang [24] 77 R-22, R-134a, R-404A 3.3–6.2 23.1–65.2 1.0–5.6 245–1022
Tang [17] 218 of 231 R-22, R-134a, R-410A 8.8 35.2–40.8 – 258–817
Chitti–Anand [25] 41 of 48 R-410A 8.0 24–36 – 161–491
Kim et al. [26] 225 R-22 8.0 48–49 2.7–9.6 229–343
Kim et al. [26] 921 Propane, n-Butane, iso-Butane, 8.0 46.6–50.2 1.2–9.8 44–204
Propylene
Wijaya–Spatz [27] 14 R-410A 7.8 46–52 – 481
Fujii [28] 158 R-11, R-12, R-113 16.0–21.4 28.4–50.1 2.2–32.7 33–577
Liebenberg [29] 48 of 50 R-22 8.11 38.4–43.2 – 304–832
Data marked with  refers to data which could not be used because they fell in a flow regime requiring experimental measurement of
(Tsat  Tw ) which were not reported.

R-404A, R-410A) and pure hydrocarbons (propane, n- e.g. a error of
0.02 at x ¼ 0:03 means that the real
butane, iso-butane and propylene). The database covers vapor quality could be from 0.01 or 0.05, which may
a very broad range of conditions: mass velocities from result in a halving or doubling the value of e, respec-
24 to 1022 kg/(m2 s), vapor qualities from 0.03 to 0.97, tively. At very high x, e and the annular liquid film
reduced pressures from 0.02 to 0.80 and tube inter- thickness d are very sensitive to small changes in the
nal diameters from 3.1 to 21.4 mm. The database in- liquid fraction (1  x); e.g. a change of just
0.01 in e
cludes heat transfer coefficients for the following flow may result in a doubling or halving of the condensate
regimes: stratified, stratified-wavy, annular, intermittent film thickness. Hence, it is particularly difficult to ac-
and mist. curately measure condensation data at vapor qualities
Regarding the experimental condensation heat less than 0.05 and above 0.95.
transfer coefficients themselves, they are ‘‘quasi-local’’ Desuperheating and subcooling. There is a less than
data obtained in short test sections that give a mean heat obvious effect of desuperheating on the test data at high
transfer coefficient for a small (or sometimes even large) vapor qualities, caused by the condensate formed in a
change in vapor quality, Dx, from inlet to outlet. The Dx desuperheater before the test section, i.e. condensate
should optimally be on the order of 0.05 or less but in formed while cooling the vapor to its saturation tem-
some tests was as large as 0.20–0.40. It would be better perature. This condensate enters the condenser test
in the future to utilize an enthalpy profile approach to section and hence the film begins with some initial value
get true local heat transfer coefficients, such as described of din > 0 rather than starting from d ¼ 0 at x ¼ 1 and
in Z€urcher et al. [30] for measuring local flow boiling e ¼ 1. This effect tends to increase the film thickness,
heat transfer coefficients. With respect to condensation which in turn decreases the heat transfer coefficient
heat transfer data, the most difficult test conditions to measured. This ‘‘desuperheating’’ condensate only in-
make accurate measurements are as follows: fluences data where its preexisting fraction of the total
Near flow regime transition zones. If a transition from condensate is significant, i.e. at high x for say
one flow pattern to another takes place within the 1:0 P x P 0:9. This may be the reason that some high
‘‘quasi-local’’ test section, the mean heat transfer coef- vapor quality heat transfer coefficients tend to plateau at
ficient for the section is an unknown average of the two high vapor qualities. Secondly, some local condensation
regimes. Also, consider that the transition from one re- test data may include subcooling since the Dx in the test
gime to the next typically occurs over a mass velocity sections during quasi-local experiments are often from
range of about 50 kg/(m2 s), which is analogous to the 0.05 to 0.3 and thus data for x < 0:05 may be averaged
transition regime in single-phase flow (where heat over a single-phase liquid zone too.
transfer coefficients are more difficult to measure). Stable operating conditions. All experimental test
Very high and low vapor qualities. As x ) 1 and as loops have a limited range in which steady-state test
x ) 0, small errors in energy balances of
2% are conditions can be maintained. At low mass velocities,
magnified. Typical errors in energy balances are on the typically a threshold is reached where fluctuations in
order of
1% to
3%, which represent identical errors pressure and flow rate become significant. In particular,
in x. Referring to Fig. 4 in Part I, at low x the void pressure fluctuations significantly influence Tsat , which is
fraction e decreases very rapidly with small changes in x; used to reduce the data, and hence the measured heat
 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3369

transfer coefficient. Thus, some databases may include

test data that fall below this threshold, giving unreliable
test data at these unstable conditions but good data
above this threshold.
Very large and very small heat transfer coefficients. A
propagation of error analysis will always give large un- Fig. 1. Illustration of wavy interface during evaporation and
certainty in heat transfer coefficients when they are very condensation.
large, i.e. at high x and high mass velocity because
Tsat  Tw becomes small with respect to the measurement
errors of the temperatures. Also, when heat transfer Reynolds number and PrL is the liquid Prandtl number
coefficients are very small, i.e. at low mass velocities, (both defined later). Comparing this expression to the
energy balances are less accurate because the change in annular flow condensing data of Cavallini et al. [22,23],
the cooling water temperature from inlet to outlet, we observed that it systematically over predicted the
DTwater , in the test section is small. data by about 15–20%. To explain this difference, as il-
Circumferential variation in heat transfer coefficients. lustrated in Fig. 1, one can imagine that for an annular
Condensation heat transfer coefficients are reported as film with a wavy interface, evaporation will tend to thin
mean values around the perimeter of the tube. Hence, in the film at its troughs and thus enhance heat transfer
stratified and stratified-wavy flows where two different while condensation will tend to fill up the troughs and
heat transfer mechanisms are involved, a sufficient reduce heat transfer. Hence, convective evaporation is
number of thermocouples in the tube wall are required not equivalent to convective condensation in annular
to capture a representative mean wall temperature to flow as might be otherwise assumed.
determine the mean heat transfer coefficient.
Thus, in summary, experimental condensation heat 4.2. Simplified flow structures for condensation in hori-
transfer data at very high and very low vapor qualities zontal tubes
are the most difficult to measure accurately and reliably.
It will be seen when comparing the new heat transfer The same simplified flow structures assumed for
model to the extensive database that these are the test evaporation inside horizontal tubes by Kattan et al. [3]
conditions that tend to give the biggest prediction errors can also be applied to condensation, where the only
because these data have a higher experimental uncer- difference is that the top to the tube in a stratified flow
tainty. will be wetted by film condensation rather than remain
dry during evaporation. Thus, the new condensation
model proposed here assumes three simplified geome-
4. New condensation model tries for describing annular flow, stratified-wavy flow
and fully stratified-wavy flow as shown in Fig. 2. For
4.1. Convective evaporation vs. convective condensation annular flow (bottom left), for the sake of simplicity a
uniform liquid film thickness of d is assumed and the
As the starting point in developing a new flow pat- actual larger thickness of the film at the bottom than the
tern based heat transfer prediction model for conden- top due to gravity is ignored. Utilizing a void fraction
sation inside plain horizontal tubes, we shall begin with equation, the cross-sectional area of the vapor phase is
analysis of the annular flow heat transfer data. Since determined and then that of the liquid phase. From the
there is a close similarity between the convection total flow of liquid plus vapor and the local vapor
mechanisms in annular film condensation and annular quality, the mass flow rate of the liquid is calculated;
film evaporation inside tubes, the condensation data then, using the liquid density and cross-sectional area
were first compared to the convective heat transfer occupied by the liquid, the mean velocity of the liquid is
correlation for the annular film during evaporation pre- determined for the film. Hence, turbulent flow heat
sented in Kattan et al. [3] transfer to the film can be correlated based on the mean
kL velocity of the liquid film, which is the standard proce-
acb ¼ 0:0133Re0:69
L PrL
0:4
ð1Þ dure for film flows undergoing condensation, evapora-
d
tion and sensible heating.
In this expression, the convective boiling heat transfer Similarly, Fig. 2 also shows the actual geometry of a
coefficient acb is that for the convective evaporation of fully stratified flow (upper left) and its equivalent ge-
the annular liquid film without any nucleate boiling ometry (upper right) with the same angle of stratification
contribution, calculated as a film flow and not as a tu- and cross-sectional area occupied by the liquid, but with
bular flow. For an annular flow, the annular liquid ring the liquid distributed as a truncated annular ring of
is assumed to be of uniform thickness around the entire uniform thickness d. Fig. 3 depicts an instantaneous
internal perimeter of the tube. ReL is the liquid film cross-sectional image of the liquid–vapor interface for
3370 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

between its maximum value of hstrat at the threshold to

fully stratified flow and its minimum value of zero at the
threshold to annular flow. Notably, these three simple
geometries have a smooth geometrical transition from
one flow structure to another and also allow the heat
transfer models to be based on the mean liquid and
mean vapor velocities. In addition, the convective heat
transfer coefficient can be applied to the stratified pe-
rimeter subtended by (2p  h) while Nusselt film con-
densation can be applied to the upper wall subtended
by h with the condensate flowing downwards into the
stratified liquid below. For annular flow, however,
convective condensation heat transfer occurs around
the entire perimeter without any Nusselt film conden-
sation.
For simplicityÕs sake, the annular flow structure is
also assumed to apply to the intermittent flow regime,
which has a very complex flow structure, and also ten-
tatively to the mist flow regime (assuming the impinging
droplets create an equivalent unsteady liquid film).
Fig. 2. Simplified flow structures for two-phase flow patterns.
Thus, while ignoring numerous details of the actual flow
structures of all these five regimes, the present geo-
metries represent their principal features. It is then a
question as to whether or not these simplified repre-
sentations are sufficient to accurately model the con-
densation process in horizontal tubes for these five flow
regimes (S, SW, I, A, MF), which will be shown to be the
case. It is pertinent to note that this new model is based
on film flow, rather than tubular flow, of the liquid. Most
previous convective condensation models assumed tu-
bular flow while these flows are in fact film flows.
Presently, no heat transfer model for the bubbly flow
regime is proposed. Bubbly flows in horizontal tubes
occur at very large mass velocities and hence are not
commonly encountered nor are heat transfer data
available for this regime.
Fig. 3. Cross-sectional image of stratified-wavy flow in a hor-
izontal tube.
4.3. Heat transfer model

an adiabatic stratified-wavy flow in a horizontal tubular Our objective here is to develop a new flow pattern/
sight glass of 13.6 mm internal diameter for ethanol and flow structure based condensation heat transfer model
air obtained using a laser sheet, a video camera and analogous to that which was proposed by Kattan et al.
image analysis by Wojtan et al. [31]. The height of the [1–3] for evaporation inside horizontal tubes. The con-
liquid on the left is a little higher than on the right be- densation model therefore uses the same flow pattern
cause of the asynchronous height of the waves on the map as for evaporation but with the new modifications
walls. Hence, the equivalent geometry assumed here in introduced in Part I. Their flow pattern map for near
Fig. 2 for stratified-wavy flow is reasonably representa- adiabatic and evaporating flows has proved to be very
tive of the real situation. accurate and reliable in comparisons to over 1000 flow
In stratified-wavy flow, the interfacial waves are pattern observations for eight different refrigerants to
small in amplitude and do not reach the top of the tube. date. In addition, the same simplified two-phase flow
Hence the top perimeter of the tube is not wetted by the structures assumed for the flow patterns in the evapo-
stratified liquid but only by the condensate that forms ration model are also assumed for the condensation
on this part of the exposed tube perimeter. Here, once model. It is also a goal here to develop a new heat
again, for simplicity the stratified liquid is assumed to transfer model with as few empirical constants as pos-
form an annular truncated ring as shown in the middle sible. Prediction methods that include a large number of
diagram at the bottom of Fig. 2. Thus, the angle h varies empirical parameters, some of the methods mentioned in
 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3371

the literature review even have correlating equations for

the empirical exponents, tend to be less reliable because
they have been tightly fit statistically to the database in
order to achieve small error bands. Instead, an accurate,
reliable heat transfer prediction method that resorts to
few empirical constants gives a good indication that the
physical model ‘‘captures’’ the trends in the heat transfer
processes.
The new condensation model assumes that two types
of heat transfer mechanisms occur within the tube:
convective condensation and film condensation. In the
present context, convective condensation refers to the
axial flow of the condensate along the channel due to
the imposed pressure gradient while film condensation
refers to the flow of condensate from the top of the tube Fig. 4. Heat transfer model showing convective and falling film
towards the bottom due to gravity. Both of these boundaries.
mechanisms have been included in some previous
models, such as that of Dobson and Chato [19]. Previous
condensation models normally have assumed only two
flow regimes: stratified flow and unstratified flow. In- tube. Hence, for annular flow with h ¼ 0, atp is equal to
stead, it is particularly important to divide the flow into ac . The stratified angle hstrat is calculated from the fol-
specific flow regimes: annular flow, stratified-wavy flow, lowing implicit geometric equation:
fully stratified flow, intermittent flow, mist flow and
d2
bubbly flow. Only the first five are included here as little AL ¼ ½ð2p  hstrat Þ  sinð2p  hstrat Þ
ð3Þ
data are available for the last one while intermittent and 8
mist flows will be treated as an annular flow for sim- where the cross-sectional area occupied by the liquid
plicityÕs sake. phase AL is
The above two heat transfer mechanisms are applied
to their respective heat transfer surface areas as shown in AL ¼ ð1  eÞA ð4Þ
Fig. 4. The convective condensation heat transfer coef-
ficient ac is applied to the perimeter wetted by the axial and the cross-sectional area occupied by the vapor is
flow of liquid film, which refers to the entire perimeter in AV ¼ eA ¼ 1  AL ð5Þ
annular, intermittent and mist flows but only part of the
perimeter in stratified-wavy and fully stratified flows. A is the total cross-sectional area of the tube and e is the
The axial film flow is assumed to be turbulent. The film local vapor void fraction, which is determined using the
condensation heat transfer coefficient af is applied to the logarithmic mean void fraction (LMe) using the Rouh-
perimeter that would otherwise be dry in an adiabatic ani and Axelsson [32] drift flux model and the homo-
two-phase flow and hence is the upper perimeter of the geneous model (see Part I) in order to cover the range
tube for stratified-wavy and fully stratified flows. af is from low to high reduced pressures.
obtained by applying the Nusselt falling film theory to For annular, intermittent and mist flows, h ¼ 0. For
the inside of the horizontal tube, which assumes the fully stratified flow, h ¼ hstrat . For stratified-wavy flow,
falling film is laminar. The effect of axial shear on this the stratified angle h is obtained by assuming a quadratic
falling film is ignored. Heat transfer coefficients for interpolation between its maximum value of hstrat at
stratified types of flow are known experimentally to be a Gstrat and its minimum value of 0 at Gwavy
function of the wall temperature difference while those
0:5
for annular flow are not. This effect will thus be included ðGwavy  GÞ
h ¼ hstrat ð6Þ
through the Nusselt falling film heat transfer equation in ðGwavy  Gstrat Þ
the present model.
The values of Gstrat and Gwavy at the vapor quality in
The general expression for the local condensing heat
question are determined from their respective transition
transfer coefficient atp is
equations in the flow pattern map (see Part I).
The convective condensation heat transfer coefficient
af rh þ ð2p  hÞrac
atp ¼ ð2Þ ac is obtained from the following turbulent film equa-
2pr
tion:
In this expression, r is the internal radius of the tube and kL
h is the falling film angle around the top perimeter of the ac ¼ cRenL PrLm fi ð7Þ
d
3372 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

where the liquid film Reynolds number ReL is based on Gð1  xÞ

uL ¼ ð11Þ
the mean liquid velocity of the liquid in AL as qL ð1  eÞ
4Gð1  xÞd Gx
ReL ¼ ð8Þ uV ¼ ð12Þ
ð1  eÞlL qV e
and PrL is the liquid Prandtl number defined as Since normally uV  uL , then ðuV  uL Þ  uV . If we
cpL lL normalized the vapor velocity with that of the liquid, we
PrL ¼ ð9Þ
kL get the slip ratio, uV =uL , typical of void fraction models,
and the interfacial shear is proportional to the term
In these expressions c, n and m are empirical constants to
(uV =uL ). Thus, interfacial roughness Ddi can be assumed
be determined from the heat transfer database and d is
to be proportional to ðuV =uL Þj where the exponent j is
the thickness of the liquid film. The best value of the
unknown. In addition, the one-dimensional Taylor in-
exponent m on PrL was determined to be m ¼ 0:5, which
stability wavelength kT for the unsupported liquid film
is slightly larger than that in the Dittus–Boelter single-
on the top of the tube is:
phase flow correlation but is the same value found earlier
by Labuntsov [33] for turbulent falling film condensation
1=2
ðqL  qV Þg pffiffiffi
on a vertical plate. The best values of c and n for Eq. (7) kT ¼ 2p 3 ð13Þ
r
were found statistically to be c ¼ 0:003 and n ¼ 0:74.
The liquid film thickness d is obtained from solving
the following geometrical expression: and this should also be related to the formation of in-
terfacial waves. If the interfacial waves have character-
ð2p  hÞ 2 istic wavelengths similar to the film thickness, then
AL ¼ ½d  ðd  2dÞ2
ð10Þ
8 substituting d for kT means that the interfacial roughness
where d is the internal diameter of the tube. When the Ddi will be approximately scaled as
liquid occupies more the one-half of the cross-section of
k
the tube in a stratified-wavy or fully stratified flow at low ðqL  qV Þgd2
Ddi / ð14Þ
vapor quality, this expression will yield a value of r
d > d=2, which is not geometrically realistic. Hence,
whenever d > d=2 then d is set equal to d=2. where the term inside the brackets is non-dimensional.
Analysis of the data demonstrated that an additional Based on this reasoning, the interfacial roughness cor-
factor influenced convective condensation. After looking rection factor fi was introduced to act on ac in Eq. (7) as
at various possibilities, the interfacial surface roughness follows by adjusting the exponents j and k based on the
was identified as the most influential of these effects for test data to nominal values of 1/2 and 1/4 but without
the following reasons. First of all, the shear of the high introducing any empirical constants:
speed vapor is transmitted to the liquid film across the  1=2  1=4
interface and hence increases the magnitude and number uV ðqL  qV Þgd2
fi ¼ 1 þ ð15Þ
of the waves generated at the interface, which in turn uL r
increases the available surface area for condensation,
The interfacial roughness correction factor fi tends to-
tending to increase heat transfer. Secondly, the inter-
wards a value of 1.0 as the film becomes very thin
facial waves are non-sinusoidal and thus tend to reduce
(roughness must be proportional to film thickness) but fi
the mean thickness of the film, again increasing heat
tends to increase as the slip ratio uV =uL increases. Fi-
transfer. These two aspects are analogous to the en-
nally, fi tends to decrease as r increases, since surface
hancement correction factor of Kutateladze [34] for in-
tension acts to smooth out the waves. For fully stratified
terfacial ripples on Nusselt film condensation on a
flow, interfacial waves are damped out and hence the
vertical plate. Interfacial roughness and wave formation
above expression becomes
are also directly relatable to entrainment of liquid
droplets into the vapor phase, which reduces the thick-  1=2  1=4  
uV ðqL  qV Þgd2 G
ness of the liquid film and increases heat transfer. Fur- fi ¼ 1 þ ð16Þ
uL r Gstrat
thermore, interfacial shear tends to create vorticies
within the liquid film, which also increase heat transfer. when G < Gstrat , which produces a smooth variation in
Applying this reasoning, the interfacial roughness atp across this flow pattern transition boundary just like
can be expected to be directly proportional to the in- for all the other transition boundaries and the ratio of
terfacial shear si , where si in turn depends on the ve- G=Gstrat acts to damp out the effect of interfacial
locity difference between the two phases, (uV  uL ) where roughness in stratified flow.
uV and uL are the mean velocities of the phases in their The film condensation heat transfer coefficient af is
respective cross-sectional areas AV and AL obtained from the theory of Nusselt [35] for laminar
 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3373

flow of a falling film on the internal perimeter of the pattern transition boundaries without any jump in the
tube, where af is the mean coefficient for this perimeter. value of atp .
Rather than integrating from the top of the tube to the
stratified liquid layer at h=2 to obtain af , which would be 4.4. Implementation
more theoretically satisfying, it was found sufficient to
simply use the mean value for condensation around the The condensation heat transfer model is implemented
perimeter from top to bottom with its analytical value of as follows:
0.728, and thus avoid a numerical integration to facili-
tate practical use of this method in designing condensers. 1. determine the local vapor void fraction using the
Hence, af is LMe method (Part 1);
2. determine the local flow pattern using the flow pat-

1=4 tern map (Part 1);
q ðq  qV ÞghLV k3L
af ¼ 0:728 L L ð17Þ 3. identify the type of flow pattern (annular, intermit-
lL dðTsat  Tw Þ
tent, mist, stratified-wavy or stratified in Part 1);
4. if the flow is annular or intermittent or mist, then
Since heat exchanger design codes are typically im-
h ¼ 0 and ac is determined with Eq. (7) and hence
plemented assuming a heat flux in each incremental zone
atp ¼ ac in Eq. (2) where d is obtained with Eq. (10)
along the exchanger, it is more convenient to convert
and fi is determined with Eq. (15);
this expression to heat flux using NewtonÕs law of
5. if the flow is stratified-wavy, then hstrat and h are calcu-
cooling, such that the heat flux version of NusseltÕs
lated using Eq. (3) or (19) and Eq. (6), then ac and af
equation where the local heat flux is q, is given by the
are calculated using Eqs. (7) and (17) or (18), and
expression
finally atp is determined using Eq. (2) where again d
is obtained with Eq. (10) and fi is determined with

1=3
q ðq  qV ÞghLV k3L Eq. (15);
af ¼ 0:655 L L ð18Þ
lL dq 6. if the flow is fully stratified, then hstrat is calculated us-
ing Eq. (3) or (19) and hstrat is set equal to h, then ac and
af are calculated using Eqs. (7) and (17) or (18), and fi-
where the leading constant 0.655 comes from 0.7284=3 . nally atp is determined using Eq. (2) where d is ob-
The difference in the accuracy of the predictions whether tained with Eq. (10) and fi is determined with Eq. (16).
using the first or second of these expressions for af is
negligible.
To completely avoid any iterative calculations, the
recent explicit expression of Biberg [36] can be used to 5. Comparison to refrigerant database
very accurately (error  0.00005 rad for 2p P hstrat P 0)
evaluate the implicit expression above for hstrat , that is Fig. 5 shows a typical example of condensation heat
Eq. (3) here and Eq. (10) in Part 1: transfer data plotted as a function of vapor quality at

(  1=3 )
pð1  eÞ þ 3p ½1  2ð1  eÞ þ ð1  eÞ1=3  e1=3

hstrat ¼ 2p  2 2 ð19Þ
1
 200 ð1  eÞe½1  2ð1  eÞ
½1 þ 4ðð1  eÞ2 þ e2 Þ

This expression gives hstrat directly from the void fraction various mass velocities. The data are those of Cavallini
and has no effect on the location of the transition curves et al. [22,23] for R-134a in an 8.0 mm tube. The heat
compared to the prior method. transfer coefficients fall monotonically from large values
The above heat transfer prediction method cannot be at high vapor quality (where the annular film thickness is
evaluated at e ¼ 1:0 because of division by zero. Fur- thinnest) to small values at low vapor qualities. The ef-
thermore, experimental condensation heat transfer test fect of mass velocity is more significant at large vapor
data will have an error in vapor quality of at least
0.01 qualities than at low vapor qualities. Hence the slope of
and hence it does not make sense that test data can be the data curves increases with increasing mass velocity.
evaluated for x > 0:99. Thus, the above condensation All of these data fall within the annular or intermittent
prediction method is applicable when 0:99 P x; when flow regimes, except at the lowest vapor qualities and
x > 0:99, then x should be reset to 0.99. Also, the lower mass velocities where they reach the stratified-wavy re-
limit of applicability is for vapor qualities x P 0:01. Our gime.
range of test data is for 0:97 > x > 0:03. This method The new model was primarily developed using the
provides for a smooth variation in atp across all the flow heat transfer database of Cavallini et al. [22,23] and then
3374 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Fig. 5. Condensing data of Cavallini et al. [22,23] for R-134a.

Fig. 6. Comparison of model without fi to Cavallini data.

 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3375

Fig. 7. Comparison of model with fi to Cavallini data.

the other independent studies were used to determine its useful as a method for the optimisation of heat ex-
general applicability. First of all, to demonstrate the changers, it is important that the method respect the
importance of the interfacial effects in the data, Fig. 6 characteristic trends in the data, i.e. the effect of indi-
depicts a comparison of the new heat transfer model vidual variables on the prediction of the local heat
without use of the interfacial roughness factor (i.e. transfer coefficient. Hence, the same data are shown in
fi ¼ 1:0) and optimised to all 425 data points of Cav- a more detailed graphical presentation below in the
allini et al. [22,23] for six refrigerants, ranging from low following graphs in which the % error f% error ¼
pressure fluids (R-236ea) to high pressure fluids (R-125, 100%  ðapred  ameas Þ=ameas g is plotted versus the im-
R-410A and R-32). For this case, the optimum values of portant parameters in the model: vapor quality x, void
c and n are 0.0016 and 0.8585. In comparison, Fig. 7 fraction e, liquid film thickness d, liquid Reynolds
shows the same data including fi where the optimal number ReL , reduced pressure pred , mass velocity G, tube
values are those cited in the previous section. As can be diameter d, flow regime, and interfacial roughness factor
visually noted, the inclusion of the interfacial roughness fi . Hence, positive values represent over prediction and
factor significantly improves accuracy. negative values represent under prediction.
Fig. 8 depicts a comparison of the new heat transfer Fig. 9 depicts the data plotted versus vapor quality
model to all data, except the hydrocarbon data of Kim where the % error is quite evenly distributed over the
et al. [26]. There are eleven fluids represented with a total range of vapor qualities. This means that the model is
of 1850 data points. Based on all the data points in Fig. correctly capturing the slope of atp vs. x as G changes.
8 from these numerous different test facilities, about The scatter is larger at very high and very low vapor
85.0% are predicted within
20%. A comparison to the qualities where measurements typically have larger er-
hydrocarbon data is shown in a later section. rors or may include desuperheating or subcooling effects
as mentioned earlier.
Fig. 10 presents the data plotted versus void fraction.
6. Parametric study on accuracy As void fraction increases rapidly with vapor quality,
refer for example to Fig. 4 (Part 1), most of the data is in
Figs. 7 and 8 provide only a statistical view of the the high void fraction range. Even so, the % errors are
accuracy of the new model. However, in order to be reasonably well distributed over the range. Data at high
3376 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Fig. 8. Comparison of model with fi to all refrigerant data.

void fractions tend to be the most difficult to predict Fig. 13 depicts the important comparison of the new
because a very small change of 0.005 in void fraction has model to test data versus reduced pressure. The lowest
a notable effect on the film thickness when the void pressure represented in the database is 78 kPa while the
fraction is larger than 0.95. Referring to Fig. 5, the in- highest is 3184 kPa. The new model works just as well at
crease in the slope of atp versus x with increasing G is the low reduced pressures as at high ones, that is from 0.02
effect of G on the void fraction via Eqs. (2) and (6) in to 0.8, while previous prediction methods are not reli-
Part 1. Other previous condensation models that used able over such a wide range.
the Zivi [37] void fraction equation, which is indepen- Fig. 14 shows the % errors plotted versus mass ve-
dent of G, therefore introduced numerous empirical locity. The range in mass velocities here is very large, 24–
correction factors to account for this trend, while the 1022 kg/(m2 s), and the new model predicts the entire
present model does not require these. range with good accuracy. The band of errors is some-
Fig. 11 shows the data plotted versus the liquid what larger at low mass velocities since these flows are in
film thickness. The values of d in the database range the stratified-wavy and fully stratified regimes, or in
from as low as 0.008 mm (only 8 lm!) up to as high as annular or intermittent flow near the transition, where
8 mm (a ratio of 1000–1). The strong deviations for the prediction of the heat transfer coefficient is sensitive
R-22 at low values are the high vapor quality data of to the calculation of the dry angle and the Gwavy flow
Kim (see comments later on his data). The experimen- pattern transition, respectively.
tal data at very high vapor qualities, which result in Fig. 15 illustrates the prediction errors as a function
the very small film thicknesses, will be affected by any of tube internal diameter. The range of internal diame-
pre-existing condensate formed during desuperheating ters represented is very broad, i.e. from 3.1 to 21.4 mm,
the vapor as mentioned earlier, which will cause the which covers nearly all the sizes of the heat transfer
model to over predict the measured heat transfer coef- tubes used in industrial practice. The predictions for
ficients. most of the tube sizes are pretty well centered around 0%
Fig. 12 depicts a comparison versus the liquid Rey- error. At the smallest diameter, the capillary effects in
nolds number, which is a key parameter in calculating the flow pattern map on flow pattern transitions and
ac . The experimental range in ReL is from 300 to 85,000. heat transfer may start to play a role, since this size may
The deviations in the data are well centered around the be in the mesoscale between macrochannels and micro-
0% error line when plotted versus this parameter. channels.
J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3377

Fig. 9. Comparison of model plotted versus vapor quality.

Fig. 10. Comparison of model plotted versus void fraction.

3378 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Fig. 11. Comparison of model plotted versus film thickness.

Fig. 12. Comparison of model plotted versus liquid Reynolds number.

J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3379

Fig. 13. Comparison of model plotted versus vapor reduced pressure.

Fig. 14. Comparison of model plotted versus mass velocity.

3380 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Fig. 15. Comparison of model plotted versus tube diameter.

Fig. 16 shows the breakdown of how the new model very old data that are available in the mist flow regime
works by individual flow regime (S ¼ fully stratified for R-12 (21 points) at 757 and 1532 kg/(m2 s) and R-22
flow, SW ¼ stratified-wavy flow, I ¼ intermittent flow, (2 points) at 1002 kg/(m2 s) from Travis et al. [12],
A ¼ annular flow and MF ¼ mist flow). First of all, it is classifying their data using the mist flow transition
seen that the method works about equally well for all the equation of the flow pattern map in Part 1. These mist
flow regimes, as indicated by how the data scatter flow heat transfer coefficients are predicted reasonably
around the 0% error line. The stratified regime is the well, under predicting their values by about 15–20%.
only one a little off center and those data are some of the However, while it is interesting to see that the annular
most difficult to predict and measure (e.g. it is difficult to flow heat transfer model seems to work surprisingly well,
maintain steady-state conditions and get good energy not enough data are available to develop a separate mist
balances at these very low mass velocities and also the flow heat transfer model at present.
variation in the falling film heat transfer coefficient Fig. 17 shows the range of interfacial roughness
around the perimeter of the tube may not be captured factor fi in the database and the % error plotted versus
correctly unless 8–10 thermocouples are used). Also, it is this parameter. The test data have also been plotted
seen that applying the annular flow heat transfer struc- versus the wall temperature difference, (Tsat  Tw ), but
ture to intermittent flow works just as well as for annular these are not shown here. The plot shows that the %
flow itself. error tends to increase as the temperature difference
Regarding mist flow, which was not originally one of becomes smaller, similar to the propagation of error in
the flow regimes planned to be modelled here, applying the experiments.
the annular flow heat transfer structure (h ¼ 0) gives a
surprisingly good prediction for these few data (nine
points) that are above the mist flow transition boundary 7. Comparison to hydrocarbons
(Gmist ). Apparently, what this means physically is that
reformation of the condensate film is very rapid on the Addressing now condensation of hydrocarbons, such
wall where liquid has been stripped off into the mist. Not data are only available from a single source, Kim et al.
many data are often measured at this high of mass ve- [26]. In addition to their tests on R-22, they tested
locity, so we have also compared the new model to some propane, n-butane, iso-butane and propylene in an
J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3381

Fig. 16. Comparison of model by flow pattern.

Fig. 17. Comparison of model plotted versus interfacial roughness factor.

3382 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

8.0 mm tube. Their experimental technique is similar to we simply eliminate all the Kim data for mass velocities
others and they quote good experimental accuracy. Fig. less than 100 kg/(m2 s), where his data deviate from
18 shows their data plotted versus vapor quality, where all others (refer to Fig. 20 with respect to predictions),
the predictions are much poorer for some of the hydro- then the model retains its high accuracy also for hydro-
carbons than for R-22, and Fig. 19 plots their data sets carbons.
for their lowest mass velocity for each of their five fluids.
It is seen that their test data exhibit an unusual levelling
off or even a minimum at high vapor qualities, which is 8. Statistical comparison
not seen in any of the other data sets in the database for
refrigerants, e.g. compare these trends to Fig. 5 at high Table 2 provides a statistical summary of the com-
vapor qualities. Hence, there seems to be an experi- parison of the new heat transfer model to the experi-
mental problem with the Kim data in the high vapor mental database. The data are classified by flow regime
quality range. Also, Fig. 20 shows all their data plotted to show their distribution. The errors reported in Cav-
versus mass velocity. It is seen that their data also ex- allini et al. [20] for the comparison of their method to
hibit a strong deviation only at low mass velocities, for parts of the current database are also shown in brackets
those below about 100–120 kg/(m2 s). Hence, again there for comparison purposes. They did not compare their
appears to be an unidentified experimental problem (for method to the hydrocarbon data nor to the Fujii data.
example, energy balances or flow instabilities as men- Statistically, the two methods give nearly the same ac-
tioned earlier) with data in this lower flow range, com- curacy while the new model here is also shown to work
pared to the earlier comparison in Fig. 14, which showed well for the data of Fujii and Liebenberg for refrigerants
good agreement with numerous independent data sets at and for the hydrocarbon data of Kim. The values in the
low G. Thus, overall, the new heat transfer model seems table for the hydrocarbons are not really representative
to work well for hydrocarbons in the range that the Kim since all the data were included, even those with the
data follow the expected trends of the other data sets, unusual trends pointed out earlier.
for which the large majority are predicted within
20%. In summary, the new heat transfer model involves
Compared to all the refrigerant data plus all KimÕs only six basic equations (Eqs. (2), (6), (7), (15) or (16),
hydrocarbon data (2771 points), the new model predicts (17) or (18) and the void fraction from Part 1) plus
75% of the data to within
20%. On the other hand, if auxiliary equations to define dimensionless numbers and

Fig. 18. Comparison to hydrocarbon data versus vapor quality.

J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3383

Fig. 19. Hydrocarbon heat transfer data at low mass velocities.

Fig. 20. Hydrocarbon data comparison plotted versus mass velocity.

Table 2

3384
Statistical information by flow regime and source
Author Fluid Present model (Cavallini et al. model)
Number of points Mean error Arithmetic error Standard deviation
S SW I A MF S SW I A MF S SW I A MF S SW I A MF
Cavallini R22 1 21 33 51 7 13 7 7 8 )13 )6 )6 0 9 7 7
)8 ()5)

J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387
106 (106) 8 (5) 8 (3)
R134a 1 19 23 31 32 11 7 8 32 1 1 8 0 13 8 5
74 (74) 9 (4) 4 (4) 10 (5)
R410A 11 20 22 9 9 13 2 1 12 10 10 7
53 (53) 11 (6) 6 (5) 10 (5)
R125 25 25 19 4 12 5 11 4 )9 1 4 4 10 6 11 2
73 (73) 9 (5) )2 (0) 10 (6)
R236ea 14 7 45 6 18 11 4 18 10 7 5 7
66 (66) 11 (5) 10 (2) 8 (7)
R32 10 19 24 8 9 12 )1 3 10 9 10 8
53 (53) 10 (5) 5 (4) 10 (4)
Dobson R22 11 62 81 92 11 17 19 12 11 )15 )19 )11 4 13 6 6
& Chato 246 (246) 15 (12) )14 ()8) 10 (12)
R134a 10 76 44 66 8 10 13 12 8 )7 )11 )11 4 10 12 8
196 (196) 11 (11) )8 ()8) 11 (9)
R410A 9 21 40 36 23 14 14 10 23 4 )14 )10 8 17 9 12
106 (106) 14 (12) )6 ()9) 16 (12)
R32/R125(60/40%) 17 46 33 14 21 13 6 )21 )13 15 6 9
96 (96) 17 (12) )13 ()6) 13 (13)
Tang R22 42 48 12 5 )12 )5 7 3
90 (73) 8 (12) )8 ()12) 6 (7)
R134a 34 42 15 4 12 3 13 4
76 (66) 9 (8) 7 (0) 10 (11)
R410A 39 11 2 7 2 3 4 1 )3 9 2 1
52 (34) 5 (14) 3 (1) 8 (16)
Zhang R22 10 13 10 7 )7 )2 10 8
23 (23) 8 (9) )4 (5) 10 (11)
R134a 18 20 8 11 1 11 11 10
38 (38) 10 (11) 7 (10) 11 (8)
R404A 9 7 14 6 )14 )6 7 7
16 (16) 11 (6) )11 ()3) 8 (7)
Chitti R410A 20 21 8 10 )4 9 11 9
&Anand 41 (12) 9 (14) 3 (7) 12 (26)
Wijaya R410A 10 4 17 17 )17 )17 2 2
14 (7) 17 (27) )17 ()27) 2 (1)
Fujii R11 3 40 3 20 3 45 14 9 11 20 45 )4 4 )5 )20 25 17 12 12 6
69 14 )2 19
 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387 3385

the flow structure. In these expressions, there is only one

empirical constant, c in Eq. (7), and there is only one
13 empirical exponent, n in Eq. (7), plus those of three

12

12

13
6

13 (12) exponents whose values were assumed, i.e. for j ( ¼ 1/2),

k ( ¼ 1/4) and 1/2 for the interpolation in Eq. (6). Also,
the exponent m on the Prandt number PrL was adjusted

11
5
11

9
13

18
16

32

17

16
from its tubular flow value of 0.4 to its film flow value of
0.5, similar to Labuntsov [33]. Thus, the use of the flow
7

4
10

12

32

20

16
pattern map to categorize the flow regimes, the simpli-
fied flow structures to describe the flow regimes, and a
9

23

reliable void fraction equation to predict liquid film

thickness results in a simple heat transfer model that is
accurate and valid over a wide range of conditions
without resorting to statistical tweaking of the modelÕs
)1

)4

12

19

45

22

equations.
5 (12)

31

23
41

9
)8

)5

3
)5

)2
10

65

12
0

9. Simulations for R-410A

1

0
65

39
)9

)6
)14

To illustrate the predicted trends in atp as a function

of vapor quality and mass velocity, the heat transfer
0

model and flow pattern map have been simulated for R-

410A condensing at 40 °C in an 8 mm diameter tube
assuming a heat flux of 40 kW/m2 . The flow pattern map
is shown in Part 1 while the heat transfer coefficients are
13

19

45

22
5

shown in Fig. 21. At the lowest flow rate, 30 kg/(m2 s),

10 (15)

the flow is in the stratified regime from inlet to outlet

and the heat transfer coefficient falls off slowly with
31
13

8
41
6

18
23
3
10

12

65

14

decreasing vapor quality. At 200 kg/(m2 s), the flow en-

ters in annular flow and then passes through intermittent
15

13

65

39

13
6

and stratified-wavy flow. At 500 kg/(m2 s), the flow en-

ters in the annular flow regime and converts to inter-
20
6

mittent flow at about x ¼ 0:55 and leaves in this same

regime. The sharp decline in atp at high vapor qualities
results from the rapid growth of the annular film
thickness. At 800 kg/(m2 s), the flow enters in the mist
5

125

248
15

62

65

24

flow regime, goes then into the annular flow regime and
225 (225)

then leaves in the intermittent regime. As can be seen,

the new heat transfer model predicts the variation in the
61
495

144

147

135
39

48
65

9
50

152

24
4

local heat transfer coefficients across flow pattern tran-

sition boundaries without any discontinuity in the value
9
28

26

35

95

76
144

of atp . This, for example was a problem in the Dobson

and Chato [19] method and also for one of the transi-
tions in the more recent Cavallini et al. [20] method
3

going into their slug flow regime. Also of note, the heat
transfer coefficient would exhibit a small peak in the SW
zone at 200 kg/(m2 s) if the same simulations were re-
peated for Fig. 21 at a low heat flux, such as 10 kW/m2 ,
iso-Butane
n-Butane
Propane

Propene

since then af around the top of the tube would be larger

R113

than ac around the bottom of the tube.

R12

R22

R22
Liebenberg

10. Condensation of mixtures

Kim

The above method is extendable to the prediction of

local condensation heat transfer coefficients of zeotropic
3386 J.R. Thome et al. / International Journal of Heat and Mass Transfer 46 (2003) 3365–3387

Fig. 21. Simulation of model for condensation of R-410A.

mixtures or for condensation in the presence of non- obtained in nine different laboratories. The new model
condensable gases by use of the Bell and Ghaly [38] has so far been tested for the following range of condi-
method. This method essentially consists of assuming tions: mass velocities from 24 to 1022 kg/(m2 s), vapor
two thermal resistances in series: the first resistance is for qualities from 0.03 to 0.97, reduced pressures from 0.02
convective heat transfer in the vapor-phase from the to 0.80 and tube internal diameters from 3.1 to 21.4 mm.
bulk vapor temperature to the temperature at the vapor– Overall, the method predicts 85% of the refrigerant heat
liquid interface and the second resistance is across the transfer coefficients in the database (1850 points) to
condensate film itself, given by 1=atp . Comparison to within
20% and predicts 75% of the refrigerant plus
such test data is beyond the present scope and will be hydrocarbon heat transfer coefficients in the database
treated in a future study. (2771 points) to within
20%.

11. Conclusions Acknowledgements

A new general flow pattern based heat transfer model A. Cavallini participated in this project as an ER-
for condensation inside horizontal, plain tubes has been COFTAC Scientific Visitor to the Laboratory of Heat
proposed here based on the same simplified flow struc- and Mass Transfer in Lausanne and we are grateful to
tures of the flow regimes used in the flow boiling model him for providing the majority of the database used in
of Kattan et al. [3]. The condensation heat transfer this project.
model includes the effect of interfacial roughness of the
liquid–vapor on heat transfer. The model resorts to very
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