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Show Scale Analysis of Pulverized Coal Combustion Min-min Zhou; Jeremy Thornock; Philip Smith; Sean T.Smith;, Department of Chemical Engineering, The University of Utah, USA Minmin.Zhou@utah.edu, +1 801-9703213 Abstract Large eddy simulation (LES) is becoming a more powerful tool than ReynoldsAveraged Navier-Stokes (RANS) for predicting the flame characteristics of coal combustion. Previous studies verified that RANS fails to capture the particle dispersion process in turbulent flows which is critical for accurate simulations. LES coupled with Direct Quadrature Method of Moments (DQMOM) was carried out to simulate pulverized coal jet flames in the large laboratory scale furnace-100 kW down-fired oxy-fuel combustion (OFC) furnace. Previous studies showed that this high-fidelity methodology can provide a deeper understanding of very complex coal flames. The effects of unresolved turbulent scales on the particle motions were analyzed by Stokes number. The analysis showed that the sub-grid scales did affect the trajectories of small particles while had little effects on large particles. Since small particles are important for flame stability, sub-grid scale velocity models are necessary to fully resolve the particle motions. Keywords: Large Eddy Simulation, Scale Analysis, Particle-laden flow Preprint submitted to Journal Name September 13, 2018 There are so many operating parameters in the combustion system which draws lots of interests for the plan engineers, such as pollutant emissions,ash deposition, material life length and so on. These time-averaged parameters results from a cumulative result of several short time or instantaneous time scale phenomena. In fact, so much of this short or instantaneous time scale phenomena have a significant influence on many fundamental outcomes of the combustion systems. Likely the instantaneous flame stability directly affects the flame shape and the subsequent formation of hazardous N ox which is a essential factor for boiler emission standards. In addition, the particle flow pattern affects a lot for ash deposition rates and the deposit thickness which usually reduces the net heat transfer to boiler tubes and determines how often the boiler shuts down and make clean-up operation. This negative impact usually need several weeks to cumulate. But this long-term phenomena is closely related with the instantaneous particle flow and ash deposition rates. Using Large eddy simulation(LES)-DQMOM model [1] to simulate this system is able to capture many temporal and spatial scales phenomena. The aforementioned CFD-based modeling provides enough information to conduct a scale analysis (spatial and time dimension) for this combustion system. This combustion system can be characterized as a turbulent multiphase reacting flows. It includes times scale for fluid flow, particle flow and wall heat transfer process. This wall heat transfer process comprised of the particle deposition, particle sistering and heat transfer process (convective, conduction and radiation), as shown in Fig. 1 Figure 1: Time scale of the combustion system Large eddy simulation (LES) is becoming a more powerful tool than Reynolds2 Averaged Navier-Stokes (RANS) for predicting the flame characteristics of coal combustion [1]. This study utilizes LES simulation results to conduct a thoroughly scale analysis. 1. Turbulence field analysis The turbulence field analysis can give rise to estimate the time scale at the LES resolved scale which lay a foundation for the further analysis of Stoke number analysis and also definitively assess the suitability of LES simulation for the coal combustion. The spatial scale analysis can provide some indications and insight of turbulent scale in this combustion system. The turbulence in this system can be indicated by the turbulent length scales(wave number),κ(1/length scale) and the turbulent kinetic energy EU (k) spectra. As this analysis using the turbulent energy spectrum, the velocity field (u(x,t)) in the target domain can be represented by a finite Fourier Transform series: κX max u(x, t) = eiκ·x û(x, t), (1) κ0 where the lowest non-zero wavenumber (κ0 ) and the largest wavenumber(κmax ) are defined as 2π/L and πN/L, respectively. The size of researched domain is defined as L and the number of grid nodes to resolve this domain refers to N. The frequency location between grid-resolved to sub-grid modeled is defined as the cut-off frequency [2]. To a fixed problem, the cut-off frequency is increased as the filter width is increased with more resolution. Therefore the dependence of resolved turbulent kinetic energy on the sub-grid model is lessened. Accordingly, LES fundamentally resolves the turbulent eddies larger than a specified filter scale(∆) while modeling the turbulence within the subgrid scale. Kolmogorov's second similarity hypothesis assumes that the statistics of inertial subrange eddies in which scale(li ) is much bigger than the dissipative eddies(θ) at the Kolmogorov scale and smaller than the integral scale, have a universal form at high Reynold number turbulent flow[3]. This scale eddies in the inertial subrange is uniquely determined by dissipation rate ,independent of molecular viscosity. This length scale(li ) is a important parameter in the determining the resolved scale in the large eddy simulation. Given a specified eddy scale l, the characteristic velocity scales and time scales are provided by: 1 u(`) = (`η) 3 ∼ u0 (`/`0 )1/3 , 3 (2) 1 u(`) = (`η) 3 ∼ u0 (`/`0 )1/3 , (3) At high Reynolds number flow, the cutoff scale usually locates at the inertial range and the filtered velocity field take for nearly all of the kinetic energy(< E >) [3] 1 1 1 < E >≈< E >= Ef + kr , where Ef = U · U , kr = U · U − U · U . (4) 2 2 2 In the practical application, the filter scale represented by the grid resolution is supposed to be imposed in the inertial subrange and ensure most of kinetic energy to be resolved at this combustion system. Other concerns about this grid resolution is about quality or reliability of LES under the used mesh size. A ratio of resolved turbulent kinetic energy(kres ) to the total total turbulent kinetic energy to assess this LES quality [4], is defined as Rresolved = kres , kres + ksgs (5) where kres refers to 12 hui ui i and the half the trace of the Reynolds Stress tensor [3]. In LES modeling, one of targets is to resolve the majority of turbulent kinetic energy field directly ( 80%) and leave a small portion of unresolved subgrid kinetic energy. Overall this analysis can answer which percentage of the turbulence is directly resolved in the turbulent spectrum. The first step of this analysis is to know the residual kinetic energy(ksgs ) in the sub-grid scale which determines subgrid velocity fluctuations. Since the filter scale usually is set in the inertial subrange, the model assumes that subgrid scale turbulent motion as a locally homogeneous and isotropic. The residual velocity is uncoupled with the resolved velocity velocity in LES, because the residual velocity built by the reconstructing model is meant to mimic some statistics quantities only and not reconstruct the complete the subgrid turbulence. In present work, there are two ways to estimate the subgrid kinetic energy(kr ). 1.1. Subgrid kinetic energy This works adapts the first two approach to estimate the subgrid kinetic energy, which both stem from the turbulent-viscosity subgrid model hypothesis. Based on Pao energy spectrum [5], the first method relates the subgrid 4 turbulent viscosity with the energy spectrum to derivate residual subgrid kinetic energy (). Pao spectrum is shown by: 2 5 4 3 E(κ) = Cε 3 κ− 3 exp[− C(κη) 3 ]. 2 The residual kinetic energy on the subgrid scale can be integrated as: Z ∞ Z ∞ 5 2 2 4 3 − 53 3 3 C 3 κκ 3 dκ C κ exp− C(κη) dκ ≈ kr = 2 κc κc (6) (7) Lesiur work [6] using the spectral analysis proposed a relationship between the turbulent viscosity and residual kinetic energy, shown by: 3 νt = 0.441Ck2 ( E(κc ) 1 )2 κc (8) If assuming a Kolmogorov constant of 1.4 in the energy cascade, pre-equation 3 constant (0.441Ck2 ) becomes to 0.067 [5]. Substituting Pao energy spectrum of Eq. into the Eq.8, a rough estimation of subgrid kinetic energy gives as [7, 6] : kr = (νt /0.0067∆f )2 (9) The accuracy of this method much depends on the assumption for Pao's energy spectrum function shape in Eq. 6. If the real flow field spectrum does not not follows up with Pao's spectrum, this method usually underestimate the subgrid energy. The second method derivation begins from the definition of filtered kinetic energy and residual kinetic energy [8] in Eq. 4 The conservation equation for filter kinetic energy(Ef ) is shown by: DEf ∂ P − [U j (2µS ij −τijr − )] = f −Pr , W here f = 2νS ij S ij , Pr = −τijr S ij . ρ Dt ∂xi (10) Dissipation rate( f ) is viscous dissipation directly from the filtered field. In the high Reynolds number flow, the filter scale usually is much larger than Kolmogorov scale and thus this viscous disspation rate is relatively small compared to the dissipation rate from the resolved scale to subgrid scale. The second term of RHS refers to the dissipation of energy by viscous forces on the filtered scale, which is usually assumed to be zero in most high Reynolds 5 number flows. This assumption simplify RHS of Eq.10 into the only first term, Pr , which represents the energy transfer from the filtered scale to the subgrid scale. The derivate of filtered energy in Eq10 RHS correspondingly is the rate of dissipation. Consequently, the rate of production of residual kinetic energy can be approximated by: < P >=< 2Cνt S̃ij S̃ij >≈ f (11) Substituting Eq.11 into aforementioned Eq.1.1 gives the residual kinetic energy [9]: 2 (12) < ksgs >≈ (2∆νt S̃ij S̃ij ) 3 The known residual turbulent kinetic energy(ksgs ) can be used for the next section about reconstructing the subgrid scale turbulence. 1.2. Subgrid velocity Other important ingredient of this analysis becomes to reconstruct the turbulent motion within the sub-grid scale. The major concerns arises from the fact whether the subgrid-scale part of the flow velocity can have a noticeable influence on the particle motion. Some of previous studies neglected this influence and justified by a low residual energy content [10, 7, 11]. In their channel flow simulation, the RMS particle velocity is only slightly affected by integrating the residual kinetic energy. But the relative particle dispersion are found to be influenced by the LES filtering operation. This influence of the sub-grid turbulence become non-negligible for a sufficiently coarse LES mesh which is determined by the fraction of residual kinetic energy. In practice, the coarse mesh size is widely applied considering the affordable computational cost, like BSF simulation [10]. Therefore effects of sub-gird turbulence on the particle dispersion is extensive studied for the pulverized coal combustion. Firstly, it is necessary to reconstruct or mimic the residual fluid velocity (u0 ) accomplished by three approaches [12]: DNS simulation, random model and deterministic model, including approximated deconvolution model(ADM) and local Laplacian of the resolved field model [13]. The most accurate method attributes to DNS simulation but it is not applicable for its non-affordable computational cost. For the random model, the SGS velocity ξi is the random variable which follows Gaussian distribution with a mean SGS velocity and standard deviation [13]. Since the LES flow field obtained with the local spatial averaging can be regarded as a perfectly deterministic, 6 instantaneous solution and subgrid motions remain statistical in nature, the deterministic structural-type model(ADM) seems a consistent approach to reconstruct a stochastic subfilter-scale motions. A sound SGS reconstructing model should make primary characteristics of fluid field to be approximated to the fully resolved computation (DNS), like the residual kinetic energy, velocity autocorrelation. This work decompose the fluid velocity as U = ũ + u0 , where ũ is the part directly resolved in LES and u' is the subgrid velocity. This work adopts the stochastic Langevin model to reconstruct the carrier fluid subgrid velocity(u'). r 4ksgs /3 vi0 0 dωi (13) dvi = − dt + τL τ where dωi is an increment of the Wiener process. Ksgs is √ the subgrid kinetic energy obtained from the previous section. τL = Csgs ∆/ kr denotes the time scale of subgrid turbulence [14], where ∆ is the filter width of LES simulation and Csgs accounts for the uncertainty of the time scale of the residual velocity autocorrelation. Here C=1 is employed in the simulations by Pozorski's work [10]. Considering the crossing-trajectory effect,the characteristic time scale τL differs in directions of the relative velocity(parallel and perpendicular ), defined as : τL τL , τL,⊥ = p , (14) τL,|| = p 1 + β 2ξ2 1 + 4β 2 ξ 2 p where p ξ = |ṽ − Up |/ 2Ksgs /3 is the normalized drift velocity and τsgs = C(∆f / 2Ksgs /3) is the time scale of subgrid turbulent motion. β = 1 is the ratio of Lagrangian to Eulerian time scale [15]. 2. Stoke Number Analysis The stokes number is defined as : Stsgs = ρp Dp2 τp umean + u0 = ∗ τsgs 18µf fD ∆|sgs (15) A complete definition of Stoke number needs to known the turbulent fluctuation velocity(u) at the subgrid scale, which is defined as s Z κc +∆κ 0 u |sgs = 2∗· E(κ)dκ kc 7 (16) Figure 2: Schematic of fluctating velocity at subgrid scale Thanks to the instantaneous TKE spectrum, Ek at the subgrid scale can extrapolated from the TKE spectrum, which can represented by the red shadow area in Figure.2 If the particle response time is much larger than the time scales of turbulent flow, effects of SGS turbulence on the particle motion is insignificant and a subgrid model to reconstruct the subgrid velocity is not required [7]. If the particle response time scale is the same order of magnitude as the turbulent flow time scale, The lower-inertia particle is sensitive to be responsive to small-scale eddy structures since the fluid velocity field is deprived of subfilter (higher) frequencies. In this scenario, the subgrid velocity is needed for better predicting the particle flows. The fluid velocity in the drag force calculation, is the resolved velocity, directly form the LES,plus a SGS velocity contribution. The SGS velocity can be obtained from the SGS velocity reconstructing model in section 1.2. A schematic of aforementioned different scenarios is shown in Fig 3, which clearly illustrates different interaction between the fluid flow and particle flow. 8 Figure 3: Schematic of particle flow within different Stokes number 3. Results and Discussion 3.1. Evaluation of LES quality The ratio of resolved turbulent kinetic energy from Eq.5 have been proposed to assess the quality of LES simulation [4]. Since only one grid system have been used in the present work, it is crucial to ensures the quality of LES simulations achieved. Figure4 shows a contour plot of the ratio of resolved turbulent kinetic energy on the mid plane of OFC. This ratio are around 0.8 at OFC almost all regions, which meets the well-accepted criterion for a good LES simulation. The relative lower value is achieved in the jet flow core, compared to other region's ratio. Because the mean velocity, velocity fluctuations and shear rate inside jet flow is higher and need a more fine grid to resolve it. But the current work adopts a uniform mesh across the domain rather than a adaptive mesh setup. Figure.5 shows the spatial distribution of the proportion of the resolved TKE at different planes. The most of ratio is about 0.75 at this industrial-scale facility except some limited locations near the burner on the Figure. 5(c). Given the affordable computational costs, BSF simulation still provide a acceptable prediction of flow fields based on the following results at Chapter ??. 9 Figure 4: The ratio of resolved turbulent kinetic energy to total turbulent kinetic energy 3.2. Energy spectra Since the turbulent kinetic energy spectrum is the most informative plot for understanding the turbulent flow field. Figure.7 and Figure.6 show the spatial turbulent kinetic energy energy spectrum at OFC and BSF's different locations, respectively. The cut-off wave number of BSF and OFC, locate in the inertial range, because the gradient of near the cut-off scale approaches to a well-known f −5/3 power curves.TKE decreases along OFC axial direction due to the occurrence of kinetic energy dissipation. Also as increasing residence time at the BSF's boiler, TKE would decay from the maximum value near the burner, to a smaller value at the boiler central domain and then continue to decrease it at the convection zone of BSF. Comparison of isotropic TKE spectrum, TKE monotonically decreases when increasing the wave number and a peak shape disappear in these two facility's spectrum. Because the production term(P) is less than the dissipation term( f ) from Eq.10 during the energy-containing range [3]. 3.3. Stokes number analysis Figure.8 and Figure.9 show the evolution of Stoke number of differ size particle along its trajectory at OFC and BSF,respectively. There are a similar mechanisms governing Stokes evaluation history between OFC and BSF. Near burner Stokes number for all three particle environments are grater than that of other locations. Since the combustion reactions occur near the burner, the system give rise to a wider distribution at x=0.2 of OFC and near burner of BSF, respectively. The distribution narrows down at BSF and OFC downstream, after the coal particle combusted. Meanwhile the particle size and density decreases correspondingly and thus particle stokes number become smaller for all three environments at DQMOM model. Downstream at 10 Figure 5: The ratio of resolved turbulent kinetic energy to total turbulent kinetic energy: (a):mid vertical cross-section,(b) mid-burner horizontal cross section,(c) mid-OFA horizontal cross section OFC and BSF, the stokes number lies around 1 or smaller than one range, which demonstrate that the subgrid scale energy can not neglected and needs to reconstruct the subgrid scale velocity for the particle flow. 4. Conclusion Large eddy simulation (LES) is becoming a more powerful tool than Reynolds-Averaged Navier-Stokes (RANS) for predicting the flame characteristics of coal combustion. Previous studies verified that RANS fails to capture the particle dispersion process in turbulent flows which is critical for accurate simulations. LES coupled with Direct Quadrature Method of 11 Figure 6: TKE spectrum at different locations along OFC's axial direction Moments (DQMOM) was carried out to simulate pulverized coal jet flames in the large laboratory scale furnace-100 kW down-fired oxy-fuel combustion (OFC) furnace and industrial scale facility (BSF). Previous studies showed that this high-fidelity methodology can provide a deeper understanding of very complex coal combustion. The effects of unresolved turbulent scales on the particle motions were analyzed by Stokes number. The analysis showed that the sub-grid scales did affect the trajectories of small particles while had little effects on large particles. 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