Unsteady Conduction

Example The equation 9T/9t = K(d T/dx ) represents the unsteady onedimensional conduction of heat. 

Parabolic The heat equation 3T/3t = 3 T/3t -i- 3 T/3y represents noneqmlibrium or unsteady states of heat conduction and diffusion. 

Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation... 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

Various numerical and graphical methods are used for unsteady-state conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berhn, 1924). These methods are very useful because any form of initial temperature distribution may be used. 

The general case is that of steady-state flow, and the thermal conductivity factor is a function of the temperature. In the unsteady state the temperature of the system changes with time, and energy is stored in the system or released from the system reduced. The storage capacity is... 

In this approach, heat transfer to a spherical particle by conduction through the surrounding fluid has been the prime consideration. In many practical situations the flow of heat from the surface to the internal parts of the particle is of importance. For example, if the particle is a poor conductor then the rate at which the particulate material reaches some desired average temperature may be limited by conduction inside the particle rather than by conduction to the outside surface of the particle. This problem involves unsteady state transfer of heat which is considered in Section 9.3.5. 

The exact mathematical solution of problems involving unsteady thermal conduction may be very difficult, and sometimes impossible, especially where bodies of irregular shapes are concerned, and other methods are therefore required. 

The equation is most conveniently solved by the method of Laplace transforms, used for the solution of the unsteady state thermal conduction problem in Chapter 9. 

Unlike at adiabatic conditions, the height of the liquid level in a heated capillary tube depends not only on cr, r, pl and 6, but also on the viscosities and thermal conductivities of the two phases, the wall heat flux and the heat loss at the inlet. The latter affects the rate of liquid evaporation and hydraulic resistance of the capillary tube. The process becomes much more complicated when the flow undergoes small perturbations triggering unsteady flow of both phases. The rising velocity, pressure and temperature fluctuations are the cause for oscillations of the position of the meniscus, its shape and, accordingly, the fluctuations of the capillary pressure. Under constant wall temperature, the velocity and temperature fluctuations promote oscillations of the wall heat flux. 

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

There is also another key parameter linked to the choice of the material for the reactor. First, the choice is obviously determined by the reactive medium in terms of corrosion resistance. However, it also has an influence on the heat transfer abilities. In fact, the heat transport depends on the effusivity relative to the material, deflned by b = (XpCp) the effusivity b appears in the unsteady-state conduction equation. 

Although catalytic wet oxidation of acetic acid, phenol, and p-coumaric acid has been reported for Co-Bi composites and CoOx-based mixed metal oxides [3-5], we could find no studies of the wet oxidation of CHCs over supported CoO catalysts. Therefore, this study was conducted to see if such catalysts are available for wet oxidation of trichloroethylene (TCE) as a model CHC in a continuous flow fixal-bed reactor that requires no subsequent separation process. The supported CoOx catalysts were characterized to explain unsteady-state behavior in activity for a certain hour on stream. 

The simulation example DRY is based directly on the above treatment, whereas ENZDYN models the case of unsteady-state diffusion, when combined with chemical reaction. Unsteady-state heat conduction can be treated in an exactly analogous manner, though for cases of complex geometry, with multiple heat sources and sinks, the reader is referred to specialist texts, such as Carslaw and Jaeger (1959). 

Unsteady-State Heat Conduction and Diffusion in Spherical and Cylindrical Coordinates... 

For laminar conditions of slow flow, as in candle flames, the heat transfer between a fluid and a surface is predominately conductive. In general, conduction always prevails, but in the unsteadiness of turbulent flow, the time-averaged conductive heat flux between a fluid and a stationary surface is called convection. Convection depends on the flow field that is responsible for the fluid temperature gradient near the surface. This dependence is contained in the convection heat transfer coefficient hc defined by... 

Figure 5.3 and Table 5.2 give solutions for 6C for convective heating of a sphere, cylinder and slab. As the degrees of freedom for heat conduction increase, it is less likely to achieve ignition as 6C increases from a slab (1), to a cylinder (2), to a sphere (3). Also as Bi —> 0, the material becomes perfectly adiabatic, and only an unsteady solution is... 

Equation (5.19) can also be obtained from the unsteady counterpart of Equation (5.5). If internal conduction is taken as zero in Equation (5.5), this adiabatic equation arises. Equation (5.19) also becomes the basis of an alternative method for determining the bulk kinetic properties. The method is called the adiabatic furnace used by Gross and Robertson [5]. The furnace is controlled to make its temperature equal to the surface temperature of the material, thus producing an adiabatic boundary condition. This method relies on the measurement of the time for ignition by varying either ra or Too, the initial temperature. 

Furthermore, we will take all other properties as constant and independent of temperature. Due to the high temperatures expected, these assumptions will not lead to accurate quantitative results unless we ultimately make some adjustments later. However, the solution to this stagnant layer with only pure conduction diffusion will display the correct features of a diffusion flame. Aspects of the solution can be taken as a guide and to give insight into the dynamics and interaction of fluid transport and combustion, even in complex turbulent unsteady flows. Incidentally, the conservation of momentum is implicitly used in the stagnant layer model since ... 

Only a finite difference numerical solution can give exact results for conduction. However, often the following approximation can serve as a suitable estimation. For the unsteady case, assuming a semi-infinite solid under a constant heat flux, the exact solution for the rate of heat conduction is... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

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