Equation of curve

Exercise 5. Taking into account exercise 4 prove that the parametric equations of curve with cusp point are the following ... 

The equation of curve AB can be obtained by integrating Equation (14.64), and the equation of curve BC can be obtained by integrating the corresponding equation ... 

Where y = J[x) denotes the equation of curve lines, x is the position of RBC center in coordinate axis. 

The curve obtained can be transformed into a curve at a different pressure by the equations of Maxwell and Bonnel (see article 4.5.3.2.c). 

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

Surface Micelles. The possibility of forming clusters of molecules or micelles in monolayer films was first proposed by Langmuir [59]. The matter of surface micelles and the issue of equilibration has been the subject of considerable discussion [191,201,205-209]. Nevertheless, many ir-a isotherms exhibit nonhorizontal lines unexplained by equations of state or phase models. To address this, Israelachvili [210] developed a model for ir-u curves where the amphiphiles form surface micelles of N chains. The isotherm... 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

In this chapter, we shall use the principle of least squares to generate the equation of a unique curve for any given set of x-y pairs of data points. The ciu ve so obtained is the best fit to the points subject to... 

When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be described by the data points. From the scatter plot we can take a basic guess as to which type of curve will best describe the X—Y relationship. To aid in the decision process, it is helpful to obtain scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear relationship, the equation has the form of number 7. To facilitate this we frequently employ special graph paper for which one or both scales are calibrated logarithmically. These are referred to as semilog or log-log graph paper, respectively. 

The Least Squares or Best-fit Line. The simplest type of approximating curve is a straight line, the equation of which can be written as in form number 1 above. It is customary to employ the above definition when X is the independent variable and Y is the dependent variable. 

It is, however, possible to calculate the tensile strength of a liquid by extrapolation of an equation of state for the fluid into the metastable region of negative pressure. Burgess and Everett in their comprehensive test of the tensile strength hypothesis, plot the theoretical curves of T /T against zjp, calculated from the equations of state of van der Waals, Guggenheim, and Berthelot (Fig. 3.24) (7], and are the critical temperature and critical... 

A standard addition calibration curve of emission versus the concentration of added sodium gives, by linear regression, an equation of... 

A calibration curve of -log(7) versus concentration gives a standardization equation of... 

Linear equations of the type u = ct — C, where c and C are constants, relate kinematic viscosity to efflux time over limited time ranges. This is based on the fact that, for many viscometers, portions of the viscosity—time curves can be taken as straight lines over moderate time ranges. Linear equations, which are simpler to use in determining and applying correction factors after caUbration, must be appHed carefully as they do not represent the tme viscosity—time relation. Linear equation constants have been given (158) and are used in ASTM D4212. 

For very small AP, flux is linear with pressure. Figure 7 shows a graph of flux versus pressure. Curve A is the pure water flux from equation 1, curve B is the theoretical permeate flux (TPE) for a typical process. As the gel layer forms, the flux deviates from the TPF following equation 7 and curve D results. Changing the hydrodynamic conditions changes K and results in a different operating curve, curve C. 

Asymptotes The hmiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that... 

Equation (14-11) is the differential equation of the operating curve, and its integral around the upper portion of the packing is the equation for the operating curve... 

Multistage CSTR This model has a particular importance because its RTD curve is beU-shaped hke those of many experimental RTDs of packed beds and some empty tubes. The RTD is found by induction by solving the equations of one stage, two stages, and so on, with the result. 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

Equation of state An equation that deseribes the properties of a given material, and distinguishes one material from another. It defines a surfaee in thermodynamie variable spaee on whieh all equilibrium states lie. In shoek-wave applieations, the initial and final states are frequently assumed to lie on the equation of state surface, and this equation ean be eombined with the jump conditions to define the Huqoniot curve whieh is material speeific. 

Equation (4.8) is often called the shock-wave equation of state since it defines a curve in the pressure-volume plane (e.g.. Fig. 4.4). 

---