chap5_transientconduction传热传质

Chapter 5 Transient Conduction l Many heat transfer problems are time dependent.The boundary conditions are changed.Objective: To develop procedures for determining the time dependence of the temperature distribution within a solid during a transient process, as well as for determining heat transfer between the solid and its surroundings.l (1)Lumped capacitance method(集总热容法 ):If temperature gradients within the solid may be neglected, a comparatively simple approach, termed the lumped capacitance method, may be used to determine the variation of temperature with time. l (2) Exact solution:Under conditions for which temperature gradients are not negligible, but heat transfer within the solid is one-dimensional, exact solutions to the heat equation may be used to compute the dependence of temperature on both location and time. (3) Finite-difference methodl 5.1 The lumped capacitance methodl Consider a hot metal forging that is initially at a uniform temperature Ti and is quenched by immersing it in a liquid of lower temperature (Figure 5.1). The essence of the lumped capacitance method is the assumption that the temperature of the solid is spatially uniform at any instant during the transient process. Figure 5.1 Cooling of a hot metal forging.Although the condition is never satisfied exactly, it is closely approximated if the resistance to conduction within the solid is small compared with the resistance to heat transfer between the solid and its surroundings.l The transient temperature response is determined by formulating an overall energy balance on the solid. l Introducing the temperature difference l It follows thatl Separating variables and integrating from the initial condition, for which l (5.5)where l Evaluating the integralsl (5.6)l Equation 5.5 may be used to determine the time required for the solid to reach some temperature T, or conversely, Equation 5.6 may be used to compute the temperature reached by the solid at some time t.l The difference between the solid and the fluid temperature must decay exponentially to zero as time t approaches infinity as shown in Fig 5.2.l The quantity (Vc/hAs) may be interpreted as a thermal time constantl where Rt is a resistance to convection heat transfer and Ct is the lumped thermal capacitance of the solid.l The total energy transfer Q occurring up to some time t5.2 Validity of the lumped capacitance methodl 1 The Biot numberThe lumped capacitance method is the simplest and most convenient method that can be used to solve transient conduction problems, it is not suitable for any conditions. It is important to develop a criterion for lumped capacitance method.l To develop a suitable criterion consider steadystate conduction though the plane wall of area A as shown in Fig 5.3. The surface energy balance l Rearranging,l (5.9)Figure 5.3 Effect of Biot number on steady-state temperature distribution in a plane wall with surface convection.The Biot number provides a measure of the temperature drop in the solid relative to the temperature difference between the surface and the fluid.The Biot number is a ratio of thermal resistances, conduction to convection.If Bi0.2, the infinite series solution, Equation 5.39a, can be approximated by the first term of the series, that isl (5.40a)l Or (5.40b) l l Where represent the middle temperaturel (5.41)l An important implication of Equation 5.40b is that the time dependence of the temperature at any location within the wall is the same as that of the mid-plane temperature.l The coefficients can be evaluated from equation 5.39b and 5.39c, respectively, or obtained from Tab5.1 for a ranger of Biot numbers.5.5.3 Total energy transferl In many situation it is useful to know the total energy that has left or entered the wall up to any time t in the transient process.l The conservation of energy requirement bounded by initial condition (t=0) and any time t0l Equating the energy transferred from the wall Q to Eout,and setting Ein=0 and Est=E(t)-E(0),l Sol Or l Introducing the quantityl which may be interpreted as the initial internal energy of the wall relative to the fluid temperature. It is also the maximum amount of energy transfer that could occur if the process were continued to time t=.l Assuming constant properties, the ratio of the total energy transferred from the wall over the time interval t to the maximum possible transfer isl (5.46)l Where o* can be determined from Equation 5.41,using Table 5.1 for values of the coefficients C1 and 1.5.5.4 Additional considerationsl (1)The foregoing results may also be applied to a plane wall of thickness L, which is insulated on one side (x*=0), and experiences convective transport on other side (x*=1).l (2)It should also be noted that the results may be used to determine the transient response of a plane wall to a sudden change in surface temperature. The process is equivalent to having an infinite convection coefficien