Characteristic value equation, definition

Chung and Wen (1968) and Wen and Fan (1975) have proposed a dimensionless equation using the dependency of the dispersion coefficient on the (particle) Reynolds number Re (Eq. 6.169) for fixed and expanded beds. It is an empirical correlation based on published experimental data and correlations from other authors that covers a wide range of Re. Owing to two different definitions of the Reynolds number, the actual appearance varies in the literature. Since the particle diameter dp, is the characteristic value of the packing, Eq. 6.168 based on the (particle) Peclet number Pe (Eq. 6.170) is used here ... 

As already mentioned, the value analysis allows to identify numerically the dynamics of relative signifieances for individual steps of reaction mechanism, which is very important. Previously the role of individual steps for the B-Zh reaction was identified, by the method of sensitivity analysis of output characteristics of the reaction, to variations in the values of the rate eonstants of these steps [25-28]. We believe the value identification of significances for individual steps in time will complement the information about them, because according to equation (3.3) the values, by definition, are somewhat different from the local sensitivities S. 

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

Substituting all the possible combinations of characteristics, i.e. values of p and q, info equation 1.10 gives rise to a number of differenf definitions of the mean size of a distribution. At minimum fluidization the drag force acting on a particle due to the flow of fluidizing gas over the particle is balanced by the net weight of fhe particle. The former is a function of surface area and the latter is proportional to particle volume. Consequently the surface-volume mean diameter, with p = 2 and = 3, is the most appropriate particle size to use in expressions for minimum fluidizing velocity. It is defined by equafion 1.11... 

The mean reaction time or reaction timescale (also called relaxation timescale relaxation denotes the return of a system to equilibrium) is another characteristic time for a reaction. Roughly, the mean reaction time is the time it takes for the concentration to change from the initial value to 1/e toward the final (equilibrium) value. The mean reaction time is often denoted as x (or Xr where subscript "r" stands for reaction). The rigorous definition of x is through the following equation (Scherer, 1986 Zhang, 1994) ... 

It can be shown that the ratio v3lvg is equal to the ratio of polymer packing densities coefficients in the amorphous and crystalline states, KJKC at Tg, because, by definition, Ka = NA V /va and Kc - NA Vi/yC)where vj is the Van der Waals volume of the chain repeat unit. The calculated values of (ATc)g correlate with the characteristic chain parameter a/o, the relationship between them being expressed by a linear equation... 

The stationary point of the system (5.2) is by a definition stable one, if all the roots of its characteristic equation (5.11) have the negative real parts. The Routh-Hurwitz criteria presented in Ref. [206] permits escaping the calculations of these roots to establish the simple relations between the coefficients ock, which allow to point out simple stability conditions. For instance, in the case of terpolymerization the positivity of both coefficients oq and oc2 is regarded to be a criteria of such stability, and as for four-component copolymerization the following non-equality a3 < oq < ot2 has also to be hold. At arbitrary number m of the components the positivity of all ak is regarded to be necessary (but not sufficient) stability condition. For the stability of the boundary SP of m-component system located inside the certain boundary 1-subsimplex of monomers Mk, M2,. .., M, the stability of the above SP in such subsimplex and negativity of all values of X, Xl+1,..., vm x (5.13) are needed. 

This helps indicate why groups of multiplicative type are important. But it should be said that solvability is definitely a necessary hypothesis. Let S for example be the group of all rotations of real 3-space. For g in S we have gtf — 1, so all complex eigenvalues of g have absolute value 1. The characteristic equation of g has odd degree and hence has at least one real root. Since det( ) = 1, it is easy to see that 1 is an eigenvalue. In other words, each rotation leaves a line fixed, and thus it is simply a rotation in the plane perpendicular to that axis (Euler s theorem). Each such rotation is clearly separable. But obviously the group is not commutative (and not solvable). Finally, since U is nilpotent, we have the following result. 

Strictly speaking, the equation K =S is an extension of Boltzmann s theory, in so far as we have ascribed a definite value to the entropy constant. According to Boltzmann, the probabihty contains an undetermined factor, which cannot be evaluated without the introduction of new hypotheses. Boltzmann and Clausius suppose that the entropy may assume any positive or negative value, and that the change in entropy alone can be determined by experiment. Of late, however, Planck, in connection with Nemst s heat theorem, has stated the hypothesis that the entropy has always a finite positive value, which is characteristic of the chemical behaviour of the substance. The probabihty must then always be greater than unity, since its logarithm is a positive quantity. The thermodynamical probabihty is therefore proportional to, but not identical with, the mathematical probabihty, which is always a proper fraction. The definition of the quantity w on p. 15 satisfies these conditions, but so far it has not been shown that this definition is sufficient under all circumstances to enable us to calculate the entropy. 

Here the superscripts (BC) and (LJ) refer to the Buckingham-Corner and Lennard-Jones potentials, respectively is the distance between the m-th atom of the molecule and the s-th atom of the surface. The coefficients C, D, B, Q depend on microscopic characteristics, such as static polarizability, ionization potential, etc. For a detailed discussion of atom-atom interaction potentials, the reader is referred to [12]. The subscripts M and S denote atomic species of the adsorbed molecule and the adsorbent surface note that in summations like (11), it is implied that for any value of m (or s) there is the definite M (or S) value which corresponds to a particular atomic species. Therefore, the internal summation in equation (11) can be performed to give the sum of atomic contributions ... 

Finally, control equations of the form proposed by Stolwijk have two characteristics the use of a set point temperature for each layer tends to hold temperatures within a rather narrow range of values, and the model contains a very large number of adjustable parameters. Thus, one should be able to fit a limited amount of data rather well even if the model is invalid. Devising definitive experiments and adapting modem techniques for parameter estimation to this problem present a real challenge for engineers and scientists. 

The value of E in relation to like the value of gj, indicates the evaporative nature of the surface so E < E q or E/Ee, < 1 reflects surface dryness or stomatal closure as well as the balance of energy exchange between the atmosphere and the underlying surface. By definition, E > Ef, can be caused only by advection. As implied above with respect to the partitioning of temperature, this may also result from the entrainment of dry air from above the convective boundary layer that develops daily over the earth surface. To further illustrate the relation between E and Ef, in terms of surface characteristics, it is helpful to write the Penman-Monteith equation (Monteith and Unsworth, 1990),... 

This semi-empirical equation has become famous in virtue of the many remarkable properties which it possesses. Only certain of these concern us at the moment. The first set is connected with the possibilities of solution. The differential equation only possesses finite, single-valued solutions for certain quite definite values of the energy, E. These values, known as characteristic or proper values (German, Eigenwerte), specify the possible quantum states of the system. 

As has been stated andillustrated, the Schrodinger equation possesses physically acceptable solutions (continuous, finite, and single-valued) only for certain definite values of the energy—the characteristic or proper values (Eigenwerte). These define the quantum states of the system. The way in which vibrational, rotational, and translational quantization follow has already been considered (p. 126). 

Equation (1.A.3) is not a well-defined mathematical function, which must have a definite value at every point x where it is defined. Dirac called it an improper function, which has the characteristic that when it occurs as a factor in an integrand the integral has a well-defined value. ... 

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