The Steady-State Size Distribution

The data in [Wi] are in terms of cell volume, so for comparison purposes it is necessary to convert the distribution by length, (7.1), to one by volume. Let / (u) be the steady-state cell-volume distribution corresponding to the distribution (7.1). Then the number of individuals with cell volume in the range V to V2, where = / Ui Vi, is given by 

Like all models in science, the one treated in this chapter makes many unrealistic assumptions. These were pointed out at the end of Section 2. The most notable deficiencies are that the model inadequately reflects the cell division process and neglects the energy required for cell maintenance. It should be pointed out, however, that the main predictions of the simple (even less realistic) models of Chapters 1 and 2 survive intact in the more complex model treated in this chapter. Therefore, it is not unreasonable to expect that many of the predictions made on the basis of Theorem 4.1 will continue to hold for more realistic models. 

As noted in Section 2, Metz and Diekmann [MD, p. 237] describe a different size-structured model, one that reflects the cell-division process quite well. They assume that cell size x varies among the individual cells of the population, from a minimum value Xmin to a maximum value that is normalized to 1. A function b x) gives the per-unit time probability of a cell of size x dividing. Small cells are not allowed to divide (f (Ar) = 0, x a). A mother cell of size x is assumed to divide into two daughter cells, one of size px and one of size (1 -p)x, with probability d p), 0 / 1. Of course, d(p) = d l —p) and/o d(p) dp =. The unit of size x -whether length, area, or volume - is not specified in [MDj. This makes their assumption that the growth rate of a cell of size x is proportional to X (and to f(S)) subject to different interpretations. The reader is referred to [MD, p. 238] for the equations and hypotheses. Their model also can be reduced to the equations considered in Chapter 1. 

It would be of considerable interest to construct and analyze a model that treats growth and consumption as in this chapter (following [Cu2], i.e., proportional to surface area) and that treats cell division as in [MD]. It seems unlikely that this marriage of the two approaches would yield a model that can easily be reduced to the ordinary differential equation models of Chapter 1. 

On the other hand, competitive success is determined solely by having the smaller break-even concentration A,. Since A, decreases (and so the /th 

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A comparison of the measured mean relative size with the model in Eq. (101) is shown in Fig. 28. The material was monosize glass beads. The data fit the model quite well, with the exception of fine 0.038-mm powder. It is evident that the steady-state size distribution is a function of the liquid content, and consequently, as shown by Sherrington (S9), there is an optimal granulating liquid for maximum granulation efficiency, that is, percentage of the product-grade material. 

In Section 4, competition between two populations is analyzed. Again, the equations can be reduced to a system that can be directly compared to the systems derived in Chapters I and 2. Section 5 explores the evolution in time of the population average length, surface area, and volume in Section 6 we formulate the conservation principle, which played such a crucial role in earlier chapters. The steady-state size distribution of a population is determined in Section 7. Our findings are summarized in a discussion section, where a comparison is made between the conclusions derived from the size-structured model and the unstructured models considered in Chapters 1 and 2. 

The reason for osdllations in conversion and surface tension become dear whea one considers particle formation and growth phenomena. If a single CSTR is started empty or by adding initiator to a full vessel of inactive emulsion, a conversion overshoot occurs. The first free radicals generated are almost entirdy utilized to ibrm new particles. Since these partides do not grow rapidly to the steady-state size distribution, radical... 

Figure 11. Computer simulation of a population using 2000 unstructured cell models. There were 20 size classes with 100 cells per class in the simulation. The steady state size distributions are considerably more stable than the simulation with 200 cells CSee Figure 10).

Another notable contribution was made by Saeman (1956) was derived equations, based on a first-order growth law, for the form of the steady-state size distribution in mixed suspensions. The assumption of size-independent growth was also made and the cumulative mass-size distribution was expressed in the form... 

What are the mechanics of a textured liquid crystal How does the steady state size distribution of domains (or density of defects) change with shear in Region I, and what is the effect on transient and steady state rheology ... 

The resulting steady-state size distribution functions were compared with experiments, and it was demonstrated that size distributions were indeed self-similar, and the functionality deduced in the development of the model was observed. 

Data from [WiJ - particularly figures 3, 6, 8, 10, 18, 19, and 21 - leave no doubt that, at least for certain populations of algae, individual cell volume varies significantly during the course of experiments in the chemo-stat. These data also suggest that steady-state size distributions are reached which have remarkably stable shapes with respect to changes in the control parameters for the chemostat (flow rates, temperature, CO2). 

Figure 7.1. Eight steady-state size distributions observed under different experimental conditions (flow rate, temperature, CO2), scaled for equal means and areas. The mean cell size for each graph is indicated next to the graph. (From [Wi, fig. 19], Copyright 1971, Academic Press. Reproduced by permission.)...

Figure 5b presents the measured variation of the steady-state temperature distribution across the microchannel array imposed by the hot and cold reservoirs computed from an ensemble of sixty instantaneous snapshots of the temperature field across the array as reported in [7]. Note that variations of fluid temperature within each microchannel were not resolved, so the spatial resolution of these measurements is 100 pim (i. e., the size of the microchan-nels). In this figure, the filled circles represent the mean temperatures of the dye mixture in the microchannel array obtained by averaging over the sixty samples, while the predicted linear temperature variation across the device (solid line in Fig. 5b) is obtained by fitting a line through... 

Since the conservation equations are in terms of mass, it is useful to convert the stoichiometric coefficients to mass instead of moles. The condition of steady state means that all properties within the control volume are independent of time. Even if the spatial distribution of properties varies within the control volume, by the steady state condition the time derivative terms of Equations (3.13) and (3.18) are zero since the control volume is also not changing in size. Only when the resultant integral is independent of time can we ignore these time derivative terms. As a consequence, the conservation of mass becomes... 

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

The sfabilify of Pf particles during the 1.2 V hold has also been investigated. At 1.2 V and 80°C in 1 M H2SO4, up to 35% of the ECA was lost after 24 h. Transmission electron microscopy analysis of the tested catalysts found a growth in the Pt particle size distribution, suggesting that small Pt particles (-2 nm) are particularly susceptible to dissolution/agglomeration xmder steady-state voltage holds at 1.2 V. 

The population balances were solved In moments form to establish the steady state conditions. Then the full size distributions were evaluated for the particular seed size distributions of the feed. Due allowance was made for the different measures of size for the two species. 

Results obtained with alloys (alloying causes variations in the distribution of the ensembles according to size) and with metals of varying particle size (due to the geometry of the curved surfaces and due to the deposition of carbon, the same effects are expected to operate here as in alloys) have lead to the conclusion that the various 3C and 5C complexes might differ in the size of ensembles which are required for the formation or the steady-state binding of the complexes. The small and big ensembles are, in the following, schematically represented by one-site and two-site ensembles. 

A steady-state (normalized) distribution function is approached asymptotically as t —> oo. This steady-state distribution, illustrated in Fig. 15.5, is approached by all initial distributions. The most frequent particle size in the steady-state distribution is 1.13(i ) and there will be no particles larger than 1.5(i ), the cut-off size. 

Voorhees s experimental study of low-volume-fraction-solid liquid+solid Pb-Sn mixtures carried out under microgravity conditions during a space shuttle flight enabled a wider range of solid-phase volume fractions to be studied without significant influence of buoyancy (flotation and sedimentation) effects [13]. The rate of approach to the steady-state particle-size distribution in 0.1-0.2 volume-fraction... 

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