(5.3)--Chapter 3 Unsteady-state heat co传热学传热学传热学.ppt

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1、Chapter 3 Unsteady heat conductionnBasic concept of unsteady heat conductionnAnalysis method of zero-dimensional problem lumped parameter methodnAnalytical solution for unsteady heat conduction of a typical one-dimensional objectnUnsteady heat conduction of semi-infinite object3.1 Basic concept of u

2、nsteady heat conductionI.Research significance Engineering application:unit start,stop,variable working conditions,etc.Rapid temperature changes can cause components to break due to thermal stress or thermal deformation.Boiler ignition can not be too urgent(boiler drum)Rapid changes in engine operat

3、ing conditions(unfavorable)Therefore,it is necessary to determine the temperature field as a function of time inside the object.II.Two categories:periodic,non-periodicnPeriodicity:the temperature of each point in object changes periodically with time.Such as:the overnight temperature change of build

4、ings,the intermittent cooling of refrigerators,the rotary air preheater.Flue gas zoneFlue gasAirTransition zoneTransition zoneAir zoneAirFlue gasNon-periodicity:the temperature of each point in the object increases or decreases with time,getting closed to the temperature of surrounding medium.Such a

5、s:quenching of workpiece,cooling of casting,temperature change of component when the thermodynamic device starts and stops.Transient unsteady heat conductionTemperature distributionTwo different stagesInformal status stage(irregular situation stage)Formal status stage(normal stage)Temperature distri

6、bution mainly depends on boundary conditions and propertiesTemperature distribution is mainly controlled by initial temperature distributionInformal status stage(initial stage),formal status stage,new steady stateThree stages of unsteady heat conduction processHeat Variation1 Heat rate introduced on

7、 the left side of plate2 Heat rate introduced on the right side of plateNotenThe amount of heat conduction at different positions in the heat transfer direction during unsteady heat conduction process is different everywhere.nThe difference in heat conduction is due to the change of internal energy

8、of object at different positions with time.nThermal resistance method generally can not be used to make quantitative analysis of unsteady heat conductionIII.Law of uniqueness for solution of heat conduction differential equationnSolution of unsteady heat conduction problem essentially comes down to

9、solving heat conduction differential equation under specific initial conditions and boundary conditions.Heat conduction differential equation and definite conditions are as follows:It can be proved mathematically that if a function satisfies the above equations and initial and boundary conditions,th

10、en the function is the only solution to this particular heat conduction problem.IV.Unsteady heat conduction under third boundary conditions1 Relationship between temperature change in an object and boundary parametersnA metal plate has thickness ,thermal conductivity of ,and initial temperature of .

11、Suddenly it is cooled in a fluid of temperature ,the surface convection heat transfer coefficient is h.Due to the difference of relative area thermal resistance of and ,the temperature field in plate changes in the following three cases.nSince the surface convection heat transfer resistance is almos

12、t negligible,the surface temperature of plate is cooled to at the beginning of the process.As time goes by,the temperature at each point in plate gradually decreases,approaching .Equivalent tonAt this time,the thermal resistance of heat conduction in plate is almost negligible,so that the temperatur

13、e of each point in the plate is nearly uniform at any time.And as time goes on,the temperature of each point overall declines and gradually approach .Equivalent tonAt this time,temperature distribution in plate at different times is between the above two extreme cases.close to Equivalent to is a fin

14、ite value.2.Biot number(Biot criterion)nIt can be seen that the relative magnitudes of the above two thermal resistances have a great influence on the temperature field of unsteady heat conduction in the object.Introduce a dimensionless number characterizing the ratio of these two thermal resistance

15、s:is characteristic length.If the half thickness of plate is used as characteristic length,that is .is thermal conductivity of solid,and molecular represents the thermal resistance per unit area inside the object.Denominator represents the heat transfer resistance per unit area on surface of objectT

16、his criterion indicates the ratio of heat conduction resistance per unit area inside the object and heat transfer resistance per unit area on surface.A brief introduction to dimensionless number Basic idea:When the problem studied is very complicated,there are many parameters involved.In order to re

17、duce these parameters,some researchers combine these parameters so that they can characterize a class of physical phenomena,or the main features of physical process,and without dimension.Therefore,such a dimensionless number is also called characteristic number,or criterion number.For example,Biot n

18、umber is also known as the Biot criterion.There will be many similar criteria later in this course.The geometric scale involved in characteristic number is called the characteristic length and is generally represented by the symbol l.For a feature number,students should grasp its definition and phys

19、ical meaning,as well as the meaning of each parameter in its definition.(1)Variation of temperature distribution and heat rate distribution with time and space(2)Unsteady heat conduction differential equation:(3)Solution method:Analytical solution,Approximate analysis method,Numerical solutionAnalyt

20、ical solution:Separation variable method,integral transformation,Laplace transformApproximate analysis method:Lumped parameter method,Integral methodNumerical solution:Finite difference method,Monte Carlo method,finite element method,molecular dynamics simulationV.Purpose of learning unsteady heat c

21、onduction3.2 Zero-dimensional modellumped parameter method1.Zero-dimensional model Isothermal or negligible temperature difference in objectI.Zero-dimensional modelThermal conductivity of object is quite large and nternal temperature difference is small.Geometric scale of the object is smallInternal

22、 thermal resistance is smallII.What situation can simplify to the zero-dimensional modelSurface heat transfer is small surface convection heat transfer coefficient is smallCombined effect of various factorsExternal thermal resistance is largeFor infinite plate For infinite long cylinderFor sphereBiB

23、iot criterion judgement If the above formula is satisfied,the deviation between the temperatures of each point in the object caused by using the lumped parameter method is less than 5%.(Thickness is 2)III.Governing equation(take third boundary conditions as an example)A solid of any shape having a v

24、olume of V and a surface area A with a uniform initial temperature t0.At the initial moment,it is suddenly placed in a fluid with constant temperature t,while t0t.The physical properties of solid and the convective heat transfer coefficient h of the solid surface are both constant.Try to find the de

25、pendence of solid temperature on time.Two methods:1.Rebuild governing equation(directly using principle of thermal balance)Thermal capacity change=convection heat transferor Change in internal energy of object=convection heat transferMake ,then2.Using heat conduction differential equation combined w

26、ith the lumped parameter method,and for three-dimensional unsteady and constant thermal conductivity:since ,then(a)By definition,the generalized heat source is the total heat transfer rate Q of the object and the external calculated by the unit volume.By Newtons formula(b)Here Q is always a positive

27、 value,but under the condition that the object is cooled,the generalized heat source should be negative,so there are:(c)Substitute(c)into equation(a):IV.Dependence of solid temperature on timeWhen the initial condition=0,separate variables from the above equation and integrate:That is(d)The index te

28、rm in the right end of the above formula can be changed as follows:has a length dimension,namely:V.Fourier criterionPhysical meaning:A relative time value that can be thought of as a dimensionless time that characterizes the depth of an unsteady process.:the deeper the thermal disturbance propagates

29、 into the interior of the object,the closer the temperature at each point in the object is to temperature of surrounding medium.The denominatorcan be thought as the time required to spread a finite amount of thermal perturbation occurring on the boundary through a solid layer of a certain thickness

30、to the area of l2.The numerator is the time interval from the moment when thermal disturbance starts to occur at the boundary to the calculating time.VI.Time constantcalled time constant,recorded asIf ,then:Physically,how fast thermocouple responding to change in fluid temperature depends on its own

31、 thermal capacity and surface heat transfer conditions(hA).When time ,the excess temperature of object has reached 36.8%of initial excess temperature value.When measuring fluid temperature with a thermocouple,the time constant of the thermocouple is an indicator of how quickly the thermocouple respo

32、nds to changes in fluid temperature.Obviously,the smaller the time constant,the faster the thermocouple can quickly reflect the change in fluid temperature.,then ,that is:thermocouple responds quickly to changes in fluid temperatureVII.Heat exchange during 1.Transient heat rateSinceThen2.Heat exchan

33、ge duringVIII.The lumped parameter method makes the partial differential equation into ordinary differential equation(),so it can also be solved for complex boundary conditions(such as radiant boundary conditions).3.3 Analytical solution of typical one-dimensional unsteady heat conductionInfinite pl

34、ate,third boundary conditions(same on both sides),constant propertiesGoverning equationUse excess temperature Geometric conditions:thickness:Initial conditions Boundary conditions:Form of solutionwhere:is Fo criterion with a characteristic scale of .Thus,the temperature distribution inside plateCool

35、ing or heating problem of long cylinder and sphereFor one-dimensional unsteady heat conduction problem of the cylinder and sphere under third boundary conditions,by analyzing in cylindrical coordinate and spherical coordinate respectively,the analytical solution of the temperature distribution can a

36、lso be obtained,with a form of fast converged infinite series.Long cylinder:J0 and J1 are the first-type Bessel functions of 0th order and 1st order,respectively.Sphere:For one-dimensional unsteady heat conduction problem of the cylinder and sphere under third boundary conditions,the analytical solu

37、tion of the temperature distribution is also a form of function of Bi,Fo and r/R.Formal status stage of unsteady state heat conductionIndependent of timeNumerical calculations show that when Fo 0.2,the deviation between calculation result by omitting the second and subsequent items in infinite serie

38、s and that by complete series is less than 1%.PlateInvestigate heat transfer If Q is the heat transferred in -Average excess temperature at time zQ0 -Maximum heat transferable by unsteady heat conductionHere A,B and function see Table 3-1 in p.127 of textbookFor infinite plate,long cylinder and sphe

39、re:and can be expressed by the general formula:Infinite platelong cylinder and sphereEngineering calculation method in formal status stagenApproximate fitting formulanGraphic solution(Heisler chat)Application range of solutionThe Heisler chat and the fitting function are only applicable to the heati

40、ng and cooling process of first boundary condition or of third boundary condition of the constant temperature medium and Fo 0.2Fitting formulawhere constant a,b,c,d see Table 3-2 in p.128;a,b,c,d see Table 3-3 in p.128.For the above formula,A,B,1,J0 can be fitted by the following formula:Heisler cha

41、tHeisler chatThree variables,need to draw separatelyTake the infinite large tablet as an example,when F00.2,take the first item of series.(1)First,(2)According to formula(3-28),draw the curves(3)Then,the temperature at any point in plate isSimilarly,heat exchange in an unsteady process can also be p

42、lotted using(3-29)and(3-31).The approximate results of analysis solution of one-dimensional unsteady heat conduction of cylinders and spheres are also plotted as line graphs,as appendices 16 and 17.(1)Applicable to case where object is heated or cooled(2)Applicable to case where one side is insulate

43、d and the other side is the third boundary condition(3)Analytical solution of Bi is the solution of first boundary conditionsInfluence of Fo and Bi on temperature field(1)Excess temperature of each point in object decreases as Fo increases(2)Under the same Fo,the larger the Bi,the smaller the m/0Bi

44、also determines the evened degree of temperature in object.Extension and discussion of the application range of analysis solution Basic idea of solving one-dimensional unsteady heat conduction problem(1)First,use Bi to check whether the conditions of the lumped parameter method are satisfied.If the

45、property belongs to unknown h or,it can be assumed first and then checked;(2)If the lumped parameter method cannot be used,the analytical solution(Heisler chat method and fitting formula method)can be used;(3)If neither(2)nor(1)can be solved,a numerical solution is used.3.4 Unsteady heat conduction

46、of semi-infinite objectsnConcept of semi-infinite object Error function Dimensionless variablenExcess temperature:nSolutionError functionMake Ifi.e.,it can be considered that there is no change in temperaturenGeometric positionTwo important parametersnTimeIf ,for a plate with thickness 2,If ,it can

47、be treated as a semi-infinite objectIf ,for a finite large actual object,the concept of a semi-infinite object is only applicable to the initial stage of unsteady heat conduction.Within the inert time,the temperature at x can be considered to be equal to t0Heat rate at any point:Make ,that is,the he

48、at rate on boundary surfaceEndothermic coefficientAccumulated heat transfer during0,Chapter SummarynConcept of unsteady heat conduction nClassification of unsteady heat conductionnPeriodicity:FeaturesnNon-periodicity(transient):characteristicsnInformal status stage(initial conditions)Formal status s

49、tage(boundary conditions)nCharacteristic number(criterion number)nCharacteristic lengthnThe lumped parameter methodnApplicable conditions:small internal thermal resistance or large external thermal resistance:nMathematical description of zero-dimensional model(governing equations,initial conditions)

50、nTemperature DistributionnFourier(Fo)criterion(form,physical meaning)nTime constant tc(form,physical meaning)nHeat conduction rate during 0tnOne-dimensional unsteady heat conductionnMathematical description(governing equations,initial conditions,boundary conditions)nApplication range of analytical s