Performance: Chemical Engineering

Ind. Eng. Chem. Res. 2009, 48, 6453–6464

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Spray Combustor Design/Performance: Chemical Engineering Contributions and the Emergence of an “Interacting Population-Balance” Perspective Daniel E. Rosner High Temperature Chemical Reaction Engineering Laboratory, Department of Chemical Engineering, Yale UniVersity, New HaVen, Connecticut 06520

Spray combustors are now widely used in many technologies, spanning commodity chemical synthesis (combustion of molten sulfur en route to sulfuric acid (H2SO4) and phosphorus en route to phosphoric acid (H3PO4), ...), and nanoparticle synthesis (eg., via “spray pyrolysis”), to energy conversion (oil-fired furnaces or boilers) and chemical propulsion (aircraft gas turbines and liquid-propellant rocket motors). While important space, weight, and pollutant constraints inevitably differ from application to application, spray-“fuel”-fed combustors share certain common performance characteristics, and there is a considerable economic incentive to develop rational yet tractable design methods for them. While this intrinsically interdisciplinary subject continues to evolve, here, we briefly review some of the principal contributions to these challenging goals published by chemical engineers since the inception of Industrial and Engineering Chemistry (I&EC). Not surprisingly, the earliest contributions focused on some of the important “unit processes” (e.g., isolated droplet evaporation rates, vapor micromixing rates in spatially homogeneous turbulent flows, and steady vaporphase laminar diffusion flames). Chemical engineers introduced “continuous mixture theory” to economically address not only phase/chemical equilibria in multicomponent mixtures, but also spatially nonuniform (nonequilibrium) flows. More-recent studies have introduced diffusion “flamelet” concepts for vapor-phase nonpremixed combustion and statistical population-balance concepts to address evolving turbulence characteristics and/or droplet size distributions. Because of the wide range of operative (length and time) scales in the full problem, in the foreseeable future, clever asymptotic methods will continue to be required to capture the essential physicochemical phenomena but still make the associated numerical simulations manageable. In this regard, a fruitful unified perspective is now emergingsone quite natural to chemical engineers. This can perhaps be best described as an interacting “multi-phase, multi-environment” approachsor simply an “interacting multivariable population balance” approach. While much remains to be done, it is hoped that this 2008/2009 perspective, and highly selective “review” of but one class of multiphase chemical reactors, will stimulate further activity along this promising path. 1. Introduction/Goals Here, the example of continuous spray combustors has been selected to illustrate some of the progress that has been made in mathematically modeling the behavior of multiphase chemical reactors, the challenges that remain, and the emergence of a helpful unifying perspective. In this particular class of devices, a liquid “fuel” is injected as a fine spray and is usually mixed and reacted with a suitable gaseous oxidizer (often air) to either produce valuable chemical intermediates or products (see below), or simply a high-pressure/enthalpy vapor mixture (to drive a turbine and/or produce thrust). The “logical” alternative of fuel “pre-vaporization” (in an upstream device) is often unattractive, for reasons of safety, maintenance, or operational complexity. My topical choice [“continuous spray combustors”si.e., liquid-fueled reactors operating in the (quasi-)steady-state (QS) mode] is based on several factors: • The technological/economic importance of such devices for large-scale commodity chemical synthesis, process heat or power generation, and chemical propulsion (aircraft gas turbines and chemical rockets). * E-mail address: [email protected].

• The comparative “invisibility” of the contributions of chemical engineers (ChEs) to the development of such devices! • The simultaneous presence of challenging multiphase, chemically reacting flows with molecular and turbulent transport, covering a computationally impractical range of spatial and time scales. Taking these in turn, we note that: While more prominent in the patent literature and internal corporate files, continuous spray combustors have had a decisive role in many industrial-scale chemical synthesis processessincluding the combustion of molten sulfur (the first chemical step in the synthesis of sulfuric acid), and the combustion of molten phosphorus (the first chemical step in the synthesis of phosphoric acid). Spray combustors have also been used as the primary source of carbon-black (pigment or rubber additive) precursors or hot/moist oxygen in the synthesis of fumed titania (pigment) (see, e.g., refs 1 and 2). Of course, spray combustors are also widely used in furnaces to supply “process heat”. Much more “visible” to the public are the higher-pressure spray combustors that are at the heart of aircraft and stationary gas turbines used for propulsion (“jet” engines; see, e.g., Figure 1) or power generation. Finally, the chemical rockets that now routinely launch communications or weather satellites and

10.1021/ie900167h CCC: $40.75  2009 American Chemical Society Published on Web 06/22/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

Figure 1. Schematic diagram of an aircraft gas turbine spray combustor (“can” type), which serves as a high-pressure hot-gas generator immediately upstream of the turbine. (After Rosner;3 also see the work of Correa.4)

science experiments into space contain even higher-pressure thrust chambers that are often bipropellant spray combustorssi.e., both the fuel (e.g., H2(l)) and oxidizer (e.g., O2(l) and/or F2(l)) are injected as liquid sprays. Perhaps less familiar are the bipropellant spray combustors used to supply hot combustion products to a turbine that drives the propellant turbopumps which feed the thrust chamber (see, e.g., ref 5). Not surprisingly, ChEs have been the dominant contributors in the first category of applications. However, most of this literature remains “proprietary” to this day, and the work of Conroy and Johnstone6 is one of the few archival publications that can be cited on the rational design or optimization of such combustors. With regard to the second class of applications, ChE contributions will be seen (below) to also have been extremely important, but because ChEs are present in these fields “at high dilution” [compared to mechanical engineers (MechEs) and aeronautical engineers (AeroEs)], rarely are such achievements even acknowledged,7 no less explicitly highlighted in centennial summaries of chemical engineering accomplishments. Thus, one of our present purposes is to simply call attention to some of these important advances. Because many of the devices mentioned previously operate for extended periods of time under rather extreme conditions (pressures of the order of 100 atm with liquid propellants that may be “cryogenic” (e.g., H2(l) or O2(l)), generating combustion products often at temperatures of >3000 K), with volumetric chemical energy release rates in excess of 10 GW/m3, the R&D challenges have been enormous, and ChEs have quietly risen to this 20th century so-called “rocket science” challenge! But, as with many technologies, much of this progress was actually due to “enlightened engineering empiricism”, which frequently leads fundamental theory and “cutting edge” mathematical modeling efforts. Yet, a grasp of the underlying physical and chemical phenomena (see Sections 2.1 and 2.2), and several clever approximate theories (some of which are mentioned in Section 3), were certainly behind many of the earliest advances, and ancillary theoretical studies (few of which were immediately published) gradually were used to reduce testing and development times/costs. At first, their archival contributions were mainly to what might be called the relevant “unit processes” (Section 2) of spray combustorssespecially the transport laws that governg individual droplets, turbulent mixing rates, and gaseous diffusion flames. More systems-oriented studies followed, including “modular models”, which shed valuable light on spray combustor operability (“stability”) limits and pollutant emissions (see Section 3.3). Still more recently, mathematical

models have been developed, not all of which are “computerintensive”, which now guide not only the preliminary design of such systems, but also the future selection of efficient modeling assumptions (this is a topic that will be emphasized in the conclusion of Section 3). Finally, in Section 4, we call attention to a fruitful theoretical perspective and formalism that has emerged from our collaborative research in this areasone that promises to shed further light on not only the rational design of continuous spray combustors, but also on more general multiphase contactors/reactors. As will be appreciated, in some sense, these recent developments can be viewed as yet another vindication of the (then controversial!) century-old vision of Ludwig Boltzmann, who was mercifully focused on the simpler but nontrivial case of single-phase, nonreacting ideal gas dynamics. The reader should be forewarned that what follows is heavily weighted by the author’s personal perspective and research involvements. Thus, while I have deliberately set out to emphasize the role of ChEs in these developments, no claim of “completeness” can be made and I apologize to any of my interdisciplinary colleagues in industry, national laboratories, or academia whose work is only implicitly present in the reference section at the end of this paper. While on the subject of caveats, we should also state up front that there are many rather important but application-specific spray combustor performance criteria that cannot possibly be explicitly addressed in this brief overviewse.g., dynamic stability (magnitude and frequency of periodic oscillations), uniformity of exit stream properties, overall stagnation pressure drop, and trace pollutant emissions (e.g., the mass (in grams) of gaseous NO emitted from an AGT combustor per kilogram of fuel burned), etc.). As always, the analyst’s challenge is to find/implement the simplest rational model that is capable of accounting for, and ultimately predicting, trends in the quantities of principal interest! 2. Chemical Engineers’ Contributions to Some of the “Unit Processes” of Spray Combustors 2.1. Isolated Droplet Drag, Heat Transfer, and Evaporation Rates: Non-QS Effects and “Continuous Mixtures”. Perhaps best known to ChEs, and widely used in many other fluid/suspended particle technologies, are the laws/economical correlations that govern isolated spherical particle momentum, heat, and mass exchange with the local fluid environment. Still widely used in complex numerical simulations of spray combus-


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tors are refinements of the early QS correlations presented in the works of Ranz and Marshall,9 McAdams,10 and Eisenklam et al.;11 these have been summarized in the more-recent treatises of Bird et al.,12 Whitaker,13 Clift et al.,14 Churchill,15 Rosner,3 and Asano.16 Normally, it is important to explicitly include the systematic reduction in the isolated sphere momentum-, heat-, and mass-transfer coefficients that are associated with the evaporation process itself (the so-called Stefan-flow “blowing” effect,17 when the liquid density far exceeds that of the surrounding vapor). An important corollary of the aforementioned fundamental work is the following approximate expression for the total diffusion-controlled evaporation (life)time of a stationary droplet of a single-component fuel of initial diameter d0 in a constant vapor environment, viz,

tvap,ref

( )[

d02 Fp ) Fg 8D ln(1 + Bm)

]

(1)

which is seen to be quadratic with regard to the initial droplet diameter. Here, D is the relevant fuel vapor Fick diffusion coefficient, and Bm is a dimensionless “driving force” for mass transfer, which is defined in terms of local fuel vapor mass fractions as Bm )

ωw - ω∞ 1 - ωw

where ωw is the fuel vapor mass fraction established at the droplet vapor/liquid interface and ω∞ prevails in the vapor phase “far” from the droplet (see discussion presented later in this paper). As will be seen, the ratio of the mean gas flow residence time in a continuous spray combustor to this characteristic fuel droplet vaporization (life)time (for example, that evaluated for the Sauter-mean droplet diameter in the injected fuel spray) will have a decisive role in determining the fraction of the spray that can evaporate, and, hence, combustor performance. Indeed, from a chemical reaction engineering perspective, we view and refer to this characteristic time ratio as a “vaporization Damko¨hler number”, which is written as (Dam)vap (see refs 3, 23, and 24) and is presented in Figures 4a and 4b, shown later in this work. Perhaps less familiar are (i) the non-quasi-steady-state (nonQS) “corrections”, which become necessary when the chamber pressures become comparable to the thermodynamic critical pressure of a single-component fuel, and/or (ii) the consequences of using multicomponent fuels (e.g., “kerosene” or deliberate fuel “blends”). Regarding the first of these, the author was able to make use of the numerical transient diffusion-controlled sphere phasechange results from the work of Duda and Vrentas25 to propose a method to predict the vaporization lifetime of isolated fuel droplets under non-QS conditions in a constant environmentsi.e., conditions for which the following dimensionless parameter is not small:24,26,27 ε≡

[( )( )

]

2 Fg ln(1 + Bm) π FL

1/2

(2)

Here, Bm is the previously mentioned dimensionless “driving force” for mass transfer. Our recent spray combustor simulations,24 which are discussed in Section 3.2 (and are focused on combustor efficiency and intensity) have made use of such a non-QS law, and clearly demonstrate the significance of such

Figure 2. Fuel droplet vaporization at chamber pressures, compared to the thermodynamic critical pressure of the fuel. Note the multiplicity of “wetbulb” temperatures and domains of droplet heating or cooling for a dodecane-like droplet in compression-heated nonvitiated air. (After Rosner and Chang.26)

effects at chamber pressures (pch) above ca. 10 atm for typical hydrocarbon liquid fuels. As an interesting byproduct of this earlier single droplet work, droplet energy balance and static stability considerations were used to clearly delineate combustion chamber conditions under which an injected fuel droplet would be driven to either (a) its stable (“wet-bulb”) temperature, which is less than both its boiling point (Tbp(p)) and its thermodynamic critical temperature (Tc), or (b) a transcritical state with vanishing latent heat (see, e.g., ref 26, loc cit). Figure 2, which has been adapted from this 1973 paper,26 illustrates the relevant domains on the pch/pc vs Tw/Tc plane for the vaporization (without “envelope flame” combustion (see discussion presented later in this paper)) of a fuel with the nominal thermophysical properties of n-dodecane (n-C12H26). Regarding the evaporation of fuel “blends”, a powerful transport formalism that is well-suited to such cases (called “continuous mixture theory”) was first introduced by Gal-Or et al. in 1975.28 In this formalism, one represents the chemical composition in each phase by a continuous “spectrum” or distribution functionsas was more common in the field of polymer reaction engineering. Equations then are derived/solved for the evolution of this distribution function (or its mathematical “moments”)sas discussed further in Section 4, presented later in this paper. While the phase equilibrium aspects of this approach were investigated by ChEs in the following decade (e.g., see refs 29-31), this formalism has been rather fully exploited in more-recent studies of individual droplet evaporation rates,32,33a,33b and, most recently, in 2008 by Laurent et al.34 Variants of this approach have also been adopted by MechEs and AeroEs in simulations of fuel droplet behavior in diesel engines (which represent an important class of non-steadystate spray combustors)35 and turbulent droplet-laden mixing layers.36,37 2.2. Laminar “Diffusion Flames” and Unsteady “Flamelets”: “Group Combustion”. Single (vapor)-phase nonpremixed flames are called “diffusion” flames, and a groundbreaking analysis of such laminar (co-)flow round fuel jet flames was published by Burke and Schumann in Industrial and

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Figure 3. Experimentally probed structure of a laminar, counterflow diffusion flame of methane (doped with jet fuel) vapor in air: Major species concentration profiles and corresponding local gas temperatures (in Kelvin) are shown. Conditions: p ) 1 atm, nominal gas strain rate ) 140 s-1. (After Bufferand et al.;43 also see the work of Tsuji.42)

Engineering Chemistry (I&EC) as early as 1928.38 They considered the so-called “flame-sheet” limit, in which the zone of intense vapor-phase chemical reaction is negligible in thickness, compared to the necessary molecular transport zones on either side of it. Needless to say, in the intervening 80 years, there have been many extensions and applications of this asymptotic approach, perhaps the most important being the notion that, in turbulent nonpremixed systems, there are distributed “flamelets”39-41 whose detailed structure (locally normal to the active reaction zone) is not unlike those probed in laboratory studies of steady counterflow laminar diffusion flames (see, e.g., refs 42 and 43 and Figures 3a and 3b). As an interesting corollary of this viewpoint, when one calculates the actual volumetric chemical heat release rates (or “space heating rate” (SHR) values) achieved in high-intensity turbulent spray combustors (e.g., chemical propulsion applications), it becomes clear that the actual volume fraction of active vapor-phase reaction zones must be quite smallsperhaps as low as 10-4 or 100 ppm. In other words, if droplet evaporation and turbulenceenhanced molecular mixing were NOT rate-limiting factors in such devices, the homogeneous chemical kinetics of fuel vapor oxidation would be capable of heat-release rates (i.e., SHR values) higher than those actually achieved (by more than 3 orders of magnitude!). We return to this evidence for the role of turbulence-enhanced “micro-mixing” limitations, and the role of “laminar flamelets”, which are discussed in Section 2.3. For application to spray combustors, a question of considerable fundamental interest has always been the location44 of such diffusion flames: i.e., are they present in the boundary layer of each evaporating droplet or do many droplets inevitably “pool” their vapor to supply a much more remote diffusion flame (which is a phenomenon aptly called “group” combustion)? A chemical engineering approach to answer this question was provided in 1976 by Labowsky and Rosner,46 who showed that the relevant dimensionless parameter that delineates these distinct regimes could be regarded as a Damko¨hler number (or Thiele/Wagner modulus) of the type already familiar to ChEs who are concerned with the access of reactants to the interior of porous catalyst support pellets (see, e.g., the review presented in ref 3, Part 6.4.4). 2.3. Turbulent Micromixing with Vapor Phase Chemical Reaction: Multienvironment PDF Methods. The aforementioned “laminar flamelet model” (LFM) opened the door to predicting local time-averaged volumetric rates of chemical energy release (and, less accurately, pollutant production) in turbulent combustors by providing a quantitative link between the local rates of turbulence-enhanced molecular mixing, and the presumably undisturbed local structure of laminar diffusion

flames (either measured or computed). In the simplest singlephase formulations (in an absence of vaporizing droplets, almostequal molecular diffusivities, and thin reaction zones (on the scale of turbulent eddies)), both of these quantities are described in terms of a single scalar “passive” variable, called the local “mixture fraction”. this parameter was perhaps first introduced into the field of turbulent fuel jet flames by Hottel et al.47 and Spalding,18 but it has precursors in the earlier theoretical studies of laminar flames by Burke and Schumann38 and the work of Zeldovich.48 Often denoted as Z and defined to be unity in the pure fuel vapor and zero in the oxidizer stream, this single scalar variable is defined in such a way that it satisfies a chemical “source-free” balance equation (partial differential equation, PDE)sagain in the absence of vaporizing droplets. Thus, we postpone explicit consideration of spray-containing flames to Section 3; here, we summarize only key features of the simpler, single-phase situation, when all of the molecular diffusivities are almost equal. Subject to these constraints, the local rate of molecular mixing, which is often simply abbreviated as the “scalar dissipation rate” (SDR), can be shown to be SDR ) 2D(grad Z)2

(3)

SDR is seen to be always positive and have units of reciprocal time. [The associated “molecular mixing time” should be distinguished from the so-called “eddy breakup” time (tt, which is defined as the ratio of the local turbulence kinetic energy to the local mechanical turbulence energy dissipation rate).] Moreover, the local reaction rate for any chemical species i [for example, with units of kg/(m3 s)] is determined to be r˙i′′′ ) -F(SDR)

( ) d2ωi dZ2

(4) LFM

(from ref 39), where the indicated 2d derivative is computed based on the laminar flamelet model. Of course, in chemically reacting turbulent flows, all local state variables, including Z, are fluctuating and considerable experimental and theoretical attention has been focused on the “statistics” of both Z and SDR (e.g., their probability density functions (PDFs)). According to eq 4, time-averaged reaction rates (and, hence, corresponding heat-release rates, -∑ir˙′′′i · hi) can be calculated by combining a knowledge of flamelet structure (in “Z space”) with the local value of 〈SDR | Zstoich〉si.e., the time-averaged value of the local molecular mixing rate conditioned on the stoichiometric Value of Z. In engineering practice, for single-phase chemically reacting turbulent flows, one usually writes/numerically solves plausible “model” PDEs for the local mean 〈Z〉, and its variance 〈Z′Z′〉, where Z′ ≡ Z - 〈Z〉, on the assumption that such results will be insensitive to the presumed shape of PDF(Z) (see, e.g., ref 49, Chapter 13 in ref 50, or ref 3). In these terms, the local time average of SDR (or the first moment of its PDF), which is written as 〈SDR〉, is considered to be simply proportional to 〈Z′Z′〉/tt, where tt is the previously mentioned local turbulent “eddy breakup” time, which corresponds to the familiar tendency of local “unmixedness” 〈Z′Z′〉 to decay exponentially in a domain of spatially uniform 〈Z〉. Finally, and more problematically, 〈SDR | Zstoich〉 and 〈SDR〉 are taken to be almost equal, on the assumption that fluctuations in SDR and Z, even locally, are only “weakly correlated”. Recent extensions of this LFM approach, which are beyond the scope of the present review, involve the consequences local flamelet extinction (above some experimentally determined

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“quench” value of SDR), and the possible re-ignition of the resulting “partially pre-mixed ”vapors (see, e.g., ref 52). For the present review, I simply note here that localized vapor-phase flamelet extinction (which is expected to also be associated with CO emissions) becomes a likely source of spray combustor inefficiencysbeyond the obvious loss mechanism of unvaporized droplets. We should not conclude this section without calling attention to the recent development/exploitation of so-called “multienvironment” (ME) turbulent mixing models that are amenable to the inclusion of finite-rate chemical kinetics (see ref 53). In such cases, a representation of the single-point PDF of reacting scalars, in terms of a finite number of discrete environments, when combined with the “QMOM” computational strategy mentioned later in Section 3.4 (in the context of multivariate population balances), leads to an Eulerian PDE formulation that is compatible with conventional CMFD software (see Section 3.4). The evolution and applications to computational fluid dynamics (CFD)-based chemical reactor design of these “MEPDF” models from Lagrangian micromixing precursors, which were first proposed in 1986 by Villermaux54 in studies of stirred chemical reactors, has been discussed and implemented more recently by Fox.49,55 This is closely related to the Gaussian quadrature approach outlined in Section 4. 2.4. Turbulent Two-Phase Dispersion/Segregation/Evaporation Phenomena. Returning to the inevitable presence of the suspended droplet phase, it is necessary to also address the following questions that are possibly relevant to the performance of high-intensity spray combustors:56 • (Q1) Does the very presence of droplets appreciably modify (amplify or diminish) the local turbulence? (See, e.g., ref 57.) • (Q2) Does the turbulence experienced by individual droplets appreciably modify their time-averaged evaporation rates? (See, e.g., ref 58.) • (Q3) Does the tendency for droplet “inertial enrichment” in a turbulent flow field bring a sufficiently large number of initially widely dispersed droplets close enough to modify their evaporation rates or produce appreciable collision coalescence? (See, e.g., refs 59-61.) • (Q4) How is the spatial dispersion of droplets in the chamber modified by turbulence? (See, e.g., refs 62 and 63.) Although interesting fundamental research results have been reported on each of these questions over the past 50 years, evidently little of this research has specifically been brought to bear on the simulation or design of spray combustors. Perhaps justifiably, other challenging modeling issues (see Sections 3 and 4) have taken precedence; however, we believe that some useful conclusions in this general area can be reached by exploiting the idealized spray combustor “platform” approach mentioned in Section 3.2 (see, e.g., ref 64). A rather different set of fundamental questions arises, with respect to the presence of small fuel droplets within the abovementioned diffusion flame structures: Must the off-line “library” of diffusion flames used in adopting the laminar “flamelet” modeling approach include so-called “spray flames” (containing a significant nonevaporated fuel content)? For an account of the early steps taken in this particular direction, see ref 65. For recent computational studies of laminar “spray flames”, see refs 66-68. 3. Idealized Mathematical Models of Spray Combustors Of course, an understanding of the aforementioned “unit processes” is usually necessary but is not sufficient for designing high-performance spray combustors, regardless of the applica-

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tion. It remains for us to address the question: How has this information been “assembled” to arrive at rational methods that accelerate the often costly design/development process? Not surprisingly, these methods have evolved considerablysespecially since the introduction/use of electronic digital computation, leading to what is presently called “computational multifluid dynamics” (see Section 3.4). Yet, as demonstrated below (in Sections 3.2 and 3.3), considerable insight has (and continues to!) come from the development/exercise of so-called “modular” mathematical modelssbased on representing an actual “nonideal” reactor as a network of simpler idealized devices, often either as “plug flow” (Dirac RTDF) or “perfectly well-mixed” in the Denbigh sense (see refs 69 and 70). Pioneers in this area, which are discussed briefly in Section 3.3, include Bragg,71 Swithenbank,72 Eisenklam et al.,11 and Villermaux.73 In the context of computational transport modeling (Section 3.4), these may be viewed as “coarse discretizations” of the real device. 3.1. Early “Plug-Flow” Models: Spray Population Balance. Plug-flow models were introduced early on, mainly to examine the consequences of injected fuel droplet spray “polydispersity” (i.e., what is the performance penalty associated with the fact that practical fuel injectors produce a “spectrum” of droplet sizes?). One of the earliest of such accounts was that of Probert in 1946,74 whose research was performed in support of the British jet engine development program. Morerealistic variants of this analysis were subsequently developed/ published by Spalding,18 Priem and Heidmann,75 Williams,41 Nuruzzaman et al.,76 and Sutton et al.77], all of whom mainly incorporated the momentum-, mass-, and heat-transfer coupling between the two co-flowing phases. Common to these analyses were the principal underlying assumptions: • (A1) The droplet volume fraction is low enough to preclude appreciable droplet-droplet interactions; • (A2) Each droplet evaporates into its local environment, according to a QS-diffusion-controlled rate law, and • (A3) Negligible vapor phase “back-mixing”. Although these idealizations certainly restrict the generality of such analyses, instructive conclusions concerning the dependence of the fraction vaporized on the aforementioned parameter (Dam)vap and initial fuel spray polydispersity could be extracted. Perhaps equally important, several of these analyses marked the explicit introduction of a droplet “population-balance” approach to this class of problems (i.e., deriving/solving an evolution equation for the droplet size distribution function, subject to these albeit restrictive simplifications). As will be seen (Section 4), in more general situations, one must consider many such interacting “populations”, each of which is more generally determined to be governed by a Boltzmann-like integro-PDE. 3.2. Well-Mixed Limit for a Spray Combustor: Multivariate Spray PBE Approach. Early in the development of stable, high-intensity spray combustors for aircraft gas turbines, it was realized that, to provide “continuous ignition”, hot gaseous combustion product recirculation, usually with swirl superimposed, was required in the near-injector region; i.e., the so-called “primary zone” (shown inside the shaded zone in Figure 1) should be operated closer to the “perfectly well-mixed” limit previously mentioned. The remainder of the air and residence time needed to achieve near-complete droplet evaporation/fuel vapor combustion was deliberately introduced downstream of this “primary zone”, creating a nearer-plug-flow “secondary” zone, and ultimately a near-plug-flow “dilution zone” used to drop the gas temperature down to levels that could be accommodated by the available turbine blade materials. Bragg71

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Figure 4. Predicted (a) dependence of jet-stirred combustor efficiency (defined using the fraction vaporized) and (b) combustion “intensity” (defined as the dimensionless volumetric chemical energy release rate) on the characteristic droplet vaporization Damko¨hler number 〈t(flow)〉/t(vap) (see Sections 2.1 and 3.2); comparison of non-QS results (solid line) with results imposing the QS approximation (dashed line). Conditions: p ) 20 atm, injector log-normal droplet size distribution (LN DSD) spread ) 2, kerosene fuel/air, fuel equivalence ratio ) 0.8 (after Rosner et al.24).

Figure 5. Comparison of predicted output (fine) and input (bold) fuel droplet volume distribution functions; chamber output using QS-approximation is shown as a dashed line). Conditions: p ) 20 atm, injector LN DSD spread ) 2, overall equivalence ratio ) 0.8; volume variable ≡ droplet volume/ feed mean droplet volume (after Rosner et al.24).

considered the optimization of two-module representations of AGT combustors for the purpose of minimizing their volume and weight for a given chemical energy release rate. Surprisingly, the fuel droplet population balance (or the socalled “spray equation”) did not have a prominent role in these early representations of the primary zonesalthough some preliminary work along these lines was published in 1960 by Courtney78 and later, in 1999, by Lefebvre.79 Indeed, this gap motivated our recent studies of adiabatic well (jet-)mixed spray combustors.24,25 Another of our goals has been to use this idealized model to systematically test the limits of certain assumptions commonly introduced into more computer-intensive simulation methods, one of which being the QS-evaporation rate assumption previously mentioned. Thus, Figures 4a and 4b, which summarize the results of instructive parametric studies,24 reveal two interesting features: (1) While the droplet vaporization-controlled combustion efficiency increases monotonically with (Dam)vap,ref, the corresponding combustion intensity passes through a local maximum. (2) At pressures above ca. 10 atm, the QS-predicted maximum intensity is seriously underestimated. Another interesting output of this model (see Figure 5) is the systematic difference between the droplet size distribution (DSD) exiting the combustor (representing a “primary zone”) and the feed DSD. Several generalizations of this idealized PBbased spray combustor model are currently in progress.23,80 Perhaps the one most relevant to this review and our broader objectives involves the performance consequences of introducing multicomponent fuel droplets. A simple example, of consider-

Figure 6. Predicted joint probability density function (PDF) for ethanol + water (EtOH + H2O) droplets in an idealized V-2 liquid propellant rocket motor chamber. Conditions: p ) 15 atm, T ) 2970 K. Fuel droplet state variables: m1 ) mass of EtOH in droplet, m2 ) mass of H2O in droplet; µ ) dimensionless total mass: m1 + m2; ω ) EtOH mass fraction: m1/(m1 + m2). Note the prevalence of large numbers of small, water-rich droplets, as a result of the loss of the higher-volatility EtOH component. (After Rosner and Arias-Zugasti.80)

able historic importance, is that of the German V-2 chemical rocket thrust chamber of WWII (see, e.g., ref 5), into which an ethanol/water mixture (75 wt % ethanol (EtOH) fuel) was sprayed, with the oxidizer being liquid oxygen. Because of the fact that the state of any droplet in such a chamber cannot be uniquely defined without introducing at least two scalar variables (e.g., droplet diameter and EtOH mass fraction), a PB analysis of such a situation must be at least ”bivariate”. We are currently calculating/studying the unusual joint PDFs that describe such situations (see, e.g., Figure 6, which displays the presence of a large number of small, water-rich droplets)sfrom which the relevant spray combustor performance characteristics follow, via the evaluation of suitable weighted integrals.80 3.3. “Modular” Models. Representing complex systems by a discrete network of simpler units is a widely used modeling strategy, with interesting precedents/examples in all branches of engineering. To cite one familiar example, biochemical engineers have made very effective use of such “compartment models” in the development of artificial organs and drug delivery systems. Returning to the subject at hand, network models of spray combustors were introduced/exploited in the 1970s by Swithenbank et al.72 and Munz and Eisenklam81 in an effort to address the formidable problems of predicting static operability limits (what is called rich or lean “blow-out”), as well as the emissions of pollutants formed by complex chemical reaction mechanisms. (Even today, these characteristics cannot be reliably predicted.) However, while network representations of single combustors containing as many as seven interconnected subunits were “identified” and examined numerically, simultaneous consideration of the fuel spray population balance was deliberately suppressed in these early studies. It seems likely that such generalizations have subsequently been implemented/ exploited by individual industrial research and development (R&D) centers; however, few studies of this type appear in the archival spray combustor literature. 3.4. Computational “Multi-fluid”/Dimensional Dynamics Approach. A modeling path that has been under sustained development, especially in the last quarter-century, is that which now is often called “Computational Multi-Fluid Dynamics” (CMFD). Here, the aforementioned “compartments” remain notional, but they increase in number by many orders of magnitudespresently by about a factor of ca. 106. Each of the field variables chosen to define the local instantaneous state of the chemically reacting mixture (such as gas velocity compo-


nents, chemical composition, ...) within each compartment, or “finite element”, is assumed to satisfy a PDE of the standard Eulerian form: accumulation rate + net convective outflow rate ) net inflow rate by diffusion + net source (see, e.g., Chapter 2 of ref 3). The earliest examples of this approach (see, e.g., refs 82-84) actually neglected the presence of evaporating droplets, instead, making limiting assumptions about their role in affecting the mixture fluid dynamics or supplying fuel vapor for turbulent mixing and vapor-phase chemical reaction. More-recent examples explicitly adopt the “multi-fluid (continuum)” viewpoint, in which all phases (perhaps including suspended solids as well as microdroplets) “co-exist” and exchange momentum, mass, and energy (see, e.g., refs 84-87). Needless to say, many interesting and nontrivial questions arise in selecting appropriate state variables, and modeling these individual interphase exchange-rate processes (Section 2.1), homogeneous chemical kinetic rate laws, and, of course, boundary conditions (see, e.g., refs 3 and 88) in a manner consistent with both physicochemical reality and numerical accuracy. Although these cannot be addressed explicitly here, this brings us to the focus of Section 4, which emphasizes the emergence of a perspective and self-consistent formalism that evidently is capable of yielding a coupled system of scalar PDEs of the aforementioned “canonical” form (Section 4.1). 4. Toward More Comprehensive PBE-Based Models In Section 3.1, we noted the rather natural introduction of a “population-balance” (PB) viewpoint for the suspended droplet phase in some of the earliest models of spray combustor performance. As noted below, an effective Gaussian quadraturebased moment method has been developed to examine such population balance equations (PBEs). However, in the present more-general context, it should be remembered that the continuum balance PDEs routinely considered to govern the continuous vapor phase actually result from the lowest moments of kinetic-theory molecular PBEs associated with the pioneering work of Maxwell and Boltzmann (see, e.g., ref 89). Also, readers familiar with statistical formulations of turbulent single-phase chemically reacting flows will recognize that single-point joint PDFs are the natural starting point for deriving an evolution equation consistent with the local “balance” PDEs (see, e.g., refs 49, 90, and 91), which is a concept that can even be broadened to include a coexisting spray (see, e.g., ref 92). Moreover, if the combustion reactions happen to produce a condensable intermediate (e.g., “soot”), or nanoparticle reaction product (e.g., SiO2 or a metal oxide), then a PBE would also govern this “aerosol phase”.93-95 Even the photons emitted/ absorbed/scattered at high pressures by high-temperature polar molecules and condensed matter in the combustion products (e.g., in deliberately “luminous” flames) will be governed by a suitable population balance equation (see, e.g., ref 13). It would appear that this “commonality” can/should be recognized and exploited in formulating future mathematical models of such devices. Fortunately, recent advances in the efficient solution of multivariate PBEs (Section 4.1) promise to enable such progress. 4.1. Interacting “Population-Balance” Perspective: “QMOM”-Based Solution Methods. Our present focus on predicting the performance of high-intensity spray combustors raises mathematical modeling issues common to many challenging technologies, transcending chemical reaction engineer-

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ing. Indeed, one finds that, in most multiphase, chemically reacting flow engineering problems of current and future engineering interest, several interacting multiVariate populations must be considered simultaneously. Thus, as noted previously, in our present class of examples, we routinely encounter a complex liquid fuel blend injected as a population of droplets into a flow field comprised of a population of vapor-phase “eddies”, containing populations of “flamelets”, omnidirectional/ frequency photons, and often a population of condensed combustion products or intermediates (e.g., participating “soot” aggregates). Each of these populations can be described by an appropriate balance equation (pseudo-continuum or often discrete) and, generally, these populations “influence each other” (i.e., there are two-way interactions among all of them, as schematically indicated in Figure 8 (see ref 96). An attractive feature of moment-based formulations (see, e.g., refs 93, 99, 100, and 101) is that, for each of these interacting populations, including intrinsically multivariate ones, we are often led to coupled Eulerian PDEs of the canonical form: “transient convective-diffusion with net source term”, as mentioned in Section 3.4, for which there is already considerable experience/ software. In discussing the general requirements for an effective algorithm to examine population balances in spatially nonuniform flow systems,101 we emphasized the essential features: • (F1) It should not presume that the associated PDF always retains some prescribed simple functional form (e.g., log-normal, beta, etc.) • (F2) It should admit realistic rate laws without arbitrarily restricting their functional form (i.e., their dependence on population state variables) • (F3) It should be amenable to extensions to describe multivariate PDFs • (F4) It should lead to a closed set of PDEs (for the moments or their surrogates, see below) of the canonical Eulerian form already being solved numerically in simpler, single-phase reactive “convective-diffusion” problems The Gaussian quadrature-based moment approach first introduced in 1997 by McGraw102 for aerosol populationssnow called QMOM (see below)spossesses these attributes. In effect, QMOM provides an adaptive representation of the actual population PDF, in terms of a finite number of Dirac functions whose positions (“abscissae”) and strengths (“weights”) evolve in accordance with the associated derived moment conservation equations. Rather than adopting the moments as new dependent variables, Marchisio and Fox103 subsequently showed that the abscissae and weights themselves could be conveniently used as “moment surrogates”sleading to what is now called “Direct”QMOM (which is abbreviated as DQMOM). A similar approach was developed independently by McGraw and Wright,104 using a Jacobian matrix transformation to track the absiccae and weights directly. In an unpublished “white paper” prepared for/transmitted to the National Science Foundation (NSF) in January 2004 by Rosner, Fox, and McGraw, the authors noted that the Gaussian quadrature-based moment methods, which were proving to be so effective in the bivariate modeling/interpretation of our experiments on nanoparticle synthesis/evolution in laminar counterflow diffusion flames (see refs 94 and 105; also see Figure 7, which provides early evidence for the success of QMOM in tracking the moments of a bivariate PDF), and then were applied/extended to the performance of industrial MSMPRtype univariate population crystallizers (see, e.g., ref 108), could be brought to bear on a much broader class of multiphase CRE-

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Figure 7. Comparison of quadrature moment (QMOM)-based dimensionless moment “surface” (for uncoupled but simultaneous Brownian continuum regime aggregate coagulation and surface-area-driven sintering) with that predicted using the more computer-intensive Monte Carlo simulation method in the long-time limit, with t(coag)/t(sinter) ) 10-2; k is the exponent on the particle volume state variable, and l is the exponent on the area state variable. (After Wright et al.105 and Rosner et al.101). For the physical significance of several “mixed moments” µkl, see the work of Rosner et al.101 The results of Monte Carlo simulations of bivariate populations of coagulating/sintering aggregates in a constant environment are contained in refs 106 and 107.

problems involving interacting multivariate populationssincluding spray combustors operated for purposes of either chemical synthesis or propulsion. Indeed, this was the precursor of two parallel NSF-supported research programs, one of which led the preparation of the Rosner et al.24 work and the present contribution, and the other led to the “flame spray pyrolysis” example briefly discussed in Section 4.2. This vision also underlies much of the recent Proceedings of the CISM Workshop, which were edited by Marchisio and Fox.109 4.2. An Instructive Recent Example: “Flame Spray Pyrolysis”. One of the many ways that combustion can be used to facilitate the continuous production of desired nanoparticle “powders” (see, e.g., refs 1, 2, 110, and 111) is to use a spray device in which the injected liquid fuel also contains one or more dissolved particle precursors (see, e.g., refs. 112 and 113). In such flame spray pyrolysis (“FSP”) cases, the reactor performance criteria must include not only particle yields, but also particle properties that transcend the size distribution function alone (i.e., particle morphology, stoichiometry, crystal phase, ...). While several of these features remain beyond the scope of presently used simulation techniques, can reactor modeling contribute to the economical design and optimization of this class of spray combustors? With this question and our “interacting population balance” perspective in mind (cf. Figure 8), here, we briefly examine an interesting example reported at the recent 2008 Centennial Meeting of the American Institute of Chemical Engineers (AIChE). Significantly, this example explicitly involves no less than three of the four interacting populations represented schematically in Figure 8. In 2008, Sung et al.114 reported preliminary simulation results for an instructive FSP example with a simple planar geometry involving three coexisting “populations”. They considered a centrally located continuous spray of evaporating heptane droplets that not only supplies the fuel vapor for “flamelet” combustion in co-flowing, turbulent gaseous O2, but also releases a precursor which is first oxidized to the condensable species TiO2 which then grows/coagulates via Brownian motion. For simplicity, the precursor release rate was taken to be directly proportional to the predicted fuel (solvent) evaporation rate. The vapor phase was treated as a transient continuum (ideal gas mixture) via “direct numerical simulation”, the spray was treated

Figure 8. Schematic of four interacting coexisting populations considered in Section 4. Circles represent (respectively, clockwise) the populations of turbulent vapor-phase “eddies”, fuel spray droplets, nanoparticles suspended in vapor phase, and photonsswith each population being “multivariate”; connecting lines indicate likely two-way couplings.

using a Lagrangian simulation of the univariate spray population balance, and the univariate TiO2 nanoparticle population balance was treated using an Eulerian QMOM technique, following six moments. The associated numerical solvers (for each coexisting phase) were closely coupled and the domain was subdivided into ca. 3 M cells. A typical “snapshot” showing the predicted instantaneous temperature field, and the associated nanoparticle number density for a jet Reynolds number of ca. 0.8 × 104 is shown in Figure 9. From an ensemble of such results (over a sufficient time interval), one can obviously compute local timeaveraged results. It is recognized that this “DNS” vapor-phase formulation, while free of “turbulence modeling” assumptions, will probably not be practical for FSP examples of much greater geometric and chemical kinetic complexity. It will also be necessary to ultimately introduce more-accurate individual rate laws (e.g., for aggregate coagulation; see e.g., ref 115) and address multivariate populations. However, the further development/ exercise of this type of simulation tool, especially when compared with the results of corresponding physicochemical experiments, should prove useful to guide future modeling effortssincluding the development and testing of generalized turbulence models to enable the “time” variable to be suppressed for “(quasi)-steady-state” turbulent flowsseither fully (as for Reynolds-time-averaged Navier-Stokes (RANS) simulations) or partially (as in more demanding “large-eddy” simulations (LES) (see, e.g., the recent work of Moin and Apte116 and Menon and Patel,117 and the recent fundamental studies of Okong’o et al.118). 4.3. Prospects. The variety and complexity of spray combustors, together with the rather-limited body of well-characterized/ documented performance data, will continue to pose formidable challenges to present and future generations of chemical reaction engineers. Yet, useful mathematical/numerical models will provide needed information about how device performance will be dependent on such factors as geometry (chamber volume, shape), inlet conditions (both liquid phase and gas phase), liquid fuel properties, and injector (“atomizer”) performance. Of course, this information not only can be used for purposes of engineering design, but also to predict the consequences of, for example, fuel substitution, nitrogen removal from the inlet air, introduction of fuel blends or additives, and even to guide the model-based control of an existing spray combustor design. In short, rational yet tractable computational models have the potential of cutting the currently high cost of new spray combustor design and development. As reviewed previously, while much has been accomplished, more powerful/general design methods are clearly needed. This


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Figure 9. “Snapshot” showing the predicted instantaneous temperature field, and the associated titania nanoparticle number density, for a jet Reynolds number of ca. 0.8 × 104. Particle precursor-seeded heptane planar spray into coflowing oxygen vapor. (After Sung et al.114)

will require continued research investments and innovations in theory (our present emphasis), as well as computation and experiment. Thus far, only relatively modest steps have been taken along the modeling path sketched in Section 4.1. It would seem that the stage is now set for the next generation of mathematical models, which will exploit these recent developments more fully. 5. Conclusions In this paper, we have traced the historical and conceptual development of rational methods to predict important aspects of high-intensity “spray combustor” performance, to select but one representative example from among a broad class of multiphase chemical reactors. Although we have been obliged to be selective in our choice of focal points and explicit citations (see the listed references), it is quite clear that our present understanding of spray combustors, irrespective of their application, owes much to the cumulative and rather unique contributions of chemical engineerssspanning the issues of nonideal mixture chemical thermodynamics and kinetics, transport properties, and turbulent micromixing processes, all the way to the dynamics of suspensions, systems aspects, and numerical simulation methods. Our own recent studies (e.g., the listed references (loc. cit) and Figures 4-7) have revealed the extent to which population-balance methods and idealized chemical contactor concepts, central to a ChE’s “toolbox”, can economically reveal the value/consequences of proposed approximations, and will hopefully guide the development of more comprehensive, yet tractable, models of these rather complex devices. But perhaps most importantly, our recent investigations/ collaborations have pointed the way toward a unifying viewpoint/ methodology, which should prove helpful in future mathematical modeling/simulation studiessthat of interacting multivariate populations which can be efficiently addressed by exploiting suitably generalized Gaussian quadrature-based “moment” methods. It will be especially gratifying if this invited manuscript, which deliberately looks to the future as well as our past, motivates creative new developments along these general lines.

Acknowledgment The author gratefully acknowledges the support of NSF (via Yale Grant No. CTS 0522944); the contributions of his research collaboratorssDrs. Manuel Arias-Zugasti, Robert McGraw (Brookhaven National Laboratories), Michael Labowsky, and former graduate students (see the listed references (loc cit)); and helpful discussions with my Yale Engineering colleaguess Profs. Alessandro Gomez (Dept. of Mechanical Engineering) and Juan Fernandez de la Mora (Dept. of Mechanical Engineering). More broadly, our “virtual center” has also benefited from the helpful comments/feedback of Prof. Rodney Fox (Dept. of Chemical Engineering, Iowa State), Bertrum Diemer, Jr. (DuPont), Venkat Raman (Dept. of Mechanical Engineering, University of Texas at Austin), and Pushkar Tandon (Corning). Finally, I want to dedicate this contribution to the memory of Prof. Sheldon K. Friedlander (Cal Tech and UCLA) who, when I was still a graduate student (initially a Guggenheim “Jet Propulsion Fellow” at Princeton), pointed out to me that “rocket science” was really “chemical reaction engineering”, and who later championed the application of population balance and moment methods in the emerging field of “aerosol reaction engineering”. Abbreviations/Acronyms AGT ) aircraft gas turbine CDF ) counterflow diffusion flame CISM ) International Center for Mechanical Sciences CFD ) computational fluid dynamics CMFD ) computational multifluid dynamics CRE ) chemical reaction engineering Dam ) vaporization-based Damko¨hler number (see Sections 2.1 and 3.2) DNS ) direct numerical simulation DQMOM ) “direct” quadrature method of moments DSD ) droplet size distribution (liquid fuel) EtOH ) ethanol FSP ) flame spray pyrolysis grad ) spatial gradient operator LES ) large-eddy simulation

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LFM ) laminar “flamelet” model LN ) log-normal (drop-size distribution) MC ) Monte Carlo ME ) multienvironment MSMPR ) mixed suspension-mixed product removal (crystallizer) np ) nanoparticle NSF ) National Science Foundation PB ) population balance PBE ) population balance equation PDE ) partial differential equation PDF ) probability density function QMOM ) quadrature method of moments QS ) quasi-steady RANS ) Reynolds-averaged Navier-Stokes R&D ) research and development RTDF ) residence time distribution function SDR ) scalar dissipation rate (see eq 3) SHR ) space heating rate (e.g., GW/m3) WWII ) World War II

Literature Cited (1) (a) Rosner, D. E. Combustion Synthesis and Materials Processing. Chem. Eng. Educ. 1997, 31, 228-235. (b) Chem. Eng. Educ. 1998, 32 (1 (Winter)), 82-83. (c) Chem. Eng. Educ. 1980, 14 (4 (Fall)), 193-212. (2) Pratsinis, S. E. Flame Aerosol Synthesis of Ceramic Powders. Prog. Energy Combust. Sci. 1998, 24, 197–2191. (3) Rosner, D. E. Transport Processes in Chemically Reacting Flow Systems, 1st Edition; Butterworths: Boston, MA, 1986. (4) Correa, S. M. Power Generation and Aero-Propulsion Gas Turbines: From Combustion Science to Combustion Technology. In Proceedings of the 27th Symposium on Combustion; The Combustion Institute: Pittsburgh, PA, 1998; pp 1793-1807. (5) Sutton, G. P. History of Liquid Propellant Rocket Engines; AIAA Press: Reston, VA, 2006. (6) Conroy, E. H., Jr.; Johnstone, H. F. Combustion of Sulfur in a Venturi Spray Burner. Ind. Eng. Chem. 1948, 41 (12), 2741–2748. (7) This is in contrast to the development of the first large-scale isotope enrichment and nuclear reactor technologies in the United States during WWIIs mainly by industrial chemical engineers working in the “shadow” of a smaller number of more outspoken (if often eloquent) physicists (see, e.g., ref 8)! (8) Ndiaye, P. Nylon and Bombs: DuPont and the march of modern America (in Fr.); Johns Hopkins University Press: Baltimore, MD, 2006. (Translated by E. Forster.) (9) Ranz, W. E.; Marshall, W. R., Jr. Chem. Eng. Progress 1952, 141146, 173–180. (10) McAdams, W. Heat Transmission; McGraw-Hill: New York, 1954. (11) Eisenklam, P.; Arunachalam, S. A.; Weston, J. A. Evaporation Rates and Drag Resistance of Burning Drops. In 11th International Symposium on Combustion; The Combustion Institute: Pittsburgh, PA, 1967; pp 715728. (12) Bird, R.; Stewart, W.; Lightfoot, E. N. Transport Phenomena, Revised, 2nd Edition; Wiley: New York, 2006. (13) Whitaker, S. Fundamental Principles of Heat Transfer; Pergamon Press: New York, 1977; Chapters 8 and 9. (14) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978. (Also available as a reprint from Dover Publications, Mineola, NY, 2005.) (15) Churchill, S. W. Viscous Flows: The Practical Use of Theory; Butterworths-Heinemann: Boston, MA, 1987. (16) Asano, K. Mass Transfer; Wiley-VCH: Weinheim, Germany, 2006. (17) Many such correlations can be found in the literature (see, e.g., the summaries of Spalding,18,19 Rosner,3 Sirignano,20 and Asano,16 usually prominently involving the dimensionless parameter Bm, which is defined in the text below)ssome based on physical experiments and others based on the correlation of “computer experiments”. However, unfortunately, there is often confusion over where the relevant thermophysical properties are to be evaluated, as well as recent claims that these systematic effects can be “suppressed” by employing certain (irrational) combinations for property estimation! Beishuizen and Roekaerts21 concluded that this transfer coefficient reduction is needed to obtain acceptable agreement with the welldefined flame spray data of Karpetis and Gomez.22

(18) Spalding, D. B. Some Fundamentals of Combustion; Butterworths: London, U.K., 1955. (19) Spalding, D. B. ConVectiVe Mass Transfer; McGraw-Hill: New York, 1963. (20) Sirignano, W. Fluid Dynamics of Drops and Sprays; Cambridge University Press: Cambridge, U.K., 1999. (21) Beishuizen, N. A.; Roekaerts, D. Investigation of Two-way Coupling and Vaporization Interaction in a Turbulent Spray Flame Using a PDF Method. Presented at the 32nd International Symposium on Combustion, Montreal, Canada, August 2008; Paper No. 5G03. (Proceedings in press, 2009.) (22) Karpetis, A.; Gomez, A. An Experimental Study of Well-Defined Turbulent Non-Premixed Spray Flames. Combust. Flame 2000, 121, 1–23. (23) Rosner, D. E.; Arias-Zugasti, M.; Labowsky, M. Interacting Roles of Fuel Evaporation, Micro-Mixing and Homogeneous Chemical Kinetics in Limiting the Performance of Continuous Spray Combustors. Presented at the AIChE Centennial Meeting, Philadelphia, PA, November 2008; Paper No. 79d. (24) Rosner, D. E.; Arias-Zugasti, M.; Labowsky, M. B. Intensity and Efficiency of Spray Fuel-Fed Well Mixed Adiabatic Combustors. Chem. Eng. Sci. 2008, 63 (August), 3909–3920. (25) Duda, J. L.; Vrentas, J. S. Heat or Mass Transfer Controlled Dissolution of an Isolated Sphere. Int. J. Heat Mass Transfer 1971, 14, 395. (26) (a) Rosner, D. E.; Chang, W. S. Transient Evaporation and Combustion of a Fuel Droplet Near Its Critical Temperature. Combust. Sci. Technol. 1973, 7, 145–158. (b) Also see Rosner, D. E. Liquid Droplet Vaporization and Combustion. In Liquid Propellant Rocket Combustion Instability, NASA Report SP-194; National Aeronautics and Space Administration: Washington, DC, 1972; Chapter 2.4, pp 74-100. (27) Crespo, A.; Linan, A. Unsteady Effects in Droplet Evaporation and Combustion. Combust. Sci. Technol. 1975, 11, 9–18. (28) Gal-Or, B.; Cullinan, H. T.; Galli, R. New Thermodynamic Transport Theory for Systems with Continuous Component Density Distributions. Chem. Eng. Sci. 1975, 30, 1085–1092. (29) (a) Cotterman, R. L.; Bender, R.; Prausnitz, J. M. Ind. Eng. Chem.-Process. Des. DeV. 1985, 24 (1), 194–203. (b) Ind. Eng. Chem.Process. Des. DeV. 1985, 24 (1), 434-443. (c) Dissertation, University of California-Berkeley, Berkeley, CA, 1985. (30) Cotterman, R. L.; Prausnitz, J. M. Continuous Thermodynamics for Phase-Equilibrium Calculations in Chemical Process Design. In Kinetic and Thermodynamic Lumping of Multicomponent Mixtures; Astarita, G., Sandler, S. I., Eds.; Elsevier: Amsterdam, 1991; pp 229-275. (31) Shibata, S. K.; Sandler, S. I.; Behrens, R. A. Phase Equilibrium Calculations for Continuous and Semi-continuous Mixtures. Chem. Eng. Sci. 1987, 42 (8), 1977–1988. (32) Hallett, W. L. H. A Simple Model for the Vaporization of Droplets with Large Numbers of Components. Combust. Flame 2000, 121, 334– 344. (33) (a) Arias-Zugasti, M.; Rosner, D. E. Multicomponent Fuel Droplet Vaporization and Combustion Using Spectral Theory for a Continuous Mixture. Combust. Flame 2003, 135, 271–284. (b) Arias-Zugasti, M.; Rosner, D. E. Soret-Transport, Unequal Diffusivity-, and Dilution Effects on Laminar Diffusion Flame Temperatures and Positions. Combust. Flame 2008, 153, 33–44. (34) Laurent, C.; Lavergne, G.; Villedieu, P. Quadrature Method of Moments for Multi-component Spray Vaporization. Presented at the 32nd International Symposium on Combustion, Montreal, Canada, August 2008; Paper No. 5G07. (Proceedings in press, 2009.) (35) Zhu, G. S.; Reitz, R. D. Int. J. Heat Mass Transfer 2002, 45, 495507. (Also see ASME Trans. 2003, 123 (April), 412-418.) (36) LeClerc, P. C.; Bellan, J. Direct Numerical Simulation of a Transitional Temporal Mixing Layer Laden with Multicomponent-Fuel Evaporating Drops Using Continuous Thermodynamics. Phys. Fluids 2004, 16 (6), 1884–1907. (37) Harstad, K.; Bellan, J. Modelling Evaporation of Jet A, JP-7 and RP-1 Drops at 1 to 15 bar. Combust. Flame 2004, 137, 163–177. (38) Burke, S. P.; Schumann, T. E. Diffusion Flames. Ind. Eng. Chem. 1928, 20 (10), 998-1003. (Also see Proc. 1st Symp. Combust. 1928, pp 1-11; reprinted by The Combustion Institute, Pittsburgh PA, 1965.) (39) Bilger, R. W. The Structure of Diffusion Flames. Combust. Sci. Technol. 1976, 13, 155. (b) Bilger, R. W. Proc. Int. Symp. Combust. 1989, 475–488. (c) Bilger, R. W. Ann. ReV. Fluid Mech. 1989, 21, 101. (40) Peters, N. Turbulent Combustion; Cambridge University Press: Cambridge, U.K., 2000. (41) Williams, F. A. Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, 1st Edition; Addison-Wesley: Reading, MA, 1965. (2nd Edition published in 1985; also published in 1994.)

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 (42) Tsuji, H. Counterflow Diffusion Flames. Prog. Energy Combust. Sci. 1982, 8, 93–119. (43) (a) Bufferand, H.; Tosatto, L. La Mantia, B.; Smooke; M. D.; Gomez, A. Combust. Flame In press, 2009. (b) Also see: 32nd International Symposium on Combustion, Montreal, Canada; August 2008, Proceedings in press, 2009. (44) Not only is the location of such thin reaction zones dictated by transport factors, so also is the flame temperature. While beyond the scope of the present review, these interesting features of diffusion flames have been considered further (see ref 33b), not only for the frequently encountered case of unequal molecular diffusivities (for Fick diffusion (fuel and oxidizer) and heat diffusion) but also including mass transport induced by temperature gradients (the Ludwig-Soret effect). Indeed, gas kinetic theory reveals that, when the fuel Fick diffusivity differs significantly from the remaining molecular diffusivities, Soret transport is likely to be simultaneously important.45 (45) Rosner, D. E.; Israel, R. S.; La Mantia, B. Heavy Species LudwigSoret Transport Effects in Air-Breathing Combustion. Combust. Flame 2000, 123, 547–560. (46) Labowsky, M. B.; Rosner, D. E. Conditions for Group Combustion of Droplets in Fuel Clouds: I. Quasi-steady Predictions. In Proceedings of the Symposium on EVaporation/Combustion of Fuel Droplets; Advances in Chemistry Series, No. 166; American Chemical Society: Washington, DC, 1976; pp 63-69. (47) (a) Hottel, H. C.; Hawthorne, W. R. In 3rd International Symposium on Combustion; William and Wilkins/The Combustion Institute: Pittsburgh, PA, 1949; pp 254-266. (b) In 3rd International Symposium on Combustion; William and Wilkins/The Combustion Institute: Pittsburgh, PA, 1949; p 226. (48) Zeldovich, Y. B. In Selected Works, Vol. 1: Chemical Physics and Hydrodynamics, Part II: Flame Propagation; Ostriker, J. P., Ed.; Princeton University Press: Princeton, NJ, 1992. (49) Fox, R. O. Computational Models for Turbulent Reacting Flows; Cambridge University Press: Cambridge, U.K., 2003. (50) Warnatz, J.; Maas, U.; Dibble, R. Combustion; Springer: Berlin, Germany, 1996; Chapter 13. (51) The quench value of SDR, which is written as (SDR)q, is a reciprocal time indicative of the fuel/oxidizer vapor intrinsic combustion kinetics. Indeed, there is a close relation between [(SDR)q]-1 and D/(Su)2, where Su is the laminar pre-mixed flame speed for the same fuel/oxidizer vapor combination (see, e.g., refs 3 and 40). (52) Fox, R. O.; Raman, V. A Multi-Environment Conditional PDF Model for Turbulent Reacting Flows. Phys. Fluids 2004, 16 (12), 4551– 4565. (53) Tang, Q.; Zhao, W.; Bockelie, M.; Fox, R. O. Multi-Environment PDF Method for Modeling Turbulent Combustion Using Realistic Chemical Kinetics. Combust. Theory Modell. 2007, 11 (6), 889–907. (54) (a) Villermaux, J. Micro-mixing Phenomena in Stirred Reactors. In Encyclopedia of Fluid Mechanics; Cheremisinoff, N. P., Ed.; Gulf Publishing Co.: Houston, TX, 1986; Vol. 2, pp 707-771. (b) Also see: Villermaux, J.; Falk, L. A generalized mixing model for initial contacting of reactive fluids. Chem. Eng. Sci. 1994, 49 (24B), 5127-5140. (55) Fox, R. O. On the Relationship Between Lagrangian Micro-mixing Models and Computational Fluid Dynamics. Chem. Eng. Process. 1998, 37, 521–535. (56) (a) Faeth, G. M. Evaporation and Combustion of Sprays. Progress Energy Combust. Sci. 1983, 9, 1-76. (b) Also see: Proceedings of the 26th Symposium on Combustion: The Combustion Institute: Pittsburgh, PA, 1996; pp 1593-1612. (57) Sundaram, S.; Collins, L. R. Numerical Study of the Modulation of Isotropic Turbulence by Suspended Particles. J. Fluid Mech. 1999, 379, 105–143. (58) Birouk, M.; Gokalb, I. Current Status of Droplet Evaporation in Turbulent Flows. Progress Energy Combust. Sci. 2006, 32, 408–423. (59) Reade, W. C.; Collins, L. R. Effect of Preferential Concentration on Turbulent Collision Rates. Phys. Fluids 2000, 12 (10), 2530–2540. (60) Tambour, Y.; Katschevsky, D. Asymptotic Analysis of Droplet Coalescence Effects in Spray Diffusion Flames in a Unidirectional Shear Flow. Atomization Sprays 1995, 5 (4-5), 357–386. (61) Law, C. K. Combustion Physics; Cambridge University Press: Cambridge, U.K., 2006; Section 13.2. (62) Mei, R.; Adrian, R. J.; Hanratty, T. J. Particle Dispersion in Isotropic Turbulence under Stokes Drag and Basset Force with Gravitational Settling. J. Fluid Mech. 1991, 225, 481–495. (63) Gousebet, G.; Berlemont, A. Eulerian and Lagrangian Approaches For Predicting the Behavior of Discrete Particles in Turbulent Flows. Prog. Energy Combust. Sci. 1999, 25, 133–159. (64) Labowsky, M. B.; Rosner, D. E. Turbulence Effects on Evaporation Rate-Controlled Spray Combustor Performance. Presented at the AIChE

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2008 National Meeting, Nov. 17, 2008; Poster No. 186v. (To be submitted to Int. J. Heat Mass Transfer.) (65) (a) Ge, H.-W.; Gutheil, E. Simulation of a Turbulent Spray Flame Using Coupled PDF Gas Phase and Spray Flamelet Modeling. Combust. Flame 2008, 153, 173–185. (b) Also see: Gutheil, E.; Sirignano, W. Combust. Flame 1998, 113, 92-105. (66) Laurent, F.; Massot, M. Multi-fluid Modeling of Laminar Polydisperse Spray Flames: Origin, Assumptions and Comparison of Sectional and Sampling Methods. Combust. Theory Modell. 2001, 5 (4), 537–572. (67) Laurent, F.; Santoro, V.; Noskov, M.; Smooke, M. D.; Gomez, A.; Massot, M. Accurate Treatment of Size Distribution Effects in Polydisperse Spray Diffusion Flames: Multi-fluid Modeling, Computations and Experiments. Combust. Theory Modell. 2004, 8 (2), 385–412. (68) Massot, M. Eulerian Multi-fluid Models for Polydisperse Evaporating Sprays. In Multiphase Reacting Flows: Modelling and Simulation; Springer: Wien, Germany, 2007; pp 79-123. (69) Denbigh, K. G.; Turner, J. C. R. Chemical Reactor Theory; Cambridge University Press: Cambridge, U.K., 1971. (70) Levenspiel, O. Chemical Reaction Engineering; John Wiley and Sons: New York, 1962. (71) (a) Bragg, S. L. Application of Reaction Rate Theory to Combustion Chamber Analysis, Report No. 16170 CF272, Aeronautical Research Council (UK), September 1953. (b) Also see: Avery, W. H.; Hart, R. W. Ind. Eng. Chem. 1953, 45 (8), 1634-1637. (72) Swithenbank, J.; Poll, I.; Vincent, M. W.; Wright, D. D. Combustor Design Fundamentals. In Proceedings of the 14th International Symposium on Combustion; The Combustion Institute: Pittsburgh, PA, 1973; pp 627638. (b) Also see: Proceedings of the 20th International Combustion Symposium, 1984; pp 541-547. (73) Lapicque, F.; Le´de´, J.; Villermaux, J. Design and Optimization of a Reactor for High Temperature Dissociation of Water and Carbon Dioxide Using Solar Energy. Chem. Eng. Sci. 1986, 41 (4), 677-684. (Proceedings of ISCRE 9.) (74) Probert, R. P. The Influence of Spray Particle Size and Distribution in the Combustion of Oil Droplets. Philos. Mag. 1946, 265 (Feb.), 94–105. (75) Priem, R. J.; Heidmann, M. F. Propellant Vaporization as a Design Criterion for Rocket Engine Combustion Chambers, NASA TR R-67; National Aeronautics and Space Administration: Washington, DC, 1960. (76) Nuruzzaman, A. S. M.; Siddall, R. G.; Beer, J. M. The Use of a Simplified Mathematical Model for Prediction of Burnout on Non-Uniform Sprays. Chem. Eng. Sci. 1971, 26, 1635–1649. (77) Sutton, R. D.; Hines, W. S.; Combs, L. P. Development and Application of a Comprehensive Analysis of Liquid-Rocket Combustion. AIAA J. 1972, 10 (2), 194–203. (78) Courtney, W. G. Combustion Intensity in a Heterogeneous Stirred Reactor. Am. Rocket Soc. J. 1960, 30, 356–357. (79) Lefebvre, A. H. Gas Turbine Combustion, 2nd Edition; Taylor and Francis: Philadelphia, PA, 1999. (80) (a) Rosner, D. E.; Arias-Zugasti, M. Bi-variate Population Balance Model of Ethanol-Fueled Spray Combustors. Presented at the AIChE Centennial Meeting, Philadelphia PA, November 2008; Paper No. 156d. (Manuscript to be submitted to AIChE J.)(b) Also see: Condensation-Induced Surface Boiling of Alcohol Fuel Droplets in Combustion Chambers. J. Propul. Power (AIAA) 2009, 25 (3), 826-828. (81) Munz, N.; Eisenklam, P. The Modeling of a High-Intensity Spray Combustion Chamber. In Proceedings of the 16th International Symposium on Combustion; The Combustion Institute: Pittsburgh, PA, 1976; pp 593604. (82) Coupland, J.; Pridden, C. H. Modeling the Flow and Combustion in a Production Gas Turbine Combustor. In Proceedings of the 5th Symposium on Turbulent Shear Flows, Cornell University, Ithaca, NY, August 7-9, 1985; pp 10.1-10.6. (83) Correa, S. M.; Shyy, W. Computational Models and Methods for Continuous Gaseous Turbulent Combustion. Prog. Energy Combust. Sci. 1987, 13 (5), 249–292. (84) (a) Jones, W. P. Turbulence Modeling and Numerical Solution Methods for Variable Density and Combusting Flows. In Turbulent Reacting Flows; Academic Press: London, U.K., 1994; Chapter 6, pp 309-374. (b) Also see: Prediction of Turbulent Flows; Hewitt, G. F., Vassilicos, J. C., Eds.; Cambridge University Press: Cambridge, U.K., 2005; Chapter 4. (85) Swithenbank, J.; Turan, A.; Felton, P. G. 3-Dimensional, 2-phase Mathematical Modeling of Gas Turbine Combustors. In Gas Turbine Combustor Design Problems; Lefebvre, A. H., Ed.; Hemisphere/McGrawHill: New York, 1980. (86) Tolpadi, A. K.; Aggarwal, S. K.; Mongia, H. C. Numer. Heat Transfer 2000, 38 (4), 325. (87) Wittig, S.; Vohringer, O.; Kim, S. High Intensity CombustorssSteady Isobaric Combustion; Wiley-VCH (and DFG): Weinheim (and Bonn), Germany, 2002.

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(88) (a) Rosner, D. E.; Papadopoulos, D. Jump, Creep and Slip Boundary Conditions at Non-Equilibrium Gas/Solid Interfaces. Ind. Eng. Chem. Res. 1996, 35 (9), 3210–3222. (b) Also see: Rosner3 (Section 2.6.1) and Rosner, D. E. Chem. Eng. Educ. 1976, 10 (4) 190-194. (89) Chapman, S.; Cowling, T. G. Mathematical Theory of Non-Uniform Gases; Cambridge University Press: Cambridge, U.K., 1939. (90) Dopazo, C.; O’Brien, E. E. An Approach to the Auto-ignition of a Turbulent Mixture. Acta Astronaut. 1974, 1, 1239–1266. (91) Pope, S. B. PDF Methods for Turbulent Reactive Flows. Prog. Energy Combust. Sci. 1985, 11, 119–192. (92) Zhu, M.; Bray, K. N. C.; Rumberg, O.; Rogg, B. PDF Transport Equations for Two Phase Reactive Flows and Sprays. Combust. Flame 2000, 122, 327–338. (93) Friedlander, S. K. Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics; Oxford University Press: New York, 2000. (94) (a) Rosner, D. E.; Pyykonen, J. J. Bivariate Moment Simulation of Coagulating and Sintering Nanoparticles in Flames. AIChE J. 2002, 48 (3), 476–491. (b) Also see: Xing, Y.; Rosner, D. E.; Koylu, U.; Tandon, P. Morphological Evolution of Nano-particles in Diffusion Flames; Measurements and Modeling. AIChE J. 1997, 43 (11A), 2641-2649. (95) Helble, J. H.; Sarofim, A. F. Factors Determining the Primary Particle Size of Flame-Generated Inorganic Aerosols. J. Colloid Interface Sci. 1989, 128, 348–362. (96) There are many interesting examples of photons interacting with particle populations; e.g., it was found that thermophoretic particle drift can influence the continuum regime coagulation dynamics of radiating soot aggregates (see ref 97, Section 2.4). Another example, particularly relevant to large-scale combustion furnaces, is the photophoretic augmentation of soot deposition rates.98 (97) Rosner, D. E.; Mackowski, D. W.; Tassopoulos, M.; Castillo, J. L.; Garcia-Ybarra, P. Effects of Heat Transfer on the Dynamics and Transport of Small Particles in Gases. Ind. Eng. Chem. Res. 1992, 31 (9), 760–769. (98) Castillo, J. L.; Mackowski, D. W.; Rosner, D. E. Photophoretic Contribution to the Transport of Absorbing Particles Across Combustion Gas Boundary Layers. Prog. Energy Combust. Sci. 1990, 16, 253–260. (99) Diemer, R. B., Jr. Moment Methods for Coagulation, Breakage and Coalescence Problems, Ph.D. Dissertation, Department of Chemical Engineering, University of Delaware, Newark, DE, Spring 1999. (100) Ramkrishna, D. Population Balances; Academic Press: New York, 2000. (101) Rosner, D. E.; McGraw, R. L.; Tandon, P. Multi-variate Population Balances via Moment and Monte Carlo Simulation Methods. Ind. Eng. Chem. Res. 2003, 42, 2699–2711. (102) McGraw, R. L. Description of Aerosol Dynamics by the Quadrature Method of Moments. Aerosol Sci. Technol. 1997, 27 (2), 255–265. (103) (a) Marchisio, D. L.; Fox, R. O. Solution of Population Balance Equations Using Direct Quadrature Methods. J. Aerosol Sci. 2005, 36, 43– 73. (b) Also see: Marchisio, D. L. Quadrature Method of Moments for Polydisperse Flows. In Multiphase Reacting Flows: Modelling and Simulation; Springer: Wien, Germany, 2007; pp 41-77.

(104) McGraw, R. L.; Wright, D. L. Chemically Resolved Aerosol Dynamics for Internal Mixtures by the Quadrature Method of Moments. J. Aerosol Sci. 2003, 34, 189–209. (105) Wright, D. L.; McGraw, R. L.; Rosner, D. E. Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Particle Sintering. J. Colloid Interface Sci. 2001, 236, 242– 251. (106) Tandon, P.; Rosner, D. E. Monte Carlo Simulation of Particle Aggregation and Simultaneous Restructuring. J. Colloid Interface Sci. 1999, 213, 273–286. (107) Rosner, D. E.; Yu, S. Monte Carlo Simulation of Aerosol Aggregation and Simultaneous Spheroidization. AIChE J. 2001, 47 (3)), 545–561. (108) Marchisio, D. L.; Pikturna, J. T.; Fox, R. O.; Vigil, R. D.; Barresi, A. A. Quadrature Method of Moments for Population Balances. AIChE J. 2003, 49, 1266–1276. (109) Marchisio, D. L., Fox, R. O., Eds. Multiphase Reacting Flows: Modelling and Simulation; Springer: Wien, Germany, 2007. [ISBN 9783-211-72463-7.] (110) Wooldridge, M. S. Gas Phase Combustion Synthesis of Particles. Prog. Energy Combust. Sci. 1998, 24, 63–87. (111) Rosner, D. E. Flame Synthesis of Valuable Nano-particles; Recent Progress/Current Needs in Areas of Rate Laws, Population Dynamics and Characterization. Ind. Eng. Chem. Res. 2005, 44, 6045–6055. (112) Kodas, T. T.; Hampden-Smith, M. J. Aerosol Processing of Materials; Wiley-VCH: New York, 1998. (113) Heine, M. C.; Pratsinis, S. E. Droplet and Particle Dynamics During Flame Synthesis of Nanoparticles. Ind. Eng. Chem. Res. 2005, 44, 6222. (114) Sung, J. ; Koo, H.; Raman, V.; Mehta, M.; Fox, R. O.; Heine, M. L.; Pratsinis, S. E. Direct Numerical Simulation of Nano-particle Evolution in Turbulent Spray Flames. Presented at the AIChE Centennial Meeting, Philadelphia, PA, November 2008; Paper No. 468g. (115) (a) Zurita-Gotor, M.; Rosner, D. E. Aggregate Size Distribution Evolution for Brownian CoagulationsSensitivitiy to an Improved Rate Constant. J. Colloid Interface Sci. 2004, 274, 502–514. (b) Also see: J. Colloid Interface Sci. 2002, 255, 10-26. (116) Moin, P.; Apte, S. V. Large Eddy Simulation of Realistic Gas Turbine Combustors. AIAA J. 2006, 44 (4), 698–708. (117) Menon, S.; Patel, N. Subgrid Modeling for Simulation of Spray Combustion in Large Scale Combustors. AIAA J. 2006, 44 (4), 709–723. (118) Okong’o, N.; Leboissetier, A.; Bellan, J. Detailed characteristics of drop-laden mixing layers: Large eddy simulation predictions compared to direct numerical simulation. Phys. Fluids 2008, 20 (10), Article No. 103305.

ReceiVed for reView January 30, 2009 ReVised manuscript receiVed May 28, 2009 Accepted June 1, 2009 IE900167H

Performance: Chemical Engineering

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