Three dimensionless parameters

The quantitative solution to the problem is given in section 11.3. The effectiveness factor T)P (< 1) which expresses the extent to which the promoting ion is fully utilized (qP=l) depends on three dimensionless parameters n, J and P n is the dimensionless dipole moment of the promoting ion, J is a dimensionless current and P, a promotional Thiele modulus, is proportional to the film thickness, L. 

Equations 8.5-34 and -35 are nonlinearly coupled through T, since kA depends exponentially on T. The equations cannot therefore be treated independently, and there is no exact analytical solution for cA(r) and T(r). A numerical or approximate analytical solution results in tj expressed in terms of three dimensionless parameters ... 

Deactivation of the Mediator Deactivation of the mediator is a commonly encountered event in the practice of homogeneous catalysis. Among the various ways of deactivating the mediator, the version sketched in Scheme 2.10 is particularly important in view of its application to the determination of the redox characteristics of transient free radicals (see Section 2.7.2).14 The current-potential responses are governed by three dimensionless parameters, 2ei = /F)(ke Cjl/v), which measures the effect of the rate-determining... 

In this case the overpotential and concentration changes across the electrode structure are related by the three dimensionless parameters, ju, s and a. 

In this case, the potential drop across the electrode structure is related by the three dimensionless parameters n, a and /. The dispersion modulus 5 related to the concentration has disappeared. The effectiveness factor will be... 

Typical cascade impactors consist of a series of nozzle plates, each followed by an impaction plate each set of nozzle plate plus impaction plate is termed a stage. The sizing characteristics of an inertial impactor stage are determined by the efficiency with which the stage collects particles of various sizes. Collection efficiency is a function of three dimensionless parameters the inertial parameter (Stokes number, Stk), the ratio of the jet-to-plate spacing to the jet width, and the jet Reynolds number. The most important of these is the inertial parameter, which is defined by Equation 2) as the ratio of the stopping distance to some characteristic dimension of the impaction stage (10), typically the radius of the nozzle or jet (Dj). 

Seven independent, dimension-specific values for filled screws were reduced to just three dimensionless parameters for throughput, pressure, and power (Fig. 2.22). 

Generally, only powders that form hard compacts under an applied pressure without exhibiting any tendency to cap or chip can be considered as readily compactible. The compactibility of pharmaceutical powders can be characterized by its tensile strength and indentation hardness, which can be used to determine three dimensionless parameters strain index, bonding index, and brittle fracture index. 

The description of the kinetics is simplified considerably by the fact that often the three rate constants, kij, k. , and kiv, are all large, allowing for the application of the steady-state approximation for the concentration of B, that is, —d[B]/dt = 0 [124], The system is conveniently discussed by introduction of three dimensionless parameters (1) the kinetic parameter X, Eq. (49) (2) the competition parameter a, Eq. (50) and (3) the concentration excess factor y, Eq. (51). In addition, it is convenient to introduce the catalytic efficiency CAT, Eq. (52), where ip,cat is the peak current during LSV observed for the catalytic system and ip rev is the peak current observed for P in the absence of A [see Fig. 18(a)]. The product ip,revK then is the maximum possible catalytic current. 

According to Wade et ah [46], the dimensionless band profile depends on three dimensionless parameters, fcg, 7 and bC fi. This last parameter characterizes the degree of column overloading. Equation 14.60 is easily rewritten as ... 

Let us return to the nonisothermal effectiveness factor. Weisz and Hicks solved Eqs. (11-46) and (11-72) numerically to obtain the concentration profile within the pellet. Then r was obtained from Eq. (11-51), which is not limited to isothermal conditions, provided is evaluated at the surface temperature. The results expressed 17 as a function of three dimensionless parameters ... 

The potential has three dimensionless parameters dy/do, d/do, and Ur/Ua where do = 1 and Ua= 1 have been chosen as units of length and energy, respectively. This potential generates three critical points in metastable region with respect to a solid phase. The algorithm of excluded volume calculation... 

There are three dimensionless parameters in addition to e. These are a parameter that Oron el al have called the gravity number, ... 

Analyze the stability for this problem. You should begin by nondimensionalizing. You should find that there are three dimensionless parameters,... 

Dimensional analysis of the problem for this reactor configuration pulls out three dimensionless parameters, the Peclet number, Pe, d1 and defined as ... 

The concentration history appears to be a function of three dimensionless parameters, a modified Thiele modulus, y, the mass Biot number, gj, and the dimensionless feed concentration, /3. The set of non-linear equations is uncoupled by introducing an effectiveness factor, r, and numerically solved. In order to reduce computer time, the effectiveness factor has been conveniently expressed as a weighted sum of its value for the zero and first order reaction rate. Different regime conditions are depicted in terms of a dimensionless parameter,... 

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

The three dimensionless parameters will be symbolized and7t3and may be formed in several different ways. Initially, a core group of r variables must be chosen which will appear in each of the pi groups and, among them, contain all of the fundamental dimensions. One way to choose a core is to exclude from it those variables whose effect one wishes to isolate. In the present problem, it would be desirable to have the mass-transfer coefficient in only one dimensionless group hence it will not be in the core. Let us arbitrarily exclude the fluid velocity and viscosity from the core. The core group now consists of DAB, D, and p, which include M, L, and t among them. 

When discussing the morphology it is useful to use the microrheology as a guide. At low stresses in a steady uniform shear flow, the deformation can be expressed by means of three dimensionless parameters the viscosity ratio, the capillarity number, and the reduced time, respectively ... 

Several mathematical models that attempt to describe FT have been reported (jj ). One such model developed by Noble and coworkers j]) combines the physical constants mentioned above into three dimensionless parameters ... 

It is possible to determine the ideal ablation factor by using the results of numerical solution of equations (13.73) and (13.74). It depends on three dimensionless parameters h, which describes the relation between electric and hydrodynamic forces acting on the drop Re, which characterizes the flow structure and d, which includes the mesh electrode parameters. The dependences Ki on these three parameters are shown in Fig. 13.23. 

A detailed data analysis has indicated that the characteristic lengths must be the same if the two dense-gas flows are to be treated similarly. That is to say, the three dimensionless parameters for two dense-gas flows with different scales cannot be the same in both flows. The similarity fails for dense-gas flows. 

The driven-oscillator model reveals several interesting points concerning the behavior of thermal-accommodation coefficients. We first note that (3.26) predicts that the accommodation coefficient is a universal function of three dimensionless parameters, the mass ratio, y, the reduced well depth, d, and the adiabaticity parameter, This latter quantity is the ratio of the duration of the collision to the oscillator period. Abrupt, impulsive collisions correspond to c values near zero, while long lasting, languid collisions correspond to large values. For small values we obtain from (3.26)... 

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