OCR Text |
Show 7„ '/> - >u (19) a + b and the variance (varT) of the temperature is assumed to be approximately equal to a fraction of the maximum possible variance. v a r 7 = ^ 7 " 2 - ( £ 7 ) 2 -7-^r-^-^ (20) {a + h)2{a + b + \) =s(r-ru)(rb-T), 0.5<5<1.0 Here, the fraction s is called the variance coefficient From the above two equations the following expressions for the B^f parameters a and b can be found: fl=lziIzZL; b=lzlJL± m s Tb-Tu s Tb-Tu The variance coefficient s is an important parameter of the NO-submodel and from a statistical point of view, its value can vary between zero and one. In Eq. (20) however, s can only vary from 0.5 to 1.0. The reason is twofold. A value of 0.6 has been recommended (Missaghi.1987) on the basis of both experimental and theoretical results. If s is allowed to take values in the range 0.5 to 1 then, both parameters a and b can vary only between zero and one. Consequently, all the possible B-pdfs have a shape similar to the shape depicted in Figure 4. For s-values near 1.0, the largest fluctuations have a very high probability to occur while the intermediate fluctuations have a negligible small probability. Thus, for s-values near 1.0, the pdf-modcl calculates relatively high statistical averages. O n the other hand, for s-values near 0.5, all fluctuations have a significant probability to occur and the model calculates lower statistical averages. The other important parameters of the NO-submodcl arc the lowest possible value (Tu) and the highest possible value (TD) of the fluctuating temperature. The choice of the unburned temperature (Tu) value is rather straightforward; it is Ihe combustion air temperature. Unfortunately there is no such an obvious choice for the burnt temperature O b ) . ,l ^ typically assumed to be either equal to the maximum value of the in-flame lime-mean temperature field of the flame computed or equal to the adiabatic flame temperature. In the authors' opinion both alternatives arc inappropriate for diffusion flames. Firstly, there is an overwhelming experimental evidence that the temperature fluctuations may exceed the maximum value of the time-mean temperature field. Secondly, in large-scale diffusion flames, where radiation losses arc significant, the maximum fluctuating temperature may be substantially lower than the adiabatic flame temperature. FIGURE 4: SHAPES O F BETA-PDF FUNCTION FOR S=0.5 A N D S=1.0 In the modeling work presented in this paper, the Tj, values are calculated throughout the flame/furnace. For this purpose, use has been made of the local stoichiometry A defined by Eq. (4). An enthalpy increase Ah which would occur if all the combustibles were burnt, can be calculated as: A/» = H^ ntyoi + HQQ I»QQ , if A £ 1, and (22) mO* Ah = Hvo,-^-JfX<\ svol where Hvoi and H^Q stand for the lower calorific values of volatiles and C O . The corresponding temperature increase A T is the solution of the algebraic equation A + A/r - ar=c,( r +A7Tr (23) Now. the unburnt and the burnt temperatures. Tu and T5 are set to ri/ = min(rcfl/,7;)-0.5 and (24) Tb =7;+max(A7\0.5) where Tai is the combustion air inlet temperature. In order to make sure that T u is always less than T and that T D is always larger than T, the value of 0.5 K appears in the above equations. PromPt-NO. The prompt-NO mechanism is complex and involves a number of radical reactions (Bowman, 1975). Considerable efforts have been recently allocated to reduce the mechanism to a few key reactions only but so far without success. A roughly estimated prompt-NO formation rate for methane combustion was given by Dc Soctc(1975) and in terms of concentration it reads (in gmolc/cm^s): Bpdfmab) s=1.0 11-11 |