Fluid-particle flow

It is quickly evident, however, that it is necessary to blend theory with experiment to achieve the engineering objectives of predicting fluid-particle flows. Fortunately, there are several semi-empirical techniques available to do so (see Di Felice, 1995 for a review). Firstly, however, it is useful to define some more terms that will be used frequently. 

FLUIDIZATION, SOLIDS HANDLING, AND PROCESSING Edited by Wen-Ching Yang INSTRUMENTATION FOR FLUID-PARTICLE FLOWS by S. L. Soo... 

Laux, H. Modeling of dilute and dense dispersed fluid-particle flow, PhD Thesis, NTNU Trondheim, Norway (1998). 

In the cross-head type of dies, the melt is split at the inlet to the manifold and recombines 180° from the inlet. Moreover, the flow is not axisymmetric, and fluid particles flowing around the mandrel have a longer distance to travel than those that do not. 

Fluid-particle flow patterns are often clarified by application of phase diagrams. Kwauk (1963) and Matsen (1983) proposed such phase diagrams for moving beds. 

Fig. 22. Fluid-particle flow phase diagram for vertical moding bed (Li, 1991). 

If we consider the random variable theory, this solution represents the residence time distribution for a fluid particle flowing in a trajectory, which characterizes the investigated device. When we have the probability distribution of the random variable, then we can complete more characteristics of the random variable such as the non-centred and centred moments. Relations (3.110)-(3.114) give the expressions of the moments obtained using relation (3.108) as a residence time distribution. Relation (3.114) gives the two order centred moment, which is called random variable variance ... 

To obtain the friction coefficient X as the transport coefficient in LRT, we would like to take a nonequilibrium steady state average of the frictional force. Thus we identify F with the phase variable B. The frictional force in our system is the force exerted by the surface on the fluid. At equilibrium, the average of this force will be zero, because there is equal likelihood of fluid particles flowing in any given direction. Thus, the first term in Eq. [210] will be zero. Under shear, the surface will exert on average a nonzero force on the fluid due to the directionality of the flow. The frictional force is given by... 

Reyes Jr JN (1989) Statistically Derived Conservation Equations for Fluid Particle Flows. Nuclear Thermal Hydraulics 5th Winter meeting, Proc ANS Winter Meeting. 

Laplace PS (1806) Traite de Mechanique Celeste. Supplement to book 10, Vol. IV. Paris Gauthier-Villars, 1806. Annotated English translation by Nathaniel Bowditch (1839). Reprinted by New York Chelsea Publishing Company, 1996 Laux H (1998) Modeling of dilute and dense dispersed fluid-particle flow. Dr Ing Thesis, Norwegian University of Science and Technology, Trondheim, Norway Leonard BP, Drummond JE (1995) Why you should not use Hybrid , Power-Law or related exponential schemes for convective modelhng - There are much better alternatives. Int J for Numerical Methods in Fluids 20 421-442. 

Laux H (1998) Modehng of Dilute and Dense Dispersed Fluid-Particle Flow. Dr Ing Thesis, Norwegian University of Science and Technology, Trondheim, Norway... 

Reeks MW (1993) On the constitutive relations for dispersed particles in nonuniform flows. I. Dispersion in simple shear flow. Phys Fluids A 5 750-761 Reyes Jr JN (1989) Statistically Derived Conservation Equations for Fluid Particle Flows. Nuclear Thermal Hydraulics 5th Winter meeting, Proc ANS Winter Meeting. 

Crowe CT (2000) On models for turbulence modulation in fluid-particle flows. Int J Multiphase Flow 26 719-727... 

Rawlings JB, Ray WH (1988) The Modeling of Batch and Continuous Emulsion Polymerization Reactors. Part 1 Model Formulation and Sensitivity to Parameters. Polymer Engineering and Science 28(5) 237-256 Reyes Jr JN (1989) Statistically derived conservation equations for fluid particle flows. Proc ANS Winter Meeting. Nuclear Thermal Hydraulics, 5th Winter Meeting... 

Dorao and Jakobsen [40, 41] did show that the QMOM is ill conditioned (see, e.g.. Press et al [149]) and not reliable when the complexity of the problem increases. In particular, it was shown that the high order moments are not well represented by QMOM, and that the higher the order of the moment, the higher the error becomes in the predictions. Besides, the nature of the kernel functions determine the number of moments that must be used by QMOM to reach a certain accuracy. The higher the polynomial order of the kernel functions, the higher the number of moments required for getting reliable predictions. This can reduce the applicability of QMOM in the simulation of fluid particle flows where the kernel functions can have quite complex functional dependences. On the other hand, QMOM can still be used in some applications where the kernel functions are given as low order polynomials like in some solid particle or crystallization problems. 

There is still no reports published evaluating the behavior of this procedure for fluid particle flows. In practical applications, the stability problems are commonly adjusted by numerous tricks that reduce the accuracy of the method. 

It is important to note that the fluid-solid drag coefficients discussed above are valid only for monodisperse particles (i.e. particles with equal diameters and material densities). Using direct numerical simulations (DNS) of the microscale equations for fluid-particle flows, several authors (Beetstra et al, 2007 Buhrer-Skinner et al, 2009 Holloway et al, 2010 Tenneti et al, 2010, 2012 Yin Sundaresan, 2009) have proposed improved drag coefficients to account for polydisperse particles. 

Desjarddjs, O., Fox, R. O. Villedieu, P. 2008 A quadrature-based moment method for dilute fluid-particle flows. Journal of Computational Physics 227, 2524-2539. 

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