Adler equation

Function q is 27r-periodic, and in the simplest case q -) = sin( ) Eq. (13.3) is called the Adler equation. One can easily see that on the plane of the parameters of the external forcing (cu, e) there exist a region eqmin < oo loq < eqmax, where Eq. (13.3) has a stable stationary solution. This solution corresponds to the conditions of phase locking (the phase 0 just follows the phase of the force, i.e. (f> = uit + constant) and frequency entrainment (the observed frequency of the oscillator Cl = (0) exactly coincides with the forcing frequency tu brackets () denote time averaging). 

In the formulation of the Barner-Adler equation of state,3 the authors placed primary emphasis on obtaining a generalized set of constants for the region of temperature and pressure most frequently encountered by the process... 

For hydrocarbon gas mixtures (including the light gases such as N2, C02, and H2S), Reid et al.55 recommend the Soave modification of the Redlich-Kwong equation, the Lee-Kesler equation, or the Lee-Erbar-Edmister equation. If the mixture contains hydrogen, the Soave equation should be used. For gas mixtures containing nonhydrocarbons, Reid et al.55 have recommended the Lee-Kesler correlation, the Yen-Alexander correlation, the Barner-Adler equation of state, or the Sugie-Lu equation of stale. 

Equations 2, 3, and 4 summarize the method proposed by the Germans for preparing parathion (compound E-605). Schrader 13) has reported that thiophosphoryl chloride was synthesized from phosphorus trichloride and sulfur by heating at 130° in a lead-lined autoclave. Woodstock and Adler 14) carried out a similar reaction at 150° to 160° C. 

In all cases the reaction products are mixtures of ethyl polyphosphates, and, on the basis of elementary analysis, they approximate the empirical formulas given in the above equations. In Equations 3 and 5 the product has been arbitrarily called hexaethyl tetraphosphate, which may contain 8 to 20% of the active tetraethyl pyrophosphate. In Equations 4 and 6 the products have been called technical tetraethyl pyrophosphate, which may contain up to 40% of pure tetraethyl pyrophosphate. Hexaethyl tetra-phosphate has also been made from phosphoric anhydride and diethyl ether by a process recently patented by Adler (1). 

A code has been written to enable the velocities of surface waves in multilayered anisotropic materials, at any orientation and propagation and including piezoelectric effects, to be calculated on a personal computer (Adler et al. 1990). The principle of the calculation is a matrix approach, somewhat along the lines of 10.2 but, because of the additional variables and boundary conditions, and because the wave velocities themselves are being found, it amounts to solving a first-order eight-dimensional vector-matrix equation. A... 

Calculate true digestibility (TD) by using the following equation (Adler-Nissen, 1984 and 1986) ... 

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

Another fractal structure of interest is considered by Adler (1986). A three-dimensional fractal suspension may be constructed from a modified Menger sponge, as shown in Fig. 7(b). A scaling argument permitted calculating the effective viscosity of such a suspension however, this viscosity should be compared with numerical results for the solution of Stokes equations in such a geometry before this rheological result is accepted unequivocally. 

This raises some complicated and important issues. How do the structures we set up to learn "what we value" affect the analyses that the "we" in that equation produce How does what emerges from analyses of "what we value" reflect broader conceptions of best practices that we might hold, and what are the relationships among these qualities that we value and broader ones Adler-Kassner and Estrem... 

The natural collision coordinates for spatial motion introduced by Marcus were made the starting point for a development (Wyatt, 1972) of reactive collision equations for AB + C -> A + BC. The treatment may be regarded as an extension of previous work (Curtis and Adler, 1952) for inelastic collisions. The kinetic energy operator was simplified by introducing two approximations appropriate for linear intermediates, and the interaction was chosen of form... 

To solve distillation problems involving multicomponent mixtures, vapor-liquid equilibrium data and enthalpy data are needed. The methods used to obtain these data may be classified as follows (1) the use of a single equation of state and (2) the use of multiple equations of state and/or correlations for the prediction of the liquid and vapor parts of the K values and the enthalpies. This classification was suggested by Adler et al.2 in an excellent paper on the industrial uses of equations of state. Although the first approach, the use of a single equation of state, is the more desirable, many industrial problems are encountered in which this approach is too inaccurate and the second approach is used. 

Case 3 T > Tci In this case, the liquid phase does not exist at any pressure. Lewis and Kay40 suggested the extrapolation of a plot of log f tPi or log P( vs. /Tr while Souders et al.61 used experimental K values and heats of solution data in their extrapolation procedure. Adler et al.2 suggested the use of the Chao-Seader (or Grayson-Streed) equations or back-calculation from experimental vapor-liquid equilibrium data. 

S. B. Adler, C. F. Spencer, H. Ozkardesh, and C.-M. Kuo Industrial Uses of Equations of State A State-of-the-Art-Review, Phase Equilibria and Fluid Properties in the Chemical Industry, T. S. Storvick and S. I. Sandler (eds.), ACS Symposium Series 60, American Chemical Society, Washington, D.C., 1977. 

For design calculations involving Refrigerant 500, a minimum-boiling azeotrope of 39.4 mol % of 1,1-difluoroethane and 60.4 mol % of difluorodichloromethane, reliable real gas thermodynamic properties are required which have been calculated from 0.2 to 100 bar and from 220 to 540 K using the recently proposed Boublik-Adler-Chen-Kreglewski equation of state and the PVT data reported in the literature. This equation of state has 21 universal constants and only five adjustable constants which have been calculated for R-500 from the PVT data, saturated vapor pressure and liquid density, and the critical constants. In order to calculate the absolute values of the real gas properties, the reference state properties, which are also reported here, are required. All properties are given in SI units. 

This variety of reactor types reflects the complex interaction between chemical reaction and mass and heat transfer. In spite of this complexity, in principle every reactor can be described by the fundamental balance equations for the preservation of mass, energy, and impetus. These form a system of five partial differential equations (PDFs), coupled through temperature, concentation, and three rate vectors [Damkohler 1936, Platzer 1996, Adler 2000]. 

One choice is to choose real values of s for both the numerical transformation of the data as well as for the s-domain fitting. The advantages are again decreased emphasis of the tail. Also, the complete model equation in the transform domain is utilized in the fitting, not just s = 0 as in ordinary moments, or a particular value(s) as in weighted moments. The disadvantages are that the numerical Laplace transformation of the data must be performed, including the question of the best choices of numerical values of s, often chosen around s = 1/t, and truncation of the tail. For this technique, see Williams, Adler, and Zolner [102],... 

The gravitational field is described in general relativity by the set of equations (4.11). The right hand side depends on the description of matter in the system of interest and the corresponding solution consists of finding that form of the fundamental tensor that satisfies (4.11). The first successful solution of cosmological interest, obtained by Schwarzschild, is text-book material, described in detail by Adler et al. (1965). The time-independent spherically symmetric line element is of particular importance as a model of the basic one-body problem of classical astronomy. This element, of the form ... 

Topologically there is no mystery. Any large-scale rotation must appear to be centred at the observer and indeed, the identity of the Einstein and Godel stress tensors shows that the held equations have two basically different solutions for the same T, one rotating and the other static. The Godel solution therefore is consistent with the general theory of relativity, but not with Mach s principle (Adler et ah, 1965, p.377). 

Another attempt to divide the solutions of (31) into two distinct classes was made by Adler and Enig. The equations (31) are autonomous and the time variable r can be eliminated by division to produce... 

Analysis of the free energy in the continuous Gaussian chain model (see Equation 3.1-189), the complete equation for molecular chciin diffusion (cf. Equation 3.1-190), and the diffusion coefficient (cf. Equation 3.1-192) (Adler and Freed, 1979) has led to equations of dynamic scaling that agree with de Gennes results. 

However, the above equation is only valid for limited range of curvature ratio. A major drawback of all three equations is as a the critical Reynolds number does not approach 2,100 for straight tube. Srinivasan et al. [Ill] used the experimental data of critical Reynolds number of Adler [86], Kubair and Kuloor [92], Storrow [112], Taylor [106] and White [85] and proposed... 

The case of general integrability with an integral of fourth degree is announced in the recent paper by Adler and van Moerbeke [215]. The theorem formulated in [215] asserts that this is the only possible (besides quadratic families) case of algebraic (see above) integrability of the Euler equations on so(4). 

---