Fluid particle

Mass transfer in a gas-liquid or a liquid-liquid reactor is mainly determined by the size of the fluid particles and the interfacial area. The diffusivity in gas phase is high, and usually no concentration gradients are observed in a bubble, whereas large concentration gradients are observed in drops. An internal circulation enhances the mass transfer in a drop, but it is still the molecular diffusion in the drop that limits the mass transfer. An estimation, from the time constant, of the time it wiU take to empty a 5-mm drop is given by Td = d /4D = (10 ) /4 x 10 = 6000s. The diffusion timescale varies with the square of the diameter of the drop, so 

Mass transfer in the continuous phase is less of a problem for liquid-liquid systems unless the drops are very small or the velocity difference between the phases is small. In gas-liquid systems, the resistance is always on the liquid side, unless the reaction is very fast and occurs at the interface. The Sherwood number for mass transfer in a system with dispersed bubbles tends to be almost constant and mass transfer is mainly a function of diffusivity, bubble size, and local gas holdup. 

Breakup can occur if the shear stress is large enough to deform the particle, that is, when r 2a j A. The surface area increases during breakup and sufficient energy must also be provided to compensate for the increase in surface energy. In turbulent flows, the available shear stress in the turbulent eddies of size A, can be estimated from [20] 

The most efficient turbulent eddies for bubble breakup are eddies of the same size as the bubbles. Large eddies will merely move the bubbles and smaller eddies do not have sufficient energy to break up the bubbles. Assuming that the most efficient eddies to break up fluid particles are eddies of the same size as the bubble, that is, A. 2d, gives the required turbulent energy dissipation 

This equation gives only the energy required to break up a bubble. The rate of breakage will also involve the number density of eddies of size A and a probability that the bubble will break up [20]. 

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Here r(t) is the stress at a fluid particle given by an integral of deformation history along the fluid particle trajectory between a deformed configuration at time f and the current reference time t. 

Figure 3.2 Defonnation of a fluid particle along its trajectory in a contracting flow...

If the position of the fluid particle at current time t is given by x t) then its motion is defined using the following position vector... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

The distance covered by a fluid particle in this flow field in a time interval of -- t can be found by integrating Equation (3.78) as... 

After the substitution from Equations (3.82), (3.83) and (3.84) into Equation (3.81) and in turn substituting from the resultant relationship into Equation (3.80) and rearranging the following equation describing the trajectory of fluid particles is found... 

In Equation (5,14), (77j ) is found by interpolating existing nodal values at the old time step and then transforming the found value to the convccted coordinate system. Calculation of the componenrs of 7 " and (/7y ) depends on the evaluation of first-order derivahves of the transformed coordinates (e.g, as seen in Equation (5.9). This gives the measure of deformation experienced by the fluid between time steps of n and + 1. Using the I line-independent local coordinates of a fluid particle (, ri) we have... 

F. A. Zenz and D. F. Othmer, Eluidi tion and Fluid Particle Systems Reinhold Publishing Corp., New York, 1960. 

Frictiona.1 Pressure Drop. The frictional pressure drop inside a heat exchanger results when fluid particles move at different velocities because of the presence of stmctural walls such as tubes, shell, channels, etc. It is calculated from a weU-known expression of... 

Dry dense medium (pneumatic fluidized-bed) separation has been used, but has not received wide attention by the industry. An area of promise for future development is the use of magnetically stabilized dense medium beds by using ferro or magnetic fluids (2,10). Laboratory and pilot-scale units such as Magstream are available. In this unit, material is fed into a rotating column of water-based magnetic fluid. Particles experience centtifugal forces and... 

Viscosity (See Sec. 5 for further information.) In flowing liquids the existence of internal friction or the internal resistance to relative motion of the fluid particles must be considered. This resistance is caUed viscosity. The viscosity of liquids usuaUv decreases with rising temperature. Viscous liquids tend to increase tlie power required by a pump, to reduce pump efficiency, head, and capacity, and to increase Friction in pipe lines. 

FIG. 17-2 Schematic phase diagram in the region of upward gas flow. W = mass flow solids, lh/(h fr) E = fraction voids Pp = particle density, Ih/ft Py= fluid density, Ih/ft Cd = drag coefficient Re = modified Reynolds uum-her. (Zenz and Othmei Fluidization and Fluid Particle Systems, Reinhold, New York, 1960. )... 

In processes where new powder feed has a much smaller particle size than the smallest granular product, the feed powder can be considered as a continuous phase which can nucleate to form new granules [Sastry Fuerstenau, Powder Tech., 7, 97 (1975)]. The size of the nuclei is then related to nucleation mechanism. In the case of nucleation by spray, the size of the nuclei is of the order of the droplet size and proportional to cos0, where 0 is binder fluid-particle contact angle (see Fig. 20-67 of Wetting section). 

Figure 2.10. (a) An Eulerian x-t diagram of a shock wave propagating into a material in motion. The fluid particle travels a distance ut, and the shock travels a distance Uti in time ti. (b) A Lagrangian h-t diagram of the same sequence. The shock travels a distance Cti in this system. 

In an oriented porous medium, the resistance to flow differs depending on the direction. Thus, if there is a pressure gradient between two points and a particular fluid particld is followed, unless the pressure gradient is parallel to oriented flow paths, the fluid particle will not travel from the original point to the point which one would expect. Instead, the particle will drift. 

The rate of change of momentum equals the sum of the forces on a fluid particle (Newton s second law). 

The fluid is regarded as a continuum, and its behavior is described in terms of macroscopic properties such as velocity, pressure, density and temperature, and their space and time derivatives. A fluid particle or point in a fluid is die smallest possible element of fluid whose macroscopic properties are not influenced by individual molecules. Figure 10-1 shows die center of a small element located at position (x, y, z) with die six faces labelled N, S, E, W, T, and B. Consider a small element of fluid with sides 6x, 6y, and 6z. A systematic account... 

It is possible to determine the x-component of the momentum equation by setting the rate of change of x-momentum of the fluid particle equal to the total force in the x-direction on the element due to surface stresses plus the rate of increase of x-momentum due to sources, which gives ... 

When a dye is injected into a fluid, the resulting streak lines provide flow visualization of fluid particles that have passed the same density of the fluid. 

Laminar flow Fluid flow in which the fluid particles move in straight lines parallel to the axis of the pipe or duct. 

In this ehapter, the transport proeesses relating to partiele eonservation and flow are eonsidered. It starts with a brief introduetion to fluid-particle hydrodynamics that deseribes the motion of erystals suspended in liquors (Chapter 3) and also enables solid-liquid separation equipment to be sized (Chapter 4). This is followed by the momentum and population balances respeetively, whieh deseribe the eomplex flows and mixing within erystallizers and, together with partieulate erystal formation proeesses (Chapters 5 and 6), enable partiele size distributions from erystallizers to be analysed and predieted (Chapters 7 and 8). 

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. 

A relationship between these four variables is required in order to prediet partiele veloeity in a variety of eireumstanees. In doing so, it is noted that as a partiele moves through a fluid it experienees drag and viee versa as the fluid moleeules move aeross and around the surfaee of the partiele. There is thus a fluid-particle interaction due to interfaeial surfaee drag. 

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