Jjh Bert Brouwers

Phase separation in centrifugal fields with emphasis on the rotational particle separator

The separation of heavy or light phases and particulate matter from fluids by centrifugation is considered. Emphasis is laid on the newly developed and patented technique of the rotational particle separator. This technique is used in areas ranging from the treatment of fluids in industrial processes to the filtering of air to protect men to respiratory allergic reactions. Principles of centrifugation, fluid forces on objects and kinematics of phases and particles in laminar flow are treated to obtain analytical expressions for parameters of separation performance. Conditions for stability of laminar flow in rotating configurations are specified; secondary flows induced by Coriolis forces and their effect on separation performance are quantified. Results of measurements are compared with theoretical predictions. Designs which have materialized in the various areas of application are discussed.

Particle collection efficiency of the rotational particle separator

The rotational particle separator is a patented technique for separating solid and/or liquid particles of 0.1 m and larger from gases. The core component is the rotating filter element which consists of a multitude of axially oriented channels which rotate as a whole around a common axis. Particles in the gas flowing in a laminar fashion through the channels are centrifuged towards the outer collecting walls of each individual channel while the purified gas leaves the channels at the exit. Solutions in closed form are presented for the probability that particles of given diameter are separated from the gas. Particle trajectories are governed by centrifugal forces and Stokes drag forces including Cunninghams correction. Solutions are given for channels of the following cross-sectional shapes: concentric rings, circles, triangles and sinusoids. Account has been taken of parabolic (Hagen?Poiseuille type) velocity distributions inside the channels and various distributions of the flow over the assembly of channels. The results compare favourably with measurements executed on six differently sized practical versions of the rotational particle separator.

Analytical solutions for non-linear conversion of a porous solid particle in a gas : I. isothermal conversion

Analytical description are presented for non-linear heterogeneous conversion of a porous solid particle reacting with a surrounding gas. Account has been taken of a reaction rate of general order with respect to gas concentration, intrinsic reaction surface area and pore diffusion, which change with solid conversion and external film transport. Results include expressions for the concentration distributions of the solid and gaseous reactant, the propagation velocity of the conversion zone inside the particle, the conversion time and the conversion rate. The complete analytical description of the non-linear conversion process is based on a combination of two asymptotic solutions. The asymptotic solutions are derived in closed form from the governing non-linear coupled partial differential equations pertaining to conservation of mass of solid and gaseous reactant, considering the limiting cases of a small and large Thiele modulus, respectively. For a small Thiele modulus, the solutions correspond to conversion dominated by reaction kinetics. For a large Thiele modulus, conversion is strongly influenced by internal and external transport processes and takes place in a narrow zone near the outer surface of the particle: solutions are derived by employing boundary layer theory. In Part II of this paper the analytical solutions are extended to non-isothermal conversion and are compared with results of numerical simulations.

Secondary flows and particle centrifugation in slightly tilted rotating pipes

A theoretical analysis is presented of viscous incompressible laminar flow in a pipe which rotates around an axis held at small angle with respect to its symmetry-axis. Analogous to the results of Barua and Benton [1, 2], solutions in closed-form are given for circulatory flows in the cross-sectional plane of the pipe due to Coriolis forces in combination with Hagen-Poiseuille flow through the pipe. The solutions are used to derive analytical expressions for trajectories of solid or liquid particles entrained in the gas and being subject to centrifugation and the said secondary flows. It is shown that despite centrifugation, particles can be locked into circulatory trajectories thus remaining suspended in the gas flowing through the pipe.

Dissipation equals production in the log layer of wall-induced turbulence

Measurements of the velocity field of wall-induced turbulence show that in the log-layer production of turbulence by shear tends to be equal to turbulent dissipation: see, e.g., Ref. 1, Chap. 7. Results of direct numerical simulation, although executed at limited Reynolds number, seem to support this observation: see, e.g., Ref. 2, part I, Chap. 7. More convincing in this respect are the recent DNS results of Hoyas and Jimenez 3 for turbulent two-dimensional 2D channel flow at Reynolds number Re=2003. Downloaded statistics of production and dissipation are approximately equal in the log-layer and in line with experimental results at similar Reynolds number. While evidence based on observation and DNS is appreciable, a theory that reveals possible equality of production and dissipation in the log-layer is lacking. Presented and discussed relationships usually refer to the outcome of experiments see, e.g., Ref. 4, Eq. 5.2.38, or they rely on qualitative arguments to disregard the contribution of other components of the kinetic energy balance see Ref. 5, Sec. 5.4. The main obstacle in the theoretical proof of the equality is a sound evaluation of the terms in the kinetic energy balance, which make the difference between production and dissipation, viz., lateral diffusion of kinetic energy and pressure, whereby the assessment of pressure diffusion is particularly problematic. In the subsequent analysis, we shall evaluate these terms on the basis of exact expressions derived from the Navier-Stokes equations. Adopting inertial sublayer asymptotics for velocities, the prevailing balances in the kinetic energy budgets are established. The starting point of our analysis is decomposition of the velocity field u i =u i x,t in a mean component u 1 =u 1 x 2 and the stochastically fluctuating components u i =u i x,t, i=1,2,3,indices 1, 2, and 3 representing the direction of the mean flow parallel to the wall, the direction perpendicular to the mean flow and the wall, and the direction perpendicular to the mean flow and parallel to the wall, respectively; x and t are space coordinate and time, respectively, ui = u 1 1i + u i , 1

On diffusion theory in turbulence

The Fokker-Planck equation for the probability density of fluid particle position in inhomogeneous unsteady turbulent flow is derived. The equation is obtained starting from the general kinematic relationship between velocity and displacement of a fluid particle and applying exact asymptotic analysis. For (almost) incompressible flow the equation reduces to the convection diffusion equation and the equation pertaining to the scalar gradient hypothesis. In this way the connection is established with eddy diffusivity models, widely used in numerical codes of computational fluid dynamics. It is further shown that within the accuracy of the approximation scheme of the diffusion limit, diffusion constants can equally be based on coarse-grained Lagrangian statistics as defined by Kolmogorov or on Eulerian statistics in a frame that moves with the mean Eulerian velocity as proposed by Burgers. The results presented for diffusion theory are the leading terms of asymptotic expansions. Truncated terms are higher-order spatial derivatives of the probability density or of the scalar mean value with coefficients based on cumulants higher than second order of fluid velocities and their derivatives. The magnitude of these terms has been assessed by employing scaling rules of turbulent flows in pipes and channels, turbulent boundary layers, turbulent jets, wakes and mixing layers, grid turbulence, convective layers and canopy turbulence. It reveals that a true diffusion limit does not exist. Although truncated terms can be of limited magnitude, a limit process by which these terms become vanishingly small and by which the diffusion approximation would become exact does not occur for any of the cases of turbulent flow considered. Applying the concepts of diffusion theory resorts to employing approximate methods of analysis.

Analytical solutions for non-linear conversion of a porous solid particle in a gas–II. Non-isothermal conversion and numerical verification

In Part I, analytical solutions were given for the non-linear isothermal heterogeneous conversion of a porous solid particle. Account was taken of a reaction rate of general order with respect to the gas reactant, intrinsic reaction surface area and effective pore diffusion, which change with solid conversion and external film transport. In this part, the analytical solutions are extended to non-isothermal conversion. Analytical solutions for the particle overshoot temperature due to heat of reaction are derived from the governing differential equation pertaining to conservation of energy, considering the limiting cases of small and large Thiele moduli. The solutions are used to assess the effect of interaction between chemical reaction rate and particle overshoot temperature on particle conversion. The analytical solutions are shown to compare favourably with numerical simulation results.

Theoretical analysis of inertially irrotational and solenoidal flow in two-dimensional radial-flow pump and turbine impellers with equiangular blades

Using the theory of functions of a complex variable, in particular the method of conformal mapping, the irrotational and solenoidal flow in two-dimensional radialflow pump and turbine impellers fitted with equiangular blades is analysed. Exact solutions are given for the fluid velocity along straight radial pump and turbine impeller blades, while for logarithmic spiral pump impeller blades solutions are given which hold asymptotically as (r1/r2)n[rightward arrow]0, in which r1 is impeller inner radius, r2 is impeller outer radius and n is the number of blades. Both solutions are given in terms of a Fourier series, with the Fourier coefficients being given by the (Gauss) hypergeometric function and the beta function respectively. The solutions are used to derive analytical expressions for a number of parameters which are important for practical design of radial turbomachinery, and which reflect the two-dimensional nature of the flow field. Parameters include rotational slip of the flow leaving radial impellers, conditions to avoid reverse flow between impeller blades, and conditions for shockless flow at impeller entry, with the number of blades and blade curvature as variables. Furthermore, analytical extensions to classical one-dimensional Eulerian-based expressions for developed head of pumps and delivered work of turbines are given.

Langevin and diffusion equation of turbulent fluid flow

A derivation of the Langevin and diffusion equations describing the statistics of fluid particle displacement and passive admixture in turbulent flow is presented. Use is made of perturbation expansions. The small parameter is the inverse of the Kolmogorov constant C0, which arises from Lagrangian similarity theory. The value of C0 in high Reynolds number turbulence is 5–6. To achieve sufficient accuracy, formulations are not limited to terms of leading order in C0−1 including terms next to leading order in C0−1 as well. Results of turbulence theory and statistical mechanics are invoked to arrive at the descriptions of the Langevin and diffusion equations, which are unique up to truncated terms of O(C0−2) in displacement statistics. Errors due to truncation are indicated to amount to a few percent. The coefficients of the presented Langevin and diffusion equations are specified by fixed-point averages of the Eulerian velocity field. The equations apply to general turbulent flow in which fixed-point Euleri...

Eulerian short-time statistics of turbulent flow at large Reynolds number

An asymptotic analysis is presented of the short-time behavior of second-order temporal velocity structure functions and Eulerian acceleration correlations in a frame that moves with the local mean velocity of the turbulent flow field. Expressions in closed-form are derived which cover the viscous and inertial subranges. They apply to general anisotropic turbulence at a large Reynolds number obeying the Kolmogorov theory. Previously published results for isotropic turbulence emerge as special cases. In the derivation use is made of the approximation of temporarily frozen turbulence proposed by Tennekes. It is shown to be valid under conditions not other than those for which the Kolmogorov hypotheses hold. The effects of intermittency appear to be marginal.