Nondimensional Equations

Diffusion in liquids can be slow, and this makes the problems hard to solve, as shown below. The best way to learn how important diffusion is for any problem is to calculate the Peclet number. Equation (11.3) is made nondimensional using the techniques illustrated in Eq. (10.42). Let c (x, y, f ) be the nondimensional concentration, and c = Csc (x, y, f ). Equation (11.3) for a constant diffusivity becomes  

You may already have picked the time standard in the flow problem. If you chose = x /u, the coefficients of the first and second terms are the same, and the coefficient 

When there is diffusion or heat conduction but no convection, typical boundary conditions are  

The boundary conditions permitted in FEMLAB are slightly more general than this see Table 11.1 for diffusion and conduction and Table 11.2 for diffusion and conduction with convection. 

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It is important to include all of the relevant physical effects in the equations to be nondimensionalized. This can be difficult because there isn t always consensus about which effects are important. Moreover, there is controversy over how to properly represent these effects in equation form. For our purposes the question is Have all of the important parameters been included in the nondimensional equations Pragmatically, the success to date of the scaling experiments using the formulation as presented adds confidence to the use of these simplifications which will be employed. Also, a limited number of tests have verified the omission of parameters specifically related to several phenomena. 

This problem can be cast in nondimensional form using the same variables as the impulsively started flow problem, Section 4.6. The nondimensional equation is... 

As in previous analyses, nondimensionalizing the equations leads to results that have general applicability. Here, however, it is not a simple task to identify a length scale and a velocity scale based on an inspection of the geometry and the boundary velocities. In fact, it was an insightful contribution of Hiemenz to identify the length and velocity scales that are required to develop a parameter-free system of nondimensional equations. They are... 

With these definitions it is a straightforward task to write the nondimensional equations as... 

The nondimensional equations emerge from these simple variable transformations as... 

The density and kinematic viscosity that appear in the normalization are taken to be constant at the far-field conditions. The nondimensional equations, assuming constant viscosity, are... 

Notice that a choice was made to scale the circumferential velocity using the inlet velocity U. The rotation-rate scale is used in the boundary-condition specification. With these variables the nondimensional equations are... 

Developing the specific functional form and the constants will require several runs of the simulation. Consider the range 0.1 < Sc < 10. The spin Reynolds number may not appear explicitly in the nondimensional equations. Thus it will have to be re-introduced to develop the correlation. 

We assume that the stellar magnetic field is dipolar ( m d), and has axial symmetry everywhere. We use cylindrical coordinates (w,, z) centered on the neutron star and aligned with the stellar rotation axis. This configuration is sketched in Figure 1. We obtain the nondimensionalized equations which construct a complete set for the dynamics of reservoir ring, as following,... 

The parametric approach, which is not strictly needed for a single Gray-Scott reaction, works very well for an arbitrary number of parallel reactions and for continuous mixtures. Figure 16 shows a case of two parallel reactions for which an isola and a mushroom coexist. Because the notions of continuous mixtures and reactions will be treated in Chapter 8, G H and in the group of papers listed in the Index of Subjects in Publications under the heading Continuous mixtures, we can be very brief and start with the nondimensional equations. Let x be the index of the mixture whose species are /4(x). The steady-state concentration of the material with index in (x, x + dx) is V(x)dx, the feed concentration a(x)dx and the conversion U(x) = 1 - V/(x)/a(x), the last being defined only for values of x for which a(x) is not zero. B, the autocatalytic agent, forms itself as an undifferentiated product whose concentration is W. The rate of the first reaction, and hence p,(x), depends on the... 

Thus we conclude that the friction coefficient for a given geometry can be expressed in terms of the Reynolds number Re and the dimensionless space variable x alone (instead of being expressed in terms of x, L, V, p, and /x). This is a very significant finding, and shows the value of nondimensionalized equations. 

Referring to the nondimensional equation of convective diffusion (3.3), it is of interest to examine the conditions under which the diffusion term, on the one hand, or convection, on the other, is the controlling mode of transport. The Peclet number, dUfD, for flow around a cylinder of diameter r/ is a measure of the relative importance of (he two term.s. For Pe 1, transport by llte flow can be neglected, and the deposition rate can be determined approximately by solving the equation of diffusion in a non flowing fluid with appropriate boundary conditions (Carslaw and Jaeger, 1959 Crank, 1975). 

The more interesting and important situation is that in which Pe 1. In this case, the nondimensionalized equation (3-196) suggests that the heat transport process is dominated by convection so that a first approximation is... 

Although we have not yet specified i and there is nothing from the nondimensionalized equation, (3-198), that provides a basis at this point to determine what it should be, what we may anticipate from a qualitative point of view is that t is likely to be much larger than a for large Pe so that the parameter... 

Rather than starting with the original dimensional equations and searching for an appropriate characteristic length scale for the near-wall region, we can determine the correct form by simply rescaling the previously nondimensionalized equation, (4-21). To do this, let us introduce a new independent spatial variable,... 

Similarly, the nondimensionalized equations for the other velocity components can be written in the form appropriate for small values of a/R ... 

Of course, the solution (4-181) is only the first approximation in the asymptotic series (4 175). In writing (4-177), we neglected certain smaller terms in the nondimensionalized equation, (4-170), because they were small compared with the terms that we kept. To obtain the governing equation for the second term in the boundary-layer region, we formally substitute the expansion, (4-175), into the governing equation, (4-170) ... 

We can obtain the governing equations for the leading-order term in any asymptotic expansion by letting the small parameter go to zero in the full nondimensionalized equations [in this case by letting - 0(and 27 e 0)in(5 61), (5-65), and (5-66)]. The governing... 

Before the details are carried out, however, it is useful to revisit the scaling that led to the nondimensionalized equation, (6-107), because we can show that a slight modification allows us to eliminate the coefficients ( 1 and A. This involves just two steps. First, we specify the length scale lc- An appropriate and convenient choice is the wavelength of the disturbance separating stable and unstable solutions in the linear stability analysis. Going back to (6-103), we see that the critical point (i.e., the point where s = 0) occurs when... 

The characteristic scales that have been used to produce these nondimensionalized equations are... 

The governing nondimensionalized equations and boundary conditions for this problem are... 

We shall see later that the solutions of the constant-surface-temperature and constant-heat-flux problems can often be quite different. However, the formal solutions of these two problems in the asymptotic limit Pe 1 are actually very similar. The problem for the constant-heat-flux case is still singular, and the scaling and nondimensionalized equations for the inner and outer regimes are identical to what we obtained already for the constant-temperature problem. The details of showing this, plus the analysis of solutions, are left to the reader by means of Problem 9-7 at the end of this chapter. We simply note here that the leading-order approximation in the inner region is still the pure conduction solution... 

Important Dimensionless Groups. The average Nusselt number for steady-state heat transfer from the body shown in Fig. 4.1a depends on the dimensionless groups that arise in the nondimensionalized equations of motion and their boundary conditions [78]. With Tw and T constant, the only dimensionless groups that appear in the boundary conditions are those associated with the body shape. Provided the simplified equations (Eqs. 4.5 1.7) are valid, the only other dimensionless groups are the Rayleigh and Prandtl numbers. Thus, for a given body shape,... 

Finally, we take these two results and combine them to derive the nondimensionalized form for the rate of change of B. We first show that the overall nondimensionalized equation is parallel in form to the fully dimensional equations and then make the appropriate substitutions and... 

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. 

What are the definitions of y, x, and that yield the above form of nondimensional equations ... 

Let us again consider the same problem of mass transfer through a membrane of Example 12.5 but this time we account for the fact that diffusivity is a function of the intramembrane concentration. The nondimensional equation can be written as... 

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