CFD modeling and multi-objective optimization of cyclone geometry using desirability function,artificial neural networks and genetic algorithms

The low-mass loading gas cyclone separator has two performance parameters, the pressure drop and the collection efficiency (cut-off diameter). In this paper, a multi-objective optimization study of a gas cyclone separator has been performed using the response surface methodology (RSM) and CFD data. The effects of the inlet height, the inlet width, the vortex finder diameter and the cyclone total height on the cyclone performance have been investigated. The analysis of design of experiment shows a strong interaction between the inlet dimensions and the vortex finder diameter. No interaction between the cyclone height and the other three factors was observed. The desirability function approach has been used for the multi-objective optimization. A new set of geometrical ratios (design) has been obtained to achieve the best performance. A numerical comparison between the new design and the Stairmand design confirms the superior performance of the new design. As an alternative approach for applying RSM as a meta-model, two radial basis function neural networks (RBFNNs) have been used. Furthermore, the genetic algorithms technique has been used instead of the desirability function approach. A multi-objective optimization study using NSGA-II technique has been performed to obtain the Pareto front for the best performance cyclone separator.     

Numerical study of the effect of changing the cyclone cone length on the gas flow field

Models of six different cyclone separator designs have been investigated numerically to study the effect of changing cone length on the gas flow field. The cone length was changed in two ways, one by fixing the cone diameter and the other by fixing the cone angle. The Reynolds stress model was applied to predict the turbulence of gas flow. The model was validated by comparing the numerical results to published experimental data. The results show that, for the cyclone without cylindrical part, the pressure drop and the value of tangential and axial velocity components increase with increasing cone length with fixed cone angle until they reach maximum value. It was found that increasing the cone length with the same cone diameter increases the back flow and the lip leakage below the vortex finder. The back flow and the lip leakage phenomena was observed to be reduced by increasing cone length with fixed cone angle. The effect of contraction ratio D_x/D, which is defined as the ratio between the vortex finder diameter and the cone diameter, on the turbulent kinetic energy and the velocity components also was investigated. As the contraction ratio decreases, the maximum value of the turbulent kinetic energy and the tangential and axial velocity components increases. The pressure drop was fitted into a correlation as a function of dimensionless quantities (Re, D_x/D, and l_c/D) for each design, so that the designers could select the appropriate dimensions according to their design limitations.     

Improving cyclone performance by proper selection of the exit pipe

Cyclone performance is determined by pressure drop and collection efficiency. This study aims to optimize the dimensions of the exit pipe to improve cyclone performance. A numerical technique was used which is based on an Eulerian–Lagrangian approach. The behavior of the cyclone was studied by solving the three-dimensional, incompressible turbulent flow governing equations. The turbulent flow was modeled by using Reynolds Stress Model. Particle trajectories were obtained by solving the particle equation of motion. The collection efficiency was obtained by releasing a specified number of particles at the inlet of the cyclone and by counting the collected particles. The model was verified by comparing the numerical results to published experimental measurements. It was found through this study that increasing exit pipe diameter decreases the pressure drop through the cyclone and affects also the collection efficiency while exit pipe length does not affect cyclone performance significantly. It was concluded that the performance of the standard cyclone can be improved by prober selection for the diameter of the exit pipe. The data obtained through this study was represented in performance maps. These maps allowed the selection of the exit pipe diameter to obtain the maximum collection efficiency while avoiding excessive pressure drop.     

Numerical study of gas–solid flow in a cyclone separator

This paper presents a numerical study of the gas–powder flow in a typical Lapple cyclone. The turbulence of gas flow is obtained by the use of the Reynolds stress model. The resulting pressure and flow fields are verified by comparing with those measured and then used in the determination of powder flow that is simulated by the use of a stochastic Lagrangian model. The separation efficiency and trajectory of particles from simulation are shown to be comparable to those observed experimentally. The effects of particle size and gas velocity on separation efficiency are quantified and the results agree well with experiments. Some factors which affect the performance of cyclone were identified. It is shown that the collision between gas streams after running about a circle and that just entering occurred around the junction of the inlet duct and the cylinder of the cyclone, resulting in a short-circuiting flow. The combination of flow source and sink was distributed near the axis of cyclone, forming a flow dipole at axial section. Particles entering at different positions gave different separation efficiency. A particle with size exceeding a critical diameter, which was condition-dependant, would stagnate on the wall of cyclone cone. This was regarded as one of the main reasons for the deposition on the inner conical surface in such cyclones used in the cement industry.     

The effect of modeling of velocity fluctuations on prediction of collection efficiency of cyclone separators

The effect of modeling of velocity fluctuations on the prediction of collection efficiency of cyclone separators has been numerically investigated using the Reynolds stress turbulence model (RSTM) and large eddy simulation (LES). The Eulerian–Lagrangian modeling approach of CFD code Fluent 6.3.26 has been employed to simulate the three dimensional, unsteady turbulent gas–solid flows in a Stairmand high efficiency cyclone. The simulated results have been compared with experimental observations available in the literature. The analysis of results shows that the RSTM and the LES have adequately predicted the mean flow field. Results of the present study demonstrate that the LES has good performance on prediction of fluctuating flow field and collection efficiency for each and every particle size. However, the performance of the RSTM is found poor in terms of prediction of velocity fluctuations and collection efficiency, especially for small particles. This relates to the precessing of the vortex core phenomenon, which is resolved more accurately by LES as compared to the RSTM simulation. The results suggest that the prediction of collection efficiency, especially for small particles is greatly influenced by the simulation of velocity fluctuations in cyclones.     

The effect of actuator parameters on the critical flow velocity of a fluidic amplifier

The effect of actuator parameters on the critical flow velocity of the fluidic amplifier in liquid-jet hammers has been investigated numerically and experimentally. The flow in the fluidic amplifier and actuators coupled with the rigid body movement of the impacting body has been simulated using a commercial CFD software package, Fluent. The flow is modeled by the RNG-based κ–ε turbulence model and the incompressible Navier–Stokes equations. Dynamic layering method and a user-defined function written in C programming language are used to update the mesh in the simulations. The results show that, increasing the piston diameter decreases rapidly the critical flow velocity as the piston diameter is less than a certain value. The critical flow velocity increases sharply as the piston rod diameter is greater than a certain value, and increases nearly linearly with mass of the impacting body, and is independent on stroke length of the impacting body. The findings of the numerical investigations agree well with corresponding experimental results.     

螺旋型旋风分离器两相流场的数值模拟

对螺旋型旋风分离器进行了两相流场的三维数值模拟．气体流场通过求解三维N-S方程得到,湍流模型采用了雷诺应力模型．计算结果表明,旋风分离器内部的流场分为两部分:螺旋通道内比较稳定的流场和筒体中心区域的复合涡结构流场．对颗粒运动轨迹的计算表明,颗粒在入口处的初始位置对颗粒分离有比较显著的影响．同时得到了不同入口速度下颗粒的分级效率曲线,并给出了气体流量对旋风分离器性能的影响,结果显示:气体流量的增加会提高分离效率,但同时导致压力损失的急剧增加．     

Blowup phenomena of solutions to the Euler equations for compressible fluid flow

The blowup phenomena of solutions is investigated for the Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form c(t)x are constructed to show the expanding and blowup properties. The solution with velocity of the form for γ?1 and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form c(t)|x|^α-1x for α≠1 and multi-space dimensions.     

On elliptic units and p-adic Galois representations attached to elliptic curves

Let K be a quadratic imaginary number field with discriminant D_K≠-3,-4 and class number one. Fix a prime p?7 which is not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

On Estrada index of trees

Let G be a graph on n vertices, and let λ₁,λ₂,…,λ_n be its eigenvalues. The Estrada index is defined as . We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices with fixed diameter. For , we also determine the tree with maximum Estrada index among the trees on n vertices with maximum degree Δ. It gives a partial solution to the conjecture proposed by Ili? and Stevanovi? in Ref. [14].     

Graphs with the second largest number of maximal independent sets

Let G be a simple undirected graph. Denote by (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. Erd?s and Moser raised the problem of determining the maximum value of among all graphs of order n and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most r cycles. In this paper we determine the second largest value of and xi(G) among all graphs of order n. Moreover, the extremal graphs achieving these values are also determined.     

(2,1)-Total labelling of outerplanar graphs

The (2,1)-total labelling number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. In this paper we prove that if G is an outerplanar graph with maximum degree Δ(G), then if Δ(G)?5, or Δ(G)=3 and G is 2-connected, or Δ(G)=4 and G contains no intersecting triangles.     

A solution of the abstract Dirichlet problem for Baire-one functions

Let X be a compact convex subset of a locally convex space. We show that any bounded Baire-one function defined on can be extended to an affine Baire-one function on X if and only if X is a Choquet simplex and satisfies a certain topological property.     

Approximation of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (H_n)_n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then is the quantum Gromov-Hausdorff limit of any sequence for the natural quantum metric structures and when the lifts of σ_n to converge pointwise to σ. This allows us in particular to approximate the quantum tori by finite-dimensional C^*-algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.     

A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow

In this paper the effects catheterization and non-Newtonian nature of blood in small arteries of diameter less than 100 μm, on velocity, flow resistance and wall shear stress are analyzed mathematically by modeling blood as a Herschel–Bulkley fluid with parameters n and θ and the artery and catheter by coaxial rigid circular cylinders. The influence of the catheter radius and the yield stress of the fluid on the yield plane locations, velocity distributions, flow rate, wall shear stress and frictional resistance are investigated assuming the flow to be steady. It is shown that the velocity decreases as the yield stress increases for given values of other parameters. The frictional resistance as well as the wall shear stress increases with increasing yield stress, whereas the frictional resistance increases and the wall shear stress decreases with increasing catheter radius ratio k (catheter radius to vessel radius). For the range of catheter radius ratio 0.3–0.6, in smaller arteries where blood is modeled by Herschel–Bulkley fluid with yield stress θ = 0.1, the resistance increases by a factor 3.98–21.12 for n = 0.95 and by a factor 4.35–25.09 for n = 1.05. When θ = 0.3, these factors are 7.47–124.6 when n = 0.95 and 8.97–247.76 when n = 1.05.     

The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation

The well-posedness of the Cauchy problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local results for data in (s?−1/8) and the global well-posedness for data in are obtained if l=2. Moreover, for l=3, the problem is locally well-posed for data in H^s (s?1/4). The main idea is to use the Fourier restriction norm method.     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.