Dimensionless form, reduction

To be effective, the reduction to dimensionless form must be complete. Going halfway will neither add to the understanding of the physical situation nor make the equations any more tractable than they are to begin with.7... 

Cast the problem in dimensionless form. The two approaches to this have been discussed in detail (see The Reduction of Equations to Dimensionless Form and An Alternative Method of Reduction in Chapter 2). Do not stop half way with only some of the variables dimensionless. 

In problems like this, it is helpful to express the equation in dimensionless form (at present, all the terms in (I) have the dimensions of force.) The advantage of a dimensionless formulation is that we know how to define small—it means much less than 1. Furthermore, nondimensionalizing the equation reduces the number of parameters by lumping them together into dimensionless groups. This reduction always simplifies the analysis. For an excellent introduction to dimensional analysis, see Lin and Segel (1988). 

The vdW eos is reducible to a dimensionless form when Equation (4.150) and Equation (4.151) are substituted into Equation (4.149). The reduction gives... 

The equation addresses the molar volume of a segment of the polymer v, the subscript m indicating a segmental molar quantity. The parameters of the equation are segmental parameters thus is the excluded volume of a segment q = bJ(Av is the density of the fluid expressed in a reduced dimensionless form, also called the packing fraction and is the attractive-pressure parameter of the segment. The eccentricity parameter of the rotators is e, equal to 1.078 for the model rotator. Results of data reduction indicate the attractive parameter to vary weakly and linearly with temperature. 

A dramatic reduction in dimensionality is often possible by converting a design equation from dimensioned to dimensionless form. Equation 1.62 contains the dependent variable a and the independent variable z. The process begins by selecting characteristic values for these variables. By characteristic value we mean some known parameter that has the same dimensions as the variable and that characterizes the system. Eor a PER, the variables are concentration and length. A characteristic value for concentration is flin and a characteristic value for length is L. These are used to define the dimensionless variables a = ala-m and zIL. The governing equation for a first-order reaction in an ideal PER becomes... 

Fig. lb shows a series of Te deposition transients at constant potentials. The transients are characterised by an initial increase in current followed by a drop at longer time. These features are consistent with 3D nucleation also followed by the diffusion limited growth. The nucleation mechanism of Te deposition on n-Si was determined from the analysis of current-time transients. For this purpose the transients were plotted in the dimensionless form by normalizing two variables i and r with respect to the maximum current imax and the time zmax at which the maximum current is observed [2]. The theoretical plots for progressive and instantaneous nucleation and experimental plot for Te(IV) reduction at E = -0.375 V are given in the inset of Fig. lb. The corresponding experimental deposition transient is in a... 

The analysis of a beam on an elastic foundation is governed by exactly the same equation as the pile under lateral loading. The analysis of this problem requires the beam equation for the link between lateral loading and lateral deflection of an elastic beam - this is another topic for reinforcement through duplication. The beam equation is a fourth order ordinary differential equation so that a certain amount of mathematical confidence is required for its solution. These are again problems which lend themselves to dimensionless analysis - and, indeed, it is through reduction of the governing equations to their dimensionless form that the appreciation of the importance of relative stiffnesses of soil and structure can be obtained. 

In above two equations, rrij is an integer constant which takes values of -1, +1, and 0 for species R(z-i) Os and inert electrolyte species respectively if the reduction current is considered positive p refers to the thickness of the compact EDL Fq refers to the radius of electrode y is the ratio between the standard rate constant of ET reaction and the mass transport coefficient of the electroactive species. It can be seen that the current density, which is given in a dimensionless form through normalization with the limiting diffusion current density (i, and the electrostatic potential distribution appear simultaneously in the two equations. Equation 2.2 could be approximated to the PB equation at low current density, while Equation 2.3 would reduce to Eq. 2.4, which is the diffusion-corrected Butler-Volmer equation and has been used to perform voltammetric analysis in conventional electrochemistry, as exp(-Zj/ rcp/F)=1, that is, electrostatic potentials in CDL are close to zero. These conditions are approximately satisfied in large electrode systems, suggesting that the voltammetric behaviour and the EDL structure can be treated separately at large electrode interface ... 

Note that there isn t anything more fundamental about one form compared to the others. Each has the same number of dimensionless groups which are made up of independent parameters which can be set by the bed design and operation and the choice of particles. However, when the number of dimensionless groups is simplified by omitting some phenomena, the reduction in number of groups could be influenced by the form chosen. 

The dimensionless model equations for an electrochemical reduction reaction in a PBE are in a form of a set of partial ... 

With the substrate biased at a potential slightly more positive than E° of A/B couple, B is oxidized to form A for both DISP1 and ECE mechanisms. However, in the latter case the reduction of C also occurs at the substrate. The numerical solution of corresponding diffusion problems (see Ref. [85] for problem formulations) yielded several families of working curves shown in Fig. 12 (DISP1 pathway) and Fig. 13 (ECE pathway). In both cases, the tip and the substrate currents are functions of the dimensionless kinetic parameter, K = ka2/D. 

Available experimental data for various flow situations have been correlated in terms of the above dimensionless variables and the results fitted by empirical equations or simply presented in graphical form. Despite the reduction in the number of significant variables achieved by the introduction of the dimensionless products, a considerable amount of experimental work has to be carried out in order to arrive at a useable correlation for most geometries. For this reason it is usually worthwhile to attempt to carry out some form of analytical or numerical solution of the problem, even if the solution is a very simplified one, because this solution may indicate the general form of the correlation equation or, at least, indicate where the major emphasis should be placed in the experimental program. 

This revolution is the application of the methods of asymptotic analysis to problems in transport phenomena. Perhaps more important than the detailed solutions enabled by asymptotic analysis is the emphasis that it places on dimensional analysis and the development of qualitative physical understanding (based upon the m. thematical structure) of the fundamental basis for correlations between dependent and independent dimensionless groups [cf. 7-10]. One major simplification is the essential reduction in detailed geometric considerations, which determine the magnitude of numerical coefficients in these correlations, but not the form of the correlations. Unlike previous advances in theoretical fluid mechanics and transport phenomena, in these developments chemical engineers have played a leading role. 

The advantage of introducing dimensionless variables has already been shown in section 1.1.4. The dimensionless numbers obtained in that section provide a clear and concise representation of the physical relationships, due to the significant reduction in the influencing variables. The dimensionless variables for thermal conduction are easy to find because the differential equations and boundary conditions are given in an explicit form. 

Two methods have proved themselves in the reproduction of heat transfer measurements. One starts from empirical correlations for the pure substances. These correlations normally contain dimensionless numbers, that now have to be formed with the properties of the binary mixture. The reduction in heat transfer because of inhibited bubble growth caused by diffusion is taken into account by the introduction of an extra term. This type of equation has been presented by... 

The values E°, pe° and are different forms to express equilibrium constants of individual oxidation-reduction reactions. The first one is measured in volts of electric voltage, and the rest of them are dimensionless values. As a rule, as equilibrium constants of oxidation-reduction... 

Equation (2) has the same form no matter whether the quantities are dimensional or dimensionless, provided that used for reducing the parameters aj Pu RuUl = Pi/pi, and t = R u while g, = Qj/V, and q, = qslipifti). Note that in Refs. 12 and 13 the source power per unit density of the medium to be heated (in J/kg/s) was assumed constant, which, as the heating increases, leads to a reduction of heat consumption because the density reduces. In this research the computation was performed with a constant (J/m /s) and a similar computation is currently performed with the source power per unit mass of medium specified. The assumption that the flow is symmetrical makes it possible to use only one half of the body in computations (6 = 0... 180 deg). The cylindrical or spherical geometry of the body is specified by the parameter t = 0 or 1, respectively. In the results obtained for a sphere, t = 1. 

In dispersive mixing, the dispersed phase undergoes size reduction and forms multiple smaller domains as a result of stresses acting at the interfaces [53, 54]. In this case, the viscous forces generated in the continuous phase overcome the restoring forces due to interfacial tension [55, 56]. Hence, the extent of deformation is determined by the ratio of the viscous and interfadal stresses, the relative magnitude of which can be expressed as a dimensionless number, the so-called capillary number [57, 58]. 

Here is the dimensionless reduction of the production width. To regain perfect flow distribution, the volumetric flow rate Q should have the following form ... 

---