Linear growth rate

FIG. 20-95 Batch dnim growth of limestone pellets hy layering with a size-independent linear growth rate [Capes, Chem. Eng., 45, CE78 (1967).]... 

The overall linear growth rate determined from these data is given by the following empirieal expression (Jones and Mydlarz, 1989)... 

First, on a purely physical basis, we determine when departures from a linear growth rate may be expected, that is when the crystal dimensions do not increase proportionally with time, but also depend on the size of the crystal (and maybe even other factors). Then we show how these limits relate to the possible values of i and g. 

Figure 17 Effect of branching on the secondary nucleation and linear growth rates from the work of Lambert and Phillips [58]. The effect of branching on the Regime l-ll transition can also be seen. Reprinted with permission from Lambert and Phillips [58]. Copyright 1994, American Chemical Society.

Figure 13 Top, plot of linear growth rates of polyfethylene adipate) spherulites as a function of crystallization temperature for indicated molecular weight fractions. Spherulites shown correspond to the indicated range of temperatures. (A) Crystallization at the lower temperature range (B) at intermediate temperatures (C) crystallization at high temperatures. Reproduced with permission from Ref. [216]. Copyright 1956,...

We define the linear growth rate Vg as the linear velocity of displacement of a crystal face relative to some fixed point in the crystal. vg may be known as a function of c and c , derived from the theory of transport control, and as a function of c and cs as well, derived from the theory of surface control. Then c may be eliminated by equating the two mathematical expressions... 

Methods used for the measurement of crystal growth rates are either a) direct measurement of the linear growth rate of a chosen crystal face or b) indirect estimation of an overall linear growth rate from mass deposition rates measured on individual crystals or on groups of freely suspended crystals 35,41,47,48). 

Because the rate of growth depends, in a complex way, on temperature, supersaturation, size, habit, system turbulence and so on, there is no simple was of expressing the rate of crystal growth, although, under carefully defined conditions, growth may be expressed as an overall mass deposition rate, RG (kg/m2 s), an overall linear growth rate, Gd(= Ad./At) (m/s) or as a mean linear velocity, // (= Ar/At) (m/s). Here d is some characteristic size of the crystal such as the equivalent aperture size, and r is the radius corresponding to the... 

In this case, the linear growth rate is usually a function of the mass and heat transfer conditions at the surface of the crystal. The linear growth rate may... 

Concentration at crystal surface in equilibrium with T, kg/kg. Slope of solution freezing curve at the freezing point, kg/(kg-h). linear growth rate, nun/h. 

NUCLEATION IN POLYMER CRYSTALLIZATION The nucleation part of the linear growth rate is... 

Various studies have confirmed that rhGH promotes increased linear growth rates in children suffering from pituitary dwarfism (classical growth hormone deficiency). Dosages are generally... 

The unit of the growth rate u in Equation 4-31 is mole per unit area per unit time. If growth rate in m/s (referred to as linear growth rate) is needed. Equation 4-31 must be multiplied by the molar volume of the crystal (VJ ... 

The total volume growth of crystals in a melt is the sum of the volume growth of all individual crystals. For a crystal nucleated at time x with linear growth rate u (which may depend on time), the radius and volume of crystal i at time t can be written as... 

Figure 16.5. Supersaturation behavior, (a) Schematic plot of the Gibbs energy of a solid solute and solvent mixture at a fixed temperature. The true equilibrium compositions are given by points b and e, the limits of metastability by the inflection points c and d. For a salt-water system, point d virtually coincides with the 100% salt point e, with water contents of the order of 10-6 mol fraction with common salts, (b) Effects of supersaturation and temperature on the linear growth rate of sucrose crystals [data of Smythe (1967) analyzed by Ohara and Reid, 1973],...

Vc = volume of holdup in the tank n = number of crystals per unit volume L = length of the crystal G = linear growth rate of the crystal t = time... 

Upon substituting for the linear growth rate G = dL/dt and rearranging,... 

The principal quantities related by these equations are tpm, d4>m/dx, L, Lpl, t, n°, and B°. Fixing a certain number of these will fix the remaining one. Size distribution data from a CSTC are analyzed in Example 16.6. In Example 16.7, the values of the predominant length Lpr and the linear growth rate G are fixed. From these values, the residence time and the cumulative and differential mass distributions are found. The effect of some variation in residence time also is found. The values of n° and B° were found, but they are ends in themselves. Another kind of condition is analyzed in Example 16.4. 

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

Adamski and Klimczyk analyzed cholesteryl pelargonate36) and caproate 37) liquid crystal to fully-ordered-crystal transitions over a temperature range of about 25 K. Again, the appearance of the fully ordered crystals was that of a spherulitic superstructure. The nucleation was time dependent, and the linear growth rate of the spherulites decreased with decreasing temperature by a factor 1/2 to 1/3, in contrast to the nonanoate and acetate. The Avrami exponent was close to 4 as judged from the measurement of the crystallized volume in the field of view under the microscope. 

The linear growth rate depends on the interface reaction and is limited by the value of ks and the availability of oxidant for the interface reaction. 

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