ABSTRACT
Notethatinamonodispersesystemofsphericalparticles,the maximumpossiblevolumefractionispi/ 18 0 74≈ . , and it correspondstothecaseofahexagonalclosepack.Butexperimentsandrecenttheoreticalestimationsshowthattypical randomclosepackvolumefractioncannotexceedabove64% (Songetal.2008).Underdiluteconditions,wecanapproximate Φm ≈ Φvρp/ρf. ›us, in the environmentally relevant case of sand particlessuspendedinclearorsaltywater(ρp/ρf ≈ 2.65), the mass loadingisoftheorderofthevolumefraction.Whereas,inthe
case of particles suspended in air (e.g., ash particles in air), mass loading will be typically three orders of magnitude larger than the corresponding volume fraction. In the other environmentallyrelevantlimitofbubblyžows,massloading(asdeŸnedearlier)willbethreeordersofmagnitudesmallerthanthevolume fraction.However,theproblemofbubblyžowsmustbetreated withcare,sincetheaddedmassofthebubbleswillmakethe e§ective mass loading to be of the order of volume fraction.
