Radial heat flow

The cross-sectional area of the wick is deterrnined by the required Hquid flow rate and the specific properties of capillary pressure and viscous drag. The mass flow rate is equal to the desired heat-transfer rate divided by the latent heat of vaporization of the fluid. Thus the transfer of 2260 W requires a Hquid (H2O) flow of 1 cm /s at 100°C. Because of porous character, wicks are relatively poor thermal conductors. Radial heat flow through the wick is often the dominant source of temperature loss in a heat pipe therefore, the wick thickness tends to be constrained and rarely exceeds 3 mm. 

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. 

Figure 9.1 Cylindrical geometry for calculation of thermal conductivity from radial heat flow.

Assume radial heat flow only, and the thermal conductivities of the insulation layers are constant. 

Another example of one-dimensional heat flow is the radial heat flow through the wall of a hollow sphere. Starting withEq. (2.36)... 

The effect of geometric parameters on the adiabaticity of a test reactor can be deduced from Table V. It can be seen that for improperly designed laboratory reactors the axial and radial heat flows can be quite appreciable even when the net heat loss is zero. From this table it follows that the radial heat flow is reduced as the bed diameter is increased, whereas the axial heat flow diminishes as the reactor length is increased. Hence, long pilot plant reactors of wide diameter will perform best as adiabatic reactors even with suboptimal design of compensation heaters. 

A (ad) deviation from true adiabacicity related to radial heat flow... 

Figure 2. Dependence of interfaee shape on heat flows Ql (through liquid), Qs (through solid), and Qr (radial heat flow) depending on temperature distribution in furnace Kl and Ks are thermal conductivities of liquid and solid, respectively. (From Ref. 1.)...

In the radial heat-flow method the specimen is in the shape of a hollow cylinder, which is positioned in the annulus between two coaxial cylinders with the internal cylinder acting as a radial heat source. The temperature profile across the specimen is determined by thermocouples placed on the inside walls of the two cylinders. This method requires a large isothermal zone in the furnace, which is difficult to achieve at high temperatures. When this technique is used for measurements on liquids, errors can occur from convective heat transfer. 

Na O -h SiO. Susa et al —- (line source method) reported thermal conductivity data for solid and liquid slags for three compositions the single value obtained by Ogino et al l3 (radial heat-flow method) is in reasonable agreement with these data. 

Under steady state conditions, i.e., when the temperature at any point does not change with time, the temperature distribution in the sample is governed by Laplace s equation (Eq. 6). The sample geometries are chosen so that the temperature is a function of only one coordinate, and simple analytical solutions to Laplace s equation can be used. There are two geometries that satisfy this condition the parallel-faced slab with heat flow normal to the surfaces and the hollow cylinder with radial heat flow. In the latter case, although the heat flow is now in two dimensions the temperature is a function of only one coordinate, namely the radius, because of symmetry. The condition is also satisfied by the sphere, but that is not relevant to the present discussion. The first case is illustrated in Fig. I which shows a parallel-faced section of thickness. v and cross-scctional area A. 

The first major source of error is a result of a failure to obtain a normal or radial heat flow, depending on the geometry. Satisfying this boundary condition is very difficult in practice. The problem has been tackled in three ways the flux is constrained by guard rings or the heat loss from the edges of the specimen is minimized and ignored or the heat loss is estimated and allowed for in the calculations. 

Whereas the longitudinal heat flow methods are most suitable for slab specimens, the radial heat flow techniques are used for loose, unconsolidated powder or granular materials. The methods can be classified as follows ... 

A SIMPLE RADIAL HEAT FLOW APPARATUS FOR FLUID THERMAL CONDUCTIVITY MEASUREMENTS. 

A RADIAL HEAT-FLOW METHOD FOR POOR CONDUCTORS. FROM PROCEEDINGS OF THE NINTH CONFERENCE ON THERMAL CONDUCTIVITY. IOWA STATE UNIVERSITY, AMES. IOWA. 

Jaeger, J.C. (1940) Radial heat flow in circular cylinders withageneralboundary condition./c Mrna/awdPrc cee(3fe>igs of Royal Society of New South Wales, 74, 342-352. 

FIGURE 3.1 Radial heat flow through cylindrical conductor. 

Total radial heat flow Heat input to inner surface = lur Q Heat output from outer surface = 2-iT(r + Ar)(Q + Accumulation of heat = 0 2-arQ 2Tt(r + Ar)(g+... 

Falthammel has calculated the effect of radial heat flow across the Bq magnetic field for a confined Z-pinch. Rather surprisingly the same general scaling laws were obtained as in Eq. (17) and (18) but with slightly different values of the constants. Xhe reason for this is that if we balance the power input IV with the radial transverse ion thermal conduction, i.e. 

FIGURE 6.39. Annular tubular reactor (radial heat flow reactor). 

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. 

Different geometries can be used, those for longitudinal heat flow and radial heat flow. 

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