Multi-Scale Methods and Complex Processes

919

89Multi-Scale Methods and Complex Processes: a Survey and Look Ahead

Angelo LuciaDepartment of Chemical EngineeringUniversity of Rhode IslandKingston, RI 02881

Abst r Ac t A comprehensive overview of numerical methodologies cur-rently available for analyzing and building understanding of complex pro-cesses is presented. Both equation-free and equation-based methods are discussed. Many multi-scale techniques, including the use of reduced order models, lifting-restricting methods, averaging strategies such as molecular dynamics, Monte Carlo simulation and the mean value theorem, stochastic sampling methods like kriging, fitting and interpolation approaches such as surrogate solution maps and gap-tooth schemes, and others are described. The importance of and ways in which information is communicated between

contents

Introduction ......................................................................................................... 920Literature Survey ................................................................................................ 920Multi-Scale Methodologies ................................................................................ 921

Reduced Order Models .............................................................................922Equation-Free Methods ............................................................................. 923Deterministic Averaging Techniques ...................................................... 924Stochastic Sampling Methods .................................................................. 925Interpolation Methods .............................................................................. 926Multiple Time Step Methods .................................................................... 926

Multi-Scale Applications ................................................................................. 927Thermodynamic & Transport Properties................................................ 927Reaction Engineering ................................................................................ 928Molecular Modeling .................................................................................. 929Advanced Materials .................................................................................. 929Process Systems Engineering ................................................................... 930

A Look to the Future ........................................................................................... 930Conclusions .......................................................................................................... 933References ............................................................................................................. 933

K10451_C089.indd 919 4/16/09 5:20:00 PM

920 Design for Energy and the Environment

scales is also discussed. Improvements that are needed for the next genera-tion of multi-scale tools for analysis, visualization, simulation, and optimiza-tion of complex processes are identified and include the need to proceed with partial knowledge, techniques for handling multiplicity at various length scales, problems with many scales, and methods for predicting behavior over very long time and length scales.

Ke y words Multi-scale methods, complex processes, communication between scales.

Introduction

A process is often called complex if it contains multiple time and length scales, complicated phenomenological descriptions, and feedback between various elements of the process that results in behavior that cannot be pre-dicted by analyzing the individual components in isolation. Mathematical models of complex processes are generally described by nonlinear algebraic and/or differential equations that must be solved numerically using multi-scale methods and often require high performance computing (clusters, supercomputers, cloud computers, and so on). At fine scales, these models are often treated as equations or as input-output black boxes with and with-out the presence of noise. Key among the attributes of any multi-scale meth-odology is the communication of information between scales.

Literature survey

Over the last decade, there has been a growing interest in the development and application of multi-scale methods in process systems engineering and many traditional processes studied by chemical engineers are complex pro-cesses. Examples include problems involving simultaneous heat and mass transfer in porous media, molecular conformation and crystal structure determination, chemical vapor deposition, and others and not all multi-scale problems are necessarily time dependent. Multi-scale modeling techniques offer valuable advantages in building better understanding and more reli-able computer tools for modeling complex system behavior. Some of the more prominent methods in multi-scale modeling and applications to com-plex processes are surveyed in this paper.

The literature on multi-scale methods in chemical process engineering is quite extensive, covering both traditional areas such as the estimation of thermodynamic and transport properties, chemical reaction engineering, advanced polymer materials, molecular conformation and crystal structure,

K10451_C089.indd 920 4/16/09 5:20:00 PM

Multi-Scale Methods and Complex Processes 921

and others to more exotic applications in advanced materials (e.g., flow of DNA through micro-fluidic devices, defects in crystals, ionic liquids, etc.). There are many application papers as well as articles that address the devel-opment of multi-scale methodologies. Some work has been exclusively com-putational while others have integrated simulation and experiment.

Multi-scale Methodologies

Any multi-scale methodology consists of models, numerical methods, and possibly experimental and/or observational data at various length/time scales as well as communication between scales. Moreover, what consti-tutes a small length-scale and what is large is generally problem dependent. Figure 1 gives a generic picture of the relationship between multiple length scales in problems of a physical nature, where small and large length scales imply small and large time scales respectively. This figure is also intended to show that discrete representations of geometry often gives rise to irregular elements that must be handled with care, that geometries can change from scale to scale, and that there is no requirement that the number and/or vari-able types at different scales be the same.

Note that the macroscopic scale is divided into rectangular control elements and that within each rectangular control element are many smaller circular elements. In addition, symmetry can often be exploited within each large or small element, if justified, to lessen computational demands. In most process engineering applications, small can range from atomistic to nano- to even millimeter length-scales while large often refers to meso- or macroscopic length scales in centimeters or meters. We also note that coarse-graining

Small scale

Large scaleControl element

Node

Figure 1 Schematic of multi-scale process

K10451_C089.indd 921 4/16/09 5:20:01 PM


refers to the process of constructing larger meshes within a fixed area. So for example, in Figure 1, where there are 32 control elements shown, coarse-graining at the larger length scale might consist of reducing the number of meshes (or control elements) from 32 to perhaps 8.

Traditional methodologies used for multi-scale modeling at small length scales include techniques from quantum chemistry (QC) such as density functional theory (DFT), molecular simulation methods such as molecular dynamics (MD), various Monte Carlo (MC) techniques (kinetic MC, lattice MC, etc.), transition state theory (TST), and computational fluid dynamics (CFD). These methods are well-developed and continue to evolve and find use in process engineering applications. Models at the large length scale are generally steady or unsteady-state conservation laws, Navier-Stokes equa-tions, and so on. Pantelides (2000) gives a good early summary of the multi-scale modeling challenges and methods in process systems engineering, describing serial, parallel, and hierarchical approaches to scale integration.

In this paper, we refer to scale integration as communication between scales. We focus on general multi-scale methods that include reduced order models, equation-free methods, deterministic averaging techniques (MD, CFD, and the mean value theorem), stochastic sampling methods (MC and kriging), interpolation techniques (surrogate solution maps, gap-tooth schemes, etc.), multiple time stepping methods, and others. We also describe the means of communication between scales for these multi-scale methods. While some view multi-scale methods as methods that just perform computations and/or analyses at different length and time scales, in the author’s view, true multi-scale methods alternate between computations at different scales and use two-way communication and either explicit or implicit coarse-graining to represent small scale information in some form of ‘average’ information at larger length and time scales.

reduced Order Models

Reduced order (or low dimension) models have been used for a very long time in process systems engineering for modeling multi-scale chemical pro-cesses, are still used today, and can be considered a form of coarse-graining. In this approach, detailed behavior at small length and (faster) time scales is ignored and approximated by explicit (closed form) or implicit (black box) algebraic representation, resulting in models that are comprised of differential-algebraic equations (DAEs). Note that algebraic representation at small time scales does not alter the fact that there is still two-way com-munication. Common examples of reduced order models are the equations of state, activity coefficient models, K-values, etc. used to model equilibrium or non-equilibrium on a stage in a distillation column. Time dependent con-tacting, mixing, mass transfer, and other effects that define phase behavior at the molecular scale are assumed to occur infinitely fast (and reach steady-state), compared to mass and energy balance dynamics on a tray, and are

K10451_C089.indd 922 4/16/09 5:20:01 PM


approximated by algebraic equations that describe either equilibrium or non-equilibrium phase behavior. In fact, any constitutive equation can be considered a reduced order model at some scale. For example, heat capac-ity polynomials in temperature can be considered reduced order molecular scale models since MD at several temperatures can be used to evaluate heat capacity. Other reduced order model approaches include coupled CFD and reduced kinetic models for combustion (Androulakis, 2000; Sirdeshpande et al., 2001; Liang et al, 2007) and modal decomposition, which eliminates fast modes in dynamical systems using techniques like center manifold the-ory (Balakotaiah and Dommeti, 1999). Classical CSTR, PFR, and other reactor models are also reduced order models (Chakraborty and Balakotaiah, 2005) since they ignore small scale flow patterns within the reactor, use assump-tions such as well mixed, no radial mixing, etc. and only model macroscopic behavior.

equation-Free Methods

Equation-free methods for multi-scale modeling have been pioneered by Kevrekidis and co-workers (Qian and Kevrekidis, 2000; Gear et al., 2002; Siettos, et al., 2003; Kevrekidis et al., 2003). As their name suggests, these methods forego closed form model descriptions and treat macroscopic mod-els as black boxes or experiments that result from ‘wrapping’ software around small scale simulation methods (MD, MC, etc.). As a result of the work of Kevrekidis, there are now equation-free, multi-scale methods in molecular dynamics (Frederix et al, 2008) and other branches of science and engineer-ing (e.g., the heterogeneous multi-scale methods of Weinan and Engquist, 2003). The key to equation-free, multi-scale modeling is the two-way commu-nication between small and large length scales. Kevrekidis calls this lifting/ restricting. Restricting is the process of communication from small or fine scale, which is considered a high dimensional map, to low dimensional or large length scale. Lifting, on the other hand, is reverse communication, from the large to the small length scale. The result of lifting/restricting yields an implicit coarse time stepping procedure at the macroscopic scale that can be written in the form

M(t + ∆t) = M(t) + R{m(L[m, ∆t])} (1)

where m denotes microscopic variables, M is a set of macroscopic variables, L and R are lifting and restriction operators respectively, m is the functional-ity defining the microscopic simulation, and t is time.

As with any multi-scale method, the creativity lies in the way in which two-way communication is designed and implemented. In this regard, lifting is the critical issue because it can be challenging to find adequate initializations at the small scale from ‘limited’ low dimensional spatial and temporal informa-tion at the large scale. In addition, spatial information at the macroscopic scale

K10451_C089.indd 923 4/16/09 5:20:01 PM


must undergo dimensional reduction and is often approximated using an interpolation scheme (e.g., gap-tooth, splines, etc.). Kevrekidis and co-workers have applied their equation-free methodology to a variety of examples in pro-cess engineering including, reaction-diffusion processes, lattice gases, liquid crystals, peptide fragments, process control and others.

Deterministic Averaging Techniques

Deterministic sampling techniques like MD are frequently used at the small length scale to obtain average bulk phase thermodynamic and transport properties. Averaging in this context exploits a fundamental axiom of sta-tistical mechanics – ergodicity (i.e., averages over time are adequately rep-resented by ensemble averages over phase space). Averaged quantities are generally represented by the equation

<P> = Σ P(pN, rN)/K (2)

where P is some property, < > denotes average, pN and rN are the momentum and position vectors for an N particle system respectively, K is the total num-ber of integration steps, and the summation in Eq. 2 is over all integration steps. Other derived properties can often be estimated from multiple MD simulations.

Other deterministic averaging methods have also been proposed. Perhaps the most straightforward is the use of the mean value theorem within the terrain/funneling methods of global optimization recently proposed by Lucia and co-workers (Lucia et al., 2004; Gattupalli and Lucia, 2007, 2008). Geometric models at two different length scales are assumed. At the small length scale, a quadratic model with considerable noise or roughness on a surface is assumed. Average gradient and second derivative information is computed using

<g> = {∫g[z(α)]dα}/α (3)

<h> = {∫h[z(α)]dα}/α (4)

by performing terrain optimizations over some small region of the surface, where the variables α, g, and h in Eqs. 3 and 4, denote some relevant length of a path on an objective function surface or terrain, the gradient, and Hessian matrix respectively. This average information is communicated to the large length scale, which is assumed to be non-quadratic (funnel- shaped) to determine funnel parameters and an estimate of the global minimum on the surface. The estimated funnel minimum is, in turn, com-municated to the small length scale in order to define the next region on the objective function surface to be explored locally. Unlike equation-free methods, the number and variable types are the same at each scale in the terrain/funneling approach.

K10451_C089.indd 924 4/16/09 5:20:01 PM


CFD is also a deterministic method that can be used to calculate ‘average’ effects due to mixing (fluid flow, mass transfer and heat transfer). In this approach, momentum, energy, and continuity equations are often solved at the small scale and then averaged to give flow patterns and transfer coef-ficients at larger length scales.

Stochastic Sampling Methods

There are many stochastic sampling methods that are used in multi-scale modeling to obtain average information. Some are used at small length/time scales while others have been employed at large length scales. Stochastic sampling methods at small length scales are frequently based on various forms of MC simulations. Stochastic averages at the small scale are calcu-lated using a simplified version of Eq. 2 given by

<P> = Σ P(rN)/K (5)

which does not account for kinetic energy. Kriging is also a stochastic averaging methodology and gets its name from

the geologist D. Krige (1951) who pioneered the technique in applications in hydrology. However kriging is quite different from variants of MC and often used to obtain approximate information at the large length scale. In kriging, a number of initial sampling points are placed in the feasible region and are assumed to be randomly distributed. A set of trial or test points are placed and both a kriging predictor, based on linearly weighted function values, and the expected variance associated with each test point are determined. The kriging predictor, z(xk), for a test point, xk, is given by

z(xk) = Σ f(xi)w(xi) (6)

where the xi’s are initial sampling points, f is the function value, w is a weighting coefficient, and where the set of weighting coefficients is determined by solv-ing a system of equations in covariance functions. The expected variance, σ(xk), is given by

σ(xk) = σmax – Σ w(xi)Cov[d(xi, xk)] (7)

where Cov denotes covariance and d(xi, xk) is the distance between points xi and xk.

From the set of test points, predictors, and expected variances, prediction and variance maps are constructed and a set of most promising regions to perform local optimizations are identified. Davis and Ierapetritou (2007) have applied kriging to nonlinear programming problems to obtain a ‘pic-ture’ of the global geometry of noisy or black box objective function surfaces. From the kriging results, local optimizations are conducted using a response surface method (RSM). While they did not alternate between kriging and RSM, due to computational expense, Davis and Ierapetritou do recognize

K10451_C089.indd 925 4/16/09 5:20:02 PM


that such an approach would produce two-way communication and a true multi-scale method.

interpolation Methods

Interpolation is often a valuable tool in resolving spatial and temporal dif-ferences in scale. Interpolating techniques such as wavelets (Reis et al., 2008), surrogate solution maps (Kulkarni et al., 2008), and gap-tooth schemes (Kevrekidis, 2000) are among the interpolation methods that have been used to capture small and/or large scale behavior.

Gap-tooth schemes, like those proposed by Kevrekidis, are intended to provide smooth spatial descriptions at the large length scale from informa-tion inferred from the small scale through simulation.

Surrogate solution maps (e.g., Kulkarni et al., 2008) are another form of inter-polation. Solutions, Y, at the small length scale for a range of explicit or implicit large scale conditions, X, are computed a priori and stored in the form of poly-nomial coefficients. Polynomial representation is used at the large length scale to lessen the computational burden using maps of the form

Y(X) = f(X) = Σ wiφi(X) (8)

where φi(X) are polynomial (or basis) functions and wi are weights or coef-ficients determined from matching values of Y(X) and X at known solu-tion points. Note that the basis functions can be orthogonal or separable, if desired. It is straightforward to couple the large and small length scale by simply using Eq. 8 to retrieve information at the small scale during large scale iterations. Two-way communication is maintained between the small and large scale in much the same way that reduced order models main-tain two-way communication. In principle, maps of any functionality and dimensionality can be constructed. However, in practice, surrogate solu-tion maps are generally restricted to low dimensionality and simple poly-nomial functions.

Multiple Time Step Methods

There are a number of multiple time step methods that have been developed over the years to address the accurate integration of coupled fast and slow dynamics in a variety of applications. Perhaps the best known approach of this type is a variant of the Car and Parrinello method (1985) where nuclear and electronic forces are separated. Other examples include Humphreys et al. (1994) and Barash et al. (2002). The basic idea is to integrate fast modes with a small time step, while inferring slow modes for several fast time steps, and integrate the slow modes intermittently with a larger time step. To illus-trate, assume that a system of differential equations

y′ = F(yf, ys, t) (9)

K10451_C089.indd 926 4/16/09 5:20:02 PM


has fast and slow modes, where y = (yf, ys) is a vector of dependent variables, the symbol ′ denotes the time derivative, and the subscripts f and s refer to fast and slow modes respectively. Integration proceeds as follows

yfk + 1 = yf

k + yf′[yfk, ys

k]∆tf (10)

ysk + 1 = ys

k + ys′[yfk, ys

k]∆ts (11)

where ∆ts = k∆tf for k >> 1. Within each fast time step, extrapolation (not function evaluation) is used to update the slow variables and this is the key to reducing computational work. Multiple time stepping methods can often be developed by exploiting physics (e.g., in MD where force splitting is exploited). Nonetheless, coupling and the multi-scale nature of the prob-lem is retained and two-way communication exists since the Eqs. 10 and 11 couple the fast and slow modes.

Multi-scale Applications

The use of multi-scale methods in various areas of chemical engineering is reviewed because of their importance to process systems engineering.

Thermodynamic & Transport Properties

Estimation of bulk phase thermodynamic and transport properties (e.g., diffusion coefficients, viscosities, etc.) is often accomplished using quan-tum chemistry such as density functional theory, molecular dynamics, and/or Monte Carlo simulation at the small length scale and then aver-aged over many configurations to predict properties at the macroscopic length scale (usually bulk phase). Greenfield and co-workers (1998, 2001, 2004, 2007) have computed transport properties of polymers using multi-scale approaches. Angstrom-scale computations in these applications include the use of MD and TST to measure temperature-dependent diffu-sion coefficients, relaxation times, viscosities, and rate constants for dif-fusion. Larger nanometer length scale computations, on the other hand, make use of kinetic MC to determine diffusivities or infer composition changes. Pricl et al. (2006) make combined use of force field models, all-atom molecular dynamics simulations, and a continuum description of protein transport dynamics to estimate macroscopic diffusion of proteins in nano-channeled silicone membranes. Economou et al. (2008), on the other hand, use ab initio DFT calculations, molecular dynamics, and the tPC-SAFT equation of state to estimate structural and thermodynamic properties of imidazolium-based ionic liquids. CFD has been used to esti-mate flow patterns, and temperature and composition distributions on

K10451_C089.indd 927 4/16/09 5:20:02 PM


sieve trays (e.g., Rahimi et al. 2006) and in packed columns for tray and packing design. Recently Egorov et al. (2005) have made combined use of CFD and rate-based models in simulating reactive separations. There are many, many other papers and examples of the computation of thermody-namic and transport properties with multi-scale methods, which due to space limitations, are not cited in this article.

reaction engineering

Many multi-scale approaches have been used in reaction engineering, ranging from simple reduced order models to CFD models of convection-diffusion- reaction equations. See Chakraborty and Balakotaiah (2005) for a good survey. Reduced models like CSTR, PRF, and so on represent the simplest models and need no further discussion. At the other extreme is the com-putationally demanding approach of computational fluid dynamics. Here convection-diffusion-reaction equations are used as the large scale model and solved using CFD. Small scale phenomena like turbulent flows are aver-aged or reduced kinetics models included and incorporated in the CFD model to create a multi-scale method. Somewhere in the middle are the tech-niques that use spatial averaging of convection-diffusion-reaction equations. Here averaging is used at the small scale to develop low order models at the large length scale suitable for process design and control activities. The basis for one such approach is Liapunov-Schmidt bifurcation analysis and the result is a coupled or two-mode model in which both local and global behav-ior are described. Chakraborty and Balakotaiah (2005) call their approach a multi-mode approach and describe the application of spatial averaging to a wide variety of reactor models including isothermal tubular reactors, loop and recycle reactors, CSTR’s, various models of non-isothermal reactors, and multiple reactor configurations but present very limited numerical compari-sons with traditional approaches. The primary limitation of the multi-mode approach is the assumption of well defined flow fields. The authors also describe situations where multiple solutions to local models can exist and give rise to numerical difficulties.

Kulkarni et al. (2008), on the other hand, take a different approach to multi-scale modeling of catalyst pellet reactors using surrogate solution maps. Orthogonal collocation over finite elements is used to construct a set of nonlinear algebraic equations for the reactor temperature and concentra-tion profiles at the large length scale (i.e., the bulk phase), which in this case is on the order of meters. However, these equations are coupled to the small (single catalyst pellet or millimeter) scale through the effectiveness factor. Effectiveness factor maps are generated as a function of anticipated values of the Thiele modulus, dimensionless heat of reaction and dimensionless activation energy. These effectiveness factor maps are used to determine an effectiveness factor for any temperature and concentration within the reac-tor. Like reduced order models, two-way communication is maintained and

K10451_C089.indd 928 4/16/09 5:20:02 PM


the method is truly multi-scale. Kulkarni et al. (2008) use their surrogate solution map approach to find solutions to the classical problem of simul-taneous heat and mass transfer in a catalyst pellet described by Weisz and Hicks (1962) and present numerical results that show that surrogate solution maps can offer significant computational advantages over nested iteration or direct numerical simulation and handle multiplicity at the small length scale in an effective manner.

Molecular Modeling

MD and MC have been used to build understanding in molecular modeling for describing reaction pathways, transition states, and so on. However, it has long been known that potential energy and free energy landscapes used in molecular conformation and crystal structure determination have many, many stationary points at the small length scale. The large scale geometry is frequently more ordered and non-quadratic (e.g., funnel-shaped, helical in the case of DNA, and so on). Multiple length scales make this class of prob-lems ideally suited for multi-scale methods. Gattupalli and Lucia (2007, 2008) have used a terrain/funneling approach to determine the molecular confor-mation of n-alkanes. Numerical comparisons with other molecular confor-mations methods that are not multi-scale methods (such as basin hopping, Wales and Doye, 1997) clearly show that exploiting the multi-scale nature of molecular conformation problems in algorithmic development leads to global optimization methodologies that have superior computational reli-ability and efficiency. See also Westerberg and Floudas (1999) who studied reaction pathways using global optimization and TST to determine large scale reaction rates.

Advanced Materials

There are seminal papers by de Pablo (2005), de Pablo and Curtin (2007) and Barrat and de Pablo (2007) that describe the role of multi-scale model-ing techniques in advanced materials science. Models in the materials sci-ence arena typically range from continuum mechanics at the macroscopic scale to atomistic models at the quantum scale. Coarse graining generally involves a hierarchy of models from fully all-atom models to bead-spring arrangements that represent chains of molecules to primitive path analysis used for modeling entanglement in polymers. Advances in rigid materi-als have progressed while multi-scale modeling of soft materials remains challenging.

Dai et al. (2005, 2006) have used multi-scale methods to analyze defect evo-lution in crystalline systems using kinetic MC and coarse-graining in which lattice sites are grouped into cells and averaged within cells to capture void morphology. Zheng et al. (2008) describe the use of hybrid multi-scale kinetic MC method for the simulation of copper electro-deposition.

K10451_C089.indd 929 4/16/09 5:20:02 PM


Process Systems engineering

A decade ago, multi-scale modeling in process systems engineering was rare. As computational power has evolved (clusters, high performance comput-ing, cloud computing), multi-scale methods have become more widespread. The previously cited papers are clear examples of this. In addition, there are papers that describe multi-scale methods in process simulation, process identification, process control, process design and optimization (Gerogiorgis and Ydstie, 2004), product/process design (Morales-Rodriguez and Gani, 2008), and others (e.g., Ingram and Cameron, 2005).

However, multi-scale modeling and multi-scale methods are not as wide-spread as one might expect in process systems engineering. Even as late as 2004, multi-scale modeling and multi-scale methods were still classified as new challenges in Computer Aided Process Engineering (CAPE). See Puijaner and Espuna (2004). Moreover, at the Symposium on Modeling of Complex Processes held in the spring of 2005 at Texas A & M, the subject of many of the multi-scale modeling papers in the process systems area was molecular simulation (e.g., Tunca and Ford, 2005; Faller, 2005) with few exceptions (e.g., the paper by Ge et al. (2005) concerned with scale-up and scale-down of labo-ratory processes to industrial operations). Nevertheless, there are papers on multi-scale methods in process systems engineering that do have a strong link to traditional macroscopic process engineering computations such as that illustrated by the work of Kevrekidis et al. (2003) and Economou et al. (2008). For example, the latter paper characterizes properties of ionic liquids and uses three length/time scales that range from the quantum scale (DFT) to the atomistic scale (MD) to the bulk phase or macroscopic scale (tPC-SAFT). Other papers on multi-scale modeling and methods that have this same strong link to traditional macroscopic or process scales include the work of Gerogiorgis and Ydstie (2004) and Kulkarni et al. (2008).

A Look to the Future

Some progress has been made in multi-scale modeling of complex processes and multi-scale methodologies and applications will continue to evolve. Needs on the horizon include (1) the use of partial information, (2) the presence of multiplicity at fine scales, and (3) integration of dynamical models over very large time and length scales. To motivate these needs, we use ocean sedimen-tary CO2 sequestration (see Fig. 2) because it embodies a wide variety of issues in multi-scale modeling relevant to energy and the environment.

In ocean sedimentary storage, carbon dioxide is pumped into sediments at ocean depths of 3 km or more, where it is known that liquid carbon dioxide is slightly denser than water (i.e., the so-called neutral buoyancy zone) for sedi-ment depths up to ~200 m. Deeper than ~200 m, water is again denser than liquid CO2. Hydrates can also form at these depths. See Dornan et al. (2007).

K10451_C089.indd 930 4/16/09 5:20:02 PM


The fundamental assumption on which sequestration is based is that a CO2 hydrate layer will form and this hydrate cap together with neutral buoyancy will trap liquid carbon dioxide (for extended periods of time on the order or hundreds to thousands of years).

The multi-scale nature of this problem is very complex. The reservoir has at least four length/time scales – the molecular scale, pore scale, sedimentary material/bulk phase scale, and reservoir scale. The associated multi-scale reservoir model is further complicated by reaction/phase equilibria, CO2 gas hydrate formation, and multi-phase flow through porous media. CO2 levels in the ocean, on the other hand, are driven by global vertical circulation of cold ocean waters which takes place on time scales of hundreds or thou-sands of years. Regional and local ocean effects must also be considered so the ocean model has several length/time scales as well.

What are the fundamental challenges and why are they multi-scale challenges?

1) Can we model the carbon exchange between ocean and reservoir? 2) Can we really store CO2 in the ocean sediments for thousands of

years? 3) If so, where do we store CO2 to avoid disruption or leaks due to

earthquakes, volcanoes, etc?

> 3 kmCO2injection

Neutral bouyancy

CO2 hydrate layer

trapped liquid CO2~ 200 m

Ocean sediments

Figure 2Ocean sedimentary CO2 sequestration

K10451_C089.indd 931 4/16/09 5:20:03 PM


The assumption that the ocean and reservoir interact in a simple source-sink manner is flawed. In order to model true behavior, each system must drive the other and thus a multi-scale systems approach is required. However, truly coupling global ocean models with CO2 or other reservoir models is challenging.

For the reservoir, knowledge of the kinetics of hydrate formation is required at the molecular scale. However, there are at least ten or more reactions occur-ring simultaneously in addition to hydrate formation. Moreover, the kinet-ics are a function of local environment as is the formation of any number of phases. At the pore length scale, there is diffusion that affects transport and reaction so there is multi-phase flow through porous media that must be included. Next, bulk phase kinetic and equilibrium considerations in a sedi-mentary environment must be accounted for. This is a complex multi-phase, multi-reaction problem in which multiple solutions can exist. At the reservoir length scale, there is a multiple moving boundary problem since there is simul-taneous accumulation of hydrate and liquid CO2. Finally all of this is coupled to the ocean, in which deep ocean currents drive the flow of water through the sediments and reservoir at the concentration of CO2 in the ocean.

Global ocean models generally consist of continuity and Navier-Stokes equations, which are simplified to include incompressible flow, constant density, and Boussinesq approximations. Nonetheless, this set of partial differential equations is extremely demanding to solve because of the large horizontal and vertical length scales involved. Spin-up time (i.e., equilibra-tion time) for global ocean models is one to two decades, eddy diffusivities are difficult to include, and turbulence is approximated from theory since the length scale for resolving ocean turbulence is on the order of millimeters. Horizontal resolution is ten to several hundred kilometers with 20 or more vertical layers, resulting in a grid of ~ 2 × 107 points. For the purposes of car-bon sequestration, realistic bottom features must also be included. Typical integration time horizons can range from 100 to 1000 years so very coarse times steps are used. Finally, the ocean and reservoir are coupled through CO2 concentration, salinity, and other factors. These variables can, in turn, affect ocean currents, ocean temperatures, and climate.

1) Partial Knowledge. Although our knowledge through observation is increasing, there is much that is not known about the ocean and the sediments (e.g., deep ocean currents cannot be measured). Thus simulation of oceans and sedimentary reservoirs requires the will and some cleverness to proceed with partial knowledge.

2) Multiplicity at Fine Scales. Similar to that which has been identi-fied by Balakotaiah and Dommeti (2000) and Kulkarni et al. (2008), reacting systems in porous media can often exhibit multiplicity. The reactions that take place in the ocean during the formation of CO2 reservoirs are no different.

K10451_C089.indd 932 4/16/09 5:20:03 PM


3) Integration for Very Large Time/Length Scales. Simulation of vertical ocean transport of CO2 requires very long time scales – on the order of 1000 years. However, behavior at much smaller length/time scales can drastically alter long term behavior. True assessment of the long term viability of carbon storage in ocean sediments must address validation, very large differences in scale, and integration over very long time scales.

conclusions

Multi-scale modeling and multi-scale methods are still relatively new. While progress has been made on several fronts, many issues remain un-resolved. As with many disciplines, multi-scale modeling and methods provide advanced computer modeling tools but also raise new questions and con-cerns. In the author’s opinion, the true value of multi-scale modeling is in modeling complex phenomena with some degree of quantitative confidence. However, this quantitative confidence can only be realized through model validation and the continued evolution of modeling techniques and support-ing numerical methods. When designed and implemented correctly these methods make ‘what if’ scenarios for complex processes possible in ways that would otherwise not be possible. Multi-scale methods are the enabling technology that makes multi-scale modeling of complex processes possible and will continue to make an increasing impact in the future.

references

Androulakis, I.P. (2000). Kinetic Mechanism Reduction Based on an Integer Programming Approach, AIChE J. 46, 361.

Balakotaiah, V., Dommeti, S.M.S. (1999). Effective Models for Packed-Bed Catalytic Reactors. Chem. Eng. Sci. 54, 1621.

Barash, D., Yang, L., Qian, X., Schlick, T. (2002). Inherent Speedup Limitations in Multiple Time Step/Particle Mesh Ewald Algorithms, J. Comput. Chem. 24, 77.

Barrat, J.-L., de Pablo, J.J. (2007). Modeling Deformation and Flow in Disordered Materials. MRS Bull. 32, 941.

Car, R., Parrinello, M. (1985). Unified Approach for Molecular Dynamics and Density Functional Theory, Phys. Rev. Lett. 55, 2471.

Chakraborty, S., Balakotaiah, V. (2005). Spatially Averaged Multi-Scale Models for Chemical Reactors. Adv. Chem. Eng. 30, 205.

K10451_C089.indd 933 4/16/09 5:20:03 PM


Dai, J. Kantes, J.M. Kapur, S.S., Seider, W.D., Sinno, T. (2005). On-Lattice Kinetic Monte Carlo Simulations of Point Defect Aggregation in Entropically-Influenced Crystalline Systems, Phys. Rev. B 72, 134105.

Dai, J. Seider, W.D., Sinno, T. (2006). Lattice Kinetic Monte Carlo Simulations of Defect Evolution in Crystals at Elevated Temperatures, Molecular Simulation 32, 305.

Davis, E., Ierapetritou, M.G. (2007). A Kriging Method for the Solution of Nonlinear Programs with Black-Box Functions, AIChE J. 53, 2001.

de Pablo, J.J. (2005). Molecular and Multiscale Modeling in Chemical Engineering – Current View and Future Perspectives. AIChE J. 51, 2371.

de Pablo, J.J., Curtin, W.A. (2007). Multiscale Modeling in Advanced Materials Research: Challenges, Novel Methods, and Emerging Applications. MRS Bull. 32, 905.

Dornan, P., Alavi, S., Woo, T.K. (2007). Free Energies of Carbon Dioxide Sequestration and Methane Recovery in Clathrate Hydrates, J. Chem. Phys. 127, 124510.

Egorov, Y., Menter, F., Kloeker, M., Kenig, E.Y. (2005). On the Combination of CFD and Rate-Based Modeling in the Simulation of Reactive Separations, Chem. Eng. Process, 44, 631.

Economou, I.G., Karakatsani, E.K., Logotheti, E.-G., Ramos, J., Vanin, A.A. (2008). Multi-scale Modeling of Structure, Dynamic and Thermodynamic Properties of Imidazolium-Based Ionic Liquids: Ab Initio DFT Calculations, Molecular Simulation and Equation of State Predictions. Oil & Gas Sci. & Tech. Rev. 63, 283.

Faller, R. (2005). Systematic Coarse-Graining of Atomistic Models for Simulation of Polymeric Systems, In El-Halwagi, M. (ed.) Symposium on Modeling of Complex Processes.

Frederix, Y., Samaey, G., Vanderkerchhove, C., Roose, D. (2007). Equation-Free Methods for Molecular Dynamics: A Lifting Procedure, Proceedings in Applied Mathematics and Mechanics.

Gattupalli, R., Lucia, A. (2007). Molecular Conformation of N-Alkanes Using Terrain/Funneling Methods, J. Global Optim. [doi:10.1007/s10898-007-9206-5].

Gattupalli, R., Lucia, A. (2008). Multi-Scale Global Optimization of All-Atom Models of N-Alkanes, Comput. Chem. Engng. [doi: 10.1016/j.compchemeng.2008.11.012].

Ge, W., Li, J., Gao, S., Song, W. (2005). Multi-Scale Approach of Multiphase Flows in Chemical Engineering, In El-Halwagi, M. (ed.) Symposium on Modeling of Complex Processes.

Gear, C.W., Kevrekidis, I.G., Theodoropoulis, C. (2002). Coarse Integration/Bifurcation Analysis via Microscopic Simulators: Micro-Galerkin Methods, Comp. Chem. Engng. 26, 941.

Gerogiorgis, D., Ydstie, B.E. (2004). Multi-Scale Modeling for Electrode Voltage Optimization in the Design of a Carbothermic Aluminum Process, In Floudas, C.A., Agrawal, R. (eds.) Foundations of Computer-Aided Process Design CACHE Corp., 265.

Greenfield, M.L. (2004). Simulation of Small Molecule Diffusion Using Continuous Space Disordered Networks, Molecular Physics 102, 421.

Greenfield, M.L., Theodorou, D.N. (1998). Molecular Modeling of Methane Diffusion in Glassy Atactic Polypropylene via Transition State Theory. Macromolecules 31, 7068.

Greenfield, M.L., Theodorou, D.N. (2001). Coarse-Grained Molecular Simulation of Penetrant Diffusion in a Glassy Polymer Using Reverse and Kinetic Monte Carlo, Macromolecules 34, 8541.

K10451_C089.indd 934 4/16/09 5:20:03 PM


Humphreys, D.D., Friesner, R.A., Berne, B.J. (1994). A Multiple Time-Step Algorithm for Macromolecules, J. Phys. Chem. 98, 6885.

Ingram, G.D., Cameron, I.T. (2005). Formulation and Comparison of Alternative Multi-Scale Models for Drum Granulation, In Puigjaner, L, Espuna, A. (eds.) European Symposium on Computer-Aided Process Engineering (ESCAPE)–15, Elsevier, Netherlands, 481.

Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulis, C. (2003). Equation-free, Coarse-Grained Multi-Scale Computations: Enabling Microscopic Simulators to Perform Systems Level Tasks, Comm. Math. Sci. 1, 715.

Krige, D. (1951). A Statistical Approach to Some Mining Valuations and Allied Problems at the Witswatersrand, M.S. Thesis, Univ. of Witswatrsrand, Johannesburg.

Kulkarni, K., Moon, J., Zhang, L., Lucia, A., Linninger, A.A. (2008). Multiscale Modeling and Solution Multiplicity in Catalytic Pellet Reactors, Ind. Eng. Chem. Res . 47, 8572.

Liang, L., Stevens, J.G., Farrell, J.T., Huynh, P.T., Androulakis, I.P., Ierapetritou, M.G. (2007). An Adaptive Approach for Coupling Detailed Chemical Kinetics and Multidimensional CFD, Fifth U.S. Combustion Meeting, San Diego, CA, paper # C09.

Lucia, A., DiMaggio, P.A., Depa, P. (2004). Funneling Algorithms for Multi-Scale Optimization on Rugged Terrains, Ind. & Eng. Chem. Res., 43, 3770.

Morales-Rodriguez, R., Gani, R. (2008). Multi-Scale Modeling Framework for Chemical Product-Process Design, In Jezowski, J., Thullie, J. (eds.) European Symposium on Computer-Aided Process Engineering (ESCAPE) – 19, Elsevier, Netherlands.

Pantelides, C.C. (2000). New Challenges and Opportunities for Process Modeling, In Gani, R., Jorgensen, S.B. (eds.) European Symposium on Computer-Aided Process Engineering (ESCAPE) – 11, Elsevier, 15.

Pricl, S., Ferrone, M., Fermeglia, M., Amato, F., Cosentino, C., Cheng, M.-C., Walczk, R., Ferrari, M. (2006). Multiscale Modeling of Protein Transport in Silicone Membrane Nanochannels. Part 1. Derivation of Molecular Parameters for Computer Simulation. Biomedical Micro-Devices 8, 277.

Puigjaner, L, Espuna, A. (2005). European Symposium on Computer-Aided Process Engineering (ESCAPE) – 15, Elsevier, Netherlands.

Qian, Y.-H., Kevrekidis, I.G. (2000). Coarse Stability and Bifurcation Analysis Using Time Steppers: A Reaction-Diffusion Example. Proc. Natl. Acad. Sci. 97, 9840.

Rahimi, M.-R., Rahimi, R., Shahraki, F., Zivdar, M. (2006). Prediction of Temperature and Concentration Distributions of Distillation Sieve Trays by CFD, in Sorensen, E. (ed.) Distillation & Absorption 2006, 220.

Reis, M.S., Saraiva, P.M., Bakshi, B.R. (2008). Multi-Scale Statistical Process Control Using Wavelets, AIChE J. 54, 2366.

Siettos, C.I., Armaou, A., Makeev, A.G., Kevrekidis, I.G. (2003). Microscopic/Stochastic Timesteppers and Coarse Control: A Kinetic Monte Carlo Example, AIChE J. 49, 1922.

Sirdeshpande, A.R., Ierapetritou, M.G., Androulakis, I.P. (2001). Design of Flexible Reduce Kinetic Mechanisms, AIChE J., 47, 2461.

Tunca, C., Ford, D.M. (2005). A Hierarchical Approach to the Molecular Modeling of Permeation Through Microporous Materials, In El-Halwagi, M. (ed.) Symposium on Modeling of Complex Processes.

K10451_C089.indd 935 4/16/09 5:20:03 PM


Wales, D.J., Doye, J.P.K. (1997). Global Optimization by Basin Hopping and Lowest Energy Structures of Lennard-Jones Clusters Containing Up To 110 Atoms, J. Phys. Chem. A 101, 5111.

Weinan, E., Engquist, B. (2003). The Heterogeneous Multi-Scale Methods, Comm. Math. Sci. 1, 87.

Weisz, P.B., Hicks, J.S. (1962). The Behavior of Porous Catalyst Particles in View of Internal Mass and Heat Diffusion Effects, Chem. Eng. Sci. 17, 265.

Westerberg, K.M., Floudas, C.A. (1999). Locating All Transition States and Studying the Reaction Pathways of Potential Energy Surfaces, J. Chem. Phys. 110, 9259.

Zhang, L., Greenfield, M.L. (2007). Relaxation Time, Diffusion, and Viscosity Analysis of Model Asphalt Systems Using Molecular Simulation, J. Chem. Phys. 127, 194502.

Zheng, Z., Stephens, R.A., Braatz, R.D., Alkire, R.C., Petzold, L.R. (2008). A Hybrid Multiscale Kinetic Monte Carlo Method for Simulation of Cooper Electrodeposition, J. Comput. Phys. 227, 5184.

K10451_C089.indd 936 4/16/09 5:20:03 PM

Documents

Multi-Scale Methods and Complex Processes