# Approach velocity

The specific cake resistance is the most troublesome parameter ideally constant, its value is needed to calculate the resistance to flow when the amount of cake deposited on the filter is known. In practice, it depends on the approach velocity of the suspension, the degree of flow consoHdation that the cake undergoes with time, the feed soHds concentration, and, most importantly, the appHed pressure drop Ap. This changes due to the compressibiHty of most cakes in practice. often decreases with the velocity and the feed concentration. It may sometimes go through a maximum when it is plotted against soHds concentration. The strongest effect on is due to pressure, conventionally expressed as [c.392]

This effect of concentration is particularly pronounced with irregularly shaped particles. A possible explanation of the variation in the specific resistance is in terms of the time available for the particles to orient themselves in the growing cake. At higher concentrations, but with the same approach velocities, less time, referred to as particle relaxation time, is available for a stable cake to form and a low resistance results. [c.393]

For cross-flow plates, net area is the column cross section less that area blocked by the downcomer or downcomers (Fig. 14-22). The vapor velocity in the net area represents an approach velocity and thus controls the level of liquid entrainment. For counterflow plates, net area is the same as the column cross section, since no downcomers are involved. [c.1372]

Approach velocity The velocity of airflow into a filter bank or heat exchanger. [c.1414]

We used the concept of sound velocity dispersion for explanation of the shift of pulse energy spectrum maximum, transmitted through the medium, and correlation of the shift value with function of medium heterogeneity. This approach gives the possibility of mathematical simulation of the influence of both medium parameters and ultrasonic field parameters on the nature of acoustic waves propagation in a given medium. [c.734]

An alternative approach is to consider ions of charge z e accelerated by the electric field strengtii, E, being subject to a frictional force, Kj, that increases with velocity, v, and is given, for simple spherical ions of [c.570]

Two more important ideas can be illustrated by means of the Langevin approach to Brownian motion. The first result comes from a fiirther integration of the velocity equation to find an expression for the fluctuating displacement of the moving particle, and for the mean square displacement as a fimction of time. By carrying out the relevant integrals and using the fluctuation-dissipation theorem, we can readily see that the mean square displacement, ((r(t) - r(0)) ), grows linearly in time t, for large times, as [c.689]

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. [c.2251]

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. [c.2251]

One can write down any number of alternative schemes for treating constrained rigid body dynamics. One frequently suggested idea is to differentiate the constraints several times to eliminate the Lagrange multipliers, then solve the resulting ordinary differential equations for the positions and velocities using a generalized leapfrog or other integration method. This approach introduces additional computational complexity, as well as a drift from the orthogonality constraint if the constraint is not enforced directly by some sort of projection. Moreover, this method is not symplectic. A symplectic approach similar to this can be developed, as in [21], by employing Dirac s concept of weak invariants, but the resulting method is again inefficient compared to the above-mentioned schemes. Various other methods have been suggested [1, 13], but none appear to be serious competitors to the symplectic approaches for long term integrations. [c.357]

Since this approach maps all possible interactions to processors, no communication is required during force calculation. Moreover, the row assignments are completed before the first step of the simulation. The computation of the bounds for each processor require O(P ) time, which is very negligible compared to N (for N S> P). The communication required at the end of each step to update the position and velocity vectors is done by reducing force vectors of length N, and therefore scales as 0 N) per node per time step. Thus the overall complexity of this algorithm is. [c.489]

To perform ah initio molecular dynamics. Car and Parrinello suggested that the electronic and nuclear dynamics could be performed simultaneously. Somewhat surprisingly, it is found that in the Car-Parrinello scheme it is not necessary for the electronic configuration to be at a minimum in coefficient space for each molecular dynamics time step, even though tins gives errors in the forces on the nuclei. It can be shown that the errors in the nuclear forces are cancelled by the associated errors in the electronic motion. One possible explanation for this rather strange (and fortuitous ) result is to consider the motion of an atom with a single occupied molecular orbital. If the nucleus starts to move with a constant velocity, then the orbital will initially lag behind the nucleus. The orbital starts to accelerate until it eventually overtakes the nucleus. Having overtaken the nucleus, the orbital starts to slow down, until the nucleus overtakes the orbital, and so on, as illustrated in Figure 11.38. An important practical feature of this molecular dynamics approach is that the fictitious masses assigned to the coefficients must be chosen so that the frequencies of electron motion are higher than those of the nuclei to avoid energy exchange. This can have the practical consequence of requiring a smaller time step, which adds to the computational cost. [c.634]

In a significant number of polymer processes the influence of fluid elasticity on the flow behaviour is small and hence it is reasonable to use the generalized Newtonian approach to analyse the flow regime. In generalized Newtonian fluids tire extra stress is explicitly expressed in terms of velocity gradients and viscosity and can be eliminated from the equation of motion. This results in the derivation of Navier-Stokes equations with velocity and pressure as tire only prime field unknowns. Solution of Navier-Stokes equations (or Stokes equation for creeping flow) by the finite element schemes is the basis of computer modelling of non-elastic polymer flow regimes. In contrast, in viscoelastic flow models the extra stress can only be given through implicit relationships with the rate of strain, and hence remains as a prime field unknown in the governing equations. In this case therefore, in conjunction with the governing equations of continuity and momentum (generally given as Cauchy s equation of motion) an appropriate constitutive equation must be solved. Numerical solution of viscoelastic constitutive equations has been the subject of a considerable amount of research in the last two decades. This has given rise to a plethora of methods [c.79]

Since HyperChem -constructed molecular systems are near 0 K(the atoms have zero velocity), a simulation usually begins by adjusting the system to a higher temperature during a heating step. Heating can take place in one step (from near 0 Kto simulation temperature), but it is better to heat to the simulation temperature slowly, in small temperature increments. Slow heating allows the system to approach equilibrium during each heating step, so the system requires less time at the simulation temperature to reach equilibrium. [c.73]

In (a), a pulse of ions is formed but, for illustration purposes, all with the same m/z value. In (b), the ions have been accelerated but, because they were not all formed in the same space, they are separated in time and velocity, with some ions having more kinetic energy than others. In (c), the ions approach the ion mirror or reflectron, which they then penetrate to different depths, depending on their kinetic energies (d). The ones with greater kinetic energy penetrate furthest. In (e), the ions leave the reflectron and travel on to the detector (f), which they all reach at the same time. The path taken by the ions is indicated by the dotted line in (f). [c.193]

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference [c.109]

In many appHcations, especially in the chemical and semiconductor fields, the closest possible approach to isothermal operation may be desired. Under these conditions, the effects of vapor velocity must be considered if the velocity of the vapor exceeds about Mach 0.1, when a noticeable temperature differential shows itself in the heat pipe. If near isothermal operation is desired, designers restrict the vapor velocity to lower levels. [c.512]

The gas and process steam mixture can then be introduced into the primary reformer. This reformer is a direct-fired chamber containing single or multiple rows of high nickel-alloy tubes HK-40, HP-Modified, Incoloy 800, or other alloys are selected according to operating pressures and temperatures. The tubes are normally 72—110 mm ID and 10—13 m long. The catalyst contains 5 to 25% Ni (lower contents also include other metal promoters) as NiO supported on calcium aluminate, alumina, calcium aluminate titanate, or magnesium aluminate. Space velocities (SV) are usually on the order of 5000 8000 h based on wet feed. Steam-to-carbon ratios are usually in the range of 3.0—5.0, outlet gas temperatures, 800—870°C, and pressure, 2.16—2.51 MPa (300—350 psig). The outlet gas composition corresponds to a 0—25°C temperature approach to steam-reforming equiUbrium. That is, an equihbrium temperature is lower than actual at start and end of mn catalyst activity. The flue gas temperatures are 980—1040°C exiting the fired section of the furnace. In the convection section, the flue gases are cooled by superheating the steam for drivers, generating steam, preheating the hydrocarbon feed for desulfurization, and preheating the feed-plus-steam mixture before entering the radiant section of the furnace. In order to obtain high overall furnace efficiency, the stack temperature can be lowered to 150—170°C by preheating combustion air for the radiant section burners. [c.419]

It is possible to prepare a system having an initial concentration for each component, and then measure a finite, but small, change in the concentration of one component, A[M] for example, over a known interval of time A/. The experimental velocity A [A] / At and the concentrations can be substituted into a proposed rate law, along with postulated values for the exponents x,j,... to determine an observed rate constant although the rate law may involve more than one If this process is repeated for a reasonable range of concentrations, and the postulated rate law having the same exponents always yields the same then it is asserted that the rate law has been verified and the rate constant has been determined, within some precision, and is vaUd for those concentration ranges. This approach is a reasonable strategy for an initial survey of a totally unknown system. Moreover, it avoids the need to know the endpoint of the reaction. It is, however, tedious and gives imprecise values for It is also wasteful, in that it extracts very Httie data from each set of initial conditions. More often, the integrated form of the rate law is fit to multiple concentration measurements recorded at different times following each set of initial conditions. [c.508]

Free-Electron Lasers. The free-electron laser (EEL) directly converts the kinetic energy of a relativistic electron beam into light (45,46). Relativistic electron beams have velocities that approach the speed of light. The active medium is a beam of free electrons. The EEL, a specialized device having probably limited appHcations, is a novel type of laser with high tunabiHty and potentially high power and efficiency. [c.11]

Whether a flame is transmitted through a flame arrester depends on the length and aperture size of the arrester, the approach velocity of the flame, the pressure rise, and the temperature of the arrester (Wilson and Flessner 1978). Wilson and Flessner state that the evidence indicates that low-speed flames can be quenched by an array of small passageways placed in a duct, provided that the effective passageway diameter (critical diameter) meets the following criterion [c.105]

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. [c.63]

As a very direct method for measuring x, the surface is extended by means of two barriers that move apart at a velocity such that d In /d/ is constant. The dilation or depletion of the film results in a higher surface tension, measured by means of a Wil-helmy slide positioned at the center between the two barriers, where no liquid motion occurs. The procedure was applied to surfactant solutions [118]. An alternative approach is that of generating longitudinal waves by means of an oscillating barrier and observing the amplitude and phase lag of the motion of a small test particle [119,120] or of the film pressure [121]. On analysis, the data yield both E and the sum of and x. One may also measure the rate of change of surface dipole orientation, as obtained from the change in contact potential, following a change in surface area [122]. [c.120]

The previous calculations, while not altogether trivial, are among the simplest uses one can make of kinetic theory arguments. Next we turn to a somewhat more sophisticated calculation, that for the mean free path of a particle between collisions witii other particles in the gas. We will use the general fonn of the distribution fiinction at first, before restricting ourselves to the equilibrium case, so as to set the stage for discussions m later sections where we describe die fomial kinetic theory. Our approach will be first to compute the average frequency with which a particle collides with other particles. The inverse of this frequency is the mean time between collisions. If we then multiply the mean time between collisions by the mean speed, given by equation (A3.1.8), we will obtain the desired result for the mean free path between collisions. It is important to point out that one might choose to define the mean free path somewhat differently, by using die root mean square velocity instead of v, for example. The only change will be in a mimerical coefficient. The important issue will be to obtain the dependence of the mean free path upon the density and temperature of the gas and on the size of the particles. The mimerical factors are not that unportant. [c.669]

Consider an ensemble of Brownian particles. The approach of P2 to as 00 represents a kmd of diflfiision process in velocity space. The description of Brownian movement in these temis is known as the Fo/c/cer-PIanc/c method [16]- For the present example, this equation can be shown to be [c.696]

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where [c.994]

The mechanism by which a transition is induced by electron impact depends on the nature of the coupling between the projectile electron and the target this in turn is influenced by the velocity and closeness of approach of the projectile to the target. There is a wide range of possibilities. A high-energy projectile electron may pass quickly by, delivering only a photon-like electric-field pulse to the target at the mstant of closest approach. Less probable are hard, billiard-ball-like collisions between tlie projectile and one target electron. At low energies, slower, more intimate collisions are characterized by many-electron interactions. Depending upon tlie mechanism, the momentum transferred from projectile to target can vary from the minimum necessary to account for the transition energy to many times more. The interaction influences the type of transition that can be induced and the way in which the projectile is scattered. It is even possible for the projectile electron to be exchanged for a target electron, thus allowing for electron-spin-changing transitions. This state of affairs is a contrast to optical excitation where the momentum transfer is a constant and only dipole-allowed transitions occur with significant probability. [c.1307]

Product angular and velocity distributions can be measured with REMPI detection, similar to Doppler probmg in a laser-induced fluorescence experiment discussed in section B2.3.3.5. With appropriate time- and space-resolved ion detection, it is possible, in principle, to detemiine the three-dimensional velocity distribution of a product (see equation (B2.3.1 bit. The time-of-arrival of a particular mass in the TOFMS will be broadened by the velocity of the neutral molecule being detected. In some modes of operation of a TOFMS, e.g. space-focusing conditions [M], the shift of the arrival time from tlie centre of a mass peak is proportional to the projection of the molecular velocity along the TOFMS axis. In addition, Doppler tuning of the probe laser allows one component of the velocity perpendicular to the TOFMS axis to be detemiined. A more general approach for the two-dimensional velocity distribution in the plane perpendicular to the TOFMS direction involves the use of imaging detectors [66]. [c.2083]

The second discussion point is how the actual quantum system is to be described should one follow the time evolution of the time-dependent Schrodinger equation (TDSE) that allows mixed states to evolve, or should one insist on selecting a pure state, taking care of (sudden) transitions between states by some additional action in order to satisfy the time evolution of probabilities of states as dictated by the TDSE The former approach was followed, among others, by Bala et al. in wave packet dynamics applied to proton transfer in phospholipase A2 [109,110] and by us in the Density Matrix Evolution (DME) method which describes the mixed time-dependent wave function on asimple, appropriately chosen, basis set. [105, 111, 112, 113,114, 106, 115]. DME is obviously not capable of giving a correct response of the classical environment to quantum transitions, but is perfectly able to describe initial late processes or quantum systems that only weakly influence their environment. In fact, DME is the common method used in the evolution of nuclear spin magnetization [116]. The latter approach has led to the surface hopping method pioneered by Pechukas [117], with a modern formulation by Tully [118]. The basic idea is that the dynamics of a pure quantum state is followed, simultaneous with the classical dynamics of the environment. At every step the probability of a transition to another quantum state is calculated and such transitions are realized on a stochastic basis. When a transition is made, velocities are scaled to conserve total energy. The method has been [c.17]

If T is large, then the coupling will be weak. If t is small, the coupling will be strong a when the coupling parameter equals the time step (t = St) then the algorithm is equivak to the simple velocity scaling method. A coupling constant of approximately 0,4 ps has be suggested as an appropriate value to use when the time step is 1 fs, giving St/r 0.0025. T advantage of this approach is that it does permit the system to fluctuate about the desir temperature. [c.399]

Simplification achieved by using a constant mesh in the modelling of the flow field in a single-blade mixer is not applicable to twin-blade mixers. Although the model equations in both simulations are identical the solution algorithm for twin-blade mixers cannot be based on the VOF method on a fixed domain and instead the Arbitrary Lagrangian-Eulerian (ALE) approach, described in Chapter 3, Section 5.2, should be used. However, the overall geometry of the plane of the rotors blades cross-section is known and all of the required mesh configurations can be generated in advance and stored in a file to speed up the calculations. Figure 5.4 shows the finite element mesh corresponding to 19 successive time steps from the start of the simulation in a typical twin-blade tangential rotor mixer. The finite element mesh configurations correspond to counter-rotating blades with unequal rotational velocities set to generate an mieven stress field for enhancing dispersive mixing efficiency. Calculation of mesh velocity, required for modification of the free surface equation (see Equation (3.73)) at each time step, is based on the following equations (Ghoreishy, 1997) [c.146]

Mass Transfer. The degree of approach to equihbrium that can be achieved in adsorption is deterrnined by the mass-transfer rates. One useful design concept is the mass-transfer zone (MTZ), an extension of the ion-exchange zone method (105). Figure 18b is a depiction of the adsorbate loading in a fixed bed during adsorption. The ordinate is loading (X) and the abscissa is distance (F) from the inlet of the bed. Between the inlet and the exhaustion point (Fg), the loading is in equihbrium with the feed gas, and this section is called the equihbrium section. From the breakthrough point (1 ) to the outlet of the bed, the adsorbate loading is stiU at the residual loading level and is unused bed. Mass transfer between the gas and the adsorbent is occurring between the breakthrough and exhaustion points, and so this zone is called the mass-transfer zone (MTZ). The length of the bed, to F, is called the mass-transfer zone length (MTZL). The MTZL is usually correlated to flow rate or flow velocity (106). [c.285]

The scale-up and design of mixer—settlers is relatively reHable because they ate practically free of interstage hackmixing and stage efficiencies ate high, typically 80 to 90%. Various studies (134—136) have shown that (/) the rate of extraction is a function of power input, and (2) mixers can be teHably scaled up by geometric similitude at constant power input pet unit mixer volume, up to a 200-fold factor of throughput (137,138). The processes taking place in the settler ate complex. In large industrial mixer-settlers, the settlers usually represent at least 75% of the total volume of the units. The flow capacity of a settler depends on the behavior of a band of dispersion at the interface. The thickness of the band is a measure of the approach to flooding (97). The thickness increases exponentially with increasing flow pet unit interfacial area, and settlers can be scaled up by factors of up to 1000 on this basis. A practical means to increase the throughput pet unit settler area is needed so that the si2e of the settler can be reduced and the inventory of solvent lowered. The efficiency of the settler can be enhanced by minimi2ing turbulence and the formation of small drops, and maintaining low values of the linear velocity along the settler to avoid entrainment of small drops from the dispersion band. [c.75]

The equiHbrium approach should not be used for species that are highly sensitive to variations in residence time, oxidant concentration, or temperature, or for species which clearly do not reach equiHbrium. There are at least three classes of compounds that cannot be estimated weU by assuming equiHbrium CO, products of incomplete combustion (PlCs), and NO. Under most incineration conditions, chemical equiHbrium results in virtually no CO or PlCs, as required by regulations. Thus success depends on achieving a nearly complete approach to equiHbrium. Calculations depend on detailed knowledge of the reaction network, its kinetics, the mixing patterns, and the temperature, oxidant, and velocity profiles. [c.58]

See pages that mention the term

**Approach velocity**:

**[c.391] [c.1390] [c.680] [c.2082] [c.2382] [c.73] [c.370] [c.400] [c.469] [c.184] [c.78] [c.95] [c.100] [c.48] [c.491]**

Industrial ventilation design guidebook (2001) -- [ c.1414 ]