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Show with the proportionality constant -rr /4 for the generic furnace with circular cross section in Fig. lOa. Assuming perfect geometric similarity, the dimensions and volume then scale wi th burner size Do in Table 1 as L1 - bs ' Do L2 - b6 • Do (16) VJurnace a4 . Dg (17) with the constants bs = 6.2, b6 = 11, and a4 = 106-rr, obtained from the ideal furnace geometry shown in Fig. 2 a. 4.1.4 V in Staging-Jet Region NOx formation associated with combustion of staged fuel is accounted for by the stagingjet region. As suggested by the schematic in Fig. lOa based on the in-flame data, the staged fuel is injected directly into the high O2 zone created by the primary air. For this reason, the mixing and combustion in the staging jets is modeled using the scaling laws for free turbulent jets. Accordingly, temperatures become significant for NOx formation only near the jet flame tip. The resulting NOx-producing region is shown in the schematic diagram in Fig. 10e. Since flame visualizations show the individual staging jets to merge at the flame tips, the resulting volume forms a torus and scales as ~taging-iet '" 8}IT From the scaling for free turbulent jets (e.g. Chen et. ai [16]) OJ = 0.44iJ (18) (19) with iJ denoting the turbulent jet diffusion flame length, given (e.g. Dahm & Dimotakis [1 7]) as (20) where c.p = mJuel/mair at stoichiometry, and d" is the staging jet momentum diameter. For methane-air chemistry <p = 17.2. The effects of molecular weight differences and temperature differences between the jet and ambient fluids essentially offset each other, giving d- ~ dJ . The volume of the staging-jet region thus scales as 0.44 2 2 Vstaging-jet = as· (-2- x 182) dJIT (21) where, for a torus-shaped volume, the constant as = 2-rr2. Assuming perfect aerodynamic and geolnetric similarity gives the variation with burner size Do in Table 1 as dJ b7 . Do iT bs ' Do (22) The proportionality constants b7 ,s are both determined from the burner geometry as 0.028 and 1, respectively. Thus the volume scales as Vstaging- jet (23) 10 |