Tangent-plane method

Step 7. Identify the root having the lowest value of f2 as the stable one-phase mixture at the proposed T, P, and From Table 8.2 we see that the stable one-phase mixture is root p. Therefore root a, which is our proposed mixture, is not a stable one-phase mixture. Further, Figure 8.18 shows that root a satisfies the requirement on the derivative (8.4.8), so the proposed mixture is not unstable. Hence, it must be metastable it might be observed, but more likely it will split into two phases. To find the compositions of those phases, we would solve the phase-equilibrium problem. Other procedures for identifying stable one-phase mixtures include the tangent-plane method which originates with Gibbs [15] and has been fully developed by Michelsen, especially for multi-component mixtures [16]. 

The exact values of E and 5E / 5n are in general unknown and the Kirchhoff or physical optics method consists in approximating the values of these two quantities on the surface and then evaluating the Helmholtz integral. We shall approximate the field at any point of the surface by the field that would be present on a tangent plane at the point. With this approximation, the field on the surface and its normal derivative are... 

When the diffusion time is short enough, the translation on the spherical surface is approximately identical with that on the tangent plane to the spherical surface. The latter is the two-dimensional diffusion process treated by the Green function method in Appendix C, and we can use Eq. (C17) again. Since the rotational diffusion coefficient Dr is related to the translational diffusion coefficient D<2) in Eq. (Cl7) by Dr = D(2)/(Le/2)2, we have... 

As an example of absolute shape criteria, the local curvature properties of a MIDCO can be used for defining absolute shape domains on it [156], and for a subsequent global shape characterization. In Figure 5.1 a MIDCO G(a) is shown as an illustration of some of the concepts discussed. The simplest method [155] is based on comparisons to a reference of a tangent plane what leads to the identification of locally convex, concave, and saddle-type domains, as mentioned previously, although much finer characterizations are also possible [156,199]. 

Wasylkiewicz et al. (1993) present a method to find regions where mixtures partition in two or more liquid phases. They base it on the Gibbs tangent plane test (Michelsen, 1982, 1993) to decide if a current composition resides in a single- or a multiple-liquid phase region. 

As it is practically impossible to compare slip resistance or static coefficients of friction of various WPC materials using the same experimental approach (Table 11.3), I have undertaken—while preparing this book for the publication—a simplified comparison using the standard incline-plane method. This method is a standard experiment in mechanical physics and involves tilting a platform to the point where movement of a material first occurs. The tangent of the angle of the plane at the point of movement is equal to the static coefficient of friction. 

The vapor-liquid equilibrium was computed from the EOS model using the reliable and robust method of Hua et al 14-16) based on interval analysis. Their method can find the correct thermodynamically stable solution to the vapor-liquid equilibrium problem with mathematical and computational certainty. Additionally, the tangent plane distance method 17,18) was used to test the predicted liquid and vapor phase compositions for global thermodynamic phase stability. 

One alternative apparently uncorrelated to the previous ones is to approximate the function using a tangent plane and use a method valid for linear functions with... 

To visualize the problem, consider the case vhth ny = 3. A sphere or an ellipsoid can be looked upon as a very large set of tangent planes (inequality constraints). The Simplex method needs a very large number of infinitesimal movements to achieve the solution and this is an example of an explosion in the number of vertices (called the tile effect) that needs to be analyzed in a three-dimensional space problem ... 

In the first-order reliability method, the limit-state surface in the standard normal space is replaced with the tangent plane at the point with minimum distance from the origin. The first-order estimate of the probability of failure, then, is... 

Optical devices are sometimes employed for the measurement of contact angle, wherein the operator must attempt to establish the tangent to the contact angle of a drop of mercury resting on a plane surface. This method has never proven sufficiently accurate because of its inherent subjectivity. Different experimenters will inevitably measure substantially different contact angles and even the same person will observe different angles on the same material on different occasions. 

The angular relationships involved in the transmission Laue method are illustrated in Fig. 8-13. Here a reference sphere is described about the crystal at C, the incident beam entering the sphere at / and the transmitted beam leaving at O. The film is placed tangent to the sphere at O, and its upper right-hand corner, viewed from the crystal, is cut off for identification of its position during the x-ray exposure. The beam reflected by the lattice plane shown strikes the film at R, and the normal to this plane intersects the sphere at P. 

Before analyzing Eq. (11), we shall briefly Illustrate the methods used and results for the corresponding deterministic system. In Eq. (11) (with 0=0), at the bifurcation point i.e. n = 0, one pair of eigenvalues is on the imaginary axis and their eigenvectors span a plane that is tangent at (u,v) = (0,0) to an invariant surface ... 

The value of reaction is zero. This can be evaluated, for e.g., using the finite element method. The arc shown in Figure 4 is not stressed by bending moment. Force F evokes normal stress in the direction of the tangent to the centre line of the shell plate of thickness t and height dh, where the centre line has the shape of a circle lying in the plane perpendicular to the axis x. For reliable operation force F must be less than or equal to the plate resistance R ... 

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