Radial Heat Flow Method

Heat flows from inner (r,) to outer (r0) radii at temperatures Ti and Ta respectively  

Thermocouples are placed at the same height along the axis, one radially extended from the other. When particulate specimens are poured in, a perforated template is used to maintain the correct positions of the thermocouples. The template is removed after the specimen material is in place. Averaging 

The central heater is made up of a platinum heater assembly within an alumina or mullite sheath.1 Voltage taps symmetrically placed about the center allow determination of power per unit length dissipated radially past the inner and outer ther- 

1 Mullite has the advantage that its lower thermal conductivity diminishes axial heat flow along the central heater more effectively. 

A tube furnace drawn over the mullite outer casing is used to heat the contents to a specified temperature, based on a furnace control thermocouple. At the same time, a constant ac voltage2 is applied across the central heater. The microprocessor waits until temperature fluctuations (within 0.1°C, over one minute) at any of the inside or outside thermocouples are eliminated.3 At that point, steady state conditions are assumed to exist. 

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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. 

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

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. 

Transient-heat-flow Methods. The axial and radial heat-flow methods discussed above depend on attaining a steady-state condition within the test specimen. This boundary condition is readily understood theoretically and easy to test experimentally. A disadvantage lies in the long, sometimes mctremely long, times required to achieve the condition. Under certain circumstances thermal conductivity can he determined by transient methods, where test times are much shorter. However, boundary conditions are difficult to analyze theoretically and often very approximate models are used for calculational purposes. 

The thermal conductivity values for polycrystalline (optically-thick) CaF obtained by. ngery —(comparative linear flow method) and by Taylor and Mills (laser pulse method) are in reasonable agreement (Figure 7). However, there is an appreciable discrepancy between the values of k obtained by the ziiQc source method —"— and the single value due to Ogino et al (radial heat source method). 

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 ... 

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. 

In the following we will discuss the difference method with consideration for temperature dependent material properties as well as for cylindrical and spherical coordinates, whereby geometric one-dimensional heat flow is assumed in the radial direction. The decisive differential equation for the temperature field is then... 

The apparatus is usually constructed with a large length-to-diameter ratio so that the heat lost from the ends is small compared with the heat transferred radially through the sample. The major source of error is not heat loss from the ends but resistance to heat flow across the interfaces between the sample and the heater and heat sink, because it is very difficult to ensure that the sample is a perfect fit in the apparatus [27. This method is more... 

In the panel steady-state method, heat flow is linear from hot to cold face of a refractory, whereas in the hot wire method, heat flow is in a radial direction from the heater wire. Therefore, if any material anisotropy exists, the latter method measures a mean conductivity of two directions (18). 

O.A. Estrada, I.D. Lopez-Gomez, and T.A. Osswald. Modeling the non-newtonian calendering process using a coupled flow and heat transfer radial basis functions collocation method. Journal of Polymer Technology, 2005. 

The heat transfer (cooling or heating) in a stirred vessel is achieved in a number of different ways. The most conventional method is the use of a coil heat exchanger. The nature and positioning of the coil depends on the nature of the flow pattern created by the stirrer. For an axial flow stirrer, a spiral coil (see Fig. 2a) is effective because it provides good liquid circulation between the coil and the wall. For radial flow stirrers, spiral coils deflect the liquid... 

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