Tube radius

Laminar Flow For films falling down vertical flat surfaces, as shown in Fig. 6-52, or vertical tubes with small film thickness compared to tube radius, laminar flow conditions prevail for Reynolds numbers less than about 2,000, where the Reynolds number is given by... 

Equation (11) shows that the pressure drop across the connecting tube increases inversely as the fourth power of the tube radius. It follows that, as it is impractical to dissipate a significant amount of the available pump pressure across a connecting tubing, there will be a limit to the reduction of (r) to minimize tube dispersion. 

Equation (12) indicates that the band variance is directly proportional to the square of the tube radius, very similar to that for a straight tube. At high linear velocities, Tijssen deduced that... 

The effect of tube radius and coil aspect ratio on the onset of radial mixing in coiled tubes was investigated using the above equations. In Table 5, the dimensions of the coiled tubes examined are given and the curves relating (H) and (u) in Figure 8. 

The length of the column is also defined by the Poiseuille equation that describes the flow of a fluid through an open tube in terms of the tube radius, the pressure applied across the tube (column), the viscosity of the fluid and the linear velocity of the fluid. Thus, for a compressible fluid. 

If the pilot reactor is turbulent and closely approximates piston flow, the larger unit will as well. In isothermal piston flow, reactor performance is determined by the feed composition, feed temperature, and the mean residence time in the reactor. Even when piston flow is a poor approximation, these parameters are rarely, if ever, varied in the scaleup of a tubular reactor. The scaleup factor for throughput is S. To keep t constant, the inventory of mass in the system must also scale as S. When the fluid is incompressible, the volume scales with S. The general case allows the number of tubes, the tube radius, and the tube length to be changed upon scaleup ... 

The same results are obtained from Equations (3.38) and (3.39), which apply to the turbulent flow of ideal gases. Thus, tube radius and length scale in the same way for turbulent liquids and gases when the pressure drop is constant. For the gas case, it is further supposed that the large and small reactors have the same discharge pressure. 

The tube radius is divided into a number of equally sized increments, Ar = R/I, where / is an integer. For reasons of convergence, we prefer to use a second-order, central difference approximation for the first partial derivative ... 

FIGURE 8.8 Elongated velocity profile resulting from a factor of 50 increase in viscosity across the tube radius. 

F r) Cumulative distribution function expressed in terms of tube radius for a monotonic velocity profile 15.29... 

The power of this technique is two-fold. Firstly, the viscosity can be measured over a wide range of shear rates. At the tube center, symmetry considerations require that the velocity gradient be zero and hence the shear rate. The shear rate increases as r increases until a maximum is reached at the tube wall. On a theoretical basis alone, the viscosity variation with shear rate can be determined from very low shear rates, theoretically zero, to a maximum shear rate at the wall, yw. The corresponding variation in the viscosity was described above for the power-law model, where it was shown that over the tube radius, the viscosity can vary by several orders of magnitude. The wall shear rate can be found using the Weissen-berg-Rabinowitsch equation ... 

The terms in Eq. (6) include the gravitational constant, g, the tube radius, R, the fluid viscosity, p, the solute concentration in the donor phase, C0, and the penetration depth, The density difference between the solution and solvent (ps - p0) is critical to the calculation of a. Thus, this method is dependent upon accurate measurement of density values and close temperature control, particularly when C0 represents a dilute solution. This method has been shown to be sensitive to different diffusion coefficients for various ionic species of citrate and phosphate [5], The variability of this method in terms of the coefficient of variation ranged from 19% for glycine to 2.9% for ortho-aminobenzoic acid. 

Figure 22 The pressure dependence of conductance A plot of the ratio of the total conductance to the free molecular flow conductance as a function of the ratio of tube radius to mean free path.

In region III near the tube center, viscous stresses scale by the tube radius and for small capillary numbers do not significantly distort the bubble shape from a spherical segment. Thus, even though surfactant collects near the front stagnation point (and depletes near the rear stagnation point), the bubble ends are treated as spherical caps at the equilibrium tension, aQ. Region... 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

In addition, the effects of pulsatile flow cannot be ignored. One measure of the impact of oscillary flow is the Wcmersley parameter (a) a= h/2tt f/v where r is the tube radius, f the frequency of oscillation and v is the kinematic viscosity of the fluid (Wcmersley, 1955). The degree of departure from parabolic flow increases with and frequency effects may become important in straight tubes when a > 1 (Ultman, 1985). For conditions of these experiments, a exceeds one to beyond the third generation. 

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