The Hydrodynamic Model

Sides and Tobias observed the coalescence of large bubbles causing fluid motions close to the electrode surface which may be important in the mass transport enhancement due to gas evolution 

The Nemst boundary layer thickness is a simple characteristic of the mass transfer but its definition is formal since no boundary layer is in fact stagnant and least of all boundary layers on gas-evolving electrodes furthermore, the Schmidt number, known to influence mass transfer, is not incorporated in the usual dimensionless form. For this reason, lines representing data from gas evolution in two different solutions can be displaced from one another because of viscosity differences. Nevertheless, the exponent in the equation 

---

In modeling an RO unit, two aspects should be considered membrane transport equations and hydrodynamic modeling of the RO module. The membrane transport equations represent the phenomena (water permeation, solute flux, etc.) taking place at the membrane surface. On the other hand, the hydrodynamic model deals with the macroscopic transport of the various species along with the momentum and energy associated with them. In recent years, a number of mathematical... 

Urrs, ULTB> aLs and otTB can be found from the hydrodynamic model, and CfTP is calculated in the same manner as discussed previously for bubbly flow. 

Schouten, J. C., and VandenBleek, C. M., Chaotic Behavior in the Hydrodynamic Model of a Fluidized Bed Reactor, Proc. llthlnt. Conf. on Fluidized Bed Comb., 1 459(1991)... 

The flow field of the impacting droplet and its surrounding gas is simulated using a finite-volume solution of the governing equations in a 3-D Cartesian coordinate system. The level-set method is employed to simulate the movement and deformation of the free surface of the droplet during impact. The details of the hydrodynamic model and the numerical scheme are described in Sections... 

The hydrodynamic model In this model the adsorbed gas is considered as a liquid film, which can glide along the surface under the influence of a pressure gradient. Gilliland, Boddour and Russel (1958) used this model to calculate their fluxes. 

The abundance of lithium in stellar atmospheres presents an important observational constraint to the hydrodynamical models of the outer layers of stars. It can be considered as a cumulative measure of the extent of matter exchange between surface and deeper layers during the stellar evolution. 

Detailed analyses of the observed spectra are helpful in order to answer questions concerning the hydrodynamic models, the mass of the envelope, the chemical composition as a function of radius and therefore the stellar evolution of the progenitor, etc.. A critical test for atmospheric calculations is the observed time dependences of the features. 

These observations would give important constraints on the distribution of the heavy elements and 56Co in the ejecta. We adopted the hydrodynamical model 11E1Y6 (Nomoto et al. 1988) and carried out Monte Carlo simulation for photon transfer. A step-like distribution of 56Co was assumed where the mass fraction of 56Co in the layers at Mr < 4.6 Mq, 4.6 - 6 M , 6-8 Mq, and 8-10 Mq are Xq0 = 0.0128, 0.0035, 0.0021, and 0.0011, respectively. Other heavy elements were distributed with mass fractions in proportion to 56Co. 

Figure 1 The calculated hard X-ray and 7-ray spectra for the the hydrodynamical model with E = 1 xlO51 erg and Menv = 6.7 M0 at a) t = 200 d, b) i = 250 d, c) t = 400 d, and d) t = 600 d. At t = 200 d, the crosses indicate the spectrum observed by Ginga and the open circles and the diamonds are obtained by Kvant. At t = 250 d, balloon-borne observation (Wilson et al. 1988) is shown.

The second is the hydrodynamic model of Flory and Fox (27) which represents the polymer molecule by an equivalent nondraining hydro-dynamic sphere. Assuming the Flory constant to be the same for linear and branched polymers, the degree of branching is given by the g3/2 rule ... 

Starting from the flame front the intensity of the vortices remains constant along each streamline, so that the region filled by combustion products is a rotational one. In some of the previous works mentioned, however, the existence of the stagnation zone behind the flame front has not been accounted for, so that the quantitative conclusions diflier essentially from those of the hydrodynamic model presently under consideration. It should be noted that the boundary streamline of the stagnation zone is a tangential velocity component discontinuity surface or a vortex sheet. As a consequence of the... 

We developed the relationship between position-dependent viscosity q(z), n, and D(ti) via generalization of the hydrodynamic model (Reynolds equation) ... 

Despite his reservations about the self-energy of an electron Madelung did offer some guidance on the hydrodynamic model for many-electron systems. Of the three possibilities that... 

The most attractive feature of the hydrodynamic model is that it obviates the statistical interpretation of quantum theory, by eliminating the need of a point particle. However, even Einstein, despite his famous insistence that "the old one does not play dice", and despite the convincing physical picture of the Schrodinger interpretation, remained convinced that an electron had to be a point particle. His first allegiance was, after all, with his own special theory of relativity that imposes an upper limit of c on the speed at which any signal can be transmitted. 

As in the hydrodynamic model the Bohmian interpretation assumes a wave function in polar form,... 

Fig. 3.3. Schematic illustration of the proposed architecture of mucus glycoproteins, (a) Subunits constitute a linear array of oligosaccharide clusters interspersed with naked stretches of protein. (b) Trypsin digestion affords glycopeptides corresponding to the oligosaccharide clusters. (c) The whole mucins are formed by an end-to-end association of subunits via disulphide bonds, (d) The hydrodynamic model of mucus glycoproteins conforms to a random coil within a spheroidal solvent domain. (After Allen et al., 1984.)...

The hydrodynamic model is based on a sufficiently high collision probability under thermodynamic equilibrium. This condition is fulfilled only at p > 1 mbar, i.e., at high pressures above the typical PLD film deposition conditions, or at the beginning of plasma expansion, at high plasma density (small target to substrate distance). 

Operating scale (laboratory vs. industrial) affects the behavior of chemical reaction systems. It is critical that we develop hydrodynamic models for those systems that are scale sensitive. This will require a collaboration between academic and industrial groups to collect data necessary for commercial-scale equipment. Once the hydrodynamic models have been developed and validated, kinetic models can be integrated with them. 

Investigation of the dependence of the viscosity of dispersions on concentration has long attracted the attention of scientists. The hydrodynamic model developed by Einstein [1] has come to be considered classical in this direction of research. This model has led to the expression ... 

The causal interpretation of quantum theory as proposed by De Broglie and Bohm is an extension of the hydrodynamic model originally proposed by Madelung and further developed by Takabayasi [36]. In Madelung s original proposal R2 was interpreted as the density p(x) of a continuous fluid with stream velocity v= VS/rri. Equation (5) then expresses conservation of fluid, while (6) determines changes of the velocity potential S in terms of the classical potential V, and the quantum potential... 

The first term in each case arises from bulk flow of gas into the floor of an isolated bubble and out the roof, as required by the hydrodynamic model of Davidson and Harrison (27). The weight of experimental evidence, from studies of cloud size (28,29), from chemical reaction studies (e.g. 30), and from interphase transfer studies (e.g. 31,32), is that this term is better described by the theory proposed by Murray (33). The latter leads to a reduction in the first term by a factor of 3. Some enhancement of the bulk flow component occurs for interacting bubbles (34,35), but this enhancement for a freely bubbling bed is only of the order of 20-30% (35), not the 300% that would be required for the bulk flow term Equations (1) and (2) to be valid. 

Gasification, I. The Hydrodynamical Model," Proceedings, Miami International Conference on Alternative Energy Sources, 1978. 

Clift (1993) has also argued that the hydrodynamic models are not sufficient to explain the stability of fluidized beds and that the interparticle forces that determine the elasticity of the bed are important to explain the stability. He has given expressions of Abdel-Ghani et al. (1991) for the mean elasticity modulus of the bed. 

Two-fluid simulations have also been performed to predict void profiles (Kuipers et al, 1992b) and local wall-to-bed heat transfer coefficients in gas fluidized beds (Kuipers et al., 1992c). In Fig. 18 a comparison is shown between experimental (a) and theoretical (b) time-averaged porosity distributions obtained for a 2D air fluidized bed with a central jet (air injection velocity through the orifice 10.0 m/s which corresponds to 40u ). The experimental porosity distributions were obtained with the aid of a nonintrusive light transmission technique where the principles of liquid-solid fluidization and vibrofluidization were employed to perform the necessary calibration. The principal differences between theory and experiment can be attributed to the simplified solids rheology assumed in the hydrodynamic model and to asymmetries present in the experiment. 

For liquids, the velocity of ultrasound depends on the compressibility and density of the liquid. For suspensions, the velocity depends also on the drag of particles in the liquid under the influence of the ultrasonic wave. At low frequencies, small particles tend to move in phase with the liquid and the ultrasonic velocity may differ widely from that in the pure liquid. As particle size and ultrasonic frequency increases, the particles tend to lag more and more behind the movement of the liquid and the ultrasonic velocity approaches that of the suspension acting as a uniform fluid. There is a transition frequency range between complete entrainment and no entrainment of the particles that can be used to obtain particle size information. The hydrodynamic model of Marker and Temple [267 ] can be used to calculate ultrasonic velocity. This model takes into account the effects of fluid viscosity, of concentration, density and elastic modulus of both particles and fluid and can predict ultrasonic velocities accurately for volume fractions between 5% and 20%. Ultrasonic velocity measurements in the 50 kHz to 50 MHz can be used to determine particle size distributions in the range of about 0.1 to 30 pm. 

In addition to the hydrostatic model, two other mechanisms have been examined osmotic model and the hydrodynamic model. 

The temperature and density variation predicted by the hydrodynamic model of Oxtoby is... 

---