Characteristic lengths for

The data were plotted, as shown in Fig. 11, using the effective diameter of Eq. (50) as the characteristic length. For fully turbulent flow, the liquid and gas data join, although the two types of systems differ at lower Reynolds numbers. Rough estimates of radial dispersion coefficients from a random-walk theory to be discussed later also agree with the experimental data. There is not as much scatter in the data as there was with the axial data. This is probably partly due to the fact that a steady flow of tracer is quite easy to obtain experimentally, and so there were no gross injection difficulties as were present with the inputs used for axial dispersion coefficient measurement. In addition, end-effect errors are much smaller for radial measurements (B14). Thus, more experimentation needs to be done mainly in the range of low flow rates. 

In formulating a population balance, crystals are assumed sufficiently numerous for the population distribution to be treated as a continuous function. One of the key assumptions in the development of a simple population balance is that all crystal properties, including mass (or volume), surface area, and so forth are defined in terms of a single crystal dimension referred to as the characteristic length. For example, Eq. (19) relates the surface area and volume of a single crystal to a characteristic length L. In the simple treatment provided here, shape factors are taken to be constants. These can be determined by simple measurements or estimated if the crystal shape is simple and known—for example, for a cube area = 6 and kY0 = 1. 

Determine the reactor s dimension (e.g., DT = tank diameter) based on geometric similarity, with the impeller diameter being the characteristic length. For example,... 

Williamson and Adams6 presented data, shown in Figure 7, for the dimensionless centre temperature (Tc — Tsurf)/(7 0 — T rf) of variously shaped bodies as a function of dimensionless time Fo. Here Tc is the centre point temperature, T0 the initial body temperature and T surf the surface temperature. Fo is the Fourier number, which is given by Fo = at/82, where a is the thermal diffusivity, t is the time and S is the characteristic length for conduction, the distance of the centre point or centreline of the body to the nearest part of the surface. The data given in Figure 7 neglects the thermal resistance at the surface. 

L is the characteristic length, for example, pipe diameter, that is, the length of the flow field v is the flow velocity r is the dynamic viscosity... 

If the electrode were partially blocked, or behaved as an array of UMEs, then the characteristic length for the scaling would be different and would depend upon whether the concentration boundary layers for the different active areas interacted. If the electrode could be represented as consisting of two or more independent active areas, then two or more characteristic relaxations would be observed in the response, and the length scale could be deduced from the dependence of the measured H values on Re. Addi-... 

Transport Limitation For the estimation of the mass transport limitation, Equation (20) has an important drawback. In many cases neither the rate constant k nor the reaction order n is known. However, the Weisz-Prater criterion, cf. Equation (21), which is derived from the Thiele modulus [4, 8], can be calculated with experimentally easily accessible values, taking < < 1 for any reaction without mass transfer limitations. However, it is not necessary to know all variable exactly, even for the Weisz-Prater criterion n can be unknown. Reasonable assumptions can be made, for example, n - 1, 2, 3, or 4 and / is the particle diameter instead of the characteristic length. For the gas phase, De can be calculated with statistical thermodynamics or estimated common values are within the range of 10-5 to 10 7 m2/s. In the liquid phase, the estimation becomes more complicated. A common value of qc for solid catalysts is 1,300 kg/m3, but if the catalyst is diluted with an inert material, this... 

Lateral capillary forces have several useful characteristics. First, the characteristic length for decay of the menisci (from a face several millimeters in width and height) is of the order of a millimeter. Objects of this size are easy to fabricate and to observe. Second, capillary forces are well understood. Their description by the Laplace equation is clear, although often mathematically intractable [ref. 2], Third, capillary forces are comparable in strength to shear... 

The parameter / is the characteristic length for a unit cell, E0 is the surface concentration of a carrier protein molecule, and a2, a3, a4, a5 are the reaction rate parameters analogous to that half saturation constants. Table 11.3 displays the experimental effective diffusion coefficients and the volume fraction of intracellular phase A. In the first four sets... 

Note that (p c J and (ptr J represent characteristic lengths for absorption and for scattering, respectively. 

For flow in a pipe, the pipe diameter is conventionally used as the characteristic length. For flow around a submerged sphere, the sphere diameter is used as the characteristic length. For flow past a... 

The characteristic length for a circular cylinder or sphere is taken to be the external diameter D. Thus, the Reynolds number is defined as Re = VD/v where V is Ihe uniform velocity of Ihe fluid as it approaches the cylinder or splicre. The critical Reynolds number for flow across a circular cylinder or sphere is about Re s 2 X 10. That is, the boundar) layer remains laminar for about Re < 2 X K) and becomes turbulent forRc 2 X l(y. ... 

Th average Nusselt number for horizontal surfaces can be determined from the simple power-law relations given in Table 9-1, The characteristic length for horizontal surfaces is calculated from... 

Kohsenow (1984) have compiled the available dala under various boundary conditions, and developed correlations for the Nusselt number and optimum spacing. The characteristic length for vertical parallel plates used as fins is usually taken to he the spacing between adjacent fins S, aliltough the n height h could also be used. The Rayleigh number is expressed as... 

Nusselt number relations for various surfaces are given in Table 9-1. All fluid properties are evaluated at the film temperature of Tf = 1(T, -T r ). The outer surface of a vertical cylinder can be treated as a vertical plate when the curvature effects are negligible. The characteristic length for a horizontal surface is Lg = AJp, where A, is the surface area and p is the perimeter. 

The crucible contents of the TGA were modelled as a dynamic system with distributed parameters assuming plate geometry. The radius of the sample was taken as the characteristic length for the heat transport. The overall rate of reaction was approximated by an irreversible reaction of T order biomass -> solids + volatiles, assuming the validity of the Arrhenius expression for the temperature dependency of the rate constant. 

Figure 11.8 Distribution of axial velocity for a region near a rotating disk. The characteristic length for mass transfer is indicated by dashed lines for Schmidt numbers of 10,000, 1,000, and 100, respectively.

The influence of the accuracy of the velocity expansion is illustrated in Figure 11.8. The characteristic length for mass transfer to a disk electrode is given by... 

The approximate solution from (2.199) offers a simple and recommended procedure for the calculation of the heating or cooling processes in any solid shape. However it is only sufficiently accurate for Biot numbers Bi < 0.1. This condition should be checked while choosing the characteristic length for the Biot number to be that of half the shortest length dimension of the body under consideration. 

The linear variation of r with P [Eq. (2)] can be understood from a simple hard-sphere picture. Once the adsorbed molecules have conformed to the surface, they produce an interlocking configuration like that envisioned by early researchers (Fig. la), but at an atomic scale. The value of a is the effective slope that the surface must be lifted up over this interlocking layer. The main factor that changed t, in the above simulations was the ratio of the characteristic length for wall adsorbate interactions, CTwa, to the nearest-neighbor spacing on the walls, d. As the ratio decreased, adsorbed atoms could penetrate more... 

---