Acceleration term

The same procedure may be applied in principle to design of forced-recirculation reboilers with shell-side vapor generation. Little is known about two-phase flow on the shell side, out a reasonable estimate of the fric tion pressure drop can be made from the data of Diehl and Unruh [Pet Refiner, 36(10), 147 (1957) 37(10), 124 (1958)]. No void-fraction data are available to permit accurate estimation of the hydrostatic or acceleration terms. Tnese may be roughly estimated by assuming homogeneous flow. 

This technique permits estimation of the volumetric flow rate at any level above a source, provided that the result is matched to the gravitational fume acceleration terms applicable near the source. The result of such an analysis is shown in Fig. 13.30. The emission flow rate from an electric arc tapping process has been estimated at any level above the steel ladle using the stopwatch technique in conjunction with the plume theory. 

In a steady state the acceleration term vanishes, and then it follows from (14.76) that... 

The essential difference in dimensioning an ejector for pneumatic conveying systems comes from the acceleration term of the solids, Eq. (1Ai37). 

One approximation considers the time it takes for the particle to travel from the entrance point, r , to the wall, rw = D/2, relative to the residence time of the fluid in the cyclone. By neglecting the acceleration term and the fluid radial velocity and assuming that the velocity of the fluid at the entrance is the same as the tangential velocity at the wall (V = Vew), Eq. (12-42) can be integrated to give the time required for the particle to travel from its initial position (r ) to the wall (Dj2). If this time is equal to or less than the residence time of the fluid in the cyclone, that particle will be trapped. The result gives the size of the smallest particle that will be trapped completely (in principle) ... 

The presence of slip also means that the acceleration term in the general governing equation [Eq. (15-45)] cannot be evaluated in the same manner as the one for homogeneous flow conditions. When the acceleration term is expanded to account for the difference in phase velocities, the momentum equation, when solved for the total pressure gradient, becomes... 

Draft Tube Pressure Drop. The pressure drop across the draft tube, AP2 3, is usually similar to that across the downcomer, APj 4, in magnitude. Thus, for a practical design basis, the total pressure drop across the draft tube and across the downcomer can be assumed to be equal. In most operating conditions, the pressure drop at the bottom section of the draft tube has a steep pressure gradient due primarily to acceleration of the solid particles from essentially zero vertical velocity. The acceleration term is especially significant when the solid circulation rate is high or when the draft tube is short. 

For low-Reynolds-number fluids the second term in the right-hand side of the Navier-Stokes equation can be neglected. Additionally, assuming that the viscous relaxation occurs more rapidly than the change of the order parameter, the acceleration term in Eq. (65) can be also omitted. Such approximations are validated in the case of polymer blends, for which they become exact in the limit of infinite polymer length, N —> oo. After these approximations, the NS equation can be easily solved in the Fourier space [160]. 

If we now consider a system which is homogeneous at the initial time (dWJdRa = 0) and if we switch on a constant electric field E, it is not difficult to show that the only modification of Eq. (203) is the introduction of an acceleration term due to this field. We thus have ... 

Note that for any statistic involving only the velocity, the reaction/diffusion term will always be zero. Likewise, for any statistic involving only the composition, the acceleration term will always be zero. 

Manipulation of this expression (as was done above for the mean velocity) leads to terms for accumulation and convection that are analogous to (6.32) and (6.33), respectively, but with (UiUj) in place of ( /, On the other hand, the acceleration term yields... 

This equation can only be solved numerically. If the acceleration term may be neglected, then ... 

At steady-state terminal velocity, the acceleration term in the force balance (B9) on a moving submerged body is zero, and the balance may be written so that gravity forces are balanced by the sum of buoyancy and resistance forces. 

The Bernoulli equation can now be written for the liquid in channel flow in the bottom part of the tube, and for the liquid in slug flow in the upper part. The acceleration terms are then neglected, and the friction factors for each type of liquid flow found from the Blasius equation and from true Reynolds numbers. The resulting equations cannot be readily evaluated because of the two hydraulic-radius terms involved in the two types of flow, and an unknown fraction defining the relative mass of liquid in each part of the tube. 

Comparison with the full Navier-Stokes equation, Eq. (1-1), shows that fluid inertia is completely neglected in Eq. (1-33). Problems arising from the nonlinearity of the convective acceleration term are thereby avoided. However, the order of the equation and hence the number of boundary conditions required are unchanged. 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

This is the fundamental equation to describe the kinetics and dynamics of polymer networks in a liquid. The left-hand side of Eq. (3.4) represents the acceleration term, whereas the first two terms of the right-hand side represent the elastic term. The last term of the right-hand side is the contribution of the friction between the network and solvent molecules. In most cases, however, the acceleration term is much smaller than the other terms. Thus one obtains... 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

A sphere of radius R is tethered by a narrow string in a steady uniform flow of incompressible viscous fluid (Fig. 3.14). Under certain circumstances (i.e., very low Reynolds number, Re = pUD/p) the creeping flow may be analyzed assuming that the viscous terms in the Navier-Stokes equations dominate over the acceleration terms. 

There are distinct similarities between second order systems and two first-order systems in series. However, in the latter case, it is possible physically to separate the two lags involved. This is not so with a true second order system and the mathematical representation of the latter always contains an acceleration term (i.e. a second-order differential of displacement with respect to time). A second-order transfer function can be separated theoretically into two first-order lags having the same time constant by factorising the denominator of the transfer function e.g. from equation 7.52, for a system with unit steady-state gain ... 

Under the centrifugal force, the acceleration term in Eq. (10.5) becomes... 

In many practical cases involving natural reservoirs, the surface area is not a simple mathematical function of z, but values of it may be known for various values of z. In such a case, Eq. (10.136) may be solved graphically by plotting values of S/(Qi - Q2) against simultaneous values of z. The area under such a curve to some scale is the numerical value of the integral. It may be observed that instantaneous values for Q have been expressed in the same manner as for steady flow. This is not strictly correct, as for unsteady flow the energy equation should also include an acceleration head. The introduction of such a term renders the solution much more difficult. In cases where the value of z does not vary rapidly, no appreciable error will be involved by disregarding this acceleration term. Therefore the equations will be written as for steady flow. 

In Eq. 6.3-9 the left-hand side represents acceleration terms, which in the case of slow motion of a viscous fluid, will be much smaller than the terms representing the viscous forces on the right-hand side. In a typical flow situation in extruders, the ratio of the inertia to viscous forces is of the order of 10 5 (17a). Thus Eq. 6.3-9 reduces to... 

If the pressure drop in a pipeline is reasonable and the linear gas velosity is well below the speed of sound, the acceleration term in the equation of notion is usually neglected, and the pressure to mass flow correlation is described by eqn (4). 

If the linear gas velocity approaches the speed of sound, the simple mathematical model used in equation (4) breaks down. The acceleration term must be taken into account, and the steady-state equation of motion for a straight pipeline with constant diameter may be written (8) ... 

By the use of extremely simple models for the gas flow in pipelines, it has been demonstrated how important the acceleration term becomes when sonic conditions are approached in a gas network. This term is usually neglected in most design computations, tut the simple examples in this paper show that this may not be justified. 

For all essential purposes then, we can ignore the average molecular acceleration term in Eq. 3.9. We consequently arrive at an expression for the average velocity (which we designate as U) in the following form... 

Equation (15) written above may be of some generality since it contains many constants empirically selected. This fact, however somewhat complicates the application to the analysis of experimental data. Therefore, to describe the rheokinetics of curing of different systems some other notation is often used for the phenomenological equation with a self-acceleration term [47] ... 

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