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Heat and Mass Transfer in Dispersed Two-Phase Flows

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Heat and Mass Transfer in Dispersed Two-Phase Flows Heat and Mass Transfer in Dispersed Two-phase Flows Kevin Moroney Department of Mathematics and Statistics University of Limerick A thesis submitted for the degree of Doctor of Philosophy (PhD) Supervised by: Dr. William Lee and Prof. Stephen B.G. O'Brien Submitted to the University of Limerick, September 2016 ii Abstract Industrial and manufacturing processes are an abundant source of rich and complex problems amenable to investigation with mathematical logic and techniques. Once a problem is identified and the right ques- tions are formulated, a range of mathematical modelling methodolo- gies can be leveraged. These methods can be used to simplify the problem to gain valuable insight into the underlying physical mech- anisms of the system under consideration. The resulting formulated model can be used to explain observed phenomenon, explore poorly understood behaviour, determine optimal parameters and predict fu- ture states of the system. In this thesis, two problems relating to industrial processes are studied. Consistently extracting the desired level of soluble material from ground coffee with hot water is a key objective in any coffee brewing method. In this study, a coffee extraction model is derived from first principles, using volume averaging techniques, with a view to relating the qual- ity of brewed coffee with the parameters of the underlying processes. Physical mechanisms for the dissolution and transport of coffee are included. The model is parametrised with experimental data for dif- ferent extraction experiments. The model is non-dimensionalised, to establish the dominant mechanisms during brewing. Numerical and asymptotic solutions are presented. The hydration process is a key step in the production of soft contact lenses. Hydration primarily involves the removal of leachable chem- icals, used as process aids during earlier manufacturing stages, from the contact lens. The leachable materials are removed by washing the lenses with a solvent. In this work, macroscopic models of heat and mass transport in the hydration process are formulated based on experimental data. The heat transport equations are leveraged to estimate the size of heating units required to maintain the specified process temperatures. Chemical transport in the hydration process is modelled with a view to optimising the system parameters. Declaration I hereby declare that this thesis is my own work and that it has not been submitted for any other academic award. The work in this thesis was conducted from September 2013 to Au- gust 2016 under the supervision of Dr. William Lee and Prof. Stephen O'Brien in the department of Mathematics and Statistics in the Uni- versity of Limerick. Kevin M. Moroney Limerick, 2016 To my brother Michael, greatly missed, fondly remembered. vi Acknowledgements There are many people whose help I wish to acknowledge here some of whom I will mention by name: • My supervisors Dr. William Lee and Professor Stephen O'Brien for their help and advice throughout my PhD. • MACSI administrator Peg Hanrahan for always having the an- swers to the questions no one else knew. • MACSI manager Dr. Sin´eadBurke and her predecessor Dr. Joanna Jordan for their constant enthusiasm and helpfulness. Also to Joanna for convincing me to do a PhD to begin with! • All in the MACSI group for providing a great environment in which to do research. • My family and friends for their support, especially my mother Helena (for making a brave attempt to proof-read this thesis!) and my aunt Catherine. • Financial support from MACSI, the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie), funded by the Science Foundation Ireland Investigator Award 12/IA/1683. Publications • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Mod- elling of coffee extraction during brewing using multiscale methods: An experimentally validated model. Chemical Engineering Science. 137: 216 - 234. 2015. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Math- ematical Modelling of the Coffee Brewing Process. Progress in Industrial Mathematics at ECMI 2014. 22: 2016. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Asymp- totic analysis of the dominant mechanisms in the coffee extraction process. SIAM Journal on Applied Mathematics. 76:6, 2196 - 2217. 2016. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Coffee extraction kinetics in a well mixed system. Journal of Mathematics in Industry. 7: 1 - 19. 2016. Conferences • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Math- ematical Modelling of the Coffee Brewing Process. British Applied Mathe- matics Colloquium. Cardiff, Wales, April 2014. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Math- ematical Modelling of the Coffee Brewing Process. ECMI (European Con- sortium for Mathematics in Industry) Conference. Taormina, Sicily, Italy, June 2014. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Mod- elling Extraction and Transport during Coffee Brewing. Irish Applied Math- ematics Research Students' Meeting. Galway, Ireland, December 2014. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Math- ematical Modelling of the Coffee Brewing Process. (Poster). UL-NUIG Research Day. Limerick, Ireland, May 2014. viii • K.M. Moroney, W.T. Lee and S.B.G. O'Brien. Mathematical Modelling of Chemical Transport during Contact Lens Hydration. British Applied Mathematics Colloquium. Oxford, UK, April 2016. • K.M. Moroney, W.T. Lee, S.B.G. O'Brien, F. Suijver and J. Marra. Asymp- totic analysis of a coffee extraction model for a fixed coffee bed. Santiago de Compostela, Spain, June 2016. ix x CONTENTS 1 Overview 1 1.1 Mathematical modelling of the coffee brewing process . .3 1.1.1 Coffee extraction in a well mixed system . .5 1.1.2 Coffee extraction in a packed coffee bed . .5 1.1.3 Infiltration into a dry coffee bed . .6 1.2 Mathematical modelling of a contact lens hydration process . .7 1.2.1 Heat transport in the contact lens hydration process . .7 1.2.2 Chemical transport in the contact lens hydration process .8 I Mathematical modelling of the coffee brewing pro- cess 11 2 Mathematical modelling of the coffee brewing process 13 2.1 Introduction . 13 2.1.1 Defining coffee quality . 14 2.1.2 Factors influencing coffee extraction . 17 2.1.3 Previous coffee research . 19 2.1.4 Current work . 22 2.2 Coffee extraction experiments . 23 2.2.1 Coffee grinds used . 23 2.2.2 Maximum extractable solubles mass from coffee grind . 25 xi CONTENTS 2.2.3 Coffee extraction kinetics during batch-wise brewing in a fixed water volume . 26 2.2.4 Coffee extraction profiles from a cylindrical brewing chamber 26 2.3 Mathematical modelling . 30 2.3.1 Basic modelling assumptions . 30 2.3.2 Coffee bed structure . 33 2.3.3 Coffee bed description . 33 2.3.3.1 Microscale point balance equations . 34 2.3.3.2 Mesoscale point balance equations . 35 2.3.3.3 Macroscale point balance equations . 35 2.3.4 Upscaling from microscale to macroscale . 36 2.4 Volume averaging procedure . 36 2.4.1 General upscaling procedure . 36 2.5 Upscaling from microscale to macroscale equations . 40 2.5.1 Equations for upscaling . 41 2.5.2 Upscaling in l-phase: microscale to mesoscale . 41 2.5.2.1 Conservation of coffee in v-phase . 41 2.5.2.2 Conservation of liquid in v-phase . 42 2.5.2.3 Conservation of coffee solid in s-phase . 43 2.5.3 Upscaling to macroscale . 43 2.5.3.1 Conservation of coffee in h-phase . 43 2.5.3.2 Conservation of liquid in h-phase . 44 2.5.3.3 Conservation of coffee in v-phase . 44 2.5.3.4 Conservation of liquid in v-phase . 45 2.5.3.5 Conservation of coffee in s-phase . 46 2.6 Developing macroscale equations . 47 2.6.1 Fluid velocity . 49 2.6.2 Total macroscopic flux . 50 2.6.3 Dispersive flux . 51 2.6.4 Coffee mass transfer terms . 52 2.6.5 Form of individual mass transfers . 56 2.6.6 Coffee bed properties . 57 2.6.6.1 Specific surface area of l-phase . 57 2.6.6.2 Specific surface area of s-phase . 58 2.6.6.3 Permeability kh .................. 58 xii CONTENTS 2.6.6.4 Permeability kv ................... 59 2.6.6.5 Average diffusion distances . 59 2.6.6.6 Coffee extraction limits . 59 2.6.7 Macroscale equations . 61 2.6.8 Dominance of advection over mechanical dispersion and dif- fusion . 63 2.7 Concentration of coffee in final beverage . 64 2.8 Numerical simulations . 64 2.8.1 Batch-wise brewing . 65 2.8.2 Cylindrical brewing chamber . 67 2.9 Conclusion . 69 3 Coffee extraction in a well mixed system 73 3.1 Introduction . 73 3.1.1 General coffee extraction model . 74 3.2 Model specialisation . 75 3.3 Non-dimensionalisation . 78 3.4 Model analysis and asymptotic solutions . 80 3.4.1 Phase plane analysis . 80 3.4.2 Perturbation solutions on the bulk diffusion (outer) timescale 82 3.4.2.1 Leading order equations . 82 3.4.2.2 Order equations . 82 3.4.2.3 Order 2 equations . 83 3.4.2.4 Outer solutions . 83 3.4.3 Perturbation solutions on the surface dissolution (inner) timescale . 84 3.4.3.1 Leading order equations . 85 3.4.3.2 Order equations . 85 3.4.3.3 Order 2 equations . 86 3.4.3.4 Inner solutions .
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