Dimensional Analysis and Correlations
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Convection Heat Transfer
Convection Heat Transfer Heat transfer from a solid to the surrounding fluid Consider fluid motion Recall flow of water in a pipe Thermal Boundary Layer • A temperature profile similar to velocity profile. Temperature of pipe surface is kept constant. At the end of the thermal entry region, the boundary layer extends to the center of the pipe. Therefore, two boundary layers: hydrodynamic boundary layer and a thermal boundary layer. Analytical treatment is beyond the scope of this course. Instead we will use an empirical approach. Drawback of empirical approach: need to collect large amount of data. Reynolds Number: Nusselt Number: it is the dimensionless form of convective heat transfer coefficient. Consider a layer of fluid as shown If the fluid is stationary, then And Dividing Replacing l with a more general term for dimension, called the characteristic dimension, dc, we get hd N ≡ c Nu k Nusselt number is the enhancement in the rate of heat transfer caused by convection over the conduction mode. If NNu =1, then there is no improvement of heat transfer by convection over conduction. On the other hand, if NNu =10, then rate of convective heat transfer is 10 times the rate of heat transfer if the fluid was stagnant. Prandtl Number: It describes the thickness of the hydrodynamic boundary layer compared with the thermal boundary layer. It is the ratio between the molecular diffusivity of momentum to the molecular diffusivity of heat. kinematic viscosity υ N == Pr thermal diffusivity α μcp N = Pr k If NPr =1 then the thickness of the hydrodynamic and thermal boundary layers will be the same. -
Laws of Similarity in Fluid Mechanics 21
Laws of similarity in fluid mechanics B. Weigand1 & V. Simon2 1Institut für Thermodynamik der Luft- und Raumfahrt (ITLR), Universität Stuttgart, Germany. 2Isringhausen GmbH & Co KG, Lemgo, Germany. Abstract All processes, in nature as well as in technical systems, can be described by fundamental equations—the conservation equations. These equations can be derived using conservation princi- ples and have to be solved for the situation under consideration. This can be done without explicitly investigating the dimensions of the quantities involved. However, an important consideration in all equations used in fluid mechanics and thermodynamics is dimensional homogeneity. One can use the idea of dimensional consistency in order to group variables together into dimensionless parameters which are less numerous than the original variables. This method is known as dimen- sional analysis. This paper starts with a discussion on dimensions and about the pi theorem of Buckingham. This theorem relates the number of quantities with dimensions to the number of dimensionless groups needed to describe a situation. After establishing this basic relationship between quantities with dimensions and dimensionless groups, the conservation equations for processes in fluid mechanics (Cauchy and Navier–Stokes equations, continuity equation, energy equation) are explained. By non-dimensionalizing these equations, certain dimensionless groups appear (e.g. Reynolds number, Froude number, Grashof number, Weber number, Prandtl number). The physical significance and importance of these groups are explained and the simplifications of the underlying equations for large or small dimensionless parameters are described. Finally, some examples for selected processes in nature and engineering are given to illustrate the method. 1 Introduction If we compare a small leaf with a large one, or a child with its parents, we have the feeling that a ‘similarity’ of some sort exists. -
Influence of Nusselt Number on Weight Loss During Chilling Process
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Procedia Engineering 32 (2012) 90 – 96 I-SEEC2011 Influence of Nusselt number on weight loss during chilling process R. Mekprayoon and C. Tangduangdee Department of Food Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand Elsevier use only: Received 30 September 2011; Revised 10 November 2011; Accepted 25 November 2011. Abstract The coupling of the heat and mass transfer occurs on foods during chilling process. The temperature of foods is reduced while weight changes with time. Chilling time and weight loss involve the operating cost and are affected by the design of refrigeration system. The aim of this work was to determine the correlations of Nu-Re-Pr and Sh-Re-Sc used to predict chilling time and weight loss, respectively for infinite cylindrical shape of food products. The correlations were obtained by estimating coefficients of heat and mass transfer based on analytical solution method and regression analysis of related dimensionless numbers. The experiments were carried out at different air velocities (0.75 to 4.3 m/s) and air temperatures (-10 to 3 ÛC) on different sizes of infinite cylindrical plasters (3.8 and 5.5 cm in diameters). The correlations were validated through experiments by measuring chilling time and weight loss of Vietnamese sausages compared with predicted values. © 2010 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of I-SEEC2011 Open access under CC BY-NC-ND license. -
Heat Transfer in Turbulent Rayleigh-B\'Enard Convection Within Two Immiscible Fluid Layers
This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Heat transfer in turbulent Rayleigh-B´enard convection within two immiscible fluid layers Hao-Ran Liu1, Kai Leong Chong2,1, , Rui Yang1, Roberto Verzicco3,4,1 and Detlef† Lohse1,5, ‡ 1Physics of Fluids Group and Max Planck Center Twente for Complex Fluid Dynamics, MESA+Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500AE Enschede, The Netherlands 2Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, PR China 3Dipartimento di Ingegneria Industriale, University of Rome “Tor Vergata”, Via del Politecnico 1, Roma 00133, Italy 4Gran Sasso Science Institute - Viale F. Crispi, 7 67100 L’Aquila, Italy 5Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 G¨ottingen, Germany (Received xx; revised xx; accepted xx) We numerically investigate turbulent Rayleigh-B´enard convection within two immiscible fluid layers, aiming to understand how the layer thickness and fluid properties affect the heat transfer (characterized by the Nusselt number Nu) in two-layer systems. Both two- and three-dimensional simulations are performed at fixed global Rayleigh number Ra = 108, Prandtl number Pr =4.38, and Weber number We = 5. We vary the relative thickness of the upper layer between 0.01 6 α 6 0.99 and the thermal conductivity coefficient ratio of the two liquids between 0.1 6 λk 6 10. Two flow regimes are observed: In the first regime at 0.04 6 α 6 0.96, convective flows appear in both layers and Nu is not sensitive to α. -
Heat Transfer by Impingement of Circular Free-Surface Liquid Jets
18th National & 7th ISHMT-ASME Heat and Mass Transfer Conference January 4-6, 2006 Paper No: IIT Guwahati, India Heat Transfer by Impingement of Circular Free-Surface Liquid Jets John H. Lienhard V Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge MA 02139-4307 USA email: [email protected] Abstract Q volume flow rate of jet (m3/s). 3 Qs volume flow rate of splattered liquid (m /s). 2 This paper reviews several aspects of liquid jet im- qw wall heat flux (W/m ). pingement cooling, focusing on research done in our r radius coordinate in spherical coordinates, or lab at MIT. Free surface, circular liquid jet are con- radius coordinate in cylindrical coordinates (m). sidered. Theoretical and experimental results for rh radius at which turbulence is fully developed the laminar stagnation zone are summarized. Tur- (m). bulence effects are discussed, including correlations ro radius at which viscous boundary layer reaches for the stagnation zone Nusselt number. Analyti- free surface (m). cal results for downstream heat transfer in laminar rt radius at which turbulent transition begins (m). jet impingement are discussed. Splattering of turbu- r1 radius at which thermal boundary layer reaches lent jets is also considered, including experimental re- free surface (m). sults for the splattered mass fraction, measurements Red Reynolds number of circular jet, uf d/ν. of the surface roughness of turbulent jets, and uni- T liquid temperature (K). versal equilibrium spectra for the roughness of tur- Tf temperature of incoming liquid jet (K). bulent jets. The use of jets for high heat flux cooling Tw temperature of wall (K). -
Capillarity and Dynamic Wetting Andreas Carlson
Capillarity and dynamic wetting by Andreas Carlson March 2012 Technical Reports Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen fredagen den 23 mars 2012 kl 10.00 i Salongen, Biblioteket, Kungliga Tekniska H¨ogskolan, Osquars Backe 25, Stockholm. c Andreas Carlson 2012 E-PRINT, Stockholm 2012 iii Capillarity and dynamic wetting Andreas Carlson 2012 Linn´eFLOW Centre, KTH Mechanics SE-100 44 Stockholm, Sweden Abstract In this thesis capillary dominated two{phase flow is studied by means of nu- merical simulations and experiments. The theoretical basis for the simulations consists of a phase field model, which is derived from the system's thermody- namics, and coupled with the Navier Stokes equations. Two types of interfacial flow are investigated, droplet dynamics in a bifurcating channel and sponta- neous capillary driven spreading of drops. Microfluidic and biomedical applications often rely on a precise control of droplets as they traverse through complicated networks of bifurcating channels. Three{dimensional simulations of droplet dynamics in a bifurcating channel are performed for a set of parameters, to describe their influence on the resulting droplet dynamics. Two distinct flow regimes are identified as the droplet in- teracts with the tip of the channel junction, namely, droplet splitting and non- splitting. A flow map based on droplet size and Capillary number is proposed to predict whether the droplet splits or not in such a geometry. A commonly occurring flow is the dynamic wetting of a dry solid substrate. -
Chapter 5 Dimensional Analysis and Similarity
Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences. -
Analysis of Mass Transfer by Jet Impingement and Heat Transfer by Convection
Analysis of Mass Transfer by Jet Impingement and Study of Heat Transfer in a Trapezoidal Microchannel by Ejiro Stephen Ojada A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering University of South Florida Major Professor: Muhammad M. Rahman, Ph.D. Frank Pyrtle, III, Ph.D. Rasim Guldiken, Ph.D. Date of Approval: November 5, 2009 Keywords: Fully-Confined Fluid, Sherwood number, Rotating Disk, Gadolinium, Heat Sink ©Copyright 2009, Ejiro Stephen Ojada i TABLE OF CONTENTS LIST OF FIGURES ii LIST OF SYMBOLS v ABSTRACT viii CHAPTER 1: INTRODUCTION AND LITERATURE REVIEW 1 1.1 Introduction (Mass Transfer by Jet Impingement ) 1 1.2 Literature Review (Mass Transfer by Jet Impingement) 2 1.3 Introduction (Heat Transfer in a Microchannel) 7 1.4 Literature Review (Heat Transfer in a Microchannel) 8 CHAPTER 2: ANALYSIS OF MASS TRANSFER BY JET IMPINGEMENT 15 2.1 Mathematical Model 15 2.2 Numerical Simulation 19 2.3 Results and Discussion 21 CHAPTER 3: ANALYSIS OF HEAT TRANSFER IN A TRAPEZOIDAL MICROCHANNEL 42 3.1 Modeling and Simulation 42 3.2 Results and Discussion 48 CHAPTER 4: CONCLUSION 65 REFERENCES 67 APPENDICES 79 Appendix A: FIDAP Code for Analysis of Mass Transfer by Jet Impingement 80 Appendix B: FIDAP Code for Fluid Flow and Heat Transfer in a Composite Trapezoidal Microchannel 86 i LIST OF FIGURES Figure 2.1a Confined liquid jet impingement between a rotating disk and an impingement plate, two-dimensional schematic. 18 Figure 2.1b Confined liquid jet impingement between a rotating disk and an impingement plate, three-dimensional schematic. -
Electrohydrodynamic Settling of Drop in Uniform Electric Field at Low and Moderate Reynolds Numbers
Electrohydrodynamic settling of drop in uniform electric field at low and moderate Reynolds numbers Nalinikanta Behera1, Shubhadeep Mandal2 and Suman Chakraborty1,a 1Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur,West Bengal- 721302, India 2Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 G¨ottingen, Germany Dynamics of a liquid drop falling through a quiescent medium of another liquid is investigated in external uniform electric field. The electrohydrodynamics of a drop is governed by inherent deformability of the drop (defined by capillary number), the electric field strength (defined by Masson number) and the surface charge convection (quantified by electric Reynolds number). Surface charge convection generates nonlinearilty in a electrohydrodynamics problem by coupling the electric field and flow field. In Stokes limit, most existing theoretical models either considered weak charge convection or weak electric field to solve the problem. In the present work, gravitational settling of the drop is investigated analytically and numerically in Stokes limit considering significant electric field strength and surface charge convection. Drop deformation accurate upto higher order is calculated analytically in small deformation regime. Our theoretical results show excellent agreement with the numerical and shows improvement over previous theoretical models. For drops falling with moderate Reynolds number, the effect of Masson number on transient drop dynamics is studied for (i) perfect dielectric drop in perfect dielectric medium (ii) leaky dielectric drop in the leaky dielectric medium. For the latter case transient deformation and velocity obtained for significant charge convection is compared with that of absence in charge convection which is the novelty of our study. -
A Review of Models for Bubble Clusters in Cavitating Flows Daniel Fuster
A review of models for bubble clusters in cavitating flows Daniel Fuster To cite this version: Daniel Fuster. A review of models for bubble clusters in cavitating flows. Flow, Turbulence and Combustion, Springer Verlag (Germany), 2018, 10.1007/s10494-018-9993-4. hal-01913039 HAL Id: hal-01913039 https://hal.sorbonne-universite.fr/hal-01913039 Submitted on 5 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Flow Turbulence and Combustion manuscript No. (will be inserted by the editor) D. Fuster A review of models for bubble clusters in cavitating flows Received: date / Accepted: date Abstract This paper reviews the various modeling strategies adopted in the literature to capture the response of bubble clusters to pressure changes. The first part is focused on the strategies adopted to model and simulate the response of individual bubbles to external pressure variations discussing the relevance of the various mechanisms triggered by the appearance and later collapse of bubbles. In the second part we review available models proposed for large scale bubbly flows used in different contexts including hydrodynamic cavitation, sound propagation, ultrasonic devices and shockwave induced cav- itation processes. -
Arxiv:2103.12893V1 [Physics.Flu-Dyn] 23 Mar 2021 (Brine), Thus Creating a Two-Phase flow System
Dynamic wetting effects in finite mobility ratio Hele-Shaw flow S.J. Jackson, D. Stevens, D. Giddings, and H. Power∗ Faculty of Engineering, Division of Energy and Sustainability, University of Nottingham, UK (Dated: July 21, 2015) In this paper we study the effects of dynamic wetting on the immiscible displacement of a high viscosity fluid subject to the radial injection of a less viscous fluid in a Hele-Shaw cell. The dis- placed fluid can leave behind a trailing film that coats the cell walls, dynamically affecting the pressure drop at the fluid interface. By considering the non-linear pressure drop in a boundary element formulation, we construct a Picard scheme to iteratively predict the interfacial velocity and subsequent displacement in finite mobility ratio flow regimes. Dynamic wetting delays the onset of finger bifurcation in the late stages of interfacial growth, and at high local capillary numbers can alter the fundamental mode of bifurcation, producing vastly different finger morphologies. In low mobility ratio regimes, we see that finger interaction is reduced and characteristic finger breaking mechanisms are delayed but never fully inhibited. In high mobility ratio regimes, finger shielding is reduced when dynamic wetting is present. Finger bifurcation is delayed which allows the primary fingers to advance further into the domain before secondary fingers are generated, reducing the level of competition. I. INTRODUCTION der of 50◦C, pressure ranging from 10 to 20 MPa and having a density between 0.4 to 0.8 times that of the During the displacement of a high viscosity fluid surrounding brine. The mobility ratio between the by a low viscosity fluid, perturbations along the in- CO2 and brine is of order 10. -
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