Filtered models for reacting gasparticle flows
Abstract
Using the kinetictheorybased twofluid models as a starting point, we develop filtered twofluid models for a gasparticle flow in the presence of an isothermal, firstorder, solidcatalyzed reaction of a gaseous species. As a consequence of the filtering procedure, terms describing the filtered reaction rate and filtered reactant dispersion need to be constituted in order to close the filtered species balance equation. In this work, a constitutive relation for filtered reaction rate is developed by performing finegrid, twofluid model simulations of an isothermal, solidcatalyzed, firstorder reaction in a periodic domain. It is observed that the clusterscale effectiveness factor, defined as the ratio between the reaction rate observed in a finegrid simulation to that observed in a coarsegrid simulation, can be substantially smaller than unity, and it manifests an inverted bell shape dependence on filtered particle volume fraction in all simulation cases. Moreover, the magnitude of the deviation in the clusterscale effectiveness factor from unity is a strong function of the mesoscale Thiele modulus and dimensionless filter size. Thus coarsegrid simulations of a reacting gasparticle flow will overestimate the reaction rate if the clusterscale effectiveness factor is not accounted for.
keywords:
reactive flows, fluidization, fluid mechanics, twofluid model1 Introduction
Gasparticle flows are commonly encountered in the chemical and energy conversion industries in the form of fluidized beds, risers, and other pneumatic conveying units. The flow behaviors of gasparticle systems are commonly analyzed using continuum models that treat the particle and fluid phases as interpenetrating continua (Gidaspow, 1994; Fan and Zhu, 1998; Jackson, 2000). Finitevolume simulations of these continuum, or ‘twofluid’ models, reveal that the scale of the gasparticle flow structures that are observed is a strong function of the grid size used in the simulation, with grid size independent results being obtained when the ratio between the grid size and particle size is (Agrawal et al., 2001; Andrews IV et al., 2005; Igci et al., 2008). Grid cell sizes of ten particle diameters are not computationally affordable when performing twofluid model simulations of large fluidized bed reactors that are often tens of meters tall and many meters in crosssection. As a result, coarsegrid simulations of large fluidized beds are common practice. However, these coarsegrid simulations effectively neglect the presence of fine scale gasparticle flow structures that are manifested by the twofluid model. To enable accurate coarsegrid simulation of the twofluid model equations one must account for the effect of the finescale gasparticle flow structures in the coarsegrid simulation (Andrews IV et al., 2005; Igci et al., 2008). This approach is embodied by the recent development of a filtered twofluid model approach for nonreacting gasparticle flows where effective closures for the mesoscale fluidparticle drag force, particle phase stress, and particle phase viscosity were extracted from finegrid simulations of the twofluid model equations (Igci et al., 2008; Igci and Sundaresan, 2011a, b; Igci et al., 2011; Parmentier et al., 2011). While filtered twofluid models for nonreacting monodisperse gasparticle flows have been developed, verified against finegrid simulations, and validated against experimental observations (Igci and Sundaresan, 2011a, b; Igci et al., 2011), the extension of these models to reacting and polydisperse gasparticle flows remains an open problem.
Finegrid twofluid model simulations of reacting gasparticle flow have been shown to yield quantitative agreement with experimental observations of conversion in solidcatalyzed ozone decomposition processes when gridindependent solutions are obtained (Syamlal and O’Brien, 2003). However, due to the computational expense of performing gridindependent simulations, coarse numerical resolution is often used to simulate reacting gasparticle flows in the continuum model framework. By not accounting for the finescale structure these coarsegrid simulations have been shown to overpredict the conversion of ozone in a bubbling fluidized bed (Zimmermann and Taghipour, 2005). In addition, a few studies have also demonstrated that smallscale gasparticle flow structures that form in devices like fluidized beds are responsible for the wide variability in mass transfer models that is present within the fluidization literature (Dong et al., 2008a, b; Kashyap and Gidaspow, 2010, 2011).
The primary objective of this work is to demonstrate the need for filtered models for reacting gasparticle flows by performing finegrid simulations of a firstorder, isothermal, solid catalyzed reaction in a periodic domain and filtering the results. It will be shown that the presence of particle clustering in finegrid simulations leads to an effective reaction rate that is substantially smaller than what would be predicted via coarsegrid simulation. We define the ratio of the reaction rate in the finegrid simulation to that in the coarsegrid simulation as the clusterscale effectiveness factor. It will be shown that this clusterscale effectiveness factor is a strong function of dimensionless filter size , and other model parameters. Here, is the filter size, is the gravitational acceleration vector, and is the terminal settling velocity of an isolated particle.
Finally, it is shown that grid resolutions finer than those used to deduce filtered models for nonreacting gasparticle flows are necessary in order to obtain gridindependent filtered closures for reacting gasparticle flows. This observation is supported by a recent work by Cloete et al. (2011). Due to this grid size dependence we propose a filtered reaction rate model based upon extrapolated effectiveness factor data obtained from finegrid simulations.
2 Microscopic twofluid model equations
The twofluid model consists of balance equations for gas and particlephase mass and momentum with an additional transport equation governing evolution of the particlephase granular temperature. The evolution equations that constitute this Eulerian framework are presented in Table 1, in addition to the constitutive relations that close the balance equations. The constitutive relations given in Table 1 were chosen to be consistent with earlier studies in filtered model development within our research group (Agrawal et al., 2001; Andrews IV et al., 2005; Igci et al., 2008; Igci and Sundaresan, 2011b; Igci et al., 2011; Igci and Sundaresan, 2011a). Here we note that while we have restricted our attention to a specific set of constitutive relations, it has been shown that the clustering and bubbling phenomena manifested by the twofluid model framework are robust to changes in constitutive relations (Glasser et al., 1998). However, we do expect that the constitutive relations comprising the microscopic twofluid models will have quantitative effects on the filtered models that arise from finegrid simulations. Therefore, while we have employed a certain set of constitutive relations in this study, filtered models can be developed from finegrid, twofluid model simulations using a different set of constitutive relations as well.
The effective diffusion coefficient and effective gasphase viscosity appearing in eqs. (5) and (9) will differ from the corresponding molecular properties as they also account for the enhanced scalar and momentum transport that occurs due to additional pseudoturbulent transport occurring as a result of microscopic interactions between individual particles and the fluid. However, twofluid model simulations have revealed that the solutions manifested by the twofluid model are insensitive to the value of (Agrawal et al., 2001). Therefore, we set equal to the molecular viscosity . In addition, the effective diffusion coefficient is kept on the order of the bulk molecular diffusivity , which is generally for many different gas species.
Here we investigate the role of clustering on a model, isothermal, solidcatalyzed, firstorder chemical reaction
(1) 
where is the particle volume fraction, and is the mass fraction of species in the gas phase. For first order reaction kinetics, the effective reaction rate constant based on the bulk concentration is related to the intrinsic reaction rate constant via
(2) 
where is the intraparticle effectiveness in the absence of mass transport limitations, is the Thiele modulus, is the volume to surface area ratio, and is the Biot number for mass transport given as (Rawlings and Ekerdt, 2009). Here, is the convective mass transport coefficient, is the particle radius, and is the intraparticle diffusivity. It is important to note that the effective reaction rate constants used in twofluid model simulations incorporate the combined effect of mass transport resistance and intraparticle effects as well. In this study all gassolid flow computations are performed for FCC catalyst particles fluidized by air the physical properties for solid and fluid phases are given in Table 2.
Evolution Equations


Gas phase stress tensor

Diffusive flux of granular energy

m  
3 Filtered twofluid model equations
The filtered twofluid models are obtained by performing a spatial average of the microscopic twofluid model equations. The consequence of the filtering approach is that the finescale gasparticle flow structure that occurs on a length scale smaller than the filter size is accounted for through residual terms that must be constituted from theoretical considerations or finegrid twofluid model simulation results. In the case of nonreacting monodisperse gasparticle flows, filtered models with accompanying constitutive relations for residual terms have been shown to yield quantitatively similar macroscopic behaviors to those observed in finegrid simulations of the same flow problem, with a dramatic savings in computational time (Igci and Sundaresan, 2011b). Here we follow, the filtering procedure given in Igci et al. (2008).
The particle volume fraction and gasspecies mass fractions obtained from finegrid simulations can be represented as , where and t represent the location and time variables, respectively. The filtered particle volume fraction is given as
(24) 
where is a weight function, and is the spatial location of the filter center. In this work will be used to indicate filtered variables. We require that , and in this study we use a top hat filter for . Filtered gasspecies mass fraction , particle phase velocity , and fluid phase velocity are given as
(25) 
(26) 
(27) 
Upon filtering the microscopic species balance equation for component we obtain the following filtered species balance equation
(28) 
In eq. (28) we invoke the definition of filtered variables given in eqs. (25)–(27). Due to the fact that the reaction we are considering in this work is isothermal and produces no volume change, the filtered momentum balance equations remain unchanged from those derived in the work of Igci et al. (2008), and as such, they will not be presented here for the sake of brevity. In eq. (28), there are several terms that appear as filtered products of microscopic variables that must be constituted to solve the filtered twofluid model equations. To facilitate filtered model simulations we must decompose the products of microscopic variables into mean and fluctuating parts. Here, we define fluctuating variables as follows
(29) 
(30) 
where , , and represent species mass fraction, gas velocity, and volume fraction fluctuations, respectively. Inserting the decomposition of mean and fluctuating parts into eq. (28), we obtain the following filtered species balance equation
(31)  
The first term appearing on the right hand side of eq. (31) can be modeled as a dispersive term given by the following constitutive equation
(32) 
where is a dispersion coefficient. Dispersion coefficients defined in this way have been presented in the research literature (Loezos and Sundaresan, 2002). The final term on the right hand side of eq. (31) represents a filtered reaction rate that must also be constituted in terms of filtered variables alone. A straightforward method for constituting this reaction rate is to define a clusterscale effectiveness factor which is given as
(33) 
However, in this paper we present filtered models and associated constitutive relations for the nonlocally corrected clusterscale effectiveness factor defined as
(34) 
where . The approach of removing nonlocal effects in the determination of filtered quantities was recently advanced by Parmentier et al. (2011), and we apply the same procedure here for determining . The nonlocally corrected effectiveness factor is constructed by removing the dominant gradient terms that contribute the evaluation of the filtered product , which is done to allow the construction of filtered constitutive relations in terms of local filtered variables alone. We present a derivation of the nonlocally corrected effectiveness factor in A. It will be shown below that both and depend on volume fraction in a qualitatively similar way, but differ quantitatively. All data presented in future sections and resulting filtered models will pertain to , thus necessitating one to track the gradients in filtered species mass fraction and volume fraction to enable the use of the resulting filtered models in coarsegrid simulations of reacting gasparticle flow.
Dimensional analysis of the parameters governing gasparticle flow suggests that the clusterscale effectiveness factor is a function of five independent dimensionless quantities, which are
(35) 
(36) 
One can also readily define additional dimensionless parameters like and the coefficient of restitution using dimensional analysis. However, earlier works of Igci et al. (2008) have shown that filtered quantities do not display any significant dependence on or , and as such, they were not included in the dimensional analysis presented here. Moreover, we found that and had a negligible effect on the clusterscale effectiveness factor, while the mesoscale Thiele modulus , Schmidt number , and the dimensionless filter size substantially alter the clusterscale effectiveness factor. Therefore, all work presented below will only interrogate the effect of , , and on . Here the term mesoscale Thiele modulus is used to distinguish it from the Thiele modulus based on intraparticle diffusivity given in eq. (2).
4 Numerical implementation
When performing periodic domain simulations of reacting gasparticle flows, one is faced with the following dilemma: due to the imposition of periodic boundary conditions, the concentration of any reactant undergoing an irreversible reaction within the periodic domain decay to zero. In order to facilitate gasparticle flow simulations in periodic domains, an alternate simulation strategy must be developed. For the special case of a firstorder reaction, it can readily be shown that one can track the evolution of rather than itself, and relate the filtered value of directly to the clusterscale effectiveness factor. The benefit of tracking the evolution of lies in the fact that it has a nonzero statistically steady value, even though will decay to zero.
Consider taking the species balance equation given by eq. (5) with the reaction rate expression given by eq. (1) and averaging it over the entire periodic domain, resulting in the following equation
(37) 
where is the volume of the periodic domain. If we now define the variable and plug into the species balance equation given by eq. (5) we obtain the following evolution equation
(38)  
Due to the fact that the reaction rate expressions in this problem are first order, we are able to track the evolution of without considering the time progression of or . The presence of nonzero source and sink terms on the right hand side of eq. (38) force statistically steady value of to be nonzero regardless of the reaction rate constant used. Therefore, in all work presented in following sections for the evolution of is solved, and the clusterscale effectiveness factor is redefined in terms of as
(39) 
All simulation results presented in this work were generated using the Multiphase Flow with Interface eXchanges (MFIX) software that relies on a variable time step, staggered grid, finitevolume method for the solution of the twofluid model equations (Syamlal, 1998). The iterative solution to the twofluid model equations is obtained using the SemiImplicit Method with Pressure Linked Equations, or the SIMPLE algorithm. Due to the strong coupling between solid and fluid phases that arises due to the fluidparticle drag force, the Partial Elimination Algorithm of Spalding was used to effectively decouple the solution of the solid and fluid phase balance equations (Spalding, 1980). In addition, a second order Superbee discretization is employed for the convective terms that are present in the conservation equations for continuity, momentum and granular energy transport to limit the effects of numerical diffusion.
5 Preliminary Study
To illustrate the behavior of the clusterscale effectiveness factor as a function of , simulation results are presented that were obtained using grid sizes that are sixteen times as large as the particle diameter. This grid resolution was chosen to follow the simulations employed in the study of Igci et al. (2008). Filtered variables were obtained from these finegrid simulations by moving filters of different sizes throughout the periodic domain. Due to the statistical homogeneity of periodic domains we are able to collect thousands of samples and average them regardless of spatial position. Moreover, by filtering these finegrid simulation results for various domainaveraged volume fractions the filteraveraged volume fraction dependence of the clusterscale effectiveness factor can be ascertained. This filtering procedure follows that outlined in the work of Igci et al. (2008). The characteristic dependence of and is presented in Figure 1. It is clear from Figure 1 that both and are strong functions of retaining an inverted bell shape that approaches unity in the limit of small and large particle volume fraction. However, there are noticeable quantitative differences between and in Figure 1 with the maximum deviation being near the minimum in both curves. The minimum value of , while the minimum value of . Therefore, at this resolution, the nonlocal correction to the clusterscale effectiveness factor contributes as much as to the value of the clusterscale effectiveness factor near the minimum in the curves given in Figure 1. While the quantitative differences between and are functions of and other model parameters, this example provides a scale of the difference between the two effectiveness factors. In this work we have decided to model because it is directly related the small scale fluctuations in and , while is dependent on nonlocal variations in and (see Appendix). Physical intuition for the decrease in the clusterscale effectiveness factor from unity can be obtained by observing the characteristic clustering patterns that are observed at different domainaveraged volume fractions, which are superimposed in Figure 1. At low volume fractions there are a few small isolated clusters throughout the periodic domain, and as such, the clusterscale effectiveness factor will begin to deviate from unity. As the volume fraction increases the frequency of these clusters increases thus making the effective contacting between gas and particle phases poor. Near the minimum value in the curve, clusters begin to span the periodic domain, and the solid phase changes from the dispersed to the continuous phase. As volume fraction continues to increase the gasparticle flow becomes more homogeneous, and due to this homogeneity the clusterscale effectiveness factor increases toward unity in the limit of high particle volume fraction.
In Figure 2 (a) the clusterscale effectiveness factor is shown as a function of for four different values at fixed values of the and . The depth of the inverted bell shape curve in is an increasing function of . The values of given in Figure 2 (a) are substantially smaller than unity, but there is a marked change in from unity. This may seem peculiar, but the small values of arise as a result of the fact that the length scale used to determine is the particle diameter . A more fitting length scale is that associated with a cluster. However, the size of a cluster emerges as a result of finegrid twofluid model simulations, and cannot be considered as an input parameter. It is for this reason that we choose the particle diameter as the relevant length scale for rather than the length scale of a cluster. In Figure 2 (b) the dependence of the clusterscale effectiveness factor is given for four different values of the dimensionless filter size at fixed and . The departure of the effectiveness factor from unity is also an increasing function of . However, Figure 2 (c) shows that the change in the clusterscale effectiveness factor from unity is a decreasing function of when keeping and constant. While one might expect that increasing the value should increase the departure in from unity, we observe a decrease in this departure due to the fact that we are demanding to remain constant in Figure 2 (c). In these simulations the variation of is achieved by varying the value of . In order to maintain a constant the effective rate constant must be changed. Therefore, varying at constant requires the variation in the effective rate constant. It is this coupled variation that brings about the somewhat surprising the dependence of on .
6 Grid Resolution Effect
When deducing a filtered model from a finegrid simulation, it is necessary to ensure that filtered statistics are independent of grid size . To that end, the dependence of on is presented for four different grid resolutions in Figure 3 (a) at fixed , , and . The dependence of on grid size is substantial, and only begins to saturate when the grid size is around particle diameters. Figure 2 (b) shows the variation of the dimensionless filtered drag coefficient as a function of for four different grid resolutions. Here the dimensionless filtered drag coefficient is defined consistent with the earlier work of Igci et al. (2008)
(40) 
where the subscript indicates that the drag coefficient is inferred from the drag force and pressure fluctuation terms directed parallel to gravity. At grid sizes of sixteen particle diameters seems to exhibit grid independence for , with some grid dependence emerging at higher volume fractions. Figures 3 (a) and (b) illustrate the sensitive grid dependence of the clusterscale effectiveness factor when compared to other filtered quantities like the fluidparticle drag coefficient. Consequently, even finer grid resolutions are required when simulating reacting gasparticle flow when compared to nonreacting systems. This observation further supports the need for the development of filtered models for accurate coarsegrid simulation of reacting systems.
Due to the sensitivity of with respect to grid size, we seek to develop a model for the clusterscale effectiveness factor extrapolated to the limit of infinite grid resolution. In Figures 4 (a) and (c), plots of versus are given for the finest two grid resolutions presented in this study. Here, is the maximum value of , and is the filtered volume fraction at which the maximum value of occurs. From inspection of Figures 4 (a) and (c) it is clear that the grid dependence of the volume fraction variation in can be removed by rescaling by and by for all . In addition, it is demonstrated in Figures 4 (b) and (d) that the grid dependence of the volume fraction variation in can be removed for all by plotting versus . Here is the volume fraction at which the value of reaches zero. Due to the gridindependent behavior observed in Figures 4 (a)(d) a filtered model utilizing a piecewise description of the variation in with particle volume fraction about is developed, while extrapolating the values of , , and to infinite resolution.
In Figure 5 (a) and (b) the grid size dependence of is presented for two different values of and a variety of different filter sizes. The values of are clearly saturating as the value of approaches zero. In order to provide an accurate value of to use in our filtered reaction rate model, the value of at infinite resolution is determined via a Richardson Extrapolation (Roache, 1998). The values of and were observed to vary linearly with grid resolution, and a linear extrapolation was performed to ascertain the infinitely resolved estimates of and .
7 Extrapolated results and filtered model
In the previous section, grid independence of the clusterscale effectiveness factor was observed by plotting against scaled volume fraction coordinates that differ depending on whether is greater than or less than . Motivated by this observation, we seek to model the volume fraction variation of via a piecewise function about . In Figures 6 (a)(c) plots of versus are presented for different values of , , and . The shape of the curve of is clearly altered by changes in and , with only mild changes in the volume fraction dependence of at different grid values of . In Figure 6 (d) the variation in with is shown to collapse by plotting all data with the same value of together.
Figures 7 (a)(c), show the variation of with for different values of , , and . The variation in the dependence of on is clear, while the and dependence is substantially weaker. In Figure 7 (d) we illustrate that the dependence of on for different , , and values can be collapsed onto a single curve provided the value of is kept constant. From Figures 6 (d) and 7 (d) it is clear that the relevant scaling parameter governing the shape of the dependence of on on both sides of is dictated by a single parameter given as . The following functional forms are used to describe the dependence of :
(41) 
(42) 
Here, and are leastsquares fit parameters that depend only on , see Figures 8 (a) and (b). Both model parameters are given by smooth functions of presented in Table 3. It should be noted that the observed fluctuation in about the model curve in Figure 8 (b) can be attributed to the uncertainty in determining from the simulation results for .
In Figure 9 (a) is presented as a function of for a variety of different and values. The variation of with , , and is evident by inspection of Figure 9 (a). However, by replotting all data as a function of alone one can collapse all values onto a single master curve in Figure 9 (b). A curve fit is presented for in terms of in Table 3.
Linear extrapolation of to infinite resolution reveals that varies between , with no discernable trend in , , or . This variation in arises as a result of the fact that the clusterscale effectiveness factor is a weak function of in the region around and thus determining the value of the is subject to error. Since no discernable trend in was observed, we recommend the use of because it represents the ensemble average of the different values obtained. Moreover, since only varies slightly in the vicinity of , small errors in will not influence the quantitative behavior our filtered model substantially. In addition, the extrapolated value of was found to vary between with no systematic dependence on , , or . As a result of this fluctuation we choose because this is consistent with earlier work in our group suggesting that filtered model corrections for gasparticle hydrodynamics become negligible at volume fractions of and higher (Igci and Sundaresan, 2011a).
The clusterscale effectiveness model developed in this work predicts that effective reaction rates observed in coarsegrid simulations be larger than those predicted in fine grid simulations if the effects of fine scale structure are not accounted for via the clusterscale effectiveness factor. Therefore, in order to accurately perform continuum model simulations of reacting gasparticle flows on coarse spatial grids, filtered models for effective reactions rates are necessary. Without these corrections, coarsegrid continuum model simulations will consistently overestimate the conversion observed in a solid catalyzed reaction (for example see Zimmermann and Taghipour (2005)). Therefore, in order to perform accurate coarsegrid numerical simulations of a firstorder, isothermal, solidcatalyzed gasphase reaction we suggest the use of the following filtered species balance equation
(43)  
where the constitutive relation for is given in Table 3, and a constitutive relation for the effective dispersion coefficient can be inferred from the work of Loezos and Sundaresan (2002). In section 5, it was shown in that the nonlocal correction to the effectiveness factor contribute much as to the observed value of . We have observed that this contribution decreases as a function of increasing grid resolution and filter size. Therefore, in the limit of large filter sizes we expect that the contribution of the nonlocal term in eq. (43) will be weak, and can thus be neglected. However, for intermediate filter sizes we recommend the inclusion of this nonlocal correction.

While in this work the evolution of species mass fraction was solved in the gas phase alone, one could envision a model for a solidcatalyzed, gas phase reaction where the species mass fraction in both gas and solid phases is tracked separately and coupled through a mass transport term between particle and fluidphases. In a model framework of this type, the effective interphase mass transfer rates will be decreased due to the presence of clustering in the gasparticle flow, and any reduction in reactant conversion that arises can be attributed to decreased mass transfer efficiency. We expect that such an observed decrease in mass transfer coefficient will be on the order of the clusterscale effectiveness factor developed in this work. As an example, consider a mass transfer operation taking place between FCC particles and air, where the effective mass transfer coefficient is calculated to be using a model developed by Gunn (1978) for fixed beds. Using one can determine the characteristic rate constant for mass transfer , where and are the surface area and volume of an FCC particle, respectively. For an FCC particle , and using the clusterscale effectiveness factor model developed in eq. (44) for a filter size of we predict that . Using this minimum value of the clusterscale effectiveness factor we predict that a nearly 20 fold decrease in the effective mass transfer coefficient! Indeed decreased effective mass transfer coefficients on this order have been found in Energy Minimization MultiScale Model (EMMS) simulations of mass transport processes in gasparticle flows (Dong et al., 2008a).
8 Conclusions
The need for the development of filtered twofluid models for reacting gasparticle flows is demonstrated by considering a model firstorder, isothermal solidcatalyzed gasphase reaction. It is shown that constitutive relations for the filtered reaction rate and filtered species dispersion must be postulated to close the filtered species balance equation. Due to the fact that the model gasphase reaction in this work is isothermal and produces no volume change, only filtered species balance equations must be developed, without the need to alter the existing filtered gas and solid momentum balance equations given in the earlier works of Igci et al. (2008) and Igci and Sundaresan (2011a) for nonreacting, monodisperse gasparticle flows. Using finegrid continuum model simulations of reacting gasparticle flows we extract the clusterscale effectiveness factor, defined as the ratio of the finegrid reaction rate to the reaction rate observed in a coarse grid simulation. The clusterscale effectiveness factor is observed to retain an inverted bell shaped dependence on volume fraction approaching unity in both the low and high volume fraction limits. At intermediate volume fractions a decrease in the clusterscale effectiveness factor from unity is observed with the magnitude of this reduction being a strong function of mesoscale Thiele modulus , and dimensionless filter size with only a weak dependence on Schmidt number .
Due to the sensitivity of the clusterscale effectiveness factor with grid size, we determine an asymptotic form for the clusterscale effectiveness factor relying on a Richardson extrapolation of the minimum in the cluster scale effectiveness factor. The extrapolated values of the clusterscale effectiveness factor are shown to collapse when plotted as a function of , with the volume fraction dependence of the clusterscale effectiveness factor collapsing for fixed . Table 3 presents a curve fit of our collapsed results for use in coarsegrid simulations of reacting gasparticle flows with firstorder reaction kinetics. The reduction in effective reaction rates observed in this study can be used to rationalize the overprediction in reactant conversion that was seen in the twofluid model simulations of Zimmermann and Taghipour (2005). Finally, we note that the filtered model developed in this work is limited to firstorder reaction kinetics, and was restricted to twodimensional periodic domain simulations. Future work should extend such analyses to other reaction kinetics, and threedimensional bounded domains. However, we expect that the characteristic volume fraction, Thiele modulus, filter size, and Schmidt number dependence of will be qualitatively similar to the results presented here.
9 Notation
– fluidparticle drag coefficient – particle radius
– fit parameter for plot of versus
– fit parameter for plot of versus
–
– Biot number for mass transport
– maximum value of – particle diameter
– effective diffusivity of reacting species in continuum model
– molecular diffusivity of reacting species
– intraparticle species diffusivity
– mesoscale dispersion coefficient
– coefficient of restitution of the particle phase
– fluidparticle drag force experienced by the fluid
– gravitational acceleration vector
– radial distribution function at contact
– weight function for filtering
– collisional dissipation of granular energy
– viscous dissipation of granular energy
– intrinsic reaction rate constant
– effective rate constant for mass transfer
– effective reaction rate constant
– mesoscale effective rate constant for mass transfer
– convective mass transport coefficient
–
– gasphase pressure
– granular energy conduction vector
– rate of production of species i
– Reynolds number for the gasphase
– Surface area of a particle
– rate of deformation tensor
– mesoscale Schmidt number
– granular temperature
– gas velocity
– particle velocity
– terminal settling velocity of an isolated particle
– volume of a periodic domain
– Volume of a particle
– generic spatial position vector
– spatial position of filter center
– generic spatial position vector
Greek letters:
– fluidparticle friction coefficient
– production of granular energy through interphase slip
– dimensional grid size
– dimensional filter size
– dimensionless filter size
– gas volume fraction
–
– intraparticle effectiveness factor
– clusterscale effectiveness factor
– nonlocally corrected clusterscale effectiveness factor
– ratio of to its domainaveraged value
– conductivity of granular energy
– bulk viscosity of particle phase
– molecular gasphase shear viscosity
– effective gasphase shear viscosity
– shear viscosity of particle phase
– gas density
– particle density
– gasphase stress tensor
– particlephase stress tensor
– particle volume fraction
– mesoscale Thiele modulus
– Thiele modulus
– mass fraction of gas species i
10 Acknowledgements
The authors would like to acknowledge the financial support from ExxonMobil Research Engineering Co. and the U.S. Department of Energy, Office of Fossil Energy’s Carbon Capture Simulation Initiative through the National Energy Technology Laboratory.
Appendix A Effectiveness factor with nonlocal contributions removed
In section 3 the cluster scale effectiveness factor is defined in two ways given by eqs. (33)(34). Here, the effectiveness factor given in eq. (34) is derived relying on the definition of the effectiveness factor given in eq. (33). Let the particle phase volume fraction and species mass fraction at any point be represented as follows
(48) 
where is a spatial variable associated with filtered and microscopic variables, respectively, neither of which are located at the filter center. Assuming the filtered variables can be given by smooth functions of space the filtered value of and can be approximated via the following Taylor series
(49) 
(50) 
The Taylor expansions given in eqs. (49) and (50) can be plugged into eq. (48) to yield expressions for and that depend only on the location of the filter center and the microscopic spatial variable . Utilizing eqs. (48)(50) the filtered product of and can be expressed as