Scaling radiation model

Kurucz, C.N., Waite, T.D., and Nickelsen, M.G., Empirical models for estimating the destruction of toxic organic compounds utilizing electron beam irradiation at full scale, Radiat. Phys. Chem., 45(5), 805-816, 1995b. 

Figure 9 Coordinate system for the radiation model of the pilot scale photoreactor. Adapted from Imoberdorf et al. (2006).

The time constants characterizing heat transfer in convection or radiation dominated rotary kilns are readily developed using less general heat-transfer models than that presented herein. These time constants define simple scaling laws which can be used to estimate the effects of fill fraction, kiln diameter, moisture, and rotation rate on the temperatures of the soHds. Criteria can also be estabHshed for estimating the relative importance of radiation and convection. In the following analysis, the kiln wall temperature, and the kiln gas temperature, T, are considered constant. Separate analyses are conducted for dry and wet conditions. 

The temperature for methane and butane calculated with the isothermal model is a factor 1.4 times greater than the average temperature measured by Lihou and Maund (1982) in their small-scale tests, although higher local maximum temperatures were measured. In this model, combustion is stoichiometric, thus leading to very high fireball temperatures which, in turn, lead to high radiation emissions. Effective surface emissions measured experimentally were one-half the value calculated from this model, because combustion is not stoichiometric and emissivity is less than unity. 

Full scale tests are particularly valuable to obtain information on fire hazard. They can be used to validate small scale tests, and to validate mathematical fire models. The most important additional dimension full scale tests add are effects, e.g. radiation from the fire itself, which are difficult to simulate in a smaller scale. Full scale tests are very expensive and time consuming. It is essential, thus, to design them in such a way as to (a) make them most relevant (b) minimize their number and (c)... 

Alternatively, one might satisfy convection near a boundary by invoking Il6 and Ilg where the heat transfer coefficient is taken from an appropriate correlation involving Re (e.g. Equation (12.38)). Radiation can still be a problem because re-radiation, n7, and flame (or smoke) radiation, II3, are not preserved. Thus, we have the art of scaling. Terms can be neglected when their effect is small. The proof is in the scaled resultant verification. An advantage of scale modeling is that it will still follow nature, and mathematical attempts to simulate turbulence or soot radiation are unnecessary. 

Since 1978, large-scale LNG spill tests have been conducted by a joint team from Lawrence Livermore National Laboratory (LLNL) and the Naval Weapons Center (NWC) (Koopman et al., 1981). The test site was located at NWC, China Lake, California. The program, sponsored primarily by the Department of Energy, had as its principal objective the acquisition of data to aid in modeling both vapor dispersion and thermal radiation effects (from LNG vapor cloud fires). 

In our numerical model, Eq.(2.8) was transformed into a six-point finite-difference equation using the alternative direction implicit method (ADIM). At the edges of the computational grid (—X,X) radiation conditions were applied in combination with complex scaling over a region x >X2, where —X X j) denotes the transverse computational window. For numerical solution of the obtained tridiagonal system of linear equations, the sweep method" was used. 

In Ref. 13, we have proved that the A transformation constructed is invertible for the classical model discussed in the previous section. Here, using the same system discussed in the previous section, we demonstrate the invertiblity of our transformation by a numerical calculation of the time evolution of the action variable J (f) for an initial condition where all the field actions are zero [20]. Due to radiation damping, J t) follows an approximately exponential decay. However, there are deviations from exponential in the exact evolution both at short and long time scales as compared with the relaxation time scale. In Fig. 1, we present numerical results. 

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