大学精品课件：Chapter 2 Steady-State Conduction-One Dimension.pptx

1、Chapter II,Steady-State Conduction-One Dimension,2-1 Introduction,The application of Fouriers law of heat conduction in some steady-state one dimensional systems.,2-1 Introduction,Steady-state A system in a steady state has numerous properties that are unchanging in time. This implies that for any p

2、roperty of the system, the partial derivation with respect to time is zero:,2-1 Introduction,Discussion The differences between steady-state system and equilibrium system? If a system fulfills the following condition: Is it a steady-state system?,2-1 Introduction,One dimension In physical terms, a o

3、ne dimension system refers to the effect of other coordinate may be so small that can be neglect.,2-2 The Plane Wall,T1,T2,Temperature profile,Integrate the one dimensional Fouriers law yields: Thus one obtain the heat flux through the plane wall:,q,2-2 The Plane Wall,The thermal conductivity is con

4、sidered to be constant in the above result. If the thermal conductivity varies with temperature, for a simple case, according to linear relation The resultant equation for the heat flow is,2-2 The Plane Wall,Multiple material By adopting the similar method as previous, one can obtain,A B C,1 2 3 4,T

5、emperature profile,2-2 The Plane Wall,The heat-transfer rate may be considered as a flow, and the combination of thermal conductivity, thickness of material, and area as a resistance to this flow. The temperature is the potential, or driving, function for this flow, and the Fourier equation may be w

6、ritten,2-2 The Plane Wall,2-2 The Plane Wall,Other examples of this viewpoint,Ohms Law in electric-circuit theory,Newtons second law,2-2 The Plane Wall,Discussion The application of this viewpoint to a system that the thermal conductivity varies with temperature.,2-3 Insulation and R Values,In class

7、ifying the performance of insulation, it is a common practice in the building industry to use a term called the R value, which is defined as,Note that this differs from the thermal-resistance concept discussed above in that a heat flow per unit area is used.,2-4 Radial Systems,Cylinders,The Fouriers

8、 law for a cylinder is written in,where is the area for heat flow,2-4 Radial Systems,Cylinders,The solation of the above equation is,The thermal-resistance in this case is,2-4 Radial Systems,Multiple-layer cylinders,2-4 Radial Systems,Spheres,In which,2-4 Radial Systems,Convection boundary condition

9、s,The form of heat-resistance can be written,2-5 The Overall Heat-transfer Coefficient,The heat transfer is expressed by,The heat transfer is written,2-5 The Overall Heat-transfer Coefficient,The overall heat transfer by combined conduction and convection is frequently expressed in terms of an overa

10、ll heat-transfer coefficient U, defined by the relation,Thus U can be written as,It represents the heat-transfer ability per unit area under certain temperature difference.,2-5 The Overall Heat-transfer Coefficient,In the following relation,The overall heat-transfer coefficient can be represent by,2

11、-5 The Overall Heat-transfer Coefficient,Hollow cylinder exposed to a convection environment,2-5 The Overall Heat-transfer Coefficient,Hollow cylinder exposed to a convection environment,The overall heat transfer should be:,2-5 The Overall Heat-transfer Coefficient,Hollow cylinder exposed to a conve

12、ction environment,The overall heat transfer coefficient should be,Inside:,outside:,The R value under this situation? Is it also surface area dependent?,2-5 The Overall Heat-transfer Coefficient,Some typical values of the overall heat-transfer coefficient,2-6 Critical Thickness of Insulation,Consider

13、 a circular pipe with insulation surrounded,The maximization condition is,which gives the result,2-6 Critical Thickness of Insulation,The central concept is that for sufficiently small values of h the convection heat loss may actually increase with the addition of insulation because of increased sur

14、face area.,Please consider whether the conception of critical thickness of insulation is applicable to plate or sphere configuration?,2-7 Heat-Source Systems,Nuclear reactors Electrical conductors Chemically reacting systems ,In the following sections, the discussion will confine to one-dimensional

15、systems, namely the temperature is a function of only one space coordinate.,2-7 Heat-Source Systems,Plane wall with heat sources,The heat generated per unit volume is ,and we assume that the thermal conductivity does not vary with temperature. From Chapter 1, the differential equation governs the he

16、at flow is,2-7 Heat-Source Systems,Plane wall with heat sources,The general solution to the above equation is,This is a parabolic distribution.,2-7 Heat-Source Systems,Plane wall with heat sources,The same result may be obtained through an energy balance, namely the total heat generated must equal t

17、he heat lost at the faces of the plane wall.,where A is the cress-sectional area of the plate.,2-7 Heat-Source Systems,Plane wall with heat sources,From the temperature distribution function we obtain the temperature gradient:,By substituting this expression into the energy balance equation,Then the

18、 same result can be obtained,2-7 Heat-Source Systems,Plane wall with heat sources,From the temperature distribution function we obtain the temperature gradient:,By substituting this expression into the energy balance equation,Then the same result can be obtained,2-8 Cylinder with Heat Sources,Consid

19、er a cylinder of radius R with uniformly distributed heat sources and constant thermal conductivity. The appropriate differential equation describing this system is,The boundary conditions are,and the heat balance condition,2-8 Cylinder with Heat Sources,The general solution of the above above equat

20、ion is,From the second boundary condition,Thus,From the first boundary condition,then,2-8 Cylinder with Heat Sources,The final solution for the temperature distribution is then,or in dimensionless form,Using this expression we can calculate the central temperature of the nuclear fuel rod.,2-8 Cylind

21、er with Heat Sources,For a hollow cylinder with uniformly distributed heat sources The appropriate boundary conditions would be,The general solution is still,Application of the new boundary conditions yields,2-9 Fins,Examples of fins,2-9 Fins,The energy balance on an element of the fin of thickness

22、dx is,2-9 Fins,Energy in left face,Energy out right face,Energy lost by convection,where A is the cross-sectional area of the fin and P is the perimeter. Thus,In which,2-9 Fins,Then we obtain,Let . The above equation becomes,with the boundary condition,The other boundary conditions depends on the ph

23、ysical situation. several cases may be considered.,Case 1 The fin is long, and temperature at the end of the fin is essentially that of the surrounding fluid. Case 2 The fin is of finite length and loses heat by convection from its end. Case 3 The end of the fin is insulated so that dT/dx=0 at x=L.,

24、2-9 Fins,Case 1 The fin is long, and temperature at the end of the fin is essentially that of the surrounding fluid.,The general solution of the equation may be written,Boundary conditions,And the solution becomes,2-9 Fins,Case 2 The fin is of finite length and loses heat by convection from its end.

25、,Boundary conditions,And the solution becomes,viz.,2-9 Fins,Case 3 The end of the fin is insulated so that dT/dx=0 at x=L.,Boundary conditions,And the solution becomes,viz.,2-9 Fins,To indicate the effectiveness of a fin in transferring a given quantity of heat, a new parameter called fin efficiency

26、 is defined by,For case 3 above, the fin efficiency becomes,2-9 Fins,where z is the depth of the fin, and t is the thickness. If the fin is sufficiently deep, the term 2z will be large compared with 2z, and,Thus,2-9 Fins,In some cases a valid method of evaluating fin performance is to compare the he

27、at transfer with the fin to that which would be obtained without the fin. The ratio of these quantities if,2-10 Thermal Contact Resistance,When two solid bodies come in contact, such as A and B in the figure, heat flows from the hotter body to the colder body. From experience, the temperature profil

28、e along the two bodies varies, approximately, as shown in the figure. A temperature drop is observed at the interface between the two surfaces in contact. This phenomenon is said to be a result of a thermal contact resistance existing between the contacting surfaces. Thermal contact resistance is de

29、fined as the ratio between this temperature drop and the average heat flow across the interface.,2-10 Thermal Contact Resistance,Performing an energy balance on the two materials, we obtain,or,2-10 Thermal Contact Resistance,Some of the fields where contact conductance is of importance are: Electron

30、ics Electronic packaging Heat sinks Brackets Industry Nuclear reactor cooling Gas turbine cooling Internal combustion engines Heat exchangers Thermal insulation Flight Hypersonic flight vehicles Thermal supervision for space vehicles,2-10 Thermal Contact Resistance,Factors influencing contact conductance 1. Contact pressure 2. Interstitial materials 3. Surface characteristics Surface roughness, waviness and flatness Surface deformations Surface cleanliness,