Mean interstitial velocity

Area-averaged circulation velocity of liquid Mean interstitial velocity of upflow, Eq. (5-9) Interstitial velocity of gas in emulsion phase II along the column axis Slip velocity of bubble relative to liquid... 

Here, the first term in parentheses is the cross-sectional void area, and the second term is the cross-sectional internal area of the gel available to the solute. The mean interstitial velocity or effective velocity through the column is... 

To the extent that dispersion in an inertia free porous medium flow arises from a nonuniform velocity distribution, its physical basis is the same as that of Taylor dispersion within a capillary. Data on solute dispersions in such flows show the long-time behavior to be Gaussian, as in capillaries. The Taylor dispersion equation for circular capillaries (Eq. 4.6.30) has therefore been applied empirically as a model equation to characterize the dispersion process in chromatographic separations in packed beds and porous media, with the mean velocity identified with the interstitial velocity. In so doing it is implicitly assumed that the mean interstitial velocity and flow pattern is independent of the flow rate, a condition that would, for example, not prevail when inertial effects become important. 

Mean Interstitial Velocity of Carrier Gas u. The interstitial velocity of the carrier gas multiplied by the pressure-gradient correction factor ... 

Linear velocity of mobile phase interstitial velocity of mobile phase Rohrschneider constant for nitromethane Mean interstitial velocity of mobile phase i.e., average linear gas velocity... 

Here, it has to be noted that for calculating the Peclet number in fixed beds, the actual velocity has to be used, i.e. the interstitial velocity, which influences the degree of mixing. In slurry bubble column reactors, the real velocity of the fluid is the bubble velocity, which is much higher than the gas superficial velocity. The mean bubble rise velocity for a batch liquid is (eq (3.201))... 

In principle, the difference between the gas-and liquid velocity should be used instead of the mean interstitial gas velocity, but in their operating conditions, the mean interstitial liquid velocity is small compared to the mean interstitial gas velocity. To a first approximation, ate was considered to be proportional to the external geometric area, ac, of the packing ... 

An expression for fia has to be derived from experimental data, by means of a power-product equation containing the gas density, the interstitial mean gas velocity, and a film-liquid Reynolds number. They found a value of-0.37 for both the exponent of pa and of ug, indicating that fiG depends on a gas-phase Reynolds number. For the exponent of the liquid film Reynolds number the value was about zero. 

In Figure 5.2-22, the pressure-gradient, s(l -fi)(AP/Z), is plotted versus the interstitial mean gas velocity, ug, at constant gas densities. Along each line for a given density, the liquid flow-rates vary considerably, but the relationship between the pressure gradient and u(, is unique, and independent of the liquid mean velocity. 

There are four unknown parameters in the theoretical impulse response for porous particles, h(t) the pellet diffusion time, tdif (which contains the effective diffusion coefficient of the pair T-C, Dtc, td.fs R p/D.f( , R is the radius of the pellet equivalent sphere), the mean residence time of the carrier-gas in the interparticle space, tc (tc = v/L with the carrier gas linear interstitial velocity, v, and column length, L), Peclet number, Pe (Pe = L.v/E, with E the effective axial dispersion coefficient) and the adsorption parameter, 5q (see below). Because matching with four unknown parameters would give highly correlated parameters, it is better to determine some parameters independently,... 

The relation given by Eq. (5-8) is obtained easily for a bulk recirculation flow of the continuous phase. We consider a bubble column of radius R in which the column liquid is in upflow centrally with a constant interstitial velocity Wju- Peripherally the liquid descends, forming a recirculation flow. For simplicity, it is assumed that gas bubbles distribute uniformly in the central upflowing liquid and ascend with it, and that no bubbles rise through the peripheral downflow. Here, the mean gas holdup of the upflow is b. and that averaged over the total column cross section is Cu. 

Tangential velocity along the bubble-emulsion interface Time-averaged interstitial velocity of liquid Interstitial mean liquid velocity, Eq. (4-15) Ascending velocity of bubbles... 

Compared with Bo , which is independent of the interstitial velocity (Eq. 7.32), Steffi is inversely proportional to interstitial velocity (Eq. 7.20). This means that the influence of mass transfer resistance will grow and surpass the influence of axial dispersion at high interstitial velocity, which is almost always the case for preparative chromatographic processes. In some extreme cases, where the mass transfer coefficients are small and the chromatographic column is operated at high flow rates, the HETP equation for the calculation of Nt can even be simplified further to ... 

Single-phase flow means that only one fluid phase is present in the void space (implying, most importantly, that no fluid-fluid interface exists). The superficial velocity i is a volume flux based on the total cross section of the material. (For flow in a packed colunm, V = Q/A, where Q is the volumetric flow rate and A is the column cross-sectional area.) The interstitial velocity, Vint = v/e, better characterizes the average fluid velocity actually present in the voids. 

Withdrawal speed of plate in dip coating Mean interstitial or effective pore velocity in porous medium,... 

In a cube of side, /, the volume of the voids is and the mean cross-sectional area is the free volmne divided by the height, i.e. sl. The volumetric flow rate through the cube is given by VqI, so that the average interstitial velocity V is given by... 

Figure 11.11 Pressure drop in packed beds and different microchannels (circular, quadratic, and parallel plates (slits)) as a function of mean velocity in the channel, respectively, interstitial velocity in the packed bed (air, 20°C).

For diffusion dominated processes, D is the molecular diffusivity in water of the contaminant of interest multiplied by the porosity and divided by the tortuosity (or hindrance factor, ffp) of the sediment (D = Dv,s/Hp). The Millington and Quirk model (as referenced in Palermo [1]) suggests Hp which for a cap with 40% porosity is about 1.4. The molecular diffusion coefficient in water is a function of temperature and can be estimated by the methods defined by Lyman [4]. In the presence of advection D is increased due to mechanical dispersion in the medium (typically modeled as some dispersivity, a, multiphed by the interstitial velocity, v). The dispersiv-ity is an empirical parameter but is related to the mean particle size in nearly... 

As a first approximation, the analysis in reference [68] uses the well-known model of Davidson and Harrison [65] in which the bubble is assumed to be a spherical cavity without particles and in which the dispersed phase is characterized by uniform concentration < )o everywhere outside the bubble. Relative interstitial fluid velocity, u, and mean particle velocity, w, can then be found on the basis of 1) a simple filtration flow model for a homogeneous porous body containing a spherical cavity, and 2) an ideal fluid model for flow around a sphere. In particular, the vertical components of these velocities along vertical axis z of the coordinate system having its origin at the bubble center are ... 

K. Universal functions introduced V, w Mean fluid interstitial velocity... 

Frenkel(47), by kinetic theory, and Jost(44,56), by statistical mechanics, showed that w Ne ol. Only a fraction of these holes, or corresponding interstitial atoms, will diffuse however, because an activation energy is necessary for diffusion. Thus the number moving is proportional to e o+2Ei)/2BT An approximate value of Z>, the diffusion coefficient, can be deduced if it is assumed that there are six ions around each hole, distant d from its ceptre, and capable of moving with the mean thermal velocity in any one of six directions. Each of the six particles may move in a single direction (to the hole), so the six of them are equivalent to a single particle free to move in all directions. Therefore the diffusion constant for a hole is... 

The dispersion coefficient Dax (iti s ) and the dimensionless numbers Bo and Pem.ax represent the spreading process of a pulse of tracer. Thus a large value of (low Bo and low Pe i,ax) means rapid spreading of the tracer curve (mixed flow), and a low value of Dax means slow spreading. For D = 0 we have no spreading and, hence, plug flow. Note that for a fixed bed we can still use Eqs. (4.10.98) and (4.10.99), but then Bo is related to the interstitial velocity (ratio of superficial velocity Us in the empty reactor to the porosity s of the packed bed with e 0.4). 

The flow in a packed bed, where the packing may be spherical, c indrical, etc., is quite complex (Figure 6.1.1(b)). However, it is often modeled as a collection of cylindrical capillaries of hydraulic radius R/, and length L, which is the packed bed length. Let e be the fractional void volume of the bed, Op be the total particle surface area per particle volume, v be the actual interstitial velocity in the void volume between particles and dp (= 2rp) be the mean particle diameter. Then the superficial velocity Vo based on the empty cross section of the packed bed is defined as... 

The conceptually simplest means by which particles can remain in a bed, subjected to a fluid flux higher than that required for minimum fluidization, is for them to separate from one another so that the bed expands, the void space around the particles increases and, as a consequence, the fluid velocity within the bed decreases. This decrease in interstitial velocity has a strong effect on the fluid-particle interaction force, causing it to fall and thereby enabling a new equilibrium condition to be established in which the particle weight is once again just supported by the fluid. The mechanism just described represents the essential feature of homogeneous fluidization. 

The results of an example calculation for a recirculating fluidized bed coal devolatilizer of 0.51 m in diameter handling coal of average size 1200 pm at 870°C and 1550 kPa are presented in Fig. 11. The calculation is based on operating the fluidized bed above the draft tube at 4 times the minimum fluidization velocity. It is also based on the selection of a distributor plate to maintain the downcomer at the minimum fluidization condition. If the two-phase theory applies, this means that the slip velocity between the gas and the particles in the downcomer must equal to the interstitial minimum fluidizing velocity as shown below. 

The simplifications taken here are, first, that the catalyst particles are usually several tens of fj,m in mean diameter, so that the interstitial gas velocity through the emulsion phase is about Umt (equal to and is... 

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