Dimensionless Coordinates

Here we present a set of suitable coordinate transformations. This is by no means the only sensible set of transformations the characteristic reference value for each variable should be chosen appropriately according to the particulars of the system under investigation. Pirst we define the dimensionless concentration of chemical species j, Cj, to be 

If the diffusion coefficients are equal then = b = 1- Later we will examine the case where species diffuse at different rates, which is why we define d as a ratio of real diffusion coefficients rather than simply as 1. 

We define the dimensionless distance X in terms of the radius of the macrodisc electrode, e  

It is now necessary to transform Eq. (2.10) and the boundary conditions into this new dimensionless coordinate system. We begin by considering the left-hand side of Eq. (2.10), dc/dt. We may make the substitution, c = 

This might not seem particularly useful, but let us consider the side of Eq. (2.10) using the chain rule for the second derivative  

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In vertical downward flow as well as in upward and downward inclined flows, the flow patterns that can be observed are essentially similar to those described above, and the definitions used can be applied. Experimental data on flow patterns and the transition boundaries are usually mapped on a two dimensional plot. Two basic types of coordinates are generally used for this mapping - one that uses dimensional coordinates such as superficial velocities, mass superficial velocities, or momentum flux and another that uses dimensionless coordinates in which some kind of dimensionless groups are used as coordinates. The dimensional coordinates maps are inherently limited to the range of data and flow conditions under which the experiments were conducted. In spite of this limitation, it is widely used because of its simplicity and ease of use. Figure 24 provides an example of such a map. 

Step 2 Calculate the value of from Equation 11 (a) or 11(c) given below. This term is a dimensionless coordinate of the concentration Cl along the centerline of the flame. From the value of, calculate the downwind... 

Fig. 9. Direct band gap for [9,2] nanotube in vicinity of band gap. Wave number is dimensionless coordinate x, with onedimensional Brillouin zone for x defined —tt < x < tt.

Since the expression in Eq. (1-93), which must be used in Eq. (1-89), involves a nonintegral power of g, the evaluation of the % and x integrations is best carried out in a -dependent coordinate -system. We may form dimensionless coordinates G0,g0 in terms of the G,g system used in Eq. (1-93) (where, again, G is defined with respect to the mean flow velocity ) ... 

The computed him thickness distribution of the cross line at the contact center is shown in Fig. 23. The abscissa axis is the dimensionless coordinate in the flow direction whereas the ordinate axis is the dimensionless him thickness. It is very clear that the him thickness distribution is similar to that of EHL predictions. 

CiC22 C2Ci2 C2C11-C1C12 OC - 2 oc - 2 C11C22-C12 11 22- 12 Y is the dimensionless coordinate, y/h) and U is the dimensionless velocity, ulUf, where Uj, is the relative speed between the upper and the lower plates. 

Fig. 22 —An enlarged cross section view of the head-disk clearance, h is the height X is the dimensionless coordinate.

The solution of Eq. (2.7.25) as the gradient of the dimensionless concentration C with respect to the dimensionless coordinate Y (perpendicular to the phase boundary) for Y = 0 is then given by the relationship... 

An alternative to Lees-Edwards boundary conditions is the formalism put forth by Parrinello and Rahman for the simulation of solids under constant stress.52,53 They described the positions of particles by reduced, dimensionless coordinates ra, where the ra can take the value 0 < ra < 1 in the central image. Periodic images of a given particle are generated by adding or subtracting integers from the individual components of r. 

Figure 7. Potentials for lower and upper surfaces in Creutz-Taube ion calculated by method of Piepho, Krausz, and Schatz, as function of dimensionless coordinate, y, with J = 0.4 eV,

This difference in Q reveals that the oxygen acts as it were a negative fuel concentration in a given stoichiometric proportion, or vice versa. This result is, of course, a consequence of the choice of the coupling function and the assumption that the fuel and oxidizer approach each other in stoichiometric proportion. It is convenient to introduce dimensionless coordinates... 

The curve which describes the concentration-time function of tracer in the exit stream of any vessel in response to an idealized instantaneous or pulse tracer injection is called the C curve. Such an input is often called a delta-function input. As with the F curve, dimensionless coordinates are chosen. Concentrations are measured in terms of the initial concentration of injected tracer if evenly distributed throughout the... 

On the whole the similarity theory reduces the problem of the amount of nitric oxide formed in explosions to one series of experiments, which in principle is sufficient for determining the characteristic curve in dimensionless coordinates, NO/[NO] as a function of fcTO[NO]r. Figure 14 represents such a curve obtained from experiments with hydrogen mixtures with equal oxygen and nitrogen content in the explosion products at p0 = 200 mm and a volume of the vessel equal to 3 liters the quantity fcm[NO]r is plotted in the logarithmic scale. In plotting the curve we made use of the expression... 

Here the dimensionless normal coordinates of the tetragonal (e) vibrations are qu and qv those of the trigonal fa) vibrations are q, qn, and q. At this simple point several different nomenclatures exist. Hereafter, the (normal) vibrational coordinates Q relate to the dimensionless coordinates q by the formula q = Q(Mco/h)1/2, where M is the effective mass of the vibrator and co its frequency [88]. The force constant is K = a M. 

In all these equations we have used the following of notation for the dimensionless variables x=tkT1 is dimensionless time q- z/H, = zi/H are dimensionless coordinates 0f and Pf are the spatial distributions of the temperature and calorimetric degree of conversion. As a first approximation it is reasonable to accept that... 

Conductive problems described by Equation 13.19 can be solved algebraically or graphically using nomograms based on dimensionless coordinates, where the dimensionless time is given by the Fourier number ... 

The solution of this differential equation, when it exists, describes the temperature profile in the solid. In order to solve this equation, we must assume that the exothermal reaction taking place in the solid follows a zero-order rate law, that is, the reaction rate is independent of the conversion. Then the variables can be changed to dimensionless coordinates. Thus, generalizing the soluhons ... 

In this section we explore the second possibility to generate multidimensional PES, i.e. a Taylor expansion in terms of normal mode coordinate with respect to the geometry of the stable structure. Including terms up to fourth order we have (using dimensionless coordinates)... 

Figure 3.1 Kinetic curves of actor and inducer (1) and acceptor (2) consumption, and intermediate substance (3) and reaction product (4) accumulation in A + Bj + B2 system (dimensionless coordinates).

The change of the equilibrium position of the phonon mode dimensionless coordinate with wave vector k induced by the displacement of lattice particles in this case can be written as [8] ... 

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