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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 Today, the development of high-tech systems isunthinkable without mathematical modeling and analysis of system behavior. As such, many fields in the modern engineering sciences (e.g. control engineering, communications engineering, mechanical engineering, and robotics) call for sophisticated mathematical methods in order to solve the tasks at hand. The series Mathematical Engineering presents new or heretofore little-known methods to support engineers in finding suitable answers to their questions, presenting those methods in such manner as to make them ideally comprehensible and applicable in practice. Therefore,theprimaryfocusis—withoutneglectingmathematicalaccuracy—on comprehensibility and real-world applicability. 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More information about this series at http://www.springer.com/series/8445 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 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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