Zero flow, extrapolation

An alternative approach to estimating the point at which dialysis sampling does not influence extracellular analyte concentrations is extrapolation to zero flow (Jacobson et al., 1985). The flow rate within or between... 

The peak hourly flow rate will now be derived. Remember that maximum values correspond to a probability of zero. Therefore, to obtain the peak hourly flow rate, read or extrapolate the flow rate corresponding to zero in the probability distribution obtained from an array arranged in descending order. 

The method of capillary Jlow measures the increase in resistance for solvent flow through a capillary (or a porous plug) due to an adsorbed polymer layer. This increase can be translated into a smaller effective capillary (or pore) radius through the Hagen-Polseuille law (1.6.4.18). The hydrodynamic radius d is supposed to be given by the difference between the "covered" and the "bare" radius. In such experiments the observed hydrodynamic thickness sometimes turns out to be flow-rate dependent. In such cases an extrapolation to zero flow rate needs to be carried out. 

The extent of the nonequilibrium region, segment BC of Fig. 1, is dependent on both the polymer-solute system (34) and experimental parameters (13, 33). As the thickness of the polymer film or the flow rate of carrier gas increases, the maximum of the retention diagram shifts towards higher temperatures (13). Equilibrium retention data can, however, be obtained by extrapolation of retention vdumes to zero flow rate. Braun and Guillet (35) have developed a model of the chromatc rq>hic behavior of polymers near Tg which reproduces most features observed experimentally. The effects of the magnitude and temperature depandence of the difihision... 

Cole and Taylor (3) determined the vapor pressures of NaB02(g) by a dynamic method with dry N2 as carrier gas. Their vapor pressure data were taken over a flow-rate range where the apparent vapor pressure Increased with decreasing flow rate, and the data were then extrapolated to zero flow rate. The vapor species of the sample was also assumed to be monomeric NaB02(g). JANAP 3rd law analyses of their reported vapor pressure data over the liquid NaB02 In the temperature range 1150 C-1350 C yield (298.15 K) = 75.37 3.32 kcal mol and the drift -20.6 cal K mol . (The 2nd law enthalpy of vaporization Is A apH (298.15 K) - 107 kcal mol ). Using the 3rd law (298.15 K), we obtain A H (298.15 K) = -154.1 4 kcal mol... 

This indicates that under these conditions, Zp can be obtained from tp only by extrapolating to zero flow rate, whereas may be used regardless of the flow rate. 

Measurement Procedure. IGC measurements were started after the thermal and flow equilibrium in the column were stable (2 to 3 h). To facilitate rapid vaporization of the probe (0.01 yL), the injector temperature was kept 30°C above the boiling point of the probe. Measurements were made at five carrier gas flow rates. The retention volumes of six injections for each probe and twenty injections of the marker (H2) at a given flow rate were averaged. The values obtained were extrapolated to zero flow rate to eliminate the flow rate dependence of the retention data. The net retention time (tR) is defined as the time difference between the first statistical moment of the solvent peak and that of the marker gas. Thus, tR was calculated by an on-line computer statistical peak analysis rather than the retention time at the peak maximum (tp,maY). This eliminated inaccuracies arising from slight peak asymmetry, which occurs even for inert and well-coated supports. The specific retention volumes (Vg°) derived from tR and tR max differed by as much as 5% for small retention times and slightly skewed peaks (11,12). 

A decay transient for the 2-nitropropane radical anion, produced in the system described above, is shown in Fig. 19 together with the calculated transient. It can be seen that, for the low flow rate used, the transient is exponential, as predicted for the first-order decay, but at higher flow rates (or slower radical decay), one finds the transients are not perfectly exponential. The cause of this is convection of radical from the ESR cavity combining with the decay signal. This problem was shown to be easily overcome by a simple method of analysis [65] whereby the measured transients are treated as exponentials, as in eqn. (17), and an "apparent rate constant is deduced for the flow rates used. A plot of "apparent rate constant against u2/3 allowed extrapolation to zero flow rate and hence the deduction of the true value for the rate constant. Results obtained in this manner for the 2-nitropropane system were shown to be in agreement with the steady-state measurements. 

The effect of the stationary phase percentage on the retention diagram of n-tetradecane on polystyrene after extrapolation to infinite dilution and for zero flow rate is shoAvn in Figure 5.25 [220]. As in Figure 5.15,... 

Fig. 5.25. — Retention diagrams of n-tetradecane on polystyrene obtained by extrapolation to infinite dilution and zero flow rate [220] the concentrations of stationary phase are indicated on the Figure.

The peak shape and retention volumes in the non-equilibrium region depend strongly on carrier gas flow rate [161, 220]. The dependence of retention volumes on flow rate at the given temperatures for n-hexa-decane on polystyrene is shown in Figure 5.26 [161]. The linear dependence of retention volume on flow rate extrapolated to zero flow rate yields... 

A. A. Harper in Newcastle upon Tyne measured total blood flow through the stomachs of anesthetized cats by conventional methods and mucosal blood flow by multiplying the aminopyrine output into the lumen by its arteriovenous difference. He found a linear relation between H secretion and mucosal flow and no discontinuity when he extrapolated mucosal flow to zero H output. This implied that the clearance method is valid at a zero secretion rate. Three years later, gastroenter-... 

One of the disadvantages of any flow-through electrochemical system is the existence of a streaming potential, which must be taken into account or eliminated. To eliminate, Estr, it is necessary to extrapolate the measured open-circuit potentials to zero flow rate so that at least four experimental points should be measured to provide a reliable linear extrapolation. Another possibility is to minimize... 

Which range should be considered The answer is the region near the origin of a plot like Fig. 2.2 for pseudoplastic materials. The slope of the tangent to a pseudoplastic curve at the origin is called the viscosity at zero rate of shear. Note that this is an extrapolation to a limit rather than an observation at zero shear (which corresponds to no flow). We shall use the symbol to indicate the viscosity of a polymer in the limit of zero shear, since the behavior is Newtonian (subscript N)in this region. 

The viscosity ratio or relative viscosity, Tj p is the ratio of the viscosity of the polymer solution to the viscosity of the pure solvent. In capillary viscometer measurements, the relative viscosity (dimensionless) is the ratio of the flow time for the solution t to the flow time for the solvent /q (Table 2). The specific (sp) viscosity (dimensionless) is also defined in Table 2, as is the viscosity number or reduced (red) viscosity, which has the units of cubic meters per kilogram (m /kg) or deciUters per gram (dL/g). The logarithmic viscosity number or inherent (inh) viscosity likewise has the units m /kg or dL/g. For Tj g and Tj p, the concentration of polymer, is expressed in convenient units, traditionally g/100 cm but kg/m in SI units. The viscosity number and logarithmic viscosity number vary with concentration, but each can be extrapolated (Fig. 9) to zero concentration to give the limiting viscosity number (intrinsic viscosity) (Table 2). 

Because it is very difficult to measure the flow characteristics of a material at very low shear rates, behaviour at zero shear rate can often only be assessed by extrapolation of experimental data obtained over a limited range of shear rates. This extrapolation can be difficult, if not impossible. From Example 3.10 in Section 3.4.7, it can be seen that it is sometimes possible to approximate the behaviour of a fluid over the range of shear rates for which experimental results are available, either by a power-law or by a Bingham-plastic equation. 

The model of a stirred tank reactor has a fraction 1-a of the flow In bypass and a fraction 1-/3 of the volume inactive. A step input of tracer is entered. Effluent concentrations were measured at high and low agitator speeds and the quantity 1-C/Cf was obtained at zero time by extrapolation. The values were 0.89 at high speed and 0.59 at low speed, (a) Find a and (3. (b) Compare... 

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