Flow-Profile Effects

FIGURE 5-11 Spectrum of Doppler-shift signal for turbulent flow at various velocities in the laboratory loop. 

FIGURE 5-12 Spectrum of Doppler-shift signal for luminar flow at various pump speeds at in the SRC-II pilot plant. 

FIGURE 5-13 Flowmeter reading vs. pump speed, obtained when pump speed was changed from 300 to 800 rpm during SRC-tests. Line represents a linear fit to the data. 

If an adjustable-gain high-pass filter has been used to eliminate the low-frequency contamination, the resulting spectrum would look like a band-pass filter spectrum. But, such a filter introduces a limit in the low range of the meter, and a nonlinearity at the lower end of the meter range. The comer frequencies or 3-dB points of such a filter will yield a frequency at the high end that can be related to the lowest possible measurable flow. 

To determine the comer frequency of the waveforms shown in Fig. 5-12 and relate it to flow, the property of a uniform spectrum is used. This property states that, for a uniform spectrum, the mean-square amplitude of the total signal energy is the product of the signal energy level and its bandwidth. In other words, the area under the curve is the product of its length and its height above the abscissa. The electronics circuit based on this concept was developed and successfully tested at the SRC-II pilot plant. 

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Figure 5-21 shows the coal/oil slurry velocity derived from the peak of the correlation function plotted against the velocity measured by flow diversion. The flow diversion gives basically the mass flow rate divided by density to give the volumetric flow rate and then the flow velocity. The data can be fitted by two linear relationships with slopes of 1.16 and 1.55, corresponding to meter factors of 0.86 and 0.64. The meter factor may be directly related to the flow profile effect (Sheen et al., 1985). 

Effect of dispersion on a sample s flow profile at different times during a flow injection analysis (a) at injection and when the dispersion is due to (b) convection ... 

Doppler Flow Meters. Doppler flow meters sense the shift in apparent frequency of an ultrasonic beam as it is reflected from air bubbles or other acoustically reflective particles that ate moving in a Hquid flow. It is essential for operation that at least some particles ate present, but the concentration can be low and the particles as small as ca 40 p.m. CaUbration tends to be influenced by particle concentration because higher concentrations result in mote reflections taking place neat the wall, in the low velocity portion of the flow profile. One method used to minimize this effect is to have separate transmitting and receiving transducers focused to receive reflections from an intercept zone neat the center of the pipe. 

The constant pattern concept has also been extended to circumstances with nonplug flows, with various degrees of rigor, including flow profiles in tubes [Sartory, Jnd. Eng. Chem. Fundam., 17, 97 (1978) Tereck et al., Jnd. Eng. Chem. Res., 26, 1222 (1987)], wall effects [Vortmeyer and Michael, Chem. Eng. ScL, 40, 2135 (1985)], channeling [LeVan and Vermeulen in Myers and Belfort (eds.). Fundamentals of Adsorption, Engineering Foundation, New York (1984), pp. 305-314, AJChE Symp. Ser No. 233, 80, 34 (1984)], networks [Aviles and LeVan, Chem. Eng. Sci., 46, 1935 (1991)], and general structures of constant cross section [RudisiU and LeVan, Jnd. Eng. Chem. Res., 29, 1054 (1991)]. 

One of the common problems associated with underwater pelletizers is the tendency of the die holes to freeze off. This results in nonuniform polymer melt flow, increased pressure drop, and irregular extrudate shape. A detailed engineering analysis of pelletizers is performed which accounts for the complex interaction between the fluid mechanics and heat transfer processes in a single die hole. The pelletizer model is solved numerically to obtain velocity, temperature, and pressure profiles. Effect of operating conditions, and polymer rheology on die performance is evaluated and discussed. 

The precise and, where needed, short setting of the residence time allows one to process oxidations at the kinetic limits. The residence time distributions are identical within various parallel micro channels in an array, at least in an ideal case. A further aspect relates to the flow profile within one micro channel. So far, work has only been aimed at the interplay between axial and radial dispersion and its consequences on the flow profile, i.e. changing from parabolic to more plug type. This effect waits to be further exploited. 

The velocity profile effect—For a convex flow velocity profile, for instance, in a steady bubbly flow, the centerline velocity is higher than the average velocity. With bubbles usually concentrating at the center, they attain a higher velocity than the liquid. 

If the fluid in the pipe is in turbulent flow, the effects of molecular diffusion will be supplemented by the action of the turbulent eddies, and a much higher rate of transfer of material will occur within the fluid. Because the turbulent eddies also give rise to momentum transfer, the velocity profile is much flatter and the dispersion due to the effects of the different velocities of the fluid elements will be correspondingly less. 

Dispersion models, as just stated, are useful mainly to represent flow in empty tubes and packed beds, which is much closer to the ideal case of plug flow than to the opposite extreme of backmix flow. In empty tubes, the mixing is caused by molecular diffusion and turbulent diffusion, superposed on the velocity-profile effect. In packed beds, mixing is caused both by splitting of the fluid streams as they flow around the particles and by the variations in velocity across the bed. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

Figure 8-7 Effect of radial mixing on flow profile and p(t).

In FFF, separation is determined by the combined action of the nonuniform flow profile and transverse field effects. The classical configuration assumes the FFF channel as two infinite parallel plates (see Figure 12.4), of which the accumulation wall lies at x=0, where x is the cross-channel axis (directed upward from the accumulation wall). Inside the channel, the carrier fluid, assumed to have a constant viscosity, has a velocity profile u(x) that takes the form... 

The classical FEE retention equation (see Equation 12.11) does not apply to ThEEE since relevant physicochemical parameters—affecting both flow profile and analyte concentration distribution in the channel cross section—are temperature dependent and thus not constant in the channel cross-sectional area. Inside the channel, the flow of solvent carrier follows a distorted, parabolic flow profile because of the changing values of the carrier properties along the channel thickness (density, viscosity, and thermal conductivity). Under these conditions, the concentration profile differs from the exponential profile since the velocity profile is strongly distorted with respect to the parabolic profile. By taking into account these effects, the ThEEE retention equation (see Equation 12.11) becomes ... 

Velocity Profile Effects Many variables can influence the accuracy of specific flow measurement methods. For example, the velocity profile in a closed conduit affects many types of flow-measuring devices. The velocity of a fluid varies from zero at the wall and at other stationary solid objects in the flow channel to a maximum at a distance from the wall. In the entry region of a conduit, the velocity field may approach plug flow and a constant velocity across the conduit, dropping to zero only at the wall. As a newtonian fluid progresses down a... 

Such regions of the bed are associated with high fluid velocities, and inertial effects increasingly influence the flow profile as Reynolds number increases (80). On the basis of these images, it is clear that any theoretical analysis of the flow within such a reactor must account for distinct populations of fast- and slow-moving liquid— channeling does not occur just at the walls of the bed. 

We shall look at the boundary conditions for this equation in the next section. I want to mention here a very important model introduced by Wes-terterp and his collaborators4 and to do so will revert to the classical form of the Taylor problem, in which there is no reaction, the tube is circular (radius R), and the flow is laminar (average velocity U). Also, we ignore the effect of molecular diffusion in the longitudinal direction and are concerned only with the effects of the lateral diffusion across the flow profile. Thus, we have... 

Pipe fittings such as elbows, tee-pieces, reducers, expanders, valves, etc. can all alter the symmetry of the flow profile. MlLLER(2) has listed the following effects ... 

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