Straight lines

This section describes how to fit a straight line to tabular data using Excel and MATLAB. The data in Table E.l represent seven measurements, in columns A and B. The goal is to find an equation, y = a + bx, that best represents this data. 

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Hammen equation A correlation between the structure and reactivity in the side chain derivatives of aromatic compounds. Its derivation follows from many comparisons between rate constants for various reactions and the equilibrium constants for other reactions, or other functions of molecules which can be measured (e g. the i.r. carbonyl group stretching frequency). For example the dissociation constants of a series of para substituted (O2N —, MeO —, Cl —, etc.) benzoic acids correlate with the rate constant k for the alkaline hydrolysis of para substituted benzyl chlorides. If log Kq is plotted against log k, the data fall on a straight line. Similar results are obtained for meta substituted derivatives but not for orthosubstituted derivatives. 

The term p is a reaction constant and is mathematically evaluated for a particular reaction by plotting log kjkQ against a. The slope of the straight lines is p, and reflects the sensitivity of the reaction under study to effects of substituents. The value of p is obviously affected by temperature, solvent changes, etc. 

ASTM proposes representing the kinematic viscosity of hydrocarbons by a straight line on graph paper, called viscometric, for which the scales are such that ... 

T = temperature a, b = straight line coefficients c = kinematic viscosity log = common logarithm (base 10)... 

The straight line and consequently its coefficients are determined starting from two known viscosity values. 

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. 

In the simplest case, for a pressure drawdown survey, the radial inflow equation indicates that the bottom hole flowing pressure is proportional to the logarithm of time. From the straight line plot ot pressure against the log (time), the reservoir permeability can be determined, and subsequently the total skin of the well. For a build-up survey, a similar plot (the so-called Horner plot) may be used to determine the same parameters, whose values act as an independent quality check on those derived from the drawdown survey. 

At the end of the project life a residual unrecovered asset value will remain. This is usually accepted in full as a capital allowance in the final year of the project. Hence the total asset value is fully recovered over the life of the field, but at a slower rate than in the straight line method. 

Assume that the only previous capex had been 120 m, spent in the previous year, with 25% straight line capital allowance, thus capital allowance in this year = 0.25 x 120 m + 0.25 X 80 m = 50 m. 

In order to ensure perpendicular beam incidence on the cylindrical specimen, the circular B-scan profiles were acquired by high frequency (narrow beam) transducers in a synthetic circular aperture array. From these profiles two-dimensional reflection tomograms were reconstructed using a filtered backprojection technique. Straight line propagation was assumed. Several artificial discontinuity types in a cylindrical Plexiglas (Perspex) specimen were compared with similar artificial discontinuities in a cylindrical A/Si-alloy [2]. Furthermore, examples of real discontinuities (an inclusion and a feed head) in the cylindrical AlSi-alloy are presented. 

By plotting InCp po) against x the damping coefficient a is the gradient of the resulting straight line. To separate the elements oq and Ota in a it is possible to measure the backscattering acoustic wave pressure Ps. 

In Figure 4 the measured attenuation values (TT) and the corresponding estimates are plotted against each other. Ideally (with error free estimates) all sample points should lie on the straight line through the origin with unit slope. Clearly there is a strong correlation between the estimates and the true values. 

Thus a plot of Ci/n versus Ci should give a straight line of slope 1/n and intercept /n b. 

A plot of P/n versus P should give a straight line, and the two constants and b may be evaluated from the slope and intercept. In turn, n may be related to the area of the solid ... 

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

The straight line in Figure A 1.7.8 shows the relationships between S and 0 expected for various models, with the straight line indicating Langmuir adsorption. 

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

Figure A3.13.2. Illustration of the analysis of the master equation in temis of its eigenvalues and example of IR-multiphoton excitation. The dashed lines give the long time straight line luniting behaviour. The fiill line to the right-hand side is for v = F (t) with a straight line of slope The intercept of the...

Figure Bl.7.9. (a) Stability diagram for ions near the central axis of a quadnipole mass filter. Stable trajectories occur only if the and values lie beneath tire curve, (b) Stability diagram (now as a fiinction of U and F) for six ions with different masses. The straight line miming tlirough the apex of each set of curves is the operating line, and conesponds to values of UIVthat will produce mass resolution (reproduced with pennission of Professor R March, Trent University, Peterborough, ON, Canada).

Flere is the volume of gas required to saturate the monolayer, V the total volume of gas adsorbed, P the sample pressure, P the saturation vapour pressure and C a constant related to the enthalpy of adsorption. The resulting shape of the isothemi is shown plotted in figure Bl.26.6 for C = 500. A plot of P/V(P - Pq) against P/Pq should give a straight line having a slope (C - )/y C and an intercept The BET surface area is... 

This nomially refers to the use of the straight-line trajectory/ (t) = (b +v t ), 0(t) = arctan(b/vt) within the classical path treatment. See Bates [18,19] for examples and fiirtlier discussion. 

In order to separate the enthalpy and the entropy of activation, the rate is measured as a fiinction of temperature. These data should give a straight line on an Eyrmg plot of log(rate/7) against (1/7) (figure... 

The polymers described so far have relatively flexible main chains which can result in complex confonnations. In some cases, tliey can double back and cross over tliemselves. There are also investigations on polymers which are constrained to remain in a confonnation corresponding, at least approximately, to a straight line, but which have amphiphilic properties tliat ensure tliat tliis line is parallel to tire water surface. Chiral molecules are one example and many polypeptides fall into tliis class [107]. Another example is cofacial phtlialocyanine polymers (figure C2.4.9). 

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