Modelling of Heat Transfer During Reflux Condensation inside Rectangular Channels and Experimental Verification

Abstract

Reflux condensation refers to the condensation process in which the vapor enters an inclined or
vertical condenser at the bottom and flows upward while the condensate flows downward due to
gravity. In this study the heat transfer during reflux condensation in a rectangular channel of a
hydraulic diameter of 7mm is modelled using the assumptions of the classical Nusselt theory for
laminar film condensation on a vertical plate. The model is validated with experimental results
for the refrigerant R134a. The film thickness distribution on the side walls of the channel is
calculated numerically. The calculated results show that low inclination angles enhance heat
transfer because of the smaller vertical flow length and hence smaller mean film thickness of
condensate. Contrary to the tube geometry with the same hydraulic diameter heat transfer
inside the rectangular channel with an inclination angle of 90◦ is underpredicted by the classical
Nusselt theory.

Full text

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MODELLING OF HEAT TRANSFER DURING REFLUX
CONDENSATION INSIDE RECTANGULAR CHANNELS AND
EXPERIMENTAL VERIFICATION
T. Klahm1, H. Auracher1, F. Ziegler1
1Berlin University of Technology, Institute for Energy Engineering, Germany
Abstract
Reflux condensation refers to the condensation process in which the vapor enters an inclined or
vertical condenser at the bottom and flows upward while the condensate flows downward due to
gravity. In this study the heat transfer during reflux condensation in a rectangular channel of a
hydraulic diameter of 7mm is modelled using the assumptions of the classical Nusselt theory for
laminar film condensation on a vertical plate. The model is validated with experimental results
for the refrigerant R134a. The film thickness distribution on the side walls of the channel is
calculated numerically. The calculated results show that low inclination angles enhance heat
transfer because of the smaller vertical flow length and hence smaller mean film thickness of
condensate. Contrary to the tube geometry with the same hydraulic diameter heat transfer
inside the rectangular channel with an inclination angle of 90◦is underpredicted by the classical
Nusselt theory.
Nomenclature
Only symbols are defined which are not included in the common list of symbols in Journal of
Heat Transfer (1999).
Symbols
B
P
U
Φ
∆
Superscript
+
Subscript
m
S
W
plate
side
top
width of the channel (m)
Point in coordinate system
Circumference of the channel (m)
Flow length(m)
Difference
Mean value
Saturation state
At the wall
Plane of the plate
Side wall
Top plate
dimensionless
1Introduction
Reflux condensation is a gravity controlled system. Vapor enters a vertical or inclined oriented
condenser at the bottom and flows upward. The condensate flows downward counter-currently
to the vapor due to gravity. Compact plate heat exchangers are increasingly used for reflux
condensation. The hydraulic diameter of flow channels formed between the plates is between
5 and 10 mm and they can be inclined. To study reflux condensation in small channels and
especially the effect of inclination Fiedler & Auracher (2004) carried out experiments with
R134a in a test tube of 7 mm inner diameter and 500 mm length as an idealised single sub-
channel of a compact condenser. Other studies on reflux condensation with vertical tubes
(Chen & Tien (1987); McNaught & Moore (1989)) and inclined tubes (McNaught & Moore
(1989); Gross (1987); Gross (1992)), respectively, are focused on tubes with larger diameters.
The reflux condensation in inclined closed two-phase thermosyphons is also a topic of research
work. Those studies are, however, mainly focused on the influence of the inclination angle on
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the maximum heat transport capability, i.e. on the maximum performance which is limited by
flooding, see e.g. Huanzhuo et al. (1997), Groll & Roesler (1992) and Chen & Tien (1987).
Fiedler & Auracher (2004) found that the inclination angle has a significant effect on heat
transfer. The optimum inclination angle from the horizontal for heat transfer was found to be
close to 45◦. Here the heat transfer for R134a is increased by a factor of nearly 2 compared to
a vertical arrangement. The reason is the nonuniform condensate film distribution around the
tube perimeter in the inclination case. The film is very thin and consequently the heat transfer
very good in the upper parts of the tube. Only at the bottom a thicker film exists where the
main part of the condensate is drained to the lower tube end. A model for the condensation in
an inclined tube supports this phenomenological explanation (Fiedler & Auracher (2004)). The
model was used to calculate the film thickness at the tube wall based on the classical Nusselt
theory (Nusselt (1916)) applied to the cylindrical geometry.
The former results for a circular tube let us assume that the positive effect of inclination on
heat transfer should be even better if a channel with a cross section is used where the ratio of
thin film area with respect to the area occupied by the film at the bottom is larger. In this
report the simple case of an inclined rectangular channel is studied where the wider walls are
vertically arranged. The same hydraulic diameter is used as by Fiedler & Auracher (2004).
2 Model approach
The rectangular channel is divided into three parts: The side walls (height H), the bottom plate
(width B) and the top plate (width B). Design-details of the test section are given in chapter
4.2. The channel is inclined (see figure 1) and the condensate that forms on the side walls flows
vertically down due to gravity. It is collected at the bottom plate of the channel. The film
thickness and the corresponding heat transfer resistance on the bottom plate is much higher
than on the side walls and the amount of heat transferred at the bottom plate is therefore
small. The top plate of the channel is idealized as an independent heat transfer surface. It is
assumed that the liquid that condenses on that plate does not mix with the liquid at the side
walls. Local heat transfer coefficients inside the channel at the side walls and at the top plate
are calculated. The heat transfer at the bottom plate is assumed to be neglegible.
Fiedler & Auracher (2004) showed that in reflux condensation the heat transfer inside an
inclined tube can be calculated utilizing the classical Nusselt theory. Consequently the same is
assumed for the rectangular channel. According to Nusselt the local heat transfer coefficient of
the film can be calculated with
h =k
δ
The thickness of the film (δ) is very small and there is no convective mixing inside it. The
interaction between vapor and the liquid film is negligible. Another assumption is that all fluid
properties are constant. In some cases where the change of a property with temperature is
large, temperature dependent fluid properties have to be used.
.(1)
According to Nusselt the film thickness is a function of flow length Φ
δ =
?4 · klµl(TS− TW)
ρl(ρl− ρg)g · ∆hlgΦ
?1/4
. (2)
Nusselt developed this equation using a vertical plate. The edge where the condensation
process started was horizontal and the liquid flow was parallel. Consequently the film thickness
along the width of the plate is constant. In the present case there are two vertical walls inside
the rectangular cross section each with height H (see figure 1). This length coincides with
the flow length Φmaxif the inclination angle is 0◦. In the case of 90◦inclination angle Φmax
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Figure 1: Definition of flow length and coordinate system
coincides with the length L of the channel and in both cases the film thickness is constant
along the corresponding width of the plate L and H, respectively. With an inclination angle
above zero two different areas of the plate exist. Even on an inclined plate the fluid flows down
vertically because it is driven by gravity. The development of the condensate film starts at the
topmost point on the plate. Figure 1 shows a coordinate system, which defines this edge as
P(z,x=0). Along this line the film thickness is zero. The flow length to the bottom on most of
the x0-area reaches the maximum flow length Φmax. The other part of the plate (z0-area) is
characterised by a condensate film starting to develop at the edge (z=0,x) but the flow length
to the bottom is smaller than the maximum length.
Using the coordinate system in figure 1, Φ is calculated in every point P(x,z). For the x0-area:
Φ(x+,z+) =x+· H
cosβ
for
?z+· L
sinβ
≥x+· H
cosβ
?
(3)
and in the z0-area by
Φ(x+,z+) =z+· L
sinβ
for
?z+· L
sinβ
<x+· H
cosβ
?
.(4)
The expressions in parenthesis denote the mathematical expression that was used to distinguish
between the x0 and the z0-areas. To generalise the results, x and z have been turned into
dimensionless parameters x+=
x
Hand z+=z
L.
Knowing Φ, the liquid film thickness across the whole plate is calculated from equation (2) and
the local heat transfer coefficient from equation (1). Calculating the integral mean of the local
values gives a mean heat transfer coefficient of the side wall hm,side. The mass flow on the side
wall in x direction ˙Mxand z direction ˙Mzis calculated by numerical integration of
d˙Mx=ρl(ρl− ρg)g · cosβ
3µl
δ3(x = H,z) · dz(5)
and
d˙Mz=ρl(ρl− ρg)g · sinβ
3µl
δ3(x,z = L) · dx,(6)
respectively. They are summarized to the overall mass flow of the side wall ˙Mside.
The plate on the top of the channel is modelled using equation (2) with Φtop= z. To accom-
modate to the case of an inclined plate the gravity constant was replaced by the component
that is in the plane of the plate:
gplate= g · sinβ(7)
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Using equation (7) and Φtop= z with equation (2) and (1) a local heat transfer coefficient was
calculated. The integral mean of the local values is given as
hm,top=4
3
?ρl(ρl− ρg)g · k3
4 · µl(TS− TW)
l∆hlg
sinβ
L
?1/4
.(8)
The mass flow on the top plate ˙Mtopis given by
˙Mtop=ρl(ρl− ρg)g · sinβ
3µl
δ3(z = L) · B(9)
The comparison with experimental values in section 5 was done using the integral mean of
hm,sideand hm,top:
hm= 2 · hm,side·Aside
with Aside= H ·L, Atop= B·L and A = 2·Aside+Atop. hmwas used to calculate Nu (equation
15). Re (equation 16) was calculated with the overall mass flow ˙Mlinside the channel which is
the sum of 2 ·˙Mside+˙Mtop.
A
+ hm,topAtop
A
(10)
3Model results
0
0.5
1
0
0.5
1
0
2
4
x 10
−5
z+
x+
Film thickness (m)
(a) β = 10◦
0
0.5
1
0
0.5
1
0
2
4
x 10
−5
z+
x+
Film thickness (m)
(b) β = 60◦
0
0.5
1
0
0.5
1
0
0.5
1
x 10
−4
z+
x+
Film thickness (m)
(c) β = 88◦
0
0.5
1
0
0.5
1
0
0.5
1
x 10
−4
z+
x+
Film thickness (m)
(d) β = 90◦
Figure 2: Film thickness distribution on one side wall of the rectangular channel, R134a at 0.7 MPa,
TS− TW= 3K
An example of the film thickness distribution on one side wall of the channel is depicted in figure
2. There are four diagrams that show the development of the film with changing inclination
angle, constant fluid properties and temperature difference. In figure 2a the inclination angle is
10◦to the horizontal. The film thickness distribution is almost the same as with a non inclined
surface with the upper edge at z+= 0. With increasing inclination angle this picture changes
not much. At 60◦it is apparent that the z0-area marked in figure 1 becomes visible. There is
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a steep increase of the film thickness in this area which flattens until the value of the x0-area
is reached. With inceasing inclination angles the z0-area grows. At an inclination angle of 88◦
both areas are almost equally sized. The z0-area increases rapidly with increasing angel and
finally it covers the whole plate at an inclination angle of 90◦.
It should be noted that the strong growth of the z0-area appears at different inclination angles
if the ratio of length and height of the side walls is different. Consider for example a channel
with square side walls. In this case the x0-area and z0-area are equally sized at an inclination
angle of 45◦. Inclination angle and aspect ratio of the sidewalls are related by
const =H
L· tanβ.(11)
Thus, any aspect ratios are included in the presented film thickness distribution. Only the
inclination angle has to be adjusted accordingly. Beside the contribution of different flow areas
figure 2 reveals an important information about the maximum film thickness.
figures 2a-d an increase of the maximum film thickness can be recognized from a to d. The film
thickness is related to the flow length which increases with inclination angle. In section 5 the
comparison with the experimental results will reveal that the decreased flow length at small
inclination angles is the main reason for the improvement of heat transfer in inclined passages.
Comparing
4 Experiments
4.1Setup
Details of the test loop are presented in Fiedler & Auracher (2004) and Klahm et al. (2007).
It consists of an evaporator where overheated vapor (temperature is kept approximately 1K
above saturation temperature) is produced, a plate heat exchanger that condenses residual
vapor which was not condensed inside the test section, a vessel to collect the liquid and control
the system pressure and a gear pump to charge the evaporator with the amount of liquid
mass flow that is needed inside the test section. The vapor mass flow rate was measured
with a Coriolis type flow meter(accuracy ±0.15%). The mass flow rate of the condensate is
determined by collecting the liquid for a known period of time (accuracy ±1%). Inlet and
outlet temperatures of the test section (see section 4.2) are determined with platine resistance
thermometers of 1 mm diameter (accuracy ±0.1◦C). The pressure difference in the test section
is determined by a differential pressure transducer with a range of 0-50 mbar and an accuracy
of ±0.25%. The pressure transducer that measures the system pressure is located at the inlet
of the test section.
4.2Test section
In figure 3a) a schematic of the rectangular test channel with the surrounding circular channel
for the cooling water is depicted. The length (L) is 500 mm and the cross sectional area 15mm x
4.6 mm (H x B) corresponding to a hydraulic diameter of 7 mm. This enables a comparison with
former results obtained with a circular test section (Fiedler & Auracher (2004)) of 7 mm inner
diameter shown in figure 3b). 6 thermocouples of type K (Ni, Cr-Ni) with 0.25 mm diameter
are installed in each of 5 cross sectional areas along the rectangular test channel (figure 3a)
which consists of two parts, sealed in the contact area. The distance of the thermocouple tips
to the inner wall is 0.25 mm. The leads are silver soldered into axial grooves along a distance
of 5 mm and then taken out through the water flow channel and the outer wall where they
are sealed to the outside. The cooling water flows countercurrently to the vapor. It enters and
leaves the outer channel via 4 tubes on each side. Its flow rate is determined with a calibrated
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? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ??
? ? ? ? ? ? ? ?
? ? ? ? ? ? ??
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ??
? ? ? ? ? ? ? ?
? ? ? ? ? ? ??
? ?? ?
Figure 3: Test section, a) Rectangular channel, b) Circular channel
impeller flow meter (accuracy ±2,5%). To enable a variation of the inclination angle, the test
section is connected by flexible tubes.
4.3Experimental procedure and data reduction
Refrigerant R134a is used as test fluid. The pressure at the inlet of the test section is 0.7 MPa
corresponding to a saturation temperature of 26.7◦C. Each test run is carried out at constant
system pressure, vapor mass flow rate (0.0005-0.0016 kg/s) and inclination angle β (30◦- 90◦
to the horizontal). By varying the temperature of the cooling water inside the cooling jacket
of the test section the condensate flow was increased. The maximum condensate flow rate is
reached when flooding occurs (Liquid is dragged upward by vapor flow, see Fiedler & Auracher
(2004)). The onset of flooding can be recognized by a sharp increase in pressure drop. All
measurements are done without reaching the onset of flooding. Vapor mass flow rate was found
to have no influence on the condensation process. It was adjusted mainly to avoid flooding or
total condensation. Thus only the inclination angle was altered between the runs.
The mean wall temperature TW is calculated as an arithmetic mean of the 30 thermocouple
temperatures inserted into the channel wall. The difference to the inner wall surface tempera-
ture is not more than 0.1 K. This difference is within the uncertainty of the measurement and
therefore not taken into account. The heat transfer coefficient is determined according to
hm=
˙Q
UL(TS− TW)
.(12)
The circumference of the rectangular channel is given by
U = 2 · (B + H).(13)
For L, B and H see figure 1. The total heat flow rate˙Q is taken from the measured condensate
flow rate ˙Ml:
˙Q =˙Ml∆hlg
.(14)
The results are presented in dimensionless Nu vs. Re-plots. As usual in studies on condensation,
the Nusselt number is defined as
Num=hm(ν2
l/g)1/3
kl
(15)
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and the Reynolds number of the condensate is given by
Re =
˙Ml
U · µl
.(16)
All fluid properties were determined from NIST´ s Refprop software (Lemmon et al. (2002)) at
test section inlet conditions.
5Comparison of theory and experiment
050100150
0
0.2
0.4
0.6
0.8
Re
Num


45° tube
30°
45°
60°
90°
90°
60°
10°
30°
45°
Model
Num=0.925⋅(Re)−1/3
Figure 4: Nu vs. Re, experimental (symbols) and theoretical (lines) results, R134a at 0.7MPa
Figure 4 shows the experimental and theoretical results for refrigerant R134a. The experimen-
tal results (symbols without lines) show that the Nusselt number decreases with increasing
Reynolds number as to be expected because of the increasing film thickness with increasing Re.
The filled triangles show the maximum Nusselt-numbers of reflux condensation inside a tube
with the same hydraulic diameter (Fiedler & Auracher (2004)). The heat transfer is signifi-
cantly better in the rectangular channel. The inclination angle has a significant effect on heat
transfer. At low inclination angles Nu is larger than at high inclination angles. The worst heat
transfer results are encountered at vertical orientation. The solid lines represent calculated
Nusselt numbers using equations (10) and (15) as outlined in section 2. The model predicts
the heat transfer at small inclination angles of 45◦and 30◦and small Reynolds-numbers quite
well. In this range the film thickness distribution changes not much as shown in figure 2a
and b. The small change of film thickness inside the z0-area has little effect on heat transfer.
Nevertheless the mean film thickness increases with increasing inclination angle because the
liquid passes the increasing vertical distance Φmax(see figure 1). Due to Nusselt’s theory the
mean film thickness grows with
film thickness and the smaller the heat transfer coefficient. At Reynolds-numbers above 80 the
model underpredicts the experimental Nusselt-numbers slightly. A possible reason may be an
increased interaction of vapor and liquid with increasing waviness of the film surface and hence
convective transport inside the film.
4√Φmax. Thus the longer the distance Φmaxthe bigger the mean
At higher inclination angles e.g. 60◦the deviation between theory and experiment reaches 10%
and the values at 90◦inclination are underpredicted by almost 50%. In figure 4 it is shown
that the numerical calculation coincides with the well known Nusselt-correlation for a vertical
wall as expected. However, the assumption that the vertical channel could be regarded as
four separate vertical walls where an undisturbed film-flow is established is not supported by
experimental results. Possible reasons are a contribution of convective heat transfer inside the
falling film due to vapor shear at the film surface because of thicker films at 90◦or a kind of
Gregorig suction effect due to the radius of the film surface in the corners of the rectangular
channel. This is under further investigation.
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6Conclusions
The dependence of heat transfer from inclination angle during reflux-condensation of R134a
inside a rectangular channel of 7mm hydraulic diameter was modeled and experimentally inves-
tigated. Heat transfer inside inclined channels is significantly better than in vertical orientation.
The increase of heat transfer coefficients results from a lower mean film thickness at the side
walls of the channel which is attributed to the smaller vertical flow path of condensate. The
numerical model and the experimental results agree in the range of 30◦to 45◦. The heat transfer
in the range of 60◦to 90◦is underpredicted by the model. Overall, heat transfer in rectangular
channels is better than in tubes with equal hydraulic diameters. Contrary to results in tubes
the Nusselt theory does not hold for vertical channels. Reasons are under investigation.
Acknowledgements
We like to thank the DFG for financial support, Clariant Inc. and Solvay Inc. for donations of
technical fluids and refrigerants, respectively.
References
Chen, S. L. & Tien, C. L. (1987), ‘General film Condensation Correlations’, Exp. Heat Transfer
1, 93–107.
Fiedler, S. & Auracher, H. (2004), ‘Experimental and Theoretical Investigation of Reflux Con-
densation in an Inclined Small Diameter Tube’, International Journal of Heat and Mass
Transfer 47(19-20), 4031–4043.
Groll, M. & Roesler, S. (1992), ‘Operation principles and performance of heat pipes and closed
two-phase thermosyphons’, Non-Equilibrium Thermodynamics 17, 91–151.
Gross, U. (1987), ‘Experimentelle Untersuchung des W¨ arme¨ uberganges bei der R¨ uckstrom-
Kondensation in einem geneigten Rohr’, Chem.-Ing.-Tech. 59(2), 168–169.
Gross, U. (1992), ‘Reflux condensation heat transfer inside a closed thermosyphon’, Int. J. Heat
Mass Transfer 35(2), 279–294.
Huanzhuo, C., Tongze, M. & Groll, M. (1997), Performance limitation of micro closed two-
phase. thermosyphons, in ‘Proc. 10th International Heat Pipe Conference’, Stuttgart.
Journal of Heat Transfer (1999), ‘Common list of symbols’, Journal of Heat Transfer
121(4), 770–773.
Klahm, T., Auracher, H. & Ziegler, F. (2007), Heat Transfer to R134a and R123 during Re-
flux Condensation in Rectangular Channels, in ‘Proc. 22nd IIR International Congress of
Refrigeration’.
Lemmon, E. W., McLinden, M. O. & Huber, M. L. (2002), ‘NIST Reference Fluid Thermody-
namic and Transport Properties (REFPROP), Version 7.0’, Software on CD-Rom.
McNaught, J. M. & Moore, M. J. C. (1989), Reflux condensation in Vertical Tubes, Technical
report, ESDU Data Item No. 89038, Heat Transfer Sub-series.
Nusselt, W. (1916), ‘Oberfl¨ achenkondensation des Wasserdampfes’, Zeitschrift des VDI 60(27).
5th European Thermal-Sciences Conference, The Netherlands, 2008