Scaling-up and -down in a Nature-Inspired Way - Industrial

Ind. Eng. Chem. Res. 2005, 44, 5011-5019

5011

Scaling-up and -down in a Nature-Inspired Way Marc-Olivier Coppens* Physical Chemistry & Molecular Thermodynamics, DelftChemTech, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

The ongoing revolution at the nanoscale can be brought to good use only if the microscopic and macroscopic worlds are properly connected. This problem has received remarkably little attention in chemical engineering: Designs focus on the molecular and nanoscales, while scale-up is typically treated empirically, lacking rational design at the mesoscale. Where structured designs are used, the term is quasi-equivalent to periodic repetition of the basic unit. Multiscale modeling is challenging when it is applied to heterogeneous systems that are inherently difficult to scale. Chemical innovation needs to be matched by chemical engineering innovation. Nature provides us with excellent examples to address the scaling issue. Promising are hierarchical designs, e.g., involving fractal networks, possibly combined with periodic assemblies. Dynamic selfassembly, source of many natural patterns, is another potential route. Although the examples discussed here focus on multiphase processes, the proposed nature-inspired chemical engineering approach applies equally well to the design of hierarchically structured products. Introduction Chemical engineering’s core task is to bridge the gap between individual molecules or building blocks and products. These products have a desired function, and this function requires a particular structure. The ongoing paradigm shift from chemical process to chemical product engineering highlights the importance of thinking “product” first.1-3 Once an interesting product has been identified on the basis of societal demands and scores of boundary conditions, how can it be made? Once promising microscopic building blocks have been discovered, how can more of them be made from raw materials, and how can their use be implemented on a practical, macroscopic scale? The variety of potential building blocks is being extended daily, to exciting complex molecules, molecular assemblies, and colloidal particles. This generation of chemical engineers now also has “nano-Legos” and “bio-Legos” to play with. Chemical innovation is the driver behind the small-scale synthesis of various building blocks with increasing precision and versatility. However, chemical engineering is the core discipline to approach the challenges of selectively and economically synthesizing building blocks in sufficient quantities and assembling them to enable their use in the form of structured products. Both challenges involve scaling issues, and radically new approaches are required to address them. This paper aims to illustrate how we can learn from Nature in guiding designs that enable the molecular and nano-revolution on the macroscopic scale. Observing Nature leads to extremely useful insights, as Nature is full of hierarchical, composite chemical engineering designs that are intrinsically scaling. Nature is a constant source of inspiration for chemists at the molecular and nanoscales. It inspires construction methods and architectural designs. It inspires the arts. Remarkably, chemical engineers seem to have largely ignored Nature, as conventional designs show little * E-mail: [email protected]. Tel.: +31-15-278 4399. Fax: +31-15-278 8713.

resemblance to Nature’s elegant and functional architecture. There are no good reasons to do so, especially at the dawn of an age where sustainability is essential to our future well being and realizing a harmonic balance with Nature is not merely an objective for poets and admirers of Rousseau or Thoreau. Multiscale Chemical Engineering Designs The responsibility of chemical engineering as a discipline to serve society is greater than ever, as we are faced with urgent requirements for sustainable energy, sustainable industrial products and processes, as well as functional materials for cheap, large-scale use in health care, food, and other essential and less-essential products to supply the needs of a fragile world. Recent years have seen numerous multidisciplinary breakthroughs linking physics, chemistry, and biology, as well as increasing computational capabilities. Breakthroughs have occurred mostly at the molecular and supramolecular nanoscale, as opening any issue of the ACS’s Chemical & Engineering News or attending an annual meeting of the AIChE amply demonstrates. This focus on the nanoscale is understandable. Because ionic and intermolecular forces extend over nanoscales, much of the key physicochemical interactions are affected by the nanoscale design, and it is the design at the nanoscale (clusters or nanoparticles, nanotubes and nanopores, nanolayers or sheets) that will affect catalytic, electronic, magnetic, and optic properties. Nonlinear interactions as opposed to simple additivity can lead to unexpected properties at the nanoscale. Tuning the building blocks to yield the right functions is the key to new products. Nevertheless, many properties (chemical and mechanical ones, in particular) are also affected by the structure above the nanoscale. Again, nonlinear coupling can lead to highly desired or undesired effects. Material strength and crack formation are examples in the mechanics of solids; turbulence-enhanced mixing and flow nonuniformity are examples in fluid mechanics; transport phenomena affect conversions and selec-

10.1021/ie0490482 CCC: $30.25 © 2005 American Chemical Society Published on Web 01/29/2005

5012

Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

tivity in heterogeneous catalysis, reactor engineering and separation technology. All of these examples underline the importance of understanding and, preferably, also designing the link between the nano- and relevant macroscales as carefully as the nanoscale itself. An engineer could essentially take two different attitudes: empirical assembly or educated, structured design. The most commonly used approach, by far, is empirical assembly. Building blocks are put together in a way that need not be entirely random or arbitrary, and would typically be guided by intuitive insight and experience, yet is far removed from a real multiscale design. Assumptions such as linearity or homogeneity are implicitly made in the original design. If linear extrapolations do not worksas they often do nots empirical rules are used. It is then up to the modelers and experimentators to try to analyze an inherently complex system. Equations might be formulated to attempt modeling the relevant phenomena using assumptions such as ideal mixing or absence of mixing. In chemical reactor engineering, for example, the continuous stirred tank reactor (CSTR) and plug-flow reactor (PFR) models are frequently used as approximations to stirred tanks and tubular reactors, possibly refined with diffusion terms and possibly extended to higher dimensions.4 One step beyond this is to account for residence time distributions and micromixing.5 To account for hydrodynamic nonidealities, computational fluid dynamics (CFD) can now be used in more sophisticated models; great advances in computer power and algorithms have made this option increasingly practical. Diffusion within porous catalysts can be taken into account in an “effective” way using continuum models or by using more detailed network models to account for the interconnectivity between pores.6 In principle, careful mathematical modeling could and should lead to sets of equations with very good predictive power. For multiphase fluid processes, such as slurry reactors and fluidized beds, hydrodynamic modeling based on careful experimentation is a considerable challenge, even without reactions, and a very active area of research, to which Professor Milorad P. Dudukovic, to whom this article is dedicated, has made major contributions.7 Nuclear magnetic resonance8 and particle tracking and tomographic methods9 can provide increasingly detailed information on the flow patterns in multiphase systems. Multiscale modeling is the methodology of choice to bridge the large ranges in length and time scales.10-13 Despite considerable progress, the challenges remain significant: Complex kinetics, frequent diffusion limitations in the tortuous pore space of heterogeneous catalysts, and hydrodynamic nonidealities that affect mass and heat transfer are all coupled together. Predicting and controlling conversions and selectivities under these conditions is not a sinecure. This is even so when intrinsic kinetic data have been obtained with the greatest care.14-17 The fundamental reason behind the difficulties is that a wide range of different scales plays a role and these scales are linked in an uncontrolled manner. Although the lack of structure complicates the analysis of unstructured systems, scale-up is even harder, because scales that are relevant to the various coupled physicochemical phenomena overlap, and the heterogeneity of many unstructured systems is scale-dependent.11 This scale-up problem exists for

both unstructured processes and products. Scale-up errors are now a great concern also in the production of fine chemicals and pharmaceutics, as a result of stricter safety and environmental regulations, increasing competition, and scarce resources.18,19 Stitt19 notes how a CSTR does not scale-up well and how the larger the CSTR gets, the worse its mixing gets; efficient alternatives, such as trickle beds and slurry reactors, also have this problem. Empirical adjustments using baffles and various mixers to amend the scaling issues are commonplace, but the essential flaw is in the initial design. Because uncontrolled heterogeneity of the environment is the prime cause behind the difficulty in modeling phenomena at the macroscopic scale, an alternative solution is to use a bottom-up approach and build structured designs, which reduces the number of unknowns. Turbulence and diffusion are nice, but better is to control them. Polydispersity and heterogeneity might not be undesirable, but it would be a stroke of luck for any arbitrary heterogeneity to give anything close to a desirable, let alone optimal solution. In our search for an optimal design to serve a given purpose, be it a product property or a process to make the desired product, it is possible to use the same methodologies as those described earlier in modeling, synthesizing, or experimentally investigating structured designs of materials and reactors. Different is that interpretation and control are likely to be much simpler, making the design more reliable and robust at any scale. There are other significant benefits of multiscale structured over unstructured assemblies, however, which should affect all but the lowest-added-value process and product designs. One benefit is realized by designs that include simple symmetries (periodicity, fractality, or a combination), which dramatically facilitates scaling by avoiding the introduction of characteristic scales other than those of the building unit or the complete structure. Because “scaling” is the topic of this paper, we will discuss and illustrate this benefit in more detail further on, with special focus on fractal scaling, omnipresent in Nature. Other multistructured assembly methods are more complicated, but extremely enticing. Physics does not stop at the nanoscale. Joining nanoscale building units with distinct properties could generate a structure with altogether different properties. It goes beyond saying that doing this in a rational way should offer advantages. Composite materials are a well-known example in product design. The simplest situations are of the type “unit A has what unit B needs, and vice versa”, but nonlinear coupling leads to nonadditive synergy, which offers the most spectacular opportunities. It should be remembered that these opportunities are not limited to the nanoscale. Synergy might originate from interfacial effects between different building blocks or because strong short-range interaction forces balance each other out, so that weaker forces become more relevant at longer ranges. Capillarity could be used in assembly on micrometer and millimeter scales. Building blocks of the same type (monofunctional assembly) or different types (multifunctional assembly) could be assembled on a two-dimensional “chip” or in three dimensions, integrating different functionalities in an organized manner. Whitesides and Grzybowski20 offer an insightful perspective on self-assembly and on how it can be used at all scales.

Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 5013

In physicochemical processes, more opportunities for efficient assembly arise when equilibria are continuously shifted by careful, organized integration. Multifunctional catalysis with carefully designed geometries and well-positioned active sites is a first example.21-23 Process intensification can be realized by coupling processes (heat integration, reactive separations); for reviews, see Westerterp,24 Dautzenberg and Mukherjee,25 and Stankiewicz and Moulijn.26,27 Although the definition of a building unit becomes blurred in this picture, interactions in a structured system are easier to take into account. Also, outside the correlation range, where nonlinear interactions stop to exist, classical scaling can again be applied; for systems where interactions are propagated in a self-similar way, fractal scaling can be used. Important is that a proper, structured design puts the engineer in control of the interactions and their range. Nature is not static. More options are available by involving time in the analysis and in the design. The complex emergence of dynamic patterns as a result of nonlinearity in a dissipative system might paradoxically facilitate scaling, if the patterns are scale-independent, whether they are periodic or chaotic. It has been known since Poincare´ that nonlinear dynamic systems can lead to multiple steady states. The Belousov-Zhabotinsky reaction is a classical example in homogeneous catalysis. Ertl, Zhdanov, Slinko, and Jaeger, among others, have provided numerous examples and analyses in heterogeneous catalysis. Amundson, Aris, and Luss, among others, have demonstrated the importance of multiplicity to reactor engineering. Further opportunities arise by imposing oscillating fields, such as fluctuating or alternating fluid flows,28 electric fields, vibrations, sound, or various electromagnetic waves. The system is constantly driven out of equilibrium and is not allowed to relax to its static equilibrium state; the frequency, amplitude, and spatial distribution of the driving fields are the parameters to trigger eigenmodes of the system and stimulate desired properties. Patterns might emerge when one or a few eigenmodes dominate, which again leads to a simpler, more easily scalable system, if the spatial distribution of the driving field can be sufficiently well maintained under scaling. Noting how crucial cycles are in the workings of Nature, from seasons, days, waves, blood flow, breathing, the functioning of cells, etc., it should be worthwhile to reach a better understanding of their reason-to-be in Nature, and to use this information as a source of inspiration in chemical product and process design. Pattern formation out of equilibrium is commonplace (numerous examples with their physical background are discussed by Cross and Hohenberg29). We suggest such dynamic self-assembly20 as a beautiful tool to facilitate scaling. One example employing dynamic self-assembly on gassolid fluidized beds is briefly discussed in a following section. We first turn to the use of static symmetries to bridge scales. Using Nature’s Symmetries To Facilitate Scaling Given building blocks with the desired propertiess whether it is nanochannels, catalytic active sites or reactor tubes -, one attractive and logical approach to scale-up is to aim for a design that maximally conserves the intrinsic properties of the building blocks from the size of these building blocks (the microscale) upward.

This might be accomplished by periodicity or by fractal networks devoid of a characteristic scale. A periodic assembly of building blocks does not introduce a new characteristic scale, other than the external size of the structure. A fractal structure, on the other hand, includes all scales between the building block and the largest, external size, yet in a similar way, and therefore again does not introduce a new characteristic scale. In both cases, a symmetry, a pattern, is the key to simplicity in scaling. Periodic structures are used on the macroscopic scale in multitubular reactors and in shell-and-tube heat exchangers and on the microscopic scale in zeolites and other ordered nanostructured materials, microarrays for high-throughput screening, structured reactors for heterogeneous catalysis (e.g., monoliths), and structured packings for separation processes.23 The choice of periodicity in these multitubular and other structured arrays is a convenient one: Because a periodic design simply repeats the same unit, which can be separately optimized to obtain desirable properties, this apparently avoids a tedious scale-up problem. This is only so, however, if each of the units can be “addressed” in the same way, i.e., if access to all units is uniform and homogeneous. For large periodic arrays, this is not obvious, and scaling becomes an issue. For fluid to enter a large multitubular array uniformly, the pressure drop over the array needs to be large enough that there are minimal variations in flow from tube to tube. Here, “fluid” can be substituted by any carrier of matter, energy, or information. As Mandelbrot showed,30 Nature is full of fractal structures (trees, root networks, lungs, the vascular network, leaves, etc.), which have benefits over periodic structures in their uniform way of linking the microscopic and macroscopic worlds. Whenever access to the individual building blocks is an issue, a fractal structure is an excellent option to consider, whether is to feed nano- or macroscopic channels that form a periodic array or to feed building blocks distributed over space. Scale-independent uniformity can be guaranteed by using a self-similar distributor. This is illustrated in Figure 1. In such a distributor, the distance between a single inlet and each outlet, feeding a building block, is the same. For fluid flow, more correctly, the hydraulic distances are the same. By using a tree-like structure, which repeatedly branches, larger arrays can be addressed simply by adding a generation. This is scaling the way Nature does it: Small and large trees essentially differ in their number of generations, while the structure is the same and the microscopic building units are the same.30 In this way, properties at the microscopic scale are preserved, and the feeding or collection of molecules, energy, or information is uniform, irrespective of the number of generations or the macroscopic size. Branching networks are omnipresent in technology as well, but typically in an intuitive, rather empirical way (e.g., transportation networks) or as a result of nonintentional self-organization (e.g., the Internet, social relations, etc.). Fractal networks emerge as a special case, automatically or purposefully, with the great advantage of being scaling. Ottino31 and Baraba´si32 recently discussed the relevance of networks in a broad context and their relation to “complexity”. In a number of cases in Nature and technology, a link between fractal trees and thermodynamic optimality has been demonstrated.33-36 Detailed biological investi-

5014


Figure 1. Fractal distributors help deliver fluid in a uniform way to an array of parallel units, e.g., (micro)reactor tubes, monolith channels, or microarrays. The outlets of the distributor typically lie on a regular, nonfractal grid; the fractality refers only to the selfsimilar hierarchy of channels in the distributor itself. Fractal distributors are size-independent and trivially scalable, just by adding new generations to the cascade, similar to a growing tree. On the left, a set of 1D distributors is shown that feeds progressively larger 1D arrays (vectors); the right images show a few generations of a 2D distributor used to feed an array of tube inlets or points on a plate. The design can be both interpolated and extrapolated. The concept can also be reversed, with the distributor acting as a collector in this case (like tree roots).

gations37-39 add to the mounting evidence that the structure of distribution networks in Nature is not a coincidence, but that the organization of the “mesodomain”, i.e., the hierarchy of structures between the micro- and macroscales, is essential to life. We do not claim that fractals are the answer to all scaling problems or that fractal networks are always the most efficient ones. On the contrary, Nature shows diversity, and many network problems are more complex than simple self-similar interpolations between the micro- and macroscales, a subject that we will touch upon in the next section. Nevertheless, fractal networks certainly facilitate scaling as a result of the absence of a characteristic scale. Furthermore, the geometric and thermodynamic efficiency of networks in Nature that are either fractal or have fractal-like characteristics is very appealing for the design of new, hierarchically structured chemical products and processes, i.e., products and processes that have a well-defined, nested structure at multiple length scales.

Nature-Inspired Networks and Fractal Injectors Figure 1 illustrates the use of a recursive distributor to uniformly feed a periodic array of building units, say, tubes. Tubes that split repeatedly are of course already used in process engineering, but to use the concept consistently over many generations is uncommon. Any tree that creates a constant (hydraulic) path length between inlet and outlet leads to a constant accessibility. The tree-like distributors do not have to be fractal per se, but fractality facilitates scaling. Also, although the distributors themselves have the scaling property of fractals, they are simply used to access a Euclidean array of integer dimension (1 or 2). A two-dimensional distributor was proposed by Kearney to be used for chromatographic columns, silos, and so on and can be extended to other processes.40 Lithography and rapid prototyping methods can be used to construct 2D distributors of virtually any size and number of generations. Also, recursive manual or robotic construction


Figure 2. Extension of the scalable, fractal distributor concept to 3D. A cubic array of building units is uniformly accessed in each case, independent of the system size. The distance from the single, central tube (inlet) to each of the building units (outlets) is constant. As in Figure 1, the function of the distributor can be reversed, so that it functions as a collector.

methods can be applied, so as to facilitate construction by making direct use of the self-similarity. A simple generalization to three dimensions is shown in Figure 2. Now, a cubic array of building units is uniformly accessed, like cells that are uniformly accessed by the circulatory network. A process is occurring inside each building unit. It is tempting to let the distance between the building units or “cells” vanish and to merge them into one volume. This would allow uniform access to all points of a (cubic) volume. Such a generalization was suggested by Kearney.40 However, under which conditions is this generalization justified? If the microscopic volumes (cells) around the outlets interact, the properties might differ from cell to cell. If, for example, a fluid exits the 3D distributor, it will react or otherwise interact with the contents of the cell, but the interactions might stretch beyond the cell. Products might also be formed that do not remain within the cell, but spread to adjoining cells. As long as the spread is isotropic, a large periodic array will still be accessed uniformly, apart from deviations near the boundaries of the volume containing the distributor, and scaling is easy. In reactor engineering, this could represent an advantage over typical distribution systems that access the three-dimensional volume via the external (twodimensional) surface only, e.g., the bottom or the top, without injection points inside the reactor volume. In most cases, however, whether it is for the distribution or collection of matter, energy, or information (for simplicity, we will continue to use the term “fluid”), a trivial generalization of the one- and two-dimensional distributors of Figure 1 cannot be applied to increase the efficiency and facilitate scaling of a three-dimensional system. This is a result of at least one or a combination of the following three effects: (i) The medium around the injector is in motion, so that changes as a result of local injection are propagated throughout the system. (ii) The fluid injected through the injector has different properties from the material in the vessel, as a result of which it moves in an anisotropic way. (iii) The channels are permeable, so

that fluid comes in and out not (only) via outlets at the ends, but also via the channel walls. The first two effects occur in almost any multiphase process involving interacting fluids. As soon as the injected fluid has a density that differs from that of the fluid in the vessel, it will rise or sink as a result of gravity. Examples are gas-liquid systems, slurries and fluidized beds, where a gas is injected via the injector, which is submerged in the reactor fluids. An example of a situation in which the third effect occurs is a nanoporous catalyst, in which a network of large meso- or macropore channels is constructed to reduce transport limitations. Submerged membranes also belong to this category. We recently carried out an optimization study to find the large-pore network in a nanoporous catalyst that maximizes the yield of a firstorder reaction for a given catalyst volume.41 In the spirit of the preceding discussion, the nanoporosity was kept constant, because these are the building blocks that we wish to connect to the macroscopic world. Possible advantages of fractal catalysts were highlighted before,42,43 and this is what led us to investigate not only the superiority over uniform networks, but also whether a self-similar structure could be optimal. We found a seemingly broad variety of optimal networks, typically nonuniform, with a structure that depended on the diffusivities in the large and the narrow pores and on their relation to the intrinsic reaction rate constant. Interestingly, however, we also found that a self-similar network structure would lead to the same efficiency per unit catalyst mass as the optimum, although it has a slightly higher porosity and is therefore slightly less efficient per unit volume. The optimal structures are reminiscent of the leaf venation architecture. It is the nonlinear interactions that lead to optima that deviate from simple self-similar fractal networks. Finding the optimal networksand preferably one that is scalablesin the most general case is a very complex problem. Where scalability is important, one objective is once again to be able to subdivide the entire system (product or process) into units that can be optimized individually, but these units might no longer have the same sizes. An efficient network might also be one that preserves the desired small-scale properties better. The entire system should also have the potential to be much more efficient, because interactions with the environment are not simply via the external surface, but also via an internal distribution/collection network, which creates more degrees of freedom. Let us consider a gas-solid fluidized bed as an example. Gas is usually distributed via the bottom plate, but it could also be injected, in part, via an injector submerged in the reactor volume. The challenge is twofold: determine the optimal injection points and flow rates and connect them. By using a fractal lung-like or tree-like structure to connect the points, the injector itself becomes scalable. If the injection points can be chosen such that the units serviced by each outlet have the same efficiency or render the system more homogeneous, the entire unit is more easily scalable. Figure 3 illustrates the concept of a fractal injector for gas-solid fluidization.44,45 Even with a simple, nonoptimized design, by injecting part of the gas as secondary gas via the injector, the bubble size is markedly decreased, and gas mixes very efficiently with the solids in the reactor. The primary gas is sufficient to fluidize the bed, but only few bubbles are formed. At

5016


Figure 3. Fractal injectors, illustration of principle: (top) 20- and (bottom) 40-cm 2D fluidized beds, with injectors, scaled in a selfsimilar way; 2D means that the thickness of the bed is small, so that bubbles can be visualized by putting a light source behind the bed. The top middle and top right panels show an air-fluidized sand bed, without air injected through the injector (top middle), and with 5/8 of the air through the injector (top right). The total flow rate is 4 times the minimum fluidization flow rate. The bottom left drawing shows two generations of a 2.6-dimensional injector, to be used in a three-dimensional reactor.

Figure 4. Schematic drawing of a reactor with secondary injection. A fraction (1 - p) of the total flow F is injected higher in the reactor. This fresh flow joins the partially reacted flow pF that has passed through the bottom fraction f of the reactor volume. Concentrations are shown by symbols Ci.

every outlet, the secondary gas breaks up existing bubbles and stabilizes a gas-solid suspension that would normally split up into bubbles if it had the chance to approach equilibrium. In other words, the system is more homogeneous, and the contact between gas and solid is markedly improved, which is especially beneficial for transport-limited chemical reactions. Scale-up occurs by increasing the number of generations in the injector. However, there are at least two important concerns: the gas residence time and the intrinsic anisotropy of the system. Indeed, the gas content increases from bottom to top, and so does the flow rate. To start addressing these issues, it is useful to consider the schematic reactor setup in Figure 4. A flow rate F is split up into a primary stream and a secondary stream, with flow rates pF and (1 - p)F, respectively. The secondary stream is injected at a certain height in the reactor, so that the reactor volume V is split up into a volume fV below the injection point and a volume (1 - f)V above the injection point. For a single reaction in a plug-flow reactor (PFR) with strictly positive reaction order, it can be shown that the highest conversion is obtained if all of the fluid is injected via the bottom, i.e., p ) 1, a simple result of the reduced residence time if part of the fluid is injected higher up. It is assumed

that there are no appreciable volume or flow changes as a result of the reaction. For a continuous stirred tank reactor (CSTR) with ideal mixing, a split of the volume (0 < f < 1) increases the conversion for reaction orders of at least 1: staging essentially leads to two CSTRs in series, hence behavior closer to plug flow. Splitting up the flow as well does not affect the conversion much; it is either the same as or slightly different from that for p ) 1.46 By induction, these conclusions can be extended to more than one secondary injection stream. This might seem disappointing: Can secondary injection not lead to major improvements? It certainly can. The assumptions were indeed that the reactor behaved in an ideal way, either CSTR or PFR, without specifying how this was achieved or how fluid staging was realized. There are several advantages to secondary injection. First, it can induce staging, without baffles. It can also turn the reactor more plug-flow-like and much more uniform in every horizontal cross section, by using several injection points in each plane, such as is realized by the fractal injector shown in Figure 3. Furthermore, enhanced micro-mixing around the injector outlets leads to more tortuous fluid trajectories and much better gassolid contact. Although simple CSTR-based models46 show that the average residence time is not or hardly affected, in reality, it is affected, because of the increased trajectory lengths.47 When there are transport limitations between bubbles and the emulsion phase, the fact that bubbles are smaller when a fractal injector is used leads to further improvements. Fluid injected in the bed encounters a rising flow that increases with increasing height. As a result, the dispersion is higher at greater heights. To maintain similar contact times in cells serviced by different outlets, the increased dispersion, together with the different local flow rates, can be taken into account. This will typically lead to a nonuniform distribution of outlets in the vertical direction and an optimal scaling relationship that involves a fractional dimension D < 3 (in a three-dimensional fluidized bed). The dimension D will also be at least 2 to service the entire cross section of the column. The precise value will depend on the process; what simplifies the procedure is that experiments and simulations could


Figure 5. Oscillating the gas inlet flow to a fluidized bed can turn the chaotic hydrodynamics into regular patterns of rising bubbles. This dynamic self-assembly is reminiscent of periodic pattern formation in Nature. The pattern is independent of bed size, hence intrinsically scaling. Shown are (right) 15- and (left, reduced in size) 40-cm-wide air-fluidized beds of sand particles, illuminated from behind. The driving frequency is 2 Hz.

be carried out on small units, with one injection point and different primary and secondary flow rates. The positions of the outlets are chosen on the basis of a maximum conservation of homogeneity and a subdivision of space into constantly performing units that are connected using a fractal structure of dimension D. Without entering into further details too specific to fluidization, this demonstrates the general possibilities for linking the micro- and macroscales using a scaling, bottom-up approach. This philosophy can be used not only on static systems, but also on more complex dynamic systems, involving fluid mixing. A next opportunity to facilitate scaling is to directly influence the dynamics by parametric forcing. Dynamic Self-Assembly as a Scaling Tool: Example of Gas-Fluidized Beds Dissipative systems, continuously driven away from equilibrium by external forces or internal feedback, can form patterns. Such dynamic self-assembly appears crucial in the workings of Nature.20 We do not aim to give an overview of this exciting fieldsits repercussions for synthesizing nanostructured systems, its importance in fluid mechanics or in understanding the living cell. We merely remark here that dynamic pattern formation can also be a very useful, as yet unexplored route to scaling in product and process engineering. A robust pattern, with a well-defined, intrinsic, characteristic length scale that relates only to the composition of the system and not to its size, is an invitation to scaling. Again, periodic and fractal patterns are ideally suited in this regard. As in the previous section, let us illustrate this by the example of gas-solid fluidized beds. A gas-fluidized bed can be described as a chaotic system. High-resolution dynamic information can be obtained by measuring pressure fluctuations. van den Bleek et al.48 showed that the best scaling results are obtained by keeping the characteristic chaotic pattern of fluidized beds as constant as possible, by preserving the bed aspect ratio and the Froude number (based on the bed height and the part of the velocity above the minimum fluidization velocity, u0 - umf). This scaling

route indeed preserves the Kolmogorov entropy or information content of the fluidized bed, which is a “natural” way to scale. The Kolmogorov entropy (or, equivalently, the average cycle frequency) can readily be calculated from the pressure fluctuation measurements; we refer to van den Bleek et al.48 for details. Using a dynamic self-assembly route, the chaotic bubble patterns of fluidized beds can be turned into periodic ones, as we recently showed.49,50 Instead of keeping the gas flow through the porous bottom distributor plate constant, it was oscillated at a frequency of a few Hertz while the minimum gas flow was kept above umf. Previous researchers reported improved fluidization performance,51,52 bubble size reduction (by using a moving distributor plate53), or a reduction of chaos.54 We found that periodic oscillations can lead to the formation of spatially ordered rows of bubbles, rising row by row, each row staggered with respect to the previous one (Figure 5). The frequency of the pattern is one-half the driving frequency. Reminiscent of pattern formation in Nature (dunes, beaches, water waves), we found that patterns in shallow air-fluidized beds of sand particles (Geldart B) show some similarity to other wellstudied systems,29 such as Rayleigh-Benard rolls and vibrated granular layers.55 Different is that patterns in a deep, quasi-two-dimensional fluidized beds persist over heights of the order of the bed width, i.e., they are not limited to the horizontal plane but are vertically extended via the rising gas, uniquely related to the multiphase nature of the system. Bubble patterns result from the dynamic self-assembly of a gas-solid suspension and uniquely depend on the gas-solid properties, not on the size of the fluidized bed: the same periodic pattern, with the same spatial wavelength of a few centimeters (for air-sand), was found in 15- and 40cm-wide fluidized beds (Figure 5). The wavelength depends in a nonlinear way on the driving frequency, with higher driving frequencies leading to closer together bubbles. We have not yet investigated the performance of the bed in terms of conversions of gassolid reactions or catalysis and are not yet able to confirm whether the significantly higher performance cited by others51 would occur under conditions where

5018


patterns form. Nevertheless, our results suggest that dynamic pattern formation is a tool worthy of exploring further for making complex nonlinear systems scalable, and thereby simpler to understand and control, not only in gas-solid fluidization but also in other composite, multiphase systems. Conclusion Periodicity and fractality are very common in Nature. Sometimes, they can emerge spontaneously by combinations of balanced forces or self-organization; in other cases, there is evidence of an optimization and a structure-function relationship that extends over several length scales. The workings of Nature are positively affected by the structured way in which the micro- and macroscopic worlds are linked. As chemical engineers striving for sustainable processes and the design of new functional materials, we have much to learn from Nature’s architecture at all scales and the role dynamics play in Nature. Although the benefits of robust scaling are known to any engineer, we have mostly focused on sophisticated designs at the smallest scales and sophisticated methods to study much less sophisticated designs at larger scales. In particular, while periodic structures are increasingly employed in chemical engineering to replace random ones, the same cannot be said of fractal structures, at least not by design. By avoiding the introduction of new, overlapping characteristic scales during the scale-up, scaling is facilitated. The nature-inspired chemical engineering philosophy discussed here should help scaling in a structured way from molecules to products and processes, in the most general sense. Rapid progress in chemistry, in materials and mechanical engineering, and in particular in nanoand microtechnology should enable the realization of designs that formerly would exist only in the mind of a theoretician. Acknowledgment I gratefully acknowledge the Dutch National Foundation for Scientific Research for support via a NWO/CW PIONIER award, as well as the Delft University of Technology, TU Delft. Literature Cited (1) Villermaux, J. Future Challenges for Basic Research in Chemical Engineering. Chem. Eng. Sci. 1993, 48 (14), 2525-2535. (2) Villadsen, J. Putting Structure into Chemical Engineering. Chem. Eng. Sci. 1997, 52, 2857-2864. (3) Cussler, E. L.; Wei, J. Chemical Product Engineering. AIChE J. 2003, 49 (5), 1071-1074. (4) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley: New York, 1990. (5) Villermaux, J. A Generalized Mixing Model for Initial Contacting of Reactive Fluids. Chem. Eng. Sci. 1994, 49, 51275140. (6) Sahimi, M.; Gavalas, G. R.; Tsotsis, T. T. Statistical and continuum models of fluid-solid reactions in porous media. Chem. Eng. Sci. 1990, 45, 1443-1502. (7) Dudukovic, M. P. Chemical Reaction Engineering, the Environment, Pollution Prevention and Sustainable Development. Int. J. Chem. React. Eng. 2004, 2, 4. http://www.bepress.com/ ijcre/vol2/P4. (8) Gladden, L. F. Magnetic Resonance: Ongoing and Future Role in Chemical Engineering Research. AIChE J. 2003, 49 (1), 2-9. (9) Chaouki, J.; Larachi, F.; Dudukovic, M. P. Non-Invasive Monitoring of Multiphase Flows; Elsevier: Amsterdam, 1997.

(10) Lerou, J. J.; Ng, K. M. Chemical Reaction Engineering: A Multiscale Approach to a Multiobjective Task. Chem. Eng. Sci. 1996, 51 (10), 1595-1614. (11) Raimondeau, S.; Vlachos, D. G. Recent Developments on Multiscale, Hierarchical Modeling of Chemical Reactors. Chem. Eng. J. 2002, 90, 3-23. (12) van der Hoef, M. A.; van Sint Annaland, M.; Kuipers, J. A. M. Computational Fluid Dynamics for Dense Gas-Solid Fluidized Beds: A Multi-Scale Modeling Strategy. Chem. Eng. Sci. 2004, 59 (22-23), 5157-5165. (13) Chen, P.; Sanyal, J.; Dudukovic, M. P. CFD Modeling of Bubble Columns Flows: Implementation of Population Balance. Chem. Eng. Sci. 2004, 59 (22-23), 5201-5207. (14) Froment, G. F. Fundamental Modeling of Complex Processes. In Chemical Reactions in Complex Mixtures, Sapre, A.V., Krambeck, F. J., Eds.; van Nostrand Reinhold: New York, 1991; pp 77-100. (15) Dumesic, I. A.; Rud, D. F.; Aparicio, L. M.; Rekoske, J. E.; Revino, A. A. The Microkinetics of Heterogeneous Catalysis; American Chemical Society: Washington, DC, 1993. (16) van Santen, R. A.; Neurock, M. Concepts in Theoretical Heterogeneous Catalytic Reactivity. Catal. Rev. Sci. Eng. 1995, 37, 557-698. (17) Broadbelt, L. J.; Snurr, R. Q. Applications of Molecular Modeling in Heterogeneous Catalysis Research. Appl. Catal. A 2000, 200 (1-2), 23-46. (18) Mills, P. L.; Ramachandran, P. A.; Chaudhari, R. V. Multiphase Reaction Engineering for Fine Chemicals and Pharmaceuticals. Rev. Chem. Eng. 1992, 8 (1-2), 1-176. (19) Stitt, E. H. Alternative Multiphase Reactors For Fine ChemicalssA World Beyond Stirred Tanks? Chem. Eng. J. 2002, 90, 47-60. (20) Whitesides, G. M.; Grzybowski, B. Self-Assembly at All Scales. Science 2002, 295, 2418-2421. (21) Bell, A. T. The Impact of Nanoscience on Heterogeneous Catalysis. Science 2004, 299, 1688-1691. (22) de Jong, K. P. Assembly of Solid Catalysts. CATTECH 1998, 2, 87-94. (23) Cybulski, A., Moulijn, J. A., Eds. Structured Catalysts and Reactors; Marcel Dekker: New York, 1998. (24) Westerterp, K. R. Multifunctional Reactors. Chem. Eng. Sci. 1992, 47, 2195-2206. (25) Dautzenberg, F. M.; Mukherjee, M. Process Intensification Using Multifunctional Reactors. Chem. Eng. Sci. 2001, 56 (2), 251-267. (26) Stankiewicz, A. I.; Moulijn, J. A. Process Intensification: Transforming Chemical Engineering. Chem. Eng. Prog. 2000, 96, 22-34. (27) Stankiewicz, A. I., Moulijn, J. A., Eds. Re-engineering the Chemical Processing PlantsProcess Intensification; Marcel Dekker: New York, 2004. (28) Matros, Yu. Sh. Performance of Catalytic Process Under Unsteady Conditions. Chem. Eng. Sci. 1990, 45, 2097-2102. (29) Cross, M. C.; Hohenberg, P. C. Pattern formation out of equilibrium. Rev. Mod. Phys. 1993, 65, 851-1112. (30) Mandelbrot, B. B. The Fractal Geometry of Nature, W. H. Freeman: San Francisco, 1983. (31) Ottino, J. M. Complex Systems. AIChE J. 2003, 49 (2), 292-299. (32) Baraba´si, A.-L. Linked; Perseus: Cambridge, U.K., 2003. (33) Bejan, A. Shape and Structure, From Engineering to Nature; Cambridge University Press: Cambridge, U.K., 2000. (34) West, G. B.; Brown, J. H.; Enquist, B. J. A General Model for the Origin of Allometric Scaling Laws in Biology. Science 1997, 276, 122-126. (35) West, G. B.; Brown, J. H.; Enquist, B. J. The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms. Science 1999, 284, 1677-1679. (36) West, G. B.; Brown, J. H.; Enquist, B. J. A General Model for the Structure and Allometry of Plant Vascular Systems. Nature 1999, 400, 664-667. (37) Weibel, E. R. Fractal Geometry: A Design Principle for Living Organisms. Am. J. Physiol. 1991, 261, L361-L369. (38) Mauroy, B.; Filoche, M.; Weibel, E. R.; Sapoval, B. An Optimal Bronchial Tree May be Dangerous. Nature 2004, 427, 633-636. (39) Roth-Nebelsick, A.; Uhl, D.; Mosbrugger, V.; Kerp, H., 2001 Evolution and Function of Leaf Venation Architecture: A Review. Ann. Bot. 2001, 87, 553-566.

Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 5019 (40) Kearney, M. M. Engineered Fractals Enhance Process Applications. Chem. Eng. Prog. 2000, 96 (12), 61-68. (41) Gheorghiu, S.; Coppens, M.-O. Optimal Bimodal Pore Networks for Heterogeneous Catalysis. AIChE J. 2004, 50 (4), 812-820. (42) Pfeifer, P.; Avnir, D. Chemistry in Noninteger Dimensions Between Two and Three. I. Fractal Theory and Heterogeneous Surfaces. J. Chem. Phys. 1983, 79, 3558-3565. (43) Sheintuch, M. Reaction Engineering Principles of Processes Catalyzed by Fractal Solids. Catal. Rev. Sci. Eng. 2001, 43, 233289. (44) Cheng, Y.; van den Bleek, C. M.; Coppens, M.-O. Hydrodynamics of Gas-Solid Fluidized Beds Using a Fractal Injector. In Proceedings of the 10th International Conference on Fluidization; Kwauk, M., Li, J., Yang, W. C., Eds.; United Engineering Foundation: New York, 2001; pp 373-380. (45) Coppens, M.-O. Structuring Fluidized Bed Operation in a Nature Inspired Way. In Proceedings of the 11th International Conference on Fluidization; Arena, U., Chicone, R., Miccio, M., Salatino, P., Eds.; Engineering Conferences International: New York, 2004; pp 403-410. (46) Coppens, M.-O. Delft University of Technology, Delft, The Netherlands. Unpublished results, 2004. (47) Villermaux, J. Trajectory Length Distribution (TLD), A Novel Concept to Characterize Mixing in Flow Systems. Chem. Eng. Sci. 1996, 51 (10), 1939-1946. (48) van den Bleek, C. M.; Coppens, M.-O.; Schouten, J. C. Application of Chaos Analysis to Multiphase Reactors. Chem. Eng. Sci. 2002, 57 (22-23), 4763-4778.

(49) Coppens, M.-O.; Regelink, M. A.; van den Bleek, C. M. Pulsation Induced Transition from Chaos to Periodically Ordered Patterns in Fluidised Beds. Presented at the 4th World Congress on Particle Technology, Sydney, Australia, Jul 21-25, 2002; Paper 355. (50) Coppens, M.-O.; van Ommen, J. R. Structuring Chaotic Fluidized Beds. Chem. Eng. J. 2003, 96, 117-124. (51) Massimilla, L.; Volpicelli, G.; Raso, G. A Study on Pulsing Gas Fluidization of Beds of Particles. Chem. Eng. Prog. Symp. Ser. 1966, 62, 63-70. (52) Wong, H. W.; Baird, M. H. I. Fluidization in a Pulsed Gas Flow. Chem. Eng. J. 1971, 2, 104-113. (53) Ko¨ksal, M.; Vural, H. Bubble Size Control in a TwoDimensional Fluidized Bed Using a Moving Double Plate Distributor. Powder Technol. 1998, 95, 205-213. (54) Pence, D. V.; Beasley, D. E. Chaos Suppression in GasSolid Fluidization. Chaos 1998, 8, 514-519. (55) Melo, F.; Umbanhowar, P.; Swinney, H. L. Transition to Parametric Wave Patterns in a Vertically Oscillated Granular Layer. Phys. Rev. Lett. 1994, 72, 172-175.

Received for review September 29, 2004 Revised manuscript received November 23, 2004 Accepted December 2, 2004 IE0490482

Scaling-up and -down in a Nature-Inspired Way - Industrial

Recommend Documents