Heat Conduction with Time Dependent Boundary Conditions

Example 8.9. Heat Conduction with Time Dependent Boundary Conditions 

Consider that one of the main advantages of the Laplace transform technique is that it can be used for time dependent boundary conditions, also. The separation of variables technique cannot be directly used and one has to use DuhameFs superposition theorem[l] for this purpose. Consider the modification of example 8.7  

Equation (8.19) is solved by slightly modifying the Maple program used for example 8.7 as  

8 Laplace Transform Technique for Partial Differential Equations 

The dimensionless temperature at the surface x =0 reaches a maximum and then decreases as a function of time  

---

Heat conduction with a constant boundary condition at x =0 was considered in example 4.1. The same technique can be applied for time dependent boundary conditions. Consider the transient heat conduction problem in a slab.[4] The governing equation is ... 

Example 8.15. Heat Conduction in a Rectangle with a Time Dependent Boundary Condition... 

Consider heat conduction in a rectangle with a time dependent boundary condition. [2] The dimensionless temperature profile is governed by ... 

It has been shown that, for property variations for which superposition of solutions is permitted, a series of solutions corresponding to a step in surface temperature can be utilized to represent an arbitrary surface temperature [22]. This approach is identical with the Duhamel method used in heat conduction problems to satisfy time-dependent boundary conditions... 

In problems involving non-steady-state heat conduction, the temperature distribution depends on a parameter x/[Kt/(pC)], where x is the distance (L) from a location, t is the time (l), K is the thermal conductivity with dimensions of energy/[(time)(area)(temperature gradient](/ 1 1 / 1), where E is energy. Here C is the heat capacity with units of EM XT X, p is density, ML 3, x is distance (L), and t is time (t). For the same boundary conditions the same temperature will be found at the same times and locations, with different materials, if x/[Kl/(pC) has the same value. 

Concentration is variable with time, Pick s second law Most interactions involving mass transfer between the packaging and food behave under non-steady state conditions and are referred to as migration. A number of solutions exist by integration of the diffusion equation 8.7 that are dependent on the so-called initial and boundary conditions of special applications. Many solutions are taken from analogous solutions of the heat conductance equation that has been known for many years ... 

The Laplace transformation has proved an effective tool for the solution of the linear heat conduction equation (2.110) with linear boundary conditions. It follows a prescribed solution path and makes it possible to obtain special solutions, for example for small times or at a certain position in the thermally conductive body, without having to determine the complete time and spatial dependence of its temperature field. An introductory illustration of the Laplace transformation and its application to heat conduction problems has been given by H.D. Baehr [2.25]. An extensive representation is offered in the book by H. Tautz [2.26]. The Laplace transformation has a special importance for one-dimensional heat flow, as in this case the solution of the partial differential equation leads back to the solution of a linear ordinary differential equation. In the following introduction we will limit ourselves to this case. 

A surface heat transfer coefficient h can be defined as the quantity of heat flowing per unit time normal to the surface across unit area of the interface with unit temperature difference across the interface. When there is no resistance to heat flow across the interface, h is infinite. The heat transfer coefficient can be compared with the conductivity the conductivity relates the heat flux to the temperature gradient the surface heat transfer coefficient relates the heat flux to a temperature difference across an unknowm distance. Some theoretical work has been done on this subject [8], but since it is rarely possible to achieve in practice the boundary conditions assumed in the mathematical formulation, it is better to regard it as an empirical factor to be determined experimentally. Some typical values are given in Table 2. Cuthbert [9] has suggested that values greater than about 6000 W/m K can be regarded as infinite. The spread of values in the Table is caused by mold pressure and by different fluid velocities. Heat loss by natural convection also depends on whether the sample is vertical or horizontal. Hall et al. [10] have discussed the effect of a finite heat transfer coefficient on thermal conductivity measurement. 

Recently Wake has applied a variational treatmrait to the stationary problem, deriving critical conditions both for the class A geometries and for the cube, square rod, and equicylinder in systems where the heat transf(H is resisted by conduction in the interior and by convection at the surface. Here the condition at the boundary becomes dO/dp + N6 = 0, where N is the Biot number hLIk The limit as bf- oo corresponds to the Frank-Kamenetskii solutions. Wake uses trigonometric, rather than polynomial, expressions for this tempoature field and proceeds to derive the conditions under which solutions of the time-dependent variational equations are just possible, associating these with a critical value of 6. Results for N = oo are listed as variational (2) in the Table. For the more rorai conditions of finite Biot numbers Wake compares his results for class A geometries with the analytical forms due to Thomas. Errors are less than 0.1 % though the computational effort required is substantial. 

The temperature instability of a two-dimensional reactive fluid of N hard disks bounded by heat conducting walls has been studied by molecular dynamics simulation. The collision of two hard disks is either elastic or inelastic (exothermic reaction), depending on whether the relative kinetic energy at impact exceeds a prescribed activation barrier. Heat removal is accomplished by using a wall boundary condition involving diffuse and specular reflection of the incident particles. Critical conditions for ignition have been obtained and the observations compared with continuum theory results. Other quantities which can be studied include temperature profiles, ignition times, and the effects of local fluctuations. 

The velocity relevant for transport is the Fermi velocity of electrons. This is typically on the order of 106 m/s for most metals and is independent of temperature [2], The mean free path can be calculated from i = iyx where x is the mean free time between collisions. At low temperature, the electron mean free path is determined mainly by scattering due to crystal imperfections such as defects, dislocations, grain boundaries, and surfaces. Electron-phonon scattering is frozen out at low temperatures. Since the defect concentration is largely temperature independent, the mean free path is a constant in this range. Therefore, the only temperature dependence in the thermal conductivity at low temperature arises from the heat capacity which varies as C T. Under these conditions, the thermal conductivity varies linearly with temperature as shown in Fig. 8.2. The value of k, though, is sample-specific since the mean free path depends on the defect density. Figure 8.2 plots the thermal conductivities of two metals. The data are the best recommended values based on a combination of experimental and theoretical studies [3],... 

An information flow diagram for a drying model appropriate for this method is shown in Figure 4.16. This model can calculate the material moisture content and temperature as a function of position and time whenever the air humidity, temperature, and velocity are known as a function of time, together with the model parameters. If the model takes into account the controlling mechanisms of heat and mass transfer, then the transport properties (moisture dif-fusivity, thermal conductivity, boundary heat and mass transfer coefficients) are included in the model as parameters. If the dependence of drying conditions (material moisture content, temperature, and thickness, as well as air humidity, temperature, and velocity) on transport properties is also considered, then the constants of the relative empirical equations are considered as model parameters. In Figure 4.16 the part of the model that contains equations... 

---