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On Dimensionless Numbers chemical engineering research and design 8 6 (2008) 835–868 Contents lists available at ScienceDirect Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd Review On dimensionless numbers M.C. Ruzicka ∗ Department of Multiphase Reactors, Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502 Prague, Czech Republic This contribution is dedicated to Kamil Admiral´ Wichterle, a professor of chemical engineering, who admitted to feel a bit lost in the jungle of the dimensionless numbers, in our seminar at “Za Plıhalovic´ ohradou” abstract The goal is to provide a little review on dimensionless numbers, commonly encountered in chemical engineering. Both their sources are considered: dimensional analysis and scaling of governing equations with boundary con- ditions. The numbers produced by scaling of equation are presented for transport of momentum, heat and mass. Momentum transport is considered in both single-phase and multi-phase flows. The numbers obtained are assigned the physical meaning, and their mutual relations are highlighted. Certain drawbacks of building correlations based on dimensionless numbers are pointed out. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Dimensionless numbers; Dimensional analysis; Scaling of equations; Scaling of boundary conditions; Single-phase flow; Multi-phase flow; Correlations Contents 1. Introduction .................................................................................................................. 836 2. Two sources of dimensionless numbers .................................................................................... 836 2.1. Source one—dimensional analysis .................................................................................. 836 2.2. Source two—scaling of equations ................................................................................... 837 3. Dimensional analysis ....................................................................................................... 838 3.1. How DA works......................................................................................................... 838 3.2. Comments on DA ..................................................................................................... 839 3.2.1. Choice of variables ........................................................................................... 839 3.2.2. Variables with independent dimensions.................................................................... 840 3.2.3. Similarity and modelling .................................................................................... 840 3.2.4. Neglecting variables in DA ................................................................................... 841 3.2.5. Limits of DA .................................................................................................. 841 3.2.6. DA versus SE.................................................................................................. 841 4. Scaling of equations ......................................................................................................... 841 5. Transport of momentum .................................................................................................... 841 5.1. Mass equation of fluid ................................................................................................ 842 5.2. Momentum equation of fluid ......................................................................................... 842 5.3. Energy equation of fluid .............................................................................................. 844 5.4. Boundary conditions: no slip and free-slip .......................................................................... 845 5.4.1. Normal component of free-slip BC .......................................................................... 845 5.4.2. Tangential component of free-slip BC ...................................................................... 845 ∗ Tel.: +420 220 390 299; fax: +420 220 920 661. E-mail address: [email protected]. Received 19 June 2007; Accepted 2 March 2008 0263-8762/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2008.03.007 836 chemical engineering research and design 8 6 (2008) 835–868 5.5. Multi-phase flow ...................................................................................................... 847 5.5.1. Microscale description ...................................................................................... 847 5.5.2. Mesoscale description (Euler/Lagrange)..................................................................... 848 5.5.3. Macroscale description (Euler/Euler) ........................................................................ 849 5.5.4. Retention time distribution.................................................................................. 850 6. Transport of heat ............................................................................................................. 852 7. Transport of mass ............................................................................................................ 853 8. Correlations .................................................................................................................. 854 9. Remark on literature ......................................................................................................... 857 10. Conclusions ................................................................................................................. 859 Acknowledgements .......................................................................................................... 859 Appendix A. Concept of intermediate asymptotics ........................................................................... 859 A.1. Motivation example .................................................................................................. 859 A.2. Two kinds of similarity ............................................................................................... 859 A.2.1. Complete similarity ......................................................................................... 859 A.2.2. Incomplete similarity........................................................................................ 860 A.3. Two kinds of self-similarity .......................................................................................... 860 A.4. Relation between DA and IA ......................................................................................... 861 A.5. Beyond IA? ............................................................................................................ 861 A.6. Broader horizons ..................................................................................................... 862 Appendix B. Suggestions for using and teaching DA.......................................................................... 863 Appendix C. New areas in chemical engineering ............................................................................. 864 C.1. Microreactors and microfluidics ..................................................................................... 864 C.1.1. Microsystems in chemical technology ...................................................................... 864 C.1.2. Prevailing forces ............................................................................................. 864 C.1.3. Governing equations and boundary conditions ............................................................ 864 C.2. Biosystems ............................................................................................................ 865 C.3. Multiscale methodology .............................................................................................. 865 1. Introduction 2. Two sources of dimensionless numbers (DN) I have always been puzzled with the plethora of dimensionless numbers (DN) occurring in the various branches of chemi- 2.1. Source one—dimensional analysis (DA) cal engineering. From sincere discussions with my peers as well as with students an impression has arose that I am not A general way how to formally describe the surrounding world the only puzzled person in this field. Consequently, the main consists of several steps. First, for the thing under study motivation of this contribution is to try to briefly review DN (‘system’), define all possible qualities of interest. Second, commonly encountered at the transport of momentum, heat, select those qualities that can be quantified, i.e. their amount and mass, as for their origin, physical meaning, interrelation in the thing can be expressed by numbers, or some other and relevance for making correlations. The selection of DN mathematical constructs. Call these measurables the phys- is neither objective nor exhaustive, being biased by working ical quantities. Choose the etalons (measuring sticks, units) mainly in the area of the multi-phase hydrodynamics. to measure each of them. Each physical quantity has four The dimensionless (nondimensional) numbers (criteria, attributes: name, notation, defining relation, and physical unit groups, products, quantities, ratios, terms) posses the follow- that determine its physical dimension (unit and dimension are ing features. They are algebraic expressions, namely fractions, often used interchangeably). The first two are only our
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