Unsteady state conduction, example

General solutions of unsteady-state conduction equations are available for certain simple shapes such as the infinite slab, the infinitely long cylinder, and the sphere. For example, the integration of Eq. (10.16) for the heating or cooling of an infinite slab of known thickness from both sides by a medium at constant... 

EXAMPLE 5.4-1. Unsteady-State Conduction and the Schmidt Numerical Method... 

EXAMPLE 5.4-2. Unsteady-State Conduction Using the Digital Computer... 

EXAMPLE 5.4-3. Unsteady-State Conduction with Convective Boundary Condition... 

Convection and Unsteady-State Conduction. For the conditions of Example 5.4-3,... 

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 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). 

The elegant solution of this first example should not tempt the reader to believe that dimensional analysis can be used to solve every problem. To treat this example by dimensional analysis, the physics of unsteady-state heat conduction had to be understood. Bridgman s (2) comment on this situation is particularly appropriate ... 

Example 1 What Is the Correlation Between the Baking Time and the Weight of a Christmas Turkey We first recall the physical situation. To facilitate this we draw a sketch (Sketch 1). At high oven temperatures the heat is transferred from the heating elements to the meat surface by both radiation and heat convection. From there it is transferred solely by the unsteady-state heat conduction that surely represents the rate-limiting step of the whole heating process. 

These apparent differences may be explained by differences in experimental conditions, differences in catalyst composition, and by transport limitations in the work of Rostrup-Nielsen and Pedersen (209). For example, the unsteady-state experiments of Bartholomew and Gardner (172) and of Wentrcek et al. (195) were conducted on presulfided catalysts in powder from at 575 K (H2/CO = 3), whereas the steady-state experiments of Rostrup-Nielsen and Pedersen were carried out in situ at 675 K (H2/CO =... 

On the basis of their initial and boundary conditions, partial differential equations may be further classified into initial-value or boundary-value problems. In the first case, at least one of the independent variables has an open region. In the unsteady-state heat conduction problem, the time variable has the range 0 r >, where no condition has been specified at r = eo therefore, this is an initial-value problem. When the region is closed for all independent variables and conditions are specified at all boundaries, then the problem is of the boundary-value type. An example of this is the three-dimensional steady-state heat conduction problem described by the equation... 

Classic examples of parabolic differential equations are the unsteady-state heat conduction equation... 

When using finite differences to solve the unsteady heat conduction problem, another approach involves writing finite difference equations at each grid point (node) only for the spatial variables while leaving the time derivative intact. This leads, generally, to a large number of simultaneous ODEs, which can be solved by, for example, a Runge-Kutta method. However, one must be careful since this set of ODEs can be stiff. Consider the same one-dimensional, unsteady state heat conduction problem as solved in Examples 8.1 and 8.2. This problem is solved by the method of lines in the next example. 

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