Temporally and spatially resolved calculations of a reacting coal jet using large eddy simulations (LES) and direct quadrature method of moments (DQMOM)

In a turbulent combusting coal jet, accurate prediction of the particle and velocity distributions are critical for predicting flame characteristics and ultimately heat flux in a coal-fired boiler. The present work combines a state-of-the-art particle model and turbulent-reacting flow model to provide a realistic simulation of the near burner region of a coal injector. The large eddy simulations (LES) technique is used to represent the turbulent coal jet the direct quadrature method of moments (DQMOM) is used to represent the particle phase. DQMOM provides an Eulerian particle-tracking method with full statistics. When, combined with the spatial and temporal resolution provided by LES, DQMOM yields a comprehensive statistical description of the behavior of the particle phase in the jet. Results are presented for a coal devolatilization experiment. Three key particle behaviors are note; particle segregation, particle clustering, and particle devolatilization. The present work identifies issues relevant to simulating reacting coal jets using LES and DQMOM, and lays the groundwork for future development.

T e m p o r a lly R e a c tin g (L E S ) an d C oal an d Spatially J et D irect U sin g R e s o lv e d L arge Q u a d r a tu re C alcu lation s E d d y o f a S im u la tio n s M e th o d o f M o m e n t s ( D Q M O M ) C h a rle s R e id , J e r e m y T h o r n o c k , P h ilip S m it h A b str a c t In a turbulent combusting coal jet, accurate prediction of the particle and velocity distributions are critical for predicting flame characteristics and ultimately heat flux in a coal-fired boiler. The present work combines a state-of-the-art particle model and turbulent-reacting flow model to provide a realistic simulation of the near burner region of a coal injector. The large eddy simulations (LES) technique is used to represent the turbulent coal jet the direct quadrature method of moments (DQMOM) is used to represent the particle phase. DQMOM provides an Eulerian particle-tracking method with full statistics. When, combined with the spatial and temporal resolution provided by LES, DQMOM yields a comprehensive statistical description of the behavior of the particle phase in the jet. Results are presented for a coal devolatilization experiment. Three key particle behaviors are note; particle segregation, particle clustering, and particle devolatilization. The present work identifies issues relevant to simulating reacting coal jets using LES and DQMOM, and lays the groundwork for future development. 1 In tro d u c tio n A major challenge in turbulence modeling is representing the large range of length and time scales. In general, modeling approaches separate these scales into two classes; resolved scales that are represented directly on the mesh and unresolved, or sub-grid scales that require modeling approximations. The Reynolds-averaging (RANS) approach, a familiar turbulence model in most industrial CFD packages, only represents the time averaged flow and as a result, models all turbulent scales. On the other hand, direct numerical simulations (DNS) seek to resolve all flow scales, yielding high fidelity results with little or no approximations. DNS is prohibitively expensive, however, for complex flows or moderate to large Reynolds numbers. Large eddy simulations (LES) present a middle ground; the large-scale flow structures are resolved without models, and only the small-scale turbulent flow structures are modeled. For flows with important time-dependent behavior, LES provides an accurate prediction of reality by resolving the large, unsteady scales, but vastly reduces the expense of DNS by modeling the small scales. In our LES calculation, the coal particles are represented using an Eulerian method, which has a stationary frame of reference relative to the computational mesh. The goal of the current Eulerian method is to represent the spatial and temporal dependence [6]of the number density function (NDF), which is the full statistical description of the particle phase. In particular, the direct quadra­ ture method of moments (DQMOM) is used to represent the coal phase [3, 2, 4]. In the DQMOM 1 2 Methodology 2 approach, the NDF is represented using a quadrature approximation that itself is a set of delta func­ tions. Computationally speaking, the DQMOM approximation results in a set of scalar equations representing the weights and weighted abscissa values of the quadrature approximation. These ad­ ditional scalars are transported in an Eulerian manner on the computational mesh. Thus, the NDF of the particle phase is represented at little additional expense to the LES calculation. The cur­ rent approach is made general enough that particle models that have been traditionally used in the Lagrangian approach are easily added to the DQMOM approach. This work is focused on representing three important characteristics that occur in coal-combustion systems; 1) particle size segregation, 2) particle clustering and 3) particle devolatilization. All of these phenomena influence operational considerations within a coal-fired boiler such as flame stability, flame lift-off, and heat flux characteristic within the boiler. Of course, none of these effect in this multiphase system occur independently. They are a result of multi-physics interactions between phases and between mixing streams. These effects are inherently unsteady processes and the mean characteristics of the flow are a function of the time dependant behavior from which the mean was derived. Thus it behooves a successful model to have some representation of the unsteady effects present in the model itself, whether directly resolved or represented with a sub-model. For this reason, LES combined with DQMOM models are used in this work as they capture the unsteady phenomenon. The combination of the two provides spatial and temporal representations of the near burner region, including the particle phase statistics. This document outlines briefly the formulation of the DQMOM used here. The formulation is incorporated into an LES code (called ARCHES), and results of a jet of devolatilizing particles are presented. Some issues related to the DQMOM are addressed. 2 M e th o d o lo g y 2 .1 LES A p p ro a ch The ARCHES (ADD REF) computational fluid dynamics (CFD) tool is used here to simulate the reacting coal jet. The ARCHES code solves the LES filtered equations of mass, species, moment, and energy balances (with radiation) on a structured, finite volume mesh. Additionally, ARCHES uses parameter reduction techniques for reaction modeling coupled with sub-grid scale mixing models. The ARCHES code is a component of the Uintah Computational Framework (UCF), a component-based problem solving environment that provides a general framework for massively-parallel simulations. The UCF manages the message passing and data handling needed for large-scale, parallel simulations performed on multi-processor computers. The simulations discussed in this work were run in parallel on processor numbers ranging from 324 to 832 which contained a set of grid cells on the order of 6M to 13M finite volumes. 2 .2 P a r tic le R e p r e s e n ta tio n 2 . 2.1 N u m b e r Density T ra n sp o rt Equation The number density function (NDF) describes the number of particles per volume as a function of several particle independent variables (e.g., particle diameter, particle composition, etc.) called internal coordinates. These internal coordinates are contained in the internal coordinate vector £ (boldface denotes vector quantities), and the internal coordinates are dependent on space and time (x ,t). The NDF transport equation is derived in the manner of Ramkrishna [6]. The velocity of the NDF in both real and internal coordinate space can be described using a set of internal coordinates 2 Methodology 3 I = [6. ,£ , • • •] and external coordinates x be written as 2 [x.,x 2 , • • •]; the time derivatives of these variables may ix i (£ ) lit = u' f x ' t) I t = 1, 2, 3 1, 2, j ( ■ x -t} (1) (2) where ui., is the velocity of the NDF in real (x) space, and G j , is the velocity of the NDF in internalcoordinate (£) space. Following Ramkrishna [6], Reynolds Transport Theorem is used to derive the NDF transport equation, f + E i-i ( K ,) / ) - £ A i=1 i=1 , , / ) + £ d j ( r * f ) + * « ;x,t), j=1 J v Jy (3) which represents the spatial and temporal evolution of the the NDF, including all particle phenomena (devolatilization, particle size change, breakage, etc.) 2 . 2.2 ( d„ f v ) = - £ ; A y j=1 J g M o m e n ts Ultimately, the NDF must be tracked in a CFD code. Nearly every CFD code is designed to run in a scalar framework were all higher-order vectors, tensors, etc. are ultimately expressed as a set of scalars. Thus, the NDF must be decomposed into a set of characteristic scalars. A convenient set of scalars are the moments of the NDF, as they describe statistical characteristics of the distribution. Additionally, the distribution can ultimately be re-constructed from its moments. The kth integer moment m k of a univariate NDF / (£; x,t) is defined as: £k/ (£; x,t) i£ mk = r / (4) / (£; x,t) i£ The first moment is the mean value of £, the second moment is the standard deviation of £, the third moment the kurtosis, the fourth moment the skewness, etc. If the internal coordinate is the particle diameter L, the 1st moment is the mean value of the particle diameter, L, the 2nd moment is the mean surface area L2 and so on. The moments of a multivariate NDF (i.e., an NDF that is a function of more than one internal coordinate) mk of a multivariate NDF /(£; x,t) are defined over all the internal coordinates as: m k = 00 (5) / (t 00 x,t) i£ 4 3 Coal Models where the integer vector k is the moment index vector for the kth (multivariate) moment, defined by k = [ki, k2, • • • , kNs], km is the m th index of the kth moment (corresponding to the mth internal coordinate), and is the number of internal coordinates. 2 . 2.3 Q u a d r a tu r e A pproxim ation: D irect Q u a d r a tu re M e th o d o f M o m e n ts Quadrature approximates the integral of an unknown function with tabulated known values as a summation of a set of N weighted abscissas. It determines a polynomial of degree 2N - 1 whose zeros are the N weighted abscissas, and approximates the unknown function using this polynomial [5]. While the unknown function does not have to be a polynomial, the quadrature approximation becomes much better if it is (and exact if the unknown function is a polynomial of degree 2N -1 or less). The general N -point quadrature formula can be written as: nb N / w (r) g (r) dr « E [wa (g (ria) g (r2a) ■ ■■)] (6) Ja a=1 The weights are common to all elements of r because the weight function w (r) is a single function common to all elements of r. The current NDF of interest (an NDF representing the coal particles) is multivariate and as such the direct quadrature method of moments (DQMOM) is used to solve Equation 3. The quadrature approximation from Equation 6 is applied to a multivariate NDF to yield, n f ( i ; x,t) ~ f N E (x,t)n \ Wa a=1 y j=1 \ 6 f e -& ( x , a) ) , J (7) where N is the number of quadrature nodes and N^ is the number of internal coordinates. This is the quadrature-approximated number density function f . Next, if the moment transform of the quadrature-approximated NDF is taken, the integral disappears, and only an algebraic expression is left. Taking the moment transform of this multivariate NDF approximation yields the quadratureapproximated multivariate moment: EN=1 (wa (nN=1& )kaj ) } mk « --------- ----------------------LL (8) wa a=i where k is the multivariate moment index vector, and kj is the j th element of k (with a total of N^ values, where N^ is the number of internal coordinates). DQMOM solves transport equations for the weights and abscissas of the quadrature approximation. These transport equations are simply scalar transport equations, and are easily implemented into a CFD code. After the weights and weighted abscissas are obtained one can reconstruct moments of the NDF from Equation 8 to obtain statics on the particle phase such as particle size, coal mass fraction, particle number and so on. For details on the full DQMOM derivation, see [4]. 3 C oal M o d e ls DQMOM and LES are used to simulate a coaxial jet of particle-laden flow in the primary inlet and particle-free gas flow in the secondary inlet. Both inlet temperatures are 298 K and introduced into a 3 Coal Models 5 1800 K environment. The velocity ratio of the primary inlet to the secondary is 0.625. The diameter of the primary jet is 0.014m and the diameter of the secondary jet is 0.032. The simulation focuses on the near burner region. The NDF in this work is a function of two internal (£) coordinates; particle size l, and particle mass of raw coal a c. Each internal coordinate was tracked using two quadrature points in the quadrature approximation. The result is a set of six extra scalar equations for the CFD solver. The initialization for each weight (equivalent to particles for each environment per volume) was wo = wi = 2x 1010 vol-1 ; the initialization for each internal coordinate was lo = 35^m, li = 75^m and a co = 0.91 x 10~10 kg, a c1 = 2.10 x 10-10 kg. 3 .1 P a r tic le D e v o la tiliz a tio n : T h e K o b a y a sh i-S a r o fim M o d el The Kobayashi and Sarofim model addresses the need to describe the pyrolysis of coal as a function of temperature in the early stages of the combustor. This model introduces a set of two, competing, parallel, first-order reactions that describe the conversion of raw coal (C) to gas phase volatiles (V ) and char residue (S). The reactions for the devolatilization of raw coal for this model is expressed as, c - ^ (1 -yi)S i + YiVi c -^ (1 -Y2)S2 + Y V where Y is some stoichiometric coefficient. The values for Yi and Y2 are determined from the volatile fraction of the proximate analysis (Yi) and the fraction devolatilized at high temperatures (Y2, often near unity). The rate expression for the depletion of raw coal in the solid phase for a particle is dac = rv + rh = -(ki + k )ac, dt 2 (9) The rate constants ki and k2 are modeled with an Arrhenius form as k = A e - E/RT, (10) where E2 ^ Ei . The values of these constants are taken from [7]. In this paper, only the mass of raw coal a c was tracked. Because of the inert environment, representation of the amount of char was not required. The conversion of raw coal to char, however, is represented in this devolatilization model. This model is the functional form of Gj,f, the velocity in phase space (specifically a c) in Equation 3. 3 .2 N D F V e lo c itie s Recently, Balachandar has applied the equilibrium Eulerian approximation to estimate the relative velocity between particles and the gas phase [1]. The relative velocity between the particles and the the gas is assumed to depend only on the particle's response time to the acceleration of the local fluid. If the particles are also assumed to be small enough that their response times are less than the Kolmogorov time scale (the smallest time scale of the fluid), then the relative velocity can be described using the following expression: 4 Results: Pyrolysis of Coal in the Near Burner Region (U 6 Up) Tp Tk - (i - uk 36 Pf + 1 (1 - p) ( l 0 (Rep) \ n 2 (11) where u, up, and Uk are the fluid, particle, and Kolmogorov-scale velocities, respectively; pp and pf are the densities of the particle and the fluid; p is a parameter associated with the density ratio, defined as, P= 3 (12) 2pP + 1 Pf where 0 (Rep) is an expression for the drag law of the particle as a function of the particle Reynolds number Rep (which, for the assumptions stated, can be treated as low Rep Stokes drag) and l and n are the particle length scale and the Kolmogorov length scale. The expression in (11) is used for the average velocity of an environment {ui)a, where in this case the particle density, particle length, and particle Reynolds number will be unique to each environment. Assuming constant density (shrinking-core) particles, the expression for {ui)a becomes: {ui) a 4 u (1 - P) f L uk 36 ( 2PP + 1 0 (Rep,a) \ n 36 V Pf 2 (13) R esu lts: P y ro ly sis o f C oal in t h e N ear B u rn er R egion A snapshot of the mixture fraction (kg hot gas/kg gas mixture) is shown in Figure 1. The total domain length shown is roughly 55 primary diameters long. As expected, the mixture fraction goes to a fully mixed state as it moves down the reactor as the hot and cold gases mix. As will be shown below, the particle behavior only partially matches the gas-phase behavior and deviations are explained by the distribution of the local drag effects on the different particles sizes. Results for a longitudinal slice across the jet radius, 20 primary diameters from the inlet, is shown in Figure 2. The plot shows the instantaneous number density of each delta function approximating the NDF per the quadrature approximation. We refer to the two delta functions approximating NDF as "environments" of the NDF. Because we have a quadrature of two, there are two environments (representing a small class of particles and a large class of particles). The results are collected from the LES simulation for a number of time steps. The black dots represent the smaller particles (Envi­ ronment 0), which started at a particle size of 35^m and a raw coal mass of 0.19 x 10~10 kg, while the red crosses are the larger class of particles (Environment 1), which started at a particle size of 75^m and a raw coal mass of 2.10 x 10~10 kg. These values are the weights (#/vol) of the two environments. Clearly, the smaller particles (black) have a higher number density near the edges of the jet, while the the larger particle (red) has a higher number density near the core of the jet. These results match the expected size segregation behavior of large and small particles in a turbulent jet. Additionally, particle clustering is observed for the Environment 0 across the radius of the jet as shown by the intermittent high concentrations of Environment 0 across the radius of the jet. Volume rendered images of the two particle fields are shown in Figures 3 and 4. These images reinforce the conclusion that the two particle classes are segregating and showing clustering effects. Analogous results are shown for instantaneous values of mass of raw coal per particle in Figure 5. While the Environment 1 (red) will naturally have a larger amount of raw coal due to the fact that the particles simply have a larger mass, we observe that the value of a c for the Environment 0 4 Results: Pyrolysis of Coal in the Near Burner Region 7 Fig. 1: Volume-rendered image of the gas mixture fraction (kg hot gas / kg mixture) throughout the reactor. Radial Location, [m] Fig. 2: Instantaneous number density results for coaxial jet at 20 diameters from the inlet. Results shown are number density for Environment 0 (black, smaller particles) and Environment 1 (red, larger particles). 5 Conclusions 8 Fig. 3: Volume-rendered image of the number density of the smaller particles. The dispersion of the small particles closely resembles the dispersion of the gas mixture fraction as the smaller particles with little inertia tend to follow the gas streamlines. (black) reaches zero in a radial location in the vicinity of 0.05 m . However, there are still a substantial number of smaller particles beyond this radius in the jet (see Figure 2). The small particles beyond the edge of the jet have completely devolatilized before the larger particles have uniformly devolatilized a significant amount. In other words, small particles move from the core of the jet to the edges, and begin to devolatilize first; the larger particles take longer to disperse, and devolatilize more slowly. Using the environments of the NDF from the calculation, we construct the time averaged statistics of the coal-particle field NDF. This includes the mean value of the coal mass for the particle distribution shown in Figure 6. The coal particles initially devolatilize slowly, but as they increase in temperature from the entrainment of hot gas, the reaction rate accelerates. After much of the raw coal has reacted and disappeared, the reaction rate slows down again. A plot of the standard deviation of length down the centerline of the reactor is shown in Figure 7. This plot demonstrates a reduction of the standard deviation until roughly 15 diameters down stream. The reduction of the standard deviation of length is attributed to the particle size segregation. Little particles disperse quickly from the center of the jet leaving behind the larger particles with high interia, effectively narrowing the distribution of the NDF. Between 15 and 20 diameters the standard deviation remains fairly constant after which is begins to rise indicating a destruction of the potential core. Afterwards, the distribution widens again as the system becomes fully mixed, widening the NDF. 5 C o n c lu s io n s A formulation was presented for tracking the moments of the NDF of coal particles as the particles devolatilize in a hot jet using DQMOM. This formulation utilizes the quadrature approximation to approximate the NDF. The model formulation was kept general, so that existing Lagrangian models for the behavior of coal particles can be implemented for future DQMOM simulations. Results were presented for a simulation of a devolatilizing coal jet using the direct quadrature method of moments. These results are very encouraging as DQMOM combined with the LES model 5 Conclusions 9 Fig. 4: Volume-rendered image of the number density of the larger particles. The jet is much less disperse compared with the number density of the smaller particles and compared with the mixture fraction of the primary inlet. Particle clustering is observed for this class of particles. Radial Location, [m] Fig. 5: Instantaneous values for a c, the total mass of raw coal in each particle. Results are shown for the small particle (black) and the large particles (red). 10 av e (a c), [kg] 5 Conclusions (a) (b) Fig. 6: Plots of the mean raw coal mass for the NDF showing a) the mass across the radius at different lengths down the reactor and b) the mass down the centerline of the reactor. Fig. 7: The number weighted standard deviation of the particle length down the centerline of the reactor. 5 Conclusions 11 produced clustering of larger particles, size segregation of different-sized particles, and differential devolatilization behavior. Future work includes introducing additional internal coordinates and adding additional particle models such as char oxidation, particle size changes, breakage, etc. to include more particle phsyics. Additionally, validation excersizes are need to test he model against experimental data. Finally, we desire to apply this model to oxy-coal systems to help assess flame stability issues in the near burner region for retrofit and scale-up purposes. R eferen ces [1] S. Balachandar. A scaling analysis for point-particle approaches to turbulent multiphase flows. Personal Communication, December 2008. [2] Rong Fan, Daniele L. Marchisio, and Rodney Fox. Application of the direct quadrature method of moments to polydisperse gas - solid fluidized beds. Powder Technology, 139:7-20, 2004. [3] Rodney O. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press, 2003. [4] Daniele Marchisio and Rodney O. Fox. Solution of population balance equations using the direct quadrature method of moments. Journal of Aerosol Science, 36:43-73, 2005. [5] William H. Press and Saul A. Teukolsky. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, 1992. [6] Doraiswami Ramkrishna. Population Balances. Academic Press (San Diego), 2000. [7] L. Douglas Smoot and Philip J. Smith. Coal Comustion and Gasification. Plenum Press, 1985.