In this paper, we study the continuum physics model equations for condensation (two phase flow problems) in vertical tubes with small diameter and obtain reduced model equations. We found that generalization of dimensional analysis to multiple spatial dimensions is an excellent tool for that purpose, so that a review of this method is also part of the paper. We obtain the nondimensional numbers of the problem and derive reduced bulk and interface equations. The problem is characterized by three length scales, tube radius R, tube length L, and initial film thickness H. For small ratio ɛ=H/L, we derive a single ordinary differential equation for the condensate film thickness as function of axial position with tube radius as parameter, which agrees well with commonly used (parametric) models from literature. Our model is based on the physical dimensions of the problem which gives a greater geometrical flexibility and a wider range of applicability. We also discuss the effect of surface tension and the limit of the model.

References

References
1.
Carey
,
V. P.
,
1992
,
Liquid-Vapor Phase-Change Phenomena
,
Hemisphere Publishing Cooperation
, New York.
2.
Kandlikar
,
S. G.
,
Shoji
,
M.
, and
Dhir
,
V. K.
,
1999
,
Handbook of Phase Change: Boiling and Condensation
,
Taylor & Francis
,
London
.
3.
Bergles
,
A. E.
,
2011
, “
Stability and Enhancement of Boiling in Microchannels
,”
ASME
9th International Conference on Nanochannels, Microchannels, and Minichannels.10.1115/ICNMM2011-58279
4.
Barbosa
,
J. R.
,
2011
, “
Recent Developments in Vapor Compression Technologies for Small Scale Refrigeration Applications
,”
ASME
9th International Conference on Nanochannels, Microchannels, and Minichannels.10.1115/ICNMM2011-58277
5.
Collier
,
J. G.
,
Thome
,
J. R.
,
1994
,
Convective Boiling and Condensation
,
3rd ed.
,
Clarendon Press
, Oxford, UK.
6.
Fiedler
,
S.
, and
Auracher
,
H.
,
2004
, “
Experimental and Theoretical Investigation of Reflux Condensation in an Inclined Small Diameter Tube
,”
Int. J. Heat Mass Transfer
,
47
(
19–20
), pp.
4031
4043
.10.1016/j.ijheatmasstransfer.2004.06.005
7.
Wang
,
B.
, and
Du
,
X.
,
1999
, “
Heat Transfer Characteristics for Flow Condensation in Vertical Thin Tube
,”
J. Shanghai Jiaotong Univ.
,
33
(
8
), pp. 970–973.
8.
Wang
,
B.
, and
Du
,
X.
,
2003
, “
Study on Transport Phenomena for Flow Film Condensation in Vertical Mini-Tube With Interfacial Waves
,”
Int. J. Heat Mass Transfer
,
46
(
11
), pp.
2095
2101
.10.1016/S0017-9310(02)00469-6
9.
Feng
,
Z.
, and
Serizawa
,
A.
,
1999
, “
Visualization of Two-Phase Flow Patterns in an Ultra-Small Tube
,” Proceedings of the 18th Multiphase Flow Symposium of Japan, Osaka, Japan, pp.
33
–36.
10.
Zhao
,
T. S.
, and
Liao
,
Q.
,
2002
, “
Theoretical Analysis of Film Condensation Heat Transfer Inside Vertical Mini Triangular Channels
,”
Int. J. Heat Mass Transfer
,
45
(
13
), pp.
2829
2842
.10.1016/S0017-9310(01)00354-4
11.
Panday
,
P. K.
,
2003
, “
Two-Dimensional Turbulent Film Condensation of Vapours Flowing Inside a Vertical Tube and Between Parallel Plates: A Numerical Approach
,”
Int. J. Refrig.
,
26
(
4
), pp.
492
503
.10.1016/S0140-7007(02)00162-7
12.
Nebuloni
,
S.
, and
Thome
,
J. R.
,
2010
, “
Numerical Modeling of Laminar Annular Film Condensation for Different Channel Shapes
,”
Int. J. Heat Mass Transfer
,
53
(
13–14
), pp.
2615
2627
.10.1016/j.ijheatmasstransfer.2010.02.054
13.
Nebuloni
,
S.
, and
Thome
,
J. R.
,
2012
, “
Numerical Modeling of the Conjugate Heat Transfer Problem for Annular Laminar Film Condensation in Microchannels
,”
ASME J. Heat Trans.
,
134
(
5
), p.
051021
.10.1115/1.4005712
14.
Narain
,
A.
, and
Phan
,
L.
,
2007
, “
Nonlinear Stability of the Classical Nusselt Problem of Film Condensation and Wave Effects
,”
ASME J. Appl. Mech.
,
74
(
2
), pp.
279
290
.10.1115/1.2198249
15.
Pan
,
Y.
,
2001
, “
Condensation Characteristics Inside a Vertical Tube Considering the Presence of Mass Transfer, Vapor Velocity and Interfacial Shear
,”
Int. J. Heat Mass Transfer
,
44
(
23
), pp.
4475
4482
.10.1016/S0017-9310(01)00087-4
16.
Benjamin
,
T. B.
,
1957
, “
Wave Formation in a Laminar Flow Down an Inclined Plane
,”
J. Fluid Mech.
,
2
(
6
), pp. 554–573.10.1017/S0022112057000373
17.
Yih
,
C. S
,
1963
, “
Stability of Liquid Flow Down an Inclined Plane
,”
Phys. Fluids
,
6
(
3
), pp.
321
334
.10.1063/1.1706737
18.
Lin
,
S. P.
,
1975
, “
Stability of Liquid Flow Down a Heated Inclined Plane
,”
Lett. Heat Mass Transfer
,
2
(
5
), pp.
361
369
.10.1016/0094-4548(75)90002-8
19.
Marschall
,
E.
, and
Lee
,
C. Y.
,
1973
, “
Stability of Condensate Flow Down a Vertical Wall
,”
Int. J. Heat Mass Transfer
,
16
(
1
), pp.
41
48
.10.1016/0017-9310(73)90249-4
20.
Ünsal
,
M.
, and
Thomas
,
W. C
,
1978
, “
Linearized Stability Analysis of Film Condensation
,”
ASME J. Heat Trans.
,
100
(
4
), pp. 629–634.10.1115/1.3450868
21.
Spindler
,
B.
,
1982
, “
Linear Stability of Liquid Films With Interfacial Phase Change
,”
Int. J. Heat Mass Transfer
,
25
(
2
), pp. 161–173.10.1016/0017-9310(82)90002-3
22.
Burelbach
,
J. P.
,
Bankoff
,
S. G.
, and
Davis
,
S. H.
,
1988
, “
Nonlinear Stability of Evaporating/Condensing Liquid Films
,”
J. Fluid Mech.
,
195
, pp. 463–494.10.1017/S0022112088002484
23.
Joo
,
S. W.
,
Davis
,
S. H.
, and
Bankoff
,
S. G.
,
1991
, “
Long-Wave Instabilities of Heated Falling Films: Two-Dimensional Theory of Uniform Layers
,”
J. Fluid Mech.
,
230
, pp. 117–146.10.1017/S0022112091000733
24.
Hwang
,
C. W.
, and
Weng
,
C. I.
,
1987
, “
Finite-Amplitude Stability Analysis of Liquid Films Down a Vertical Wall With and Without Interfacial Phase Change
,”
Int. J. Multiphase Flow.
,
13
(
6
), pp. 803–814.10.1016/0301-9322(87)90067-X
25.
Fieg
,
G. P.
, and
Roetzel
,
W.
,
1994
, “
Calculation of Laminar Film Condensation In/On Inclined Elliptical Tubes
,”
Int. J. Heat Mass Transfer
,
37
(
4
), pp.
619
624
.10.1016/0017-9310(94)90133-3
26.
Mosaad
,
M.
,
1999
, “
Combined Free and Forced Convection Laminar Film Condensation on an Inclined Circular Tube With Isothermal Surface
,”
Int. J. Heat Mass Transfer
,
42
(
21
), pp.
4017
4025
.10.1016/S0017-9310(99)00032-0
27.
Wang
,
B.
, and
Du
,
X.
,
2000
, “
Study on Laminar Film-Wise Condensation for Vapor Flow in an Inclined Small/Mini-Diameter Tube
,”
Int. J. Heat Mass Transfer
,
43
(
10
), pp.
1859
1868
.10.1016/S0017-9310(99)00256-2
28.
Alekseenko
,
S. V.
,
Nakoryakov
,
V. E.
, and
Pokusaev
,
B. G
,
1994
,
Wave Flow of Liquid Films
,
Begel House
, New York.
29.
Yoshimura
,
P. N.
,
Nosoko
,
T.
, and
Nagata
,
T.
,
1996
, “
Enhancement of Mass Transfer Into a Falling Laminar Liquid Film by Two-Dimensional Surface Waves—Some Experimental Observations and Modeling
,”
Chem. Eng. Sci.
,
51
(
8
), pp. 1231–1240.10.1016/0009-2509(95)00387-8
30.
Dziubek
,
A.
,
2012
, “
Equations for Two-Phase Flows: A Primer
,”
Meccanica
,
47
(
8
), pp.
1819
1836
.10.1007/s11012-012-9555-0
31.
Neemann
,
K.
, and
Schade
,
H.
,
2001
, “
Dimensionsanalyse, Grundlagen und Anwendungen
,” TUB Herman-Föttinger-Institut für Strömungsmechanik, TU Berlin, Germany.
32.
Schlichting
,
H.
,
1979
,
Boundary Layer Theory
,
McGraw-Hill
,
New York
.
33.
Nusselt
,
W.
, “
Die Oberflächenkondensation des Wasserdampfes
,”
Z. Ver. Dtsch. Ing.
,
60
(
27
), pp.
541
575
.
34.
Bejan
,
A
,
2004
,
Convection Heat Transfer
,
3rd ed.
,
Wiley
,
New York
, p.
462ff
.
35.
Dalkilic
,
A. S.
, and
Wongwises
,
S.
,
2009
, “
Intensive Literature Review of Condensation Inside Smooth and Enhanced Tubes
,”
Int. J. Heat Mass Transfer
,
52
(
15–16
), pp. 3409–3426.10.1016/j.ijheatmasstransfer.2009.01.011
36.
Wei
,
X.
,
Fang
,
X.
, and
Shi
,
R.
,
2012
, “
A Comparative Study of Heat Transfer Coefficients for Film Condensation
,”
Energy Sci. Technol.
,
3
(
1
), pp.
1
9
.
37.
Shah
,
M. M.
,
2009
, “
An Improved and Extended General Correlation for Heat Transfer During Condensation in Plain Tubes
,”
HVAC&R Res.
,
15
(
5
), pp. 889–913.10.1080/10789669.2009.10390871
38.
Chen
,
S. L.
,
Gerner
,
F. M.
, and
Tien
,
C. L.
,
1987
, “
General Film Condensation Correlations
,”
Exp. Heat Transfer
,
1
(2), pp.
93
–107.10.1080/08916158708946334
39.
Dullin
,
H. R.
,
Gottwald
,
G. A.
, and
Holm
,
D. D.
,
2003
, “
Camassa-Holm, Korteweg-de Vries-5 and Other Asymptotically Equivalent Equations for Shallow Water Waves
,”
Fluid Dyn. Res.
,
33
(1–2), pp. 73–95.10.1016/S0169-5983(03)00046-7
You do not currently have access to this content.