Cylindrical coordinates, radial conduction

The theoretical model assumes a line heat source dissipating heat radially into an infinite solid, initially at uniform temperature. The fundamental heat conduction equation in cylindrical coordinates, assuming uniform radial heat transfer, is [3] ... 

A circular waveguide with inner radius a is shown in Fig. 4.14. Here the axis of the waveguide is aligned with the z axis of a circular-cylindrical coordinate system, where p and are the radial and azimuthal coordinates, respectively. If the walls are perfectly conducting and the dielectric material is lossless, the equations for the TE , modes are... 

For the plug flow region and the refractory wall, the discretized equations are the same as for the active region of the bed except that they are in cylindrical coordinates. For the radial direction, only the conduction component is considered and the coefficient of the algebraic (discretized) equation is equal to, for example,... 

The corresponding cylindrical coordinate equation to (5-17) for only radial 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. 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

One-Dimensional Conduction In the absence of energy source terms, Q is constant with distance, as shown in Fig. 5-la. For steady conduction, the integrated form of (5-1) for a planar system with constant k and A is Eq. (5-2) or (5-3). F or the general case of variables k (k is a function of temperature) and A (cylindrical and spherical systems with radial coordinate r, as sketched in Fig. 5-2), the average heat-transfer area and thermal conductivity are defined such that... 

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