Heat and Mass Correlations

Total Page:16

File Type:pdf, Size:1020Kb

Heat and Mass Correlations Heat and Mass Correlations Alexander Rattner, Jonathan Bohren November 13, 2008 Contents 1 Dimensionless Parameters 2 2 Boundary Layer Analogies - Require Geometric Similarity 2 3 External Flow 3 3.1 External Flow for a Flat Plate . 3 3.2 Mixed Flow Over a plate . 4 3.3 Unheated Starting Length . 4 3.4 Plates with Constant Heat Flux . 4 3.5 Cylinder in Cross Flow . 4 3.6 Flow over Spheres . 5 3.7 Flow Through Banks of Tubes . 6 3.7.1 Geometric Properties . 6 3.7.2 Flow Correlations . 7 3.8 Impinging Jets . 8 3.9 Packed Beds . 9 4 Internal Flow 9 4.1 Circular Tube . 9 4.1.1 Properties . 9 4.1.2 Flow Correlations . 10 4.2 Non-Circular Tubes . 12 4.2.1 Properties . 12 4.2.2 Flow Correlations . 12 4.3 Concentric Tube Annulus . 13 4.3.1 Properties . 13 4.3.2 Flow Correlations . 13 4.4 Heat Transfer Enhancement - Tube Coiling . 13 4.5 Internal Convection Mass Transfer . 14 5 Natural Convection 14 5.1 Natural Convection, Vertical Plate . 15 5.2 Natural Convection, Inclined Plate . 15 5.3 Natural Convection, Horizontal Plate . 15 5.4 Long Horizontal Cylinder . 15 5.5 Spheres . 15 5.6 Vertical Channels . 16 5.7 Inclined Channels . 16 5.8 Rectangular Cavities . 16 5.9 Concentric Cylinders . 17 5.10 Concentric Spheres . 17 1 JRB, ASR MEAM333 - Convection Correlations 1 Dimensionless Parameters Table 1: Dimensionless Parameters k α Thermal diffusivity ρcp τs Cf 2 Skin Friction Coefficient ρu1=2 α Le Lewis Number - heat transfer vs. mass transport DAB hL Nu Nusselt Number - Dimensionless Heat Transfer kf P e P e = RexP r Peclet Number ν µC P r = p Prandtl Number - momentum diffusivity vs. thermal diffusivity α k ρu x u x Re 1 = 1 Reynolds Number - Inertia vs. Viscosity µ ν ν Sc Schmidt Number momentum vs. mass transport DAB h L Sh m Sherwood Number - Dimensionless Mass Transfer DAB h Nu St = L Stanton Number - Modified Nusselt Number ρV cp ReLP r hm ShL Stm = Stanton mass Number - Modified Sherwood Number V ReLSc 2 Boundary Layer Analogies - Require Geometric Similarity Table 2: Boundary Layer Analogies Nu Sh = P rn Scn Heat and Mass Analogy Applies always for same geometry, n is positive hL hmL n = n kP r DABSc C j = f = StP r2=3 Chilton Colburn Heat H 2 0:6 < P r < 60 C j = f = St Sc2=3 Chilton Colburn Mass M 2 m 0:6 < Sc < 3000 2/17 JRB, ASR MEAM333 - Convection Correlations 3 External Flow T + T These typically use properties at the film temperature T = s 1 f 2 3.1 External Flow for a Flat Plate T + T These use properties at the film temperature T = s 1 f 2 Table 3: Flat Plate Isothermal Laminar Flow 5:0 Flat plate Boundary Layer Thickness δ = p Re < 5E5 u1=vx p Local Shear Stress τs = 0:332u1 ρµu1=x Re < 5E5 −0:5 Local Skin Friction Coefficient Cf;x = 0:664Rex Re < 1 hxx 0:5 1=3 Re < 5E5 Local Heat Transfer Nux = = 0:332Re P r k x P r ≥ 0:6 hm;xx 0:5 1=3 Re < 5E5 Local Mass Transfer Shx = = 0:332Rex Sc DAB Sc ≥ 0:6 −0:5 Average Skin Friction Coefficient Cf;x = 1:328Rex Re < 1 Isothermal h x Average Heat Transfer Nu = x = 0:664Re0:5P r1=3 Re < 5E5 x k x P r ≥ 0:6 hm;xx 0:5 1=3 Re < 5E5 Average Mass Transfer Shx = = 0:664Rex Sc DAB Sc ≥ 0:6 Liquid Metals 0:5 Nux = 2Nux Nux Nux = 0:565P e x P r ≤ 0:05 P ex ≥ 100 0:3387Re0:5P r1=3 All Prandtl Numbers Nu Nu = x x x 1=4 1 + (0:0468=P r)2=3 P ex ≥ 100 5 Table 4: Turbulent Flow Over an Isothermal Plate Rex > 5 · 10 −0:2 8 Skin Friction Coefficient Cf;x = 0:0592Rex 5E5 < Re < 10 −0:2 8 Boundary Layer Thickness δ = 0:37xRex 5E5 < Re < 10 8 0:8 1=3 5E5 < Re < 10 Heat Transfer Nux = StRexP r = 0:0296Re P r x 0:6 < P r < 60 8 0:8 1=3 5E5 < Re < 10 Mass Transfer Shx = StRexSc = 0:0296Re Sc x 0:6 < P r < 3000 3/17 JRB, ASR MEAM333 - Convection Correlations 3.2 Mixed Flow Over a plate xc If transition occurs at L ≥ 0:95 The laminar plate model may be used for h. Once the critical transition point 0:8 0:5 has been found, we define A = 0:037Rex;c − 0:664Rex;c These typically use properties at the film temperature T + T T = s 1 f 2 Table 5: Mixed Flow Over an Isothermal Plate 0:8 1=3 0:6 < P r < 60 Average Heat Transfer NuL = (0:037ReL − A)P r 5 8 5 · 10 < ReL < 10 −0:2 2A 5 8 Average Skin Friction Coefficient CfL = 0:074Re − 5 · 10 < ReL < 10 ReL 0:8 1=3 0:6 < Sc < 60 Average Mass Transfer ShL = (0:037ReL − A)Sc 5 8 5 · 10 < ReL < 10 3.3 Unheated Starting Length T + T Here the plate has T = T until x = ζ These typically use properties at the film temperature T = s 1 s 1 f 2 Table 6: Unheated Starting Length Nu j laminar Local Heat Transfer Nu = x ζ=0 x 1=3 5 [1 − (ζ=x)0:75] 0 < ReL < 5 · 10 Nu j turbulent Local Heat Transfer Nu = x ζ=0 x 1=9 5 8 1 − (ζ=x)9=10 5 · 10 < ReL < 10 p=(p+1) L h p+1 i p = 2 Laminar Flow Average Heat Transfer NuL = NuLjζ=0 1 − (ζ=L) p+2 L−ζ p = 8 Turbulent Flow 3.4 Plates with Constant Heat Flux R For average heat transfer values, it is acceptable to use the isothermal results for T = 0 L(Ts − T1)dx Table 7: Constant Heat Flux 5 0:5 1=3 0 < ReL < 5 · 10 Local Heat Transfer Laminar Nux = 0:453Re P r x P r > 0:6 5 0:8 1=3 ReL > 5 · 10 Local Heat Transfer Turbulent Nux = 0:0308Re P r x 0:6 < P r < 60 3.5 Cylinder in Cross Flow ρV D VD For the cylinder in cross flow, we use ReD = µ = ν These typically use properties at the film temperature T + T T = s 1 f 2 4/17 JRB, ASR MEAM333 - Convection Correlations Table 8: Cylinder in Cross Flow 0:7 < P r < 60 m 1=3 NuD = CReD P r C; m are found as functions of ReD on P426 0:7 < P r < 500 6 0:25 1 < ReD < 10 m n P r NuD = CReD P r All properties evaluated at P rs T1 except P rs Uses table 7.4 P428 " #4=5 0:62Re0:5P r1=3 Re 5=8 Nu = 0:3 + D 1 + d D 1=4 P r > 0:2 1 + (0:4=P r)2=3 282; 000 3.6 Flow over Spheres Table 9: Flow over Spheres 0:71 < P r < 380 4 1=4 3:5 < P r < 6:6 · 10 0:5 2=3 0 µ NuD = 2 + (0:4ReD + 0:06ReD )P r :4 1:0 < (µ/µs) < 3:2 µs All properties except µs are evaluated at T1 0:5 1=3 NuD = 2 + 0:6ReD P r For Freely Falling Drops Infinite Stationary Medium NuD = 2 Red ! 0 5/17 JRB, ASR MEAM333 - Convection Correlations 3.7 Flow Through Banks of Tubes 3.7.1 Geometric Properties Table 10: Tube Bank Properties ρV D Re = max D µ Aligned OR ST Vmax = Vi ST + D ST − D Staggered and S > D 2 ST ST + D Vmax = Vi Staggered and SD < 2(SD − D) 2 Figure 1: Tube bank geometries for aligned (a) and staggered (b) banks 6/17 JRB, ASR MEAM333 - Convection Correlations 3.7.2 Flow Correlations Table 11: Flow through banks of tubes More than 10 rows of tubes 2000 < ReD;max < 40; 000 m 1=3 NuD = 1:13C1ReD;maxP r P r > 0:7 Coefficients come from table 7.5 on P438 C2 comes from Table 7.6 on P439 2000 < ReD;max < 40; 000 NuDj(NL<10) = C2NuDj(NL≥10) P r > 0:7 Coefficients come from table 7.5 on P438 C; m comes from Table 7.7 on P440 0:25 6 m 0:36 P r 1000 < ReD;max < 2 · 10 NuD = CReD;maxP r P rs 0:7 < P r < 500 More than 20 rows For the above correlation C2 comes from Table 7.8 on P440 NuDj(NL<20) = C2NuDj(NL≥20) 2000 < ReD;max < 40; 000 P r > 0:7 Table 12: Flow through banks of tubes 2 (Ts − Ti) − (Ts − T o) Log Mean Temp. ∆Tlm = ln Ts−Ti Ts−To T − T πDNh¯ Dimensionless Temp Correlation s o = exp − Ts − Ti ρV NT ST cP N - total number of tubes, NT - total number of tubes in transverse plane 0 ¯ Heating Per Unit Length q = NhπD∆Tlm 7/17 JRB, ASR MEAM333 - Convection Correlations 3.8 Impinging Jets 00 Heat and mass transfer is measured against the fluid properties at the nozzle exit q = h(Ts − Te) The Reynolds Ac;e and Nusselt numbers are measured using the hydraulic diameter of the nozzle Dh = P The Reynolds number uses the nozzle exit velocity. All correlations use the target cell region Ar which is affected by the nozzle.
Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Archives of Thermodynamics
    POLISH ACADEMY OF SCIENCES THERMODYNAMICS AND COMBUSTION COMMITTEE archives of thermodynamics QUARTERLY Vol. 38 2017 Wydawnictwo IMP No. 3 Gdańsk Editorial Office Archives of Thermodynamics The Szewalski Institute of Fluid Flow Machinery, Fiszera 14, 80-231 Gdańsk, Poland, Tel.: (+48) 58-341-12-71 int. 141, E-mail: [email protected] http://www.imp.gda.pl/archives-of-thermodynamics/ http://at.czasopisma.pan.pl/ © by the Polish Academy of Science Publication funding of this journal is provided by resources of the Polish Academy of Sciences and the Szewalski Institute of Fluid-Flow Machinery PASci Terms of subscription outside Poland Annual subscription rate outside Poland (2017) is 120 EUR. Price of single issue is 30 EUR. Back of previously published volumes are available on request. Subscription orders should be sent directly to IMP PAN Publishers, The Szewalski Institute of Fluid-Flow Machinery PASci, ul. Fiszera 14, 80-231 Gdansk, Poland; fax: (+48) 58-341-61-44; e-mail: [email protected]. Payments should be transferred to the bank account of IMP PAN: IBAN 28 1130 1121 0006 5498 9520 0011 at Bank Gospo- darstwa Krajowego; Code SWIFT: GOSKPLPW Warunki prenumeraty w Polsce Roczna prenumerata (2017 r.) wynosi 168 PLN. Cena pojedynczego numeru wynosi 42 PLN. Osiągalne są również wydania archiwalne. Zamówienia z określeniem okresu prenumeraty, nazwiskiem i adresem odbiorcy należy kierować bezpośrednio do Wydawcy (Instytut Maszyn Przepływowych im. R. Szewalskiego PAN, ul. Fiszera 14, 80-231 Gdańsk, Aleksandra Nowaczewska, e-mail: [email protected]).
    [Show full text]
  • 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.
    [Show full text]
  • Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer 1. Nusselt Number Average Nusselt Number: Nul = Convective
    Jingwei Zhu http://jingweizhu.weebly.com/course-note.html Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer 1. Nusselt number Average Nusselt number: convective heat transfer ℎ퐿 Nu = = L conductive heat transfer 푘 where L is the characteristic length, k is the thermal conductivity of the fluid, h is the convective heat transfer coefficient of the fluid. Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer; some examples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area. The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean-average of the bulk fluid temperature T∞ and wall surface temperature Tw. Local Nusselt number: hxx Nu = x k The length x is defined to be the distance from the surface boundary to the local point of interest. 2. Prandtl number The Prandtl number Pr is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. That is, the Prandtl number is given as: viscous diffusion rate ν Cpμ Pr = = = thermal diffusion rate α k where: ν: kinematic viscosity, ν = μ/ρ, (SI units : m²/s) k α: thermal diffusivity, α = , (SI units : m²/s) ρCp μ: dynamic viscosity, (SI units : Pa ∗ s = N ∗ s/m²) W k: thermal conductivity, (SI units : ) m∗K J C : specific heat, (SI units : ) p kg∗K ρ: density, (SI units : kg/m³).
    [Show full text]
  • Who Was Who in Transport Phenomena
    l!j9$i---1111-1111-.- __microbiographies.....::..._____:__ __ _ ) WHO WAS WHO IN TRANSPORT PHENOMENA R. B YRON BIRD University of Wisconsin-Madison• Madison, WI 53706-1691 hen lecturing on the subject of transport phenom­ provide the "glue" that binds the various topics together into ena, I have often enlivened the presentation by a coherent subject. It is also the subject to which we ulti­ W giving some biographical information about the mately have to tum when controversies arise that cannot be people after whom the famous equations, dimensionless settled by continuum arguments alone. groups, and theories were named. When I started doing this, It would be very easy to enlarge the list by including the I found that it was relatively easy to get information about authors of exceptional treatises (such as H. Lamb, H.S. the well-known physicists who established the fundamentals Carslaw, M. Jakob, H. Schlichting, and W. Jost). Attention of the subject, but that it was relatively difficult to find could also be paid to those many people who have, through accurate biographical data about the engineers and applied painstaking experiments, provided the basic data on trans­ scientists who have developed much of the subject. The port properties and transfer coefficients. documentation on fluid dynamicists seems to be rather plen­ tiful, that on workers in the field of heat transfer somewhat Doing accurate and responsible investigations into the history of science is demanding and time-consuming work, less so, and that on persons involved in diffusion quite and it requires individuals with excellent knowledge of his­ sparse.
    [Show full text]
  • Flow, Heat Transfer, and Pressure Drop Interactions in Louvered-Fin Arrays
    Flow, Heat Transfer, and Pressure Drop Interactions in Louvered-Fin Arrays N. C. Dejong and A. M. Jacobi ACRC TR-I46 January 1999 For additional information: Air Conditioning and Refrigeration Center University of Illinois Mechanical & Industrial Engineering Dept. 1206 West Green Street Prepared as part ofACRC Project 73 Urbana. IL 61801 An Integrated Approach to Experimental Studies ofAir-Side Heat Transfer (217) 333-3115 A. M. Jacobi, Principal Investigator The Air Conditioning and Refrigeration Center was founded in 1988 with a grant from the estate of Richard W Kritzer, the founder of Peerless of America Inc. A State of Illinois Technology Challenge Grant helped build the laboratory facilities. The ACRC receives continuing support from the Richard W. Kritzer Endowment and the National Science Foundation. The following organizations have also become sponsors of the Center. Amana Refrigeration, Inc. Brazeway, Inc. Carrier Corporation Caterpillar, Inc. Chrysler Corporation Copeland Corporation Delphi Harrison Thermal Systems Eaton Corporation Frigidaire Company General Electric Company Hill PHOENIX Hussmann Corporation Hydro Aluminum Adrian, Inc. Indiana Tube Corporation Lennox International, Inc. rvfodine Manufacturing Co. Peerless of America, Inc. The Trane Company Thermo King Corporation Visteon Automotive Systems Whirlpool Corporation York International, Inc. For additional information: .Air.Conditioning & Refriger-ationCenter Mechanical & Industrial Engineering Dept. University ofIllinois 1206 West Green Street Urbana IL 61801 2173333115 ABSTRACT In many compact heat exchanger applications, interrupted-fin surfaces are used to enhance air-side heat transfer performance. One of the most common interrupted surfaces is the louvered fin. The goal of this work is to develop a better understanding of the flow and its influence on heat transfer and pressure drop behavior for both louvered and convex-louvered fins.
    [Show full text]
  • Dimensionless Numbers
    Dimensionless Numbers 3.185 November 24, 2003 Note: you are not responsible for knowing the different names of the mass transfer dimensionless numbers, just call them, e.g., “mass transfer Prandtl number”, as many people do. Those names are given here because some people use them, and you’ll probably hear them at some point in your career. Heat Transfer Mass Transfer Biot Number M.T. Biot Number Ratio of conductive to convective H.T. resistance Ratio of diffusive to reactive or conv MT resistance Determines uniformity of temperature in solid Determines uniformity of concentration in solid Bi = hL/ksolid Bi = kL/Dsolid or hD L/Dsolid Fourier Number Ratio of current time to time to reach steady­state Dimensionless time in temperature curves, used in explicit finite difference stability criterion 2 2 Fo = αt/L Fo = Dt/L Knudsen Number: Ratio of gas molecule mean free path to process lengthscale Indicates validity of line­of­sight (> 1) or continuum (< 0.01) gas models λ kT Kn = = √ L 2πσ2 P L Reynolds Number: Ratio of convective to viscous momentum transport Determines transition to turbulence, dynamic pressure vs. viscous drag Re = ρU L/µ = U L/ν Prandtl Number M.T. Prandtl (Schmidt) Number Ratio of momentum to thermal diffusivity Ratio of momentum to species diffusivity Determines ratio of fluid/HT BL thickness Determines ratio of fluid/MT BL thickness Pr = ν/α = µcp/k Pr = ν/D = µ/ρD Nusselt Number M.T. Nusselt (Sherwood) Number Ratio of lengthscale to thermal BL thickness Ratio of lengthscale to diffusion BL thickness Used to calculate heat transfer
    [Show full text]
  • A Similarity Analysis for Heat Transfer in Newtonian and Power Law Fluids Using the Instantaneous Wall Shear Stress
    A Similarity Analysis for Heat Transfer in Newtonian and Power Law Fluids Using the Instantaneous Wall Shear Stress Trinh, Khanh Tuoc [email protected] B.H.P. Wilkinson [email protected] Institute of Food Nutrition and Human Health Massey University, New Zealand N.K.Kiaka [email protected] Department of Applied sciences The Papua New Guinea University of technology Abstract This paper presents a technique that collapses existing experimental data of heat transfer in pipe flow of Newtonian and power law fluids into a single master curve. It also discusses the theoretical basis of heat, mass and momentum analogies and the implications of the present analysis to visualisations of turbulence. Key words: Heat transfer, transition, turbulence, Newtonian, power law, instantaneous wall shear stress, similarity plot 1 Introduction The study of heat and mass transfer has been dominated from an early stage by the similar form of the equations of heat, mass and momentum. Following Boussinesq (1877), the transport flux (e.g. of heat) can be defined in terms of an eddy viscosity d q k E (1) h dy where is the temperature y the normal distance from the wall k the thermal conductivity Eh the eddy thermal diffusivity q the rate of heat transfer flux the fluid density Equation (1) may be rearranged as y q q w dy (2) E 0 1 h k which is very similar to the equation for momentum transport y U w dy (3) E 0 1 Where U U u* , y yu* yu* and Cp w u* q w have been normalised with the friction velocity u* w and the fluid apparent viscosity .
    [Show full text]
  • Mass Or Heat Transfer Inside a Spherical Gas Bubble at Low to Moderate Reynolds Number Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud
    Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud To cite this version: Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud. Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number. International Journal of Heat and Mass Transfer, Elsevier, 2013, vol. 67, pp.1096-1105. 10.1016/j.ijheatmasstransfer.2013.08.069. hal-00919752 HAL Id: hal-00919752 https://hal.archives-ouvertes.fr/hal-00919752 Submitted on 17 Dec 2013 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. Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 10540 To link to this article : DOI:10.1016/j.ijheatmasstransfer.2013.08.069 URL : http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.08.069 To cite this version : Colombet, Damien and Legendre, Dominique and Cockx, Arnaud and Guiraud, Pascal Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number.
    [Show full text]
  • 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 .................................................................................................................
    [Show full text]