Dimensional exponent

Koziol and Mackowiak (55a) found the Kister and Haas correlation to give good agreement with experimental data. They developed a new dimensionless correlation [thus overcoming the need for a dimensional exponent in Eq. (6.27)] for spray regime entrainment at very low liquid rates (0.1-1.5 gpm/in). Their correlation, however, postulates that entrainment rises with tower diameter at the same steep rate at which it rises with hole diameter. This postulate conflicts with the industiy s experience that entrainment does not increase upon tower diameter scale-up. 

The formula nevertheless reflect the essential physics in nonadiabatic transition. The exponential dependence on parameters reflects the non-perturbative nature of this problem. The non-dimensional exponent hv Fi-F2 " hich represents adiabaticity, ensures the correct limiting behavior (the adiabatic limit v —> 0 and its opposite v — 00 correspond to P = 0 and P = 1, respectively). One also sees the correct scale in this problem for example, whether the passage velocity v is fast or not should be measured in unit V / h Fi — F2 ). Qualitative accuracy of this formula is known to be rather robust see for example, Ref. [445], in which they compare exact result with LZ formula as well as their proposed (surface hopping) method. These virtues are favorable for making rough estimate as well as constructing new guiding principle of one s study see, for example, Ref. [404] to see that the LZ estimate is still conceptually important even in the state-of-art studies. 

A non-dimensional 77 parameter results when the sum of the dimensional exponents is zero for the base dimensions included in the equation system the exponents ei,62,63, , therefore, must satisly the following linear and homogeneous equation system... 

For minor tasks with a few P parameters it is possible in a simple manner to determine the targeted non-dimensional 77 parameters by solving the equation system (m) the known dimensional exponents (a,/ ,7, ) are introduced into the expression (1), whereupon the targeted exponents (ei,e2,e3 ) are determined. 

Solution. To determine the non-dimensional 71 parameters, the homogeneous equation system according to eqn. (m) for the seven parameters is drawn up. The dimensional exponents (a, / , ) of this equation system can be directly seen from the dimensions of the P parameters, which are... 

The individual values of the exponents are detennined by the symmetry of the Hamiltonian and the dimensionality of the system. 

Kim H K and Chan M H W 1984 Experimental determination of a two-dimensional liquid-vapor critical exponent Phys. Rev. Lett. 53 170-3... 

Figure C2.10.1. Potential dependence of the scattering intensity of tire (1,0) reflection measured in situ from Ag (100)/0.05 M NaBr after a background correction (dots). The solid line represents tire fit of tire experimental data witli a two dimensional Ising model witli a critical exponent of 1/8. Model stmctures derived from tire experiments are depicted in tire insets for potentials below (left) and above (right) tire critical potential (from [15]).

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

Those exponents which we have discussed expUcitly are identified by equation number in Table 4.3. Other tabulated results are readily rationalized from these. For example, according to Eq. (4.24) for disk (two-dimensional) growth on contact from simultaneous nucleations, the Avrami exponent is 2. If the dimensionality of the growth is increased to spherical (three dimensional), the exponent becomes 3. If, on top of this, the mechanism is controlled by diffusion, the... 

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

An appropriate set of iadependent reference dimensions may be chosen so that the dimensions of each of the variables iavolved ia a physical phenomenon can be expressed ia terms of these reference dimensions. In order to utilize the algebraic approach to dimensional analysis, it is convenient to display the dimensions of the variables by a matrix. The matrix is referred to as the dimensional matrix of the variables and is denoted by the symbol D. Each column of D represents a variable under consideration, and each tow of D represents a reference dimension. The /th tow andyth column element of D denotes the exponent of the reference dimension corresponding to the /th tow of D ia the dimensional formula of the variable corresponding to theyth column. As an iEustration, consider Newton s law of motion, which relates force E, mass Af, and acceleration by (eq. 2) ... 

In the example, the exponents of dimensions in the dimensional formula of the variable F are 1, 1 and —2, and hence the first column is (1,1, —2). Likewise, the second and third columns of D correspond to the exponents of dimensions in the dimensional formulas of the variables M and, respectively. 

As indicated earlier, the vaUdity of the method of dimensional analysis is based on the premise that any equation that correcdy describes a physical phenomenon must be dimensionally homogeneous. An equation is said to be dimensionally homogeneous if each term has the same exponents of dimensions. Such an equation is of course independent of the systems of units employed provided the units are compatible with the dimensional system of the equation. It is convenient to represent the exponents of dimensions of a variable by a column vector called dimensional vector represented by the column corresponding to the variable in the dimensional matrix. In equation 3, the dimensional vector of force F is [1,1, —2] where the prime denotes the matrix transpose. 

Since are dimensionless products having dimensional vectors equal to the zero vector, the exponents of the R j = 1, 2,. .., m) must add up to zero, giving (eq. 10) ... 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

Various proposed values for the constants can be found in the literature [8]. Despite double-layer model predictions [148,149] that exponents Jt and y are both unity, and a dimensional analysis model [204] giving x as 1.88 andy as 0.88, test work on a practical scale [202,203] has indicated that both exponents are approximately equal to 2. This implies that a is roughly independent of pipe diameter and that the ratio //3 s 4/jt s 1. 

Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the ir-theorem is tc = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number Re = w Upl/i. Thus, we may write ... 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

For example, the time average definition of the Lyapunov exponent for one-dimensional maps, A = lim v->oo (which is often difficult to calculate in prac-... 

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