Growth rate equation

EMBRYO GROWTH RATE EQUATIONS FOR VARIOUS CASES... 

It Is Important to use highly accurate surface area values of a crystal particle In growth rate equations, when the equations Include a term of the surface area. In practice, however. It Is difficult to measure the surface area unless the crystal possesses a simple geometry (e.g., sphere, cube, octahedron, etc.), or unless the B.E.T. method of measuring surface area Is applied. Therefore, a characteristic diameter Is usually defined, and the area calculated from the diameter Is used. 

Details of the mathematical treatment of the fluxes can be found in reference 69. After the oxidant flux is related to oxide growth rate, equation 60 is derived... 

In order to avoid other complicated factors concealing the major problems being examined, the overall crystal-growth rate equation, Eq. (12.3), is assumed to be valid for the system under consideration and is used as the basis for the interpretation of data. 

This same equation can be written as a function of the specific growth rate (Equation 16). 

The proposal made by Megee et al. (1972) is valid whenever limiting substrates are consumed simultaneously and both are essential for growth. This means that growth is interrupted when complete consumption of any substrate occurs, even though the other substrate is in excess. Dunn et al. (1992) proposed the idea of enhancing substrates that are non-essential, but if available they can improve specific cell growth rate (Equation 20). 

This expression establishes a variation of the real yield factor as a function of specific growth rate (Equation 27). 

In terms of differential calculus, we can write this as a growth rate equation for motion of cation interstitials... 

Getting back to the expression for the growth rate (Equation 10-41), we specified that this applies to low undercoolings, because as the temperature is reduced die mobility of the chains decreases and their motion essentially freezes around the glass transition temperature, so crystallization must decrease dramatically as the T is approached The expression for the growth rate can accommodate this by including an additional transport term. We then get (Equation 10-42) ... 

For now, let s just focus on the secondary nucleation part of the equation for the growth rate (Equation 10-43) ... 

Garside, J. Janssen-Van Rosmalen, R. Bennema, P. Verification of crystal growth rate equations. J. Cryst. Growth 1975, 29, 353-366. 

Embryo Growth Rate Equations for a Simple Nth Order... 

The growth of the crystal particle in the solution occurs via two processes. The first is the transport of monomers from the bulk solution onto the crystal surface and the second is the reaction of monomers on the surface. In this section, we derive separate equations describing these mechanisms, and then combine them to obtain the general growth rate equation. 

This is a growth rate equation obtained by considering only the mass transport process (9). Although simphfied, this equation is a good approximation when Cs varies httie with r as is the case on the micrometer scale. The hyperbola shape of the graph in Figure 6.3 is derived from this equation (Problem 2). However, on the nanometer scale, Cs is very sensitive to r and Equation (6.11) is far from accurate. 

However, because kp and kj in this equation are also functions of r, we have to know these functions to complete the derivation of the growth rate equation. 

Table III. Fatigue Crack Growth Rate Equation Parameters for Various Metallurgical and Test Variables...

C = fatigue crack growth rate equation coefficient da/dN = fatigue crack growth rate E = Young s modulus FZ = fusion zone of weld HAZ = heat-aflfected zone of weld... 

Similar generalizations of the differential form of the specific growth rate equation have been derived by Kargi and Shuler (1979), combining the differential forms of the four aforementioned specific growth rate equations. The following expression was the result of this work ... 

Ice crystals are often carried or descend into regions where the relative humidity is below saturation with respect to ice and begin to evaporate common ice subsaturated regions are near the bases of thunderstorm anvils and at the base of cirrus clouds. The growth rate equations given by Eqs. (9) to (11) can be used to compute the rate... 

FIGURE 11 Variation of a term in the ice crystal growth rate equation with temperature at two atmospheric pressures. 

Table 6.3 Growth rate equations and growth rate data for linear polyethyl-...

The nucleation and growth rate values obtained in this study were in substantial agreement with those of other researchers[53]. Further, nucleation and growth rate equations formulated in this MSMPR experiment correlate well with those obtained fiom an MSMPR study carried out by Al-Tarazi and co-workers [52], in terms of the values of the constant parameters and the correlation coefficient (R ). 

According to growth rate equations and considering that adsorption lowers the edge or surface energies and the size of the critical two-dlmen-... 

Microgravity Thin Film Growth Rate Equation... 

This differential equation was integrated using the Runge-Kutta technique with initial conditions /=0 at r=0. This returned a set of data points for r, / that were then curve-fitted to produce a microgravity growth rate equation, I = -2.60 10 + 5.01 10" t + 6.56 10", shown graphically in Figure 4. 

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