Problem 8-27 Circular reference

Usually circular references occur unintentionally, because the user incorrectly entered a cell reference in an equation. But occasionally a problem can be solved by intentionally creating a circular reference. 

Another example of a problem that can be solved by intentional circular reference and iteration is the calculation of the pH of a solution of a weak acid. Combining the equations for mass balance and K, we obtain the following equation for the [H" "] of a weak acid of concentration C mol/L. 

As an example of a weak acid problem that can readily be solved by circular reference and iteration, consider the calculation of the pH of a 0.1000 M solution of sodium bisulfate, NaHS04. The bisulfate ion is a relatively strong weak acid with Ka = 1.2 X 10 . (Roughly speaking, weak acids with values greater than 10 will require the quadratic or successive approximations approach.) The spreadsheet fragment in Figure 10-13 illustrates a case of a cell that contains a circular reference. Cell C5 contains the formula =SQRT( C 3 (A5-C5)). 

Step 8 Turn on the iteration feature for the circular reference and the problem is solved. 

Example 3 Calculation of View Factor Evaluate the view factor between two parallel circular tubes long enough compared with their diameter D or their axis-to-axis separating distance C to make the problem two-dimensional. With reference to Fig. 5-18, the crossed-strings method yields, per unit of axial length,... 

Poorly Conducting Samples. Samples with low ion concentrations (less than a few mM) will result in poor conductivity of the sample solution. Low conductivity not only causes a slow response time from the electrode but also creates a diffusion potential between the reference electrolyte and the measuring solution which results in an inaccurate pH reading. Using circular ground junctions that create optimal contact between the reference electrolyte and measuring solution can eliminate the problem. Adding conductive-free ions such as a few drops of... 

The problem of determining where a hydraulic jump will occur is a combined application. In the case of supercritical flow on a mild slope, for instance, the tail water depth y2 is determined by the uniform flow depth jo for that slope. The rate of flow and the application of Eq. (10.133) then fix yu and the length of the M3 curve required to reach this depth from the upstream control may be computed from Eq. (10.123). Similarly, in the case of subcritical flow on a steep slope, the initial depth is equal to y0, the tail water depth is given by Eq. (10.133), and the length of the Si curve to the jump from the downstream control is computed from Eq. (10.123). For application of the hydraulic jump to design problems, and for analysis of the jump in circular and other nonrectangular sections, the reader is referred to more extensive treatises on the subject [42],... 

Similar problems affect the definition of hypertension in relation to the establishment of health-associated reference values and exclusion criteria based on laboratory examinations. It has been argued that we might be lost in a circular process when we use laboratory tests to assess the health of subjects who are subsequently to be used as healthy control subjects for laboratory tests. But actually there is no difference, in this context, between measuring the height, weight, and blood pressure and performing selected laboratory tests, provided that these laboratory tests are neither those for which we shall produce reference values nor tests that are significantly correlated with them. ... 

The problem of start-up flow for a circular cylinder has received a great deal of attention over the years because of its role in understanding the inception and development of boundary-layer separation. An insightful paper with a comprehensive reference list of both analytical and numerical studies is S. I. Cowley, Computer extension and analytic continuation of Blasius expansion for impulsive flow past a circular cylinder, J. Fluid Mech. 135, 389-405 (1983). 

In a synodic reference frame, the Hamiltonian of the planar, circular, restricted three-body problem is given by... 

Let us consider two bodies Pi, P2 with masses 1 — fi, fi, moving on circular orbits around the barycenter O. Let us normalize to unity their distance. In the framework of the circular, restricted three-body problem, let us consider the motion of a third body S, moving in the same plane of the primaries. Let its coordinates be ( 1, 2) in the synodic reference frame. 

To conclude this section we might assert that the main techniques for assessing the problem of Rydberg and valence excited states of organic molecules are, in practice, far-UV absorption spectra, photoelectron spectra, and advanced quantum chemical calculations. It should be added that the electronic spectra can also be obtained from electron-impact (energy loss) and circular dichroism spectra. The reader is referred to Robin s Volumes I to III [S, 23] he systematically considers the three kinds of spectra when they are available. 

FT-IR spectroscopy is particularly useful for probing the structure of membrane proteins. Until recently, a lack of adequate experimental techniques has been the reason for the poor understanchng of the secondary structure of most membrane proteins. X-ray diffraction requires high quality crystals and these are not available for many membrane proteins. Circular dichroism (CD) has been widely used for studying the conformation of water-soluble proteins, but problems arise in its use for membrane proteins. The light scattering effect may distort CD spectra and lead to substantial errors in their interpretation. In addition, the reference spectra used for the analysis of CD spectra are based on globular proteins in aqueous solution and may not be applicable to membrane proteins in the hydrophobic environment of lipid bilayers. 

This rarely presents problems, since engineering tolerances can be much closer than necessary for the required accuracy in resistivity. The electrodes are often of a circular, concentric design but may also be linear and parallel. The ratio of resistivity resistance is defined by the electrode dimensions and is often referred to as the "cell constant." For volume resistivity it has the dimension of a length (electrode area,electrode spacing). For surface resistivity it is a dimensionless number (electrode length.-electrode spacing). 

Allied to the problem of the burning velocity of a flame is that of the quenching distance—the minimum separation of two walls which contain the flame for it to propagate rather than be quenched. If a flame is burning on a circular burner, the distance is referred to as the quenching or extinction diameter. 

If one claims that analysis destroys the form of the substance and that synthesis restores this form, this does not suffice to tell us what this form actually is. If, on the other hand, substantial form is something non-physical that disappears and then reappears, then this does not explain how a chemical procedure can affect a non-physical form . If substantial form simply means the substance itself, then we end up with a circular explanation The substance has been altered because the substance has been altered. Any of these three choices leaves us epistemically unsatisfied. To be fair, there is another, more interesting way of understanding substantial form that serves as a precursor to Boyle s mechanistic structuralism. Substantial form can be understood as meaning the inner structure that gives the substance its essential properties. The problem with this understanding of substantial form, however, is that it stUl does not teU us what this inner structure is. Boyle latches on to this idea of inner structure but seeks to understand it in strictly mechanistic and corpuscularian terms, without any reference to substantial form. 

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