Numerical Solutions of an Unsteady 2-D Incompressible Flow with Heat
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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. -
General@ Electric Space Sciences Laboratory Theoretical Fluid Physics Section
4 3 i d . GPO PRICE $ CFSTI PRICE(S) $ R65SD50 Microfiche (MF) , 7.3, We53 July85 THE STRUCTURE OF THE VISCOUS HYPERSONIC SHOCK LAYER SPACE SCIENCES LABORATORY . MISSILE AND SPACE DIVISION GENERAL@ ELECTRIC SPACE SCIENCES LABORATORY THEORETICAL FLUID PHYSICS SECTION THE STRUCTURE OF THE VISCOUS HYPERSONIC SHOCK LAYER BY L. Goldberg . Work performed for the Space Nuclear Propulsion Office, NASA, under Contract No. SNPC-29. f This report first appeared as part of Contract Report DIN: 214-228F (CRD), October 1, 1965. Permission for release of this publication for general distribution was received from SNPO on December 9, 1965. R65SD50 December, 1965 MISSILE AND SPACE DIVISION GENERAL ELECTRIC CONTENTS PAGE 1 4 I List of Figures ii ... Abstract 111 I Symbols iv I. INTRODUCTION 1 11. DISCUSSION OF THE HYPER NIC CO TTINUI 3 FLOW FIELD 111. BASIC RELATIONS 11 IV . BOUNDARY CONDITIONS 14 V. NORMALIZED SYSTEM OF EQUATIONS AND BOUNDARY 19 CONDITIONS VI. DISCUSSION OF RESULTS 24 VII. CONCLUSIONS 34 VIII. REFERENCES 35 Acknowledgements 39 Figures 40 1 LIST OF FIGURES 'I PAGE 1 1 1. Hypersonic Flight Regimes 40 'I 2. Coordinate System 41 3. Profiles Re = 15, 000 42 S 3 4. Profiles Re = 10 43 S 3 5. Profiles Re = 10 , f = -0.4 44 S W 2 6. Profiles Re = 10 45 S 2 7. Profiles Re = 10 , f = do. 4 46 S W Profiles Re = 10 47 8. s 9. Normalized Boundary Layer Correlations 48 10. Reduction in Skin Friction and Heat Transfer with Mass 49 Transfer 11. Normalized Heat Transfer 50 12. Normalized Skin Friction 51 13. -
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. -
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. -
Inertia-Less Convectively-Driven Dynamo Models in the Limit
Inertia-less convectively-driven dynamo models in the limit of low Rossby number and large Prandtl number Michael A. Calkins a,1,∗, Keith Julienb,2, Steven M. Tobiasc,3 aDepartment of Physics, University of Colorado, Boulder, CO 80309 USA bDepartment of Applied Mathematics, University of Colorado, Boulder, CO 80309 USA cDepartment of Applied Mathematics, University of Leeds, Leeds, UK LS2 9JT Abstract Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, Pr, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection as long as the buoyancy force is not too strong. We develop asymptotic dynamo models for the case of small Rossby number and large Prandtl number in which inertia is absent on the convective scale. The rele- vant diffusivity parameter for this limit is the compositional Roberts number, q = D/η, which is the ratio of compositional and magnetic diffusivities. Dy- namo models are developed for both order one q and the more geophysically relevant low q limit. For both cases the ratio of magnetic to kinetic energy densities, M, is asymptotically large and reflects the fact that Alfv´en waves have been filtered from the dynamics. Along with previous investigations of asymptotic dynamo models for Pr = O(1), our results show that the ratio M is not a useful indicator of dominant force balances in the momentum equa- tion since many different asymptotic limits of M can be obtained without changing the leading order geostrophic balance. -
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. -
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). -
The Analogy Between Heat and Mass Transfer in Low Temperature Crossflow Evaporation
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AUT Scholarly Commons The analogy between heat and mass transfer in low temperature crossflow evaporation Reza Enayatollahi*, Roy Jonathan Nates, Timothy Anderson Department of Mechanical Engineering, Auckland University of Technology, Auckland, New Zealand. Corresponding Author: Reza Enayatollahi Email Address: [email protected] Postal Address: WD308, 19 St Paul Street, Auckland CBD, Auckland, New Zealand Phone Number: +64 9 921 9999 x8109 Abstract This study experimentally determines the relationship between the heat and mass transfer, in a crossflow configuration in which a ducted airflow passes through a planar water jet. An initial exploration using the Chilton-Colburn analogy resulted in a coefficient of determination of 0.72. On this basis, a re-examination of the heat and mass transfer processes by Buckingham’s-π theorem and a least square analysis led to the proposal of a new dimensionless number referred to as the Lewis Number of Evaporation. A modified version of the Chilton-Colburn analogy incorporating the Lewis Number of Evaporation was developed leading to a coefficient of determination of 0.96. 1. Introduction Heat and mass transfer devices involving a liquid interacting with a gas flow have a wide range of applications including distillation plants, cooling towers and aeration processes and desiccant drying [1-5]. Many studies have gone through characterising the heat and mass transfer in such configurations [6-9]. The mechanisms of heat and mass transfer are similar and analogical. Therefore, in some special cases where, either the heat or mass transfer data are not reliable or may not be available, the heat and mass transfer analogy can be used to determine the missing or unreliable set of data. -
Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels1
Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels1 We investigate the constant-wall-temperature convective heat-transfer characteristics of a model gaseous flow in two-dimensional micro and nano-channels under hydrodynamically Nicolas G. and thermally fully developed conditions. Our investigation covers both the slip-flow Hadjiconstantinou regime 0рKnр0.1, and most of the transition regime 0.1ϽKnр10, where Kn, the Knud- sen number, is defined as the ratio between the molecular mean free path and the channel Olga Simek height. We use slip-flow theory in the presence of axial heat conduction to calculate the Nusselt number in the range 0рKnр0.2, and a stochastic molecular simulation technique Mechanical Engineering Department, known as the direct simulation Monte Carlo (DSMC) to calculate the Nusselt number in Massachusetts Institute of Technology, the range 0.02ϽKnϽ2. Inclusion of the effects of axial heat conduction in the continuum Cambridge, MA 02139 model is necessary since small-scale internal flows are typically characterized by finite Peclet numbers. Our results show that the slip-flow prediction is in good agreement with the DSMC results for Knр0.1, but also remains a good approximation beyond its ex- pected range of applicability. We also show that the Nusselt number decreases monotoni- cally with increasing Knudsen number in the fully accommodating case, both in the slip-flow and transition regimes. In the slip-flow regime, axial heat conduction is found to increase the Nusselt number; this effect is largest at Knϭ0 and is of the order of 10 percent. Qualitatively similar results are obtained for slip-flow heat transfer in circular tubes. -
Fluid Dynamics Analysis and Numerical Study of a Fluid Running Down a Flat Surface
29TH DAAAM INTERNATIONAL SYMPOSIUM ON INTELLIGENT MANUFACTURING AND AUTOMATION DOI: 10.2507/29th.daaam.proceedings.116 FLUID DYNAMICS ANALYSIS AND NUMERICAL STUDY OF A FLUID RUNNING DOWN A FLAT SURFACE Juan Carlos Beltrán-Prieto & Karel Kolomazník This Publication has to be referred as: Beltran-Prieto, J[uan] C[arlos] & Kolomaznik, K[arel] (2018). Fluid Dynamics Analysis and Numerical Study of a Fluid Running Down a Flat Surface, Proceedings of the 29th DAAAM International Symposium, pp.0801-0810, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-902734- 20-4, ISSN 1726-9679, Vienna, Austria DOI: 10.2507/29th.daaam.proceedings.116 Abstract The case of fluid flowing down a plate is applied in several industrial, chemical and engineering systems and equipments. The mathematical modeling and simulation of this type of system is important from the process engineering point of view because an adequate understanding is required to control important parameters like falling film thickness, mass rate flow, velocity distribution and even suitable fluid selection. In this paper we address the numerical simulation and mathematical modeling of this process. We derived equations that allow us to understand the correlation between different physical chemical properties of the fluid and the system namely fluid mass flow, dynamic viscosity, thermal conductivity and specific heat and studied their influence on thermal diffusivity, kinematic viscosity, Prandtl number, flow velocity, fluid thickness, Reynolds number, Nusselt number, and heat transfer coefficient using numerical simulation. The results of this research can be applied in computational fluid dynamics to easily identify the expected behavior of a fluid that is flowing down a flat plate to determine the velocity distribution and values range of specific dimensionless parameters and to help in the decision-making process of pumping systems design, fluid selection, drainage of liquids, transport of fluids, condensation and in gas absorption experiments. -
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. -
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.