Free Convection Film Flows and Heat Transfer

8/8/2019 Free Convection Film Flows and Heat Transfer

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Deyi Shang

Free ConvectionFilm Flowsand Heat TransferW ith 109 Figures and 69 Tables


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Contents

1 Introduction 11.1 Scope 11.2 Application Backgrounds 11.3 Previous Developments 2

1.3.1 For Accelerating Boundary Layers and Film Flowof Newtonian Fluids 2

1.3.2 For Gravity-Driven Film Flow of Non-NewtonianPower-Law Fluids 61.4 Recent Development 7

1.4.1 A Novel System of Analysis Models 71.4.2 A New Approach for the Treatment

of Variable Thermophysical Properties 81.4.3 Hydrodynamics and Heat and Mass Transfer 91.4.4 Recent Experimental Measurements

of Velocity Field in Boundary Layer 12References 13

Part I Laminar Free Convection2 Basic Conservation Equations for Laminar Free Convection 21

2.1 Continuity Equation 222.2 Momentum Equation (Navier-Stokes Equations) 242.3 Energy Equation 272.4 Basic Equations of Free Convection Boundary Layer 30

2.4.1 Continuity Equation 302.4.2 Momentum Equations (Navier-Stokes Equations) 312.4.3 Energy Equations 34


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X Contents3 Brief Review of Previous Method for Analysis

of Laminar Free Convection 373.1 Falkner-Skan Transformation for Fluid Laminar Forced

Convection 383.2 Falkner-Skan Transformation for Fluid Laminar Free

Convection 423.2.1 For Boussinesq Approximation 423.2.2 Consideration of Variable Thermophysical Properties . . 44

3.3 Some Previous Methods for Treatment of VariableThermophysical Properties 45

References 474 Laminar Free Convection of Monatomic and Diatomic

Gases, Air, and Water Vapor 494.1 Introduction 504.2 Governing Partial Differential Equations 514.3 Similarity Transformation of the Governing Equations 52

4.3.1 Assumed Dimensionless Variables with VelocityComponent Method 52

4.3.2 The Similarity Transformation 534.4 Treatment of Variable Thermophysical Properties 58

4.4.1 Temperature Parameters 584.4.2 Temperature Parameter Method 624.5 Heat Transfer Analysis 644.6 Numerical Results 654.7 Effect of Variable Thermophysical Properties on Heat Transfer 684.8 Summary 704.9 Remarks 714.10 Calculation Example 73References 74

5 Laminar Free Convection of Polyatomic Gas 775.1 Introduction 785.2 Variable Thermophysical Properties 795.3 Governing Partial Differential Equations and their

Similarity Transformations 795.4 Heat Transfer Analysis 855.5 Numerical Solutions 865.6 Curve-Fit Formulas for Heat Transfer 885.7 Summary 925.8 Remarks 925.9 Calculation Example 94References 95


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Contents XI6 Lam inar Free C onv ection of Liquid 976.1 Introduction 986.2 Governing Partial Deferential Equations and theirSimilarity Transformation 996.2.1 Governing Par tia l Differential Equations 996.2.2 Dimensionless Transformation Variables 1006.2.3 Similarity Transformation 1006.2.4 Identical Buoyancy Factor 1026.3 Trea tment of Variable Thermophysical Properties 1026.4 Heat Transfer Analysis 1046.5 Numerical Solutions 1046.6 A Curve-Fit Formula for Heat Transfer 109

6.7 Summary I l l6.8 Remarks I l l6.9 Calculation Examples 113References 1157 H eat Transfer D evia tion of Laminar Free Co nvectionCaused by Boussinesq Approximation 1177.1 Introduc tion 1187.2 Governing Equations of Fluid Laminar Free Convection

under Boussinesq Approximation 1197.2.1 For Fluid Lam inar Free Convection 1197.2.2 For Gas Laminar Free Convection 1217.3 Heat Transfer Deviation of Liquid Lam inar Free ConvectionCaused by Boussinesq Approximation 1217.3.1 Boussinesq Solutions for Lam inar Free Convection 1217.3.2 Models for Predic ted Deviation on Heat TransferCaused by Boussinesq Approximation 1227.3.3 Pred iction of Heat Transfer Deviation E^xfor Water Lam inar Free Convection 1247.4 Heat Transfer Deviation of Gas Lam inar Free ConvectionCaused by Boussinesq Approximation 1287.4.1 Boussinesq Solutions for Gas Lam inar Free Convection . 1287.4.2 Models on Predic ted Deviation of Heat Transfer ofGas Laminar Free Convection Caused by BoussinesqApproximation 1297.4.3 Prediction Results of Deviation E^x for Gas LaminarFree Convection 1307.5 Sum mary 1347.6 Rem arks 1347.7 Calculation example 136References 138


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XII Contents8 Expe rimental M easur em ents of Free Con vection w ithLarge Temperature Difference 1398.1 Introduction 140

8.2 Experim ental Measurem ents of Velocity Field for AirLaminar Free Convection 1418.2.1 Experimental Devices and Instruments 1418.2.2 Measurement Results 1438.2.3 Governing Equations 1438.2.4 The Num erical Solutions 1468.3 Experim ental Measurements of Velocity Fieldfor Water Lam inar Free Convection 1478.3.1 Main Experimental Apparatu s 1478.3.2 The Results of Experiment 1488.3.3 Governing Equations 1488.3.4 Num erical Solutions 1528.4 Rem arks 153References 160

9 Re lationship on Laminar Free Con vection and H eatTransfer Between Inclined and Vertical Cases 1619.1 Introduction 1639.2 Fluid Free Convection on inclined pla te 1649.2.1 Physical Model and Basic Equations 1649.2.2 Similarity Transformation of the Basic Equations 1659.2.3 Relationships of Momentum, Heat, and Mass Transferbetween Inclined and Vertical Cases 1669.3 Gas Free Convection on Inclined Plate 1739.4 Summary 1749.5 Rem arks 1749.6 Calculation Example 175Appendix A. Derivation of Equations (9.1)-(9.3) 1771 Derivation of equation (9.1) 1772 Derivation of equation (9.2) 1793 Derivation of equation (9.3) 181References 182

Part II Film Boiling and Condensation10 Laminar Film Boiling of Satu rated Liquid 18710.1 Introduction 18910.2 Governing Partia l Differential Equations 19010.3 Similarity Transformation 19110.3.1 Similarity Transformation Variables 19110.3.2 Similarity Transformation 192


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Contents XIII10.4 Numerical Calculation 19710.4.1 Trea tment of Variable Thermophysical Properties 19710.4.2 Numerical Calculation 19810.4.3 Numerical Results 20010.5 Heat Transfer 20110.5.1 Heat Transfer Analysis 20110.5.2 Curve-fit Equation for Heat Transfer 20310.6 Mass Transfer 20510.6.1 Mass Transfer Analysis 20510.6.2 Curve-Fit Formulae for Mass Transfer 20710.7 Remarks 20710.8 Calculation Exam ple 209References 213

11 Laminar Film Boiling of Su bcooled Liquid 21511.1 Introduction 21611.2 Governing Par tia l Differential Equations 21711.3 Similarity Transformation 21911.3.1 Transformation Variables 21911.3.2 Similarity Transformation 22011.4 Numerical Calculation 22511.4.1 Treatment of Variable Thermophysical Properties 22511.4.2 Numerical Calculation 22711.5 Heat and Mass transfer 23211.5.1 Heat Transfer Analysis 23211.5.2 Curve-Fit Equations for Heat Transfer 23311.5.3 Mass Transfer Analysis 23411.6 Summary 23811.7 Rem arks 23811.8 Calculation Exam ple 243References 245

12 Laminar Film Cond ensation of Satu rated Vapor 24712.1 Introd uction 24812.2 Governing Partia l Differential Equations 25012.3 Similarity Variables 25112.4 Similarity Transformation of Governing Equations 25212.5 Numerical Solutions 25312.5.1 Treatment of Variable Thermophysical Properties 25312.5.2 Calculation Procedure 25512.5.3 Solution 25512.6 Heat and Mass Transfer 25612.6.1 Analysis for Heat and Mass Transfer 25612.6.2 Curve-Fit Equations for Heat and Mass Transfer 26012.7 Rem arks 265


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XIV Contents12.8 Calcu lation Example 265Appendix A. Derivation of Similarity Transformation of GoverningEquations (12.1)-(12.5) 270References 276

13 Effects of Various Ph ysical Con dition s on FilmCondensations 27713.1 Introduction 27913.2 Review of Governing Equations for Film Condensation ofSaturated Vapor 28013.2.1 Par tia l Differential Equations 28013.2.2 Similarity Variables 28113.2.3 Transformed Dimensionless Differential Equations 28213.3 Different Physical Assum ptions 28313.3.1 Assumption a (with Boussinesq Approximationof Condensate Film) 28313.3.2 Assumption b (Ignoring Shear Force at Liquid-Vapor Interface) 28413.3.3 Assumption c (Ignoring Inertia Forceof the Condensate Film) 28513.3.4 Assumption d (Ignoring Thermal Convection

of the Condensate Film) 28513.4 Effects of Various Physical Conditions on Velocity andTem perature Fields 28513.5 Effects of Various Physical Conditions on Heat Transfer 28713.6 Effects of Various Physical Conditions on CondensateFilm Thickness 28813.7 Effect of Various Physical Conditions on Mass Flow Rateof the Condensation 29313.8 Rem arks 29813.8.1 Effects of Boussinesq Approximation 29813.8.2 Effects of Shear Force at th e Liquid-Vapor Interface . . . 29813.8.3 Effect of Ine rtial Force of the Condensate Film 29913.8.4 Effects of Thermal Convection of the Condensate Film . 299References 30014 Laminar Film C ond ensa tion of Su perh eated Vapor 30114.1 Introduction 303

14.2 Governing Partial Differential Equations with Two-Phase Film 30414.3 Similarity Transformation 30514.3.1 Transformation Variables 30514.3.2 Ordinary Differential Equations 30614.4 Treatm ent of Variable Therm ophysical Pro per ties 30814.5 Numerical Solutions 31014.5.1 Calculation Procedure 310


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Contents XV14.5.2 Numerical Solution 31114.6 Heat Transfer 31314.6.1 Heat Transfer 31314.7 Condensate Mass Flow Rate 316

14.8 Summary 32114.9 Rem arks 32114.10Calculation Exam ple 326References 329Part III Falling Film Flow of Non-Newtonian Fluids15 H ydro dyn am ics of Falling Film Flow of N on -N ew ton ianPower-Law Fluids 33315.1 Principal Types of Power-Law Fluids 33415.1.1 Newtonian Fluids 33415.1.2 Power-Law Flu ids 33415.2 Introduction of Studies on Hydrodynamics of Gravity-DrivenFilm Flow of Non-Newtonian Power-Law Fluids (FFNF) 33615.3 Physical Model and Governing Partial Differential Equations. . 33815.4 A New Similarity Transformation 340

15.5 Num erical Solutions 34215.6 Local Skin-Friction Coefficient 34415.7 Mass Flow Rate 34615.8 Length of Boundary Layer Region 34815.9 Critical Film Thickness 34915.10Effect of Wall Inclination 35015.11Summary 35115.12Remarks 35415.13Calculation Exam ple 354References 358

16 Pseu dosim ilarity and Bou nda ry Layer Thickness forNon-Newtonian Fal l ing Fi lm Flow 36116.1 Introduction 36216.2 Physical Model and Governing Partial Differential Equations. . 36316.3 Similarity Transformation 36516.4 Local Prand tl Num ber 36816.5 Pseudosim ilarity for Energy Equation 36916.6 Critical Local Prand tl Number 37116.7 Analysis of Boundary Layer Thickness 37216.7.1 Precautions for Prx > Pr* 37216.7.2 Precautions for Prx < Pr* 37316.8 Rem arks 375References 377


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XVI Contents17 H eat Transfer of th e Falling Film F low 37917.1 Introduction 38017.2 Governing Equations 38117.3 Heat Transfer Analysis 383

17.4 Num erical Solution for Heat Transfer 38517.5 Local Similarity vs. Local Pseudosimilarity 38917.6 Summary 39117.7 Rem arks 39117.8 Calculation Example 394References 397A Tables with Therm ophysical Pro perties 399References 405Index 407

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Free Convection Film Flows and Heat Transfer