Nondimensionalization dimensionless parameters

Once the model fluid and its pressure and temperature are chosen, which sets the gas density and viscosity, there is only one unique set of parameters for the model which gives similarity when using the full set of dimensionless parameters. The dependent variables, as nondimensionalized by Eq. (18), will be the same in the respective dimensionless time and spatial coordinates of the model as the commercial bed. The spatial variables are nondimensionalized by the bed diameter so that the dimensional and spatial coordinates of the model is proportional to the two-thirds power of the kinematic viscosity, as given by Eq. (69)... 

Gliksman s approach The result of the conversion of equations into nondimensional ones is a set of dimensionless parameters (Froude number, velocity, particle size, diameter ratios, etc.) that should be matched in both small and large systems. It is not necessary for the values of the parameters to be equal in each system. Instead, the dimensionless number ratios have to remain the same. To achieve this, the particle size and/or the particle density of the solids have to be changed appropriately in the small unit. It usually results in a smaller particle size in the small unit compared to the large one. 

Instead, the procedure of nondimensionalizing the governing DE and boundary conditions is first considered. For now, the motivation for this procedure is not considered, except to say that it will result in a reduction in the number of parameters that characterize the problem (and its solution) from four dimensional parameters (G, //, d, and U) to a single dimensionless parameter. To nondimensionalize, we define a dimensionless velocity,... 

It should be noted that these equations are completely characterized by the two dimensionless parameters e and f22/f2i. In particular, the Reynolds number, Re = ucic/v = a2i2 /v, does not appear in spite of the fact that it would have appeared in the full equations, (3—56)—(3—58), if these had been nondimensionalized in the same way. From a mathematical point of view, this is because the viscous terms turned out to be identically equal to zero in (3-56), whereas the inertia and pressure terms were zero in (3-57) - compare (3-66) and (3-67). Thus the form of the velocity and pressure fields in the Couette flow problems is completely independent of Re. In this sense, the Couette flow problem is very similar to a unidirectional flow. 

The precedent that we have followed in previous problems is to nondimensionalize the governing equations and boundary conditions in order to identify important dimensionless parameters. In this case, it is easy to scale the temperatures so that they vary between 0 and 1, e.g.,... 

We now seek a new nondimensionalization of the dimensional variable (z ) such that the viscous term remains in the problem as Re —> oo. To do this we introduce a rescaling of z by means of the dimensionless parameter Re, i.e.,... 

The reader may again be curious whether there are consequences of making the wrong choice for c, Uc, or Tc when there is more than one possibility available We will need to discuss this point in some detail, once we see how we intend to use the nondimensional-ized versions of our governing equations (and boundary conditions) in the development of asymptotic approximations. For now, we simply assume that the appropriate choices have been made. Then, for each additional dimensional scale that appears in a particular problem, we get one more dimensionless parameter, in addition to the two that will appear based on... 

To define a dimensionless surface concentration, use T = T / r0. The dimensionless parameters that should appear on nondimensionalization are... 

We are interested in the case in which p2 > P, where the system is potentially unstable. Although we have established the principle of nondimensionalizing as a way of identifying dimensionless parameters, in the present case it is not clear what choice to make for any of the characteristic variables, and so we proceed in this case with the dimensional equations and boundary conditions. 

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

If the nonparticipating gas convection is two-dimensional and laminar, and if the fluid is considered to be Boussinesq, appropriate nondimensionalization of the descriptive equations yields the following dimensionless parameters [194] ... 

Equation 12.49 is the basic nondimensional equation describing the mole fraction of A in a fixed-bed reactor containing an exponentially decaying catalyst as a function of position and time in terms of two dimensionless parameters, B" and A. The performance of this reactor can be best judged by solving the equation for the reactor exit, that is, for z = 1. The solution for a first-order reaction (m = 1) is given in Table 12.7 (Sadana and Doraiswamy, 1971). It is also possible to assume various other forms of catalyst decay. Solutions are included in the table for two other forms, one of them linear. 

When dealing with simple equations (as in the previous three models), the dimensional equations are solved without recourse to the process of nondimen-sionalisation. Now, we must deal with partial differential equations, and to simplify the notation during the analysis and also to deduce the proper dimensionless parameters, it is necessary to reduce the equations to nondimensional form. To achieve this, we introduce the following nondimensional variables and parameters ... 

With these definitions of variables and parameters, the nondimensional mass balance equations take the uncluttered form containing only two (dimensionless) parameters... 

Equation 9.45 is the basic nondimensional equation describing the mole fraction of A in a fixed-bed reactor containing an exponentially decaying catalyst as a function of position and time in terms of two dimensionless parameters j8 and X. The performance of this reactor can be best judged... 

To facilitate reference, the dimensionless parameters and functions are listed on the left hand side of the following table. Definitions and working equations are given on the right hand side of the table. Before using the following table, it is noted that based on the non random interaction solution given in Appendix A and the nondimensional parameters of frequency 0 - w/wi, interaction frequency ratio Oj- - wi/ws and nondimens ional time r - w t, the dimensionless complex frequency response for the dam acceleration... 

Before performing averaging, (6.5) must be transformed to the standard form [56]. To that end, some simplifications are necessary. In the following sections, first the equation of motion is simplified and then converted into a nondimensionalized form. Next, a small parameter, e, is introduced and the new dimensionless parameters are ordered to reach an approximate weakly nonlinear equation of motion accurate up to 0(e). 

If the particular extracting technique applied to a solution depends on the volatility of the solute between air and water, a parameter to predict this behavior is needed to avoid trial and error in the laboratory. The volatilization or escaping tendency (fugacity) of solute chemical X can be estimated by determining the gaseous, G, to liquid, L, distribution ratio, KD, also called the nondimensional, or dimensionless, Henry s law constant, If. 

Substituting these nondimensional numbers into eqs 11-18, and after some rearrangement, the general dimensionless representation of the problem is obtained as depicted in Table 1. These equations are valid not only for spherical pellet geometry, but also for the infinite cylinder and the infinite flat plate (slab). The dimensionless numbers x, , Bim and Bib must then be calculated on the basis of the respective characteristic length, i.e. the cylinder radius or the plate thickness. Moreover, the parameter b in eqs 32 and 33 is a factor depending on the pellet geometry. It is 2 for the sphere, 1 for the cylinder, and 0 for the flat plate. 

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 model (1) has four parameters R, K, A, and B. As usual, there are various ways to nondimensionalize the system. For example, both A and K have the same dimension as A, and so either N/A or N/K could serve as a dimensionless population level. It often takes some trial and error to find the best choice. In this case, our heuristic will be to scale the equation so that all the dimensionless groups are pushed into the logistic part of the dynamics, with none in the predation part. This turns out to ease the graphical analysis of the fixed points. 

To nondimensionalize (3-88) and the boundary conditions (3-89), it is also necessary to identify a characteristic time scale tc in order to define a dimensionless time 7 = t/tc. The characteristic time scale tc should be proportional to the time period over which the velocity profile evolves from its initial form, uz = 0, to the final steady state, (3-44). However, it is not immediately obvious how this time scale depends on the dimensional parameters of the problem. Let us leave tc undefined for the moment and write (3-88) and (3-89) in dimensionless form in terms of tc, that is,... 

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