**B. WSGG linear absorption coefficients and blackbody weights**

Analogous to the tabulation in Appendix 7, the computed linear absorption coefficients and the corresponding weights for the gray gases are given in this appendix for all the 6 WSGG models for each of the two oxy-fuel environments. These values are used when solving the

65% CO2 90% CO2

65% CO2 90% CO2

65% CO2 90% CO2

65% CO2 90% CO2

*i ki*(1/*m*) *ai ki*(1/*m*) *ai* 0 0.26177 0 0.31687 0.05677 0.30533 0.04006 0.33408 0.58148 0.25560 0.41427 0.20004 5.64642 0.13281 5.18028 0.10602 100.07946 0.04449 123.52189 0.04298

Nongray EWB and WSGG Radiation Modeling in Oxy-Fuel Environments 509

Table 12. Linear absorption coefficients and blackbody weights for the 5-gas/quadratic

Table 13. Linear absorption coefficients and blackbody weights for the 5-gas/quadratic

*i ki*(1/*m*) *ai ki*(1/*m*) *ai* 0 0.31064 0 0.32763 0.09370 0.33218 0.06288 0.35561 1.08144 0.25582 1.02333 0.22449 99.99991 0.10136 100.00000 0.09227 Table 14. Linear absorption coefficients and blackbody weights for the 4-gas/linear WSGGM

*i ki*(1/*m*) *ai ki*(1/*m*) *ai* 0 0.31812 0 0.39788 0.05225 0.22831 0.05105 0.23703 0.69574 0.26925 0.68033 0.25810 7.71486 0.15584 14.04069 0.08263 188.01466 0.02849 294.45477 0.02436 Table 15. Linear absorption coefficients and blackbody weights for the 5-gas/cubic WSGGM

*i ki*(1/*m*) *ai ki*(1/*m*) *ai* 0 0.28678 0 0.34849 0.06146 0.33543 0.05633 0.36697 0.86869 0.23910 0.87767 0.17687 9.13846 0.10048 9.82222 0.07032 116.15385 0.03822 131.11111 0.03734

WSGGM in (22) – Two oxy-fuel environments

WSGGM in (24) – Two oxy-fuel environments

in (25) – Two oxy-fuel environments

in (27) – Two oxy-fuel environments


Table 9. Idealized box/EWB spectrum for the oxy-fuel environment with 65% CO2


Table 10. Idealized box/EWB spectrum for the oxy-fuel environment with 90% CO2

RTEs given in Equation (9). The linear absorption coefficient for the clear gas is *k*0=0; its blackbody weight (*a*0) is obtained from the requirement that *<sup>a</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>∑</sup>*N*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *ai*.


Table 11. Linear absorption coefficients and blackbody weights for the 4-gas/quadratic WSGGM in (22) – Two oxy-fuel environments

Numerical Simulations / Book 1

 0.00 − 448.47 0.2097897 0.00346421 12 2 410.00 − 3 048.95 0.0000000 0.13252660 448.47 − 845.47 0.4276537 0.01647795 13 3 048.95 − 3 334.04 0.1683664 0.05929771 845.47 − 885.53 0.5403818 0.00261648 14 3 334.04 − 3 985.96 0.3584664 0.12793486 885.53 − 921.00 0.3225178 0.00247182 15 3 985.96 − 4 471.05 0.1683664 0.08419003 921.00 − 969.29 0.4931395 0.00360145 16 4 471.05 − 4 929.89 0.0000000 0.06916402 969.29 − 1 074.53 0.6162794 0.00879137 17 4 929.89 − 4 982.64 0.0633557 0.00727657 1 074.53 − 1 150.71 0.5035513 0.00716945 18 4 982.64 − 5 470.11 0.1754496 0.06076172 1 150.71 − 1 258.43 0.3804114 0.01127985 19 5 470.11 − 5 717.36 0.1120939 0.02650485 1 258.43 − 1 944.35 0.1706217 0.09986805 20 5 717.36 − 6 975.54 0.0000000 0.09647500 1 944.35 − 2 279.00 0.4116662 0.06230296 21 6 975.54 − 7 524.46 0.1135142 0.02613845 2 279.00 − 2 410.00 0.2410445 0.02595104 22 7 524.46 − 100 000 0.0000000 0.06573557

*i η* (1/cm) *ki*(1/*m*) *ai i η* (1/cm) *ki*(1/*m*) *ai*

Table 9. Idealized box/EWB spectrum for the oxy-fuel environment with 65% CO2

*i η* (1/cm) *ki*(1/*m*) *ai i η* (1/cm) *ki*(1/*m*) *ai*

Table 10. Idealized box/EWB spectrum for the oxy-fuel environment with 90% CO2

blackbody weight (*a*0) is obtained from the requirement that *<sup>a</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>∑</sup>*N*−<sup>1</sup>

WSGGM in (22) – Two oxy-fuel environments

RTEs given in Equation (9). The linear absorption coefficient for the clear gas is *k*0=0; its

*i ki*(1/*m*) *ai ki*(1/*m*) *ai* 0 0.29433 0 0.37459 0.11695 0.41272 0.09837 0.41704 2.51559 0.23307 2.66557 0.15639 70.56945 0.05988 88.92354 0.05198

Table 11. Linear absorption coefficients and blackbody weights for the 4-gas/quadratic

65% CO2 90% CO2

*<sup>i</sup>*=<sup>1</sup> *ai*.

 0.00 − 440.64 0.1841169 0.00329548 12 2 410.00 − 3 193.84 0.0000000 0.16280362 440.64 − 839.06 0.4063678 0.01624495 13 3 193.84 − 3 319.79 0.1324004 0.02608835 839.06 − 893.36 0.5304856 0.00355165 14 3 319.79 − 4 000.21 0.3281601 0.13349028 893.36 − 964.10 0.3082347 0.00513915 15 4 000.21 − 4 326.16 0.1324004 0.05756875 964.10 − 1 059.44 0.4417534 0.00785136 16 4 326.16 − 4 931.64 0.0000000 0.09340562 1 059.44 − 1 067.07 0.2576366 0.00067467 17 4 931.64 − 5 073.92 0.0809341 0.01929669 1 067.07 − 1 080.94 0.3902864 0.00124357 18 5 073.92 − 5 468.36 0.1404420 0.04830074 1 080.94 − 1 155.90 0.2661685 0.00710566 19 5 468.36 − 5 626.08 0.0595080 0.01724289 1 155.90 − 1 930.42 0.1326498 0.10819446 20 5 626.08 − 7 033.12 0.0000000 0.10905531 1 930.42 − 2 132.93 0.3772371 0.03674431 21 7 033.12 − 7 466.88 0.0571616 0.02063116 2 132.93 − 2 410.00 0.2445874 0.05394936 22 7 466.88 − 100 000 0.0000000 0.06812197


Table 12. Linear absorption coefficients and blackbody weights for the 5-gas/quadratic WSGGM in (22) – Two oxy-fuel environments


Table 13. Linear absorption coefficients and blackbody weights for the 5-gas/quadratic WSGGM in (24) – Two oxy-fuel environments


Table 14. Linear absorption coefficients and blackbody weights for the 4-gas/linear WSGGM in (25) – Two oxy-fuel environments


Table 15. Linear absorption coefficients and blackbody weights for the 5-gas/cubic WSGGM in (27) – Two oxy-fuel environments

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Table 16. Linear absorption coefficients and blackbody weights for the 4-gas/cubic WSGGM in (28) – Two oxy-fuel environments

#### **8. References**


18 Numerical Simulations / Book 1

*i ki*(1/*m*) *ai ki*(1/*m*) *ai* 0 0.52282 0 0.66567 0.42019 0.28898 0.40334 0.20536 9.63050 0.16303 13.92300 0.10516 242.96000 0.02517 351.06000 0.02381 Table 16. Linear absorption coefficients and blackbody weights for the 4-gas/cubic WSGGM

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**8. References**

65% CO2 90% CO2


**24**

*Poland* 

Bohdan Mochnacki

*Institute of Mathematics* 

*Czestochowa University of Technology* 

**Numerical Modeling of Solidification Process** 

This chapter is devoted to the problems connected with numerical modeling of moving boundary ones. In particular the solidification and cooling processes proceeding in the system casting-mould are considered. The subject matter of solidification process modeling is very extensive and only the selected problems from this scope will be here discussed. From the mathematical point of view the thermal processes proceeding in the domain considered (both in macro and micro/macro scale) are described by a system of partial differential equations (energy equations) supplemented by the geometrical, physical, boundary and initial conditions. The typical solidification model bases on the Fourier-Kirchhoff type equations, but one can formulate the more complex (coupled with the basic model) ones concerning the heat convection in a molten metal sub-domain, the changes of local chemical constitution of solidifying alloy (segregation process) etc. Here we limit oneself only to the tasks connected with the predominant heat conduction process, and (as the examples of various problems) the macro models of alloys solidification and the micro/macro models of pure metals crystallization will be presented, at the same time the

In order to construct a numerical model and an adequate computer program simulating the course of problem considered, one must accept a certain mathematical description of the process (the governing equations). The next step is the transformation of this mathematical model into a form called the re-solving system constructed on the basis of a selected

After transformation of the algorithm developed into a computer program and supplementing it with suitable pre- and post-processing procedures (input data loading, graphic presentation of results, print-outs etc.), and carrying out of computations, one obtains the results including information concerning the transient temperature field, kinetics of solidification process, the temporary shapes of sub-domains etc. They may have a form of numerical print-outs, e.g., giving the temperature field at distinguished set of points or the

Numerical modeling of the heat and mass transfer in solidifying metal is a typical interdisciplinary problem and requires particular knowledge in the field of foundry practice, mathematics (a course in mathematical analysis offered in technical schools is quite sufficient here), thermodynamics (in particular heat transfer), numerical methods and, in the

volumetric fraction of solid state at the neighbourhood of these points.

final stage, also programming and operation of computer equipment.

**1. Introduction** 

numerical method.

direct and inverse problems will be discussed.

