Channel radius

Fig. 12A, has a 10° libration. This gives a channel size which would be optimal for an ionic radius between that of Rb+ and Cs+. Therefore enhanced discrimination is not expected between Rb+ and Cs+, but the energy required to librate further inward to make contact with smaller ions in the series can be expected to enhance selectivity between these ions. Work is currently in progress to calculate the change in channel energy as a function of libration angle or of the equivalent, the effective channel radius S6). The implications of a peptide libration mechanism for enhancing ion selectivity can also be pursued experimentally as outlined below. 

For an incompressible fluid, the density variation with temperature is negligible compared to the viscosity variation. Hence, the viscosity variation is a function of temperature only and can be a cause of radical transformation of flow and transition from stable flow to the oscillatory regime. The critical Reynolds number also depends significantly on the specific heat, Prandtl number and micro-channel radius. For flow of high-viscosity fluids in micro-channels of tq < 10 m the critical Reynolds number is less than 2,300. In this case the oscillatory regime occurs at values of Re < 2,300. 

In order to show how specific guidelines for the reactor layout can be derived, the maximum allowable micro-channel radius giving a temperature rise of less than 10 K was computed for different values of the adiabatic temperature rise and different reaction times. For this purpose, properties of nitrogen at 300 °C and 1 atm and a Nusselt number of 3.66 were assumed. The Nusselt number is a dimensionless heat transfer coefficient, defined as... 

Figure 9. Model of convective diffusion inside a tube channel (radius R) towards the walls of the tube considered as perfect active surfaces. A Poiseuille profile for the velocity (equation (35)) is also schematically shown with arrows on the right-hand side. The maximum velocity v is reached at the centre of the tube (r = 0)...

Based on a semidifferential control volume that spans the channel radius, develop the overall and species continuity equations for flow along the tube. Show how the species equations may be written is a form that uses the substantial-derivative operator. 

Based on a differential control volume that spans the channel radius (r — r, ) but is differential in dO, develop a relationship between the friction factor / and the channel pressure gradient dp/dO. Restate this relationship in nondimensional form. 

Figure 5.24 illustrates an elbow section in a cylindrical channel where the radius of curvature of the section R is comparable to the channel radius r,-. Analysis of the flow field in this section may be facilitated by the development of a specialized orthogonal curvilinear coordinate system, (r, 6, a). The unit vectors are illustrated in the figure. Referenced to the cartesian system, the angle 6 is measured from the x axis in the x-y plane. The angle a is measured from and is normal to the x-y plane. The distance r is measured radially outward from the center of the toroidal channel. 

Discuss the behavior of the continuity equation (and by analogy the other governing equations) as the toroid radius R becomes large compared to the channel radius r,. Under these circumstances, show how the curvilinear equation becomes closer and closer to the regular cylindrical-coordinate equations. 

Multiplying throughout by zs/poMo and defining a Reynolds number based on the channel radius Re, = Powoa/mo, the nondimensional axial-momentum equation becomes... 

The first choice, where the ratio tends to zero, would lead to an inviscid flow as all diffusion terms be negligible. For our purpose this is an uninteresting alternative because, without viscous effects, it would be impossible to support the no-slip condition at the tube wall. The third alternative, where the ratio tends toward infinity would lead to a conclusion that there are no convective effects—again, an uninteresting alternative. Thus we are left with the alternative that the ratio is order one. After choosing the channel radius ro as the characteristic radial length scale, the axial length-scale zs scales as... 

An additional complication that occurs with oscillating flow is the existence of several regimes of laminar and turbulent flow that are functions of frequency as well as Reynolds number, as shown in Figure 3 for the case of smooth circular tubes [2]. These flow regimes are the subject of much research [3]. They are shown as a function of the peak Reynolds number Nr,peak and the ratio of channel radius R to the viscous penetration depth Sv This ratio is sometimes referred to as the dynamic Reynolds number and is similar to the Womersley number Wo = D 28y). In the weakly turbulent regime... 

The two equations above indicate that the pressure drop increases and the average velocity decreases dramatically with decreasing channel radius. Thus, it is obvious, why electrokinetic pumping procedures find widespread application in fluid systems with micrometer dimensions. 

In nuclear physics, because of the nature of the short range interactions involved, this boundary has a definite physical identity. In atomic physics, there is no clear physical value for a, but one introduces the sphere for computational convenience (see, e.g., [371] for further discussion of this point), and it is generally advantageous to define as small a radius a as possible for the problem in hand. We call a the channel radius. 

Fig. 19. Model calculation CO and O2 concentration gradients as function of channel radius (different scale for washcoat and gas volume) and CO resp. O coverages.

Thus for Dm = 7 x 10 , the minimum time for Taylor s effect to begin to develop is 6.2 s for = 0.025 cm. This minimum time limit is just about met in FIA, since tubes of more than 0.5 mm ID are seldom used, while typical residence times are 20 s or more. Further decrease in channel radius and flow rate will lead to progressively more intensive redistribution of sample material in the radial direction by molecular diffusion and therefore decrease in the flow rate in a straight narrow tube will lead to a decrease in dispersion (cf. Rule 3, Section 2.2.3). 

The reactor volume scales with the square of the channel radius. 

The radial mixing speed scales as the inverse square of the channel radius—if diffusion is the dominant radial mass transfer force. 

The channel model of an arc includes three parameters to be determined plasma temperature Tm, arc channel radius ro, and electric field E. Electric current I and discharge tube radius R are experimentally controlled parameters. To find r, and E, the channel model has only two equations, (4-59) and (4-60). Steenbeck suggested the principle of minimum power (see Section 4.2.4) as the third equation to complete the system. The minimum power principle has been proved for arcs by Rozovsky (1972). According to the principle of minimum power, temperature and arc chaimel radius ro should minimize the specific discharge power w and electric field E = w/1 at fixed values of current 1 and discharge tube radius R. The minimization ( )/=const = 0 gives the third equation of the model ... 

The model equations of Diamond and Bossert [44], suppose a channel of length L, cross-section area A, and circumference S. An equivalent channel radius, r, may be defined by... 

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