comprehensive review of pure vapour condensation outside
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Comprehensive review of pure vapour condensationoutside of horizontal smooth tubes
Clément Bonneau, Christophe Josset, Vincent Melot, Bruno Auvity
To cite this version:Clément Bonneau, Christophe Josset, Vincent Melot, Bruno Auvity. Comprehensive review of purevapour condensation outside of horizontal smooth tubes. Nuclear Engineering and Design, Elsevier,2019, 349, pp.92-108. 10.1016/j.nucengdes.2019.04.005. hal-02372722
Comprehensive Review of Pure Vapour Condensation
Outside of Horizontal Smooth Tubes
Clement BONNEAU1,2, Christophe JOSSET2, Vincent MELOT1, and Bruno
AUVITY∗2
1Naval Group Nantes-Indret
2Laboratoire de Thermique et Energie de Nantes (CNRS UMR 6607), Ecole
Polytechnique de l’Universite de Nantes
March 27, 2019
Abstract
The thermal design of an industrial shell-and-tube condenser requires the use of heat transfer
coefficients, usually obtained from tables or correlations. Willing to develop a numerical model for
design purposes, the present authors noticed the surprising diversity of correlations for the shell-
side heat transfer coefficient in the case of pure vapour condensation outside of horizontal smooth
tubes. In order to shed light on this specific topic, a bibliographic study was therefore initiated.
This comprehensive review is meant to provide the designers with means to understand how each
correlation was obtained, from the assumptions to the resolution method. Thus two main phenom-
ena are well accounted for in this paper: vapour shear stress and condensate inundation. Indeed,
the review lists the most important contributions to this field and details their interconnections.
Consequently, the present authors conclude this paper with their recommendations.
Keywords: Condensation, Condenser, Tube bundle, Heat transfer coefficient
∗[email protected]; Corresponding author
1
*Title Page
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Comprehensive Review of Pure Vapour Condensation
Outside of Horizontal Smooth Tubes
Clement BONNEAU1,2, Christophe JOSSET2, Vincent MELOT1, and Bruno
AUVITY∗2
1Naval Group Nantes-Indret
2Laboratoire de Thermique et Energie de Nantes (CNRS UMR 6607), Ecole
Polytechnique de l’Universite de Nantes
March 27, 2019
Abstract
The thermal design of an industrial shell-and-tube condenser requires the use of heat transfer
coefficients, usually obtained from tables or correlations. Willing to develop a numerical model for
design purposes, the present authors noticed the surprising diversity of correlations for the shell-
side heat transfer coefficient in the case of pure vapour condensation outside of horizontal smooth
tubes. In order to shed light on this specific topic, a bibliographic study was therefore initiated.
This comprehensive review is meant to provide the designers with means to understand how each
correlation was obtained, from the assumptions to the resolution method. Thus two main phenom-
ena are well accounted for in this paper: vapour shear stress and condensate inundation. Indeed,
the review lists the most important contributions to this field and details their interconnections.
Consequently, the present authors conclude this paper with their recommendations.
Keywords: Condensation, Condenser, Tube bundle, Heat transfer coefficient
∗[email protected]; Corresponding author
1
*Manuscript
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1 Nomenclature
Dimensionless numbers
F = PrFr·Ja
Fr = U2
gDoFroude number
H = JaPr
Condensation number
Ja = cp∆T
∆hvJakob number
Nu = hDλ
Nusselt number
P = PrJa
ρvρl
Pr = µcpλ
Prandtl number
R =√
ρlµlρvµv
ρµ-ratio
Re = UDo
νReynolds number
Re = UvDo
νlTwo-phase flow Reynolds number
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Latin letters
cp Thermal capacity (J.kg−1.K−1)
Do Tube outer diameter (m)
f Friction factor (-)
Fd Tube spacing parameter
g Gravitational acceleration (m.s−2)
Hf Enthalpy flow rate per unit length (W.m−1)
h Heat transfer coefficient (W.m−2.K−1)
∆hv Phase change enthalpy (J.kg−1)
j Condensation mass flux (kg.s−1.m−2)
n Tube row number
pt Tube pitch (m)
p Pressure (Pa)
Q Heat flux (W )
q Surface heat flux (W.m−2)
R Tube outer radius (m)
r Radial coordinate from the surface of the surface (m)
S Heat transfer surface (m2)
T Temperature (K)
U Vapour orthoradial velocity (m.s−1)
u Condensate orthoradial velocity (m.s−1)
V Vapour radial velocity (m.s−1)
v Condensate radial velocity (m.s−1)
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Greek letters
α Volume fraction
Γ Global condensate mass rate (kg.s−1)
γ Local condensate mass rate (kg.s−1)
δ Local condensate film thickness (m)
θ Angle measured clockwisely from top of the tube (rad)
λ Thermal conductivity (W.m−1.K−1)
µ Dynamic viscosity (kg.m−1.s−1)
ν Kinematic viscosity (m2.s−1)
ρ Density (kg.m−3)
τ Surface shear stress (Pa)
χ Coefficient in Fujii et al. [1]
Exponents & Subscripts
c Critical angle
GR Gravity component
l Liquid property
lv At liquid-vapour interface
n Relating to the n-th tube
sat At saturation
SH Shear stress component
t Turbulent
v Vapour property
w Wall
θ At the distance corresponding to θ-angle
x Spatial mean of variable x
x∗ Dimensionless value of variable x
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2 Introduction
2.1 General introduction
Thermo-economic optimization of thermodynamic cycles is a topic of current interest [2] due to growing
concerns regarding energy use. A key role in these cycles is played by the condensers [3], since they are
responsible for most exergy loss [4] - [8]. It is therefore crucial to properly describe this equipment’s
thermalhydraulic performances. In this context, Naval Group, which is a world leader in naval defence5
and an innovative player in energy, intends to improve its condenser design tools in order to manufacture
more efficient products. Whether they are embedded on military ships, where their size matters, or used
in onshore facilities, where their cost matters, it is crucial to properly design the condensers with margins
as reasonable as possible.
The most common surface condenser met is the shell-and-tube one, which consists of several circular10
pipes within a cylindrical shell enclosed by tubes sheets at each end, with the condensation phenomenon
occurring on the outside of these tubes (i.e. shellside). Shell-and-tube condensers are found in many
applications, since they offer a wide operating range in terms of pressure, fluids and power. Besides,
their modularity makes them easy to design, which may be the reason of their success over the 20th
century. In spite of that, for economic reasons, their efficiency had to be improved, not only through15
their designers experience feedback, but through a deeper understanding of condensation phenomena. In
the meantime, most condenser designers would use the formulation provided by Nusselt in the early 20th
century for condensation of pure stagnant vapour outside a horizontal smooth tube [9], which proved
itself to be precise enough to predict condensers performance within an acceptable margin of error. The
study of condensation outside of smooth tubes progressively ended in the 1980’s to the advantage of20
enhanced surface tubes. Nowadays, manufacturers are more interested in new technologies such as plate
condensers, which offer a wide field of research.
However, plate condensers still have limitations that make them unsuitable for specific applications.
For instance, in the electro-nuclear sector, they are nowhere to be found, for they cannot meet the high
power requirements. Besides, there are mechanical issues, because there usually is a strong pressure25
difference between the vapour side and the cooling water side. These mechanical constraints are well
handled in shell-and-tube condensers, since it is related to the tube thickness. However, for plate
exchangers, the design is a bit more complicated. The size limitation is mainly due to the current
manufacturers’ equipment. These reasons have led some manufacturers to keep designing shell-and-tube
5
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condensers. Therefore, shell-and-tubes condensers are commonly used in refrigeration, air-conditioning30
and heat pump equipment of medium to large capacity.
As mentioned before, studies of tubes with enhanced surfaces, both inside and outside, have pro-
gressively appeared over the 1980s. These new tube geometries have proved themselves quite efficient,
but mostly for single-phase flows. As for condensers, the shellside tube enhancement has brought a
very small benefit at best [10], since the main thermal resistance comes from the tube-side flow. Fur-35
thermore, when considering industrial aspects, only low finned tube may be considered, because of the
several baffle plates electronuclear condensers have: the tube assembly would scrap these fins.
Nowadays, most condenser designs are still based on the standards of the Heat Exchange Institute
(HEI) [11], which are based on formulation and data by Orrok (1910, [12]). In the meantime, numerical
modelling of industrial condensers appeared in the early 1980s [13] - [25]. These approaches are all40
based on the resistance summation method, which is often recommended instead of the HEI method
[26] [27]. Such a local approach is probably better-suited than a global one, when considering a detailed
performance characterisation within an optimisation process. Coupling this method with the porous
media approach, which is an homogenisation method used to obtain a sufficient description of the
vapour flow, a complete CFD tool is obtained such as the one developed by the present authors [28]45
[29].
What struck the present authors in above mentioned publications, is the constant change in resistance
correlations from an author to another, and even from a publication to another of the same author. Plus,
no justification is provided by the authors regarding the reasons that lead to their choice. Since the heat
transfer is the core of such modelling, it is of major importance to correctly choose the most appropriate50
correlation. Indeed, the porous media CFD modelling requires a deep understanding of implemented
correlations, otherwise no optimization process can be run.
Therefore the present authors intend to clarify the meaning of the various existing correlations for
the heat transfer coefficient in the case of pure vapour condensation outside of horizontal smooth tubes.
Though several authors have already partially reviewed this case [30] [31] [32], they appeared to be55
incomplete, as they deal with several modes of condensation, and to be sometime inaccurate in the
cited literature. The purpose of the present article is therefore to describe how these correlations were
obtained and how they are interconnected. It is meant to help CFD engineers, researchers and designers
choose the appropriate correlations for their condenser modelling.
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2.2 Modelling strategy60
The study of condensation in industrial shell-and-tube condensers is complex since it is a combination
of elementary phenomena. The methodology followed in the present article is to start from a single tube
with numerous assumptions (i.e. Nusselt’s work), and to progressively lift them. As shown in Figure 1
where tubes are presented in white and condensate in black, there are two main phenomena that drive
the condensation process. On the horizontal axis, there is the increasing impact of the vapour flow65
around the tube, which causes shear stress on the condensate film, and on the vertical axis, there is
the influence of tube bundle effects on local condensation, due to condensate inundation. A few general
situations can be highlighted:
• Condensation around a single tube in stagnant vapour (a) (§3)
• Condensation around a single tube under shear stress, with a laminar condensate film (b) (§4)70
• Condensation around a single tube under shear stress, with a turbulent condensate film (c) (§4)
• Condensation in a bundle of tubes in stagnant vapour, with a laminar condensate film (d) (§5)
• Condensation in a bundle of tubes in stagnant vapour, with a turbulent condensate film (e) (§5)
• Condensation in a bundle of tubes under shear stress. This situation lies within the bottom-right
corner in Figure 1, where both shear stress and condensate inundation are present at the same75
time. It is the most complex situation, hence the lack of graphical representation, but also the
most common in industrial condensers.
3 Condensation on a single tube with vapour at rest
3.1 Nusselt’s theory: The cornerstone
A hundred years ago, german professor Wilhelm Nusselt studied analytically the condensation phe-80
nomenon on both a vertical plate and a horizontal tube. This pioneering work [9] has been used ever
since, mentioned in every publication on surface condensation. As for the horizontal tube study, here
are the hypotheses:
1. The surrounding vapour is at rest.
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2. The vapour is saturated.85
3. The wall temperature Tw is uniform.
4. The condensate flow around the tube surface is laminar and purely orthoradial.
5. The condensate film is not subject to shear stress at the liquid/vapour interface.
6. The heat is transferred through the condensate film only by conduction.
7. The condensate properties are taken at saturation90
In order to understand the further developments undertaken by reseachers after Nusselt’s paper, it
seems necessary to demonstrate his analytical work. Starting with the equation of momentum with the
assumptions of a 2D stationary purely orthogonal flow within a slender film, one obtain the equation of
movement for the condensate film:
νl∂2u
∂r2+ g sin(θ) = 0 (1)
The pressure gradient term is here neglected since it is much smaller than the body force effect.95
Then this equation is solved using the following boundary conditions:u(r = 0) = 0 (no slip boundary)(∂u
∂r
)δ
= 0 (no shear stress at vapour-liquid interface)
with δ the local film thickness.
We obtain:
u(r, θ) =g
2νlsin(θ)r(2δ − r) (2)
and the mean velocity over the thickness of the film is:
u =1
δ
∫ δ
0
u dr
=g
3νlsin(θ)δ2 (3)
As presented in Figure 2, a mass balance is achieved over a portion of the condensate film, which100
leads to the following equation:
d(uδρl) = jθ ·Rdθ (4)
8
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where u is the mean velocity of the condensate over the thickness of the film and jθ is the condensation
mass flux in the case of pure conduction defined by:
jθ =λl∆T
δ∆hv(5)
where ∆T = Tsat − Tw is the temperature difference between the saturated vapour and the tube wall.
This equation is obtained by a simple energy balance equating the phase change enthalpy (left member)105
with the conductive heat transfer through the film (right member) :
jθ∆hv =λl∆T
δ(6)
Substituting (3) and (5) into (4), we obtain:
d
dθ(δ3 sin(θ)) =
3νlRλl∆T
ρlg∆hv· 1
δ(7)
The solution of this differential equation is:
δ =
(3νlRλl∆T
ρlg∆hv· 4
3(sin(θ))4/3
∫ θ
0
(sin(ω))1/3dω
)1/4
(8)
Under the assumption that heat is transferred by pure conduction, the local heat transfer coefficient
h is:110
h =λ
δ(9)
Finally, integrating h over the tube, we obtain:
h =1
π
∫ π
0
hdθ (10)
= 0.728
(ρlgλ
3l∆hv
νlDo∆T
)1/4
(11)
This heat transfer coefficient is often written with the constant equal to 0.725. This is due to a lack
of precision in Nusselt’s calculations, who probably used a handmade Riemann integral back in the day.
The Nusselt number can therefore be expressed as:
Nu = 0.728
(ρ2l gD
3o∆hv
µlλl∆T
)1/4
(12)
9
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This work is the cornerstone of all scientific production on filmwise condensation outside horizon-115
tal tubes. The advantage of this method is that it is purely analytical, though limited by certain
assumptions. However, it is not limited to a specific fluid, which is often the case with experimental
correlations.
3.2 Pressure gradient effect
Another form of this equation is commonly found, namely :120
Nu = 0.728
(ρl(ρl − ρv)gD3
o∆hvµlλl∆T
)1/4
(13)
This comes from the mechanical equilibrium equation which takes into account the pressure gradient,
which is hydrostatic. Therefore the equation (1) is slightly changed :
µl∂2u
∂r2+ (ρl − ρv)g · sin(θ) = 0 (14)
It appears that the Nusselt number obtained from (13) is equal to one obtained from (12) when
ρl ρv.
Belghazi et al. [33] confronted their experimental results for condensation of HFC134a on tubes125
of the upper row of a bundle with equation (13) and obtained discrepancies of about 10%. So did
Fernandez-Seara et al. [10] who obtained similar discrepancies with ammonia.
3.3 Enthalpy convection effect
In the previous analysis, it was assumed that heat was only transmitted by conduction through the
condensate film. In the heat balance, the impact of the thermal capacity cp is neglected. In 1952,130
Bromley [34] included this thermal capacity in Nusselt’s work, since he assumed that heat couldn’t only
be transmitted by conduction, which case provided with a linear temperature profile (red plain line
Figure 3), as if the condensate was still. According to Bromley, for a given thickness of the condensate
film and temperature difference, the temperature profile should be curved (red dotted line Figure 3).
In the case of negligible viscous dissipation and negligible compressibility effects (Bejan, 2004 [35]),135
10
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the stationary energy equation within the condensate film is:
ρlcp∇ ·(T ~V)
= λl∇ · ∇T (15)
which becomes
∇ ·(ρlcpT ~V − λl∇T
)= 0 (16)
Then, using the divergence theorem over the domain shown in Figure 4 and neglecting the orthoradial
component of the temperature gradient:
− λl(∂T
∂r
)r=0
Rdθ + λl
(∂T
∂r
)r=δ
Rdθ +Hf − (Hf + dHf ) = 0 (17)
with Hf the enthalpy flow rate per unit length entering the shaded domain:140
Hf =
∫ δ
0
ρlucp (T − Tsat) dr (18)
The incoming heat flux on the right side of the domain comes from the condensation mass rate:
λl
(∂T
∂r
)r=δ
= jθ ·∆hv (19)
In order to obtain the proper temperature profile, Bromley solved the same equations as Nusselt did,
except that he corrected (6) by using the above equations. This calculation requires the temperature
gradient at the wall, which is not known a priori. Therefore, Bromley proceeded with an iterative
procedure and initialised with a linear temperature profile. At the end of the first iteration, he obtained:145
Nu = 0.728
(1 +
3
8Ja
)(
1 +7
30Ja
)3/4
(ρl(ρl − ρv)gD3
o∆hvµlλl∆T
)1/4
(20)
Where Ja is the Jakob number. For cp = 0, (20) is equal to (13). Then, Bromley repeated the same
procedure, starting from the new temperature gradient at the wall. At the end of the second iteration,
11
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he obtained the same equation as above with
(1 +
3
8Ja
)(
1 +7
30Ja
)3/4replaced by:
1 +
[1
2− 1
8
(1 + 0.052322 · Ja1 + 0.233333 · Ja
)]· Ja
1 +
[1
3− 1
10
(1 + 0.051786 · Ja1 + 0.233333 · Ja
)]· Ja
3/4(21)
This second approximation is really close to the first one, therefore no further calculation was
achieved. Bromley proposed a simpler correction:150
√1 + 0.4 · Ja (22)
This correction is often applied to the phase change enthalpy, namely :
∆h′v = ∆hv (1 + 0.4 · Ja)2 (23)
Nu/Nucp=0
1st approx. 2nd approx. SimplifiedJa (20) (21) (22)
0.01 1.0020 1.0020 1.00200.1 1.0197 1.0198 1.01980.5 1.093 1.095 1.0951.0 1.175 1.180 1.1832.0 1.31 1.33 1.343.0 1.43 1.45 1.485.0 1.61 1.64 1.73
Table 1: Comparison of the 3 expressions of the ratio Nu/Nucp=0 for different Ja numbers
These three expressions are compared in Table 1 for a wide range of Ja numbers. The simpli-
fied version is a good approximation for Ja below unity, but differs when above. However, for steam
condensation, Ja rarely exceeds 0.1.
In 1956, Rohsenow [36] did the same analysis than Bromley [34] for a vertical plate but took into155
account the effect of the crossflow within the condensate film. The force balance for the vertical plate
12
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is similar to the one from the tube and therefore the velocity is:
w(x, z) =g(ρl − ρv)
µl
(xδ(z)− x2
2
)(24)
For the red control volume in Figure 5, the mass flowrate dm across the face at x is:
dm(x) =
∫ x
0
ρlw(x, z + dz)dx−∫ x
0
ρlw(x, z)dx
=gρl(ρl − ρv)
µl
x2
2δ (25)
Then the heat balance at a distance x from the wall:
λl∂T
∂x= jθ∆hv + cpρlw(x)T (x) + cpdm(x)T (x) (26)
Repeating the same procedure as Bromley’s, the obtained simplified correction after a few iterations160
is:
∆h′v = ∆hv (1 + 0.68 · Ja) (27)
It is close to Bromley’s correction, which is more obvious when (23) is expanded into Taylor series
for small values of Ja:
∆h′v = ∆hv (1 + 0.8 · Ja) (28)
There is no apparent restriction in applying Rohsenow’s correction for a horizontal tube.
In 1959, a boundary-layer analysis for laminar film condensation was performed by Sparrow & Gregg165
for both a vertical plate [37] and a horizontal tube [38]. Their analysis took into account both the inertia
forces and energy convection. They considered the following system of equations:
∂u
∂x+∂v
∂y= 0 (Mass) (29)
u∂u
∂x+ v
∂u
∂y= g · sin
(xr
)+ νl
∂2u
∂y2(Momentum) (30)
u∂T
∂x+ v
∂T
∂y=
λlρlcp
∂2T
∂y2(Energy) (31)
The system of coordinates is described in Figure 6.
Then they reduced these partial differential equations to ordinary differential equations, which have
13
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been solved numerically for Prandtl numbers ranging from 0.003 to 100 and for Jakob numbers ranging170
from 0 to 1. However, no formulation of the Nusselt number depending on Pr and Ja could be expressed.
For small value of Ja, the Nusselt number tend to:
Nu = 0.733
(ρ2l gD
3o∆hv
µlλl∆T
)1/4
(32)
which is really close to (12).
Sadasivan & Lienhard (1987 [39]) noticed that Bromley and Rohsenow’s developments involved the
implicit assumption that the Pr was infinite. However, Sparrow & Gregg demonstrated that the Nusselt175
number depended upon Pr and Ja, and that for high values of Pr, their solution was close to Nusselt’s
solution with Rohsenow’s correction. So the authors assumed that the general solution was of the form:
∆h′v = ∆hv (1 + C(Pr) · Ja) (33)
where C was a parameter to determine.
For Pr ranging from 0.6 to 1000 and Ja below 0.8, Sadasivan & Lienhard obtained the following
formulation using a best curve fitting method:180
∆h′v = ∆hv
[1 +
(0.683− 0.228
Prl
)Ja
](34)
3.4 Surface temperature variation
All previous work assumed an isothermal surface at the tube wall for practical reasons. Nevertheless,
due to an uneven film thickness around the tube, the wall temperature is higher at the top of the tube,
where the film is thinner, and lower at the bottom, where the film thickness becomes near infinite. This
phenomenon has been studied by Memory & Rose [40], who assumed a cosine temperature distribution:185
Tw = A · cos(θ)∆T + Tw (35)
Where A ∈ [0; 1].
This assumption is based on the conclusion of Lee et al. [41], who experimentally observed such a
distribution. Then Memory & Rose solved Nusselt equation for various values of A in the range [0; 1]
and obtained the Nusselt constant equal to 0.7280 quite surprisingly. These first four significant digits
14
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remained constant for all values of A.190
3.5 Uniform heat flux
Instead of assuming a cosine wall temperature, another approach was chosen by Fujii et al. [42]. The
authors used the same assumptions as Nusselt, except the uniform wall temperature replaced by a
uniform surface heat flux q. They obtained the following set of equations:
d2u
dr2= − g
νlsin(θ) (Momentum) (36)
d
dθ(uδρl) =
qR
∆hv(Heat balance) (37)
Using the same boundary conditions, the same mean velocity u is obtained. Therefore, substituting195
(3) into (37):d
dθ(δ3 sin(θ)) =
3µlDoq
2ρ2l g∆hv(38)
The solution of this equation is:
δ =
(3µlDoq
2ρ2l g∆hv· 1
sinc(θ)
)1/3
(39)
Then the mean heat transfer coefficient is obtained:
h = 0.693
(λ3l ρ
2l g∆hv
µlDoq
)1/3
(40)
and the corresponding Nusselt number:
Nu = 0.693
(ρ2l gD
2o∆hv
µlq
)1/3
(41)
The analysis of Butterworth [43] regarding this work is quite confusing, since it suggests that the200
only difference between (12) and (41) is the coefficient changing from 0.728 to 0.693, whereas the whole
expression is different.
As mentioned by the authors, the assumption of uniform wall temperature makes sense when the
cooling side is the most resistive, but it is no longer appropriate when both sides are of closer magnitudes.
15
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4 Condensation on a single tube under vapour flow205
In the previous section, the vapour surrounding the tube was at rest and the condensate film was set
in motion by gravity. The following section will present the phenomenology of filmwise condensation
under vapour crossflow.
Indeed, for high vapour velocities, the condensate film undergoes a shear stress at liquid-vapour
interface. This shear stress enhances the heat transfer by locally thinning the liquid film. For a vertical210
downflow as represented in Figure 7, the film is thinner on the upper half while it is thicker on the
lower half. The thickening is caused by the separation of the vapour boundary-layer after the separation
point, which is located at an angle usually comprised between 80 and 180 from the top of the tube.
At this very point, the shear stress at the liquid-vapour interface changes sign, hence the recirculation
flow as presented in Figure 8. Below this separation point, the vapour flows in the direction opposite to215
the gravity, which causes the thickening and considerably lowers the heat transfer.
This separation is caused by the vapour pressure gradient, which is more important when the vapour
is flowing, due to the dynamic component of the pressure. This gradient is positive on the forward half,
which tend to thin the film, while it is negative on the rear half, provoking the opposite effect.
4.1 Approach of Sugawara et al. (1956)220
Though condensation of a vapour flow on a single horizontal tube has been treated by Fuks [44] in the
USSR, the first semi-analytical study was conducted by Sugawara et al. [45] from Japan. They kept
most of Nusselt assumptions, except the fact that the vapour is not at rest. Therefore, they obtained the
same equations (1) & (4), but the momentum equation is solved using different boundary conditions,
namely:225 u(r = 0) = 0 (no slip boundary)
µl
(∂u
∂r
)δ
=1
2fρvU
2∞ (shear stress at vapour-liquid interface)
with f the dimensionless friction factor.
Consequently:
u(r, θ) =g
2νlsin(θ)r(2δ − r) +
fρvU2∞
2µlr (42)
and the mean velocity over the thickness of the film is:
u =g
3νlsin(θ)δ2 +
fρvU2∞
4µlδ (43)
16
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Substituting (43) into (4):
ρlg
3
d
dθ
(δ3 sin(θ)
)+ρvU
2∞
4
d
dθ
(fδ2)
=νlRλl∆T
∆hv· 1
δ(44)
The friction factor is estimated by using experimental results for a single phase flow around a cylinder.230
For this experiment, boundary-layer separation occured at 83. So behind the separation point, the
authors have neglected the friction force, solving the same equation as (1):
ρlg
3
d
dθ
(δ3 sin(θ)
)=νlRλl∆T
∆hv· 1
δ(45)
Then (44) and (45) are solved numerically by Runge-Kutta’s method. Unfortunately, neither a local
nor a mean Nusselt number was expressed from these results, only diagrams.
This method seems to neglect the fact that the vapour drag has an undesirable effect on condensation235
over the rear half of the tube. Therefore, the results may be too optimistic.
4.2 Approach of Shekriladze & Gomelauri (1966)
In 1966, Shekriladze & Gomelauri [46] noticed that previous work ([44] [45] [47]) used the assumption
that the shear stress at the liquid-vapour interface was the same as at a dry tube surface without
condensation, thus ignoring the momentum transfer caused by the mass of the condensing vapour.240
The authors took it into account in their study, under similar assumptions to Nusselt’s. However,
they assumed that the tube was in a vertical downward potential flow of vapour and inertia forces were
neglected. In a potential flow with a velocity far from the tube U∞, the velocity varies from 0 at the
front and back stagnation points to 2U∞ on the sides:
U(θ) = 2U∞sin(θ) (46)
The following set of equations describe the condensate film flow:245
µl∂2u
∂r2= 0 (Momentum) (47)
d(uδρl) = jθ ·Rdθ (Heat balance) (48)
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
with the boundary conditions:u(r = 0) = 0 (no slip boundary)
µl
(∂u
∂r
)δ
= τδ (shear stress at vapour-liquid interface)
with the local shear stress defined by Shekriladze & Gomelauri as:
τδ = jθ (U(θ)− Ulv) (49)
with Ulv U(θ) the velocity at liquid-vapour interface (neglected).
The local heat transfer coefficient obtained from this set of equations is:
h(θ) =sin(θ)√
1− cos(θ)
√λ2l ρlU∞µlDo
(50)
Hence the mean heat transfer coefficient:250
h =2√
2
π
√λ2l ρlU∞µlDo
(51)
Introducing the two-phase flow Reynolds number:
Re =ρlDU∞µl
(52)
The corresponding Nusselt number is:
NuSH = 0.900Re1/2
(53)
Being unable to analytically solve (47) with added gravitational term, the authors used an asymptotic
model, which formulation approaches their previous result for high vapour velocities (i.e. NuSH) and
the basic Nusselt formulation for low or nil vapour velocity (i.e. NuGR), namely:255
Nu =
[Nu2SH
2+
(Nu4SH
4+Nu4GR
)1/2]1/2
(54)
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Introducing the dimensionless number F :
F =Pr
Fr · Ja(55)
The previous equation may be expressed as a function of F :
Nu = 0.637[1 + (1 + 1.68F )1/2
]1/2Re
1/2(56)
This dimensionless number describes if the condensate film is subject to either vapour drag (F −→ 0)
or gravity (F −→ ∞).
In (56), the original coefficient 0.725 from Nusselt was used, instead of 0.728.260
However, they did not take into account the effect of the pressure gradient, which results in a
too optimistic Nusselt number. Well aware of this phenomenon, Shekriladze & Gomelauri proposed
a correction of (56). According to the authors, the boundary-layer separation point only happens
beyond 82, or 65% of the heat transfer takes place on the tube surface comprised between 0 and 82.
Therefore, they decided to neglect the heat transfer on the surface lying beyond this angle, where the265
boundary-layer may be separated, thus obtaining a 35% lower Nusselt number:
Nu = 0.414[1 + (1 + 1.68F )1/2
]1/2Re
1/2(57)
However, Butterworth [48] noticed that reducing the whole Nusselt number by 35% also affect the
solution for stagnant vapour case and thus proposed the following correction:
Nu =
[(0.65NuSH)2
2+
((0.65NuSH)4
4+Nu4GR
)1/2]1/2
(58)
which results in:
Nu = 0.414[1 + (1 + 9.42F )1/2
]1/2Re
1/2(59)
This is the conservative version of (56), which safely underestimates the heat transfer coefficient.270
Figure 9 describes the evolution of Nu/Re1/2
for equations (12), (56) and (59) as a function of F .
For F −→∞, all formulations approach the Nusselt solution.
It should be noted that the above equations are particularly conservative, given the fact that behind
the separation point the heat transfer is assumed to be nil while it is just lowered. Furthermore, the
19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
considered angle of separation of 82 used to limit the heat transfer surface is the smallest one observed,275
which is less than half of the tube surface. Depending on the actual angle of separation, the Nusselt
number is likely to lie between equations (56) and (59).
4.3 Approach of Fujii et al. (1972)
Unsatisfied with previous work, Japanese searchers Fujii et al. [1] extended the analysis to the vapour
boundary-layer. Therefore, no assumption is made regarding the shear stress at the interface. Beyond280
the boundary layer, the vapour flow is assumed to be a vertical downward potential flow. They obtained
the following set of equations for the condensate film:
1
r
∂u
∂θ+∂v
∂r= 0 (Continuity) (60)
νl∂2u
∂r2+ g · sin(θ) = 0 (Momentum) (61)
d(uδρl) = jθ ·Rdθ (Heat balance) (62)
and for the vapour boundary-layer:
1
r
∂U
∂θ+∂V
∂r= 0 (Continuity) (63)
U
r
∂U
∂θ+ V
∂V
∂r= νv
∂2U
∂r2+
2U2∞r
sin(2θ) (Momentum) (64)
with the boundary conditions:u(r = 0) = 0 (no slip boundary)
µl
(∂u
∂r
)δ
= µv
(∂U
∂r
)δ
(shear stress at vapour-liquid interface)
ρl
(u
R
∂δ
∂θ− v)δ
= ρv
(U
R
∂δ
∂θ− V
)δ
(mass transfer at vapour-liquid interface)
For convenience of solving, the authors decided to neglect the pressure term in (61). They solved285
the sets of equation with both analytical and numerical tools. The analytical part shed light on new
dimensionless numbers: the ρµ-ratio R and the condensation number H. Then, by means of Runge-
Kutta-Gill method and for different values of R, H and Fr, the equations were solved.
Then they correlated the obtained results for high vapour flow. For F −→ 0, a dependence of Nu
to RH was highlighted as shown in Figure 10. For large values of RH, the Nusselt number approach290
20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Shekriladze & Gomelauri solution (53) (— · line), while for small values of RH, the authors correlated
the following Nusselt number (– – line):
NuSH = 0.90 (RH)−1/3 Re1/2
(65)
A general solution for all RH was obtained using an asymptotic model of the form:
(Nu31 +Nu32)1/3 (66)
Introducing the coefficient χ:
χ = 0.90
(1 +
1
RH
)1/3
(67)
The obtained Nusselt number is:295
NuSH = χRe1/2
(68)
The results for small oncoming vapour velocity tended to Nusselt’s equation (12) with the original
coefficient of 0.725. Then to connect NuSH (68) and NuGR (12), another asymptotic model was used,
namely:
Nu =(Nu4SH +Nu4GR
)1/4(69)
which expressed as a function of F :
Nu =(χ4 + 0.276F
)1/4Re
1/2(70)
Fujii et al. confronted this correlation to experimental results of condensation of steam on a single300
horizontal tube [1] and on banks of horizontal tubes [49]. They noticed a fair agreement, except for the
case of the tube bank with in-line arrangement, where heat transfer was about 20% lower than expected.
This discrepancy may be justified by the specific flow pattern. Therefore, the authors added a coefficient
that equals either 1 or 0.8.
However, Fujii and coworkers made the assumption that the vapour flow outside the boundary-layer305
was a potential one, which causes(∂U∂r
)δ
to always be positive, therefore preventing the boundary-layer
from separating.
Their solution was conservatively modified by Lee & Rose [50] from UK, whose analysis neglects heat
transfer behind the calculated separation point. Though little information is available on their method,
21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the same Lee mentioned in his PhD thesis [51] the following equation, which is due to Prandtl:310
−vδU∞
√Rev = 4.36
√− cos(θc) (71)
where θc > π/2 is the critical angle at which separation occurs.
Then, using this angle, they correlated again the value of NuSH and obtained the following coeffi-
cient χ:
χ = 0.88
(1 +
0.74
RH
)1/3
(72)
Concerning NuGR, the Nusselt equation was used with the more precise coefficient 0.728, hence the
mean Nusselt number:315
Nu =(χ4 + 0.281F
)1/4Re
1/2(73)
Equations (70) and (73) are compared in Figure 11 for different values of RH. They both approach
Nusselt solution for F −→ ∞ for all values of RH. Moreover, the formulation of Fujii et al. coincides
with the formulation of Shekriladze & Gomelauri for RH −→∞.
4.4 Approach of Rose (1984)
In all previous studies, the pressure gradient within the condensate film, arising from the vapour flow320
around the tube, has been omitted in the momentum equation. The effect of this pressure gradient
upon the condensate film has been studied by Rose [52], whose analysis was based upon the same
assumptions as Shekriladze & Gomelauri [46], except he took into account the body force and pressure
gradient within the condensate film. Though the asymptotic value of the surface shear stress was known
to be inaccurate, it was adopted owing to its simplicity and the fact that Rose’s aim was to investigate325
the effect of the pressure term. The following set of equations was solved:
µl∂2u
∂r2+ ρlg · sin(θ)− 1
R
dp
dθ= 0 (Momentum) (74)
d(uδρl) = jθ ·Rdθ (Heat balance) (75)
22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
with the boundary-layer conditions:u(r = 0) = 0 (no slip boundary)
µl
(∂u
∂r
)δ
= τδ (shear stress at vapour-liquid interface)
Equation (74) may be integrated to give:
u(r, θ) =1
µl
[τδr −
(ρlg · sin(θ)− 1
R
dp
dθ
)(r2
2− δr
)](76)
Assuming a potential flow outside the vapour boundary-layer yields:
τδ = 2jθU∞ sin(θ) (77)
330
dp
dθ= −2ρvU
2∞ sin(2θ) (78)
Injecting (77) and (78) into (76) gives:
u(r, θ) =1
µl
[2jθU∞ sin(θ)r −
(ρlg · sin(θ) +
2ρvU2∞
Rsin(2θ)
)(r2
2− δr
)](79)
Then the mean velocity over the thickness of the film:
u =
(ρlg · sin(θ) +
2ρvU2∞
Rsin(2θ)
)δ2
3µl+jθU∞δ
µlsin(θ) (80)
Injecting the expression of u and the definition of jθ in (75):
1
δ
λl∆T
∆hv=
ρlµlR
d
dθ
[(ρlg · sin(θ) +
2ρvU2∞
Rsin(2θ)
)δ3
3+λl∆TU∞∆hv
δ sin(θ)
](81)
Two dimensionless groups can be formed: the dimensionless film thickness δ∗
δ∗ = δ
(U∞Rνl
)1/2
(82)
23
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
and the dimensionless parameter P which relates to the inclusion of the pressure gradient335
P =ρv∆hvνlλl∆T
(83)
=ρvρlH−1 (84)
=ρvρl· PrJa
(85)
Re-arranging (81):
1
δ∗=
d
dθ
[(F
2· sin(θ) + 2P · sin(2θ)
)δ∗3
3+ δ∗ sin(θ)
](86)
with the boundary condition due to the symmetry at the top of the tube:
dδ∗
dθ= 0 at θ = 0 (87)
The last term inside the brackets in (86) results from the shear stress at vapour-liquid interface,
while the second one relates to the inclusion of the pressure gradient in the analysis. If both of these
terms are omitted, then equation (86) reduces to the simple equation (7) from Nusselt [9].340
Then, Rose considered the possibility that the velocity gradient at the wall may be nil or negative,
which would lead to a separation of the condensate boundary-layer. Therefore the surface shear stress
upstream this separation point is negative:
(∂u
∂r
)r=0
≤ 0 (88)
which becomes:
1 +δ∗2
2
(F
2+ 4P · cos(θc)
)≤ 0 (89)
This inequality is satisfied for θ > π/2 and P > F/8, which means that a solution of (86) only345
exists for θ ∈ [0; θc], unless θc = π, in which case the equation may be solved over the whole tube.
Then the authors numerically solved (86) for F = 103, 102, 10, 1, 10−1, 10−2, 10−3, 0 and for P =
0, 0.001, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10.
The present authors have solved this differential equation (86) using Runge-Kutta fourth-order
method.350
24
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4.4.1 P = 0
In this case, the pressure gradient is omitted, which corresponds to the analysis of Shekriladze & Gome-
lauri [46]. Rose correlated his results and obtained:
Nu =0.9(1 + (RH)−1)1/3 + 0.728F 1/2
(1 + 3.44F 1/2 + F )1/4Re
1/2(90)
4.4.2 0 < P ≤ F/8
When the pressure gradient is taken into account (i.e. P > 0), Figure 12 shows that the condensate355
film is thinner on the upper part (θ < π/2) while it is thicker on the lower half. Rose noted only small
differences for the mean Nusselt number for P = 0 and for P > 0. Even for the limiting cases were
P was close to F/8, the maximum discrepancy was around 5%. Therefore, Rose advocated the use of
equation (90).
4.4.3 P > F/8360
In this case, no solution can be obtained beyond the separation point (i.e. beyond θc). As shown in
Figure 13 and Figure 14, the closer is θc from π/2, the thinner is the condensate film on the upper part.
The author’s suggestion was to obtained an accurate solution over the upper part of the tube, where no
boundary-layer separation can occur, and to neglect heat transfer over the lower half, thus obtaining a
conservative solution.365
Let Nuπ/2 be the mean Nusselt number for the upper half of the tube. On one hand, when the
pressure gradient is omitted (i.e. P = 0), Nuπ/2 is given by:
Nuπ/2,P=0 =1.273 + 0.866F 1/2
(1 + 3.51F 0.53 + F )1/4Re
1/2(91)
On the other hand, when it is included, for the case of large oncoming vapour velocity (i.e. F −→ 0),
Nuπ/2 is given by:
Nuπ/2,P 6=0 = 1.273(1 + 1.81P )0.209(1 + (RH)−1)1/3Re1/2
(92)
Combining equations (91) and (92), for the upper half of the tube:370
Nuπ/2 =1.273(1 + 1.81P )0.209(1 + (RH)−1)1/3 + 0.866F 1/2
(1 + 3.51F 0.53 + F )1/4Re
1/2(93)
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Finally, equation (93) is then applied to whole surface of the tube, without considering any other
heat transfer. Thus the mean Nusselt number should be divided by 2, but in order to maintain the
convergence towards Nusselt’s solution for F −→∞, the coefficient before F at the numerator is taken
equal to 0.728:
Nu =0.636(1 + 1.81P )0.209(1 + (RH)−1)1/3 + 0.728F 1/2
(1 + 3.51F 0.53 + F )1/4Re
1/2(94)
Rose concluded his article by stating that the pressure gradient effects should be more notable for low375
vapour flow for refrigerants, due to their higher vapour density, than for steam at equivalent operating
conditions. However, the pressure gradient would have a significant effect on steam condensation at
high pressures.
Figure (15) shows the pressure gradient effect on Nu/Re1/2
for different values of P . The noticeable
discontinuities in Rose curves (dotted lines) are located at F = 8P , and are due to the switch from380
equation (90) to (94).
In the same paper, Rose confronted his analytical results against experimental ones obtained from
the literature for steam, R113 and R21. The results were in really good agreement, which supports the
use these formulations.
4.5 Approach of Homescu & Panday (1999)385
In previous cited literature, both vapour and liquid phases were solved assuming laminar flows. In 1999,
Homescu & Panday [53] studied the influence of turbulence in the case of forced convection condensation
on a horizontal tube. They also retained in their analysis the pressure gradient, inertia and enthalpy
convection terms. Lacking information on local flow structure, the authors assumed that the flow is
turbulent all around the tube. Using the above mentioned assumptions, they obtained the following set390
of equations for the condensate film, in the coordinate system defined in Figure 6:
∂u
∂x+∂v
∂y= 0 (Continuity) (95)
ρl
(u∂u
∂x+ v
∂u
∂y
)= g · sin θ − ∂p
∂x+
∂
∂y
[(µ+ µt)l
∂u
∂y
](Momentum) (96)
ρlcp
(u∂T
∂x+ v
∂T
∂y
)=
∂
∂y
[(k + kt)l
∂T
∂y
](Energy) (97)
26
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
and for the vapour boundary-layer:
∂U
∂x+∂V
∂y= 0 (Continuity) (98)
ρv
(U∂U
∂x+ V
∂U
∂y
)= −∂p
∂x+
∂
∂y
[(µ+ µt)v
∂U
∂y
](Momentum) (99)
(100)
with the boundary conditions:
u(y = 0) = 0 (no slip boundary)
T (y = 0) = Tw (isothermal wall)
u(y = δ) = U(y = δ) (no slip interface)
T (y = δ) = Tsat (thermal continuity)
(µ+ µt)l
(∂u
∂y
)δ
= (µ+ µt)v
(∂U
∂y
)δ
(shear stress at vapour-liquid interface)
ρl
(u∂δ
∂x− v)δ
= ρv
(U∂δ
∂x− V
)δ
(momentum transfer at vapour-liquid inter-
face)
Regarding the turbulence modelling, the authors used the mixing length concept, where µt is defined
by:395
µt = ρLm
∣∣∣∣∂u∂y∣∣∣∣ (101)
where Lm is the mixing length.
The authors tested several combinations of models and concluded that the combination of Pletcher’s
model for the vapour phase and Kato’s model best fitted the experimental results. These models are
well enough described in the article [53].
Then, they numerically solved this set of equations using a finite difference scheme. To represent400
their numerical results, the following equation was obtained by the authors:
Nut = 0.291
[0.75
(1 + (RH)−1
)1/3+ 0.25F 1/4 +
(1 + 0.8F )1/2
(0.25F 1/2 + 1.75F )1/4
]3/2Re
1/2(102)
This Nusselt number describes both the cases of laminar and turbulent condensation. Its evolution
is shown in Figure 16 along with Fujii et al. formulation. Unlike any other correlation, this one does
not approach Nusselt’s curve for F −→ ∞. It approaches the curve 0.173F 3/8 instead of 0.728F 1/4.
27
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Therefore, this correlation should not be used for F > 10 in the current formulation. An asymptotic405
model could be considered to correct it.
4.6 Vapour flow inclination effect
In 1974, Honda & Fujii [54] studied analytically the case of inclined vapour flow within a plane orthogonal
to the axis of the tube. As described in Figure 17, the oncoming vapour velocity direction has a angle
φ with respect to the vertical. Based on the asymptotic shear stress of Shekriladze & Gomelauri [46],410
the authors considered the following set of equations:
µl∂2u
∂r2+ ρlg · sin(θ) = 0 (Momentum) (103)
d(uδρl) = jθ ·Rdθ (Heat balance) (104)
with the boundary-layer conditions:u(r = 0) = 0 (no slip boundary)
µl
(∂u
∂r
)δ
= τδ (shear stress at vapour-liquid interface)
Then they numerically solved the obtained differential equation for several values of F . Finally, they
concluded that the average Nusselt number is slightly affected by the vapour flow orientation. Only the
case of upward oncoming vapour velocity (φ > 5π/6) and F about unity has a significant effect on the415
Nusselt number, since the condensate tend to flood the lower part of the tube.
4.7 Surface temperature variation
Carrying on with their previous work (see 3.4), Memory et al. [55] applied the same method in order
to observe the effect of temperature distribution in the case of flowing vapour. For large oncoming
velocities, their approach is based on Shekriladze & Gomelauri method [46]. They solved the differential420
equation for several values of A (see (35)) and obtained the mean Nusselt numbers listed in Table 2.
For A = 0 (i.e. isothermal case), the original constant of Shekriladze & Gomelauri is obtained, but for
increasing values of A, the heat transfer coefficient is slightly improved.
28
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A NuRe−1/2
0.0 0.9000.2 0.9060.4 0.9240.6 0.9530.8 0.9921.0 1.040
Table 2: Dependence of mean Nusselt number on a forced convection film condensation
4.8 Synthesis
This section sheds light on the most prominent publications related to condensation under shear stress425
and provides a better understanding of physical phenomena taken into account in each work. The vapour
flow tends to thin the condensate film and therefore decrease the thermal resistance of this film. From
this review, it would seem appropriate to use the formulation of Rose [52] (equations (90) and (94)) for
it is the most complete approach, considering the phenomena taken into account, and the fact that it
remains valid for stagnant vapour. However, it should be noted that it results from an interpolation,430
and therefore should not be used if parameters are out of the initially considered domain (F < 103 and
P < 10). Nevertheless, this domain may be expanded by solving the differential equation for the range
of interest.
As the vapour flows deeper inside a tube bundle, its velocity decreases, which decreases the con-
densation rate. Therefore, the tubes on the outer layer of the bundle have the best condensation rate,435
which is something a condenser designer will benefit by modifying the tubes layout. This is known as
geometry effects.
5 Condensation on a bundle of tubes
As soon as a condenser contains several tubes in a vertical bank, the inundation phenomenon must
be accounted for. It is encountered in every single industrial condenser with horizontal tubes. This440
phenomenon is quite simple, since it only results from condensate streaming down from the bottom of
a tube to the top of the tube beneath. The condensate is mostly set in motion by gravity and therefore
keeps flooding the tubes up until it reaches the condenser well at the bottom of the device. Then the
surrounding condensate film of flooded tubes is thicker and the thermal resistance is increased. This
also leads to a higher subcooling of the condensate, which also decreases the global performance of445
29
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the device. Thus inundation is an undesired phenomenon, that must be dealt with when designing a
condenser.
Moreover, this phenomenon depends on many factors, such as the tube bundle geometry, the tube
layout and the spacing. The more tubes a vertical bank contains, the more flooded are the lower tubes
(see Figure 18(a)). Besides, if the tubes are staggered, then the condensate may not fall directly on the450
tube beneath as shown in Figure 18(b), but on an intermediate tube, which is named lateral drainage.
When the flow rate is high enough, the drainage splashes on the top of the tube (see Figure 18(c)),
and therefore modifies the thickness of the film, which becomes ”wavy”. This thickness distribution
actually gives a better heat transfer coefficient than an even distribution, which partially compensates
the thickening of the film.455
5.1 Correlations based on row number
The first analysis of inundation found in literature dates back from 1949 with Jakob [56]. Starting from
Nusselt’s theory [9], he adapted the analysis for a vertical bank of isothermal tubes. Let γn be the local
condensation mass rate (on the n-th tube) and Γn be the global condensation mass rate (over the n
tubes) defined as:460
γn =πDohn∆T
∆hv(105)
Γn =nπDohn∆T
∆hv(106)
Equation (12) may be written as a function of γ1, which is the condensation mass rate of the top
tube (or first tube row):
h1λl
[µ2l
ρ2l g
]1/3= 1.523
(4γ1µl
)−1/3(107)
This expression is sometimes used to define a relation between a Nusselt number (left side of the
equation) and the condensate Reynolds number (right side of the equation) [43].465
Assuming that the condensate film falls vertically from one tube to another as a continuous sheet
(see Figure 19(a)), then the mean heat transfer coefficient over n tubes is:
hnλl
[µ2l
ρ2l g
]1/3= 1.523
(4Γnµl
)−1/3(108)
Then, the inundation factor may be obtained. It represents the decrease of heat transfer due to
30
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inundation:hnh1
= n−1/4 (109)
and for the n-th tube:470
hnh1
= n3/4 − (n− 1)3/4 (110)
Thus, under Jakob assumptions, the heat transfer coefficient decreases with the fourth root of 1/n.
For example, for a vertical bank of 16 tubes, the mean heat transfer coefficient is only 50% of the heat
transfer coefficient of the top tube. This formulation is sometimes associated to Nusselt himself, though
nothing seems to prove that he made these calculations.
According to Kern [57], the condensate is more likely to fall as droplets or columns, rather than as475
sheets. Such a drainage improves the splashing, and therefore the heat transfer coefficient. The author
suggested to replace n by n2/3 in equation (109):
hnh1
= n−1/6 (111)
and for the n-th tube:hnh1
= n5/6 − (n− 1)5/6 (112)
This formulation is often cited for the design of condensers for it gives pertinent results, while Jakob
formulation is too conservative. These two formulations are meant for vertical banks of tubes, where480
tubes are numbered as shown in Figures 20(a) and 20(b) for inline arrangements. When considering a
staggered arrangement, tubes should be numbered as shown in Figure 20(c).
However, in numerous tube bundles with a triangular layout, lateral drainage may be observed
depending on the pitch to diameter ratio p/D. This phenomenon has been studied by Eissenberg &
Noritake [58] in 1970. When pure lateral drainage is occurring, as shown in Figure 21, the authors485
obtained the following theoretical result:
(hnh1
)lat
= 0.6 + 0.4353n−1/5 (113)
Though no explanation could be found about how this expression was obtained, the present authors
obtained similar results with the following calculations. Let Nu[0;π/2] and Nu[π/2;π] be respectfully the
mean Nusselt numbers over the upper half and the lower half obtained from (10) with different intervals
31
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of integration:490
Nu[0;π/2] = 0.866F 1/4 (114)
Nu[π/2;π] = 0.590F 1/4 (115)
Assuming that the condensate drains over the following tube at θ = π/2, then the Nusselt number
for the upper half of this second tube remains unchanged, while the Nusselt number for the lower half
decreases due to inundation, thus:
NunNu1
=1
2× 0.866F 1/4
0.728F 1/4+
1
2× 0.590F 1/4 n−1/5
0.728F 1/4(116)
= 0.5949 + 0.4052n−1/5 (117)
which is close to (113), especially when considering that Eissenberg & Noritake used the original con-495
stants in their calculations instead of the more precise ones. The coefficient 1/5 for the inundation will
be discussed later.
However, according to Figure 20(c), n should actually be n/2 since the condensate drains over an
intermediate tube: (hnh1
)lat
= 0.5949 + 0.4655n−1/5 (118)
which is still close to (113).500
Finally, it was assumed that both left and right halves of the tube were flooded from π/2 to π. It
would seem more appropriate to consider that the pattern shown in Figure 21 repeats itself for every
column. In this case, only a quarter of the tube is flooded, hence the following inundation factor:
(hnh1
)lat
= 0.7974 + 0.2327n−1/5 (119)
For these formulations, it should noted that the inundation factors is not equal to 1 for n = 1 and
should therefore be set to this value for the first row.505
Then Eissenberg & Noritake suggested that a classical vertical drainage would give the following
inundation factor: (hnh1
)vert
= n−1/5 (120)
This expression is more conservative than Kern’s, but less than Jakob’s. Though they claimed it was
obtained experimentally, no data could be found. However, the inundation factor used in (113) probably
32
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originated from here.510
Then the authors defined a spacing parameter Fd, which accounts for the tube layout and ratio
pt/Do. Though brief, this parameter is defined in Table 3. Then, this parameter will take into account
the proportion of lateral drainage and vertical drainage, using a convex combination:
hnh1
= Fd
(hnh1
)lat
+ (1− Fd)(hnh1
)vert
(121)
= 0.6Fd + (1− 0.5647Fd) · n−1/5 (122)
Tube layout pt/Do Fd
Inline - 0
≥ 1.40 0
Staggered = 1.33 0.5
≤ 1.25 1
Table 3: Tube spacing parameter Fd definition
The local inundation factor is:
hnh1
= 0.6Fd + (1− 0.5647Fd)(n4/5 − (n− 1)4/5
)(123)
According to equation (121), pure lateral drainage should be encountered for pt/Do ≤ 1.25 and pure515
vertical drainage for pt/Do ≥ 1.40. In between, a combination of both inundation modes should be
observed.
Figure 22 shows the local inundation factors for the above mentioned correlations over 40 tubes. For
every correlation, a sharp decrease is noticeable over the first five tubes.
Finally, Short & Brown [59] have obtained experimental results with steam and Freon-11 on a vertical520
bank of 20 tubes. Their results tend to prove that inundation is negligible due to the counteracting
mixing action of the drainage. They concluded that ”The average condensate film heat transfer coefficient
for a bank of twenty tubes is, to a good approximation, very nearly equal to the value predicted by the
Nusselt theory for the top tube in the bank”. Butterworth [60] [43] had a different interpretation of these
results and claimed that the following inundation factor could be deduced:525
hnh1
= 1.24n−1/4 (124)
33
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and for the n-th tube:hnh1
= 1.24(n4/5 − (n− 1)4/5
)(125)
However, this correlation should not be used in any calculation considering its confused origin.
Chen [61] performed an analytical study of laminar film condensation over a vertical bank of tubes.
He took into account the additional condensation between tubes and the effect of heat capacity:
hnh1
= n−1/4 [1 + 0.2 Ja(n− 1)]
(1 + 0.68 Ja+ 0.02 JaH
1 + 0.95H − 0.15 JaH
)1/4
(126)
Though often mentioned, this work is never used for condenser design. Chen also noticed that the tem-530
perature difference could modify the inundation factor. He concluded that the smaller the temperature
difference, the more important was the inundation. Asbik et al. [62] later adapted Chen’s developments.
More recently, Murase et al. [63] obtained experimental results shown in Figure 23. They are close
to Kern correlation and the present authors propose the following correlation from these results:
hnh1
= n−1/7 (127)
and for the n-th tube:535
hnh1
= n6/7 − (n− 1)6/7 (128)
In 2012, Ma et al. [64] developed an experimental procedure to artificially obtain the inundation
factor of a vertical bank of tubes using an actually smaller one. Though no correlation was produced,
it appears in Figure 24 that the experimental results lie above Kern line, which means it is slightly
conservative.
5.2 Correlations based on mass rate540
If previous correlations are based on the tube row number, which corresponds to the position of the
tube within the bundle, the following ones are based on the condensate mass rate received by each tube.
The latter are therefore based on experimental results and are expressed as:
hnh1
=
(Γnγn
)−s(129)
where Γn is the condensation mass rate of the first n tubes, γn is the condensation mass rate of the
34
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n-th tube and s a coefficient. It is important to notice that Γn is the condensate mass rate draining from545
the n-th tube, and not the condensate mass rate draining onto the n-th tube. This way, the inundation
factor equals 1 for n = 1. Γn is therefore defined as:
Γn =n∑i=1
γi (130)
From experimental measurements performed on a rectangular bundle of 72 tubes divided into 11
rows with steam condensation, Fuks [65] obtained a value of s = 0.07. According to Bontemps [66],
Fuks did not manage to separate shear stress effects from inundation effects, therefore resulting in a low550
inundation factor.
In 1968, Grant & Osment [67] performed an experimentation on an oval bundle of 139 staggered
tubes with p/Do = 1.5 with low pressure steam. Using Fuks formulation, they obtained a coefficient
s = 0.223. This formulation is commonly found in condensation literature.
The formulation of Wilson [68] is sometimes mentioned with a coefficient s = 0.16, which was555
obtained by calculation to fit experimental data.
More recently, Hu & Zhang [69] have proposed a new inundation correlation with a variable coefficient
s ranging from 0 at the top of the tube bundle to 0.37 at the bottom of it. According to the authors,
the nil value at the top ensures an inundation factor equal to 1, which is frivolous since the ratio Γ1/γ1
already equals 1. Regarding the value 0.37, it was obtain from the fitting of experimental results.560
5.3 Synthesis
Using equations (105) and (106), the present authors have managed to compare both kinds of formu-
lation. An iterative procedure was performed to calculate the mass rate of a vertical bank of tubes,
in order to obtain the inundation factor of the n-th tube. Results are presented in Figure 25 with the
inundation factors of Grant & Osment and Fuks.565
The sharp decrease is also noticeable in the Grant & Osment dashed line. It is of great importance
to notice that both the Kern formulation and the Grant & Osment formulation give similar results,
since they are often recommended in condenser design literature. Therefore, we would recommend to
use either of these two formulations : equations (111) and (112) for Kern or (129) with s = 0.223 for
Grant & Osment.570
If one is looking for the average inundation factor for a whole condenser, we would advise to calculate
35
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the average inundation factor for each column, and then to calculate the mean value over the columns
weighted by the number of tubes in each column.
6 Overall synthesis
In the case of industrial condensers, tube bundles may contains from hundreds to several thousands of575
tubes which leads to an important condensate flow. Besides, considering that each tube acts as a sink
for the vapour phase, the more tubes, the higher the vapour mass flow rate within the tube bundle.
Such a vapour flow is most likely to disturb the condensate as shown in Figure 26, making it modelling
quite impossible.
If the vapour shear stress and the condensate inundation phenomena are both well enough described580
when not occuring concomitantly, current literature is of little help when they do. The most frequent
method consists of choosing a heat transfer coefficient for a single tube, and then to multiply it by an
inundation factor.
Regarding the heat transfer coefficient on a single tube, the most appropriate correlation advised by
the present authors is the one from Rose [52]. Indeed, it takes into account several phenomena without585
being too conservative. However, the designer should be careful with this correlation, since it has been
obtained for P ≤ 10. Were P be out of this range, the correlation of Fujii et al. [1] should be preferred.
As for the inundation factor, it appears that both the Kern equation [57] and Grant & Osment
[67] are often used in condenser design. They give similar results, really close to experimental results
being only slightly conservative. One is based on the tube row number while the other is based on the590
condensate mass flow rate.
However, if the purpose of the modelling is to safely provide a thermal design of the condenser,
one should use more conservative correlations for both the heat transfer coefficient and the inundation
factor.
Once the local phenomena occurring near a tube in the bundle are acquired, the designer then needs595
to figure out the tube bundle layout and how it will affect the vapour flow. This is a whole other matter
that is rarely discussed by the industrials, since it is a valuable knowledge.
36
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tube. International Journal of Heat and Mass Transfer, 27(1):39–47, 1984.
[53] Daniel Homescu and Prabodh Kumar Panday. Forced convection condensation on a horizontal tube:725
Influence of turbulence in the vapor and liquid phases. Journal of Heat Transfer, 121:874–885, 11
1999.
41
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[54] Hiroshi Honda and Tetsu Fujii. Effect of the direction of on-coming vapor on laminar filmwise con-
densation on a horizontal cylinder. In Proceedings of the 5th International Heat Transfer Conference
in Tokyo, pages 235–246, 1974.730
[55] Stephen B. Memory, Wah Cheng Lee, and John W. Rose. Forced convection film condensation on
a horizontal tube - effect of surface temperature variation. International Journal of Heat and Mass
Transfer, 36(6):1671–1676, 1993.
[56] Max Jakob. Heat Transfer, chapter Theory of Film Condensation of Vapor at Rest on Cylindric
Surfaces, pages 667–673. John Wiley & Sons, 1949.735
[57] Donald Quentin Kern. Process Heat Transfer, chapter Condensation of Single Vapors - Development
of Equation for Calculations, pages 263–268. McGraw-Hill Book Company, 1950.
[58] David Martin Eissenberg and H. M. Noritake. Computer model and correlations for prediction of
horizontal-tube condenser performance in sea water distillation plants. Technical report, Oak Ridge
National Laboratory, 10 1970.740
[59] Byron Elliott Short and Howard E. Brown. Condensation of vapors on vertical banks of horizontal
tubes. In Proceedings of the General Discussion on Heat Transfer, pages 27–31. Institution of
Mechanical Engineers, 9 1951.
[60] David Butterworth. Power Condenser Heat Transfer Technology: Computer Modeling, Design,
Fouling, chapter Inundation without Vapour Shear, pages 271–277. Hemisphere Publishing, 1981.745
[61] Michael Ming Chen. An analytical study of laminar film condensation: Part 2 - single and multiple
horizontal tubes. Journal of Heat Transfer, 83:55–60, 2 1961.
[62] Mohamed Asbik, Abdallah Daıf, Prabodh Kumar Panday, and Ahmed Khmou. Numerical study
of laminar condensation of downward flowing vapour on a single horizontal cylinder or a bank of
tubes. The Canadian Journal of Chemical Engineering, 77:54–61, 1999.750
[63] T. Murase, Hua Sheng Wang, and John W. Rose. Effect of inundation for condensation of steam on
smooth and enhanced condenser tubes. International Journal of Heat and Mass Transfer, 49:3180–
3189, 2006.
42
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[64] ZhiXian Ma, JiLi Zhang, and DeXing Sun. New experimental method for inundation effect of film
condensation on horizontal tube bundle. Science China, 55(10):2856–2863, 2012.755
[65] S. N. Fuks. Heat transfer with condensation of steam flowing in a horizontal tube bundle. Thermal
Engineering (Teploenergetika), 4:35–39, 1957.
[66] Andre Bontemps. Condensation en film d’une vapeur pure dans un faisceau de tubes lisses hori-
zontaux. Technical report, Commissariat a l’Energie Atomique - Grenoble, 9 1998.
[67] Ian Douglas Raffan Grant and Bernard David John Osment. The effect of condensate drainage760
on condenser performance. Technical Report 350, National Engineering Laboratory, East Kilbride,
Glasgow, 1968.
[68] J. L. Wilson. The design of condensers by digital computers. In Decision Design and the Computer,
35, pages 21–27. Institution of Mechanical Engineers, 1972.
[69] Hong Gang Hu and Chao Zhang. A new inundation correlation for the prediction of heat transfer765
in steam condensers. Numerical Heat Transfer, Part A - Applications, 54(1):34–46, 2008.
Figure 1: Different kinds of vapour and condensate flow in a shell-and-tube condenser
43
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 2: Surface mass rate balance over a portion of condensate
Figure 3: Qualitative profile of condensate film temperature
Figure 4: Heat and mass transfer within a portion of condensate film
44
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 5: Crossflow within the condensate film
Figure 6: Coordinate system used by Sparrow & Gregg [38]
Figure 7: Tube under vapour flow
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 8: Separation point and recirculation flow
Figure 9: Evolution of Nu/Re1/2
for (12), (56) and (59)
46
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 10: Average Nusselt number for large oncoming vapour velocity
Figure 11: Evolution of Nu/Re1/2
for (70) and (73) for different values of RH
47
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 12: Calculated thickness of the condensate film
48
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 13: Calculated thickness of the condensate film
49
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 14: Calculated thickness of the condensate film
Figure 15: Evolution of Nu/Re1/2
for (70), (90) and (94) for different values of P
50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 16: Evolution of Nu/Re1/2
for (70) and (102) for different values of RH
Figure 17: Inclination of vapour flow
51
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 18: Different kinds of inundation (a) inline laminar, (b) triangular laminar, (c) inline turbulent(Bontemps, 1998 [66])
Figure 19: Condensation on vertical bank of tubes (a) continuous sheet, (b) droplets (Butter-worth, 1983 [43])
Figure 20: Tube numbering (a) single column, (b) inline layout, (c) triangular layout (Bon-temps, 1998 [66])
52
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 21: Pure lateral drainage (Eissenberg & Noritake, 1970 [58])
Figure 22: Local inundation factor
53
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 23: Local inundation factor
Figure 24: Local inundation factor for different fluids (Ma et al., 2012 [64])
54
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 25: Local inundation factor with mass rate based correlations
Figure 26: Condensate flow under lateral vapour flow (Bontemps, 1998 [66])
55
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horizontal tube: Influence of turbulence in the vapor and liquid phases. Journal of Heat
Transfer, 121:874-885, 11, 1999.
[54] Hiroshi Honda and Tetsu Fujii. Effect of the direction of on-coming vapor on laminar
filmwise condensation on a horizontal cylinder. In Proceedings of the 5th International Heat
Transfer Conference in Tokyo, pages 235-246, 1974.
[55] Stephen B. Memory, Wah Cheng Lee, and John W. Rose. Forced convection film
condensation on a horizontal tube - effect of surface temperature variation. International
Journal of Heat and Mass Transfer, 36(6):1671-1676, 1993.
[56] Max Jakob. Heat Transfer, chapter Theory of Film Condensation of Vapor at Rest on
Cylindric Surfaces, pages 667-673. John Wiley & Sons, 1949.
[57] Donald Quentin Kern. Process Heat Transfer, chapter Condensation of Single Vapors –
Development of Equation for Calculations, pages 263-268. McGraw-Hill Book Company,
1950.
[58] David Martin Eissenberg and H. M. Noritake. Computer model and correlations for
prediction of horizontal-tube condenser performance in sea water distillation plants. Technical
report, Oak Ridge National Laboratory, 10, 1970.
[59] Byron Elliott Short and Howard E. Brown. Condensation of vapors on vertical banks of
horizontal tubes. In Proceedings of the General Discussion on Heat Transfer, pages 27-31.
Institution of Mechanical Engineers, 9 1951.
[60] David Butterworth. Power Condenser Heat Transfer Technology: Computer Modeling,
Design, Fouling, chapter Inundation without Vapour Shear, pages 271-277. Hemisphere
Publishing, 1981.
[61] Michael Ming Chen. An analytical study of laminar film condensation: Part 2 - single
and multiple horizontal tubes. Journal of Heat Transfer, 83:55-60, 2 1961.
[62] Mohamed Asbik, Abdallah Daïf, Prabodh Kumar Panday, and Ahmed Khmou.
Numerical study of laminar condensation of downward flowing vapour on a single horizontal
cylinder or a bank of tubes. The Canadian Journal of Chemical Engineering, 77:54-61, 1999.
[63] T. Murase, Hua Sheng Wang, and John W. Rose. Effect of inundation for condensation
of steam on smooth and enhanced condenser tubes. International Journal of Heat and Mass
Transfer, 49:3180-3189, 2006.
[64] ZhiXian Ma, JiLi Zhang, and DeXing Sun. New experimental method for inundation
effect of film condensation on horizontal tube bundle. Science China, 55(10):2856-2863,
2012.
[65] S. N. Fuks. Heat transfer with condensation of steam flowing in a horizontal tube bundle.
Thermal Engineering (Teploenergetika), 4:35-39, 1957.
[66] André Bontemps. Condensation en film d'une vapeur pure dans un faisceau de tubes
lisses horizontaux. Technical report, Commissariat à l'Energie Atomique - Grenoble, 9 1998.
[67] Ian Douglas Raffan Grant and Bernard David John Osment. The effect of condensate
drainage on condenser performance. Technical Report 350, National Engineering Laboratory,
East Kilbride, Glasgow, 1968.
[68] J. L. Wilson. The design of condensers by digital computers. In Decision Design and the
Computer, 35, pages 21-27. Institution of Mechanical Engineers, 1972.
[69] Hong Gang Hu and Chao Zhang. A new inundation correlation for the prediction of heat
transfer in steam condensers. Numerical Heat Transfer, Part A - Applications, 54(1):34-46,
2008.