Mean nucleation time

With growing overpotentials, the nucleation rate J increases much faster than the propagation rate [see Eqs. (13) or (20) and (29)] so that the mean nucleation time r becomes much shorter than the propagation time Tp, i.e., JS) < The deposition of each layer proceeds with formation of a... 

Figure 7.9 EJfect of mean residence time on mean crystal size parameter i = hjg, the relative nucleation to growth order index)...

As we can see, from the results in Fig. 2, a metastable hadronic star can have a mean-life time many orders of magnitude larger than the age of the universe Tuniv = (13.7 0.2) x 109 yr = (4.32 0.06) x 1017 s (Spergel et al. 2003). As the star accretes a small amount of mass (of the order of a few per cent of the mass of the sun), the consequential increase of the central pressure lead to a huge reduction of the nucleation time and, as a result, to a dramatic reduction of the HS mean-life time. 

To summarize, in the present scenario pure hadronic stars having a central pressure larger than the static transition pressure for the formation of the Q -phase are metastable to the decay (conversion) to a more compact stellar configuration in which deconfined quark matter is present (i. e., HyS or SS). These metastable HS have a mean-life time which is related to the nucleation time to form the first critical-size drop of deconfined matter in their interior (the actual mean-life time of the HS will depend on the mass accretion or on the spin-down rate which modifies the nucleation time via an explicit time dependence of the stellar central pressure). We define as critical mass Mcr of the metastable HS, the value of the gravitational mass for which the nucleation time is equal to one year Mcr = Miis t = lyr). Pure hadronic stars with Mh > Mcr are very unlikely to be observed. Mcr plays the role of an effective maximum mass for the hadronic branch of compact stars. While the Oppenheimer-Volkov maximum mass Mhs,max (Oppenheimer Volkov 1939) is determined by the overall stiffness of the EOS for hadronic matter, the value of Mcr will depend in addition on the bulk properties of the EOS for quark matter and on the properties at the interface between the confined and deconfined phases of matter (e.g., the surface tension a). 

The size distribution is solely determined by the mean residence time and the rates of nucleation and growth. In general, the total number of particles present in the system can be calculated by the following integral ... 

The last column is of the summation wjL , at corresponding values of crystal length L. The volumetric shape factor is 0 = 0.866, the density is 1.5 g/mL, and the mean residence time was 2.0 hr. The linear growth rate G and the nucleation rate B will be found. 

If the nucleation time is much larger then the propagation time, each nucleus has sufficient time to spread over the surface before the next nucleus is formed. Under these conditions, each layer is formed by one nucleus only. The current i corresponding to the development and decay of the peripheral edge of each layer is not stable, and fluctuates with the nucleation and spreading of the layers. Fig. 5.9. The mean current density is given by the nucleation fi-equency l/ nuc = /A and is independent of the propagation rate v ... 

See Section 9.3.1 for the mathematics of size distributions, i is a characteristic time scale it would be the mean residence time in the crystallizer mentioned. A plot of log f(r) versus r is a straight line of slope — 1/Lc t. This generally implies a wide size distribution. The number average radius r10 is simply given by Lcx. In a steady-state crystallizer, the average size is thus independent of nucleation rate. 

Mean retention time defined by VjQ Zero size population density Nucleation rate Concentration of a solution Rate constant from Eq. (4.28)... 

Since the mean residence time, r, is known for a particular set of operating conditions, the growth rate, G, can be determined from the slope of the semilog plot of the CSD, and the nucleation rate, B, can be obtained from the intercept, n°(recall B = n G). 

The distribution of residence times (Equations (8.1) and (8.2)) which was discussed earlier represents another difference between batch or PFTs and CSTRs. Figure 8.1 shows residence time distributions for a single CSTR and differrait numbers of equal-size CSTRs connected in series. All curves are for the same total residence time r = 1.0, where r = n in which n is the number of CSTRs and the mean residence time of each one. If all particles are nucleated in the first CSTR of a CSTR series the curves shown in Figure 8.1 would also represent the age distribution of the particles in the process effluent stream. The particle... 

Figures 6 and 7 present the solids yield for canola and sunflower proteins during isoelectric precipitations, respectively. Sunflower protein yields from the flow-type precipitators increased with increases in mean residence times. This means that slower processes of particle growth by aggregation and diffusion follow an initial rapid nucleation process. About two minutes are required before the final yield is reached according to the results obtained from the tubular precipitator operating in the laminar flow regime and the batch precipitator. For canola proteins, mns in an MSMPR precipitator showed little changes in the yield with the mean residence time. This is because the mean residence times were longer (between 1.5 and 7.5 min) allowing the reaction to go to completion.

According to Mersmann [16], the supersaturation in cooling crystallization is usually small, i.e., relative supersaturation, a, is less than 0.1 and the relative solubility expressed by the ratio of solubility (kg/m ) and crystal density is usually higher than 0.01. As a result, primary nucleation does not take place and the nuclei are formed only as attrition fragments. If a coarse product is desired, the attrition rate should be low and the crystal growth rate at the maximum allowable level with respect to crystal purity, all of which depends mainly on the mean residence time of the slurry. 

For the plant constructors, the benefit of the population balance is mainly derived from the transparency of the interplay of the kinetic variables of nucleation rate and crystal growth rate, the mean retention time, the fines dissolving, and the classification of product crystals during withdrawal with respect to the resulting mean particle size of the crystallization process. Here, reference is made to the publications, such as that of Toyokura and Sakai [12]. 

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