Heat flow radial systems

Analysis of Heat Transfer. In the vertical Bridgman-Stockbarger system shown in Figure la, the axial temperature gradient needed to induce solidification is created by separating hot and cold zones with a diabatic zone in which radial heat flow from the ampoule to the furnace is suppressed. Analyses of conductive heat transfer have focused on this geometry. 

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. 

Consider a long cylinder of inside radius r, outside radius r0, and length L, such as the one shown in Fig. 2-3. We expose this cylinder to a temperature differential Tt - Ta and ask what the heat flow will be. For a cylinder- with length very large compared to diameter, it may be assumed that the heat flows in a radial direction, so that the only space coordinate needed to specify the system is r. Again, Fourier s law is used by inserting the proper area relation. The area for heat flow in the cylindrical system is ... 

In separation processes and chemical reactors, flow through cylindrical ducts filled with granular materials is important. In such systems conduction, convection, and radiation all contribute to the heat flow, and thermal conduction in axial ke x and radial ke r directions may be quite different, leading to highly anisotropic thermal conductivity. For a bed of uniform spheres, the axial and radial elements are approximated by... 

When the Shutdovm Cooling System (SCS) is in operation, the thermal/hydraulic configuration is different in the lower plenum area. The primary coolant exiting from the core is still collected in the lower plenum. From there it flows radially inward via narrow vertical channels between the central reflector column supports at the lower plenum elevation to a central chamber. The central chamber collects the primary coolant and directs it downward through an opening in the metal core support structure to the shutdown cooling heat exchanger. 

The actual experimental system uses abridge circuit and a lock-in amplifier, and the AT values are measured as a function of frequency. The theoretical value of AT is given by the solution of the diffusion equation for a radial heat flow from the surface electrode. At low frequencies in which the thermal penetration depth is much larger than the PS layer thickness, AT is the summation of two components one from the PS layer, ATps, and the other from the c-Si substrate, ATs- The ATps value depends on the thermal conductivity of the PS layer and the experimental parameters. Since the ATs value is obtained from the known thermal parameters of c-Si, a value of the PS layer can be determined from the analysis of the measured AT At high frequencies, on the other hand, the thermal penetration depth is smaller than the PS layer thickness, and then the contribution of the substrate to AT is negligible. Under this situation, the experimental AT data simply relate to the D value that is given as a/C. So the C value of the PS layer can be deduced from a measured at low frequencies. Owing to the insensitivity to errors from black-body radiation, this method makes it possible to determine the thermal constants more precisely rather than the method based on simple thermal flow measurements. 

A special case arises with polymer melts. Here excellent surffice contact can be obtained, particularly if the melt is pumped into the apparatus from an external reservoir. The radial-heat-flow method may become the method of choice for this situation, but it suffers from two potentially serious disadvantages the complexities of the required external pumping system and the large quantity of sample needed. 

Close-Clearance Stirrers For some pseiidoplastic fluid systems stagnant fluid may be found next to the -essel walls in parts remote from propeller or turbine impellers. In such cases, an anchor impeller maybe used (Fig, 18-6), The fluid flow is principally circular or helical (see Fig, 18-7) in the direction of rotation of the anchor. Whether substantial axial or radial fluid motion also occurs depends on the fluid iscosity and the design of the upper blade-supporting spokes. Anchor agitators are used particularly to obtain irnpro ed heat transfer in high-consistency fluids,... 

Binary liquid metal systems were used in liquid-metal magnetohydrodynamic generators and liquid-metal fuel cell systems for which boiling heat transfer characteristics were required. Mori et al. (1970) studied a binary liquid metal of mercury and the eutectic alloy of bismuth and lead flowing through a vertical, alloy steel tube of 2.54-cm (1-in) O.D., which was heated by radiation in an electric furnace. In their experiments, both axial and radial temperature distributions were measured, and the liquid temperature continued to increase when boiling occurred. A radial temperature gradient also existed even away from the thin layer next to the... 

These results verified that heat transfer in the melt was conduction dominated, except at intense convection levels because of the low-Prandtl-number characteristic of semiconductor melts. The shape of the melt-crystal interface changes with convection only at these higher convection levels. The flows are cellular, with the direction and magnitude of each cell determined by the radial temperature gradients induced by the thermal boundary conditions. In the idealized system studied, the mismatch in boundary conditions at the junction of the hot zone and the adiabatic region (Figure 16) causes the temperature to increase radially and drive a flow up along the... 

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