Heat conduction in a rod

Consider one-dimensional heat conduction in a rod with a length of L. Obtain the function that minimizes the entropy production. 

D is replaced by DeB for porous solids. Equation (2.369) agrees in terms of its form with (2.53) for heat conduction in a rod. In this case the temperature profile was calculated with the boundary conditions of constant temperature at the beginning of the rod and vanishing heat flow at the end of the rod. In the current problem these correspond to the boundary conditions (2.370) and (2.371). The solution of (2.369) to (2.371) therefore corresponds to the relationship (2.59) found earlier for heat conduction in rods. The solution is... 

Example 6.3 Finite Difference Solution for Heat Conduction in a Rod... 

Recall the problem of Example 6.2 involving heat conduction in a rod. The requisite ODEs and boundary conditions are... 

Heat Conduction in a Solid Circular Cylinder with Heat Generation—Fuel Rod.. 

Let us consider the problem of simulating the unsteady-state heat conduction in a long rod as shown in Figure 8.14. 

Recall Example 6.1 involving heat conduction in a thin rod of length 1 (when the... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Consider a circular rod. The radial heat conduction in nondimensional form is described... 

It has the same form as equation (2.51) from section 2.2.2, which is valid for heat conduction in the axial direction of a rod. Using the abbreviation... 

The recursion is performed on new flow data using gradient techniques, which are then followed by the determination of control input. The RePOD method can also be used in isolation to compute the modal basis, potentially resulting in substantial savings in computational resources and time. The algorithm was applied to two classes of problems [15]. In the first, the problem of heat conduction through a one-dimensional rod with a sinusoidal heat perturbation source located inside the rod was considered. The objective is to reduce the temperature... 

One-dimensional rod model and a planar model of heat conduction in one direction... 

In this edition of the text, 1 have attempted to address these two aspects. Chapter 1 has been expanded to include two popular approaches to model development. Each approach is discussed in the format of an example while using a real application from research. Both applications are solved as examples in Chapter 7. Further, a model of a one-dimensional rod is introduced in Chapter 6 and a planar model of heat conduction in one direction is introduced in Chapter 7. The solutions to the two examples of modeling approaches developed in Chapter 1 are verified with independently derived experimental data in Chapter 7. In addition, a figure comparing model results to independently derived experimental data is included for mie example of mass transfer in a membrane separator discussed in Chapter 7. 

The goal of the conduction heat transfer is to determine the temperature field in a medium (such as fuel rod) and the rate of heat transfer to and from the medium. Typically, the media is subjected to nonimiform temperature distribution which is a result of either a heat source within the medium or heat flux from the boundary of the medium. In this section various forms of the heat conduction equation that govern the temperature field in a medium and its associated boimdary conditions are given. Some examples of the heat conduction in nuclear fuel rod and other components are presented. 

A metal rod initially at ambient air temperature, 25 °C, is connected on one side to a wall at constant temperature 100 °C, and the other side is surrounded by air. Heat conduction through the rod and losses by convection to the surroundings are balanced by accumulation of thermal energy in the rod. The energy balance is given by... 

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