An important particular case, of principal interest here, is that of particle ‘growth'(+/-). It appears that all previous applications of the PBE approach to particle populations evolving in the presence of either growth or dissolution (‘negative growth') have made use of quasi-steady (QS-) power-law rate laws. This is not only true for deliberately idealized situations (such as growth proportional to (particle volume)1/3 in a well-stirred nutrient-containing vessel; Friedlander(2000), but also transient liquid fuel spray evaporation in a more complex Diesel engine environment. In such applications QS-power-law relations are known to be inaccurate, even in the low particle volume fraction limit (negligible interparticle interactions). This is because the single particle diffusion-controlled mass transfer process even in a constant local environment is intrinsically transient, leading to more complex dv/dt vs. v relationships (where v is the instantaneous particle volume)(see, eg., Rosner(2000, 2006)). In the present paper we examine some of the consequences of non-QS behavior, and introduce multivariable methods to systematically account for such effects when they are apt to have engineering consequences. As an interesting, yet tractable class of ‘model' problems, we examine both single-module and two-module representations of dilute spray devices----viz. a well-stirred vessel---perhaps fed by a high pressure ‘plug-flow' vessel. In each steady-flow vessel the droplet population evaporates in accord with a rational non-QS diffusion-controlled rate law. Under such circumstances, we ask: 1) How well can we predict the exit droplet size distributions, and their important ‘moments', and 2) Will the effects of non-QS individual particle behavior somehow ‘average-out' provided the feed is ‘polydispersed' and one invokes a suitable non-QS overall droplet ‘lifetime'?
Our experience with these, and related, prototypical examples supports the following broader recommendations: i) systematic introduction of more accurate rate laws will be essential to meet the quantitative demands of next-generation PBE-based CRE-simulation models for spray devices and/or high-value particulate synthesis equipment, and, ii) Quadrature-based moment methods (ie. ‘QMOMs'; see, eg., Rosner, McGraw and Tandon(2003)) are able to incorporate realistic rate laws and economically generate their effects on important ‘moments' characterizing the joint distribution functions of such particulate products. In ‘multi-modal' situations, where the pdf-itself is of special interest, Spectral/Orthogonal Collocation (SOC-) methods (Rosner et al.(2005)) are recommended. ________________________________________________________ a Based, in part, on work supported by NSF Grant #CTS 0522944 ; Submitted 5/1/06 for AIChE 11/06 b L W Jones Prof ChE, Director HTCRE Lab, Yale U c Permanent Address: UNED-Madrid, Dept. Fluid Physics Friedlander, S.K.(2000), Fundamentals of Aerosol Dynamics, Oxford U. Press (2d ed.) Rosner, D. E.,(2000), Transport Processes in Chemically Reacting Flow Systems, DOVER Rosner, D.E. (2006), Int. J. Chem Reactor Engrg.(Berkeley press) vol 4 No.1