Slope variance

Levenspiel and Smith Chem. Eng. Sci., 6 (227), 1957] have reported the data below for a residence time experiment involving a length of 2.85 cm diameter pyrex tubing. A volume of KMn04 solution that would fill 2.54 cm of the tube was rapidly injected into a water stream with a linear velocity of 35.7 cm/sec. A photoelectric cell 2.74 m downstream from the injection point is used to monitor the local KMn04 concentration. Use slope, variance, and maximum concentration approaches to determine the dispersion parameter. What is the mean residence time of the fluid ... 

Figure 7. Comparison of slope variance for 3-in. thick sulfur-asphalt and asphalt concrete pavements...

The overall structural adequacy of pavements is given in probabilistic terms of the present serviceability index developed from the AASHTO road test (17). This index, which is a measure of the momentary ability of a pavement to serve traffic, is based on such factors as rut depth, slope variance, cracking, and patching of the pavement. The relationship between serviceability index and these pavement distress modes is given by the AASHTO road test equation (16) ... 

The subjectivity of rating the pavement condition by users in terms of PSR has led to the development of the present serviceability index (PSI). The same sections of the AASHTO study were also surveyed at the same time and physical measurements (slope variance in wheel paths, cracking, area of patching and rut depth) were carried out (AASHTO 1962). The PSR estimated from objective physical measurements was termed PSI. The equations derived to determine PSR in an objective way in terms of PSI were as follows ... 

Carey W.N., H.C. Huckins, and R.C. Leathers. 1962. Slope Variance as a Measure of Roughness and the CHLOE Profilometer. HRB Special Report 73. Washington, DC Transportation Research... 

The variance of the regression times the diagonal elements of the inverse coefficient mahix gives the variance of the intercept and slope. 

Equations 5.13 for the slope, h, and 5.14 for the y-intercept, ho, assume that indeterminate errors equally affect each value of y. When this assumption is false, as shown in Figure 5.11b, the variance associated with each value of y must be included when estimating [3o and [3i. In this case the predicted slope and intercept are... 

Column type Peak variance (o) Calibration slope (D) Specific resolution (R,p)... 

Traditionally, column efficiency or plate counts in column chromatography were used to quantify how well a column was performing. This does not tell the entire story for GPC, however, because the ability of a column set to separate peaks is dependent on the molecular weight of the molecules one is trying to separate. We, therefore, chose both column efficiency and a parameter that we simply refer to as D a, where Di is the slope of the relationship between the log of the molecular weight of the narrow molecular weight polystyrene standards and the elution volume, and tris simply the band-broadening parameter (4), i.e., the square root of the peak variance. 

One-way analysis of variance, 229-230, 230f—231f Operational model derivation of, 54-55 description of, 45—47, 46f function for variable slope, 55 for inverse agonists, 221 of agonism, 47f orthosteric antagonism, 222 partial agonists with, 124, 220-221 Opium, 147 Orphan receptors, 180 Orthosteric antagonism... 

This variance grows without bound with the size of the CCD and dictates that the smallest practical number of pixels should be used. A 2 x 2 array (or quadcell) with = i, is thus the configuration that dominates many existing slope sensors (Tyler and Fried, 1982). 

Figure 7 shows that for the maximum likelihood estimator the variance in the slope estimate decreases as the telescope aperture size increases. For the centroid estimator the variance of the slope estimate also decreases with increasing aperture size when the telescope aperture is less than the Fried parameter, ro (Fried, 1966), but saturates when the aperture size is greater than this value. 

Figure 7. The variance in the slope estimate versus aperture size for the centroid and maximum likelihood estimators for turbulence defined by ro = 0.25.

In order to formulate the statistical problem generally, let us return to the Arrhenius graph (Figure 5) and ask the question of how to estimate the position of the common point of intersection, if it exists (162). That is, in the coordinates x = T and y = log k, a family of 1 straight lines is given with the slopes bj (i = 1,2,..3) and with a common point of intersection (xq, yo). The ith line is determined by mj points (m > 2) with coordinates (xy, yjj) where j = 1,2,..., mj. Instead of the true coordinates yy, only the values yy = yy + ey are available, ey being random variables with a zero average value and a constant variance,. If the hypothesis of a common point of intersection is accepted, ey may be identified with the experimental error. 

Two slopes are compared in a similar manner as are two means the simplest case is obtained when both calibrations are carried out using identical calibration concentrations (as is usual when SOPs are followed) the average variance V u is used in a t-test ... 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

In this study the reader is introduced to the procedures to be followed in entering parameters into the CA program. For this study we will keep Pm = 1.0. We will first carry out 10 runs of 60 iterations each. The exercise described above will be translated into an actual example using the directions in Chapter 10. After the 10-run simulation is completed, determine (x)6o, y)60, and d )6o, along with their respective standard deviations. Do the results of this small sample bear out the expectations presented above Next, plot d ) versus y/n for = 0, 10,20, 30,40, 50, and 60 iterations. What kind of a plot do you get Determine the trendline equation (showing the slope and y-intercept) and the coefficient of determination (the fraction of the variance accounted for by the model) for this study. Repeat this process using 100 runs. Note that the slope of the trendline should correspond approximately to the step size, 5=1, and the y-intercept should be approximately zero. 

One possibility is that although averages for polystyrene standards require correction, those for PMMA would not According to symmetrical axial dispersion theory (5) the correction depends upon both the slope of the calibration curve (different for each polymer type) and the variance of the chromatogram of a truly monodisperse sample. Furthermore, the calibration curve to be utilized can be obtained from a broad standard as well as from monodisperse samples. The broad standard method may itself incorporate some axial dispersion correction depending upon how the standard was characterized. 

Mjj and My or [q] for the broad MWD standard are taken as known quantities. Fy(v) is the normalized chromatogram for the broad MWD standard obtained with a mass detector. D2 is the slope of the molecular weight calibration curve at the peak position of the chromatogram (the equation of the tangent is given by M(v) = Dj exp(-D2v). is the variance of the single-species chromatogram... 

By way of illustration, the regression parameters of a straight line with slope = 1 and intercept = 0 are recursively estimated. The results are presented in Table 41.1. For each step of the estimation cycle, we included the values of the innovation, variance-covariance matrix, gain vector and estimated parameters. The variance of the experimental error of all observations y is 25 10 absorbance units, which corresponds to r = 25 10 au for all j. The recursive estimation is started with a high value (10 ) on the diagonal elements of P and a low value (1) on its off-diagonal elements. 

The voltammograms at the microhole-supported ITIES were analyzed using the Tomes criterion [34], which predicts ii3/4 — iii/4l = 56.4/n mV (where n is the number of electrons transferred and E- i and 1/4 refer to the three-quarter and one-quarter potentials, respectively) for a reversible ET reaction. An attempt was made to use the deviations from the reversible behavior to estimate kinetic parameters using the method previously developed for UMEs [21,27]. However, the shape of measured voltammograms was imperfect, and the slope of the semilogarithmic plot observed was much lower than expected from the theory. It was concluded that voltammetry at micro-ITIES is not suitable for ET kinetic measurements because of insufficient accuracy and repeatability [16]. Those experiments may have been affected by reactions involving the supporting electrolytes, ion transfers, and interfacial precipitation. It is also possible that the data was at variance with the Butler-Volmer model because the overall reaction rate was only weakly potential-dependent [35] and/or limited by the precursor complex formation at the interface [33b]. 

In addition to the aforementioned slope and variance methods for estimating the dispersion parameter, it is possible to use transfer functions in the analysis of residence time distribution curves. This approach reduces the error in the variance approach that arises from the tails of the concentration versus time curves. These tails contribute significantly to the variance and can be responsible for significant errors in the determination of Q)L. 

Illustration 11.2 indicates the use of the slope and variance methods for evaluating JuL. 

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