Straight-line relationship

The same definition of viscosity applies to oil as gas (see Section 5.2.6), but sometimes the kinematic viscosity is quoted. This is the viscosity divided by the density (u = i7p), and has a straight line relationship with temperature. 

Once the production potential of the producing wells is insufficient to maintain the plateau rate, the decline periodbegins. For an individual well in depletion drive, this commences as soon as production starts, and a plateau for the field can only be maintained by drilling more wells. Well performance during the decline period can be estimated by decline curve analysis which assumes that the decline can be described by a mathematical formula. Examples of this would be to assume an exponential decline with 10% decline per annum, or a straight line relationship between the cumulative oil production and the logarithm of the water cut. These assumptions become more robust when based on a fit to measured production data. 

The particle sizes of fillers are usually collected and ordered to yield size distributions which are frequendy plotted as cumulative weight percent finer than vs diameter, often given as esd, on a log probabiUty graph. In this manner, most unmodified fillers yield a straight-line relationship or log normal distribution. Inspection of the data presented in this manner can yield valuable information about the filler. The coarseness of a filler is often quantified as the esd at the 99.9% finer-than value. Deviations from linearity at the high and low ends of the plot suggest that either fractionation has occurred to remove coarse or fine particles or the data are suspect in these ranges. 

The apparent bypass can be estimated by assuming it is approximately equal to the water spHt, ie, the percentage of water in the feed that reports to the underflow. The water spHt has been found to foUow a straight-line relationship with the inverse of the feed water rate for cyclones having diameters greater than 7.5 cm and standard geometries. However, for cyclones of smaller diameters, the apparent bypass appears to be much greater than the water spht, and is typically proportional to the square root of the water spHt. 

Quantitative controllable variables are ftequentiy related to the response (or performance) variable by some assumed statistical relationship or model. The minimum number of conditions or levels per variable is determined by the form of the assumed model. For example, if a straight-line relationship can be assumed, two levels (or conditions) may be sufficient for a quadratic relationship a minimum of three levels is required. However, it is often desirable to include some added points, above the minimum needed, so as to allow assessment of the adequacy of the assumed model. 

Equation (4.4) indicates that at low temperatures, a graph of Cp.m/T against T2 should be a straight line with an intercept of zero at T = 0. Figure 4.3 shows such a graph for glucose, where the lowest measurement is at approximately 7.5 K (T2 = 56 K2). A straight-line relationship is obtained below T= 14 K... 

The total vapor pressure line in Figure 6.5 for ( yic-C6HmCH3 +. Y2C-C6H12) at T = 308.15 K is reproduced as the upper line in Figure 8.13. This line is often known as the bubble-pressure curve. We will refer to it as the liquid line. The straight line relationship for this line as predicted by equation (8.16) is evident. 

In the polystyrene case, carefully handled conventional viscometric techniques were used to obtain [t]] values and subsequently the needed K and a values. The data are illustrated in Figure 2. The log-log plot of the reported molecular weights of the standards and the measured [n] values (plotted as [n] meas.) gives the expected straight line relationship in the molecular weight range of interest. A K value of 1.25 x 10 dl/g and an a value of 0.72, which define the solid line shown in the figure, were then used with the SEC... 

Backhurst and Harker present the equilibrium data as straight line relationships for the temperatures 293 K, 298 K and 303 K. This data was curve fitted to the form... 

Equation (6.10) is a linear function with slope = kJKf vv and v-in(creept = k6. Hence a plot of fcobs as a function of [/] will yield the same straight-line relationship as seen for the mechanism of scheme B. Therefore the observation of a linear relationship between kobs and [/] cannot unambiguously be taken as evidence of a one-step slow binding mechanism. 

The equilibrium data from Table 10.2 are plotted in Figure 10.6a. It can be seen that this does not form a straight-line relationship overall. Figure 10.6a shows the operating line of maximum slope that touches the equilibrium line at xR = 0.1576 kg AA/kg Water. The slope of this line is the ratio of feed to extraction flowrates. If the liquids are immiscible, then the flowrate of solute-free feed (F) is equal to the flowrate of the... 

The power-steam flow relationship in Figure 23.10b can be represented over a reasonable range by a linear relationship, as shown in Figure 23.10c7-9. The straight-line relationship is given by7-9 ... 

Looking back at one of our earliest examples—Fig. 13.3 (p. 361) in which log K for the ionisation of ArC02H is plotted against log k for the base-catalysed hydrolysis of ArC02Et—the straight line implies that there is also a linear relationship between the AG° values for the former reaction and the AG+ values for the latter. Such a straight line relationship between these two series of AG terms is to be expected only if, for each series, one or other of the following conditions is satisfied ... 

A straight line relationship between LJpr and Gr as shown in Fig. 8 implies that the volumetric solid loading (j) is approximately constant because Ar is constant and er can be assumed to be approximately constant when the downcomer is not fluidized. More than 85% of the gas supplied through the central No. 7 and No. 8 flows in those experiments ends up in the draft tube as can be seen from the gas bypassing data presented in Fig. 7. 

Best fit plot of data from Table 2.7 obtained by least squares regression analysis. (Important note This graph implies a straight line relationship down to zero concentration. It is, however, unsafe to use the extrapolated portion as there are no experimental data for this part of the curve). 

Probit Equation The probit equation has been used in an attempt to quantitatively correlate hazardous material concentration, duration of exposure, and probability of effect/injury, for several types of exposures. The objective of such use is to transform the typical sigmoidal (S-shaped) relationship between cause and effect to a straight-line relationship (Mannan, Lees Loss Prevention in the Process Industries, 3d ed., p. 9/68, 2005). 

In some cases, curved lines may be used as standard curves if they give a suitable equation for the line and an acceptable r2 is obtained (see below). However, it is always preferable to have a straight-line relationship between concentration and absorbance. 

The general equation of a straight line relationship between an independent variable x and a dependent variable y is given by ... 

The mathematical equation representing a straight-line relationship between variables x and y is as follows ... 

Rogne (1970) has measured the reactivity of some of the same nucleophiles toward benzenesulfonyl chloride in water at 25°. When log km for reaction of these nucleophiles with PhSOjCl is plotted vs. the log values for the same nucleophiles from Table 10, one obtains a good straight line relationship with a slope of about 0.8. This shows that the reactivity pattern observed with PhSOjSOjPh and shown in Table 10 is representative of what will be observed generally in nucleophilic substitution at the sulfonyl sulfur of reactive sulfonyl substrates. 

It is usual to record data in absorbance units and, although a straight line relationship is theoretically valid, the effective linear range does not usually exceed 1.0 absorbance unit and in many cases may be only up to 0.5 absorbance units. In order to increase the versatility of an instrument, some manufacturers incorporate a concentration mode in which it is possible to alter the sensitivity of the instrument and so work over various concentration ranges. If a direct read-out of concentration is used, the problem of non-linearity becomes more serious and in order to try to overcome this, some instruments incorporate a curvature correction device. 

An alternative to using the proportional correction factor discussed in the previous section is to use two sets of guessed and calculated values to construct a straight line relationship and extrapolate this line to the solution. Figure 4.5 shows the method graphically. 

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