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Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and TechnologyEmpirical and Practical Relations for Forced-Convection Heat Transfer 1 Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology26-1 Introduction The discussion and analyses of Chapter 5 have shown how forced-convection heat transfer may be calculated for several cases of practical interest;however,the problems considered were those that could solved in an analytical fashion.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology3But,it is not always possible to obtain analytical solutions to convection problems,and the individual is forced to resort to experimental methods to obtain design information,as well as to secure the more elusive date that increase the physical understanding of the process.What we have to do:Generalize the results of ones experiments in form of some empirical correlation Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology4 DifficultiesWhich variables should we measure?What functional form should the data be organized into?Its hard and expensive to do the experiments,so,how many experiments should we do?Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology5Similarity ConsiderationsPurpose:to do research on the relationship between similar physical phenomena.For similar physical phenomena:at corresponding time with corresponding location on the physical quantity related with the phenomenon correspondence proportional.For same type of phenomena:Phenomenon described by differential equations with the same form and content.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology6Characteristics for physical phenomena similarityThe same characteristic numbers are equal There is somewhat relationship between different characteristics.For example Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology7The conditions for physical phenomena similaritySame identified characteristic numbers are equalIts similar for Monodromy conditions,which includes initial conditions,boundary conditions initial conditions,boundary conditions andand Geometric Geometric conditionsconditions Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology8How to get dimensionless groupsHow to get dimensionless groupsSimilarity Considerations:To establish the column proportion coefficient between the two phenomena,relationship between the export of these similarity coefficient and obtain a dimensionless quantity based on Known mathematical description of the physical phenomena.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology9Phenomenon1：Phenomenon2：mathematical description Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology10Establish similar multiplesRelationship between them Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology11Berkeley numberTo obtain dimensionless groupsdimensionless groups Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology12Dimensional AnalysisIn dimensional analysis,dimensional groups such as the Reynolds and Prandtl numbers are derived from purely dimensional and functional considerations.Fundamental Basis Theorem of ,A consistent dimensionless equation showing the relationship between the n physical quantities could be transferred to a relationship which contains(n-r)independent dimensionless groups.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology13Advantages of dimensionless analysisSimpleWe can still obtain dimensionless groups without knowing the Differential EquationsFundamental quantity in the SI UnitsLength m，MASSkg，times，ELECTRIC CURRENTA，thermodynamic temperatureK，amount of substancemol，luminous intensitycd Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology14Now we come back to the difficultiesWhich variables should we measure?Only variables that are contained in characteristic numbersWhat functional form should the data be organized into?Arrange the data according to the relationship between the characteristic numbersIts hard and expensive to do the experiments,so,how many experiments should we do?Modular Experiments under the guidance of the similar consideration Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology15 Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology166-2 Empirical Relations For Pipe And Tube FlowCases of Undeveloped FlowThe cases of undeveloped laminar flow systems where the fluid properties vary widely with temperature,and turbulent-flow systems are considerably more complicated but are of very important practical interest in heat exchangers and associated heat-transfer equipment.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology17For design and engineering purposes,empirical correlations are usually of greatest practical utility.For laminar flowFor laminar flow,the length of the undeveloped partundevelopeddeveloped(Average from 0 to x)Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology18For turbulent flowFor turbulent flow,the length of the undeveloped partundevelopeddeveloped(Average from 0 to x)Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology19Further consideration to Bulk temperature Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology20For tube in Figure 6-1 the total energy added can be expressed in terms of bulk-temperature by In differential equation,The Tw and Tb here are the wall and bulk temperature at the particular x location.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology21The total heat transfer can also be expressed as 6-3where A is the total surface area for heat transfer.Because both Tw and Tb can vary along the length of the tube,a suitable averaging process must be adopted for use with Equation(6-3).In chapter 10 well discuss different methods for taking proper account of temperature variations in heat exchangers.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology22A tradition expression for calculation of heat transfer in fully developed turbulent flow in smooth tubes For heating of the fluid For cooling of the fluid Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology23Conditions Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology24Wide temperature differencesThese property variations may be evidenced by a change in the velocity profile as indicated in the figure.1.Inothermal flow2.Gas heating,Liquid cooling3.Liquid heating,gas cooling Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology25Some relations take property variations into accountGas heating Gas coolingLiquid HeatingLiquid Cooling Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology26Conditions Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology27Conditions Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology28If the channel through which the fluid flows is nor circular cross the section,it is recommended that the heat-transfer correlations be based on the hydraulic hydraulic diameterdiameter.Various SectionsDefinitionHydraulic diameterA is cross-sectional area of the flow P is the wetted perimeter Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology29The hydraulic diameter should be used in calculating the Nusselt and Reynolds numbers,and in establishing the friction coefficient for use with Reynolds analogy.Average Nusselt number for uniform heat flux in flow direction and uniform wall temperature at particular flow cross sectionAverage Nusselt number for uniform wall temperature Product of friction factor and Reynolds number based on hydraulic diameter Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology30Constant axial wall heat fluxConstant axial wall temperatureHeat transfer and fluid friction for fully developed flow in ducts of various cross sectionsGeometryTriangleSquareRegular HexagonCircleRectangle with b=2a Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology31Air at 2atm and 200 is heated as it flows through a tube with a diameter of 1 in(2.54cm)at velocity of 10 m/s.Calculate the heat transfer per unit length of tube is constant-heat-flux condition is maintained at the wall and the wall and the wall temperature is 20,above the air temperature,all along the length of the tube.How much would the bulk temperature increase a 3-m length of the tube?Example Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology32SolutionWe first calculate the Reynolds number to determine if the flow is laminar or turbulent,and then select the appropriate empirical correlation to calculate the heat transfer.The properties of air at a bulk temperature of 200 are Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology33So the flow is turbulent.We therefore use Equation(6-4a)to calculate the heat-transfer coefficient Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology34The heat-flow per unit length is then Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology35We can now make an energy balance to calculate the increase in bulk temperature in a 3.0-m length of the tube Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology36So the heat transfer per unit length is Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology376-3 Flow Across Cylinders And SpheresBoundary-layer SeparationLook at Figure 6-7,it is necessary to include the pressure gradient in the analysis because this influences the boundary-layer velocity and causes separated flow region to develop on the back side of the cylinder when the free stream velocity is sufficiently large.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology38 Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology39Figure 68 Velocity distributions indicating flow separation on a cylinder in cross flowBoundary LayerSeparation Region Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology40In case of cylinder,one might measure x distance from the front stagnation point of the cylinder.Thus the pressure in the boundary layer should follow that of the free stream for potential flow around a cylinder,provided this behavior would not contradict some basic principle.As the increase flow progresses along the front side of the cylinder,the pressure would decrease and then increase along the back side of the cylinder,resulting in an increase in free-stream velocity on the front side of the cylinder and a decrease on the back side.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology41The detailed behavior of the heat transfer from a heated cylinder to air are summarized in Figure 6-11The change of heat-transfer coefficient through circular Cylinders Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology42lAt the lower Re numbers,a minimum point in the heat-transfer coefficient occurs at approximately the point of separation.lThere is a subsequent increase in the heat-transfer coefficient on the rear side of the cylinder,resulting from the turbulent eddy motion in the separated flow.lAt higher Re numbers,two minimum points are observed Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology43Because of the complicated nature of the flow-separation,it is not possible to calculate analytically the average coefficients in cross flow.Knudsen and Katz suggested that the correlation be extended to liquids by inclusion of The resulting correlation for average heat-transfer coefficients in cross flow over circular cylinders is The relationship of Flow Across Cylinders 6-17 Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology44The constant C and n are tabulated in the Table 6-2ReCn0.440.9890.3304400.9110.3854040000.6830.4664000400000.1930.618400004000000.02660.805，Bulk temperature Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology45Still a more comprehensive relation is given by Churchill and Bernstein which is applicable over the complete range of available data.For Bulk temperature Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology46Noncircular CylindersEquation6-17 is employed in order to obtain an empirical correlation for gases,and the constants for use with this equation are summarized in Table 6-3.The data upon which Table 6-3 is based were for gases with Pr 0.7 and were modified by same 1.11 factor employed for the information presented in Table 6-2 Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology47Table 6-3constants for heat transfer from non circular cylinders for use with Equation(6-17)FlatSquareRegular Hexagon Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology48SpheresA single Equation for gases and liquids flowing past spheresAt free-stream temperature Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology49ExampleAir at 1 atm and 35 flows across a 5.0-cm-diameter cylinder at a velocity of 50m/s.The cylinder surface is maintained at a temperature of 150.Calculate the heat loss per unit length of the cylinder Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology50SolutionWe first determine the Re number and then find the applicable constants from Table 6-2 for use with Equation(6-17).The properties of air are evaluated at the film length of cylinder.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology51From Table 6-2,we have Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology526-4 Flow Across Tube BanksDifferent Arrangement of BanksUsually,there are two kinds of arrangement of banks,they are In line&StaggeredFigure 6-14In-lineStaggered Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology53For Staggered tube banks,they have better heat-transfer performance,but hard to clean and have more resistance loss.6-17On the basis of a correlation of the results of various investigators,Grimson was able to represent data in form of Equation 6-17 Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology54Number of the Rows10 The Re number is based on the maximum velocity occurring in the tube bank,that is,the velocity through the minimum-flow area.The value of C and n are listed in the following table Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology55In-lineStaggered Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology56Zhukauskas suggested a group of formula which could be used for a wide range of Pr numbersThe bulk temperatureThe Re numbers are based on Outside diameter of the tube and on the velocity through the minimum-flow area.Range of Pr number Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology57For number of tube banks 16 rows,he suggested a ratio Equations for heat-transfer in tube banks of 16 rows or more(IN-LINE arrangement)EquationsRange of Re number Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology58Equations for heat-transfer in tube banks of 16 rows or more(Staggered arrangement)Raito for tube banks 16 rows EquationsRange of Re NumberNumber of RowsIn-lineStaggered Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology596-5 Liquid-Metal Heat TransferLets first consider the simple flat plate with a liquid metal flowing across it.The Prandtl number for liquid metals is very low,of the order of 0.01,so that the thermal boundary-layer thickness should be substantially larger than the hydrodynamic-boundary layer thickness.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology60This situation results from the high values of thermal conductivity for conductivity for liquid metals and is depicted in Figure 6-15.Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology61Since the ratio of is small,the velocity profile has a very blunt shape over most of the thermal boundary layer.As a first approximation,then,we might assume a slug-flow model for calculation of the heat transfer;that is,we take Throughout the thermal boundary layer for purposes of computing the energy-transport term in the integral energy equation Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology62The conditions on the temperature profile are the same as those in section 5-6,so that we use the cubic parabola as before:Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology63Inserting Equations gives That may be integrated to give Chapter 6College of Nuclear Chapter 6College of Nuclear Science and TechnologyCollege of Nuclear Science and Technology64The solution to this differential equations is For a plate heated over its entire length.The heat-transfer coefficie
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