Rate plots

Routine production tests are performed, approximately once per month on each producing well, by diverting the production through the test separator on surface to measure the liquid flowrate, water cut, and gas production rate. The wellhead pressure (also called the flowing tubing head pressure, FTHP) is recorded at the time of the production test, and a plot of production rate against FTHP is made. The FTHP is also recorded continuously and used to estimate the well s production rate on a daily basis by reference to the FTHP vs production rate plot for the well. 

Of the models Hsted in Table 1, the Newtonian is the simplest. It fits water, solvents, and many polymer solutions over a wide strain rate range. The plastic or Bingham body model predicts constant plastic viscosity above a yield stress. This model works for a number of dispersions, including some pigment pastes. Yield stress, Tq, and plastic (Bingham) viscosity, = (t — Tq )/7, may be determined from the intercept and the slope beyond the intercept, respectively, of a shear stress vs shear rate plot. 

If the reaetion rate depends on more than one speeies, use the method of exeess eoupled either with the half-life method or the differential method. If the method of exeess is not suitable, an initial rate plot may be eonstrueted by varying the eoneentration of one reaetant while the eoneentrations of the others are held eonstant. This proeess is repeated until the orders of reaetion of eaeh speeies and the speeifie reaetion rate are evaluated. At level 5, the least-squares analysis ean be employed. 

However, as the pH—rate plot shows, at very low pH the observed rate actually decreases. Because, as the preceding argument shows, rate-determining dehydration should result in a pH-dependent rate at low pH, this decreased rate must mean that the rds has changed. This is reasonable, for at pH values well below the pKg of hydroxylamine, the decreasing proportion of the hydroxylamine in the unprotonated form will decrease the rate of the initial addition. At some pH, then, the rate of the addition step will fall below that of the dehydration step, and the observed rate curve will lie lower than the rate predicted for the dehydration. 

Where a large collection of data exists then it may be effectively condensed in the form of diagrams. A popular method is the use of iso-corrosion rates plotted on co-ordinates of temperature and concentration for one material and one chemical. Because of the large amount of data on the common acids there are many examples of this type of diagram, e.g. the work of Berg who has chosen metals and alloys that are readily available. He has excluded many metals and alloys on the grounds that they are either Non-resistant or can be substituted by cheaper materials. ... 

The last step is facilitated by the use of calibration curves of analytical-line intensity versus counting rate plotted at several levels of background intensity. All steps are rapid and simple. 

Fig. 3.12. Schematic growth rate plot showing regimes with sharp breaks in slope, as well as the smooth transition predicted by the Frank model...

FIGURE 38.7 Viscosity versus shear rate plots of virgin natural mbber-low-density polyethylene (NR-LDPE) blends (N2) and reclaimed rubber (RR)-LDPE blends (A2). (Reprinted from Nevatia, P., Baneijee, T.S., Dutta, B., Jha, A., Naskar, A.K., and Bhowmick, A.K., J. Appl. Polym. Sci., 83, 2035, 2002. With permission from Wiley InterScience.)... 

On this basis = 0.0170 sec , = 0.645 sec , and K = 0.739 mole.P at 25 °C. The corresponding activation parameters were determined also by Es-penson. By a method involving extrapolation of the first-order rate plots at various wavelengths to zero time, the absorption spectrum of the intermediate was revealed (Fig. 1). Furthermore, the value of K obtained from the kinetics was compatible with that derived from measurements on the acid dependence of the spectrum of the intermediate. Rate data for a number of binuclear intermediates are collected in Table 2. Espenson shows there to be a correlation between the rate of decomposition of the dimer and the substitution lability of the more labile metal ion component. The latter is assessed in terms of the rate of substitution of SCN in the hydration sphere of the more labile hydrated metal ion. 

The rate of reduction of Tl(III) by Fe(II) was studied titrimetrically by John-son between 25 °C and 45 °C in aqueous perchloric acid (0.5 M to 2.0 M) at i = 3.00 M. At constant acidity the rate data in the initial stages of reaction conform to a second-order equation, the rate coefficient of which is not dependent on whether Tl(III) or Fe(II) is in excess. The second-order character of the reaction confirms early work on this system . A non-linearity in the second-order plots in the last 30 % of reaction was noted, and proved to be particularly significant. Ashurst and Higginson observed that Fe(III) retards the oxidation, thereby accounting for the curvature of the rate plots in the last stages of reaction. On the other hand, the addition of Tl(l) has no significant effect. On this basis, they proposed the scheme... 

Figure 20.2. Reciprocal temperature-rate plot for the determination of the energy of activation for the hydrogenation of 1-hexene over Wilk/STA.

Figure 9 The trimazozin release rate plotted versus the overall membrane thickness for an asymmetrical membrane coating on tablets. (From Ref. 20.)...

If the shear stress versus shear rate plot is a straight line through the origin (or a straight line with a slope of unity on a log-log plot), the fluid is Newtonian ... 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

Prepare rate plots of the data shown in Table 8. lusing an Avrami rate law with n values of 2, 3, and 4. 

Figure 2. First-order rate plots for the consumption of CO in a 100-mL batch reactor (catalyst solution is 5 mL of aqueous diglyme with .5M HtO, 1.0M H2SO,lf T = 100°C and Pco (initial) = 0.9 atm). Slopes of the three linear plots are 2 X 10 2, 4.4 X JO 2, and 9.3 X JO 2 hr1 for the respective Ru (CO)I2 initial concentrations of (I) 0.006M, (II) 0.012U, and (III) 0.024U.

Figure 10.1 Initial-rate plot for S + E - P + E showing interpretation of rp0TOU in terms of rate Parameters constant cE , 7 pH...

Fig. 7. Experimental flow rates plotted vs. theoretical flow rates calculated using the Hagen-Poiseuille equation. The 100 data points include flow rates calculated using five different solvents, four different tube lengths and five different values of overpressure (see Supporting information for data used to generate this plot). The dashed lines indicate the variation expected if the inner diameter was 25% less (1 0.32) or 25% more (1 2.44) than the claimed value.

Figure 2. Dissolution of p-Cl-PHMP and PBPhs in NaOH solutions at 20.0°C. (a) log(Rate) plotted against log(NaOH), + p-Cl-PHMP, PBPh-1 (b) log(Rate) plotted against log(ANaOH), p-Cl-PHMP, A PBPh-1, PBPh-2.

Thus the rate constant, k2, can be obtained from the integrated rate plot only if the excited-state absorption coefficient, e4i0, and the path length, i, are known. 

Figure 3 Observed in vitro dissolution data for three ER formulations (panel a) fast ( target 80%=12hr), medium (o target 80%=lbhr), and slow ( target 80% = 20hr). Also shown are the predicted lines corresponding to fitting the data to the double Weibull equation (fitted parameter values are listed in Table 2). The associated rate plot for the three formulations is shown in panel b (fast,--------- medium,--------- slow,---).

Fig. 3. Linearized initial rate plots for alcohol dehydrogenation, 285°C.

Fig. 4. Linearized initial rate plot for alcohol dehydration, Eq, (16).

An intrinsic parameter is one that is inherently present in or arises naturally from a reaction-rate model. These parameters, which are of a simpler functional form than the entire rate model, facilitate the experimenter s ability to test the adequacy of a proposed model. Using these intrinsic parameters, this section presents a method of preparing linear plots for high conversion data, which is entirely analogous to the method of the initial-rate plots discussed in Section II. Hence, these plots provide a visual indication of the ability of a model to fit the high conversion data and thus allow a more... 

The Q (initial reaction rate) plots were presented in Fig. 19. Note again that the data may be correlated well by straight lines for both models. The C2 values are correlated by the solid lines of Fig 20. Note that the dual-site values can again be correlated by a straight line, but that the single-site values of C2 show a definite curvature. Alternatively, the 0.975 atm value of the single-site C2 could be rejected, and the three high-pressure points... 

An intriguing observation is that a plot of AVJ, the activation volume as a function of salt concentration, almost invariably seems to be a mirror image of the rate plot. A typical diagram is shown in Fig. 9. However, it can be shown that the functional form of the rate constant necessarily must be reflected in the form of the activation volume in this case. According to what has been said hitherto, an equation... 

Figure 15. Illustration of possible variations in isotopic fractionation between Fe(III),q and ferric oxide/ hydroxide precipitate (Aje(,n),q.Fenicppt) and precipitation rate. Skulan et al. (2002) noted that the kinetic AF (ni)aq-Feiricppt fractionation produced during precipitation of hematite from Fe(III), was linearly related to precipitation rate, which is shown in the dashed curve (precipitation rate plotted on log scale). The most rapid precipitation rate measured by Skulan et al. (2002) is shown in the black circle. The equilibrium Fe(III),-hematite fractionation is near zero at 98°C, and this is plotted (black square) to the left of the break in scale for precipitation rate. Also shown for comparison is the calculated Fe(III),q-ferrihydrite fractionation from the experiments of Bullen et al. (2001) (grey diamond), as discussed in the previous chapter (Chapter lOA Beard and Johnson 2004). The average oxidation-precipitation rates for the APIO experiments of Croal et al. (2004) are also noted, where the overall process is limited by the rate constant ki. As discussed in the text, if the proportion of Fe(III),q is small relative to total aqueous Fe, the rate constant for the precipitation of ferrihydrite from Fe(III), (Ai) will be higher, assuming first-order rate laws, although its value is unknown.

Fig. 5. Typical rate plots of carbon remaining versus burning time ( ) normal sample (O) initial flattening due to carbon overload (partial inaccessibility). Redrawn from Weisz and Goodwin (1963).

The relations of molar extinction coefficients and oxygen absorption rates, plotted vs. time, are illustrated in Figures 3 and 4 (solvents, benzene and butyric acid, respectively). The variation of coefficients was parallel to the rate of oxygen absorption—i.e. the larger the coefficient, the higher the oxygen absorption rate. It is considered therefore that the catalyst at its maximum absorption coefficient is in a desirable form for oxidizing acrolein. 

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