Lecture notes on Heat and Mass transfer

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Section 5 Heat and Mass Transfer James G. Knudsen, Ph.D., Professor Emeritus of Chemical Engineering, Oregon State University; Member, American Institute of Chemical Engineers, American Chemical Society; Registered Professional Engineer (Oregon). (Conduction and Convection; Condensation, Boil- ing; Section Coeditor) Hoyt C. Hottel, S.M., Professor Emeritus of Chemical Engineering, Massachusetts Insti- tute of Technology; Member, National Academy of Sciences, American Academy of Arts and Sci- ences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Adel F. Sarofim, Sc.D., Lammot du Pont Professor of Chemical Engineering and Assis- tant Director, Fuels Research Laboratory, Massachusetts Institute of Technology; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Phillip C. Wankat, Ph.D., Professor of Chemical Engineering, Purdue University; Mem- ber, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. (Mass Transfer Section Coeditor) Kent S. Knaebel, Ph.D., President, Adsorption Research, Inc.; Member, American Insti- tute of Chemical Engineers, American Chemical Society, International Adsorption Society. Pro- fessional Engineer (Ohio). (Mass Transfer Section Coeditor) HEAT TRANSFER HEAT TRANSFER BY CONVECTION Modes of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 The Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 HEAT TRANSFER BY CONDUCTION Individual Coefficient of Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . 5-12 Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Overall Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Three-Dimensional Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . 5-8 Representation of Heat-Transfer Film Coefficients . . . . . . . . . . . . . . 5-12 Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 Natural Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Steady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 Nusselt Equation for Various Geometries . . . . . . . . . . . . . . . . . . . . . . 5-12 One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 Simplified Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Conduction through Several Bodies in Series . . . . . . . . . . . . . . . . . . . 5-9 Simultaneous Loss by Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Conduction through Several Bodies in Parallel. . . . . . . . . . . . . . . . . . 5-10 Enclosed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Several Bodies in Series with Heat Generation . . . . . . . . . . . . . . . . . 5-10 Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14 Example 1. Steady-State Conduction with Heat Generation . . . . . . . 5-10 Analogy between Momentum and Heat Transfer. . . . . . . . . . . . . . . . 5-14 Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Unsteady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16 One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16 Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11 Example 2. Calculation of j Factors in an Annulus . . . . . . . . . . . . . . . 5-17 Conduction with Change of Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11 Jackets and Coils of Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19 The contribution to the section on Interphase Mass Transfer of Mr. William M. Edwards (editor of Sec. 14), who was an author for the sixth edition, is acknowledged. 5-15-2 HEAT AND MASS TRANSFER Nonnewtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19 Self Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Liquid Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19 Tracer Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Mass-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Problem Solving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 HEAT TRANSFER WITH CHANGE OF PHASE Continuity and Flux Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Condensation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Flux Expressions: Simple Integrated Forms of Fick’s First Law . . . . 5-47 Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Stefan-Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47 Boiling (Vaporization) of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 Diffusivity Estimation—Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-48 Boiling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 Binary Mixtures—Low Pressure—Nonpolar Components . . . . . . . . 5-48 Boiling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 Binary Mixtures—Low Pressure—Polar Components . . . . . . . . . . . . 5-49 Self-Diffusivity—High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49 Supercritical Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49 HEAT TRANSFER BY RADIATION Low-Pressure/Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . 5-50 General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23 Diffusivity Estimation—Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-50 Nomenclature for Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23 Stokes-Einstein and Free-Volume Theories . . . . . . . . . . . . . . . . . . . . 5-50 Nature of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-24 Dilute Binary Nonelectrolytes: General Mixtures . . . . . . . . . . . . . . . 5-50 Blackbody Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-24 Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Radiative Exchange between Surfaces and Solids . . . . . . . . . . . . . . . . . 5-25 Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-51 Emittance and Absorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25 Dilute Binary Mixtures of a Nonelectrolyte in Water . . . . . . . . . . . . . 5-52 Black-Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-27 Dilute Binary Hydrocarbon Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 5-52 Example 3: Calculation of View Factor . . . . . . . . . . . . . . . . . . . . . . . . 5-29 Dilute Binary Mixtures of Nonelectrolytes with Water as the Example 4: Calculation of Exchange Area. . . . . . . . . . . . . . . . . . . . . . 5-29 Solute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-52 Non-Black Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29 Dilute Dispersions of Macromolecules in Nonelectrolytes . . . . . . . . 5-52 Example 5: Radiation in a Furnace Chamber . . . . . . . . . . . . . . . . . . . 5-31 Concentrated, Binary Mixtures of Nonelectrolytes . . . . . . . . . . . . . . 5-52 Emissivities of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-32 Binary Electrolyte Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-53 Gaseous Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-33 Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-54 Example 6: Calculation of Gas Emissivity and Absorptivity . . . . . . . . 5-34 Diffusion of Fluids in Porous Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-54 Flames and Particle Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35 Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-54 Radiative Exchange between Gases or Suspended Matter Mass-Transfer Principles: Dilute Systems . . . . . . . . . . . . . . . . . . . . . . 5-54 and a Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36 Mass-Transfer Principles: Concentrated Systems . . . . . . . . . . . . . . . . 5-56 Example 7: Radiation in Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36 HTU (Height Equivalent to One Transfer Unit) . . . . . . . . . . . . . . . . 5-57 Single-Gas-Zone/Two-Surface-Zone Systems . . . . . . . . . . . . . . . . . . . 5-37 NTU (Number of Transfer Units) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-57 The Effect of Nongrayness of Gas on Total-Exchange Area . . . . . . . 5-37 ˆ ˆ Definitions of Mass-Transfer Coefficients k and k . . . . . . . . . . . . . 5-57 G L Example 8: Effective Gas Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38 Simplified Mass-Transfer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-57 Treatment of Refractory Walls Partially Enclosing a Radiating Mass-Transfer Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-58 Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-39 ˆ ˆ Combustion Chamber Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-40 Effects of Total Pressure on k and k . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 G L ˆ ˆ Example 9: Radiation in a Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-41 Effects of Temperature on k and k . . . . . . . . . . . . . . . . . . . . . . . . . . 5-64 G L ˆ ˆ Effects of System Physical Properties on k and k . . . . . . . . . . . . . . 5-66 G L ˆ ˆ Effects of High Solute Concentrations on k and k . . . . . . . . . . . . . 5-69 G L MASS TRANSFER ˆ ˆ Influence of Chemical Reactions on k and k . . . . . . . . . . . . . . . . . . 5-69 G L General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42 Effective Interfacial Mass-Transfer Area a . . . . . . . . . . . . . . . . . . . . . 5-74 ˆ ˆ Fick’s First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42 Volumetric Mass-Transfer Coefficients K a and K a . . . . . . . . . . . . . 5-78 G L Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient . . . . 5-46 Chilton-Colburn Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-79HEAT AND MASS TRANSFER 5-3 Nomenclature and Units Specialized heat transfer nomenclature used for radiative heat transfer is defined in the subsection “Heat Transmission by Radiation.” Nomenclature for mass trans- fer is defined in the subsection “Mass Transfer.” Symbol Definition SI units U.S. customary units a Proportionality coefficient Dimensionless Dimensionless 2 2 a Cross-sectional area of a fin m ft x a¢ Proportionality factor 2 2 A Area of heat transfer surface; A for inside; A for outside; A for m ft i o m mean; A for average; A , A , and A for points 1, 2, and 3 avg 1 2 3 respectively; A for bare surface of finned tube; A for finned portion B f of tube; A for external area of unfinned portion of finned tube; A uf of for external area of finned tube before fins are attached, equals A ; o A for effective area of finned surface; A for total external area of oe T finned tube; A for surface area of dirt (scale) deposit d b Proportionality coefficient b¢ Proportionality factor b Height of fin m ft f - 0.5 B Material constant = 5D c , c , etc. Constants of integration 1 2 c, c Specific heat at constant pressure; c for specific heat of solid; c for J/(kg K) Btu/(lb F) p s g specific heat of gas C Thermal conductance, equals kA/x, hA, or UA; C , C , C , C , J/(s K) Btu/(h F) 1 2 3 n thermal conductance of sections 1, 2, 3, and n respectively of a composite body C Correlating constant; proportionality coefficient Dimensionless Dimensionless r d Depth of divided solids bed m ft m D Diameter; D for outside; D for inside; D for root diameter of finned m ft o i r tube D Diameter of a coil or helix m ft c D Equivalent diameter of a cross section, usually 4 times free area m ft e divided by wetted perimeter; D for equivalent diameter of window w D Diameter of a jacketed cylindrical vessel m ft j D Outside diameter of tube bundle m ft otl D Diameter of packing in a packed tube m ft p D Inside diameter of heat-exchanger shell m ft s D Solids-processing vessel diameter m ft t D , D Diameter at points 1 and 2 respectively; inner and outer diameter of m ft 1 2 annulus respectively E Eddy conductivity of heat J/(s m K) Btu/(h ft F) H E Eddy viscosity Pa s lb/(ft h) M f Fanning friction factor; f for inner wall and f for outer wall of annulus; f Dimensionless Dimensionless 1 2 k for ideal tube bank; skin friction drag coefficient F Entrance factors 2 2 F Dry solids feed rate kg/(s m ) lb/(h ft ) a 3 2 3 2 F Gas volumetric flow rate m (s m of bed area) ft /(h ft of bed area) g F Fraction of total tubes in cross-flow; F for fraction of cross-flow area c bp available for bypass flow F Factor, ratio of temperature difference across tube-side film to overall Dimensionless Dimensionless t mean temperature difference F Factor, ratio of temperature difference across shell-side film to overall Dimensionless Dimensionless s mean temperature difference F Factor, ratio of temperature difference across retaining wall to overall Dimensionless Dimensionless w mean temperature difference between bulk fluids F Factor, ratio of temperature difference across combined dirt or scale Dimensionless Dimensionless D films to overall mean temperature difference between bulk fluids F Temperature-difference correction factor T 2 8 2 g, g Acceleration due to gravity 981 m/s (4.18)(10 ) ft/h L 2 8 2 g Conversion factor 1.0 (kg m)/(N s ) (4.17)(10 )(lb ft)/(lbf h ) c 2 2 G Mass velocity, equals Vr or W/S; G for vapor mass velocity kg/(m s) lb/(h ft ) v 2 2 G Mass velocity through minimum free area between rows of tubes kg/(m s) lb/(h ft ) max normal to the fluid stream 2 2 G Minimum fluidizing mass velocity kg/(m s) lb/(h ft ) mf 2 2 h Local individual coefficient of heat transfer, equals dq/(dA)(D T) J/(m s K) Btu/(h ft F) 2 2 h , h Film coefficient based on arithmetic-mean temperature difference and J/(m s K) Btu/(h ft F) am lm logarithmic-mean temperature difference respectively 2 2 h Film coefficient delivered at base of fin J/(m s K) Btu/(h ft F) b 2 2 h Effective combined coefficient for simultaneous gas-vapor cooling and J/(m s K) Btu/(h ft F) cg vapor condensation 2 2 h + h Combined coefficient for conduction, convection, and radiation J/(m s K) Btu/(h ft F) c r between surface and surroundings5-4 HEAT AND MASS TRANSFER Nomenclature and Units (Continued) Symbol Definition SI units U.S. customary units 2 2 h , h Film coefficient for dirt or scale on outside or inside respectively of a J/(m s K) Btu/(h ft F) do di surface 2 2 h Film coefficient for finned-tube exchangers based on total external J/(m s K) Btu/(h ft F) f surface 2 2 h Effective outside film coefficient of a finned tube based on inside area J/(m s K) Btu/(h ft F) fi 2 2 h Film coefficient for air film of an air-cooled finned-tube exchanger J/(m s K) Btu/(h ft F) fo based on external bare surface 2 2 h , h Effective film coefficient for dirt or scale on heat-transfer surface J/(m s K) Btu/(h ft F) F s 2 2 h , h Film coefficient for heat transfer for inside and outside surface J/(m s K) Btu/(h ft F) i o respectively 2 2 h Film coefficient for ideal tube bank; h for shell side of baffled J/(m s K) Btu/(h ft F) k s exchanger; h for coefficient at liquid-vapor interface sv 2 2 h Condensing coefficient on top tube; h coefficient for N tubes in a J/(m s K) Btu/(h ft F) 1 N vertical row 2 2 h¢ Film coefficient for enclosed spaces J/(m s K) Btu/(h ft F) 2 2 h Film coefficient based on log-mean temperature difference J/(m s K) Btu/(h ft F) lm 2 2 h Heat-transfer coefficient for radiation J/(m s K) Btu/(h ft F) r 2 2 h Coefficient of total heat transfer by conduction, convection, and J/(m s K) Btu/(h ft F) T radiation between the surroundings and the surface of a body subject to unsteady-state heat transfer 2 2 h Equivalent coefficient of retaining wall, equals k/x J/(m s K) Btu/(h ft F) w j Ordinate, Colburn j factor, equals f/2; j for heat transfer; j for Dimensionless Dimensionless H H1 inner wall of annulus; j for outer wall of annulus; j for heat H2 k transfer for ideal tube bank J Mechanical equivalent of heat 1.0(N m)/J 778(ft lbf)/Btu J , J , J , J Correction factors for baffle bypassing, baffle configuration, baffle b c l r leakage, and adverse temperature gradient respectively 2 k Thermal conductivity; k , k , k , thermal conductivities of bodies 1, 2, J/(m s K) (Btu ft)/(h ft F) 1 2 3 and 3 2 k Thermal conductivity of vapor; k for liquid thermal conductivity; k J/(m s K) (Btu ft)/(h ft F) v 1 s for thermal conductivity of solid 2 k , k Mean thermal conductivity J/(m s K) (Btu ft)/(h ft F) avg m 2 k Thermal conductivity of fluid at film temperature J/(m s K) (Btu ft)/(h ft F) f 2 k Thermal conductivity of retaining-wall material J/(m s K) (Btu ft)/(h ft F) w K¢ Property of non-Newtonian fluid l Baffle cut; l for baffle spacing m ft c s L Length of heat-transfer surface m ft L Flow rate kg/s lb/h o L Undisturbed length of path of fluid flow m ft u L Thickness of dirt or scale deposit m ft F L Depth of fluidized bed m ft H L Diameter of agitator blade m ft p m Ratio, term, or exponent as defined where used M Molecular weight kg/mol lb/mol M Weight of fluid kg lb n Position ratio or number Dimensionless Dimensionless n Number of tubes in parallel in a heat exchanger t n Number of rows in a vertical plane r n¢ Flow-behavior index for nonnewtonian fluids n Number of baffle-type coils b N Speed of agitator rad/s r/h r N Number of tubes in a vertical row; or number of tubes in a bundle; N b for number of baffles; N for total number of tubes in exchanger; N T c for number of tubes in one cross-flow section; N for number of cw cross-flow rows in each window N Biot number, h D x/k B T N Proportionality coefficient, dimensionless group Dimensionless Dimensionless d 3 2 2 N Grashof number, L r gbD t/m Gr N Nusselt number, hD/k or hL/k Nu N Peclet number, DGc/k Pe N Prandtl number, cm /k Pr N Reynolds number, DG/m Re N Stanton number, N /N N St Nu Re Pr N Number of sealing strips ss 2 p Pressure kPa lbf/ft abs p Perimeter of a fin m ft f p, p¢ Center-to-center spacing of tubes in tube bundle (tube pitch); p for m ft n tube pitch normal to flow; p for tube pitch parallel to flow pHEAT AND MASS TRANSFER 5-5 Nomenclature and Units (Continued) Symbol Definition SI units U.S. customary units 2 D p Pressure of the vapor in a bubble minus saturation pressure of a flat kPa lbf/ft abs liquid surface 2 P Absolute pressure; P for critical pressure kPa lbf/ft c P¢ Spacing between adjacent baffles on shell side of a heat exchanger m ft (baffle pitch) 2 D P , D P Pressure drop for ideal-tube-bank cross-flow and ideal window kPa lbf/ft bk wk respectively; D P for shell side of baffled exchanger s q Rate of heat flow, equals Q/q W, J/s Btu/h 3 3 q¢ Rate of heat generation J/(s m ) Btu/(h ft ) 2 2 (q/A) Maximum heat flux in nucleate boiling J/(s m ) Btu/(h ft ) max Q Quantity of heat; rate of heat transfer J/s Btu/h Q Quantity of heat; Q for total quantity J Btu T r Radius; cylindrical and spherical coordinate; distance from midplane m ft to a point in a body; r for inner wall of annulus; r for outer wall of 1 2 annulus; r for inside radius of tube; r for distance from midplane i m or center of a body to the exterior surface of the body r Inside radius Dimensionless Dimensionless j R Thermal resistance, equals x/kA, 1/UA, 1/hA; R , R , R , R for (s K)/J (h F)/Btu 1 2 3 n thermal resistance of sections 1, 2, 3, and n of a composite body; R T for sum of individual resistances of several resistances in series or parallel; R and R for dirt or scale resistance on inner and outer di do surface respectively R Ratio of total outside surface of finned tube to area of tube having j same root diameter 2 2 S Cross-sectional area; S for minimum cross-sectional area between m ft m rows of tubes, flow normal to tubes; S for tube-to-baffle leakage area tb for one baffle; S for shell-to-baffle area for one baffle; S for area sb w for flow through window; S for gross window area; S for window wg wt area occupied by tubes S Slope of rotary shell r s Specific gravity of fluid referred to liquid water t Bulk temperature; temperature at a given point in a body at time q K F t , t , t Temperature at points 1, 2, and n in a system through which heat is K F 1 2 n being transferred t¢ Temperature of surroundings K F t¢ , t¢ Inlet and outlet temperature respectively of hotter fluid K F 1 2 t† , t† Inlet and outlet temperature respectively of colder fluid K F 1 2 t Initial uniform bulk temperature of a body; bulk temperature of a K F b flowing fluid t , t High and low temperature respectively on tube side of a heat K F H L exchanger t Surface temperature K F s t Saturated-vapor temperature K F sv t Wall temperature K F w t Temperature of undisturbed flowing stream K F ∞ T , T High and low temperature respectively on shell side of a heat K F H L exchanger T Absolute temperature; T for bulk temperature; T for wall K R b w temperature; T for vapor temperature; T for coolant temperature; v c T for temperature of emitter; T for temperature of receiver e r D T, D t Temperature difference; D t , D t , and D t temperature difference K F, R 1 2 3 across bodies 1, 2, and 3 or at points 1, 2, and 3; D T , D t for overall o o temperature difference; D t for temperature difference between b surface and boiling liquid D t , D t Arithmetic- and logarithmic-mean temperature difference respectively K F am lm D t Mean effective overall temperature difference K F om D T , D t Greater terminal temperature difference K F H H D T , D t Lesser terminal temperature difference K F L L D T , D t Mean temperature difference K F m m u Velocity in x direction m/s ft/h u Friction velocity m/s ft/h 2 2 U Overall coefficient of heat transfer; U for outside surface basis; U¢ for J/(s m K) Btu/(h ft F) o overall coefficient between liquid-vapor interface and coolant 2 2 U , U Overall coefficient of heat transfer at points 1 and 2 respectively J/(s m K) Btu/(h ft F) 1 2 2 2 U , U , U , U Overall coefficients for divided solids processing by conduction, J/(s m K) Btu/(h ft F) co cv ct ra convection, contact, and radiation mechanism respectively 2 2 U Mean overall coefficient of heat transfer J/(s m K) Btu/(h ft F) m v Velocity in y direction m/s ft/h 3 3 V Volume of rotating shell m ft r5-6 HEAT AND MASS TRANSFER Nomenclature and Units (Concluded) Symbol Definition SI units U.S. customary units V Velocity m/s ft/h V¢ , V Velocity m/s ft/s s V Face velocity of a fluid approaching a bank of finned tubes m/s ft/h F 3 3 V , V Specific volume of gas, liquid m /kg ft /lb g l V¢ Maximum velocity through minimum free area between rows of tubes m/s ft/h max normal to the fluid stream w Velocity in z direction m/s ft/h w Flow rate kg/s lb/h W Total mass rate of flow; mass rate of vapor generated; W for total kg/s lb/h F rate of vapor condensation in one tube W Weight rate of flow kg/(s tube) lb/(h tube) r W , W Total mass rate of flow on tube side and shell side respectively of a kg/s lb/h 1 o heat exchanger x Vapor quality, x for inlet quality, x for outlet quality kg/s lb/h q i o x Coordinate direction; length of conduction path; x for thickness of m ft s scale; x , x , and x at positions 1, 2, and 3 in a body through which 1 2 3 heat is being transferred X Factor Dimensionless Dimensionless y Coordinate direction m ft + y Wall distance Dimensionless Dimensionless Y Factor Dimensionless Dimensionless z Coordinate direction m ft z Distance (perimeter) traveled by fluid across fin m ft p Z Ratio of sensible heat removed from vapor to total heat transferred Dimensionless Dimensionless H Greek symbols 2 2 a Thermal diffusivity, equals k/r c; a for effective thermal diffusivity of m /s ft /h e powdered solids - 1 - 1 b Volumetric coefficient of thermal expansion K F b¢ Contact angle of a bubble 2 - n¢ 2 - n¢ g Fluid consistency kg/(s m) lb/(ft s ) G Mass rate of flow of a falling film from a tube or surface per unit kg/(s m) lb/(h ft) perimeter, equals w/p D for vertical tube, w/2L for horizontal tube d Correction factor, ratio of nonnewtonian to newtonian shear rates s d Cell width m ft d Diametral shell-to-baffle clearance m ft sb 2 2 e Eddy diffusivity; e for eddy diffusivity of momentum; e for eddy m /s ft /h M H diffusivity of heat e Fraction of voids in porous bed v h Fluidization efficiency q Time sh q Baffle cut b l Latent heat (enthalpy) of vaporization (condensation) J/kg Btu/lb l Radius of maximum velocity m ft m m Viscosity; m for viscosity at wall temperature; m for viscosity at bulk Pa s lb/(h ft) w b temperature; m for viscosity at film temperature; m , m , and m for f G g v viscosity of gas or vapor; m , m for viscosity of liquid; m for viscosity L l w at wall; m for viscosity of fluid at inner wall of annulus l 2 2 n Kinematic viscosity m /s ft /h 3 3 r Density; r , r for density of liquid; r , r for density of gas or vapor; kg/m lb/ft L l G v r for density of solid s s Surface tension between a liquid and its vapor N/m lbf/ft Term indicating summation of variables 2 2 t Shear stress t for shear stress at the wall N/m lbf/ft w f Velocity-potential function f Particle sphericity p F Viscous-dissipation function w Angle of repose of powdered solid rad rad W Fin efficiency Dimensionless DimensionlessGENERAL 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When the forced velocity is relatively low, it should be realized Conduction is the transfer of heat from one part of a body to that “free-convection” factors, such as density and temperature differ- another part of the same body, or from one body to another in physi- ence, may have an important influence. cal contact with it, without appreciable displacement of the particles Radiation is the transfer of heat from one body to another, not in of the body. contact with it, by means of wave motion through space. HEAT TRANSFER BY CONDUCTION FOURIER’S LAW direction of the flow of heat, i.e., the temperature gradient. The factor k is called the thermal conductivity; it is a characteristic property of Fourier’s law is the fundamental differential equation for heat transfer the material through which the heat is flowing and varies with tem- by conduction: perature. dQ/dq=- kA(dt/dx) (5-1) THREE-DIMENSIONAL CONDUCTION EQUATION where dQ/dq (quantity per unit time) is the rate of flow of heat, A is the area at right angles to the direction in which the heat flows, and Equation (5-1) is used as a basis for derivation of the unsteady-state three-dimensional energy equation for solids or static fluids: - dt/dx is the rate of change of temperature with the distance in theHEAT TRANSFER BY CONDUCTION 5-9 One-Dimensional Conduction Many heat-conduction prob- ¶ t ¶ ¶ t ¶ ¶ t ¶ ¶ t cr = k + k + k + q¢ (5-2) lems may be formulated into a one-dimensional or pseudo-one- 1 2 1 2 1 2 ¶q ¶ x ¶ x ¶ y ¶ y ¶ z ¶ z dimensional form in which only one space variable is involved. Forms of the conduction equation for rectangular, cylindrical, and spherical where x, y, z are distances in the rectangular coordinate system and q¢ coordinates are, respectively, is the rate of heat generation (by chemical reaction, nuclear reaction, 2 or electric current) in the solid per unit of volume. Solution of Eq. ¶ t q¢ =- (5-4a) (5-2) with appropriate boundary and initial conditions will give the 2 ¶ x k temperature as a function of time and location in the material. Equa- tion (5-2) may be transformed into spherical or cylindrical coordinates 1 d dt q¢ to conform more closely to the physical shape of the system. r =- (5-4b) 1 2 r dr dr k THERMAL CONDUCTIVITY 1 d dt q¢ 2 r =- (5-4c) 2 1 2 r dr dr k Thermal conductivity varies with temperature but not always in the same direction. The thermal conductivities for many materials, as a These are second-order differential equations which upon integra- function of temperature, are given in Sec. 2. Additional and more tion become, respectively, comprehensive information may often be obtained from suppliers of 2 t=- (q¢ x /2k) + c x + c (5-5a) 1 2 the materials. Impurities, especially in metals, can give rise to varia- tions in thermal conductivity of from 50 to 75 percent. In using ther- 2 t=- (q¢ r /4k) + c ln r + c (5-5b) 1 2 mal conductivities, engineers should remember that conduction is not 2 t=- (q¢ r /6k) - (c /r) + c (5-5c) the sole method of transferring heat and that, particularly with liquids 1 2 and gases, radiation and convection may be much more important. Constants of integration c and c are determined by the boundary 1 2 The thermal conductivity at a given temperature is a function of the conditions, i.e., temperatures and temperature gradients at known apparent, or bulk, density. Thus, at 0 C (32 F), k for asbestos wool is locations in the system. 0.09 J/(m?s?K) 0.052 Btu/(hr?ft? F) when the bulk density is 400 For the case of a solid surface exposed to surroundings at a differ- 3 3 kg/m (24.9 lb/ft ) and is 0.19 (0.111) for a density of 700 (43.6). ent temperature and for a finite surface coefficient, the boundary In determining the apparent thermal conductivities of granular condition is expressed as solids, such as granulated cork or charcoal grains, Griffiths (Spec. h (t - t¢ ) =- k(dt/dx) (5-6) Rep. 5, Food Investigation Board, H. M. Stationery Office, 1921) T s surf found that air circulates within the mass of granular solid. Under a Inspection of Eqs. (5-5a), (5-5b), and (5-5c) indicates the form of certain set of conditions, the apparent thermal conductivity of a char- temperature profile for various conditions and geometries and also coal was 9 percent greater when the test section was vertical than reveals the effect of the heat-generation term q¢ upon the tempera- when it was horizontal. When the apparent conductivity of a mixture ture distributions. of cellular or porous nonhomogeneous solid is determined, the In the absence of heat generation, one-dimensional steady-state observed temperature coefficient may be much larger than for the conduction may be expressed by integrating Eq. (5-1): homogeneous solid alone, because heat is transferred not only by x t 2 2 dx the mechanism of conduction but also by convection in the gas pock- q E =- E k dt (5-7) ets and by radiation from surface to surface of the individual particles. x A t 1 1 If internal radiation is an important factor, a plot of the apparent con- Area A must be known as a function of x. If k is constant, Eq. (5-7) is ductivity as ordinate versus temperature should show a curve concave expressed in the integrated form upward, since radiation increases with the fourth power of the absolute temperature. Griffiths noted that cork, slag, wool, charcoal, q = kA (t - t )/(x - x ) (5-8) avg 1 2 2 1 and wood fibers, when of good quality and dry, have thermal conduc- x 2 tivities about 2.2 times that of still air, whereas a highly cellular form 1 dx 3 3 where A = E (5-9) avg of rubber, 112 kg/m (7 lb/ft ), had a thermal conductivity only 1.6 x - x x A 1 2 1 times that of still air. In measuring the apparent thermal conductivity Examples of values of A for various functions of x are shown in the avg of diathermanous substances such as quartz (especially when exposed following table. to radiation emitted at high temperatures), it should be remembered that a part of the heat is transmitted by radiation. Bridgman Proc. Am. Acad. Arts Sci., 59, 141 (1923) showed that Area proportional to A avg the thermal conductivity of liquids is increased by only a few percent Constant A = A under a pressure of 100,330 kPa (1000 atm). The thermal conductiv- 1 2 xA - A 2 1 ity of some liquids varies with temperature through a maximum. It is ln (A /A ) 2 1 often necessary for the engineer to estimate thermal conductivities; 2 x w Aw Aw methods are indicated in Sec. 2. Ï 2 1 Equation (5-2) considers the thermal conductivity to be variable. If k is expressed as a function of temperature, Eq. (5-2) is nonlinear and Usually, thermal conductivity k is not constant but is a function of difficult to solve analytically except for certain special cases. Usually in temperature. In most cases, over the ranges of values used the relation complicated systems numerical solution by means of computer is pos- is linear. Integration of Eq. (5-7), with k linear in t, gives sible. A complete review of heat conduction has been given by Davis x 2 and Akers Chem. Eng., 67(4), 187, (5), 151 (1960) and by Davis dx q E = k (t - t ) (5-10) avg 1 2 Chem. Eng., 67(6), 213, (7), 135 (8), 137 (1960). x A 1 where k is the arithmetic-average thermal conductivity between avg STEADY-STATE CONDUCTION temperatures t and t . This average probably gives results which are 1 2 correct within the precision of the data in the majority of cases, For steady flow of heat, the term dQ/dq in Eq. (5-1) is constant and though a special integration can be made whenever k is known to be may be replaced by Q/q or q. Likewise, in Eq. (5-2) the term ¶ t/¶q is greatly different from linear in temperature. zero. Hence, for constant thermal conductivity, Eq. (5-2) may be Conduction through Several Bodies in Series Figure 5-1 expressed as illustrates diagrammatically the temperature gradients accompanying 2 t = (q¢ /k) (5-3) the steady conduction of heat in series through three solids.5-10 HEAT AND MASS TRANSFER For fuel, at x = 0, t = 570 C (1050 F), dt/dx = 0 (this follows if the tempera- ture is finite at the midplane). For fuel and cladding, at x = x , t = t , 1 f c k (dt/dx) = k (dt/dx) f c For cladding, at x = x , 2 t - 400 =- (k /42,600)(dt/dx) c c For the fuel, the first integration of Eq. (10-4a) gives dt /dx=- (q¢ /k )x + c f f 1 which gives c = 0 when the boundary condition is applied. Thus the second inte- 1 gration gives 2 t =- (q¢ /2k )x + c f f 2 from which c is determined to be 570 (1050) upon application of the boundary 2 condition. Thus the temperature profile in the fuel is 2 t =- (q¢ /2k )x + 570 f f FIG. 5-1 Temperature gradients for steady heat conduction in series through The temperature profile in the cladding is obtained by integrating Eq. (10-4a) three solids. twice with q¢= 0. Hence (dt /dx) = c and t = c x + c c 1 c 1 2 There are now three unknowns, c , c , and q¢ , and three boundary conditions by 1 2 Since the heat flow through each of the three walls must be the which they can be determined. same, At x = x , 1 q = (k A D t /x ) = (k A D t /x ) = (k A D t /x ) (5-11) 2 1 1 1 1 2 2 2 2 3 3 3 3 q¢ x /2k + 570 = c x + c - k q¢ x /k = k c 1 f 1 1 2 f 1 f c 1 Since, by definition, individual thermal resistance At x = x , 2 c x + c - 200 =- (k /42,600)c R = x/kA (5-12) 1 2 2 c 1 9 3 8 3 From which q¢= (2.53)(10 ) J/(m s)(2.38)(10 )Btu/(h ft ) then D t = qR D t = qR D t = qR (5-13) 1 1 2 2 3 3 5 c =- (1.92)(10 ) 1 c = 724 Adding the individual temperature drops, noting that q is uniform, 2 Two-Dimensional Conduction If the temperature of a material q(R + R + R ) =D t +D t +D t = D t (5-14) 1 2 3 1 2 3 is a function of two space variables, the two-dimensional conduction or q = D t/R = (t - t )/R (5-15) T 1 4 T equation is (assuming constant k) 2 2 2 2 where R is the overall resistance and is the sum of the individual T ¶ t/¶ x +¶ t/¶ y =- q¢ /k (5-18) resistances in series, then When q¢ is zero, Eq. (5-18) reduces to the familiar Laplace equation. R = R + R ++ R (5-16) T 1 2 n The analytical solution of Eq. (10-18) as well as of Laplace’s equation is possible for only a few boundary conditions and geometric shapes. When a wall is constructed of several layers of solids, the joints at Carslaw and Jaeger (Conduction of Heat in Solids, Clarendon Press, adjacent layers may not perfectly exclude air spaces, and these addi- Oxford, 1959) have presented a large number of analytical solutions of tional resistances should not be overlooked. differential equations applicable to heat-conduction problems. Gen- Conduction through Several Bodies in Parallel For n resis- erally, graphical or numerical finite-difference methods are most tances in parallel, the rates of heat flow are additive: frequently used. Other numerical and relaxation methods may be q=D t/R +D t/R ++D t/R (5-17a) 1 2 n found in the general references in the “Introduction.” The methods may also be extended to three-dimensional problems. 1 1 1 q = + ++ D t (5-17b) 1 2 R R R 1 2 n UNSTEADY-STATE CONDUCTION q = (C + C ++ C )D t = C D t (5-17c) 1 2 n When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the where R to R are the individual resistances and C to C are the indi- 1 n 1 n three-dimensional unsteady-state conduction equation. It involves the vidual conductances; C = kA/x. rate of change of temperature with respect to time ¶ t/¶q . Solutions to Several Bodies in Series with Heat Generation The simple most practical problems must be obtained through the use of digital Fourier type of equation indicated by Eq. (5-15) may not be used computers. Numerous articles have been published on a wide variety when heat generation occurs in one of the bodies in the series. In this of transient conduction problems involving various geometrical shapes case, Eq. (5-5a), (5-5b), or (5-5c) must be solved with appropriate and boundary conditions. boundary conditions. One-Dimensional Conduction The one-dimensional transient conduction equations are (for constant physical properties) Example 1: Steady-State Conduction with Heat Generation A plate-type nuclear fuel element, consisting of a uranium-zirconium alloy 2 2 ¶ t/¶q=a (¶ t/¶ x ) + q¢ /cr (rectangular coordinates) (5-19a) - 3 - 4 (3.2)(10 ) m (0.125 in) thick clad on each side with a (6.4)(10 )-m- (0.025-in-) thick layer of zirconium, is cooled by water under pressure at 200 C (400 F), ¶ t a ¶ ¶ t q¢ 2 2 = r + (cylindrical coordinates) (5-19b) the heat-transfer coefficient being 42,600 J/(m s K) 7500 Btu/(h ft F). If the 1 2 ¶q r dr ¶ r cr temperature at the center of the fuel must not exceed 570 C (1050 F), deter- mine the maximum rate of heat generation in the fuel. The zirconium and zir- ¶ t a ¶ ¶ t q¢ 2 2 conium alloy have a thermal conductivity of 21 J/(m s K) 12 Btu/(h ft )( F/ft). = r + (spherical coordinates) (5-19c) 2 1 2 Solution. Equation (5-4a) may be integrated for each material. The heat ¶q r ¶ r ¶ r cr generation is zero in the cladding, and its value for the fuel may be determined These equations have been solved analytically for solid slabs, cylin- from the integrated equations. Let x = 0 at the midplane of the fuel. Then x = 1 - 3 - 3 ders, and spheres. The solutions are in the form of infinite series, and (1.6)(10 ) m (0.0625 in) at the cladding-fuel interface and x = (2.2)(10 ) m 2 usually the results are plotted as curves involving four ratios Gurney (0.0875 in) at the cladding-water interface. Let the subscripts c, f refer to and Lurie, Ind. Eng. Chem., 15, 1170 (1923) defined as follows with cladding and fuel respectively. The boundary conditions are: q¢= 0:HEAT TRANSFER BY CONDUCTION 5-11 2 Brown and Marco, Schack, and Stoever (see “Introduction: General Y = (t¢- t)/(t¢- t ) X = kq /r cr (5-20a,b) b m References”). For a solid of infinite thickness (Fig. 5-5) and with m = 0, m = k/h r n = r/r (5-20c,d) T m m z 2 2 Y = E exp (- z ) dz (5-21) Since each ratio is dimensionless, any consistent units may be 0 Ïpw employed in any ratio. The significance of the symbols is as follows: where z = 1/Ï2 wX and the “error integral” may be evaluated from stan- t¢= temperature of the surroundings; t = initial uniform temperature b dard mathematical tables. of the body; t = temperature at a given point in the body at the time q Various numerical and graphical methods are used for unsteady- measured from the start of the heating or cooling operations; k = uni- state conduction problems, in particular the Schmidt graphical form thermal conductivity of the body; r= uniform density of the method (Foppls Festschrift, Springer-Verlag, Berlin, 1924). These body; c = specific heat of the body; h = coefficient of total heat trans- T methods are very useful because any form of initial temperature dis- fer between the surroundings and the surface of the body expressed as tribution may be used. heat transferred per unit time per unit area of the surface per unit dif- Two-Dimensional Conduction The governing differential equa- ference in temperature between surroundings and surface; r = dis- tion for two-dimensional transient conduction is tance, in the direction of heat conduction, from the midpoint or 2 2 midplane of the body to the point under consideration; r = radius of m ¶ t ¶ t ¶ t q¢ =a + + (5-22) a sphere or cylinder, one-half of the thickness of a slab heated from 1 2 22 ¶q ¶ x ¶ y cr both faces, the total thickness of a slab heated from one face and insu- McAdams (Heat Transmission, 3d ed., McGraw-Hill, New York, lated perfectly at the other; and x = distance, in the direction of heat 1954) gives various forms of transient difference equations and meth- conduction, from the surface of a semi-infinite body (such as the sur- ods of solving transient conduction problems. The availability of com- face of the earth) to the point under consideration. In making the inte- puters and a wide variety of computer programs permits virtually grations which lead to the curves shown, the following factors were routine solution of complicated conduction problems. assumed constant: c, h , k, r, r , t¢ , x, and r . T m Conduction with Change of Phase A special type of transient The working curves are shown in Figs. 5-2 to 5-5 for cylinders of problem (the Stefan problem) involves conduction of heat in a mate- infinite length, spheres, slabs of infinite faces, and semi-infinite rial when freezing or melting occurs. The liquid-solid interface moves solids respectively, with Y plotted as ordinates on a logarithmic scale with time, and in addition to conduction, latent heat is either gener- versus X as abscissas to an arithmetic scale, for various values of the ated or absorbed at the interface. Various problems of this type are ratios m and n. To facilitate calculations involving instantaneous rates of discussed by Bankoff in Drew et al. (eds.), Advances in Chemical cooling or heating of the semi-infinite body, Fig. 5-5 shows also a curve Engineering, vol. 5, Academic, New York, 1964. of dY/dX versus X. Similar plots to a larger scale are given in McAdams, FIG. 5-4 Heating and cooling of a solid slab having a large face area relative to FIG. 5-2 Heating and cooling of a solid cylinder having an infinite ratio of the area of the edges. length to diameter. FIG. 5-5 Heating and cooling of a solid of infinite thickness, neglecting edge effects. (This may be used as an approximation in the zone near the surface of a FIG. 5-3 Heating and cooling of a solid sphere. body of finite thickness.)5-12 HEAT AND MASS TRANSFER HEAT TRANSFER BY CONVECTION Individual Coefficient of Heat Transfer Because of the COEFFICIENT OF HEAT TRANSFER complicated structure of a turbulent flowing stream and the impracti- In many cases of heat transfer involving either a liquid or a gas, con- cability of measuring thicknesses of the several layers and their tem- vection is an important factor. In the majority of heat-transfer cases peratures, the local rate of heat transfer between fluid and solid is met in industrial practice, heat is being transferred from one fluid defined by the equations through a solid wall to another fluid. Assume a hot fluid at a tempera- dq = h dA (t - t ) = h dA (t - t ) (5-24) i i 1 3 o o 5 7 ture t flowing past one side of a metal wall and a cold fluid at t flow- 1 7 ing past the other side to which a scale of thickness x adheres. In such s where h and h are the local heat-transfer coefficients inside and out- i o a case, the conditions obtaining at a given section are illustrated dia- side the wall, respectively, and temperatures are defined by Fig. 5-6. grammatically in Fig. 5-6. The definition of the heat-transfer coefficient is arbitrary, depend- For turbulent flow of a fluid past a solid, it has long been known ing on whether bulk-fluid temperature, centerline temperature, or that, in the immediate neighborhood of the surface, there exists a some other reference temperature is used for t or t . Equation (5-24) 1 7 relatively quiet zone of fluid, commonly called the film. As one is an expression of Newton’s law of cooling and incorporates all the approaches the wall from the body of the flowing fluid, the flow tends complexities involved in the solution of Eq. (5-23). The temperature to become less turbulent and develops into laminar flow immediately gradients in both the fluid and the adjacent solid at the fluid-solid adjacent to the wall. The film consists of that portion of the flow which interface may also be related to the heat-transfer coefficient: is essentially in laminar motion (the laminar sublayer) and through dt dt which heat is transferred by molecular conduction. The resistance of dq = h dA (t - t ) = - k = - k (5-25) i i 1 3 1 2 1 2 dx fluid dx solid the laminar layer to heat flow will vary according to its thickness and can range from 95 percent of the total resistance for some fluids to Equation (5-25) holds for the liquid only if laminar flow exists imme- diately adjacent to the solid surface. The integration of Eq. (5-24) will about 1 percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core give out out each offer a resistance to heat transfer which is a function of the dq dq A =E or A =E (5-26) turbulence and the thermal properties of the flowing fluid. The rela- i o in in h D t h D t i i o o tive temperature difference across each of the layers is dependent which may be evaluated only if the quantities under the integral can upon their resistance to heat flow. be expressed in terms of a single variable. If q is a linear function of D t The Energy Equation A complete energy balance on a flowing and h is constant, then Eq. (5-26) gives fluid through which heat is being transferred results in the energy hA(D t -D t ) equation (assuming constant physical properties): in out q = (5-27) ¶ t ¶ t ¶ t ¶ t ln (D t /D t ) in out cr + u + v + w 1 2 where the D t factor is the logarithmic-mean temperature differ- ¶q ¶ x ¶ y ¶ z ence between the wall and the fluid. 2 2 2 ¶ t ¶ t ¶ t Frequently experimental data report average heat-transfer coeffi- = k ++ + q¢+F (5-23) 1 2 2 22 cients based upon an arbitrarily defined temperature difference, the ¶ x ¶ y ¶ z two most common being where F is the term accounting for energy dissipation due to fluid vis- h A(D t -D t ) lm in out cosity and is significant in high-speed gas flow and in the flow of highly q = (5-28a) viscous liquids. Except for the time term, the left-hand terms of Eq. (5- ln(D t /D t ) in out 23) are the so-called convective terms involving the energy carried by h A(D t +D t ) am in out the fluid by virtue of its velocity. Therefore, the solution of the equa- q = (5-28b) 2 tion is dependent upon the solution of the momentum equations of flow. Solutions of Eq. (5-23) exist only for several simple flow cases and where h and h are average heat-transfer coefficients based upon lm am geometries and mainly for laminar flow. For turbulent flow the diffi- the logarithmic-mean temperature difference and the arithmetic- culties of expressing the fluid velocity as a function of space and time average temperature difference, respectively. coordinates and of obtaining reliable values of the effective thermal Overall Coefficient of Heat Transfer In testing commercial conductivity of the flowing fluid have prevented solution of the equa- heat-transfer equipment, it is not convenient to measure tube tem- tion unless simplifying assumptions and approximations are made. peratures (t or t in Fig. 5-6), and hence the overall performance is 3 4 expressed as an overall coefficient of heat transfer U based on a con- venient area dA, which may be dA , dA , or an average of dA and dA ; i o i o whence, by definition, dq = U dA (t - t ) (5-29) 1 7 U is called the “overall coefficient of heat transfer,” or merely the “overall coefficient.” The rate of conduction through the tube wall and scale deposit is given by kdA (t - t ) avg 3 4 dq== h dA (t - t ) (5-30) d d 4 5 x Upon eliminating t , t , t from Eqs. (5-24), (5-29), and (5-30), the com- 3 4 5 plete expression for the steady rate of heat flow from one fluid through the wall and scale to a second fluid, as illustrated in Fig. 5-6, is t - t 1 7 dq== U dA (t - t ) (5-31) 1 7 1 x 1 1 + + + h dA kdA h dA h dA i i avg d d o o FIG. 5-6 Temperature gradients for a steady flow of heat by conduction and Normally, dirt and scale resistance must be considered on both sides of the convection from a warmer to a colder fluid separated by a solid wall. tube wall. The area dA is any convenient reference area.HEAT TRANSFER BY CONVECTION 5-13 Representation of Heat-Transfer Film Coefficients There tally and are given in Table 5-1. Fluid properties are evaluated at t = f are two general methods of expressing film coefficients: (1) dimen- (t + t¢ )/2. For vertical plates and cylinders and 1 N 40, Kato, s Pr sionless relations and (2) dimensional equations. Nishiwaki, and Hirata Int. J. Heat Mass Transfer, 11, 1117 (1968) rec- The dimensionless relations are usually indicated in either of two ommend the relations forms, each yielding identical results. The preferred form is that sug- 0.36 0.175 N = 0.138N (N - 0.55) (5-33a) Nu Gr Pr gested by Colburn Trans. Am. Inst. Chem. Eng., 29, 174–210 (1933). 9 for N 10 , and It relates, primarily, three dimensionless groups: the Stanton number Gr 0.25 0.25 0.25 h/cG, the Prandtl number cm /k, and the Reynolds number DG/m . For N = 0.683N N N /(0.861 + N ) (5-33b) Nu Gr Pr Pr Pr more accurate correlation of data (at Reynolds number 10,000), two 9 for N 10 . Gr additional dimensionless groups are used: ratio of length to diameter Simplified Dimensional Equations Equation (5-32) is a L/D and ratio of viscosity at wall (or surface) temperature to viscosity at dimensionless equation, and any consistent set of units may be used. bulk temperature. Colburn showed that the product of the Stanton Simplified dimensional equations have been derived for air, water, number and the two-thirds power of the Prandtl number (and, in addi- and organic liquids by rearranging Eq. (5-32) into the following form tion, power functions of L/D and m /m for Reynolds number 10,000) w by collecting the fluid properties into a single factor: is approximately equal to half of the Fanning friction factor f/2. This m 3m - 1 product is called the Colburn j factor. Since the Colburn type of h = b(D t) L (5-34) equation relates heat transfer and fluid friction, it has greater utility Values of b in SI and U.S. customary units are given in Table 5-1 for than other expressions for the heat-transfer coefficient. air, water, and organic liquids. The classical (and perhaps more familiar) form of dimensionless Simultaneous Loss by Radiation The heat transferred by radi- expressions relates, primarily, the Nusselt number hD/k, the Prandtl ation is often of significant magnitude in the loss of heat from surfaces number cm /k, and the Reynolds number DG/m . The L/D and viscosity- to the surroundings because of the diathermanous nature of atmo- ratio modifications (for Reynolds number 10,000) also apply. spheric gases (air). It is convenient to represent radiant-heat transfer, The dimensional equations are usually expansions of the dimen- for this case, as a radiation film coefficient which is added to the sionless expressions in which the terms are in more convenient units film coefficient for convection, giving the combined coefficient for and in which all numerical factors are grouped together into a single convection and radiation (h + h ). In Fig. 5-7 values of the film coef- c r numerical constant. In some instances, the combined physical proper- ficient for radiation h are plotted against the two surface tempera- r ties are represented as a linear function of temperature, and the tures for emissivity = 1.0. dimensional equation resolves into an equation containing only one or Table 5-2 shows values of (h + h ) from single horizontal oxidized c r two variables. pipe surfaces. Enclosed Spaces The rate of heat transfer across an enclosed NATURAL CONVECTION space is calculated from a special coefficient h¢ based upon the tem- perature difference between the two surfaces, where h¢= (q/A)/ Natural convection occurs when a solid surface is in contact with a fluid (t - t ). The value of h¢ L/k may be predicted from Eq. (5-32) by s1 s2 of different temperature from the surface. Density differences provide using the values of a and m given in Table 5-3. the body force required to move the fluid. Theoretical analyses of nat- For vertical enclosed cells 10 in high and up to 2-in gap width, ural convection require the simultaneous solution of the coupled equa- Landis and Yanowitz (Proc. Third Int. Heat Transfer Conf., Chicago, tions of motion and energy. Details of theoretical studies are available 1966, vol. II, p. 139) give in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-Hill, New York, 1958; and Jakob, Heat q d 0.84 0.28 = 0.123(d /L) (N N ) (5-35) Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957) but have generally Gr Pr 12 A k D t been applied successfully to the simple case of a vertical plate. Solution 3 3 7 of the motion and energy equations gives temperature and velocity for 2 · 10 N N (d /L) 10 , where q/A is the uniform heat flux and Gr Pr fields from which heat-transfer coefficients may be derived. The gen- D t is the temperature difference at L/2. Equation (5-35) is applicable eral type of equation obtained is the so-called Nusselt equation: for air, water, and silicone oils. 3 2 m For horizontal annuli Grugal and Hauf (Proc. Third Int. Heat hL L r gbD t cm = a (5-32a) Transfer Conf., Chicago, 1966, vol. II, p 182) report 12 2 k m k 0.25 hd d d m N = a(N N ) (5-32b) Nu Gr Pr = 0.2 + 0.145 N exp - 0.02 (5-36) Gr 1 2 1 2 k D D 1 1 Nusselt Equation for Various Geometries Natural-convection coefficients for various bodies may be predicted from Eq. (5-32). The for 0.55 d /D 2.65, where N is based upon gap width d and D is 1 Gr 1 various numerical values of a and m have been determined experimen- the core diameter of the annulus. TABLE 5-1 Values of a, m, and b for Eqs. (5-32) and (5-34) b, organic b, air at b, water at liquid at Configuration Y = N N am 21C70F21C70F21C70 F Gr Pr 4 1 Vertical surfaces 10 1.36 ⁄5 4 9 L = vertical dimension 3 ft 10 Y 10 0.59 d 1.37 0.28 127 26 59 12 9 10 0.13 s 1.24 0.18 - 5 Horizontal cylinder 10 0.49 0 - 5 - 3 1 L = diameter 8 in 10 Y 10 0.71 ⁄25 - 3 1 10 Y 1 1.09 ⁄10 4 1 1 Y 10 1.09 ⁄5 4 9 10 Y 10 0.53 d 1.32 0.27 9 10 0.13 s 1.24 0.18 5 7 Horizontal flat surface 10 Y 2 · 10 (FU) 0.54 d 1.86 0.38 7 10 2 · 10 Y 3 · 10 (FU) 0.14 s 5 10 3 · 10 Y 3 · 10 (FD) 0.27 d 0.88 0.18 NOTE: FU = facing upward; FD = facing downward. b in SI units is given in C column; b in U.S. customary units, in F column.5-14 HEAT AND MASS TRANSFER siderable effort has been directed toward deriving some simple rela- tionship between momentum and heat transfer. The methodology has been to use easily observed velocity profiles to obtain a measure of the diffusivity of momentum in the flowing stream. The analogy between heat and momentum is invoked by assuming that diffusion of heat and diffusion of momentum occur by essentially the same mechanism so that a relatively simple relationship exists between the diffusion coef- ficients. Thus, the diffusivity of momentum is used to predict temper- ature profiles and thence by Eq. (10-25) to predict the heat-transfer coefficient. The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For high- Prandtl-number fluids the empirical analogy of Colburn Trans. Am. Inst. Chem. Eng., 29, 174 (1933) has been very successful. A j factor for momentum transfer is defined as j = f/2, where f is the fric- tion factor for the flow. The j factor for heat transfer is assumed to be equal to the j factor for momentum transfer 2/3 j = h/cG(cm /k) (5-37) More involved analyses for circular tubes reduce the equations of motion and energy to the form (n+e )du t g M c =- (5-38a) dy r (a+e )dt q/A H =- (5-38b) dy cr FIG. 5-7 Radiation coefficients of heat transfer h . To convert British thermal r where e is the eddy diffusivity of heat and e is the eddy diffusivity H M units per hour-square foot-degrees Fahrenheit to joules per square meter– 2 of momentum. The units of diffusivity are L /q . The eddy viscosity second–kelvins, multiply by 5.6783; C = ( F - 32)/1.8. is E =re , and the eddy conductivity of heat is E =e cr . Values M M H H of e are determined via Eq. (5-38a) from experimental velocity- M distribution data. By assuming e /e = constant (usually unity), Eq. H M FORCED CONVECTION (10-38b) is solved to give the temperature distribution from which the Forced-convection heat transfer is the most frequently employed heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38b) are in accurately defining the thickness of the var- mode of heat transfer in the process industries. Hot and cold fluids, separated by a solid boundary, are pumped through the heat-transfer ious flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assis- equipment, the rate of heat transfer being a function of the physical properties of the fluids, the flow rates, and the geometry of the tance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao Am. Inst. Chem. Eng. J., 10, system. Flow is generally turbulent, and the flow duct varies in com- plexity from circular tubes to baffled and extended-surface heat 269 (1964). Internal and External Flow Two main types of flow are consid- exchangers. Theoretical analyses of forced-convection heat transfer have been limited to relatively simple geometries and laminar flow. ered in this subsection: internal or conduit flow, in which the fluid completely fills a closed stationary duct, and external or immersed Analyses of turbulent-flow heat transfer have been based upon some mechanistic model and have not generally yielded relationships which flow, in which the fluid flows past a stationary immersed solid. With internal flow, the heat-transfer coefficient is theoretically infinite at were suitable for design purposes. Usually for complicated geometries only empirical relationships are available, and frequently these are the location where heat transfer begins. The local heat-transfer coef- ficient rapidly decreases and becomes constant, so that after a certain based upon limited data and special operating conditions. Heat- transfer coefficients are strongly influenced by the mechanics of flow length the average coefficient in the conduit is independent of the length. The local coefficient may follow an irregular pattern, however, occurring during forced-convection heat transfer. Intensity of turbu- lence, entrance conditions, and wall conditions are some of the factors if obstructions or turbulence promoters are present in the duct. For immersed flow, the local coefficient is again infinite at the point which must be considered in detail as greater accuracy in prediction of coefficients is required. where heating begins, after which it decreases and may show various irregularities depending upon the configuration of the body. Usually Analogy between Momentum and Heat Transfer The inter- relationship of momentum transfer and heat transfer is obvious from in this instance the local coefficient never becomes constant as flow proceeds downstream over the body. examining the equations of motion and energy. For constant fluid properties, the equations of motion must be solved before the energy When heat transfer occurs during immersed flow, the rate is depen- dent upon the configuration of the body, the position of the body, the equation is solved. If fluid properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. Con- proximity of other bodies, and the flow rate and turbulence of the TABLE 5-2 Values of (h + h ) c r 2 Btu/(h ft F from pipe to room) For horizontal bare standard steel pipe of various sizes in a room at 80 F Temperature difference, F Nominal pipe diameter, in 30 50 100 150 200 250 300 350 400 450 500 550 600 650 700 1 2.16 2.26 2.50 2.73 3.00 3.29 3.60 3.95 4.34 4.73 5.16 5.60 6.05 6.51 6.98 3 1.97 2.05 2.25 2.47 2.73 3.00 3.31 3.69 4.03 4.43 4.85 5.26 5.71 6.19 6.66 5 1.95 2.15 2.36 2.61 2.90 3.20 3.54 3.90 10 1.80 1.87 2.07 2.29 2.54 2.82 3.12 3.47 3.84 2 2 Bailey and Lyell Engineering, 147, 60 (1939) give values for (h + h ) up to D t of 1000 F. C = ( F - 32)/1.8; 5.6783 Btu/(h ft F) = J/(m s /K). c r sHEAT TRANSFER BY CONVECTION 5-15 TABLE 5-3 Values of a and m for Eq. (5-32) For laminar flow in vertical tubes a series of charts developed by Pigford Chem. Eng. Prog. Symp. Ser. 17, 51, 79 (1955) may be used 3 Configuration N N (d /L) am Gr Pr to predict values of h . am 4 5 - 5/36 Vertical spaces 2 · 10 to 2 · 10 0.20 (d /L) d Annuli Approximate heat-transfer coefficients for laminar flow in 5 7 1/9 2 · 10 to 10 0.071 (d /L) s annuli may be predicted by the equation of Chen, Hawkins, and Sol- 4 5 - 1/4 Horizontal spaces 10 to 3 · 10 0.21 (d /L) d berg Trans. Am. Soc. Mech. Eng., 68, 99 (1946): 5 7 3 · 10 to 10 0.075 s 0.4 0.8 0.14 D D m e 2 b 0.45 0.5 0.05 d= cell width, L = cell length. (N ) = 1.02N N N (5-41) Nu am Re Pr Gr 12 12 12 L D m 1 1 Limiting Nusselt numbers for slug-flow annuli may be predicted stream. The heat-transfer coefficient varies over the immersed body, (for constant heat flux) from Trefethen (General Discussions on Heat since both the thermal and the momentum boundary layers vary in Transfer, London, ASME, New York, 1951, p. 436): thickness. Relatively simple relationships are available for simple con- 2 2 8(m - 1)(m - 1) figurations immersed in an infinite flowing fluid. For complicated (N ) = (5-42) Nu lm 4 4 2 configurations and assemblages of bodies such as are found on the 4m ln m - 3m + 4m - 1 shell side of a heat exchanger, little is known about the local heat- where m = D /D . The Nusselt and Reynolds numbers are based on 2 1 transfer coefficient; empirical relationships giving average coefficients the equivalent diameter, D - D . 2 1 are all that are usually available. Research that has been conducted on Limiting Nusselt numbers for laminar flow in annuli have been cal- local coefficients in complicated geometries has not been extensive culated by Dwyer Nucl. Sci. Eng., 17, 336 (1963). In addition, theo- enough to extrapolate into useful design relationships. retical analyses of laminar-flow heat transfer in concentric and Laminar Flow Normally, laminar flow occurs in closed ducts eccentric annuli have been published by Reynolds, Lundberg, and when N 2100 (based on equivalent diameter D = 4 · free area ‚ Re e McCuen Int. J. Heat Mass Transfer, 6, 483, 495 (1963). Lee Int. J. perimeter). Laminar-flow heat transfer has been subjected to exten- Heat Mass Transfer, 11, 509 (1968) presented an analysis of turbulent sive theoretical study. The energy equation has been solved for a vari- heat transfer in entrance regions of concentric annuli. Fully devel- ety of boundary conditions and geometrical configurations. However, oped local Nusselt numbers were generally attained within a region of true laminar-flow heat transfer very rarely occurs. Natural-convection 4 5 30 equivalent diameters for 0.1 N 30, 10 N 2 · 10 , 1.01 Pr Re effects are almost always present, so that the assumption that molecu- D /D 5.0. 2 1 lar conduction alone occurs is not valid. Therefore, empirically Parallel Plates and Rectangular Ducts The limiting Nusselt derived equations are most reliable. number for parallel plates and flat rectangular ducts is given in Table Data are most frequently correlated by the Nusselt number (N ) Nu lm 5-4. Norris and Streid Trans. Am. Soc. Mech. Eng., 62, 525 (1940) or (N ) , the Graetz number N = (N N D/L), and the Grashof NU am Gz Re Pr report for constant wall temperature (natural-convection effects) number N . Some correlations consider Gr 1/3 only the variation of viscosity with temperature, while others also con- (N ) = 1.85N (5-43) Nu lm Gz sider density variation. Theoretical analyses indicate that for very long for N 70. Both Nusselt number and Graetz numbers are based on Gz tubes (N ) approaches a limiting value. Limiting Nusselt numbers Nu lm equivalent diameter. For large temperature differences it is advisable for various closed ducts are shown in Table 5-4. 0.14 to apply the correction factor (m /m ) to the right side of Eq. (5-43). b w Circular Tubes For horizontal tubes and constant wall tem- For rectangular ducts Kays and Clark (Stanford Univ., Dept. perature, several relationships are available, depending on the Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for 4 Graetz number. For 0.1 N 10 , Hausen’s Allg. Waermetech., 9, Gz heating and cooling of air in rectangular ducts of various aspect ratios. 75 (1959), the following equation is recommended. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if 0.8 0.14 0.19N m Gz b the equivalent diameter (= 4 · free area/wetted perimeter) is used as (N ) = 3.66 + (5-39) Nu lm 0.46712 the characteristic length. See also Kays and London, Compact Heat 1 + 0.117N m Gz w Exchangers, 3d ed., McGraw-Hill, New York, 1984. For N 100, the Sieder-Tate relationship Ind. Eng. Chem., 28, Gz Immersed Bodies When flow occurs over immersed bodies such 1429 (1936) is satisfactory for small diameters and D t’s: that the boundary layer is completely laminar over the whole body, 1/3 0.14 laminar flow is said to exist even though the flow in the mainstream is (N ) = 1.86N (m /m ) (5-40) Nu am Gz b w turbulent. The following relationships are applicable to single bodies A more general expression covering all diameters and D t’s is immersed in an infinite fluid and are not valid for assemblages of 1/3 obtained by including an additional factor 0.87(1 + 0.015N ) on the Gr bodies. right side of Eq. (5-40). The diameter should be used in evaluating In general, the average heat-transfer coefficient on immersed bod- N . An equation published by Oliver Chem. Eng. Sci., 17, 335 Gr ies is predicted by (1962) is also recommended. m 1/3 N = C (N ) (N ) (5-44) Nu r Re Pr Values of C and m for various configurations are listed in Table 5-5. r TABLE 5-4 Values of Limiting Nusselt Number The characteristic length is used in both the Nusselt and the Reynolds in Laminar Flow in Closed Ducts numbers, and the properties are evaluated at the film temperature = (t + t )/2. The velocity in the Reynolds number is the undisturbed w ∞ Limiting Nusselt number N 4.0 Gr free-stream velocity. Constant wall Constant heat Heat transfer from immersed bodies is discussed in detail by Configuration temperature flux Eckert and Drake, Jakob, and Knudsen and Katz (see “Introduction: Circular tube 3.66 4.36 General References”), where equations for local coefficients and the Concentric annulus Eq. (10-42) effects of unheated starting length are presented. Equation (5-44) Equilateral triangle 3.00 may also be expressed as Rectangles 2/3 m - 1 Aspect ratio: N N = C N = f/2 (5-45) St Pr r Re 1.0 (square) 2.89 3.63 where f is the skin-friction drag coefficient (not the form drag coeffi- 0.713 3.78 0.500 3.39 4.11 cient). 0.333 4.77 Falling Films When a liquid is distributed uniformly around the 0.25 5.35 periphery at the top of a vertical tube (either inside or outside) and 0 (parallel planes) 7.60 8.24 allowed to fall down the tube wall by the influence of gravity, the fluid5-16 HEAT AND MASS TRANSFER TABLE 5-5 Laminar-Flow Heat Transfer over Immersed Bodies Eq. (5-44) Configuration Characteristic length N N C m Re Pr r 3 5 Flat plate parallel to flow Plate length 10 to 3 · 10 0.6 0.648 0.50 Circular cylinder axes perpendicular to flow Cylinder diameter 1 – 4 0.989 0.330 4 – 40 0.911 0.385 40 – 4000 0.6 0.683 0.466 3 4 4 · 10 – 4 · 10 0.193 0.618 4 5 4 · 10 – 2.5 · 10 0.0266 0.805 3 5 Non-circular cylinder, axis Square, short diameter 5 · 10 – 10 0.104 0.675 3 5 Perpendicular to flow, characteristic Square, long diameter 5 · 10 – 10 0.250 0.588 3 5 Length perpendicular to flow Hexagon, short diameter 5 · 10 – 10 0.6 0.155 0.638 3 4 Hexagon, long diameter 5 · 10 – 2 · 10 0.162 0.638 4 5 2 · 10 – 10 0.0391 0.782 4 Sphere Diameter 1 – 7 · 10 0.6 – 400 0.6 0.50 Replace N by N - 2.0 in Eq. (5-44). Nu Nu does not fill the tube but rather flows as a thin layer. Similarly, when a 4, 91 (1934) fits both the laminar extreme and the fully turbulent liquid is applied uniformly to the outside and top of a horizontal tube, extreme quite well. it flows in layer form around the periphery and falls off the bottom. In 2/3 0.14 D m b 2/3 1/3 (N ) = 0.116(N - 125)N 1 + (5-49) both these cases the mechanism is called gravity flow of liquid layers Nu am Re Pr 3 12 41 2 L m w or falling films. For the turbulent flow of water in layer form down the walls of between 2100 and 10,000. It is customary to represent the probable vertical tubes the dimensional equation of McAdams, Drew, and magnitude of coefficients in this region by hand-drawn curves (Fig. Bays Trans. Am. Soc. Mech. Eng., 62, 627 (1940) is recommended: 5-8). Equation (5-40) is plotted as a series of curves ( j factor versus Reynolds number with L/D as parameters) terminating at Reynolds 1/3 h = bG (5-46) lm number = 2100. Continuous curves for various values of L/D are then hand-drawn from these terminal points to coincide tangentially with where b = 9150 (SI) or 120 (U.S. customary) and is based on values of the curve for forced-convection, fully turbulent flow Eq. (5-50c). G= W /p D ranging from 0.25 to 6.2 kg/(m?s) 600 to 15,000 lb/(h?ft) F of wetted perimeter. This type of water flow is used in vertical vapor- Turbulent Flow in-shell ammonia condensers, acid coolers, cycle water coolers, and other process-fluid coolers. Circular Tubes Numerous relationships have been proposed for The following dimensional equations may be used for any liquid predicting turbulent flow in tubes. For high-Prandtl-number fluids, flowing in layer form down vertical surfaces: relationships derived from the equations of motion and energy 3 2 1/3 1/3 1/3 through the momentum-heat-transfer analogy are more complicated 4G k r g cm 4G For 2100 h = 0.01 (5-47a) lm and no more accurate than many of the empirical relationships that 12 2 12 12 m m k m have been developed. For N 10,000, 0.7 N 170, for properties based on the bulk 2 4/3 2/3 1/3 1/4 1/9 Re Pr 4G k r cg m 4G temperature and for heating, the Dittus-Boelter equation Boel- For 2100 h = 0.50 (5-47b) am 12 1/3 12 12 m Lm m m w ter, Cherry, Johnson and Martinelli, Heat Transfer Notes, McGraw- Hill, New York (1965) may be used: Equation (5-47b) is based on the work of Bays and McAdams Ind. 0.8 0.4 0.14 Eng. Chem., 29, 1240 (1937). The significance of the term L is not N = 0.0243 N N (m /m ) (5-50a) Nu Re Pr b w clear. When L = 0, the coefficient is definitely not infinite. When L is For cooling, the relationship is large and the fluid temperature has not yet closely approached the wall temperature, it does not appear that the coefficient should nec- 0.8 0.3 0.14 N = 0.0265 N N (m /m ) (5.50b) Nu Re Pr b w essarily decrease. Within the finite limits of 0.12 to 1.8 m (0.4 to 6 ft), The Colburn correlation is this equation should give results of the proper order of magnitude. 2/3 0.14 - 0.2 For falling films applied to the outside of horizontal tubes, the j = N N (m /m ) = 0.023N (5-50c) H St Pr w b Re Reynolds number rarely exceeds 2100. Equations may be used for In Eq. (5-50c), the viscosity-ratio factor may be neglected if properties falling films on the outside of the tubes by substituting p D/2 for L. are evaluated at the film temperature (t + t )/2. For water flowing over a horizontal tube, data for several sizes of b w pipe are roughly correlated by the dimensional equation of McAdams, Drew, and Bays Trans. Am. Soc. Mech. Eng., 62, 627 (1940). 1/3 h = b (G /D ) (5-48) am 0 where b = 3360 (SI) or 65.6 (U.S. customary) and G ranges from 0.94 to 4 kg/m s) 100 to 1000 lb/(h ft). Falling films are also used for evaporation in which the film is both entirely or partially evaporated (juice concentration). This principle is also used in crystallization (freezing). The advantage of high coefficient in falling-film exchangers is par- tially offset by the difficulties involved in distribution of the film, maintaining complete wettability of the tube, and pumping costs required to lift the liquid to the top of the exchanger. Transition Region Turbulent-flow equations for predicting heat transfer coefficients are usually valid only at Reynolds numbers greater than 10,000. The transition region lies in the range 2000 N 10,000. Re FIG. 5-8 Graphical representation of the Colburn j factor for the heating and No simple equation exists for accomplishing a smooth mathematical cooling of fluids inside tubes. The curves for N below 2100 are based on Eq. Re transition from laminar flow to turbulent flow. Of the relationships pro- (5-40). L is the length of each pass in feet. The curve for N above 10,000 is rep- Re resented by Eq. (5-50c). posed, Hausen’s equation Z. Ver. Dtsch. Ing. Beih. Verfahrenstech., No.HEAT TRANSFER BY CONVECTION 5-17 For the transition and turbulent regions, including diameter For large temperature differences different equations are nec- to length effects, Gnielinski Int. Chem. Eng., 16, 359 (1976) rec- essary and usually are specifically applicable to either gases or liquids. ommends a modification of an equation suggested by Petukhov and Gambill (Chem. Eng., Aug. 28, 1967, p. 147) provides a detailed Popov High Temp., 1, 69 (1963). This equation applies in the ranges review of high-flux heat transfers to gases. He recommends 6 0 D/L 1, 0.6 N 2000, 2300 N 10 . 0.8 0.4 Pr Re 0.021N N Re Pr N = (5-59) Nu 2/3 0.14 0.29 + 0.0019 (L/D) ( f/2)(N - 1000) N D m (T /T ) Re Pr b w b N = 1 + (5-51a) Nu 0.5 2/3 1 12 21 2 1 + 12.7( f/2) (N - 1) L m for 10 L/D 240, 110 T 1560 K (200 T 2800 R), 1.1 b b Pr w (T /T ) 8.0, and properties evaluated at T . For liquids, Eq. (5-50c) w b b The Fanning friction f is determined by an equation recommended by is generally satisfactory. Filonenko Teploenergetika, 1, 40 (1954) Annuli For diameter ratios D /D 0.2, Monrad and Pelton’s 1 2 - 2 equation Trans. Am. Inst. Chem. Eng., 38, 593 (1942) is recom- f = 0.25 (1.82 log N - 1.64) (5-51b) 10 Re mended for either or both the inner and outer tube: Any other appropriate friction factor equation for smooth tubes may 0.8 1/3 0.53 N = 0.020N N (D /D ) (5-60a) be used. Nu Re Pr 2 1 Approximate predictions for rough pipes may be obtained from Equation (5-51a) may also be used for smooth annuli as follows: Eq. (5-50c) if the right-hand term is replaced by f/2 for the rough (N ) D Nu ann 1 pipe. For air, Nunner (Z. Ver. Dtsch. Ing. Forsch., 1956, p. 455) =f (5-60b) 12 (N ) D obtains Nu tube 2 The hydraulic diameter D - D is used in N , N , and D/L is used (N ) f 2 1 Nu Re Nu rough rough = (5-52) for the annulus. The function on the right of Eq. (5-60b) is given by (N ) f Nu smooth smooth Petukhov and Roizen High Temp., 2, 65 (1964) as follows: - 0.16 Dippery and Sabersky Int. J. Heat Mass Transfer, 6, 329 (1963) Inner tube heated 0.86 (D /D ) 1 2 0.6 present a complete discussion of the influence of roughness on heat Outer tube heated 1 - 0.14 (D /D ) 1 2 transfer in tubes. If both tubes are heated, the function is the sum of the above two Dimensional Equations for Various Conditions For gases at functions divided by 1 + D /D Stephan, Chem. Ing. Tech., 34, 207 1 2 ordinary pressures and temperatures based on cm /k = 0.78 and m= (1962). The Colburn form of relationship may be employed for the - 5 (1.76)(10 ) Pa s 0.0426 lb/(ft h) individual walls of the annulus by using the individual friction factor for each wall see Knudsen, Am. Inst. Chem. Eng. J., 8, 566 (1962): 0.8 0.8 0.2 h = bcr (V /D ) (5-53) 2/3 j = (N ) N = f /2 (5-61a) H1 St 1 Pr 1 - 3 - 2 where b = (3.04)(10 ) (SI) or (1.44)(10 ) (U.S. customary). For air at 2/3 j = (N ) N = f /2 (5-61b) atmospheric pressure H2 St 2 Pr 2 0.8 0.2 Rothfus, Monrad, Sikchi, and Heideger Ind. Eng. Chem., 47, 913 h = b(V /D ) (5-54) (1955) report that the friction factor f for the outer wall bears the 2 - 4 where b = 3.52 (SI) or (4.35)(10 ) (U.S. customary). For water based same relation to the Reynolds number for the outer portion of the 2 on a temperature range of 5 to 104 C (40 to 220 F) annular stream 2(r -l )Vr /r m as the friction factor for circular 2 m 2 0.8 0.2 tubes does to the Reynolds number for circular tubes, where r is the 2 h = 1057 (1.352 + 0.02t) (V /D ) (5-55a) radius of the outer tube and l is the position of maximum velocity in m in SI units with t= C, or the annulus, estimated from 0.8 0.2 2 2 h = 0.13(1 + 0.011t)(V /D ) (5-55b) r - r 2 1 l = (5-62) m 2 ln (r /r ) in U.S. customary units with t= F. 2 1 For organic liquids, based on c = 2.092 J/kg K)0.5 Btu/(lb F), To calculate the friction factor f for the inner tube use the relation 1 - 3 k = 0.14 J/(m s K) 0.08 Btu/(h ft F), m = (1)(10 ) Pa s (1.0 cP), and 2 b f r (l - r ) 2 2 m 1 3 3 r= 810 kg/m (50 lb/ft ), f = (5-63) 1 2 r (r -l ) 1 2 m 0.8 0.2 h = b(V /D ) (5-56) There have been several analyses of turbulent heat transfer in annuli: - 2 for example, Deissler and Taylor (NACA Tech. Note 3451, 1955), where b = 423 (SI) or (5.22)(10 ) (U.S. customary). Within reasonable limits, coefficients for organic liquids are about one-third of the values Kays and Leung Int. J. Heat Mass Transfer, 6, 537 (1963), Lee Int. J. Heat Transfer, 11, 509 (1968), Sparrow, Hallman and Siegel Appl. obtained for water. Entrance effects are usually not significant industrially if L/D Sci. Res., 7A, 37 (1958), and Johnson and Sparrow Am. Soc. Mech. Eng. J. Heat Transfer, 88, 502 (1966). The reader is referred to these 60. Below this limit Nusselt recommended the conservative equation for 10 L/D 400 and properties evaluated at bulk temperature for details of the analyses. For annuli containing externally finned tubes the heat-transfer 0.8 1/3 - 0.054 N = 0.036N N (L/D) (5-57) Nu Re Pr coefficients are a function of the fin configurations. Knudsen and Katz (Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958) It is common to correlate entrance effects by the equation present relationships for transverse finned tubes, spined tubes, and h /h = 1 + F(D/L) (5-58) m longitudinal finned tubes in annuli. Noncircular Ducts Equations (5-50a) and (5-50b) may be em- where h is predicted by Eq. (5-50a) or (5-50b), and h is the mean m ployed for noncircular ducts by using the equivalent diameter D = coefficient for the pipe in question. Values of F are reported by Boel- e 4 · free area per wetted perimeter. Kays and London (Compact Heat ter, Young, and Iverson NACA Tech. Note 1451, 1948 and tabulated Exchangers, 3rd ed., McGraw-Hill, New York, 1984) give charts for by Kays and Knudsen and Katz (see “Introduction: General Refer- various noncircular ducts encountered in compact heat exchangers. ences”). Selected values of F are as follows: Vibrations and pulsations generally tend to increase heat-transfer Fully developed velocity profile 1.4 coefficients. Abrupt contraction entrance 6 90 right-angle bend 7 Example 2: Calculation of j Factors in an Annulus Calculate 180 round bend 6 the heat-transfer j factors for both walls of an annulus for the following condi- Equation (5-62) predicts the point of maximum velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964) and Macagno and McDougall Am. Inst. Chem. Eng. J., 12, 437 (1966) give more accurate equations for predicting the point of maximum velocity for turbulent flow.5-18 HEAT AND MASS TRANSFER tions: D = 0.0254 m (1.0 in); D = 0.0635 m (2.5 in); water at 15.6 C (60 F); For the general case, the treatment suggested by Kern (Process 1 2 - 6 2 - 5 2 m /r= (1.124)(10 ) m /s (1.21)(10 ) ft /s; velocity = 1.22 m/s (4 ft/s). Heat Transfer, McGraw-Hill, New York, 1950, p. 512) is recom- 2 2 0.0635 - 0.0254 mended. Because of the wide variations in fin-tube construction, it is - 4 2 2 l== (4.621)(10 ) m (0.716 in ) m 2 convenient to convert all film coefficients to values based on the inside 4 ln (0.0635/0.0254) bare surface of the tube. Thus to convert the film coefficient based on 2 2 - 4 2(r -l )Vr 20.0318 - (4.621)(10 )(1.22) 2 m 4 Re== = (3.74)(10 ) 2 outside area (finned side) to a value based on inside area Kern gives - 6 r m (0.0318)(1.124)(10 ) 2 the following relationship: From Eq. (5-51b), f = 0.0055. Hence 2 2/3 h = (W A + A )(h /A ) (5-67) j = (N ) N = 0.00275 fi f o f i H2 St 2 Pr From Eq. (5-63), in which h is the effective outside film coefficient based on the inside fi - 4 2 (0.0055)(0.0318)(4.621)(10 ) - 0.0127 area, h is the outside film coefficient calculated from the applicable f f== 0.00754 1 2 - 4 equation for bare tubes, A is the surface area of the fins, A is the sur- (0.0127)0.0318 - (4.621)(10 ) f o face area on the outside of the tube which is not finned, A is the inside i 2/3 from which j = (N ) N = 0.00377. H St 1 Pr 1 area of the tube, and W is the fin efficiency defined as These results indicate that for this system the heat-transfer coefficient on the inner tube is about 40 percent greater than on the outer tube. W= (tanh mb )/mb (5-68) f f in which Coils For flow inside helical coils, Reynolds number above 10,000, multiply the value of the film coefficient obtained from the 1/2 - 1 - 1 m = (h p /ka ) m (ft ) (5-69) f f x applicable equation for straight tubes by the term (1 + 3.5 D /D ). i c For flow inside helical coils, Reynolds number less than 10,000, and b = height of fin. The other symbols are defined as follows: p is f f 1/2 substitute the term (D /D ) for (L/D ) where the latter appears in the c i i the perimeter of the fin, a is the cross-sectional area of the fin, and k x applicable equation for straight tubes (frequently as part of the Graetz is the thermal conductivity of the material from which the fin is made. number). Fin efficiencies and fin dimensions are available from manufactur- For flat spiral (pancake) coils, in which the ratio D /D varies for c i ers. Ratios of finned to inside surface are usually available so that the each turn, a different value of coefficient will be obtained for each terms A , A , and A may be obtained from these ratios rather than f o i turn; a weighted average based on length per turn is used. from the total surface areas of the heat exchangers. For flow outside helical coils use the equation for flow normal to a Banks of Tubes For heating and cooling of fluids flowing normal bank of tubes, in-line flow. to a bank of circular tubes at least 10 rows deep the following equa- Finned Tubes (Extended Surface) When the film coefficient on tions are applicable: the outside of a metal tube is much lower than that on the inside, as Colburn type: when steam condensing in a pipe is being used to heat air, externally 2/3 h cm a finned (or extended) heating surfaces are of value in increasing sub- == j (5-70) 12 0.4 stantially the rate of heat transfer per unit length of tube. The data on cG k (D G /m ) max o max extended heating surfaces, for the case of air flowing outside and at Nusselt type: right angles to the axes of a bank of finned pipes, can be represented 0.6 1/3 approximately by the dimensional equation derived from hD D G cm o max = a (5-71) 0.6 0.6 V p¢ 12 12 F k m k h = b (5-64) f 0.412 D p¢- D 0 0 The dimensionless constant a in these equations varies depending - 3 where b = 5.29 (SI) or (5.39)(10 ) (U.S. customary); h is the film coef- f upon conditions. ficient of heat transfer on the air side; V is the face velocity of the air; F p¢ is the center-to-center spacing, m, of the tubes in a row; and D is 0 the outside diameter, m, of the bare tube (diameter at the root of the Conditions, Reynolds number 3000 Value of a fins). Flow normal to apex of diamond, staggered arrangement In atmospheric air-cooled finned tube exchangers, the air-film coef- No leakage 0.330 ficient from Eq. (5-64) is sometimes converted to a value based on Normal leakage in baffled exchanger 0.198 outside bare surface as follows: Flow normal to flat side of diamond, not staggered (in-line) arrangement A + A f uf A T h = h = h (5-65) No leakage 0.260 fo f f A A of o Normal leakage in baffled exchanger 0.156 in which h is the air-film coefficient based on external bare surface; fo h is the air-film coefficient based on total external surface; A is total f T external surface, and A is external bare surface of the unfinned tube; For Reynolds number less than 3000, Eq. (5-70) would give con- o servative results, but greater accuracy (if desired) may be obtained by A is the area of the fins; A is the external area of the unfinned por- f uf tion of the tube; and A is area of tube before fins are attached. using the following equation. of Fin efficiency is defined as the ratio of the mean temperature dif- 2/3 h cm a ference from surface to fluid divided by the temperature difference == j (5-72) 12 m cG k (D G /m ) from fin to fluid at the base or root of the fin. Graphs of fin efficiency max o max for extended surfaces of various types are given by Gardner Trans. in which the constant a and exponent m are as follows: Am. Soc. Mech. Eng., 67, 621 (1945). Heat-transfer coefficients for finned tubes of various types are given in a series of papers Trans. Am. Soc. Mech. Eng., 67, 601 Reynolds Tube number m pitch Leakage a (1945). For flow of air normal to fins in the form of short strips or pins, 100–300 0.492 Staggered None 0.695 Norris and Spofford Trans. Am. Soc. Mech. Eng., 64, 489 (1942) cor- Normal 0.416 relate their results for air by the dimensionless equation of In-line None 0.548 Pohlhausen: Normal 0.329 1–100 0.590 Staggered None 1.086 2/3 - 0.5 h c m z G m p p max Normal 0.650 = 1.0 (5-66) 12 12 In-line None 0.855 c G k m p max Normal 0.513 for values of z G /m ranging from 2700 to 10,000. p maxHEAT TRANSFER BY CONVECTION 5-19 The following dimensional equations (5-73 to 5-77) are based on LIQUID METALS flow normal to a bank of staggered tubes without leakage. Multiply the Liquid metals constitute a class of heat-transfer media having Prandtl values obtained for h by 0.6 for normal leakage and, in addition, by numbers generally below 0.01. Heat-transfer coefficients for liquid 0.79 for in-line (not staggered) tube arrangement. metals cannot be predicted by the usual design equations applicable 1/3 2/3 0.6 0.6 c k r V max to gases, water, and more viscous fluids with Prandtl numbers greater h = b (5-73) 0.267 0.4 m D than 0.6. Relationships for predicting heat-transfer coefficients for 0 liquid metals have been derived from solution of Eqs. (5-38a) and where b = 0.33 (SI) or 0.261 (U.S. customary). For gases at ordinary - 5 (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy pressures and temperatures, based on cm /k = 0.78; m= (1.76)(10 ) conductivity of heat is kN (E /m ) » k for small N . Thus in the solu- Pr M Pr Pa?s 0.0426 lb/(ft?h), tion of Eqs. (5-38a) and (5-38b) the knowledge of the thickness of var- 0.6 G max ious layers of flow is not critical. In fact, assumption of slug flow and h = bc (5-74) 0.4 constant conductivity (=k) across the duct gives reasonable values of D 0 heat-transfer coefficients for liquid metals. - 3 where b = (4.82)(10 ) (SI) or 0.109 (U.S. customary). For air at For constant heat flux: atmospheric pressure 0.8 0.6 N = 5 + 0.025(N N ) (5-81) Nu Re Pr V max h = b (5-75) 0.4 For constant wall temperature: D 0 - 3 0.8 where b = 5.33 (SI) or (5.44)(10 ) (U.S. customary). For water based N = 7 + 0.025(N N ) (5-82) Nu Re Pr on a temperature range 7 to 104 C (40 to 220 F) For 0.003 N 0.05 and constant heat flux, Sleicher and Rouse Int. Pr 0.6 V max J. Heat Mass Transfer, 18, 677 (1975) obtained the correlation h = 986(1.21 + 0.0121t) (5-76 a) 0.4 D 0 0.85 0.93 N = 6.3 + 0.0167 N N (5-83) Nu Re Pr in SI units and t in C. For parallel plates and annuli with D /D 1.4 and uniform heat 2 1 0.6 V max flux, Seban Trans. Am. Soc. Mech. Eng., 72, 789 (1950) obtained the h = 1.01(1 + 0.0067t) (5-76 b) 0.4 D equation 0 0.8 in U.S. customary units and t in F. For organic liquids, based on N = 5.8 + 0.020(N N ) (5-84) Nu Re Pr c = 2.22 J/(kg?K) 0.53 Btu/(lb? F), k = 0.14 J/(m?s?K) 0.08 Btu/ 0.53 - 3 3 3 For annuli only, application of a factor of 0.70(D /D ) is recom- (h?ft? F), m = (1)(10 ) Pa?s (1.0 cP), r= 810 kg/m (50 lb/ft ), 2 1 b mended for Eqs. (5-81) and (5-82). For more accurate semiempirical 0.6 V max relationships for tubes, annuli, and rod bundles, refer to Dwyer Am. h = b (5-77) 0.4 D Inst. Chem. Eng. J., 9, 261 (1963). 0 Hsu Int. J. Heat Mass Transfer, 7, 431 (1964) and Kalish and where b = 400 (SI) or 0.408 (U.S. customary). Dwyer Int. J. Heat Mass Transfer, 10, 1533 (1967) discuss heat transfer to liquid metals flowing across banks of tubes. Hsu recom- JACKETS AND COILS OF AGITATED VESSELS mends the equations See Sec. 18. 1/2 N = 0.81N N (f /D) (for uniform heat flux) (5-85) Nu Re Pr 1/2 N = 0.096N N (f /D) (for cosine surface temperature) Nu Re Pr NONNEWTONIAN FLUIDS (5-86) A wide variety of nonnewtonian fluids are encountered industrially. where the heat-transfer coefficient is based on the average circumfer- They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior ential temperature around the tubes, the Reynolds number is based and may or may not be thixotropic. For design of equipment to handle on the superficial velocity through the tube bank, D is the tube out- or process nonnewtonian fluids, the properties must usually be mea- side diameter, and f is a velocity potential function having the follow- sured experimentally, since no generalized relationships exist to pre- ing values: dict the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (Non- Newtonian Flow and Heat Transfer, Wiley, New York, 1967). The gen- f /D square f /D equilateral eralized shear-stress rate-of-strain relationship for nonnewtonian D/p¢ pitch triangular pitch fluids is given as 0 2.00 2.00 d ln (D D P/4L) 0.1 2.02 2.02 n¢= (5-78) 0.2 2.07 2.06 d ln (8V/D) 0.3 2.16 2.15 0.4 2.30 2.27 as determined from a plot of shear stress versus velocity gradient. 0.5 2.52 2.45 For circular tubes, N 100, n¢ 0.1, and laminar flow Gz 0.6 2.84 2.71 1/3 1/3 (N ) = 1.75 d N (5-79) 0.7 3.34 3.11 Nu lm s Gz 0.8 4.23 3.80 where d = (3n¢+ 1)/4n¢ . When natural-convection effects are consid- s ered, Metzer and Gluck Chem. Eng. Sci., 12, 185 (1960) obtained the following for horizontal tubes: Equations (5-85) and (5-86) are useful in calculating tube-surface 0.4 1/3 0.14 temperatures. N N D g Pr Gr b 1/3 (N ) = 1.75 d N + 12.6 (5-80) Further information on liquid-metal heat transfer in tube banks is Nu lm s Gz 3 12 4 12 L g w given by Hsu for spheres and elliptical rod bundles Int. J. Heat Mass where properties are evaluated at the wall temperature, i.e., g= Transfer, 8, 303 (1965) and by Kalish and Dwyer for oblique flow n¢- 1 n¢ across tube banks Int. J. Heat Mass Transfer, 10, 1533 (1967). For g K¢ 8 and t = K¢ (8V/D) . c w Metzner and Friend Ind. Eng. Chem., 51, 879 (1959) present additional details of heat transfer with liquid metals for various sys- tems see Dwyer (1968 ed., Na and Nak supplement to Liquid Metals relationships for turbulent heat transfer with nonnewtonian fluids. Relationships for heat transfer by natural convection and through Handbook) and Stein (“Liquid Metal Heat Transfer,” in Advances in Heat Transfer, vol. 3, Academic, New York, 1966). laminar boundary layers are available in Skelland’s book (op. cit.).5-20 HEAT AND MASS TRANSFER HEAT TRANSFER WITH CHANGE OF PHASE where b = 2954 (SI) or 6978 (U.S. customary). For organic vapors at In any operation in which a material undergoes a change of phase, provision must be made for the addition or removal of heat to provide normal boiling point, k = 0.138 J/(m?s?K) 0.08 Btu/(h?ft? F), r= 3 3 - 3 720 kg/m (45 lb/ft ), m = (0.35)(10 ) Pa?s (0.35 cP), for the latent heat of the change of phase plus any other sensible heat- b ing or cooling that occurs in the process. Heat may be transferred by 1/3 h = b(D/W ) (5-91) F any one or a combination of the three modes—conduction, convec- tion, and radiation. The process involving change of phase involves where b = 457 (SI) or 1080 (U.S. customary). mass transfer simultaneous with heat transfer. Horizontal Tubes For the following cases Reynolds number 2100 and is calculated by using G= W /2L. F Colburn type: CONDENSATION h cm 4.4 = (5-92) Condensation Mechanisms Condensation occurs when a satu- cG k 4G /m rated vapor comes in contact with a surface whose temperature is 2 2 1/3 below the saturation temperature. Normally a film of condensate is G W r g F 2 2 G== kg/(s?m ) lb/(h?ft ) formed on the surface, and the thickness of this film, per unit of 2 1/312 2 (3mG /r g) 12L m breadth, increases with increase in extent of the surface. This is called Nusselt type: film-type condensation. 3 2 1/4 3 2 1/3 Another type of condensation, called dropwise, occurs when the hD D r gl D r g = 0.73 = 0.76 (5-93) wall is not uniformly wetted by the condensate, with the result that 12 12 k kmD t mG the condensate appears in many small droplets at various points on the Dimensional: surface. There is a growth of individual droplets, a coalescence of adjacent droplets, and finally a formation of a rivulet. Adhesional force 3 2 1/3 h = b(k r L/m W ) (5-94) b F is overcome by gravitational force, and the rivulet flows quickly to the where b = 205.4 (SI) or 534 (U.S. customary). For steam at atmo- bottom of the surface, capturing and absorbing all droplets in its path and leaving dry surface in its wake. spheric pressure Film-type condensation is more common and more dependable. 1/3 h = b(L/W ) (5-95) F Dropwise condensation normally needs to be promoted by introduc- ing an impurity into the vapor stream. Substantially higher (6 to 18 where b = 2080 (SI) or 4920 (U.S. customary). For organic vapors at times) coefficients are obtained for dropwise condensation of steam, normal boiling point but design methods are not available. Therefore, the development of 1/3 h = b(L/W ) (5-96) F equations for condensation will be for the film type only. The physical properties of the liquid, rather than those of the vapor, where b = 324 (SI) or 766 (U.S. customary). are used for determining the film coefficient for condensation. Nus- Figure 5-9 is a nomograph for determining coefficients of heat selt Z. Ver. Dtsch. Ing., 60, 541, 569 (1916) derived theoretical rela- transfer for condensation of pure vapors. tionships for predicting the film coefficient of heat transfer for Banks of Horizontal Tubes (N 2100) In the idealized case of Re condensation of a pure saturated vapor. A number of simplifying N tubes in a vertical row where the total condensate flows smoothly assumptions were used in the derivation. from one tube to the one beneath it, without splashing, and still in The Reynolds number of the condensate film (falling film) is laminar flow on the tube, the mean condensing coefficient h for the N 4G /m , where G is the weight rate of flow (loading rate) of condensate entire row of N tubes is related to the condensing coefficient for the per unit perimeter kg/(s?m) lb/(h?ft). The thickness of the conden- top tube h by 1 2 1/3 sate film for Reynolds number less than 2100 is (3mG /r g) . - 1/4 h = h N (5-97) N 1 Condensation Coefficients Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theory. It is generally known and con- Vertical Tubes For the following cases Reynolds number 2100 ceded that the film coefficients for steam and organic vapors calcu- and is calculated by using G= W /p D. The Nusselt equation for F lated by the Nusselt theory are conservatively low. Dukler Chem. the heat-transfer coefficient for condensate films may be written in Eng. Prog., 55, 62 (1959) developed equations for velocity and tem- the following ways (using liquid physical properties and where L is the perature distribution in thin films on vertical walls based on expres- cooled length and D t is t - t ): sv s sions of Deissler (NACA Tech. Notes 2129, 1950; 2138, 1952; 3145, Colburn type: 1959) for the eddy viscosity and thermal conductivity near the solid h cm 5.35 boundary. According to the Dukler theory, three fixed factors must be = (5-87) known to establish the value of the average film coefficient: the termi- cG k 4G /m nal Reynolds number, the Prandtl number of the condensed phase, 2 2 1/3 G W r g and a dimensionless group N defined as follows: F d 2 2 where G== kg/(s?m ) lb/(h?ft ) 2 1/312 2 1.173 0.16 2/3 2 0.553 0.78 (3mG /r g) 29.6D m N = (0.250m m /(g D r r ) (5-98) d L G L G Nusselt type: Graphical relationships of these variables are available in Document 3 2 1/4 3 2 1/3 6058, ADI Auxiliary Publications Project, Library of Congress, Wash- hL L r gl L r g = 0.943 = 0.925 (5-88) ington. If rigorous values for condensing-film coefficients are desired, 12 12 k kmD t mG - 5 especially if the value of N in Eq. (5-98) exceeds (1)(10 ), it is sug- d Dimensional: gested that these graphs be used. For the case in which interfacial shear is zero, Fig. 5-10 may be used. It is interesting to note that, 3 2 1/3 h = b(k r D/m W ) (5-89) b F according to the Dukler development, there is no definite transition Reynolds number; deviation from Nusselt theory is less at low where b = 127 (SI) or 756 (U.S. customary). For steam at atmospheric 3 Reynolds numbers; and when the Prandtl number of a fluid is less pressure, k = 0.682 J/(m?s?K) 0.394 Btu/(h?ft? F), r= 960 kg/m 3 - 3 (60 lb/ft ), m = (0.28)(10 ) Pa?s (0.28 cP), b 1/3 2 h = b(D/W ) (5-90) If the vapor density is significant, replace r with r (r -r ). l l v F

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