Flow Profiling

The time-to-distance transfonnation requires fast mixing and a known flow profile, ideally a turbulent flow with a well-defined homogeneous composition perpendicular to the direction of flow ( plug-flow ), as indicated by tire shaded area in figure B2.5.1. More complicated profiles may require numerical transfomiations. 

The electroosmotic flow profile is very different from that for a phase moving under forced pressure. Figure 12.40 compares the flow profile for electroosmosis with that for hydrodynamic pressure. The uniform, flat profile for electroosmosis helps to minimize band broadening in capillary electrophoresis, thus improving separation efficiency. 

Schematic showing a comparison of the flow profiles for (a) GC and HPLC, and (b) electrophoresis.

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 ... 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Fig. 1. Flow profiles, where N is velocity (a) laminar, and (b) turbulent for fluids having Reynolds numbers of A, 2 x 10, and B, 2 x 10 .

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. 

Both wetted-sensor and clamp-on Doppler meters ate available for Hquid service. A straight mn of piping upstream of the meter and a Reynolds number of greater than 10,000 ate generally recommended to ensure a weU-developed flow profile. Doppler meters ate primarily used where stringent accuracy and repeatabiHty ate not requited. Slurry service is an important appHcation area. 

Electroosmotic flow in a capillary also makes it possible to analyze both cations and anions in the same sample. The only requirement is that the electroosmotic flow downstream is of a greater magnitude than electrophoresis of the oppositely charged ions upstream. Electro osmosis is the preferred method of generating flow in the capillary, because the variation in the flow profile occurs within a fraction of Kr from the wall (49). When electro osmosis is used for sample injection, differing amounts of analyte can be found between the sample in the capillary and the uninjected sample, because of different electrophoretic mobilities of analytes (50). Two other methods of generating flow are with gravity or with a pump. 

Convergence was achieved in 3 iterations. Converged values of temperatures, total flows, and component flow rates are tabulated in Table 13-14. Computed reboiler duty is 1,295,000 W (4,421,000 Btu/h). Computed temperature, total vapor flow, and component flow profiles, shown in Fig. 13-54, are not of the shapes that might be expected. Vapor and liquid flow rates for nC4 change dramatically from stage to stage. 

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)]. 

Once a flow profile has been established, samphng strategy can be considered. Since samphng collection can be simphfied and greatly reduced depending on flow characteristics, it is best to complete the flow-profile measurement before sampling or measuring pollutant concentrations. 

ON-LINE MONITORING OF HYDRODYNAMIC FLOW PROFILES IN MICROFLUIDIC CHANNELS BASED UPON MICROELECTROCHEMISTRY... 

OPTIMISATION OF MICROBAND ELECTRODE SIZES AND LOCATIONS WITHIN A RECTANGULAR MICROFLUIDIC CHANNEL FOR ELECTROCHEMICAL MONITORING OF HYDRODYNAMIC FLOW PROFILES... 

In this work, we determine constraints on the dimensionless parameters of the system (dimensionless electrode widths, gap size and Peclet number), first qualitatively and then quantitatively, which ensure that the proposed flow reconstmction approach is sufficiently sensitive to the shape of the flow profile. The results can be readily applied for identification of hydrodynamic regimes or electrode geometries that provide best performance of our flow reconstmction method. 

Figure 8.3. VeJocily flow profile in a tube for a fluid with zero yield stress and assuming no slip at...

The graphical integration method is based on graphical presentation of the average flow profile. For a circular duct, the cross-section is virtually divided into several concentric ring elements. The spatial mean velocity of such an element is determined as an arithmetical mean of local velocities along the circumference of the corresponding radius. For a circular cross-section the flow rate can be expressed as... 

Figure 9.14 Optimal flow profiles for a five stage KNO3 crystallization process Sheikh and Jones, 1997)...

The flow profiles of electrodriven and pressure driven separations are illustrated in Figure 9.2. Electroosmotic flow, since it originates near the capillary walls, is characterized by a flat flow profile. A laminar profile is observed in pressure-driven systems. In pressure-driven flow systems, the highest velocities are reached in the center of the flow channels, while the lowest velocities are attained near the column walls. Since a zone of analyte-distributing events across the flow conduit has different velocities across a laminar profile, band broadening results as the analyte zone is transferred through the conduit. The flat electroosmotic flow profile created in electrodriven separations is a principal advantage of capillary electrophoretic techniques and results in extremely efficient separations. 

Figure 9.2 Pressure-driven (a) and electrodriven (b) flow profiles. Laminar flow in pressure-driven systems results in a bullet-shaped profile, wliile the profile of electroosmotic flow is plug-shaped, wliich reduces band broadening.

Estimate the cash-flow profile for each alternative. The cash-flow profile should include the costs and revenues if they differ, for the alternative being considered during each period in the planning horizon. For public projects, revenues may be replaced by estimates of benefits for the public as a whole. If revenues can be assumed to be constant for all alternatives, only costs in each period are estimated. Cash-flow profiles should be specific to each alternative. We shall denote revenues for an alternative x in period t as B (t,x), and costs as C (t,x). By convention, cash flows are usually assumed to occur at the end of the time period, and initial expenditures to occur at the beginning of the planning horizon, that IS, m year 0. 

Perform sensitivity and uncertainty analysis. Calculation of life-cycle costs and net benefits assumes that cash-flow profiles and the value of MARR are reasonably accurate. In most cases, uncertain assumptions and estimates are made in developing cash flow profile forecasts. Sensitivity analysis can be performed by testing how the outcome changes as the assumptions and input values change. 

Viscous forces within the fluid will always prevent a completely unhindered discharge, but in extrusion practice an additional die head resistance is used to generate backflow and mixing, so that a more uniform product is obtained. The flow profile along the channel is then of some intermediate form, such as that shown in Figure 8.18c. 

To maintain a laminar flow profile over a long distance the entrance length (Lg) of the flow channel has to be small compared to the total length. The entrance length is given by... 

On the basis of the flow profiles described above [Eqs. (12) and (13)] it is possible to estimate a flow-averaged shear stress for a given set of flow conditions. 

For laminar flow in channels of rectangular cross-section, the velocity profile can be determined analytically. For this purpose, incompressible flow as described by Fq. (16) is assumed. The flow profile can be expressed in form of a series expansion (see [100] and references therein), which, however, is not always useful for practical applications where often only a fair approximation of the velocity field over the channel cross-section is needed. Purday [101] suggested an approximate solution of the form... 

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