The Limits of Scale

Consider the scaleup of a stirred tank reactor. It is common practice to scale using geometric similarity so that the larger reactor will have the same shape as the small reactor. Suppose the volume scaleup factor is given by Eq. (1). 

1) The symbols used in this chapter are listed at the end of the text, under Notation . 

Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 2005 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN 3-527-31014-2 

The lesson from this is that even a simple phenomenon - such blending of two fluids - requires longer and longer times as the size of the process increases. Eventually, the operation becomes unfeasible. 

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In this type of bioreactor, there is a compartment where cells remain attached to a surface or immobilized on or inside a biocompatible bed. Culture medium has to be pumped through this compartment for cells to have access to nutrients and dissolved oxygen. The main disadvantages of these bioreactors, developed for the cultivation of adherent cells that require a surface for proliferation, are the impossibility of obtaining homogeneous samples of the cell population and the limitations of scale-up. 

When accurate data are not available in the literature or when past experience does not give an adequate design basis, pilot-plant tests may be necessary in order to design effective plant equipment. The results of these tests must be scaled up to the plant capacity. A chemical engineer, therefore, should be acquainted with the limitations of scale-up methods and should know how to select the essential design variables. 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

R. de la Torre and M. Earre, Neurotoxicity of MDMA (ecstasy) the limitations of scaling from animals to humans. Trends Pharmacol Set, 2004,25, 505-508. 

In homopolymers all tire constituents (monomers) are identical, and hence tire interactions between tire monomers and between tire monomers and tire solvent have the same functional fonn. To describe tire shapes of a homopolymer (in the limit of large molecular weight) it is sufficient to model tire chain as a sequence of connected beads. Such a model can be used to describe tire shapes tliat a chain can adopt in various solvent conditions. A measure of shape is tire dimension of tire chain as a function of the degree of polymerization, N. If N is large tlien tire precise chemical details do not affect tire way tire size scales witli N [10]. In such a description a homopolymer is characterized in tenns of a single parameter tliat essentially characterizes tire effective interaction between tire beads, which is obtained by integrating over tire solvent coordinates. 

The previous application — in accord with most MD studies — illustrates the urgent need to further push the limits of MD simulations set by todays computer technology in order to bridge time scale gaps between theory and either experiments or biochemical processes. The latter often involve conformational motions of proteins, which typically occur at the microsecond to millisecond range. Prominent examples for functionally relevant conformatiotial motions... 

Industrial scale polymer forming operations are usually based on the combination of various types of individual processes. Therefore in the computer-aided design of these operations a section-by-section approach can be adopted, in which each section of a larger process is modelled separately. An important requirement in this approach is the imposition of realistic boundary conditions at the limits of the sub-sections of a complicated process. The division of a complex operation into simpler sections should therefore be based on a systematic procedure that can provide the necessary boundary conditions at the limits of its sub-processes. A rational method for the identification of the subprocesses of common types of polymer forming operations is described by Tadmor and Gogos (1979). 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

Synthetic Heme. Synthetic compounds that biad or chelate O2 have been produced. These compounds are commercially attractive because manufacture and Hcensure might be developed as a dmg, rather than as a biological product. It has been shown that synthetic hemes can be used to transfuse animals (53). Although synthetic O2 carriers would avoid the limited hemoglobin supply problem, the synthetic procedures are very tedious, and the possibihty of scale up seems remote. 

The present author was worried about the lack of knowledge concerning the quality of the kinetic models used in the industry. A model is by definition a small, scaled-down imitation of the real thing. (Men should remember tliis when their mothers-in-law call them model husbands.) In the industry all we require from a kinetic model is that it describe the chemical rate adequately by using traditional mathematical forms (Airhenius law, power law expressions and combinations of these) within the limits of its applications. Neither should it rudely violate the known laws of science. 

When using dimensional analysis in computing or predicting performance based on tests performed on smaller-scale units, it is not physically possible to keep all parameters constant. The variation of the final results will depend on the scale-up factor and the difference in the fluid medium. It is important in any type of dimensionless study to understand the limit of the parameters and that the geometrical scale-up of similar parameters must remain constant. 

The above measurements all rely on force and displacement data to evaluate adhesion and mechanical properties. As mentioned in the introduction, a very useful piece of information to have about a nanoscale contact would be its area (or radius). Since the scale of the contacts is below the optical limit, the techniques available are somewhat limited. Electrical resistance has been used in early contact studies on clean metal surfaces [62], but is limited to conducting interfaces. Recently, Enachescu et al. [63] used conductance measurements to examine adhesion in an ideally hard contact (diamond vs. tungsten carbide). In the limit of contact size below the electronic mean free path, but above that of quantized conductance, the contact area scales linearly with contact conductance. They used these measurements to demonstrate that friction was proportional to contact area, and the area vs. load data were best-fit to a DMT model. 

While at high densities we observe perfect exponential scaling of p x), at lower dilute densities (with sufficiently long chains ) one observes results consistent with Eq. (16b). The insert in Fig. 5 shows that the MWD at dilute densities agrees with the additional power-law dependence p x) oc in the limit of small x, confirming the theoretical predictions [33,34]. 

The properties of the periodic surfaces studied in the previous sections do not depend on the discretization procedure in the hmit of small distance between the lattice points. Also, the symmetry of the lattice does not seem to influence the minimization, at least in the limit of large N and small h. In the computer simulations the quantities which vary on the scale larger than the lattice size should have a well-defined value for large N. However, in reality we work with a lattice of a finite size, usually small, and the lattice spacing is rather large. Therefore we find that typical simulations of the same model may give diffferent quantitative results although quahtatively one obtains the same results. Here we compare in detail two different discretization... 

The limitation of these instruments is that they only indicate overall corrosion rate. Their sensitivity is affected by deposition of corrosion products, mineral scales or accumulation of hydrocarbons. Corrosivity of a system can be measured only if the continuous component of the system is an electrolyte. 

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