Transient conduction from cylinder

External Transient Conduction From Long Cylinders... 

Sdmmary of the soluiiuns for one-dimensional transient conduction in a plane wall of thickness 21., a cylinder of radius r and a sphere of radius r subjected to convention from all surfaces. ... 

EX A MPLE 53-4. Two-Dimensional Conduction in a Short Cylinder Repeat Example 5.3-3 for transient conduction in a can of pea puree but assume that conduction also occurs from the two flat ends. 

Consider a sltori cylinder of height a and radius r initially at a uniform temperature T,. There is no heat generation in the cylinder. At time t = 0. the cylinder is subjected to convection from all surfaces to a medium at temperature l with a heal transfer coefficieiu h. The temperature within the cylinder will change with a as well as r and time f since heal transfer occurs from Ihe top and bottom of the cylinder as weU as its side surfaces. That is, T = 7 (r,, v, f) and thus this is a two-dimensional transient heal conduction problem. When the properties are assumed to be constant, it can be shown that the solution of this two-dimensional problem can be expressed as... 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

A newer method is the transient hot-wire method, where an electric current is passed through a metal wire immersed in the fluid. The resistance of the wire is affected by its temperature, which in turn is affected by the dissipation of heat from the wire s surface, which depends on the thermal conductivity of the fluid. These instruments require sophisticated data analysis, but that is no longer an obstacle with the ready availability of personal computers. The absence of convection is relatively easy to verify. The best research instruments can achieve an accuracy of better than 1%. Measurements on conducting fluids (such as polar liquids) are more difficult because of the need to electrically insulate the wire. Other geometries, such as needle-shaped cylinders and thin strips, are also sometimes used for transient measurements. 

In Table 5.1 all available experimental thermal conductivity data sources at high temperatures (above 200 °C) and high pressures are presented. As one can see from this table, all data were derived by the parallel-plate and the coaxial-cylinder techniques, except only two datasets for LiBr by Bleazard et al. (1994) and DiGuilio and Teja (1992) which were obtained by the transient hot-wire technique. We further note that almost all investigators quote an uncertainty of better than 2%. In this section a brief analyses of these methods is presented. The theoretical bases of the methods, and the working equations employed is presented, together with a brief description of the experimental apparatus and the measurements procedure of each technique. For a more thorough discussion of the various techniques employed, the reader is referred to relevant literature (Kestin and Wakeham, 1987 Wakeham et al., 1991 Assael et al, 1991, and Wakeham and Assael, in press). 

The essential difference between a steady state and a transient state is that the temperature at a particular location changes with time under transient conditions. A line heat source probe has been recommended by many researchers [28,29,52,53]. The method is simple, fast, and requires a relatively small sample. A schematic representation of the thermal conductivity probe, the direct current (dc) supply, and the temperature measuring system is shown in Figure 24.5 [54]. The probe is inserted into a sample of a uniform temperature and is heated at a constant rate. The temperature adjacent to the line heat source is recorded. Various modifications of the line heat source probe can be found in the literature. The probe attached to a 20-cm diameter aluminum cylinder as a sample holder is one of them (Figure 24.6) [35]. Other modifications are related to placement of thermocouples directly on the heating element [55,56] or at a fixed distance from it [43]. 

The thermal conductivity excess contribution Most of the available measurements of the thermal conductivity of nitrogen at elevated pressures have been carried out either in concentric cylinder or in transient hot-wire instruments. Both of these instruments can yield data of high accuracy and as such have to be considered as primary. Seven sets of data (see Millat Wakeham 1995) were selected for the purpose of developing the excess thermal conductivity correlation. The primary data set consists of 864 data points, covering a temperature range 81 K < T < 973 K and densities up to 32 mol. The ascribed uncertainty of the data varies from 0.8% to 3.0%. 

Tests at Babcock and Wilcox (B W) (Reference 22) simulated the approximate flow decay and power transient in a single full-size, electrically-heated fuel assembly mockup. The FIs were approximately 10 percent higher than those predicted by FLOWTRAN-FI using OSV as a predictor of FI. While the coolant flow recovered after about 0.1 seconds in the B W tests, due to system hydraulic "bounce-back," there was generally no recovery from FI once it occurred. The heater cylinders in the B W tests were Monel, as opposed to an aluminum alloy used for the SRS fuel cylinders. The good conductivity of the SRS fuel might possibly provide some margin to recovery from FI. 

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