Hydrodynamic transfer function

Through the von Karman transformation, the steady-state Navier-Stokes equations for a rotating disk can be expressed in terms of three coupled, nonlinear, ordinary differential equations as  

The steady flow field created by an infinite disk rotating at a constant angular velocity in a fluid with constant physical properties was presented in Chapter 11. 

For the unsteady situation, the instantaneoiis value of rotation speed O can be defined by 

Following the development in Chapters 10 and 11 for a current or potential response to a perturbation, the radial, angular, and axial velocity components can be expressed as 

As discussed in Section 1.2.2, each complex function may be written as the sum of a real function and an imaginary function. The set of the three coupled equations 

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No other approximation was made and, in particular, the hydrodynamic transfer function was quantitatively calculated (see Chapter 2). In [62] measurements were done with the same system as that of Blauch and Anson [12]. The results obtained for 0.1 Hz[Pg.239]

Remember 15.3 The phase shift of the hydrodynamic transfer function f tends toward —45° as the dimensionless frequency p = co/Ci tends toward infinity. 

Hydrodynamic Transfer function model (Laplace transform s))... 

The archetypal, stagewise extraction device is the mixer-settler. This consists essentially of a well-mixed agitated vessel, in which the two liquid phases are mixed and brought into intimate contact to form a two phase dispersion, which then flows into the settler for the mechanical separation of the two liquid phases by continuous decantation. The settler, in its most basic form, consists of a large empty tank, provided with weirs to allow the separated phases to discharge. The dispersion entering the settler from the mixer forms an emulsion band, from which the dispersed phase droplets coalesce into the two separate liquid phases. The mixer must adequately disperse the two phases, and the hydrodynamic conditions within the mixer are usually such that a close approach to equilibrium is obtained within the mixer. The settler therefore contributes little mass transfer function to the overall extraction device. 

Fig. 10.5(a). Transfer function, H, relating modulated current response to modulated flow rate for tube electrode (Reference [20]), rectangular electrode embedded in a wall (Reference [12]) and modulated RDE (amplitude only). The dimensionless modulation frequency see text) is the ratio of the time scale for diffusion across the concentration boundary layer to the timescale for modulation of the hydrodynamics. 

Theoretical developments show that it is possible to deduce hydrodynamic information from the limiting current measiuement, either in quasi-steady state where /(f) cx py t) or, at higher frequency, in terms of spectral analysis. In the latter case, it is possible to obtain the velocity spectra from the mass-transfer spectra, where the transfer function between the mass-transfer rate and the velocity perturbation is known. However, in most cases, charge transfer is not infinitely fast, and the analysis also requires knowledge of the convective-diffusion impedance, i.e., the transfer function between a concentration modulation at the interface and the resulting flux of meiss under steady-state convection. 

When the Schmidt number is infinitely large, W,- is reduced to f p) h + jh) and appears as the product of a hydrodynamic transfer fimction f p) and a mass-transport transfer function Zc = fi -I- jt2- The mass-transport transfer function Zc is presented in Figure 15.3. It is eeisily verified that W,- approaches 0.5 when the frequency tends toward zero, in agreement with the exponent of the rotation speed in the Levich equation. This value of 0.5 is also verified if the complete expression of W,- is used. The complex function 2W,- is presented in Figure 15.4 in... 

In Table 2.3 some examples of transfer functions and the combination for the hydrodynamic modelling are given. 

Table 2.3 Transfer functions of one-dimensional hydrodynamic models and model combinations.

Transfer function depends on the hydrodynamical mode of drop motion in the settling section of the settler. In the elementary case of drop sedimentation, when drop interactions are not taken into account, the expressions for (r) are easy to obtain when the emulsion motion is perpendicular or parallel to gravity [4]. In the former case the term settler refers to a horizontal one and... 

For flow parallel to an electrode, a maximum in the value of the mass-transfer rate occurs at the leading edge of the electrode. This is not only the case in flow over a flat plate, but also in pipes, annuli, and channels. In all these cases, the parallel velocity component in the mass-transfer boundary layer is practically a linear function of the distance to the electrode. Even though the parallel velocity profile over the hydrodynamic boundary layer (of thickness h) or over the duct diameter (with equivalent diameter de) is parabolic or more complicated, a linear profile within the diffusion layer (of thickness 8d) may be assumed. This is justified by the extreme thinness of the diffusion layer in liquids of high Schmidt number ... 

The heat transfer from tubes in the freeboard was also measured for the 20 MW model. Figure 45 shows a comparison of the measured overall heat transfer coefficient in the 20 MW pilot plant versus that predicted from the scale model test. When the bed height is lowered, uncovering some tubes, the heat transfer is reduced because there are fewer particles contacting the tube surface. Although the scale model did not include proper scaling for convective heat transfer, the rate of change of the overall heat transfer should be a function of the hydrodynamics. 

Several vendors offer NO redncing catalyst additives. All have seen commercial success in reducing NO. However, the resnlts have not been consistent. The additives do not function like SO redncing additives that absorb the desired pollutant and transfer it to the reactor. Cnrrent generation NO additives affect the availability of nitrogen species to be oxidized and rednced, which is highly dependent on the bed hydrodynamics. As such, performance depends on the application. 

This leads to largely differing areas of interest for the two fields. Models to determine the concentration of the OH0 as a function of the water matrix are needed in drinking water applications, while models that incorporate the influence of the hydrodynamics and mass transfer on the reaction rate are necessary for waste water applications. 

The mass transfer factor has also been correlated as a function of the Reynolds number only and thus taking account only of hydrodynamic conditions. If e is the voidage of the packed bed and the total volume occupied by all of the catalyst pellets is Vp, then the total reactor volume is Vp/(l - e). Hence the rate of mass transfer of component A per unit volume of reactor is NASx(l - e)/Vp. If we now consider a case in which only external mass transfer controls the overall reaction rate we have ... 

Detailed analysis of the results published by Casper and Schulz 2) and measurements with the new chromatograph mentioned above 3) have shown that irrevesible thermodynamics, including two different kinetic effects, has to be applied to explain the resolution of the PDC-column 4 5 9) and to obtain the MWD of narrowly distributed polystyrene samples 6 8). In this way, not only the MWD is obtained, but also kinetic constants and thermodynamic functions of the polymer transfer between sol and gel, as well as hydrodynamic and kinetic spreading parameters of the system investigated, can be calculated from PDC-measurements performed at different constant column temperatures, with the same sample injected. The usual static quantities (such as the exponent of the partition function, ratio of the gel/sol volumes, etc.) proposed by Casper and Schulz can then be obtained by extrapolating the results to the theta temperature of the system. In addition, spreading phenomena alone can directly be... 

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