Dynamics of Liquid Solidification: Thermal Resistance of Contact Layer

146 Pages·2017·5.299 MB·English

by  Zygmunt Lipnicki (auth.)

Mathematical Engineering
Zygmunt Lipnicki
Dynamics
of Liquid
Solidification
Thermal Resistance of Contact Layer
Mathematical Engineering
Series editors
Jörg Schröder, Essen, Germany
Bernhard Weigand, Stuttgart, Germany
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Zygmunt Lipnicki
Dynamics of Liquid
fi
Solidi cation
Thermal Resistance of Contact Layer
123
ZygmuntLipnicki
University of ZielonaGóra
ZielonaGóra, Lubuskie
Poland
ISSN 2192-4732 ISSN 2192-4740 (electronic)
Mathematical Engineering
ISBN978-3-319-53431-2 ISBN978-3-319-53432-9 (eBook)
DOI 10.1007/978-3-319-53432-9
LibraryofCongressControlNumber:2017931050
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Acknowledgements
I would like to thank Prof. Bernhard Weigand from Stuttgart University for
long-term scientific cooperation. Thanks to it came into being a number of joint
publications, which are the basis of this book.
I also thank to Prof. Waldemar Wołczyński from the Polish Academy of
SciencesinKrakówandProf.JerzyGodziszewskifromUniversityofZielonaGóra
for their valuable advice and assistance in editing this book.
I would also like to dedicate my book to my wife Krystyna and my daughter
Monika.
v
Contents
1 Solidification ... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 The General Solidification Problem ... .... .... .... ..... .... 1
1.2 Equations Describing the Solidification Problem . .... ..... .... 2
2 Solidification of PCMs ... ..... .... .... .... .... .... ..... .... 7
2.1 Heat Storage Based on the Solidification Process. .... ..... .... 7
2.1.1 Multidimensional Solidification Heat Transfer.. ..... .... 8
2.1.2 Solutions of the Practical Solidification Problem..... .... 9
2.2 The Geometry of Outer PCMs... .... .... .... .... ..... .... 11
2.3 The Criterion for the Selection of Accumulator Design ..... .... 12
3 The Model of Solidification of a Liquid with the Contact Layer........ 15
3.1 Equations Describing the Problem with the Contact Layer... .... 15
3.2 Strict and Approximate Solution of the One-Dimensional
Solidification Problem ..... .... .... .... .... .... ..... .... 18
3.2.1 Strict Solution of the One-Dimensional Solidification
Problem.. .... ..... .... .... .... .... .... ..... .... 18
3.2.2 Approximate Solution of the One-Dimensional
Solidification Problem.... .... .... .... .... ..... .... 22
4 Solidification on a Rectangular Geometrics.... .... .... ..... .... 25
4.1 The Immovable Layer of Liquid Solidification... .... ..... .... 25
4.2 Forced Convection Effects .. .... .... .... .... .... ..... .... 28
4.2.1 The Effect of the Boundary Layer on the Solidification
Process .. .... ..... .... .... .... .... .... ..... .... 29
4.2.2 Solidification on a Plate with Variable Temperature.. .... 38
4.3 Free Convection Effect..... .... .... .... .... .... ..... .... 45
4.3.1 Quasi-Stationary Convection in a Vertical Channel... .... 46
4.3.2 Quasi-Stationary Convection in a Horizontal Channel. .... 50
4.3.3 The Simplified Model of Convection
with Solidification... .... .... .... .... .... ..... .... 55
vii
viii Contents
5 Solidification in an Annular Space... .... .... .... .... ..... .... 63
5.1 The Solidification of a Non-heated Liquid .. .... .... ..... .... 63
5.2 Solidification of a Superheated Liquid . .... .... .... ..... .... 67
5.2.1 Theoretical Solution of the Solidification Problem ... .... 68
5.2.2 The Beginning of a Solidification Process—Asymptotic
Solution.. .... ..... .... .... .... .... .... ..... .... 73
5.3 The Experiment . .... ..... .... .... .... .... .... ..... .... 74
5.3.1 Research Equipment . .... .... .... .... .... ..... .... 74
5.3.2 The Course of the Experiment . .... .... .... ..... .... 74
5.3.3 The Results of the Research and Discussion... ..... .... 78
5.3.4 Conclusions... ..... .... .... .... .... .... ..... .... 82
6 Solidification of a Liquid Flowing into the Channel. .... ..... .... 83
6.1 The General Analysis. ..... .... .... .... .... .... ..... .... 83
6.2 The Solidification of a Liquid Flowing into a Channel
with Weakly Conducting Walls .. .... .... .... .... ..... .... 87
6.3 The Solidification of a Liquid Flowing into a Channel
with Conducting Walls..... .... .... .... .... .... ..... .... 92
7 Role of the Contact Layer in a Solidification Process.... ..... .... 97
7.1 Studies of the Thermal Contact Resistance.. .... .... ..... .... 97
7.2 A Theoretical Analysis of Thermal Contact Resistance
on the Basis of Experimental Investigations. .... .... ..... .... 101
7.2.1 Thermal Contact Resistance for Solidification
on a Plate .... ..... .... .... .... .... .... ..... .... 101
7.2.2 Thermal Contact Resistance for Solidification
on a Cylinder . ..... .... .... .... .... .... ..... .... 103
7.3 Development of the Contact Layer During the Solidification
Process.... .... .... ..... .... .... .... .... .... ..... .... 105
7.3.1 Development of Contact Layer and Its Role
in Phase-Change Process on a Rectangular Plate .... .... 108
7.3.2 Development of a Contact Layer in Solidification
on an Annular Surface ... .... .... .... .... ..... .... 111
8 Phase Heat Accumulator.. ..... .... .... .... .... .... ..... .... 119
8.1 Selected Review of the Units Used for Heat Storage.. ..... .... 119
8.2 Cooperation of the Heat Accumulator Storage
with the Heat Pump .. ..... .... .... .... .... .... ..... .... 120
8.3 Example of Calculations.... .... .... .... .... .... ..... .... 123
Summary. .... .... .... .... ..... .... .... .... .... .... ..... .... 127
Annexe 1 (Zalba i in., 2003; Domański, Moszyński, 1983;
and Others)... .... .... .... ..... .... .... .... .... .... ..... .... 129
References.... .... .... .... ..... .... .... .... .... .... ..... .... 133
About the Author
Zygmunt Lipnicki is the professor of the Institute of
EnvironmentalEngineeringatZielonaGóraUniversity
in Poland. He received his doctoral degree in 1985 at
Poznań Technical University and his habilitation
degree from Poznań Technical University in 2000.
Dr. Lipnicki’s teaching focuses primarily on fluid
dynamics, thermodynamics and heat transfer. His cur-
rent research programmes include analytical and
experimental studies of thermal resistance of contact
layer in solidification process.
ix
Introduction
Theresistanceofheatflowingthroughatransitionlayerwhichoccursontheborder
of different heat-exchanging phases is a phenomenon common both in technology
and in nature. It has been a subject of many scientific papers discussed in detail in
the works of Madhusudana (1996) and Furmański and Wiśniewski (2002). In the
processofsolidificationoccurring,forexample,incastingmoulds,acontactlayeris
formed between the wall of the container and the solidified liquid determining the
intensity of heat exchange between the wall and the liquid. This particular matter
was the research subject of many authors (Beck 1969; Longa 1985; Wang and
Matthys 1994; Sahai 1998; Loulou et al. 1999a, b).
The transition layer is a spatial element bounded by two not very precisely
defined surfaces contacting with other bodies. The literature uses a less precise
name of this phenomenon: contact layer. This name will be also applied in this
book.
Toagreatdegree,thisbookisasummaryofitsauthor’sworkspublishedinthe
last years concerning the research on thermal resistance of a transition layer in the
solidification process of the flowing liquid phase (Lipnicki 1999, 2003; Lipnicki
et al. 2005; Lipnicki and Bydałek 2008; Lipnicki and Kuczma 2009; Lipnicki and
Weigand 2011, 2012; Lipnick et al. 2014a, b; Lipnicki and Pantoł 2014; Weigand
andLipnicki2016).ItisthesummingupofthebookpublishedinPolish(Lipnicki
2012) and in a great part can be considered its second edition. However, this one
includesnewelements.Someslightmistakes werespottedandremoved,andmore
chapters and new approach were included.
Storageandreleaseofheatduringaphase transition ofmeltingorsolidification
are interesting and important phenomena that can be practically applied in the
energy sector (Domański and Moszyński 1983; Alexiades and Solomon 1993;
Jaworski 2001; Wnuk 2001; Suguwara et al. 2008; Mehling and Cabeza 2008;
KrasowskiandLipnicki2009; LipnickiandKrasowski 2010). Themainfeature of
those phenomena is a relatively small volume of the heat-accumulating units and
stabletemperatureoftransitionprocesses.Moreover,byapplyingPCMmaterialsof
high transformation energy accompanying heat storage or release during a
xi