The Inertial Term

In conclusion, the barrier height to FC change transfer, as envisioned by Weiss,32 may be correct. In a 1.0M electrolyte, it would be about 50 kJ/mole (0.5 eV, 20kT, 12 kcal/mole). This appears to be of the correct order.90,177 However, an alternative view is discussed below. 

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In the equation shown above, the first term—including p for density and the square of the linear velocity of u—is the inertial term that will dominate at high flows. The second term, including p. for viscosity and the linear velocity, is the viscous term that is important at low velocities or at high viscosities, such as in liquids. Both terms include an expression that depends on void fraction of the bed, and both change rapidly with small changes in e. Both terms are linearly dependent on a dimensionless bed depth of L/dp. 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

Consider the limit of high particle Reynolds numbers where the inertial term in the Ergun equations dominates. 

The 2-component Navier-Stokes equation is equation A.25. Each of the inertial terms is zero, the reasons being respectively that the flow is steady, vr = 0, the flow is axisymmetric and the flow is fully developed. The second and third viscous terms vanish because the flow is axisymmetric and fully-developed. The flow being horizontal, gz = 0. 

It is to be noted that on expressing the motion of the center in terms of the motion of the base of the bubble, the various terms of Eq (47) have been split into two parts, viz. (i) that associated with the movement of the base due to the free motion of the bubble (this would be the only term if the bubble does not change its size during its movement), and (ii) that associated with the movement of the bubble center due to expansion. The inertial term has split into left-hand-side terms and the second term on the right-hand side, whereas the viscous drag term has split into the third and fourth terms on the right-hand side of Eq. (48). 

If the inertial terms on the right-hand side of equation 3.108 are neglected, then ... 

The first term on the left is called the inertial term, and the second arises from the temporal variation in the velocity at any given position. For low velocities, the former may be neglected. 

Under conditions for which the inertial term can be neglected compared to the other terms in the equation, Equation (28a) becomes... 

In addition, the simple phenomenological relation (6.1.4), with a constant electro-osmotic coefficient lc, was replaced by a more elaborate one, accounting for the w dependence on the flow rate and the concentrations Ci, C2 via a stationary electro-osmotic calculation. This approach was further adopted by Meares and Page [7] [9] who undertook an accurate experimental study of the electro-osmotic oscillations at a Nuclepore filter with a well-defined pore structure. They compared their experimental findings with the numerically found predictions of a theoretical model essentially identical to that of [5], [6]. It was observed that the actual numerical magnitude of the inertial terms practically did not affect the observable features of the system concerned. 

It should be noted that if the inertial terms are omitted, the above relations will return to the original form of the EMMS model (Li and Kwauk, 1994). For specified conditions Us, Gs, and g, this set of 10... 

If we neglect the effect inertia has on the stretching fiber and drop the inertial term in the above equation, it can be solved as... 

Neglecting the inertial terms and using eqn. (8.96), the momentum equations are... 

The other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

From equation (14)i we could obtain the Darcy s law, if we neglect the inertial terms and the mass exchange and make suitable constitutive hypotheses on fields m,bf and Tf. The equation of balance (15)i for the volume fraction generalize the classical Langmuir s evolution equation, while the balance (15)2 for the microstretch Us includes the Wilmanski s porosity balance as well as the equation which rules the changes of internal surfaces area of the pores (see [8, 11, 1], respectively). The energy balance equations do not appear at all because the process is assumed to be isothermal. 

The inertial terms on the left side of the equation are balanced by the viscous forces minus the driving force on the right side of the equation. The solution... 

The second term in the right-hand side of Eq. (5.392) represents the fluid inertia, whereas the first term represents the viscous contribution to the pressure drop. At low Reynolds numbers, the Ergun equation can be simplified by neglecting the inertial term. Under this condition, Eq. (5.392) can be expressed as... 

Where Ro is the lumen radius, I if the fiber length, u is the longitudinal convective velocity, P is pressure, X is the dimensionless axial space coordinate, R is the dimensionless radial coordinate. In writing Equation 14.24 it was assumed that flow is laminar and that entrance effects can be ignored. In addition, the axial stress terms have been neglected since the aspect ratio of the hollow fiber (Ro/L) is typically less than 0.01. The inertial terms have been neglected also, which is valid if the radial Reynolds number (ReR— puRo/fi) is much less than 1 [11], The boundary conditions for the solution ... 

For very viscous fluids the inertial term pv7 dv,/dz) is negligible, thus... 

Relative Importance of the Various Terms in the Analysis of the Isothermal Fiber Spinning of a Newtonian Melt Use the data and result of Problem 14.1(b) to evaluate the importance in the isothermal fiber spinning of a Newtonian melt analysis (nylon 6-6 at 285°C) of the inertial terms and gravity relative to the viscous stress terms. Using Eq. E14.1-2 for FD, evaluate the importance of the air-drag force term. 

The viscous effects dominate the fluid-particle interaction of small particles (below 100 pm), thus the inertial term of the Ergun equation can be neglected. Hence, the minimum fluidisation velocity can be obtained from ... 

The inertial terms in the momentum equation were neglected in the above analysis, i.e., the terms ... 

This indicates that the inertial terms will only have a significant effect on the heat transfer rate if Ja is large and Pr is small. This is illustrated by the results given in Fig. 11.22 which shows the variation of ... 

In the case of the transition from region I to region II, which is gradual, the function/i becomes negative as the numerator exceeds the denominator (which remains positive). This generally happens when the inertial terms B and C are small, term A approaches unity, and thus the transition is decided by the comparison between GIF and Vz. It will be shown later... 

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