The scaling relations

One way of tackling the problem is to build a model bed in which the fluidization quality of the proposed plant can be simulated and studied. Only the fluidization characteristics need be considered, so that the model may be operated without the heat transfer and chemical reaction processes required of the envisaged commercial unit it may therefore be operated under ambient conditions of temperature and pressure (or perhaps under somewhat elevated pressure) and so be constructed cheaply, perhaps using transparent material through which the behaviour may be directly observed. The particles, fluid and operating conditions must be chosen so as to ensure equivalence of the cold model to the final plant it is the scaling relations that provide the criteria for making these choices. 

The v-direction component of the Navier-Stokes equation for momentum conservation in a Newtonian fluid of constant density and viscosity is 

To render eqn (13.1) dimensionless, it is first necessary to select convenient reference levels for all the variables x, y, z, t, and p. The choice is quite arbitrary. For the distance variables (x, y and z) the reference level L would typically consist of a key equipment dimension - a tube or tank diameter, or the length of a submerged object, etc. for velocity an average value or an entering volumetric flux V and for the remaining variables t and p) appropriate combinations of the other reference level can be constructed for example, LjV for t, and pfV for p. 

Many physical systems may be constructed and operated in a manner that results in Re and Fr having the same numerical values such systems are referred to as being dynamically similar. If, in addition, the dimensionless boundary conditions are the same, which is usually the case if the systems are geometrically similar (that is to say, one represents a scale model of the other), then the flow behaviour of matched systems, expressed in terms of the dimensional variables, will be identical. This has provided the basis for countless cold-modelling studies, firmly establishing the procedure at the forefront of experimental process research. 

We start with the one-dimensional, two-phase formulation reported in Chapter 8 eqns (8.21)-(8.24), together with the constitutive relations, eqns (8.4) and (8.18). The primary reference levels may be chosen with regard to particle characteristics dp for distance and m, for velocity. On this basis, all the variables may be related to their dimensionless counterparts  

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In Fig. 21(a) we plot the variation of R with increasing system density Cobs 3.nd, for comparison, also give the respective change for a system of moving medium (dynamic host matrix) of equal density. This result is in good agreement with recent predictions [89]. If one defines an effective Flory exponent from the scaling relation Rg oc it is then evident from... 

Dendrites can grow at constant speed at arbitrarily small undercooling A, but usually a non-zero value of the anisotropy e is required. The growth pattern evolving from a nucleus acquires a star-shaped envelope surrounding a well-defined backbone. The distances between the corners of the envelope increase with time. For small undercooling we can use the scaling relation for the motion of the corners as for free dendrites [103-106] with tip... 

Colby et al. [35] proposed an interesting experimental approach to measure the static exponents. They noticed that it is hard to accurately measure the chemical extent of reaction, p, and thus eliminated this variable (more precisely the distance from the gel point p — pc ) from the scaling relations. For example combining Eqs. 2-5 and 2-6 yields the following relation between the weight average molecular weight, Mw, and the characteristic radius, Rchar ... 

Similar relations between different scaling exponents were also developed by Stauffer [37] by combining two of the scaling relations at a time to eliminate Ip - Pel-... 

Precise knowledge of the critical point is not required to determine k by this method because the scaling relation holds over a finite range of p at intermediate frequency. The exponent k has been evaluated for each of the experiments of Scanlan and Winter [122]. Within the limits of experimental error, the experiments indicate that k takes on a universal value. The average value from 30 experiments on the PDMS system with various stoichiometry, chain length, and concentration is k = 0.214 + 0.017. Exponent k has a value of about 0.2 for all the systems which we have studied so far. Colby et al. [38] reported a value of 0.24 for their polyester system. It seems to be insensitive to molecular detail. We expect the dynamic critical exponent k to be related to the other critical exponents. The frequency range of the above observations has to be explored further. 

Figure 35. The scaling relations (4)—(7) do not hold in the intermediate regime of the phase separation. The crossover between early and intermediate regime occurs when the order parameter saturates inside the domains (the order parameter is nonconserved).

The power of the scaling relation is the same as the instantaneous quench, except for a time-lag proportional to the cube of the quench duration r. 

The scaling relations for the brush height and the crossover boundaries between the various regimes constitute the simplest approach towards charged brushes. We have already pointed out a few limitations of the presented results, which have to do with nonlinear stretching and finite-volume effects. 

In order to estimate the degree of branching, we need information to establish a baseline concerning the behavior of linear PVAc in teinns of the scaling relations R = k M, R, h] = M , etc., which are essentially available in the... 

Fio. 9. Experimental measurements of CAST soot hydrodynamic resistance factor as a function of Peclet number and aggregate mobility diameter. The continuous lines are plotted using the scaling relation form Eqs. (8) and (9). 

If you have a successful gradient separation and want to transfer it from column 1 to column 2, whose dimensions are different, the scaling relations are ... 

Fig. 2.51 Effect of reciprocating shear (strain amplitude, k = 200%) on the ODT of an /pep = 0.55 PEP-PEE diblock (Koppi etal. 1993). Here y denotes the shear rate.The equilibrium order-disorder transition (, A) and disordered state stability limit (A.O) are shown. The upper curve is a fit to the scaling relation Tom y2- The lower curve represents the. scaling rs(A) A-i,3Todt> where A = y/y, with y an adjustable, parameter. Points given by and O were obtained at fixed temperature by varying y, while those represented by A and A were determined by varying the temperature at fixed y.

As mentioned previously, the most relevant to polymer science seems to be the product homogeneous kernel Kitj=(ij)a>, (the homogeneity degree A=2co). According to van Dongen and Ernst [69], exponents in the scaling relation given by Eq. (135) are... 

Once the scaling relation of Eq. (39) is known, the molar mass distribution can, at least in principle, be obtained from a Laplace inversion of the multi-exponential decay function as defined in Eq. (40). At this point, the differences between PCS and TDFRS stem mainly from the different statistical weights and from the uniform noise level in heterodyne TDFRS, which does not suffer from the diverging baseline noise of homodyne PCS caused by the square root in Eq. (38). 

Fig. 15. Rates for the two components as a function of exposure time. The solid lines have been calculated from the scaling relation D = 2.76 x 1(T4 cmV1 x M0 525...

The two modes could be recovered for all values of xp. Despite the exposure time dependent average rate (T) in Fig. 13, the two rates and F2 are almost independent of Xp, and there is a good agreement between the measured values and the ones calculated from the scaling relation, as shown in Fig. 15. 

Bimodal distribution Once the rate distribution P(I) and the scaling relation in Eq. (39) are known, the molar mass distribution c(M), expressed in weight fractions, is obtained for the three basic types of experiment discussed ... 

P(r) can be transformed into a distribution of the particle size as defined by the hydrodynamic radius Rh. But only for TDFRS, and not for PCS, a particle size distribution in terms of weight fractions can be obtained without any prior knowledge of the fractal dimension of the polymer molecule or colloid, which is expressed by the scaling relation of Eq. (39). This can be seen from the following simple arguments ... 

The best results are attained within the template molecules approach. In order to arrive at approximations to the scaling functions we select a homologous series of molecules, G,. For this series, the reduced topological invariants are functions of only the index i. We formally invert the functions k(i) to obtain direct relations between the reduced invariants. One can say that the homologous series forms a template for the scaling relations. 

Finally, we should come back to Eq. (2) in the introduction. We learn that the scaling relations are a unique property of the Huckel Hamiltonian. Even the simplest extension, such as introduction of the variable P approximation, invalidates the scaling. This is certainly not a good news from the point of view of the practical applications. Still, the author believes that the appealing simplicity of the Huckel Hamiltonian makes it worthwhile to investigate, even if just for the pure fun of mathematical adventure. 

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