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(f)Tj /T1_3 7.97 Tf 6.513 3.959 Td (2)Tj ET q 0.926495 0 0 0.926495 232.0575635 529.9638618 cm 0.398 w 0 0 m 16.218 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 232.0576 520.5043 Tm (d\021)Tj /T1_3 7.97 Tf 11.486 3.152 Td (2)Tj /T1_4 10.909 Tf 8.352 4.331 Td (\000)Tj /T1_5 9.963 Tf 10.91 14.284 Td (\022)Tj /T1_2 10.909 Tf 9.084 -6.904 Td (d)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 269.885 553.602 Td (f)Tj ET q 0.926495 0 0 0.926495 268.4475077 529.9638618 cm 0.398 w 0 0 m 11.486 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 268.4475 520.5043 Tm (d\021)Tj /T1_5 9.963 Tf 12.681 21.767 Td (\023)Tj /T1_3 7.97 Tf 7.334 -2.483 Td (2)Tj /T1_4 10.909 Tf 7.157 -11.801 Td (\000)Tj /T1_0 10.909 Tf 10.909 0 Td (\000)Tj /T1_2 10.909 Tf 0.9265 0 0 0.9265 22.4927 21.3654 Tm 310.367 546.222 Td [("\f)-53(P)-139(r)]TJ 27.328 7.38 Td [(d)167(f)]TJ ET q 0.926495 0 0 0.926495 334.8503308 529.9638618 cm 0.398 w 0 0 m 11.486 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 334.8503 520.5043 Tm (d\021)Tj /T1_0 10.909 Tf 15.712 7.483 Td (=)Tj /T1_4 10.909 Tf 11.515 0 Td (\000)Tj /T1_2 10.909 Tf 0.9265 0 0 0.9265 22.4927 21.3654 Tm 372.853 546.222 Td [("P)-139(r)]TJ /T1_0 10.909 Tf 20.647 0 Td [(\(1)-222(+)]TJ /T1_2 10.909 Tf 23.03 0 Td (\f)Tj /T1_0 10.909 Tf 6.747 0 Td (\000\))Tj /T1_2 10.909 Tf 12.878 0 Td (;)Tj /T1_0 10.909 Tf 81.118 0 Td (\(2\))Tj -283.655 -26.24 Td (1)Tj ET q 0.926495 0 0 0.926495 238.3076988 498.8150999 cm 0.398 w 0 0 m 6.818 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 238.3077 489.3556 Tm (\000)Tj /T1_2 10.909 Tf 9.209 14.932 Td (d)Tj /T1_3 7.97 Tf 5.678 3.959 Td (2)Tj /T1_2 10.909 Tf 4.732 -3.959 Td (y)Tj /T1_6 7.97 Tf 5.349 -1.777 Td (f)Tj ET q 0.926495 0 0 0.926495 246.8397913 498.8150999 cm 0.398 w 0 0 m 21.204 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 249.1495 489.3556 Tm (d\021)Tj /T1_3 7.97 Tf 11.486 3.151 Td (2)Tj /T1_0 10.909 Tf 10.844 4.332 Td (+)Tj /T1_2 10.909 Tf 10.909 0 Td (Le)Tj /T1_6 7.97 Tf 12.504 -1.689 Td (F)Tj /T1_2 10.909 Tf 7.034 1.689 Td (f)Tj 7.71 7.449 Td (dy)Tj /T1_6 7.97 Tf 11.027 -1.777 Td (f)Tj ET q 0.926495 0 0 0.926495 305.1904464 498.8150999 cm 0.398 w 0 0 m 16.471 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 307.5002 489.3556 Tm (d\021)Tj /T1_0 10.909 Tf 18.205 7.483 Td (=)Tj /T1_2 10.909 Tf 11.515 0 Td (Le)Tj /T1_6 7.97 Tf 12.504 -1.689 Td (F)Tj /T1_2 10.909 Tf 8.228 9.069 Td (w)Tj /T1_6 7.97 Tf 7.81 -1.689 Td (F)Tj ET q 0.926495 0 0 0.926495 354.2437241 498.8150999 cm 0.398 w 0 0 m 14.843 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 358.4491 489.3556 Tm (a)Tj 11.5 7.483 Td (;)Tj /T1_0 10.909 Tf 143.163 0 Td (\(3\))Tj -290.908 -26.172 Td (1)Tj ET q 0.926495 0 0 0.926495 231.5869041 467.7302661 cm 0.398 w 0 0 m 6.818 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 231.5869 458.2698 Tm (\000)Tj /T1_2 10.909 Tf 9.209 14.863 Td (d)Tj /T1_3 7.97 Tf 5.678 3.959 Td (2)Tj /T1_2 10.909 Tf 4.732 -3.959 Td (y)Tj /T1_6 7.97 Tf 5.349 -1.688 Td (O)Tj ET q 0.926495 0 0 0.926495 240.1189965 467.7302661 cm 0.398 w 0 0 m 22.918 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 243.2228 458.2698 Tm (d\021)Tj /T1_3 7.97 Tf 11.486 3.152 Td (2)Tj /T1_0 10.909 Tf 11.701 4.331 Td (+)Tj /T1_2 10.909 Tf 10.909 0 Td (Le)Tj /T1_6 7.97 Tf 12.504 -1.688 Td (O)Tj /T1_2 10.909 Tf 7.159 1.688 Td (f)Tj 7.711 7.38 Td (dy)Tj /T1_6 7.97 Tf 11.026 -1.688 Td (O)Tj ET q 0.926495 0 0 0.926495 300.1744024 467.7302661 cm 0.398 w 0 0 m 18.185 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 303.2772 458.2698 Tm (d\021)Tj /T1_0 10.909 Tf 19.062 7.483 Td (=)Tj /T1_2 10.909 Tf 11.515 0 Td (S)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 344.591 479.05 Td (Le)Tj /T1_6 7.97 Tf 12.505 -1.688 Td (F)Tj /T1_2 10.909 Tf 8.229 9.068 Td (w)Tj /T1_6 7.97 Tf 7.81 -1.688 Td (F)Tj ET q 0.926495 0 0 0.926495 360.9645189 467.7302661 cm 0.398 w 0 0 m 14.843 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 365.1699 458.2698 Tm (a)Tj 11.5 7.483 Td (;)Tj /T1_0 10.909 Tf 135.909 0 Td (\(4\))Tj -318.649 -33.551 Td (0)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 207.111 445.499 Td (=)Tj /T1_2 10.909 Tf 13.574 7.38 Td (")Tj ET q 0.926495 0 0 0.926495 226.1539374 436.6445059 cm 0.398 w 0 0 m 6.818 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 226.1539 427.185 Tm (\000)Tj /T1_2 10.909 Tf 9.21 14.863 Td (d)Tj /T1_3 7.97 Tf 5.678 3.959 Td (2)Tj /T1_2 10.909 Tf 4.732 -3.959 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.636 Td (g)Tj ET q 0.926495 0 0 0.926495 234.6869564 436.6445059 cm 0.398 w 0 0 m 20.346 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 236.5983 427.185 Tm (d\021)Tj /T1_3 7.97 Tf 11.486 3.151 Td (2)Tj /T1_0 10.909 Tf 10.416 4.332 Td (+)Tj /T1_2 10.909 Tf 10.909 0 Td ("f)Tj 12.798 7.38 Td (d\022)Tj /T1_6 7.97 Tf 10.799 -1.636 Td (g)Tj ET q 0.926495 0 0 0.926495 278.854826 436.6445059 cm 0.398 w 0 0 m 15.613 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 280.7662 427.185 Tm (d\021)Tj /T1_0 10.909 Tf 17.17 7.483 Td (+)Tj /T1_2 10.909 Tf 10.909 0 Td ("q)Tj 11.544 7.38 Td (w)Tj /T1_6 7.97 Tf 7.81 -1.689 Td (F)Tj ET q 0.926495 0 0 0.926495 317.4766966 436.6445059 cm 0.398 w 0 0 m 14.843 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 321.6811 427.185 Tm (a)Tj /T1_0 10.909 Tf 13.925 7.483 Td (+)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 347.759 445.499 Td (\000)Tj /T1_2 10.909 Tf (n)Tj /T1_6 7.97 Tf 13.366 -1.637 Td (g)Tj /T1_0 10.909 Tf 4.814 1.637 Td (\()Tj /T1_2 10.909 Tf (\022)Tj /T1_6 7.97 Tf 9.364 -1.637 Td (s)Tj /T1_4 10.909 Tf 6.838 1.637 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.637 Td (g)Tj /T1_0 10.909 Tf 4.814 1.637 Td (\))Tj /T1_2 10.909 Tf (;)Tj /T1_0 10.909 Tf 114.288 0 Td (\(5\))Tj -288.096 -33.552 Td [(0)-278(=)-277(\(1)]TJ /T1_4 10.909 Tf 32.121 0 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (")Tj /T1_0 10.909 Tf (\))Tj /T1_2 10.909 Tf 10.525 7.38 Td (d)Tj /T1_3 7.97 Tf 5.678 3.959 Td (2)Tj /T1_2 10.909 Tf 4.732 -3.959 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.636 Td (s)Tj ET q 0.926495 0 0 0.926495 284.4425173 405.5596721 cm 0.398 w 0 0 m 19.946 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 286.1686 396.0992 Tm (d\021)Tj /T1_3 7.97 Tf 11.487 3.152 Td (2)Tj /T1_4 10.909 Tf 10.215 4.331 Td (\000)Tj /T1_0 10.909 Tf 10.909 0 Td (\000)Tj /T1_2 10.909 Tf 0.9265 0 0 0.9265 22.4927 21.3654 Tm 324.024 411.947 Td (n)Tj /T1_6 7.97 Tf 6.548 -1.636 Td (g)Tj /T1_0 10.909 Tf 4.814 1.636 Td (\()Tj /T1_2 10.909 Tf (\022)Tj /T1_6 7.97 Tf 9.364 -1.636 Td (s)Tj /T1_4 10.909 Tf 6.838 1.636 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.636 Td (g)Tj /T1_0 10.909 Tf 4.814 1.636 Td (\))Tj /T1_2 10.909 Tf (;)Tj /T1_0 10.909 Tf 144.841 0 Td (\(6\))Tj -439.604 -26.149 Td [(in)-363(whic)27(h)]TJ /T1_2 10.909 Tf 44.597 0 Td (")Tj /T1_0 10.909 Tf 9.052 0 Td [(is)-363(the)-364(medium)-363(p)-28(orosit)28(y)83(,)-371(\000)]TJ /T1_4 10.909 Tf 127.702 0 Td (\021)Tj /T1_0 10.909 Tf 12.518 2.879 Td (\026)Tj /T1_2 10.909 Tf -0.455 -2.879 Td (\025)Tj /T1_6 7.97 Tf 6.364 -1.636 Td (s)Tj /T1_2 10.909 Tf 4.414 1.636 Td (=)Tj /T1_0 10.909 Tf 5.909 2.879 Td (\026)Tj /T1_2 10.909 Tf -0.454 -2.879 Td (\025)Tj /T1_6 7.97 Tf 6.363 -1.636 Td (g)Tj /T1_4 10.909 Tf 8.392 1.636 Td (\035)Tj /T1_0 10.909 Tf 14.487 0 Td (1,)Tj /T1_2 10.909 Tf 12.532 0 Td (Le)Tj /T1_6 7.97 Tf 12.504 -1.636 Td (i)Tj /T1_0 10.909 Tf 7.346 1.636 Td [(is)-364(th)1(e)]TJ /T1_2 10.909 Tf 30.415 0 Td (i)Tj /T1_0 10.909 Tf [(-sp)-28(ecies)-363(Lewis)-364(n)28(um)28(b)-28(er,)-371(de\fned)]TJ -301.686 -13.549 Td (as)Tj /T1_2 10.909 Tf 13.537 0 Td (Le)Tj /T1_6 7.97 Tf 12.504 -1.636 Td (i)Tj /T1_4 10.909 Tf 6.65 1.636 Td (\021)Tj /T1_2 10.909 Tf 11.754 0 Td (\013)Tj /T1_6 7.97 Tf 6.978 -1.636 Td (i)Tj /T1_2 10.909 Tf 3.382 1.636 Td (=D)Tj /T1_6 7.97 Tf 14.486 -1.636 Td (i)Tj /T1_0 10.909 Tf 3.381 1.636 Td [(,)-350(the)-346(ratio)-347(b)-27(et)27(w)28(een)-346(the)-347(thermal)-346(and)-347(mass)-346(di\013usivities,)]TJ /T1_2 10.909 Tf 263.179 0 Td [(P)-139(r)]TJ /T1_4 10.909 Tf 17.013 0 Td (\021)Tj /T1_2 10.909 Tf 11.753 0 Td [(\027)-64(=\013)]TJ /T1_0 10.909 Tf 22.337 0 Td [(is)-346(the)-347(Prandtl)]TJ -386.954 -13.549 Td [(n)28(um)28(b)-28(er,)-502(a)-467(ratio)-468(b)-28(et)28(w)28(een)-468(the)-468(viscous)-468(di\013usion)-467(rate)-468(and)-468(the)-468(thermal)-467(di\013usion)-468(rate,)-501(and)]TJ /T1_2 10.909 Tf 435.299 0 Td (\021)Tj /T1_0 10.909 Tf 10.911 0 Td (is)Tj -446.21 -13.549 Td [(the)-345(mo)-28(di\fed)-345(v)28(ertical)-345(co)-28(ordinate.)-479(The)-345(c)28(hem)-1(ical)-345(reaction)-345(is)-345(accoun)28(ted)-345(through)-345(the)-345(reaction)-345(rate)]TJ T* (term,)Tj /T1_2 10.909 Tf 28.298 0 Td (w)Tj /T1_6 7.97 Tf 7.81 -1.689 Td (F)Tj /T1_0 10.909 Tf 9.641 1.689 Td [(and)-239(the)-239(non-dimensional)-239(heat)-239(released,)]TJ /T1_2 10.909 Tf 184.91 0 Td (q)Tj /T1_0 10.909 Tf 5.261 0 Td [(,)-258(while)]TJ /T1_2 10.909 Tf 33.3 0 Td (S)Tj /T1_0 10.909 Tf 9.925 0 Td [(is)-239(the)-239(non-dimensional)-239(stoic)28(hiometric)]TJ -279.145 -13.55 Td [(co)-28(e\016cien)28(t.)]TJ 16.937 -13.549 Td [(The)-425(l)1(as)-1(t)-424(term)-425(in)-424(Eqs.)-718(\(5\))-425(and)-424(\(6\))-425(accoun)28(ts)-424(for)-425(the)-424(heat)-425(exc)28(hange)-425(b)-27(e)-1(t)28(w)28(een)-425(solid)-424(and)-425(gas)]TJ -16.937 -13.549 Td [(phases,)-333(and)-334(is)-333(of)-333(the)-334(order)-333(of)-333(\000)]TJ /T1_2 10.909 Tf 151.909 0 Td (>>)Tj /T1_0 10.909 Tf 20 0 Td (1.)Tj -154.972 -13.549 Td [(The)-447(v)56(ap)-28(orization)-447(rate,)-776(_)]TJ /T1_2 10.909 Tf 114.396 0 Td (m)Tj /T1_0 10.909 Tf [(,)-475(is)-448(related)-447(with)]TJ /T1_2 10.909 Tf 94.925 0 Td (f)Tj /T1_4 10.909 Tf 6.516 0 Td (j)Tj /T1_3 7.97 Tf 3.03 -3.226 Td (0)Tj /T1_0 10.909 Tf 9.609 3.226 Td (through)Tj /T1_4 10.909 Tf 42.484 0 Td (\000)Tj /T1_2 10.909 Tf (a)Tj /T1_3 7.97 Tf 14.251 3.959 Td (1)Tj /T1_6 7.97 Tf (=)Tj /T1_3 7.97 Tf (2)Tj /T1_2 10.909 Tf 16.214 -3.959 Td (f)Tj /T1_4 10.909 Tf 6.515 0 Td (j)Tj /T1_3 7.97 Tf 3.031 -3.226 Td (0)Tj /T1_0 10.909 Tf 9.83 3.226 Td [(=)-767(_)]TJ /T1_2 10.909 Tf 13.583 0 Td (m)Tj /T1_0 10.909 Tf [(,)-476(i)1(n)-448(wh)1(ic)27(h)]TJ /T1_2 10.909 Tf 64.217 0 Td (a)Tj /T1_0 10.909 Tf 10.643 0 Td [(is)-447(the)]TJ -426.181 -13.549 Td [(non-dimensional)-419(strain-rate.)-702(Belo)28(w)-419(the)-420(gas-liquid)-419(in)28(terface)-419(the)-419(follo)28(wing)-419(go)27(v)28(erning)-419(equations)]TJ 0 -13.55 TD [(are)-333(v)55(alid)-333(\(mass,)-333(energy)-334(for)-333(liquid)-333(and)-333(for)-334(solid,)-333(resp)-28(ectiv)28(ely\):)]TJ /T1_2 10.909 Tf 204.61 -24.508 Td (%)Tj /T1_6 7.97 Tf 5.641 -1.777 Td (l)Tj /T1_2 10.909 Tf 3.12 1.777 Td (v)Tj /T1_6 7.97 Tf 5.288 -1.777 Td (l)Tj /T1_0 10.909 Tf 6.15 1.777 Td [(=)-578(_)]TJ /T1_2 10.909 Tf 11.515 0 Td (m;)Tj /T1_0 10.909 Tf 203.28 0 Td (\(7\))Tj /T1_2 10.909 Tf -297.954 -31.157 Td ("J)Tj 13.381 7.38 Td (d)Tj /T1_3 7.97 Tf 5.679 3.959 Td (2)Tj /T1_2 10.909 Tf 4.732 -3.959 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.777 Td (l)Tj ET q 0.926495 0 0 0.926495 238.089046 204.2258232 cm 0.398 w 0 0 m 18.652 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 239.3343 194.7663 Tm (dz)Tj /T1_3 7.97 Tf 11.231 3.151 Td (2)Tj /T1_4 10.909 Tf 9.696 4.332 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td ("M)Tj 18.056 7.38 Td (d\022)Tj /T1_6 7.97 Tf 10.799 -1.777 Td (l)Tj ET q 0.926495 0 0 0.926495 285.5589438 204.2258232 cm 0.398 w 0 0 m 13.919 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 286.8042 194.7663 Tm (dz)Tj /T1_0 10.909 Tf 16.801 7.483 Td (=)Tj /T1_4 10.909 Tf 11.515 0 Td (\000)Tj /T1_0 10.909 Tf 0.9265 0 0 0.9265 22.4927 21.3654 Tm 322.084 194.641 Td (\000)Tj /T1_3 7.97 Tf 6.816 4.504 Td (2)Tj /T1_2 10.909 Tf 8.369 -4.504 Td (n)Tj /T1_6 7.97 Tf 6.548 -1.778 Td (l)Tj /T1_0 10.909 Tf 3.12 1.778 Td (\()Tj /T1_2 10.909 Tf (\022)Tj /T1_6 7.97 Tf 9.364 -1.637 Td (s)Tj /T1_4 10.909 Tf 6.838 1.637 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.778 Td (l)Tj /T1_0 10.909 Tf 3.12 1.778 Td (\))Tj /T1_2 10.909 Tf (;)Tj /T1_0 10.909 Tf 134.984 0 Td (\(8\))Tj -281.193 -31.431 Td (\(1)Tj /T1_4 10.909 Tf 12.121 0 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (")Tj /T1_0 10.909 Tf (\))Tj /T1_2 10.909 Tf 10.525 7.381 Td (d)Tj /T1_3 7.97 Tf 5.678 3.958 Td (2)Tj /T1_2 10.909 Tf 4.732 -3.958 Td (\022)Tj /T1_6 7.97 Tf 5.122 -1.637 Td (s)Tj ET q 0.926495 0 0 0.926495 272.3082123 175.1060853 cm 0.398 w 0 0 m 19.946 0 l S Q BT /T1_2 10.909 Tf 0.9265 0 0 0.9265 274.1529 165.6456 Tm (dz)Tj /T1_3 7.97 Tf 11.231 3.152 Td (2)Tj /T1_0 10.909 Tf 10.95 4.331 Td (=)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 305.327 163.21 Td (\000)Tj /T1_3 7.97 Tf 6.813 4.505 Td (2)Tj /T1_2 10.909 Tf 8.369 -4.505 Td (n)Tj /T1_6 7.97 Tf 6.548 -1.777 Td (l)Tj /T1_0 10.909 Tf 3.12 1.777 Td (\()Tj /T1_2 10.909 Tf (\022)Tj /T1_6 7.97 Tf 9.363 -1.636 Td (s)Tj /T1_4 10.909 Tf 6.839 1.636 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.777 Td (l)Tj /T1_0 10.909 Tf 3.12 1.777 Td (\))Tj /T1_2 10.909 Tf (;)Tj /T1_0 10.909 Tf 151.744 0 Td (\(9\))Tj -439.604 -24.027 Td [(in)-250(whic)28(h)]TJ /T1_2 10.909 Tf 42.125 0 Td (J)Tj /T1_4 10.909 Tf 10.129 0 Td (\021)Tj /T1_0 10.909 Tf 11.97 2.878 Td (\026)Tj /T1_2 10.909 Tf -0.455 -2.878 Td (\025)Tj /T1_6 7.97 Tf 6.364 -1.778 Td (l)Tj /T1_2 10.909 Tf 3.12 1.778 Td (=)Tj /T1_0 10.909 Tf 5.909 2.878 Td (\026)Tj /T1_2 10.909 Tf -0.454 -2.878 Td (\025)Tj /T1_6 7.97 Tf 6.363 -1.637 Td (s)Tj /T1_0 10.909 Tf 7.144 1.637 Td (and)Tj /T1_2 10.909 Tf 20.305 0 Td (M)Tj /T1_4 10.909 Tf 14.803 0 Td (\021)Tj /T1_0 10.909 Tf 14.789 0 Td (_)Tj /T1_2 10.909 Tf -3.274 0 Td (m)Tj /T1_0 10.909 Tf (\()Tj /T1_2 10.909 Tf (c)Tj /T1_6 7.97 Tf 18.541 -1.778 Td (l)Tj /T1_2 10.909 Tf 3.121 1.778 Td (=c)Tj /T1_6 7.97 Tf 10.175 -1.637 Td (p)Tj /T1_0 10.909 Tf 4.761 1.637 Td [(\).)-417(The)-250(heat)-250(exc)28(hange)-251(b)-27(et)28(w)27(een)-250(solid)-250(and)-250(liquid)-250(is)-250(considered)]TJ -175.436 -13.55 Td [(to)-333(b)-28(e)-333(of)-334(the)-333(order)-333(of)-334(\000)]TJ /T1_3 7.97 Tf 107.182 3.959 Td (2)Tj /T1_2 10.909 Tf 7.763 -3.959 Td (>>)Tj /T1_0 10.909 Tf 20 0 Td 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6.666 0 Td (\000)Tj /T1_2 10.909 Tf 10.909 0 Td (\022)Tj /T1_6 7.97 Tf 5.121 -1.777 Td (f)Tj /T1_0 10.909 Tf 8.476 1.777 Td (=)Tj /T1_2 10.909 Tf 11.515 0 Td (y)Tj /T1_6 7.97 Tf 5.348 -1.688 Td (F)Tj /T1_0 10.909 Tf 7.034 1.688 Td (\()Tj /T1_2 10.909 Tf (\021)Tj /T1_6 7.97 Tf 9.659 -1.777 Td (f)Tj /T1_0 10.909 Tf 5.445 1.777 Td [(\))-278(=)]TJ /T1_2 10.909 Tf 18.788 0 Td (y)Tj /T1_6 7.97 Tf 5.348 -1.688 Td (O)Tj /T1_0 10.909 Tf 7.159 1.688 Td (\()Tj /T1_2 10.909 Tf (\021)Tj /T1_6 7.97 Tf 9.659 -1.777 Td (f)Tj /T1_0 10.909 Tf 5.445 1.777 Td [(\))-278(=)-278(0)]TJ /T1_2 10.909 Tf 24.242 0 Td (;)Tj ET /Artifact <>BDC q 0.926495 0 0 0.926495 299.816765 44.2607662 cm /GS0 gs /Fm1 Do Q EMC Q endstream endobj 3 0 obj <>stream /
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/T1_5 7.97 Tf 5.348 -1.688 Td (F)Tj /T1_1 10.909 Tf 7.034 1.688 Td (\()Tj /T1_0 10.909 Tf (z)Tj /T1_2 10.909 Tf 12.825 0 Td [(!)-278(\0001)]TJ /T1_1 10.909 Tf (\))Tj /T1_2 10.909 Tf 40 0 Td (\000)Tj /T1_1 10.909 Tf 10.909 0 Td [(1)-278(=)]TJ /T1_0 10.909 Tf 20 0 Td (y)Tj /T1_5 7.97 Tf 5.349 -1.688 Td (O)Tj /T1_1 10.909 Tf 7.159 1.688 Td (\()Tj /T1_0 10.909 Tf (z)Tj /T1_2 10.909 Tf 12.825 0 Td [(!)-278(\0001)]TJ /T1_1 10.909 Tf [(\))-278(=)-277(0)]TJ /T1_0 10.909 Tf (;)Tj /T1_1 10.909 Tf 78.81 0 Td (\(10\))Tj -434.149 -20.025 Td [(in)-405(whic)27(h)]TJ /T1_0 10.909 Tf 45.514 0 Td (\021)Tj /T1_5 7.97 Tf 5.417 -1.777 Td (f)Tj /T1_1 10.909 Tf 9.868 1.777 Td [(is)-406(th)1(e)-406(\rame)-406(p)-27(osition,)]TJ /T1_0 10.909 Tf 106.8 0 Td (\022)Tj /T1_5 7.97 Tf 5.121 -1.777 Td (f)Tj /T1_1 10.909 Tf 9.869 1.777 Td [(is)-405(the)-406(\rame)-406(t)1(e)-1(mp)-27(erature,)]TJ /T1_0 10.909 Tf 127.406 0 Td (\022)Tj /T1_5 7.97 Tf 5.121 -1.688 Td (B)Tj /T1_1 10.909 Tf 11.702 1.688 Td [(is)-406(th)1(e)-406(b)-28(oiling)-405(temp)-28(erature)]TJ -326.818 -13.549 Td [(of)-388(the)-388(liquid)-388(fuel,)]TJ /T1_0 10.909 Tf 88.293 0 Td (\022)Tj /T1_5 7.97 Tf 5.121 -1.636 Td (s)Tj /T1_3 7.97 Tf (0)Tj /T1_1 10.909 Tf 12.881 1.636 Td [(is)-388(the)-388(solid)-388(phase)-388(temp)-28(erature)-388(at)-388(the)-388(in)28(terface)-388(and)]TJ /T1_0 10.909 Tf 250.792 0 Td (y)Tj /T1_5 7.97 Tf 5.349 -1.688 Td (F)Tj /T1_3 7.97 Tf 6.535 0 Td (0)Tj /T1_1 10.909 Tf 8.965 1.688 Td [(is)-388(the)-388(fuel)-388(mass)]TJ -377.936 -13.549 Td [(fraction)-333(at)-334(the)-333(in)28(terface.)]TJ 16.937 -13.549 Td [(Mass)-333(and)-334(energy)-333(m)28(ust)-334(b)-27(e)-334(conserv)28(ed)-333(at)-334(th)1(e)-334(in)28(terface)-333(follo)27(wing:)]TJ 131.492 -21.652 Td (1)Tj ET q 0.926495 0 0 0.926495 231.3404564 598.1520408 cm 0.398 w 0 0 m 6.818 0 l S Q BT /T1_1 10.909 Tf 0.9265 0 0 0.9265 231.3405 588.6925 Tm (\000)Tj 16.25 14.863 Td (1)Tj ET q 0.926495 0 0 0.926495 239.8725489 598.1520408 cm 0.398 w 0 0 m 19.537 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 239.8725 588.6925 Tm (Le)Tj /T1_5 7.97 Tf 12.504 -1.689 Td (F)Tj /T1_0 10.909 Tf 12.438 16.552 Td (dy)Tj /T1_5 7.97 Tf 11.026 -1.689 Td (F)Tj ET q 0.926495 0 0 0.926495 262.9811872 598.1520408 cm 0.398 w 0 0 m 18.06 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 266.0266 588.6925 Tm (d\021)Tj /T1_4 9.963 Tf 15.968 22.165 Td (\f)Tj 0 -5.977 TD (\f)Tj 0 -5.978 TD (\f)Tj T* (\f)Tj /T1_5 7.97 Tf 3.321 -6.852 Td (\021)Tj /T1_3 7.97 Tf 4.536 0 Td (=0)Tj /T1_7 5.978 Tf 10.82 2.27 Td (+)Tj /T1_1 10.909 Tf 9.617 7.833 Td (=)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 318.637 619.82 Td (\(1)Tj /T1_2 10.909 Tf 12.116 0 Td (\000)Tj /T1_0 10.909 Tf 10.909 0 Td (y)Tj /T1_5 7.97 Tf 5.349 -1.689 Td (F)Tj /T1_3 7.97 Tf 6.535 0 Td (0)Tj /T1_1 10.909 Tf 4.732 1.689 Td (\))Tj /T1_0 10.909 Tf 7.256 0 Td (f)Tj /T1_2 10.909 Tf 6.516 0 Td (j)Tj /T1_3 7.97 Tf 3.03 -3.226 Td (0)Tj /T1_0 10.909 Tf 6.55 3.226 Td (;)Tj /T1_1 10.909 Tf 130.189 0 Td 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BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 358.8827 551.8968 Tm (a)Tj /T1_3 7.97 Tf 5.767 3.167 Td (1)Tj /T1_5 7.97 Tf 0.9265 0 0 0.9265 22.4927 21.3654 Tm 373.077 575.789 Td (=)Tj /T1_3 7.97 Tf (2)Tj /T1_4 9.963 Tf 11.982 18.442 Td (Z)Tj /T1_3 7.97 Tf 9.963 -3.155 Td (0)Tj /T1_6 7.97 Tf -4.428 -19.483 Td (\000)Tj /T1_5 7.97 Tf (z)Tj /T1_8 5.978 Tf 10.526 -1.405 Td (b)Tj /T1_1 10.909 Tf 4.27 10.246 Td (\()Tj /T1_0 10.909 Tf (\022)Tj /T1_5 7.97 Tf 9.363 -1.636 Td (s)Tj /T1_2 10.909 Tf 6.838 1.636 Td (\000)Tj /T1_0 10.909 Tf 10.91 0 Td (\022)Tj /T1_5 7.97 Tf 5.121 -1.777 Td (l)Tj /T1_1 10.909 Tf 3.12 1.777 Td (\))Tj /T1_0 10.909 Tf [(dz)-44(:)]TJ /T1_1 10.909 Tf 71.077 0 Td (\(12\))Tj -417.213 -26.7 Td [(The)-305(\rame)-305(t)1(e)-1(mp)-27(erature,)]TJ /T1_0 10.909 Tf 115.854 0 Td (\022)Tj /T1_5 7.97 Tf 5.121 -1.777 Td (f)Tj /T1_1 10.909 Tf 5.445 1.777 Td [(,)-310(is)-305(obtained)-305(from)-304(the)-305(matc)28(hing)-305(condition)-305(b)-27(et)27(w)28(een)-305(the)-304(zones)-1(,)-310(and)]TJ -143.357 -13.549 Td [(the)-322(\rame)-321(p)-28(osition,)]TJ /T1_0 10.909 Tf 92.129 0 Td (\021)Tj /T1_5 7.97 Tf 5.417 -1.777 Td (f)Tj /T1_1 10.909 Tf 5.444 1.777 Td [(,)-324(is)-322(obtained)-321(from)-322(the)-322(mass)-321(conserv)55(ation)-321(at)-322(the)-322(in)28(terface.)-440(The)-322(solid)-322(phase)]TJ -102.99 -13.549 Td [(temp)-28(erature)-359(at)-359(the)-359(in)28(terface)-359(is)-359(obtained)-359(b)28(y)-359(the)-359(con)28(tin)27(u)1(it)27(y)-359(of)-359(the)-359(solid)-359(phase)-359(heat)-359(\rux)-359(an)1(d)-359(the)]TJ 0 -13.549 TD [(fuel)-333(mass)-334(fraction)-333(at)-333(the)-334(in)28(terface)-333(is)-334(ob)1(tained)-334(through)-333(the)-333(mass)-334(conserv)56(ation.)]TJ /T1_9 14.346 Tf 0 -34.373 TD [(3)-1125(Metho)-31(dology)]TJ /T1_1 10.909 Tf 0 -24.352 TD [(The)-399(non-dimensional)-399(reaction)-399(rate)-398(term)]TJ /T1_0 10.909 Tf 197.667 0 Td (w)Tj /T1_5 7.97 Tf 7.81 -1.688 Td (F)Tj /T1_1 10.909 Tf 11.385 1.688 Td [(that)-399(app)-28(ears)-398(in)-399(Eqs.)-641(\(3\),)-416(\(4\))-398(and)-399(\(5\))-399(is)-399(strongly)]TJ -216.862 -13.549 Td [(non-linear.)-444(This)-334(fact)-333(imp)-28(oses)-333(di\016culties)-334(in)-333(solving)-333(the)-333(go)27(v)28(erning)-333(equations.)]TJ 16.937 -13.549 Td [(In)-481(order)-481(to)-481(a)28(v)28(oid)-481(this)-481(strong)-481(non-linearit)28(y)83(,)-517(Eqs.)-888(\(3\),)-518(\(4\))-480(and)-481(\(5\))-481(are)-481(com)27(b)1(ined)-481(in)-481(suc)27(h)]TJ -16.937 -13.549 Td [(a)-469(w)28(a)27(y)-469(that)-469(the)-469(reaction)-469(term)-469(is)-469(eliminated)-469(from)-469(the)-469(equations.)-852(De\fning)-469(the)-469(new)-469(v)56(ariables)]TJ /T1_0 10.909 Tf 0 -13.55 TD (Z)Tj /T1_2 10.909 Tf 11.818 0 Td (\021)Tj /T1_0 10.909 Tf 12.076 0 Td [(S)-58(y)]TJ /T1_5 7.97 Tf 12.666 -1.688 Td (F)Tj /T1_2 10.909 Tf 9.682 1.688 Td (\000)Tj /T1_0 10.909 Tf 11.133 0 Td (y)Tj /T1_5 7.97 Tf 5.349 -1.688 Td (O)Tj /T1_1 10.909 Tf 9.807 1.688 Td [(+)-243(1)-364(and)]TJ /T1_0 10.909 Tf 42.108 0 Td (H)Tj /T1_2 10.909 Tf 13.545 0 Td (\021)Tj /T1_1 10.909 Tf 12.076 0 Td (\()Tj /T1_0 10.909 Tf (S)Tj /T1_1 10.909 Tf 14.209 0 Td [(+)-243(1\))]TJ /T1_0 10.909 Tf 20.83 0 Td (\022)Tj /T1_5 7.97 Tf 5.121 -1.636 Td (g)Tj /T1_0 10.909 Tf 4.814 1.636 Td (=q)Tj /T1_1 10.909 Tf 13.364 0 Td (+)Tj /T1_0 10.909 Tf 11.133 0 Td (y)Tj /T1_5 7.97 Tf 5.349 -1.688 Td (F)Tj /T1_1 10.909 Tf 9.682 1.688 Td (+)Tj /T1_0 10.909 Tf 11.133 0 Td (y)Tj /T1_5 7.97 Tf 5.348 -1.688 Td (O)Tj /T1_1 10.909 Tf 11.132 1.688 Td [(and)-364(considering)-364(the)-364(Lewis)-365(n)28(um)28(b)-28(er)-364(for)-364(the)]TJ -252.375 -13.549 Td 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7.483 Td (=)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 340.364 356.065 Td (0)Tj /T1_0 10.909 Tf (;)Tj /T1_1 10.909 Tf 171.455 0 Td (\(13\))Tj /T1_0 10.909 Tf -309.186 -30.655 Td (")Tj ET q 0.926495 0 0 0.926495 209.4288497 318.5451147 cm 0.398 w 0 0 m 6.818 0 l S Q BT /T1_1 10.909 Tf 0.9265 0 0 0.9265 209.4288 309.0856 Tm (\000)Tj /T1_0 10.909 Tf 9.209 14.863 Td (d)Tj /T1_3 7.97 Tf 5.678 3.959 Td (2)Tj /T1_0 10.909 Tf 4.733 -3.959 Td (H)Tj ET q 0.926495 0 0 0.926495 217.9609421 318.5451147 cm 0.398 w 0 0 m 20.365 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 219.8825 309.0856 Tm (d\021)Tj /T1_3 7.97 Tf 11.486 3.151 Td (2)Tj /T1_1 10.909 Tf 10.425 4.332 Td (+)Tj /T1_0 10.909 Tf 10.909 0 Td ("f)Tj 12.798 7.38 Td (dH)Tj ET q 0.926495 0 0 0.926495 262.1473417 318.5451147 cm 0.398 w 0 0 m 15.633 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 264.068 309.0856 Tm (d\021)Tj /T1_1 10.909 Tf 17.179 7.483 Td (+)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 288.829 318.03 Td (\000)Tj /T1_0 10.909 Tf (n)Tj /T1_5 7.97 Tf 13.366 -1.636 Td (g)Tj /T1_1 10.909 Tf 4.814 1.636 Td (\()Tj /T1_0 10.909 Tf (\022)Tj /T1_5 7.97 Tf 9.364 -1.636 Td (s)Tj /T1_2 10.909 Tf 6.838 1.636 Td (\000)Tj /T1_0 10.909 Tf 10.909 0 Td (\022)Tj /T1_5 7.97 Tf 5.122 -1.636 Td (g)Tj /T1_1 10.909 Tf 4.813 1.636 Td (\))Tj 5.438 7.38 Td (\()Tj /T1_0 10.909 Tf (S)Tj /T1_1 10.909 Tf 13.985 0 Td [(+)-222(1\))]TJ ET q 0.926495 0 0 0.926495 346.29625 318.5451147 cm 0.398 w 0 0 m 34.591 0 l S Q BT /T1_0 10.909 Tf 0.9265 0 0 0.9265 359.8833 309.0856 Tm (q)Tj /T1_1 10.909 Tf 24.152 7.483 Td (=)Tj 0.9265 0 0 0.9265 22.4927 21.3654 Tm 399.83 318.03 Td (0)Tj /T1_0 10.909 Tf (:)Tj /T1_1 10.909 Tf 111.989 0 Td (\(14\))Tj -434.15 -25.258 Td [(The)-292(n)1(e)-1(w)-291(v)56(ariables)-292(are)-291(the)-292(mixture)-291(fraction)-292(\(Z\))-291(and)-291(the)-292(excess)-292(en)28(thalp)28(y)-291(\(H\).)-292(This)-291(form)28(ulation)-292(in)]TJ 0 -13.549 TD [(whic)28(h)-334(th)1(e)-334(strong)-333(non-linear)-333(term)-334(do)-27(e)-1(s)-333(not)-333(app)-28(ear)-333(is)-334(called)-333(the)-333(Sc)28(h)27(v)56(ab-Ze'ldo)28(vic)28(h)-334(form)28(ulation.)]TJ 16.937 -13.549 Td [(In)-315(the)-315(pro)-27(ce)-1(ss)-314(of)-315(obtaining)-315(the)-315(go)28(v)28(erning)-315(equations)-315(in)-315(their)-315(non-dimensional)-314(form,)-319(the)-315(large)]TJ -16.937 -13.55 Td [(parameter)-341(\000)-341(app)-28(ears.)-468(This)-341(parameter,)-343(the)-341(ratio)-341(b)-28(et)28(w)27(een)-341(solid)-341(and)-341(gas)-341(phases)-341(thermal)-342(cond)1(uc-)]TJ T* [(tivities,)-318(motiv)56(ates)-315(the)-314(use)-314(of)-314(the)-314(p)-28(erturbation)-314(theory)-314(in)-314(order)-314(to)-315(seek)-314(solutions)-314(for)-314(the)-314(problem.)]TJ /T1_9 14.346 Tf 0 -34.372 TD [(4)-1125(Analysis)]TJ /T1_1 10.909 Tf 0 -24.352 TD [(The)-333(prop)-28(osed)-333(problem)-333(presen)28(ts)-333(di\013eren)28(t)-333(b)-28(oundary)-333(la)28(y)27(ers:)-444(a)-333(macroscopic)-333(b)-28(oundary)84(,)-333(o)27(wn)-333(to)-333(the)]TJ 0 -13.549 TD [(viscous)-379(\r)1(o)27(w,)-389(a)-379(Darcy)-378(b)-28(oundary)-378(la)28(y)27(er,)-389(o)27(wn)-378(to)-378(the)-379(\ruid)-378(attac)28(hmen)28(t)-379(at)-378(the)-379(p)-27(ore)-379(w)28(alls,)-390(and)-378(t)28(w)27(o)]TJ 0 -13.55 TD [(thermal)-333(b)-28(oundary)-333(la)28(y)27(ers.)]TJ 16.937 -13.549 Td [(The)-259(existence)-259(of)-259(b)-28(oundary)-259(la)28(y)27(ers)-259(in)-259(the)-259(problem)-259(pro)28(vides)-259(the)-259(ground)-259(for)-259(utilizing)-259(the)-260(metho)-27(d)]TJ -16.937 -13.549 Td [(of)-496(matc)27(hed)-496(asymptotic)-496(e)-1(xp)1(ansion)-497([11].)-933(In)-497(the)-496(gas-solid)-497(region)-496(three)-496(di\013eren)27(t)-496(length)-496(scales)]TJ 0 -13.549 TD [(are)-443(observ)28(ed:)-663(outer)-443(zone)-442(\(go)28(v)27(erned)-442(b)28(y)-443(the)-443(Darcy)-442(\ro)28(w)-443(and)-443(thermal)-442(equilibrium)-443(b)-28(et)28(w)28(een)-443(gas)]TJ T* [(and)-376(solid\),)-386(\rame)-376(zone)-376(\(where)-376(the)-376(exothermic)-376(reaction)-376(consumes)-376(o)28(xidan)28(t)-376(and)-376(fu)1(e)-1(l)-375(and)-376(releases)]TJ T* [(heat\))-407(and)-406(inner)-407(zone)-407(\(in)-406(whic)28(h)-407(viscous)-407(e\013ects)-407(arise,)-425(thermal)-406(non-equilibrium)-407(is)-407(observ)28(ed)-407(an)1(d)]TJ 0 -13.55 TD [(the)-407(di\013usion)-407(of)-407(the)-407(v)55(ap)-27(orized)-408(li)1(quid)-407(is)-408(dominan)28(t\).)-665(The)-408(an)1(alyz)-1(ed)-407(case)-407(is)-407(di\013eren)28(t)-407(from)-407(\\free")]TJ 0 -13.549 TD [(impinging)-458(\rames,)-489(as)-458(they)-458(o)-28(ccur)-457(in)-458(the)-458(inner)-458(zone)-458([5],)-489(and)-458(in)-458(the)-458(presen)28(t)-458(w)28(ork)-458(the)-458(\rame)-458(is)]TJ T* 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