chap2-introduction to conduction传热传质

Chap 2 Introduction to conductionObjectives:1. To develop a deeper understanding of Fouriers law. What form does it take for different geometries?2. To develop the general equation, termed the heat equation, which governs the temperature distribution in a medium. temperature distribution determine the heat flux 2.1 The conduction rate equationOrigin: Fouriers law is phenomenological. Evaluating this expression in the limit as ,we obtain for the heat rateor for the heat fluxk-the thermal conductivity (w/m.k)Recall that the minus sign is necessary because heat is always transferred in the direction of decreasing temperature.The heat flux is directional quantity. The direction of heat flow will always be normal to a surface of constant temperature, called an isothermal (等温 ) surface. The heat flux is vector quantity, so the more general statement of the Fouriers law is as follows:is 3-D del operator and is the scalar(标量 ) temperature field. An alternative form iswhere is the heat flux in a direction n,which is normal (perpendicular) to an isotherm.The heat flux vector can be resolved into components such that, in Cartesian coordinates, the general expression isFrom Equation 2.3,it follows thatEach of these expressions relates the heat flux across a surface to the temperature gradient in a direction perpendicular to the surface. 2.2 The thermal properties of matterTransport propertiesn2.2.1 Thermal conductivityDefinition: Thermal conductivity associated with conduction in the x-direction is defined asFor an isotropic(各向同性 ) material:Usually , as illustrated in Fig 2.4.The solid stateAccordingly, transport of thermal energy is due to two effects: the migration of free electrons and lattice vibrational waves. For pure metals, is much lager thanFor alloys, is no longer negligibleFor nonmetallic solids, can be very large, such as diamond.The conductivity vary with the temperature. Fig 2.5 . Insulation SystemThermal insulations are comprised of low thermal conductivity materials combined to achieve an even lower system thermal conductivity.Foamed insulation, reflective insulationThe fluid stateGas The effect of temperature, pressure, and chemical species on the thermal conductivity of a gas may be explained in terms of the kinetic theory of gases. n, the number of particles per unit volume;c, the mean molecular speed c;, and the mean free path . The thermal conductivity of a gas increases with increasing temperature and decreasing molecular weight, these trends are shown in fig 2.6. The thermal conductivity is independent of pressure.LiquidMolecular conditions associated with the liquid state are more difficult to describe, and physical mechanisms for explaining the thermal conductivity are not well understood. Values of the thermal conductivity are generally tabulated as a function of temperature for the saturated state of the liquid. (Table A.5, Fig 2.7) n2.2.2 Other relevant propertiesThermophysical properties include two distinct categories: Transport and thermodynamic properties. Transport properties the thermal conductivity k;the kinetic viscosity , (for momentum transfer). Thermodynamic properties(equilibrium state)density and specific heat cp. volumetric heat capacity, cp (J/m3.K), measures the ability of a material to store thermal energy. Thermal diffusivity In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity , which has units of m2/s. It means the ability of a material to conduct thermal energy relative to its ability to store thermal energy.Materials of large will respond quickly to changes in their thermal environment, while materials of small will respond more sluggishly, taking longer to reach a new equilibrium condition. 2.3 The heat diffusion equationA major objective in a conduction analysis is to determine temperature field (temperature distribution) in a medium resulting from conditions imposed on its boundaries. The conduction heat flux at any point in the medium or on its surface may be computed from Fouriers law.Cartesian coordinatesConsider a homogeneous medium within which there is no bulk motion (advection) and the temperature distribution T(x,y,z) is expressed in Cartesian coordinates.Step1: Define a differential control volume, dx.dy.dz, as shown in fig 2.8.Step 2: Consider energy processesThe conduction heat rates perpendicular to the control surface at the x,y,z coordinate locations are indicated by the terms qx, qy, qz, respectively. The conduction heat rates at the opposite surfaces can then be expressed as a Taylor series expansion where, neglecting higher order terms,Energy sourcethe rate at which energy is generated per unit volume (W/m3). Energy storage (source or sink)No change in phase, it may be expressed asis the time rate of change of the sensible (thermal) energy of the medium per unit volume.Difference between Eg and Est.Step 3:to express conservation of energy using forgoing rate expression. The general form of the first law is (rate basis)The conduction rates constitute the energy inflow and outflow. Substituting the above equationsThe may be evaluated from Fouriers lawSubstituting Equations 2.12 into Equation 2.11 and dividing out the dimensions of the control volume (dxdydz), the heat diffusion equationThis equation, usually known as the heat equation, provides the basic tool for heat conduction analysis.If the thermal conductivity is a constant,Under steady state condition,Equation 2.13 reduces to1-d, no energy generation,steady stateimplication: the heat flux is a constant Cylindrical coordinatesl In cylindrical coordinates,the general form of heat flux vector isl Heat flux components in the radial,circumferential,and axial direction,respectively.Spherical coordinatesl In spherical coordinates, the general form of heat flux vector and Fouriers law isHeat flux components in the radial, polar, and azimuthal direction, respectively.2.3 boundary and initial conditionsTo determine the temperature distribution in a medium, it is necessary to solve the appropriate form of the heat equation. However, such a solution depends on the physical conditions existing at the boundaries and, if the situation is time dependent, on condition