External coefficient

For the external coefficient, correlations and numerical approximations are available. However, for the internal rate coefficient the experience in using the model can probably dictate some values for the coefficient ks [51]. 

INTERNAL AND EXTERNAL MASS-TRANSFER COEFFICIENTS. The overall coefficient depends on the external coefficient and on an effective internal coefficient Diffusion within the particle is actually an unsteady-state process, and the value of decreases with time, as solute molecules must penetrate further and further into the particle to reach adsorption sites. An average coefficient can be used to give an approximate fit to uptake data for spheres ... 

The effective diffusion coefficient depends on the particle porosity, the pore diameter, the tortuosity, and the nature of the diffusing species. For gas-filled pores, the above factors can be allowed for to make a reasonable estimate of the effective diffusivity in the gas phase. However, diffusion of adsorbed molecules along the pore walls, called surface diffusion, often contributes much more to the total flux than diffusion in the gas phase. This is particularly evident in the adsorption of water vapor on silica gel and the adsorption of hydrocarbon vapors on carbon, where the measured values of correspond to internal and external coefficients of comparable magnitude or even to external film control, For adsorption of solutes from aqueous solutions, surface migration is much less important, and the internal diffusion resistance generally dominates the transfer process. 

Johnson et al. (B12) followed Friedlander s (F2) solution based on the Stokes velocity profiles around solid particles, and numerically calculated the external coefficients for Ar,. < 1. Only a slight difference in the Nusselt number was observed when the velocity profiles of Stokes and Hadamard were postulated. These calculations showed that the transfer coefficient ratio of drops and solids increases from 1 for (Ape), = 1 to 3 for (Ape) = 10 . In the absence of oscillation, similar results may be expected at moderately higher Reynolds numbers (Cl, H3). 

The mass transfer coeflRcient Aot composed of the external coefficient in the gas... 

The coefficient of friction of the bulk material is another very important property. One can distinguish both internal and external coefficient of friction. The internal coefficient of friction is a measure of the resistance present when one layer of particles slides over another layer of particles of the same material. The external coefH-cient of friction is a measure of the resistance present at an interface between the polymeric particles and a wall of a different material of construction. The coefficient of friction is simply the ratio of the shear stress at the interface to the normal stress at the interface. Friction itself is the tangential resistance offered to the sliding of one solid over another. 

Measurement of the external coefficient of friction of particulate polymers is very difficult because of the very large number of variables that influence the coefficient of friction. Many investigators have made elaborate measurements on the external coefficient of friction [24-32]. The result of this work is that many variables have been identified that affect the frictional behavior however, most measurement techniques do not yield accurate and reproducible results that can be used in the analysis of the extrusion process. The most elaborate measurements and the most meaningful results have probably been obtained at the DKl in Darmstadt, Germany [95]. It is possible to obtain reproducible results by very careful experimental techniques and special surface preparation of the metal wall. However, the frictional coefficients determined in this fashion are hardly representative of the frictional process conditions occurring in an extruder. 

A typical plot of external coefficient of friction versus temperature at various pressures is shown in Fig. 6.4(a). 

If the transport of particulate polymer occurs by plug flow, then the only frictional coefficient of importance is the external coefficient of friction. This condition is usu-... 

The particle shape can generally be established by simple visual observation or by using a microscope. The transport characteristics of particulate solids are quite sensitive to the particle shape. Both the internal and external coefficient of friction can change substantially with variations in particle shape even if the major particle dimensions remain unchanged. Small differences in the pelletizing process can cause major problems in a downstream extrusion process. Variations in the ratio of regrind to virgin polymer can cause variations in the extrusion process. 

Important bulk properties are bulk density, compressibility, internal coefficient of friction, external coefficient of friction, particle size and particle size distribution, and particle shape and particle shape distribution. 

The internal coefficient of friction is the friction between the plastic particles themselves. The external coefficient of friction is the friction between the plastic particles and another surface, such as the barrel surface. For efficient conveying in the feed hopper, it is required that both the internal and external frictions should be low. For efficient conve5ung along the extruder screw, it is required that the barrel friction to be high and the screw friction should be low this is discussed later. 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Figure C2.5.8. Plot of the folding times Tp as a fimction of cr nfor tlie 22 sequences. This figure shows tlrat under tire external conditions when tire NBA is tire most populated tlrere is a remarkable correlation between ip and The correlation coefficient is 0.94. It is clear tlrat over a four orders of magnitude of folding times Xp = expf-a, / Oq) where CTq is a constant. The filled and open circles correspond to different contact interactions used in C2.5.1. The open squares are for A = 36.

The application of a small external electric field A to a semiconductor results in a net average velocity component of the carriers (electrons or holes) called the drift velocity, v. The coefficient of proportionality between E and is known as the carrier mobility p. At higher fields, where the drift velocity becomes comparable to the thennal... 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

Table 4. Correlations for Convective Heat-Transfer and Friction Coefficients for External Flow...

Fig. 35. Correlations for calculating heat-transfer coefficients for (a) turbine external jackets, internal cods, and baffle cods, and (b) for close-clearance...

External Dilute-Phase Upflow Cooler. The external ddute-phase upflow design (68) offers some control in the range of heat removal duties but generates relatively low heat-transfer coefficients [60—170 W/(m K)]- This design substantially increases the surface area requirement and thereby reduces the ultimate duty that can be achieved from a single bundle. In addition, poor mechanical rehabdity has been continuously experienced because of excessive erosion at the lower tube sheets as a result of the high catalyst fluxes and gas velocities imposed. 

For annuli containing externally Hnned tubes the heat-transfer coefficients are a function of the fin configurations. Knudsen and Katz (Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958) present relationships for transverse finned tubes, spined tubes, and longitudinal finned tubes in annuli. 

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