The　coupling　of　Eulerian　and　Lagrangian　methods　in　the　Eulerian-Lagrangian　Spray　Atomization　(ELSA)　approach　is　critical.　This　study　proposes　an　equation　for　the　primary　breakup　particle　diameter　D　of　a　diesel　fuel　spray　and　adopts　it　as　a　key　transition　criterion　for　coupling.　A　three-dimensional　diesel　spray　is　modeled　by　the　large-eddy　simulation　(LES)　approach.　This　improved　ELSA　simulation　was　conducted　using　various　transition　criteria　for　particle　diameter　D-cr.　The　results　show　that　fuel　spray　experiences　two　stages:　stage　I,　when　a　liquid　column　appears　without　a　dispersed　phase,　and　stage　II,　when　primary　breakup　occurs　with　many　discrete　particles.　Although　D-cr　has　little　influence　on　the　macro-spray　characteristics,　such　as　top　penetration　distance　S　and　spray　cone　angle　theta,　it　has　significant　effects　on　discrete　particles,　such　as　their　number,　average　diameter,　distribution　and　location,　and　spray　cone　area.　D-cr　should　be　determined　on　the　basis　of　actual　operating　conditions.