Princeton-ChemRC.rar_flux_mechanism reduction_化学计算_数值算法/人工智能下载-pudn.com

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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/628a94aa16e0ca71412d396b/bg2.jpg"><div class="c x0 ye w2 h3"><div class="t m0 xd h4 yf ff2 fs1 fc1 sc0 ls0 ws0">Author's personal copy</div></div><div class="c x0 y10 w0 h5"><div class="t m0 xe h6 y11 ff3 fs2 fc2 sc0 ls0 ws0">A<span class="_ _4"> </span>path<span class="_ _4"> </span>ﬂux<span class="_ _4"> </span>analysis<span class="_ _4"> </span>method<span class="_ _4"> </span>for<span class="_ _4"> </span>the<span class="_ _4"> </span>reduction<span class="_ _4"> </span>of<span class="_ _4"> </span>detailed<span class="_ _4"> </span>chemical</div><div class="t m0 xe h6 y12 ff3 fs2 fc2 sc0 ls0 ws0">kinetic<span class="_ _4"> </span>mechanisms</div><div class="t m0 xe h7 y13 ff3 fs3 fc2 sc0 ls0 ws0">Wenting<span class="_ _3"> </span>Sun</div><div class="t m2 xf h8 y14 ff3 fs4 fc2 sc0 ls0 ws0">a</div><div class="t m0 x10 h7 y13 ff3 fs3 fc2 sc0 ls0 ws0">,<span class="_ _3"> </span>Zheng<span class="_ _5"> </span>Chen</div><div class="t m2 x11 h8 y14 ff3 fs4 fc2 sc0 ls0 ws0">b</div><div class="t m0 x12 h7 y13 ff3 fs3 fc2 sc0 ls0 ws0">,<span class="_ _5"> </span>Xiaolong<span class="_ _3"> </span>Gou</div><div class="t m2 x13 h8 y14 ff3 fs4 fc2 sc0 ls0 ws0">c</div><div class="t m0 x14 h7 y13 ff3 fs3 fc2 sc0 ls0 ws0">,<span class="_ _5"> </span>Yiguang<span class="_ _3"> </span>Ju</div><div class="t m2 x15 h8 y14 ff3 fs4 fc2 sc0 ls0 ws0">a,</div><div class="t m2 x16 h9 y15 ff4 fs4 fc2 sc0 ls0 ws0">*</div><div class="t m3 xe ha y16 ff3 fs5 fc2 sc0 ls0 ws0">a</div><div class="t m4 x17 hb y17 ff5 fs6 fc2 sc0 ls0 ws0">Department<span class="_ _6"> </span>of<span class="_ _6"> </span>Mechanical<span class="_ _6"> </span>and<span class="_ _6"> </span>Aerospace<span class="_ _6"> </span>Engineering,<span class="_ _6"> </span>Princeton<span class="_ _6"> </span>University,<span class="_ _6"> </span>Princeton,<span class="_ _6"> </span>NJ<span class="_ _6"> </span>08544,<span class="_ _6"> </span>USA</div><div class="t m3 xe ha y18 ff3 fs5 fc2 sc0 ls0 ws0">b</div><div class="t m4 x17 hb y19 ff5 fs6 fc2 sc0 ls0 ws0">State<span class="_ _6"> </span>Key<span class="_ _6"> </span>Laboratory<span class="_ _6"> </span>for<span class="_ _6"> </span>Turbulence<span class="_ _6"> </span>and<span class="_ _6"> </span>Complex<span class="_ _6"> </span>Systems,<span class="_ _6"> </span>College<span class="_ _6"> </span>of<span class="_ _6"> </span>Engineering,<span class="_ _6"> </span>Peking<span class="_ _6"> </span>University,<span class="_ _6"> </span>Beijing<span class="_ _6"> </span>100871,<span class="_ _6"> </span>China</div><div class="t m3 xe ha y1a ff3 fs5 fc2 sc0 ls0 ws0">c</div><div class="t m4 x17 hb y1b ff5 fs6 fc2 sc0 ls0 ws0">School<span class="_ _6"> </span>of<span class="_ _6"> </span>Power<span class="_ _6"> </span>Engineering,<span class="_ _6"> </span>Chongqing<span class="_ _6"> </span>University,<span class="_ _6"> </span>Chongqing<span class="_ _6"> </span>400044,<span class="_ _6"> </span>China</div><div class="t m0 xe hc y1c ff6 fs7 fc2 sc0 ls1 ws0">article<span class="_ _7"> </span>info</div><div class="t m4 xe hb y1d ff5 fs6 fc2 sc0 ls0 ws0">Article<span class="_ _6"> </span>history:</div><div class="t m4 xe hd y1e ff3 fs6 fc2 sc0 ls0 ws0">Received<span class="_ _6"> </span>11<span class="_ _6"> </span>August<span class="_ _6"> </span>2009</div><div class="t m4 xe hd y1f ff3 fs6 fc2 sc0 ls0 ws0">Received<span class="_ _6"> </span>in<span class="_ _6"> </span>revised<span class="_ _6"> </span>form<span class="_ _6"> </span>13<span class="_ _6"> </span>October<span class="_ _6"> </span>2009</div><div class="t m4 xe hd y20 ff3 fs6 fc2 sc0 ls0 ws0">Accepted<span class="_ _6"> </span>18<span class="_ _6"> </span>March<span class="_ _6"> </span>2010</div><div class="t m4 xe hd y21 ff3 fs6 fc2 sc0 ls0 ws0">Available<span class="_ _6"> </span>online<span class="_ _6"> </span>8<span class="_ _6"> </span>April<span class="_ _6"> </span>2010</div><div class="t m4 xe hb y22 ff5 fs6 fc2 sc0 ls0 ws0">Keywords:</div><div class="t m4 xe hd y23 ff3 fs6 fc2 sc0 ls0 ws0">PFA</div><div class="t m4 xe hd y24 ff3 fs6 fc2 sc0 ls0 ws0">Chemical<span class="_ _6"> </span>kinetic<span class="_ _6"> </span>mechanism</div><div class="t m4 xe hd y25 ff3 fs6 fc2 sc0 ls0 ws0">Reduced<span class="_ _6"> </span>chemistry</div><div class="t m4 xe hd y26 ff3 fs6 fc2 sc0 ls0 ws0">n-decane</div><div class="t m4 xe hd y27 ff3 fs6 fc2 sc0 ls0 ws0">n-heptane</div><div class="t m0 x18 hc y28 ff6 fs7 fc2 sc0 ls2 ws0">abstract</div><div class="t m5 x18 he y29 ff3 fs8 fc2 sc0 ls0 ws0">A<span class="_ _8"> </span>direct<span class="_ _8"> </span>path<span class="_ _8"> </span>ﬂux<span class="_ _8"> </span>analysis<span class="_ _8"> </span>(PFA)<span class="_ _8"> </span>method<span class="_ _8"> </span>for<span class="_ _8"> </span>kinetic<span class="_ _8"> </span>mechanism<span class="_ _8"> </span>reduction<span class="_ _8"> </span>is<span class="_ _8"> </span>proposed<span class="_ _8"> </span>and<span class="_ _9"> </span>validated<span class="_ _8"> </span>by</div><div class="t m5 x18 he y2a ff3 fs8 fc2 sc0 ls0 ws0">using<span class="_ _9"> </span>high<span class="_ _0"> </span>temperature<span class="_ _9"> </span>ignition,<span class="_ _9"> </span>perfect<span class="_ _0"> </span>stirred<span class="_ _9"> </span>reactors,<span class="_ _0"> </span>and<span class="_ _9"> </span>steady<span class="_ _9"> </span>and<span class="_ _0"> </span>unsteady<span class="_ _9"> </span>ﬂame<span class="_ _9"> </span>propagations</div><div class="t m5 x18 he y2b ff3 fs8 fc2 sc0 ls0 ws0">of<span class="_ _6"> </span>n-heptane<span class="_ _a"> </span>and<span class="_ _a"> </span>n-decane/air<span class="_ _6"> </span>mixtures.<span class="_ _a"> </span>The<span class="_ _6"> </span>formation<span class="_ _a"> </span>and<span class="_ _6"> </span>consumption<span class="_ _a"> </span>ﬂuxes<span class="_ _6"> </span>of<span class="_ _a"> </span>each<span class="_ _6"> </span>species<span class="_ _a"> </span>at<span class="_ _6"> </span>multi-</div><div class="t m5 x18 he y2c ff3 fs8 fc2 sc0 ls0 ws0">ple<span class="_ _a"> </span>reaction<span class="_ _a"> </span>path<span class="_ _6"> </span>generations<span class="_ _6"> </span>are<span class="_ _a"> </span>analyzed<span class="_ _6"> </span>and<span class="_ _a"> </span>used<span class="_ _a"> </span>to<span class="_ _6"> </span>identify<span class="_ _a"> </span>the<span class="_ _6"> </span>important<span class="_ _a"> </span>reaction<span class="_ _a"> </span>pathways<span class="_ _6"> </span>and<span class="_ _a"> </span>the</div><div class="t m5 x18 he y2d ff3 fs8 fc2 sc0 ls0 ws0">associated<span class="_ _a"> </span>species.<span class="_ _6"> </span>The<span class="_ _a"> </span>formation<span class="_ _a"> </span>and<span class="_ _6"> </span>consumption<span class="_ _a"> </span>path<span class="_ _a"> </span>ﬂuxes<span class="_ _6"> </span>used<span class="_ _a"> </span>in<span class="_ _a"> </span>this<span class="_ _6"> </span>method<span class="_ _a"> </span>retain<span class="_ _a"> </span>ﬂux<span class="_ _6"> </span>conserva-</div><div class="t m5 x18 he y2e ff3 fs8 fc2 sc0 ls0 ws0">tion<span class="_ _6"> </span>information<span class="_ _6"> </span>and<span class="_ _6"> </span>are<span class="_ _a"> </span>used<span class="_ _6"> </span>to<span class="_ _6"> </span>deﬁne<span class="_ _6"> </span>the<span class="_ _6"> </span>path<span class="_ _6"> </span>indexes<span class="_ _6"> </span>for<span class="_ _6"> </span>the<span class="_ _6"> </span>ﬁrst<span class="_ _6"> </span>and<span class="_ _6"> </span>the<span class="_ _a"> </span>second<span class="_ _6"> </span>generation<span class="_ _6"> </span>reaction</div><div class="t m5 x18 he y2f ff3 fs8 fc2 sc0 ls0 ws0">paths<span class="_ _8"> </span>related<span class="_ _9"> </span>to<span class="_ _8"> </span>a<span class="_ _9"> </span>targeted<span class="_ _8"> </span>species.<span class="_ _9"> </span>Based<span class="_ _9"> </span>on<span class="_ _8"> </span>the<span class="_ _9"> </span>indexes<span class="_ _8"> </span>of<span class="_ _9"> </span>each<span class="_ _9"> </span>reaction<span class="_ _8"> </span>path<span class="_ _9"> </span>for<span class="_ _8"> </span>the<span class="_ _9"> </span>ﬁrst<span class="_ _8"> </span>and<span class="_ _9"> </span>second</div><div class="t m5 x18 he y30 ff3 fs8 fc2 sc0 ls0 ws0">generations,<span class="_ _0"> </span>different<span class="_ _0"> </span>sized<span class="_ _9"> </span>reduced<span class="_ _0"> </span>chemical<span class="_ _0"> </span>mechanisms<span class="_ _0"> </span>which<span class="_ _0"> </span>contain<span class="_ _9"> </span>different<span class="_ _0"> </span>number<span class="_ _0"> </span>of<span class="_ _0"> </span>species</div><div class="t m5 x18 he y31 ff3 fs8 fc2 sc0 ls0 ws0">are<span class="_ _3"> </span>generated.<span class="_ _5"> </span>The<span class="_ _3"> </span>reduced<span class="_ _3"> </span>mechanisms<span class="_ _5"> </span>of<span class="_ _3"> </span>n-heptane<span class="_ _3"> </span>and<span class="_ _5"> </span>n-decane<span class="_ _3"> </span>obtained<span class="_ _5"> </span>by<span class="_ _3"> </span>using<span class="_ _3"> </span>the<span class="_ _5"> </span>present</div><div class="t m5 x18 he y32 ff3 fs8 fc2 sc0 ls0 ws0">method<span class="_ _8"> </span>are<span class="_ _8"> </span>compared<span class="_ _6"> </span>to<span class="_ _8"> </span>those<span class="_ _8"> </span>generated<span class="_ _8"> </span>by<span class="_ _8"> </span>the<span class="_ _8"> </span>direct<span class="_ _8"> </span>relation<span class="_ _8"> </span>graph<span class="_ _8"> </span>(DRG)<span class="_ _6"> </span>method.<span class="_ _8"> </span>The<span class="_ _8"> </span>reaction<span class="_ _8"> </span>path</div><div class="t m5 x18 he y33 ff3 fs8 fc2 sc0 ls0 ws0">analysis<span class="_ _a"> </span>for<span class="_ _a"> </span>n-decane<span class="_ _a"> </span>is<span class="_ _b"> </span>conducted<span class="_ _a"> </span>to<span class="_ _b"> </span>demonstrate<span class="_ _a"> </span>the<span class="_ _a"> </span>validity<span class="_ _a"> </span>of<span class="_ _b"> </span>the<span class="_ _a"> </span>present<span class="_ _a"> </span>method.<span class="_ _a"> </span>The<span class="_ _b"> </span>comparisons<span class="_ _a"> </span>of</div><div class="t m5 x18 he y34 ff3 fs8 fc2 sc0 ls0 ws0">the<span class="_ _a"> </span>ignition<span class="_ _6"> </span>delay<span class="_ _6"> </span>times,<span class="_ _6"> </span>ﬂame<span class="_ _6"> </span>propagation<span class="_ _a"> </span>speeds,<span class="_ _6"> </span>ﬂame<span class="_ _6"> </span>structures,<span class="_ _6"> </span>and<span class="_ _6"> </span>unsteady<span class="_ _a"> </span>spherical<span class="_ _6"> </span>ﬂame<span class="_ _6"> </span>prop-</div><div class="t m5 x18 he y35 ff3 fs8 fc2 sc0 ls0 ws0">agation<span class="_ _6"> </span>processes<span class="_ _8"> </span>showed<span class="_ _6"> </span>that<span class="_ _6"> </span>with<span class="_ _8"> </span>either<span class="_ _6"> </span>the<span class="_ _6"> </span>same<span class="_ _8"> </span>or<span class="_ _6"> </span>signiﬁcantly<span class="_ _6"> </span>less<span class="_ _8"> </span>number<span class="_ _6"> </span>of<span class="_ _6"> </span>species,<span class="_ _8"> </span>the<span class="_ _6"> </span>reduced</div><div class="t m5 x18 he y36 ff3 fs8 fc2 sc0 ls0 ws0">mechanisms<span class="_ _6"> </span>generated<span class="_ _6"> </span>by<span class="_ _a"> </span>the<span class="_ _6"> </span>present<span class="_ _6"> </span>PFA<span class="_ _6"> </span>are<span class="_ _6"> </span>more<span class="_ _6"> </span>accurate<span class="_ _a"> </span>than<span class="_ _6"> </span>that<span class="_ _6"> </span>of<span class="_ _6"> </span>DRG<span class="_ _6"> </span>in<span class="_ _6"> </span>a<span class="_ _a"> </span>broad<span class="_ _6"> </span>range<span class="_ _6"> </span>of<span class="_ _6"> </span>initial</div><div class="t m5 x18 he y37 ff3 fs8 fc2 sc0 ls0 ws0">pressures<span class="_ _9"> </span>and<span class="_ _0"> </span>temperatures.<span class="_ _9"> </span>The<span class="_ _9"> </span>method<span class="_ _0"> </span>is<span class="_ _8"> </span>also<span class="_ _0"> </span>integrated<span class="_ _9"> </span>with<span class="_ _9"> </span>the<span class="_ _0"> </span>dynamic<span class="_ _9"> </span>multi-timescale<span class="_ _9"> </span>method</div><div class="t m5 x18 he y38 ff3 fs8 fc2 sc0 ls0 ws0">and<span class="_ _8"> </span>a<span class="_ _8"> </span>further<span class="_ _8"> </span>increase<span class="_ _8"> </span>of<span class="_ _8"> </span>computation<span class="_ _6"> </span>efﬁcien<span class="_ _c"></span>cy<span class="_ _6"> </span>is<span class="_ _8"> </span>achieved.</div><div class="t m5 x19 he y39 ff7 fs8 fc2 sc0 ls0 ws0">Ó<span class="_ _8"> </span><span class="ff3">2010<span class="_ _6"> </span>The<span class="_ _8"> </span>Combustion<span class="_ _8"> </span>Institute.<span class="_ _8"> </span>Published<span class="_ _8"> </span>by<span class="_ _8"> </span>Elsevier<span class="_ _8"> </span>Inc.<span class="_ _8"> </span>All<span class="_ _8"> </span>rights<span class="_ _8"> </span>reserved.</span></div><div class="t m5 xe hf y3a ff8 fs9 fc2 sc0 ls0 ws0">1.<span class="_ _8"> </span>Introduction</div><div class="t m5 x1a h10 y3b ff3 fs9 fc2 sc0 ls0 ws0">Numerical<span class="_ _5"> </span>simulation<span class="_ _5"> </span>plays<span class="_ _3"> </span>an<span class="_ _5"> </span>increasingly<span class="_ _5"> </span>important<span class="_ _5"> </span>role<span class="_ _3"> </span>in</div><div class="t m5 xe h10 y3c ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _6"> </span>study<span class="_ _8"> </span>of<span class="_ _8"> </span>combustion<span class="_ _6"> </span>as<span class="_ _8"> </span>a<span class="_ _8"> </span>complementary<span class="_ _8"> </span>or<span class="_ _6"> </span>even<span class="_ _8"> </span>replacement</div><div class="t m5 xe h10 y3d ff3 fs9 fc2 sc0 ls0 ws0">tool<span class="_ _3"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>traditional,<span class="_ _3"> </span>high-cost<span class="_ _3"> </span>experiments<span class="_ _3"> </span>for<span class="_ _3"> </span>both<span class="_ _3"> </span>scientiﬁc</div><div class="t m5 xe h10 y3e ff3 fs9 fc2 sc0 ls0 ws0">discoveries<span class="_ _a"> </span>and<span class="_ _a"> </span>engineering<span class="_ _a"> </span>designs.<span class="_ _a"> </span>However,<span class="_ _a"> </span>combustion<span class="_ _b"> </span>of<span class="_ _a"> </span>fossil</div><div class="t m5 xe h10 y3f ff3 fs9 fc2 sc0 ls0 ws0">fuels<span class="_ _8"> </span>involves<span class="_ _8"> </span>hundreds<span class="_ _9"> </span>of<span class="_ _8"> </span>species<span class="_ _8"> </span>and<span class="_ _9"> </span>thousands<span class="_ _8"> </span>of<span class="_ _9"> </span>reactions<span class="_ _8"> </span>and</div><div class="t m5 xe h10 y40 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _8"> </span>employment<span class="_ _6"> </span>of<span class="_ _8"> </span>large<span class="_ _9"> </span>detailed<span class="_ _8"> </span>mechanisms<span class="_ _6"> </span>in<span class="_ _8"> </span>numerical<span class="_ _8"> </span>simu-</div><div class="t m5 xe h10 y41 ff3 fs9 fc2 sc0 ls0 ws0">lation<span class="_ _8"> </span>demands<span class="_ _9"> </span>huge<span class="_ _9"> </span>amount<span class="_ _9"> </span>of<span class="_ _9"> </span>CPU<span class="_ _9"> </span>time.<span class="_ _9"> </span>Although<span class="_ _9"> </span>the<span class="_ _9"> </span>available</div><div class="t m5 xe h10 y42 ff3 fs9 fc2 sc0 ls0 ws0">computational<span class="_ _8"> </span>power<span class="_ _8"> </span>is<span class="_ _8"> </span>growing<span class="_ _8"> </span>rapidly,<span class="_ _8"> </span>a<span class="_ _9"> </span>direct<span class="_ _8"> </span>numerical<span class="_ _8"> </span>simu-</div><div class="t m5 xe h10 y43 ff3 fs9 fc2 sc0 ls0 ws0">lation<span class="_ _7"> </span>of<span class="_ _d"> </span>turbulent<span class="_ _d"> </span>combustion<span class="_ _7"> </span>of<span class="_ _d"> </span>hydrocarbon<span class="_ _7"> </span>fuels<span class="_ _d"> </span>with<span class="_ _7"> </span>a</div><div class="t m5 xe h10 y44 ff3 fs9 fc2 sc0 ls0 ws0">detailed<span class="_ _a"> </span>chemical<span class="_ _a"> </span>kinetic<span class="_ _a"> </span>mechanism<span class="_ _b"> </span>at<span class="_ _6"> </span>practical<span class="_ _a"> </span>engine<span class="_ _a"> </span>conditions</div><div class="t m5 xe h10 y45 ff3 fs9 fc2 sc0 ls0 ws0">remains<span class="_ _3"> </span>challenging.<span class="_ _3"> </span>Therefore,<span class="_ _5"> </span>in<span class="_ _3"> </span>order<span class="_ _3"> </span>to<span class="_ _3"> </span>make<span class="_ _3"> </span>the<span class="_ _3"> </span>numerical</div><div class="t m5 xe h10 y46 ff3 fs9 fc2 sc0 ls0 ws0">simulation<span class="_ _0"> </span>of<span class="_ _5"> </span>reactive<span class="_ _0"> </span>ﬂow<span class="_ _5"> </span>computationally<span class="_ _0"> </span>affordable<span class="_ _5"> </span>and<span class="_ _0"> </span>com-</div><div class="t m5 xe h10 y47 ff3 fs9 fc2 sc0 ls0 ws0">prehensively<span class="_ _e"> </span>accurate,<span class="_ _e"> </span>the<span class="_ _e"> </span>development<span class="_ _e"> </span>of<span class="_ _e"> </span>computational</div><div class="t m5 xe h10 y48 ff3 fs9 fc2 sc0 ls0 ws0">approaches<span class="_ _d"> </span>for<span class="_ _d"> </span>rigorous<span class="_ _d"> </span>reduction<span class="_ _f"> </span>of<span class="_ _f"> </span>detailed<span class="_ _d"> </span>mechanisms<span class="_ _d"> </span>is</div><div class="t m5 xe h10 y49 ff3 fs9 fc2 sc0 ls0 ws0">essential.</div><div class="t m5 x1a h10 y4a ff3 fs9 fc2 sc0 ls0 ws0">Various<span class="_ _9"> </span>mechanism<span class="_ _9"> </span>reduction<span class="_ _9"> </span>methods<span class="_ _9"> </span>have<span class="_ _9"> </span>been<span class="_ _9"> </span>proposed<span class="_ _9"> </span>to</div><div class="t m5 xe h10 y4b ff3 fs9 fc2 sc0 ls0 ws0">generate<span class="_ _0"> </span>a<span class="_ _5"> </span>reduced<span class="_ _5"> </span>kinetic<span class="_ _5"> </span>mechanism.<span class="_ _0"> </span>The<span class="_ _5"> </span>ﬁrst<span class="_ _5"> </span>approach<span class="_ _0"> </span>is<span class="_ _5"> </span>the</div><div class="t m5 x1b h10 y4c ff3 fs9 fc2 sc0 ls0 ws0">sensitivity<span class="_ _0"> </span>and<span class="_ _0"> </span>rate<span class="_ _0"> </span>analyses<span class="_ _0"> </span><span class="fc3">[1–3]</span>.<span class="_ _0"> </span>Although<span class="_ _0"> </span>this<span class="_ _0"> </span>method<span class="_ _0"> </span>is<span class="_ _0"> </span>very</div><div class="t m5 x1b h10 y4d ff3 fs9 fc2 sc0 ls0 ws0">effective,<span class="_ _5"> </span>it<span class="_ _3"> </span>provides<span class="_ _5"> </span>neither<span class="_ _3"> </span>the<span class="_ _5"> </span>timescales<span class="_ _3"> </span>of<span class="_ _3"> </span>different<span class="_ _5"> </span>reaction</div><div class="t m5 x1b h10 y4e ff3 fs9 fc2 sc0 ls0 ws0">groups<span class="_ _8"> </span>nor<span class="_ _9"> </span>the<span class="_ _9"> </span>possible<span class="_ _9"> </span>quasi-steady<span class="_ _9"> </span>state<span class="_ _9"> </span>(QSS)<span class="_ _9"> </span>and<span class="_ _9"> </span>partial<span class="_ _9"> </span>equi-</div><div class="t m5 x1b h10 y4f ff3 fs9 fc2 sc0 ls0 ws0">librium<span class="_ _f"> </span>groups<span class="_ _7"> </span>without<span class="_ _f"> </span>good<span class="_ _d"> </span>human<span class="_ _d"> </span>experience.<span class="_ _d"> </span>The<span class="_ _d"> </span>second</div><div class="t m5 x1b h10 y50 ff3 fs9 fc2 sc0 ls0 ws0">approach<span class="_ _a"> </span>is<span class="_ _6"> </span>the<span class="_ _6"> </span>reaction<span class="_ _a"> </span>Jacobian<span class="_ _6"> </span>analysis<span class="_ _6"> </span>which<span class="_ _a"> </span>includes<span class="_ _6"> </span>the<span class="_ _6"> </span>com-</div><div class="t m5 x1b h10 y51 ff3 fs9 fc2 sc0 ls0 ws0">putational<span class="_ _f"> </span>singular<span class="_ _d"> </span>perturbation<span class="_ _f"> </span>(CSP)<span class="_ _f"> </span>method<span class="_ _d"> </span><span class="fc3">[4–6]<span class="_ _d"> </span></span>and<span class="_ _f"> </span>the</div><div class="t m5 x1b h10 y52 ff3 fs9 fc2 sc0 ls0 ws0">intrinsic<span class="_ _5"> </span>low<span class="_ _3"> </span>dimensional<span class="_ _3"> </span>manifold<span class="_ _5"> </span>(ILDM)<span class="_ _3"> </span><span class="fc3">[7]</span>.<span class="_ _5"> </span>The<span class="_ _3"> </span>CSP<span class="_ _3"> </span>method</div><div class="t m5 x1b h10 y53 ff3 fs9 fc2 sc0 ls0 ws0">obtains<span class="_ _3"> </span>eigen-values<span class="_ _10"> </span>and<span class="_ _10"> </span>eigen-vectors<span class="_ _3"> </span>of<span class="_ _10"> </span>independent<span class="_ _3"> </span>reaction</div><div class="t m5 x1b h10 y54 ff3 fs9 fc2 sc0 ls0 ws0">modes,<span class="_ _0"> </span>so<span class="_ _5"> </span>that<span class="_ _5"> </span>the<span class="_ _0"> </span>fast<span class="_ _5"> </span>and<span class="_ _5"> </span>slow<span class="_ _0"> </span>modes<span class="_ _5"> </span>can<span class="_ _5"> </span>be<span class="_ _0"> </span>ranked<span class="_ _5"> </span>by<span class="_ _5"> </span>using</div><div class="t m5 x1b h10 y55 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _8"> </span>eigen-values.<span class="_ _8"> </span>In<span class="_ _8"> </span>the<span class="_ _9"> </span>ILDM<span class="_ _8"> </span>method,<span class="_ _9"> </span>the<span class="_ _8"> </span>low<span class="_ _8"> </span>dimensional<span class="_ _8"> </span>mani-</div><div class="t m5 x1b h10 y56 ff3 fs9 fc2 sc0 ls0 ws0">fold<span class="_ _4"> </span>is<span class="_ _11"> </span>identiﬁed<span class="_ _4"> </span>based<span class="_ _11"> </span>on<span class="_ _4"> </span>the<span class="_ _11"> </span>decomposition<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _11"> </span>Jacobian</div><div class="t m5 x1b h10 y57 ff3 fs9 fc2 sc0 ls0 ws0">matrix<span class="_ _a"> </span>of<span class="_ _a"> </span>the<span class="_ _6"> </span>chemical<span class="_ _a"> </span>source<span class="_ _6"> </span>term<span class="_ _a"> </span>with<span class="_ _6"> </span>the<span class="_ _a"> </span>constraints<span class="_ _a"> </span>of<span class="_ _6"> </span>element</div><div class="t m5 x1b h10 y58 ff3 fs9 fc2 sc0 ls0 ws0">and<span class="_ _8"> </span>enthalpy<span class="_ _8"> </span>conservations.<span class="_ _8"> </span>Although<span class="_ _8"> </span>these<span class="_ _9"> </span>approaches<span class="_ _8"> </span>can<span class="_ _8"> </span>effec-</div><div class="t m5 x1b h10 y59 ff3 fs9 fc2 sc0 ls0 ws0">tively<span class="_ _9"> </span>identify<span class="_ _0"> </span>the<span class="_ _9"> </span>timescales<span class="_ _9"> </span>of<span class="_ _0"> </span>different<span class="_ _9"> </span>reaction<span class="_ _0"> </span>groups<span class="_ _9"> </span>and<span class="_ _0"> </span>the</div><div class="t m5 x1b h10 y5a ff3 fs9 fc2 sc0 ls0 ws0">QSS<span class="_ _5"> </span>species,<span class="_ _5"> </span>it<span class="_ _3"> </span>requires<span class="_ _5"> </span>signiﬁcant<span class="_ _5"> </span>computation<span class="_ _5"> </span>time<span class="_ _3"> </span>to<span class="_ _5"> </span>conduct</div><div class="t m5 x1b h10 y5b ff3 fs9 fc2 sc0 ls0 ws0">Jacobian<span class="_ _0"> </span>decomposition<span class="_ _5"> </span>and<span class="_ _0"> </span>mode<span class="_ _5"> </span>projection<span class="_ _0"> </span>and<span class="_ _5"> </span>is<span class="_ _5"> </span>signiﬁcantly</div><div class="t m5 x1b h10 y5c ff3 fs9 fc2 sc0 ls0 ws0">more<span class="_ _9"> </span>complicated<span class="_ _0"> </span>to<span class="_ _0"> </span>implement.<span class="_ _9"> </span>To<span class="_ _0"> </span>achieve<span class="_ _9"> </span>efﬁcient<span class="_ _0"> </span>calculations</div><div class="t m5 x1b h10 y5d ff3 fs9 fc2 sc0 ls0 ws0">of<span class="_ _6"> </span>fast<span class="_ _6"> </span>mode<span class="_ _6"> </span>species,<span class="_ _6"> </span>a<span class="_ _6"> </span>third<span class="_ _6"> </span>approach,<span class="_ _8"> </span>the<span class="_ _6"> </span>parameterization<span class="_ _6"> </span>meth-</div><div class="t m5 x1b h10 y5e ff3 fs9 fc2 sc0 ls0 ws0">od,<span class="_ _8"> </span>was<span class="_ _8"> </span>proposed<span class="_ _9"> </span><span class="fc3">[8–11]</span>.<span class="_ _8"> </span>One<span class="_ _8"> </span>example<span class="_ _8"> </span>is<span class="_ _9"> </span>the<span class="_ _8"> </span>in<span class="_ _9"> </span>situ<span class="_ _8"> </span>adaptive<span class="_ _8"> </span>tab-</div><div class="t m5 x1b h10 y5f ff3 fs9 fc2 sc0 ls0 ws0">ulation<span class="_ _11"> </span>(ISAT)<span class="_ _11"> </span><span class="fc3">[8]</span>.<span class="_ _11"> </span>The<span class="_ _f"> </span>basic<span class="_ _11"> </span>idea<span class="_ _11"> </span>of<span class="_ _f"> </span>ISAT<span class="_ _11"> </span>is<span class="_ _f"> </span>to<span class="_ _11"> </span>integrate<span class="_ _11"> </span>the</div><div class="t m5 x1b h10 y60 ff3 fs9 fc2 sc0 ls0 ws0">chemical<span class="_ _0"> </span>source<span class="_ _5"> </span>term<span class="_ _0"> </span>and<span class="_ _5"> </span>store<span class="_ _5"> </span>the<span class="_ _5"> </span>information<span class="_ _0"> </span>in<span class="_ _5"> </span>a<span class="_ _0"> </span>binary<span class="_ _5"> </span>tree</div><div class="t m5 x1b h10 y61 ff3 fs9 fc2 sc0 ls0 ws0">data<span class="_ _0"> </span>structure<span class="_ _0"> </span>as<span class="_ _5"> </span>the<span class="_ _0"> </span>simulation<span class="_ _0"> </span>is<span class="_ _5"> </span>being<span class="_ _0"> </span>performed<span class="_ _5"> </span>in<span class="_ _0"> </span>situ.<span class="_ _5"> </span>ISAT</div><div class="t m4 xe hd y62 ff3 fs6 fc2 sc0 ls0 ws0">0010-2180/$<span class="_ _6"> </span>-<span class="_ _6"> </span>see<span class="_ _6"> </span>front<span class="_ _6"> </span>matter<span class="_ _6"> </span><span class="ff7">Ó<span class="_ _6"> </span></span>2010<span class="_ _6"> </span>The<span class="_ _6"> </span>Combustion<span class="_ _6"> </span>Institute.<span class="_ _6"> </span>Published<span class="_ _6"> </span>by<span class="_ _6"> </span>Elsevier<span class="_ _6"> </span>Inc.<span class="_ _6"> </span>All<span class="_ _6"> </span>rights<span class="_ _6"> </span>reserved.</div><div class="t m4 xe hd y63 ff3 fs6 fc2 sc0 ls0 ws0">doi:<span class="fc3">10.1016/j.combustﬂame.2010.03.006</span></div><div class="t m4 x1c hd y64 ff4 fs6 fc2 sc0 ls0 ws0">*<span class="_ _5"> </span><span class="ff3">Corresponding<span class="_ _a"> </span>author.<span class="_ _6"> </span>Fax:<span class="_ _8"> </span>+1<span class="_ _a"> </span>609<span class="_ _6"> </span>258<span class="_ _6"> </span>6233.</span></div><div class="t m4 x1a hd y65 ff5 fs6 fc2 sc0 ls0 ws0">E-mail<span class="_ _6"> </span>address:<span class="_ _6"> </span><span class="ff3 fc3">yju@princeton.edu<span class="_ _6"> </span><span class="fc2">(Y.<span class="_ _6"> </span>Ju).</span></span></div><div class="t m4 x1d hd y66 ff3 fs6 fc2 sc0 ls0 ws0">Combustion<span class="_ _6"> </span>and<span class="_ _a"> </span>Flame<span class="_ _8"> </span>157<span class="_ _6"> </span>(2010)<span class="_ _a"> </span>1298–1307</div><div class="t m0 x1e h11 y67 ff9 fs9 fc2 sc0 ls0 ws0">Contents<span class="_ _8"> </span>lists<span class="_ _0"> </span>available<span class="_ _8"> </span>at<span class="_ _9"> </span><span class="fc3">ScienceDirect</span></div><div class="t m6 x1f h12 y68 ff3 fsa fc2 sc0 ls0 ws0">Combustion<span class="_ _4"> </span>and<span class="_ _4"> </span>Flame</div><div class="t m5 x20 h11 y69 ff9 fs9 fc2 sc0 ls3 ws0">journal<span class="_ _9"> </span>homepage:<span class="_ _8"> </span>www.elsevi<span class="_ _c"></span><span class="ls4">er.com/locate/combustﬂame</span></div></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/628a94aa16e0ca71412d396b/bg3.jpg"><div class="c x0 ye w2 h3"><div class="t m0 xd h4 yf ff2 fs1 fc1 sc0 ls0 ws0">Author's personal copy</div></div><div class="c x0 y10 w0 h5"><div class="t m5 x21 h10 y6a ff3 fs9 fc2 sc0 ls0 ws0">is<span class="_ _8"> </span>efﬁcient<span class="_ _9"> </span>if<span class="_ _8"> </span>the<span class="_ _9"> </span>ﬂame<span class="_ _8"> </span>structures<span class="_ _9"> </span>are<span class="_ _9"> </span>close<span class="_ _8"> </span>to<span class="_ _9"> </span>those<span class="_ _8"> </span>created<span class="_ _9"> </span>in<span class="_ _8"> </span>the</div><div class="t m5 x21 h10 y6b ff3 fs9 fc2 sc0 ls0 ws0">tabulated<span class="_ _3"> </span>database.<span class="_ _10"> </span>However,<span class="_ _3"> </span>for<span class="_ _3"> </span>non-premixed<span class="_ _10"> </span>turbulent<span class="_ _3"> </span>com-</div><div class="t m5 x21 h10 y6c ff3 fs9 fc2 sc0 ls0 ws0">bustion<span class="_ _6"> </span>involving<span class="_ _8"> </span>a<span class="_ _8"> </span>large<span class="_ _6"> </span>kinetic<span class="_ _8"> </span>mechanism<span class="_ _8"> </span>with<span class="_ _6"> </span>a<span class="_ _8"> </span>broad<span class="_ _8"> </span>temper-</div><div class="t m5 x21 h10 y6d ff3 fs9 fc2 sc0 ls0 ws0">ature<span class="_ _9"> </span>and<span class="_ _9"> </span>concentration<span class="_ _9"> </span>gradient<span class="_ _9"> </span>distributions,<span class="_ _9"> </span>time<span class="_ _9"> </span>consumption</div><div class="t m5 x21 h10 y6e ff3 fs9 fc2 sc0 ls0 ws0">in<span class="_ _0"> </span>table<span class="_ _0"> </span>buildup<span class="_ _0"> </span>and<span class="_ _0"> </span>the<span class="_ _0"> </span>difﬁculties<span class="_ _0"> </span>in<span class="_ _0"> </span>data<span class="_ _0"> </span>retrieval<span class="_ _0"> </span>from<span class="_ _0"> </span>a<span class="_ _0"> </span>high</div><div class="t m5 x21 h10 y6f ff3 fs9 fc2 sc0 ls0 ws0">dimensional<span class="_ _3"> </span>table<span class="_ _3"> </span>lookup<span class="_ _3"> </span>lead<span class="_ _10"> </span>to<span class="_ _3"> </span>reduced<span class="_ _3"> </span>advantage<span class="_ _3"> </span>of<span class="_ _10"> </span>ISAT<span class="_ _3"> </span>in</div><div class="t m5 x21 h10 y70 ff3 fs9 fc2 sc0 ls0 ws0">comparison<span class="_ _10"> </span>to<span class="_ _3"> </span>direct<span class="_ _10"> </span>integration<span class="_ _3"> </span><span class="fc3">[12]</span>.<span class="_ _10"> </span>The<span class="_ _10"> </span>fourth<span class="_ _3"> </span>method<span class="_ _10"> </span>is<span class="_ _10"> </span>to</div><div class="t m5 x21 h10 y71 ff3 fs9 fc2 sc0 ls0 ws0">use<span class="_ _5"> </span>the<span class="_ _3"> </span>reaction<span class="_ _5"> </span>rate<span class="_ _5"> </span>or<span class="_ _3"> </span>reaction<span class="_ _5"> </span>path<span class="_ _3"> </span>relations<span class="_ _5"> </span>as<span class="_ _5"> </span>a<span class="_ _3"> </span>measure<span class="_ _5"> </span>of</div><div class="t m5 x21 h10 y72 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _5"> </span>degree<span class="_ _3"> </span>of<span class="_ _5"> </span>interaction<span class="_ _5"> </span>among<span class="_ _3"> </span>species.<span class="_ _5"> </span>The<span class="_ _5"> </span>advantages<span class="_ _3"> </span>of<span class="_ _5"> </span>this</div><div class="t m5 x21 h10 y73 ff3 fs9 fc2 sc0 ls0 ws0">kind<span class="_ _8"> </span>of<span class="_ _9"> </span>method<span class="_ _9"> </span>are<span class="_ _9"> </span>simple<span class="_ _9"> </span>and<span class="_ _9"> </span>fast,<span class="_ _9"> </span>requiring<span class="_ _9"> </span>minimal<span class="_ _9"> </span>user<span class="_ _8"> </span>inter-</div><div class="t m5 x21 h10 y74 ff3 fs9 fc2 sc0 ls0 ws0">action<span class="_ _9"> </span>with<span class="_ _9"> </span>the<span class="_ _9"> </span>reduction<span class="_ _9"> </span>process.<span class="_ _9"> </span>Bendtsen<span class="_ _0"> </span>et<span class="_ _8"> </span>al.<span class="_ _0"> </span><span class="fc3">[13]<span class="_ _8"> </span></span>adopts<span class="_ _0"> </span>the</div><div class="t m5 x21 h10 y75 ff3 fs9 fc2 sc0 ls0 ws0">reaction<span class="_ _3"> </span>matrix<span class="_ _5"> </span>with<span class="_ _3"> </span>each<span class="_ _3"> </span>of<span class="_ _3"> </span>whose<span class="_ _5"> </span>elements<span class="_ _3"> </span><span class="ff5">P</span></div><div class="t m7 x22 h13 y76 ff5 fsb fc2 sc0 ls0 ws0">ij</div><div class="t m5 x23 h10 y11 ff3 fs9 fc2 sc0 ls0 ws0">deﬁned<span class="_ _3"> </span>as<span class="_ _5"> </span>the</div><div class="t m5 x21 h10 y77 ff3 fs9 fc2 sc0 ls0 ws0">net<span class="_ _5"> </span>production<span class="_ _5"> </span>rate<span class="_ _5"> </span>of<span class="_ _5"> </span>species<span class="_ _3"> </span><span class="ff5">i<span class="_ _5"> </span></span>from<span class="_ _5"> </span>all<span class="_ _5"> </span>reactions<span class="_ _5"> </span>involving<span class="_ _5"> </span><span class="ff5">j<span class="_ _3"> </span></span>to</div><div class="t m5 x21 h10 y78 ff3 fs9 fc2 sc0 ls0 ws0">establish<span class="_ _0"> </span>a<span class="_ _0"> </span>reduced<span class="_ _9"> </span>reaction<span class="_ _0"> </span>pathway<span class="_ _0"> </span>(or<span class="_ _0"> </span>mechanism).<span class="_ _0"> </span>The<span class="_ _0"> </span>set<span class="_ _0"> </span>of</div><div class="t m5 x21 h10 y79 ff3 fs9 fc2 sc0 ls0 ws0">important<span class="_ _6"> </span>species<span class="_ _6"> </span>is<span class="_ _8"> </span>selected<span class="_ _6"> </span>by<span class="_ _8"> </span>going<span class="_ _6"> </span>through<span class="_ _8"> </span>the<span class="_ _6"> </span>reaction<span class="_ _6"> </span>matrix</div><div class="t m5 x21 h10 y7a ff3 fs9 fc2 sc0 ls0 ws0">following<span class="_ _0"> </span>the<span class="_ _0"> </span>reaction<span class="_ _0"> </span>path<span class="_ _0"> </span>that<span class="_ _0"> </span>connects<span class="_ _5"> </span>one<span class="_ _0"> </span>species<span class="_ _0"> </span>to<span class="_ _0"> </span>another</div><div class="t m5 x21 h10 y7b ff3 fs9 fc2 sc0 ls0 ws0">that<span class="_ _0"> </span>is<span class="_ _0"> </span>most<span class="_ _0"> </span>strongly<span class="_ _0"> </span>coupled<span class="_ _0"> </span>with<span class="_ _9"> </span>it.<span class="_ _0"> </span>The<span class="_ _0"> </span>reduced<span class="_ _0"> </span>mechanism<span class="_ _0"> </span>is</div><div class="t m5 x21 h10 y7c ff3 fs9 fc2 sc0 ls0 ws0">then<span class="_ _0"> </span>completed<span class="_ _9"> </span>by<span class="_ _0"> </span>including<span class="_ _0"> </span>a<span class="_ _9"> </span>number<span class="_ _0"> </span>of<span class="_ _0"> </span>reactions<span class="_ _9"> </span>such<span class="_ _0"> </span>that<span class="_ _0"> </span>for</div><div class="t m5 x21 h10 y7d ff3 fs9 fc2 sc0 ls0 ws0">each<span class="_ _8"> </span>of<span class="_ _9"> </span>the<span class="_ _9"> </span>selected<span class="_ _9"> </span>species,<span class="_ _9"> </span>a<span class="_ _9"> </span>certain<span class="_ _9"> </span>percentage<span class="_ _8"> </span>of<span class="_ _0"> </span>the<span class="_ _8"> </span>total<span class="_ _9"> </span>pro-</div><div class="t m5 x21 h10 y7e ff3 fs9 fc2 sc0 ls0 ws0">duction<span class="_ _5"> </span>or<span class="_ _5"> </span>consumption<span class="_ _5"> </span>rate<span class="_ _5"> </span>(threshold<span class="_ _3"> </span>value)<span class="_ _0"> </span>of<span class="_ _3"> </span>that<span class="_ _5"> </span>species<span class="_ _5"> </span>is</div><div class="t m5 x21 h10 y7f ff3 fs9 fc2 sc0 ls0 ws0">kept<span class="_ _0"> </span>in<span class="_ _0"> </span>the<span class="_ _0"> </span>reduced<span class="_ _9"> </span>mechanism.<span class="_ _0"> </span>Similar<span class="_ _0"> </span>selection<span class="_ _0"> </span>procedures<span class="_ _0"> </span>are</div><div class="t m5 x21 h10 y80 ff3 fs9 fc2 sc0 ls0 ws0">used<span class="_ _6"> </span>in<span class="_ _6"> </span>directed<span class="_ _6"> </span>relation<span class="_ _6"> </span>graph<span class="_ _8"> </span>(DRG)<span class="_ _6"> </span>method<span class="_ _6"> </span><span class="fc3">[14,15]<span class="_ _6"> </span></span>and<span class="_ _8"> </span>directed</div><div class="t m5 x21 h10 y81 ff3 fs9 fc2 sc0 ls0 ws0">relation<span class="_ _3"> </span>graph<span class="_ _10"> </span>method<span class="_ _3"> </span>with<span class="_ _3"> </span>error<span class="_ _3"> </span>propagation<span class="_ _3"> </span>(DRGEP)<span class="_ _10"> </span><span class="fc3">[16]<span class="_ _3"> </span></span>by</div><div class="t m5 x21 h10 y82 ff3 fs9 fc2 sc0 ls0 ws0">using<span class="_ _0"> </span>the<span class="_ _0"> </span>absolute<span class="_ _0"> </span>and<span class="_ _5"> </span>net<span class="_ _0"> </span>reaction<span class="_ _0"> </span>rates,<span class="_ _0"> </span>respectively.<span class="_ _0"> </span>However,</div><div class="t m5 x21 h10 y83 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _6"> </span>use<span class="_ _8"> </span>of<span class="_ _8"> </span>absolute<span class="_ _8"> </span>reaction<span class="_ _8"> </span>rates<span class="_ _8"> </span>in<span class="_ _6"> </span>DRG<span class="_ _8"> </span>makes<span class="_ _8"> </span>the<span class="_ _8"> </span>relation<span class="_ _8"> </span>index</div><div class="t m5 x21 h10 y84 ff3 fs9 fc2 sc0 ls0 ws0">not<span class="_ _a"> </span>conservative<span class="_ _6"> </span>(the<span class="_ _a"> </span>interaction<span class="_ _6"> </span>coefﬁcient<span class="_ _a"> </span>or<span class="_ _6"> </span>relation<span class="_ _6"> </span>index<span class="_ _a"> </span>is<span class="_ _6"> </span>the</div><div class="t m5 x21 h10 y85 ff3 fs9 fc2 sc0 ls0 ws0">ratio<span class="_ _8"> </span>of<span class="_ _8"> </span>species<span class="_ _9"> </span>ﬂux).<span class="_ _8"> </span>DRGEP<span class="_ _8"> </span><span class="fc3">[16]<span class="_ _9"> </span></span>employed<span class="_ _8"> </span>the<span class="_ _8"> </span>absolute<span class="_ _8"> </span>net<span class="_ _9"> </span>reac-</div><div class="t m5 x21 h10 y86 ff3 fs9 fc2 sc0 ls0 ws0">tion<span class="_ _3"> </span>ﬂux<span class="_ _10"> </span>to<span class="_ _3"> </span>include<span class="_ _3"> </span>error<span class="_ _10"> </span>propagation<span class="_ _3"> </span>across<span class="_ _3"> </span>multi-generations.</div><div class="t m5 x21 h10 y87 ff3 fs9 fc2 sc0 ls0 ws0">However,<span class="_ _8"> </span>for<span class="_ _8"> </span>the<span class="_ _8"> </span>indirect<span class="_ _8"> </span>relations,<span class="_ _9"> </span>DRGEP<span class="_ _8"> </span>only<span class="_ _8"> </span>pick<span class="_ _8"> </span>up<span class="_ _9"> </span>the<span class="_ _8"> </span>stron-</div><div class="t m5 x21 h10 y88 ff3 fs9 fc2 sc0 ls0 ws0">gest<span class="_ _a"> </span>reaction<span class="_ _a"> </span>path<span class="_ _6"> </span>which<span class="_ _a"> </span>cannot<span class="_ _a"> </span>identify<span class="_ _6"> </span>the<span class="_ _a"> </span>species<span class="_ _a"> </span>ﬂux<span class="_ _6"> </span>physically</div><div class="t m5 x21 h10 y89 ff3 fs9 fc2 sc0 ls0 ws0">when<span class="_ _10"> </span>the<span class="_ _4"> </span>intermediate<span class="_ _4"> </span>species<span class="_ _4"> </span>are<span class="_ _4"> </span>more<span class="_ _4"> </span>than<span class="_ _4"> </span>one<span class="_ _4"> </span>in<span class="_ _4"> </span>parallel.</div><div class="t m5 x21 h10 y8a ff3 fs9 fc2 sc0 ls0 ws0">Moreover,<span class="_ _9"> </span>another<span class="_ _0"> </span>problem<span class="_ _8"> </span>of<span class="_ _0"> </span>DRGEP<span class="_ _9"> </span>is<span class="_ _9"> </span>that<span class="_ _0"> </span>the<span class="_ _9"> </span>deﬁnition<span class="_ _9"> </span>of<span class="_ _0"> </span>the</div><div class="t m5 x21 h10 y8b ff3 fs9 fc2 sc0 ls0 ws0">interaction<span class="_ _3"> </span>coefﬁcient<span class="_ _3"> </span>fails<span class="_ _3"> </span>to<span class="_ _10"> </span>identify<span class="_ _3"> </span>the<span class="_ _3"> </span>relation<span class="_ _3"> </span>between<span class="_ _3"> </span>the</div><div class="t m5 x21 h10 y8c ff3 fs9 fc2 sc0 ls0 ws0">species<span class="_ _6"> </span>that<span class="_ _8"> </span>have<span class="_ _6"> </span>both<span class="_ _6"> </span>fast<span class="_ _8"> </span>production<span class="_ _6"> </span>and<span class="_ _6"> </span>consumption<span class="_ _8"> </span>rate,<span class="_ _6"> </span>such</div><div class="t m5 x21 h10 y8d ff3 fs9 fc2 sc0 ls0 ws0">as<span class="_ _6"> </span>species<span class="_ _8"> </span>having<span class="_ _6"> </span>catalytic<span class="_ _6"> </span>effect.<span class="_ _8"> </span>A<span class="_ _6"> </span>typical<span class="_ _8"> </span>example<span class="_ _6"> </span>is<span class="_ _6"> </span>the<span class="_ _8"> </span>NO</div><div class="t m7 x24 h13 y8e ff5 fsb fc2 sc0 ls0 ws0">x</div><div class="t m5 x19 h10 y8f ff3 fs9 fc2 sc0 ls0 ws0">cat-</div><div class="t m5 x21 h10 y90 ff3 fs9 fc2 sc0 ls0 ws0">alytic<span class="_ _0"> </span>effect<span class="_ _0"> </span>on<span class="_ _5"> </span>ignition<span class="_ _0"> </span>enhancement<span class="_ _0"> </span><span class="fc3">[17]</span>,<span class="_ _0"> </span>in<span class="_ _5"> </span>which<span class="_ _0"> </span>NO</div><div class="t m7 x25 h13 y91 ff5 fsb fc2 sc0 ls0 ws0">x</div><div class="t m5 x26 h10 y92 ff3 fs9 fc2 sc0 ls0 ws0">plays<span class="_ _0"> </span>a</div><div class="t m5 x21 h10 y93 ff3 fs9 fc2 sc0 ls0 ws0">role<span class="_ _0"> </span>of<span class="_ _0"> </span>catalyst<span class="_ _0"> </span>and<span class="_ _0"> </span>has<span class="_ _0"> </span>a<span class="_ _0"> </span>small<span class="_ _0"> </span>net<span class="_ _0"> </span>reaction<span class="_ _0"> </span>rate<span class="_ _0"> </span>but<span class="_ _0"> </span>contributes</div><div class="t m5 x21 h10 y94 ff3 fs9 fc2 sc0 ls0 ws0">signiﬁcantly<span class="_ _3"> </span>to<span class="_ _10"> </span>the<span class="_ _3"> </span>acceleration<span class="_ _3"> </span>of<span class="_ _10"> </span>ignition<span class="_ _3"> </span>delay<span class="_ _3"> </span>time.<span class="_ _10"> </span>Another</div><div class="t m5 x21 h10 y95 ff3 fs9 fc2 sc0 ls0 ws0">example<span class="_ _a"> </span>is<span class="_ _a"> </span>that<span class="_ _a"> </span>a<span class="_ _6"> </span>reaction<span class="_ _a"> </span>path<span class="_ _a"> </span>which<span class="_ _a"> </span>has<span class="_ _6"> </span>very<span class="_ _a"> </span>long<span class="_ _a"> </span>reaction<span class="_ _a"> </span>chains</div><div class="t m5 x21 h10 y96 ff3 fs9 fc2 sc0 ls0 ws0">with<span class="_ _9"> </span>equal<span class="_ _0"> </span>reaction<span class="_ _9"> </span>rates.<span class="_ _0"> </span>The<span class="_ _9"> </span>deﬁnition<span class="_ _0"> </span>of<span class="_ _9"> </span>interaction<span class="_ _0"> </span>coefﬁcient</div><div class="t m5 x21 h10 y97 ff3 fs9 fc2 sc0 ls0 ws0">of<span class="_ _0"> </span>DRGEP<span class="_ _0"> </span>may<span class="_ _5"> </span>also<span class="_ _0"> </span>be<span class="_ _0"> </span>problematic<span class="_ _0"> </span><span class="fc3">[18]</span>.<span class="_ _0"> </span>In<span class="_ _5"> </span>order<span class="_ _0"> </span>to<span class="_ _0"> </span>improve<span class="_ _0"> </span>the</div><div class="t m5 x21 h10 y98 ff3 fs9 fc2 sc0 ls0 ws0">accuracy<span class="_ _3"> </span>of<span class="_ _3"> </span>DRG<span class="_ _3"> </span>and<span class="_ _3"> </span>DRGEP<span class="_ _10"> </span>method,<span class="_ _5"> </span>a<span class="_ _10"> </span>combinati<span class="_ _1"></span>on<span class="_ _3"> </span>of<span class="_ _3"> </span>DRG<span class="_ _3"> </span>or</div><div class="t m5 x21 h10 y99 ff3 fs9 fc2 sc0 ls0 ws0">DRGEP<span class="_ _11"> </span>with<span class="_ _4"> </span>species<span class="_ _11"> </span>sensitivity<span class="_ _11"> </span>was<span class="_ _11"> </span>adopted<span class="_ _4"> </span><span class="fc3">[18,19]</span>.<span class="_ _11"> </span>Unfortu-</div><div class="t m5 x21 h10 y9a ff3 fs9 fc2 sc0 ls0 ws0">nately,<span class="_ _8"> </span>the<span class="_ _9"> </span>addition<span class="_ _8"> </span>of<span class="_ _9"> </span>sensitivity<span class="_ _9"> </span>analysis<span class="_ _8"> </span>is<span class="_ _9"> </span>very<span class="_ _9"> </span>computationally</div><div class="t m5 x21 h10 y9b ff3 fs9 fc2 sc0 ls0 ws0">expensive.<span class="_ _0"> </span>Furthermore,<span class="_ _5"> </span>all<span class="_ _0"> </span>the<span class="_ _5"> </span>above<span class="_ _0"> </span>methods<span class="_ _5"> </span>are<span class="_ _5"> </span>based<span class="_ _0"> </span>on<span class="_ _5"> </span>the</div><div class="t m5 x21 h10 y9c ff3 fs9 fc2 sc0 ls0 ws0">analysis<span class="_ _3"> </span>of<span class="_ _5"> </span>one-generation<span class="_ _3"> </span>direct<span class="_ _5"> </span>ﬂux/reaction<span class="_ _3"> </span>rates.<span class="_ _3"> </span>In<span class="_ _3"> </span>order<span class="_ _5"> </span>to</div><div class="t m5 x21 h10 y9d ff3 fs9 fc2 sc0 ls0 ws0">improve<span class="_ _9"> </span>the<span class="_ _0"> </span>prediction<span class="_ _9"> </span>of<span class="_ _0"> </span>reaction<span class="_ _9"> </span>ﬂuxes,<span class="_ _9"> </span>a<span class="_ _0"> </span>multi-generation<span class="_ _9"> </span>ﬂux</div><div class="t m5 x21 h10 y9e ff3 fs9 fc2 sc0 ls0 ws0">analysis<span class="_ _10"> </span>including<span class="_ _10"> </span>both<span class="_ _10"> </span>consumption<span class="_ _10"> </span>and<span class="_ _10"> </span>production<span class="_ _10"> </span>pathways</div><div class="t m5 x21 h10 y9f ff3 fs9 fc2 sc0 ls0 ws0">are<span class="_ _8"> </span>needed.</div><div class="t m5 x27 h10 ya0 ff3 fs9 fc2 sc0 ls0 ws0">In<span class="_ _9"> </span>this<span class="_ _0"> </span>paper,<span class="_ _9"> </span>we<span class="_ _0"> </span>developed<span class="_ _9"> </span>a<span class="_ _0"> </span>path<span class="_ _9"> </span>ﬂux<span class="_ _0"> </span>analysis<span class="_ _8"> </span>(PFA)<span class="_ _0"> </span>method</div><div class="t m5 x21 h10 ya1 ff3 fs9 fc2 sc0 ls0 ws0">based<span class="_ _9"> </span>on<span class="_ _0"> </span>multi-generation<span class="_ _9"> </span>ﬂuxes<span class="_ _0"> </span>to<span class="_ _9"> </span>reduce<span class="_ _0"> </span>the<span class="_ _9"> </span>detailed<span class="_ _0"> </span>chemical</div><div class="t m5 x21 h10 ya2 ff3 fs9 fc2 sc0 ls0 ws0">kinetic<span class="_ _6"> </span>mechanism<span class="_ _6"> </span>with<span class="_ _6"> </span>improved<span class="_ _6"> </span>model<span class="_ _8"> </span>reduction<span class="_ _6"> </span>accuracy<span class="_ _6"> </span>com-</div><div class="t m5 x21 h10 ya3 ff3 fs9 fc2 sc0 ls0 ws0">pared<span class="_ _6"> </span>to<span class="_ _6"> </span>DRG<span class="_ _6"> </span>method<span class="_ _8"> </span>with<span class="_ _6"> </span>the<span class="_ _6"> </span>same<span class="_ _8"> </span>size<span class="_ _6"> </span>of<span class="_ _6"> </span>a<span class="_ _6"> </span>reduced<span class="_ _8"> </span>mechanism.</div><div class="t m5 x21 h10 ya4 ff3 fs9 fc2 sc0 ls0 ws0">At<span class="_ _8"> </span>ﬁrst,<span class="_ _6"> </span>the<span class="_ _8"> </span>methodology<span class="_ _8"> </span>of<span class="_ _6"> </span>the<span class="_ _8"> </span>PFA<span class="_ _8"> </span>method<span class="_ _8"> </span>is<span class="_ _8"> </span>presented<span class="_ _6"> </span>and<span class="_ _8"> </span>ana-</div><div class="t m5 x21 h10 ya5 ff3 fs9 fc2 sc0 ls0 ws0">lyzed.<span class="_ _9"> </span>The<span class="_ _9"> </span>method<span class="_ _0"> </span>are<span class="_ _9"> </span>tested<span class="_ _9"> </span>by<span class="_ _0"> </span>the<span class="_ _9"> </span>simulations<span class="_ _9"> </span>of<span class="_ _9"> </span>ignition<span class="_ _0"> </span>delay</div><div class="t m5 x21 h10 ya6 ff3 fs9 fc2 sc0 ls0 ws0">time<span class="_ _a"> </span>of<span class="_ _6"> </span>homogeneous<span class="_ _a"> </span>mixtures,<span class="_ _a"> </span>the<span class="_ _6"> </span>extinction<span class="_ _a"> </span>curves<span class="_ _6"> </span>in<span class="_ _a"> </span>a<span class="_ _6"> </span>perfectly</div><div class="t m5 x21 h10 ya7 ff3 fs9 fc2 sc0 ls0 ws0">stirred<span class="_ _5"> </span>reactor<span class="_ _5"> </span>(PSR)<span class="_ _5"> </span>and<span class="_ _5"> </span>compared<span class="_ _5"> </span>with<span class="_ _5"> </span>the<span class="_ _5"> </span>results<span class="_ _5"> </span>of<span class="_ _5"> </span>DRG<span class="_ _5"> </span>and</div><div class="t m5 x21 h10 ya8 ff3 fs9 fc2 sc0 ls0 ws0">detailed<span class="_ _10"> </span>mechanism.<span class="_ _10"> </span>Then<span class="_ _10"> </span>the<span class="_ _4"> </span>reaction<span class="_ _10"> </span>path<span class="_ _10"> </span>analyses<span class="_ _10"> </span>are<span class="_ _10"> </span>con-</div><div class="t m5 x21 h10 ya9 ff3 fs9 fc2 sc0 ls0 ws0">ducted<span class="_ _0"> </span>for<span class="_ _9"> </span>n-decane<span class="_ _0"> </span>and<span class="_ _9"> </span>methane<span class="_ _0"> </span>ignition<span class="_ _0"> </span>problem<span class="_ _9"> </span>with<span class="_ _0"> </span>the<span class="_ _0"> </span>NO</div><div class="t m7 x28 h13 yaa ff5 fsb fc2 sc0 ls0 ws0">x</div><div class="t m5 x21 h10 yab ff3 fs9 fc2 sc0 ls0 ws0">catalytic<span class="_ _6"> </span>effect<span class="_ _6"> </span>to<span class="_ _6"> </span>validate<span class="_ _8"> </span>the<span class="_ _6"> </span>method.<span class="_ _6"> </span>The<span class="_ _8"> </span>accuracy<span class="_ _6"> </span>of<span class="_ _6"> </span>the<span class="_ _6"> </span>present</div><div class="t m5 x21 h10 yac ff3 fs9 fc2 sc0 ls0 ws0">model<span class="_ _a"> </span>is<span class="_ _6"> </span>also<span class="_ _6"> </span>demonstrated<span class="_ _a"> </span>by<span class="_ _6"> </span>comparing<span class="_ _6"> </span>the<span class="_ _a"> </span>steady<span class="_ _6"> </span>and<span class="_ _6"> </span>unsteady</div><div class="t m5 x21 h10 yad ff3 fs9 fc2 sc0 ls0 ws0">ﬂame<span class="_ _0"> </span>propagation<span class="_ _5"> </span>speeds<span class="_ _5"> </span>and<span class="_ _5"> </span>structures<span class="_ _5"> </span>with<span class="_ _5"> </span>those<span class="_ _5"> </span>of<span class="_ _0"> </span>DRG<span class="_ _5"> </span>and</div><div class="t m5 x21 h10 yae ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _0"> </span>detailed<span class="_ _0"> </span>mechanism.<span class="_ _0"> </span>Finally,<span class="_ _0"> </span>the<span class="_ _0"> </span>reduced<span class="_ _0"> </span>mechanism<span class="_ _0"> </span>is<span class="_ _0"> </span>inte-</div><div class="t m5 x21 h10 yaf ff3 fs9 fc2 sc0 ls0 ws0">grated<span class="_ _5"> </span>with<span class="_ _5"> </span>the<span class="_ _5"> </span>hybrid<span class="_ _3"> </span>multi-time<span class="_ _0"> </span>scale<span class="_ _3"> </span>(HMTS)<span class="_ _5"> </span>method<span class="_ _5"> </span>for<span class="_ _5"> </span>the</div><div class="t m5 x21 h10 yb0 ff3 fs9 fc2 sc0 ls0 ws0">modeling<span class="_ _6"> </span>of<span class="_ _6"> </span>the<span class="_ _8"> </span>unsteady<span class="_ _6"> </span>spherically<span class="_ _8"> </span>propagating<span class="_ _6"> </span>ﬂames<span class="_ _6"> </span>of<span class="_ _8"> </span>n-dec-</div><div class="t m5 x21 h10 yb1 ff3 fs9 fc2 sc0 ls0 ws0">ane-air<span class="_ _a"> </span>mixtures.<span class="_ _a"> </span>A<span class="_ _a"> </span>signiﬁcant<span class="_ _a"> </span>increase<span class="_ _a"> </span>of<span class="_ _a"> </span>computation<span class="_ _a"> </span>efﬁciency<span class="_ _a"> </span>is</div><div class="t m5 x29 h10 yb2 ff3 fs9 fc2 sc0 ls0 ws0">achieved.<span class="_ _a"> </span>The<span class="_ _6"> </span>present<span class="_ _6"> </span>method<span class="_ _6"> </span>is<span class="_ _a"> </span>an<span class="_ _6"> </span>extension<span class="_ _6"> </span>of<span class="_ _6"> </span>the<span class="_ _6"> </span>previous<span class="_ _a"> </span>work</div><div class="t m5 x29 h10 yb3 ff3 fs9 fc2 sc0 ls0 ws0">of<span class="_ _8"> </span>Ref.<span class="_ _9"> </span><span class="fc3">[13]</span>,<span class="_ _9"> </span>DRG<span class="_ _9"> </span><span class="fc3">[14,15]<span class="_ _9"> </span></span>and<span class="_ _8"> </span>DRGEP<span class="_ _9"> </span><span class="fc3">[16]</span>.</div><div class="t m5 x15 hf yb4 ff8 fs9 fc2 sc0 ls0 ws0">2.<span class="_ _8"> </span>Path<span class="_ _9"> </span>ﬂux<span class="_ _9"> </span>analysis<span class="_ _9"> </span>method</div><div class="t m5 x2a h10 yb5 ff3 fs9 fc2 sc0 ls0 ws0">The<span class="_ _0"> </span>goal<span class="_ _5"> </span>of<span class="_ _0"> </span>mechanism<span class="_ _0"> </span>reduction<span class="_ _0"> </span>is<span class="_ _5"> </span>to<span class="_ _0"> </span>identify<span class="_ _5"> </span>species<span class="_ _0"> </span>which</div><div class="t m5 x15 h10 yb6 ff3 fs9 fc2 sc0 ls0 ws0">are<span class="_ _6"> </span>important<span class="_ _6"> </span>to<span class="_ _6"> </span>the<span class="_ _6"> </span>target<span class="_ _6"> </span>species.<span class="_ _a"> </span>Therefore,<span class="_ _8"> </span>it<span class="_ _a"> </span>is<span class="_ _8"> </span>crucial<span class="_ _a"> </span>to<span class="_ _8"> </span>deﬁne</div><div class="t m5 x15 h10 yb7 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _5"> </span>importance<span class="_ _3"> </span>index<span class="_ _5"> </span>(interaction<span class="_ _5"> </span>coefﬁcient)<span class="_ _5"> </span>of<span class="_ _3"> </span>each<span class="_ _5"> </span>species<span class="_ _3"> </span>in</div><div class="t m5 x15 h10 yb8 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _9"> </span>mechanism.<span class="_ _0"> </span>In<span class="_ _9"> </span>the<span class="_ _0"> </span>DRG<span class="_ _9"> </span>method,<span class="_ _0"> </span>the<span class="_ _0"> </span>direct<span class="_ _9"> </span>interaction<span class="_ _9"> </span>coefﬁ-</div><div class="t m5 x15 h10 yb9 ff3 fs9 fc2 sc0 ls0 ws0">cient<span class="_ _8"> </span>is<span class="_ _9"> </span>deﬁned<span class="_ _9"> </span>as<span class="_ _9"> </span><span class="fc3">[14]</span>:</div><div class="t m8 x29 h14 yba ff5 fsc fc2 sc0 ls0 ws0">r</div><div class="t m9 x2b h15 ybb ff3 fsd fc2 sc0 ls0 ws0">DRG</div><div class="t m9 x2b h15 ybc ff3 fsd fc2 sc0 ls0 ws0">AB</div><div class="t m8 x2c h16 yba ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t m8 x2d h17 ybd ffb fsc fc2 sc0 ls0 ws0">P</div><div class="t m9 x2e h15 ybe ff5 fsd fc2 sc0 ls0 ws0">i<span class="ffa">¼<span class="ff3">1<span class="ffc">;</span></span></span>I</div><div class="t ma x2f h18 ybf ffd fse fc2 sc0 ls0 ws0">m</div><div class="t m9 x30 h19 yc0 ff5 fsd fc2 sc0 ls0 ws0">A<span class="ffc">;</span>i</div><div class="t mb x31 h1a ybf ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x32 h19 yc0 ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x33 h1b ybf ffd fsc fc2 sc0 ls0 ws0">d</div><div class="t m9 x34 h19 yc1 ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m9 x34 h19 yc2 ff5 fsd fc2 sc0 ls0 ws0">B</div><div class="t m8 x35 h17 yc3 ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x35 h17 yc4 ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x35 h17 yc5 ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x36 h17 yc3 ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x36 h17 yc4 ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x36 h17 yc5 ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x37 h17 yc6 ffb fsc fc2 sc0 ls0 ws0">P</div><div class="t m9 x38 h15 yc7 ff5 fsd fc2 sc0 ls0 ws0">i<span class="ffa">¼<span class="ff3">1<span class="ffc">;</span></span></span>I</div><div class="t ma x30 h18 yc8 ffd fse fc2 sc0 ls0 ws0">m</div><div class="t m9 x39 h19 yc9 ff5 fsd fc2 sc0 ls0 ws0">A<span class="ffc">;</span>i</div><div class="t mb x3a h1a yc8 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x3b h19 yc9 ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x3c h17 yca ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x3c h17 ycb ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x34 h17 yca ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x34 h17 ycb ffb fsc fc2 sc0 ls0 ws0"></div><div class="t m8 x3d h1c ycc ffa fsc fc2 sc0 ls0 ws0">ð<span class="ff3">1</span>Þ</div><div class="t m8 x3e h1b ycd ffd fsc fc2 sc0 ls0 ws0">d</div><div class="t m9 x16 h19 yce ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m9 x16 h19 ycf ff5 fsd fc2 sc0 ls0 ws0">B</div><div class="t m8 x2a h16 yd0 ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t m8 x3f h1c yd1 ff3 fsc fc2 sc0 ls0 ws0">1<span class="_ _12"> </span>if<span class="_ _5"> </span>the<span class="_ _0"> </span><span class="ff5">i</span>th<span class="_ _0"> </span>elementary<span class="_ _9"> </span>reaction</div><div class="t m8 x40 h1c yd2 ff3 fsc fc2 sc0 ls0 ws0">involves<span class="_ _9"> </span>species<span class="_ _0"> </span>B</div><div class="t m8 x3f h1c yd3 ff3 fsc fc2 sc0 ls0 ws0">0<span class="_ _e"> </span>otherwise</div><div class="t m8 x41 h17 yd4 ffb fsc fc2 sc0 ls0 ws0">8</div><div class="t m8 x41 h17 yd5 ffb fsc fc2 sc0 ls0 ws0">></div><div class="t m8 x41 h17 yd6 ffb fsc fc2 sc0 ls0 ws0"><</div><div class="t m8 x41 h17 yd7 ffb fsc fc2 sc0 ls0 ws0">></div><div class="t m8 x41 h17 yd8 ffb fsc fc2 sc0 ls0 ws0">:</div><div class="t mb x29 h1a yd9 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x42 h19 yda ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x2a h16 yd9 ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t mb x41 h1a yd9 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x43 h19 ydb ff5 fsd fc2 sc0 ls0 ws0">f<span class="_ _13"> </span><span class="ffc">;</span>i</div><div class="t m8 x2e h16 yd9 ffa fsc fc2 sc0 ls0 ws0"></div><div class="t mb x44 h1a yd9 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x2f h19 ydb ff5 fsd fc2 sc0 ls0 ws0">b<span class="ffc">;</span>i</div><div class="t m8 x3d h1c yd9 ffa fsc fc2 sc0 ls0 ws0">ð<span class="ff3">2</span>Þ</div><div class="t mc x29 h10 ydc ff3 fs9 fc2 sc0 ls0 ws0">where</div><div class="t md x43 h1d ydc ffd fs10 fc2 sc0 ls0 ws0">m</div><div class="t m7 x45 h13 ydd ff5 fsb fc2 sc0 ls0 ws0">A,i</div><div class="t mc x46 h10 ydc ff3 fs9 fc2 sc0 ls0 ws0">is<span class="_ _5"> </span>the<span class="_ _5"> </span>stoichiometric<span class="_ _5"> </span>coefﬁcient<span class="_ _5"> </span>of<span class="_ _5"> </span>species<span class="_ _5"> </span><span class="ff5">A<span class="_ _5"> </span></span>in<span class="_ _3"> </span>the<span class="_ _0"> </span><span class="ff5">i</span>th</div><div class="t mc x29 h10 yde ff3 fs9 fc2 sc0 ls0 ws0">reaction.</div><div class="t me x2e h1e ydf ffd fs11 fc2 sc0 ls0 ws0">x</div><div class="t m7 x47 h13 ye0 ff5 fsb fc2 sc0 ls0 ws0">f,i</div><div class="t mc x48 h10 ydf ff3 fs9 fc2 sc0 ls0 ws0">,</div><div class="t me x3c h1e ydf ffd fs11 fc2 sc0 ls0 ws0">x</div><div class="t m7 x39 h13 ye0 ff5 fsb fc2 sc0 ls0 ws0">b,i</div><div class="t mc x49 h10 ydf ff3 fs9 fc2 sc0 ls0 ws0">,<span class="_ _9"> </span>and</div><div class="t me x4a h1e ydf ffd fs11 fc2 sc0 ls0 ws0">x</div><div class="t m7 x4b h13 ye0 ff5 fsb fc2 sc0 ls0 ws0">i</div><div class="t mc x4c h10 ydf ff3 fs9 fc2 sc0 ls0 ws0">are<span class="_ _8"> </span>the<span class="_ _9"> </span>forward,<span class="_ _9"> </span>backward,<span class="_ _9"> </span>and<span class="_ _9"> </span>net<span class="_ _9"> </span>reac-</div><div class="t mc x29 h10 ye1 ff3 fs9 fc2 sc0 ls0 ws0">tion<span class="_ _a"> </span>rate<span class="_ _a"> </span>of<span class="_ _a"> </span>the<span class="_ _a"> </span><span class="ff5">i</span>th<span class="_ _a"> </span>reaction,<span class="_ _a"> </span>respectively.<span class="_ _a"> </span><span class="ff5">I<span class="_ _6"> </span></span>is<span class="_ _a"> </span>the<span class="_ _a"> </span>total<span class="_ _a"> </span>number<span class="_ _a"> </span>of<span class="_ _a"> </span>ele-</div><div class="t mc x29 h10 ye2 ff3 fs9 fc2 sc0 ls0 ws0">mentary<span class="_ _0"> </span>reactions.<span class="_ _0"> </span>The<span class="_ _0"> </span>magnitude<span class="_ _9"> </span>of<span class="_ _5"> </span><span class="ff5">r</span></div><div class="t mf x4d h1f ye3 ff3 fs12 fc2 sc0 ls5 ws0">DRG</div><div class="t mf x4d h1f ye4 ff3 fs12 fc2 sc0 ls5 ws0">AB</div><div class="t mc x4e h10 ye5 ff3 fs9 fc2 sc0 ls0 ws0">shows<span class="_ _0"> </span>the<span class="_ _0"> </span>dependence/</div><div class="t mc x29 h10 ye6 ff3 fs9 fc2 sc0 ls0 ws0">importance<span class="_ _8"> </span>of<span class="_ _9"> </span>species<span class="_ _9"> </span><span class="ff5">B<span class="_ _9"> </span></span>to<span class="_ _9"> </span>species<span class="_ _9"> </span><span class="ff5">A<span class="_ _9"> </span></span><span class="fc3">[14]</span><span class="ff5">.</span></div><div class="t m5 x2a h10 ye7 ff3 fs9 fc2 sc0 ls0 ws0">To<span class="_ _5"> </span>initiate<span class="_ _5"> </span>the<span class="_ _5"> </span>selection<span class="_ _5"> </span>process,<span class="_ _5"> </span>a<span class="_ _5"> </span>set<span class="_ _5"> </span>of<span class="_ _5"> </span>pre-selected<span class="_ _5"> </span>species</div><div class="t m5 x29 h10 ye8 ff3 fs9 fc2 sc0 ls0 ws0">(e.g.<span class="_ _5"> </span><span class="ff5">A</span>)<span class="_ _5"> </span>and<span class="_ _5"> </span>a<span class="_ _5"> </span>threshold<span class="_ _3"> </span>value</div><div class="t m10 x4f h1d ye9 ffd fs10 fc2 sc0 ls0 ws0">e</div><div class="t m5 x50 h10 ye9 ff3 fs9 fc2 sc0 ls0 ws0">need<span class="_ _5"> </span>to<span class="_ _5"> </span>be<span class="_ _5"> </span>speciﬁed.<span class="_ _5"> </span>If<span class="_ _3"> </span><span class="ff5">r</span></div><div class="t m11 x51 h1f yea ff3 fs12 fc2 sc0 ls0 ws0">DRG</div><div class="t m11 x51 h1f yeb ff3 fs12 fc2 sc0 ls0 ws0">AB</div><div class="t m5 x52 h20 ye9 ffc fs9 fc2 sc0 ls0 ws0"><</div><div class="t m12 x53 h21 ye9 ffd fs13 fc2 sc0 ls0 ws0">e</div><div class="t m5 x54 h10 ye9 ff3 fs9 fc2 sc0 ls0 ws0">,</div><div class="t m5 x29 h10 yec ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _5"> </span>relation<span class="_ _3"> </span>between<span class="_ _5"> </span><span class="ff5">B<span class="_ _5"> </span></span>and<span class="_ _3"> </span><span class="ff5">A<span class="_ _5"> </span></span>is<span class="_ _3"> </span>considered<span class="_ _5"> </span>to<span class="_ _5"> </span>be<span class="_ _3"> </span>negligible.<span class="_ _5"> </span>On</div><div class="t m5 x29 h10 yed ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _8"> </span>other<span class="_ _9"> </span>hand,<span class="_ _9"> </span>species<span class="_ _9"> </span><span class="ff5">B<span class="_ _9"> </span></span>is<span class="_ _8"> </span>selected<span class="_ _9"> </span>when<span class="_ _9"> </span><span class="ff5">r</span></div><div class="t m11 x55 h1f yee ff3 fs12 fc2 sc0 ls0 ws0">DRG</div><div class="t m11 x55 h1f yef ff3 fs12 fc2 sc0 ls0 ws0">AB</div><div class="t m5 x56 h22 yf0 ffe fs9 fc2 sc0 ls0 ws0">P</div><div class="t m12 x57 h21 yf0 ffd fs13 fc2 sc0 ls0 ws0">e</div><div class="t m5 x58 h10 yf0 ff3 fs9 fc2 sc0 ls0 ws0">.</div><div class="t m5 x2a h10 yf1 ff3 fs9 fc2 sc0 ls0 ws0">However,<span class="_ _9"> </span>because<span class="_ _0"> </span>the<span class="_ _9"> </span>direct<span class="_ _0"> </span>interaction<span class="_ _9"> </span>coefﬁcient<span class="_ _0"> </span>is<span class="_ _9"> </span>not<span class="_ _0"> </span>con-</div><div class="t m5 x29 h10 yf2 ff3 fs9 fc2 sc0 ls0 ws0">servative,<span class="_ _6"> </span>one<span class="_ _6"> </span>drawback<span class="_ _6"> </span>of<span class="_ _6"> </span>this<span class="_ _6"> </span>method<span class="_ _6"> </span>is<span class="_ _6"> </span>that<span class="_ _6"> </span>only<span class="_ _6"> </span>the<span class="_ _6"> </span>ﬁrst<span class="_ _6"> </span>gener-</div><div class="t m5 x29 h10 yf3 ff3 fs9 fc2 sc0 ls0 ws0">ation<span class="_ _9"> </span>(directed<span class="_ _0"> </span>relation)<span class="_ _9"> </span>of<span class="_ _0"> </span>the<span class="_ _9"> </span>pre-selected<span class="_ _9"> </span>species<span class="_ _0"> </span>is<span class="_ _9"> </span>considered.</div><div class="t m5 x29 h10 yf4 ff3 fs9 fc2 sc0 ls0 ws0">In<span class="_ _5"> </span>the<span class="_ _3"> </span>point<span class="_ _5"> </span>view<span class="_ _5"> </span>of<span class="_ _3"> </span>reaction<span class="_ _5"> </span>ﬂux,<span class="_ _5"> </span>both<span class="_ _3"> </span>the<span class="_ _5"> </span>ﬁrst<span class="_ _5"> </span>generation<span class="_ _5"> </span>and</div><div class="t m5 x29 h10 yf5 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _8"> </span>second<span class="_ _9"> </span>generation<span class="_ _9"> </span>or<span class="_ _9"> </span>the<span class="_ _8"> </span>higher<span class="_ _9"> </span>generations<span class="_ _9"> </span>are<span class="_ _9"> </span>important.</div><div class="t m5 x2a h10 yf6 ff3 fs9 fc2 sc0 ls0 ws0">In<span class="_ _6"> </span>this<span class="_ _6"> </span>study,<span class="_ _6"> </span>instead<span class="_ _6"> </span>of<span class="_ _6"> </span>using<span class="_ _6"> </span>the<span class="_ _6"> </span>absolute<span class="_ _6"> </span>reaction<span class="_ _6"> </span>rate,<span class="_ _6"> </span>we<span class="_ _6"> </span>use</div><div class="t m5 x29 h10 yf7 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _0"> </span>production<span class="_ _9"> </span>and<span class="_ _0"> </span>consumption<span class="_ _9"> </span>ﬂuxes<span class="_ _0"> </span>to<span class="_ _0"> </span>identify<span class="_ _0"> </span>the<span class="_ _9"> </span>important</div><div class="t m5 x29 h10 yf8 ff3 fs9 fc2 sc0 ls0 ws0">reaction<span class="_ _10"> </span>pathways.<span class="_ _10"> </span>The<span class="_ _10"> </span>production<span class="_ _10"> </span>and<span class="_ _10"> </span>consumption<span class="_ _10"> </span>ﬂuxes,<span class="_ _10"> </span><span class="ff5">P</span></div><div class="t m7 x59 h13 yf9 ff5 fsb fc2 sc0 ls0 ws0">A</div><div class="t m5 x29 h10 yfa ff3 fs9 fc2 sc0 ls0 ws0">and<span class="_ _8"> </span><span class="ff5">C</span></div><div class="t m7 x5a h13 yfb ff5 fsb fc2 sc0 ls0 ws0">A</div><div class="t m5 x5b h10 yfa ff5 fs9 fc2 sc0 ls0 ws0">,<span class="_ _8"> </span><span class="ff3">of<span class="_ _9"> </span>species<span class="_ _9"> </span></span>A<span class="_ _9"> </span><span class="ff3">can<span class="_ _9"> </span>be<span class="_ _9"> </span>calculated<span class="_ _8"> </span>as<span class="_ _9"> </span>following:</span></div><div class="t m8 x29 h14 yfc ff5 fsc fc2 sc0 ls0 ws0">P</div><div class="t m9 x16 h19 yfd ff5 fsd fc2 sc0 ls0 ws0">A</div><div class="t m8 x5c h16 yfc ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t m8 x41 h17 yfe ffb fsc fc2 sc0 ls0 ws0">X</div><div class="t m9 x5d h15 yff ff5 fsd fc2 sc0 ls0 ws0">i<span class="ffa">¼<span class="ff3">1<span class="ffc">;</span></span></span>I</div><div class="t m8 x2e h1c yfc ff3 fsc fc2 sc0 ls0 ws0">max<span class="ffa">ð</span></div><div class="t ma x30 h18 yfc ffd fse fc2 sc0 ls0 ws0">m</div><div class="t m9 x39 h19 yfd ff5 fsd fc2 sc0 ls0 ws0">A<span class="ffc">;</span>i</div><div class="t mb x3a h1a yfc ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x3b h19 yfd ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x34 h1c yfc ffc fsc fc2 sc0 ls0 ws0">;<span class="_ _b"> </span><span class="ff3">0<span class="ffa">Þ</span></span>:<span class="_ _14"> </span><span class="ffa">ð<span class="ff3">3</span>Þ</span></div><div class="t m8 x15 h14 y100 ff5 fsc fc2 sc0 ls0 ws0">C</div><div class="t m9 x16 h19 y101 ff5 fsd fc2 sc0 ls0 ws0">A</div><div class="t m8 x5c h16 y102 ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t m8 x5d h17 y103 ffb fsc fc2 sc0 ls0 ws0">X</div><div class="t m9 x5d h15 y104 ff5 fsd fc2 sc0 ls0 ws0">i<span class="ffa">¼<span class="ff3">1<span class="ffc">;</span></span></span>I</div><div class="t m8 x2e h1c y102 ff3 fsc fc2 sc0 ls0 ws0">max<span class="ffa">ð</span></div><div class="t ma x31 h18 y102 ffd fse fc2 sc0 ls0 ws0">m</div><div class="t m9 x3a h19 y105 ff5 fsd fc2 sc0 ls0 ws0">A<span class="ffc">;</span>i</div><div class="t mb x3b h1a y102 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x5e h19 y105 ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x5f h1c y102 ffc fsc fc2 sc0 ls0 ws0">;<span class="_ _b"> </span><span class="ff3">0<span class="ffa">Þ</span></span>:<span class="_ _15"> </span><span class="ffa">ð<span class="ff3">4</span>Þ</span></div><div class="t m5 x2a h10 y106 ff3 fs9 fc2 sc0 ls0 ws0">And<span class="_ _0"> </span>the<span class="_ _0"> </span>ﬂux<span class="_ _0"> </span>of<span class="_ _0"> </span>species<span class="_ _0"> </span><span class="ff5">A<span class="_ _0"> </span></span>related<span class="_ _0"> </span>with<span class="_ _0"> </span>species<span class="_ _0"> </span><span class="ff5">B<span class="_ _0"> </span></span>can<span class="_ _0"> </span>be<span class="_ _0"> </span>calcu-</div><div class="t m5 x15 h10 y107 ff3 fs9 fc2 sc0 ls0 ws0">lated<span class="_ _8"> </span>as:</div><div class="t m8 x29 h14 y108 ff5 fsc fc2 sc0 ls0 ws0">P</div><div class="t m9 x16 h15 y109 ff3 fsd fc2 sc0 ls0 ws0">AB</div><div class="t m8 x60 h16 y108 ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t m8 x61 h17 y10a ffb fsc fc2 sc0 ls0 ws0">X</div><div class="t m9 x61 h15 y10b ff5 fsd fc2 sc0 ls0 ws0">i<span class="ffa">¼<span class="ff3">1<span class="ffc">;</span></span></span>I</div><div class="t m8 x62 h1c y108 ff3 fsc fc2 sc0 ls0 ws0">max<span class="ffa">ð</span></div><div class="t ma x63 h18 y108 ffd fse fc2 sc0 ls0 ws0">m</div><div class="t m9 x64 h19 y10c ff5 fsd fc2 sc0 ls0 ws0">A<span class="ffc">;</span>i</div><div class="t mb x32 h1a y108 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x65 h19 y10c ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x36 h1b y108 ffd fsc fc2 sc0 ls0 ws0">d</div><div class="t m9 x66 h19 y10d ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m9 x66 h19 y10e ff5 fsd fc2 sc0 ls0 ws0">B</div><div class="t m8 x4a h1c y108 ffc fsc fc2 sc0 ls0 ws0">;<span class="_ _b"> </span><span class="ff3">0<span class="ffa">Þ</span></span>:<span class="_ _16"> </span><span class="ffa">ð<span class="ff3">5</span>Þ</span></div><div class="t m8 x29 h14 y10f ff5 fsc fc2 sc0 ls0 ws0">C</div><div class="t m9 x16 h15 y110 ff3 fsd fc2 sc0 ls0 ws0">AB</div><div class="t m8 x60 h16 y111 ffa fsc fc2 sc0 ls0 ws0">¼</div><div class="t m8 x61 h17 y112 ffb fsc fc2 sc0 ls0 ws0">X</div><div class="t m9 x61 h15 y113 ff5 fsd fc2 sc0 ls0 ws0">i<span class="ffa">¼<span class="ff3">1<span class="ffc">;</span></span></span>I</div><div class="t m8 x62 h1c y111 ff3 fsc fc2 sc0 ls0 ws0">max<span class="ffa">ð</span></div><div class="t ma x49 h18 y111 ffd fse fc2 sc0 ls0 ws0">m</div><div class="t m9 x67 h19 y114 ff5 fsd fc2 sc0 ls0 ws0">A<span class="ffc">;</span>i</div><div class="t mb x65 h1a y111 ffd fsf fc2 sc0 ls0 ws0">x</div><div class="t m9 x68 h19 y114 ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m8 x4a h1b y111 ffd fsc fc2 sc0 ls0 ws0">d</div><div class="t m9 x69 h19 y115 ff5 fsd fc2 sc0 ls0 ws0">i</div><div class="t m9 x69 h19 y116 ff5 fsd fc2 sc0 ls0 ws0">B</div><div class="t m8 x4b h1c y111 ffc fsc fc2 sc0 ls0 ws0">;<span class="_ _b"> </span><span class="ff3">0<span class="ffa">Þ</span></span>:<span class="_ _17"> </span><span class="ffa">ð<span class="ff3">6</span>Þ</span></div><div class="t m5 x2a h10 y117 ff3 fs9 fc2 sc0 ls0 ws0">Here<span class="_ _8"> </span><span class="ff5">P</span></div><div class="t m7 x6a h23 y118 ff3 fsb fc2 sc0 ls0 ws0">AB</div><div class="t m5 x48 h10 y117 ff3 fs9 fc2 sc0 ls0 ws0">and<span class="_ _8"> </span><span class="ff5">C</span></div><div class="t m7 x32 h23 y118 ff3 fsb fc2 sc0 ls0 ws0">AB</div><div class="t m5 x36 h10 y117 ff3 fs9 fc2 sc0 ls0 ws0">denote<span class="ff5">,<span class="_ _8"> </span></span>respectively,<span class="_ _9"> </span>the<span class="_ _8"> </span>production<span class="_ _9"> </span>and<span class="_ _9"> </span>con-</div><div class="t m5 x15 h10 y119 ff3 fs9 fc2 sc0 ls0 ws0">sumption<span class="_ _8"> </span>rates<span class="_ _9"> </span>of<span class="_ _9"> </span>species<span class="_ _9"> </span><span class="ff5">A<span class="_ _9"> </span></span>due<span class="_ _8"> </span>to<span class="_ _9"> </span>the<span class="_ _9"> </span>existence<span class="_ _9"> </span>of<span class="_ _9"> </span>species<span class="_ _8"> </span><span class="ff5">B</span>.</div><div class="t m5 x2a h10 y11a ff3 fs9 fc2 sc0 ls0 ws0">In<span class="_ _a"> </span>order<span class="_ _6"> </span>to<span class="_ _a"> </span>consider<span class="_ _a"> </span>the<span class="_ _6"> </span>conservative<span class="_ _a"> </span>ﬂux<span class="_ _6"> </span>information,<span class="_ _a"> </span>we<span class="_ _a"> </span>intro-</div><div class="t m5 x15 h10 y11b ff3 fs9 fc2 sc0 ls0 ws0">duce<span class="_ _3"> </span>a<span class="_ _3"> </span>different<span class="_ _3"> </span>deﬁnition<span class="_ _5"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>interaction<span class="_ _3"> </span>coefﬁcients,<span class="_ _3"> </span>which</div><div class="t m5 x15 h10 y11c ff3 fs9 fc2 sc0 ls0 ws0">contain<span class="_ _8"> </span>the<span class="_ _9"> </span>ﬂux<span class="_ _9"> </span>information<span class="_ _9"> </span>for<span class="_ _9"> </span>both<span class="_ _9"> </span>the<span class="_ _9"> </span>ﬁrst<span class="_ _9"> </span>and<span class="_ _9"> </span>second<span class="_ _9"> </span>genera-</div><div class="t m5 x15 h10 y11d ff3 fs9 fc2 sc0 ls0 ws0">tion.<span class="_ _a"> </span>Note<span class="_ _a"> </span>that<span class="_ _a"> </span>although<span class="_ _a"> </span>in<span class="_ _a"> </span>the<span class="_ _a"> </span>present<span class="_ _a"> </span>studies,<span class="_ _a"> </span>only<span class="_ _a"> </span>two<span class="_ _a"> </span>generation</div><div class="t m5 x15 h10 y11e ff3 fs9 fc2 sc0 ls0 ws0">ﬂuxes<span class="_ _8"> </span>are<span class="_ _8"> </span>considered,<span class="_ _8"> </span>the<span class="_ _8"> </span>method<span class="_ _9"> </span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>extended<span class="_ _8"> </span>to<span class="_ _9"> </span>any<span class="_ _8"> </span>genera-</div><div class="t m5 x15 h10 y11f ff3 fs9 fc2 sc0 ls0 ws0">tions.<span class="_ _a"> </span>Nevertheless,<span class="_ _b"> </span>with<span class="_ _a"> </span>the<span class="_ _a"> </span>increase<span class="_ _a"> </span>of<span class="_ _a"> </span>the<span class="_ _a"> </span>number<span class="_ _a"> </span>of<span class="_ _a"> </span>generations,</div><div class="t m5 x15 h10 y120 ff3 fs9 fc2 sc0 ls0 ws0">the<span class="_ _6"> </span>computation<span class="_ _a"> </span>time<span class="_ _6"> </span>is<span class="_ _6"> </span>proportional<span class="_ _6"> </span>to<span class="_ _6"> </span>(species<span class="_ _6"> </span>number)</div><div class="t m7 x6b h23 y121 ff3 fsb fc2 sc0 ls0 ws0">generations</div><div class="t m5 x54 h10 y122 ff3 fs9 fc2 sc0 ls0 ws0">.</div><div class="t m5 x2a h10 y123 ff3 fs9 fc2 sc0 ls0 ws0">The<span class="_ _8"> </span>interaction<span class="_ _9"> </span>coefﬁcients<span class="_ _8"> </span>for<span class="_ _9"> </span>production<span class="_ _9"> </span>and<span class="_ _9"> </span>consumption<span class="_ _8"> </span>of</div><div class="t m5 x29 h10 y124 ff3 fs9 fc2 sc0 ls0 ws0">species<span class="_ _8"> </span><span class="ff5">A<span class="_ _9"> </span></span>via<span class="_ _9"> </span><span class="ff5">B<span class="_ _9"> </span></span>of<span class="_ _9"> </span>ﬁrst<span class="_ _8"> </span>generation<span class="_ _9"> </span>are<span class="_ _9"> </span>deﬁned<span class="_ _9"> </span>as:</div><div class="t m4 xd hb y125 ff5 fs6 fc2 sc0 ls0 ws0">W.<span class="_ _6"> </span>Sun<span class="_ _6"> </span>et<span class="_ _a"> </span>al.<span class="_ _b"> </span>/<span class="_ _13"> </span>Combustion<span class="_ _6"> </span>and<span class="_ _6"> </span>Flame<span class="_ _6"> </span>157<span class="_ _6"> </span>(2010)<span class="_ _6"> </span>1298–1307</div><div class="t m4 x6c hd y126 ff3 fs6 fc2 sc0 ls0 ws0">1299</div></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div>