Volume as a function of time

An analysis of the mechanism of crystallization can best be carried out at a constant temperature. A sample is rapidly cooled down from the melt to a temperature below Tm and kept constant at that level. The crystallization process can then be studied by measuring the volume as a function of time in a dilatometer. The result is a curve as shown in Figure 4.9. The volume approaches to an equilibrium value at which the maximum possible crystallization is reached, and the rate of crystallization can, for example, be expressed as l/to.5, in which fo.5 is the time in which half of the route is... 

Related Calculations. Filtrate volume as a function of time can be calculated by dividing each value of L in Fig. 14.7 by cL and multiplying by the filter area. 

As an extension of Exercise 15, consider the reversible, elementary, gas phase reaction of A and B to form C occurring at 300 K in a variable volume (constant pressure) reactor with an initial volume of 1.0 L. For a reactant charge to the reactor of 1.0 mol of A, 2.0 mol of B, and no C, find the equilibrium conversion of A. Plot the composition in the reactor and the reactor volume as a function of time. 

Integration of the Darcy equation to obtain flow rate as a function of the pressure drop across the cake Ap, and the thickness L, or the volume of solids per unit area Overall material balances involving slurry, filtrate, and cake and determination of pressure and filtrate volume as a function of time. 

Integration of Equation (22.2) to obtain filtrate volume as a function of time t requires ... 

Figure 8 shows the volume as a function of time for four overdriven single shock wave simulations in the [110] direction of a 25688 atom perfect Lennard-Jones face centered cubic crystal. Elastic compression is characterized by VjV 0.9 and plastic compression occurs for smaller volumes. As the shock speed decreases, the amount of time the molecular dynamics system spends in the elastically compressed state increases. This plot illustrates how the final thermodynamic state in the shock is a function of the simulation duration when slow chemical reactions or phase transitions occur. For example, on the 10-20 ps timescale, the 2.8 km/sec shock has an elastically compressed final state on the 100 ps timescale, this simulation has a plastically compressed final state. 

Semi batch reactor volume as a function of time... 

The above-mentioned techniques all have in common that they are of macroscopic scale, often enabling diagrams of separated volume as a function of time to be displayed. 

Eq.( 10.2-24) contains two factors. The first factor (the square bracket term in the RHS) is the capacity factor and it decreases with temperature, while the second factor is the fractional uptake and it increases with temperature due to the increase in the diffusivity. The net result of these two factors is that the rate of adsorption decreases with temperature as the decrease in the capacity with respect to temperature overcompensates the increase in the fractional uptake. This is due to the higher heat of adsorption (Q) than the activation energy for diffusion (E ). Figure 10.2-3 shows typical plots of the adsorbed amount per unit volume as a function of time for three values of temperatures, 273, 298 and 333 K. The parameters used in generating these plots are ... 

Figure 8.2 Volume as a function of time after memory effect thermal history. Curve 1 is a pure quench. Curves 2 to 4 correspond to quenches and annealing at 10, 15, and 25°C, respectively. (From Kovacs, A.J., Glass transition in amorphous polymers a phenomenological study, Adv. Polym. Sci., 3, 394,1963. With permission.)...

An interesting kinetic study deals with the solution-mediated phase transformation of COT and COD into the thermodynamically stable COM [50]. The experimental conditions were adjusted so that either COT or mixtures of COD and COM crystallized initially as confirmed by X-ray diffraction powder patterns. The systems were then aged in contact with the mother liquid, and the transformation of COT or COD into COM was followed by monitoring the total crystal volume as a function of time (by Coulter counter) and determining (by thermo-gravimetric analysis) the relative proportion of the crystal hydrates at fixed time intervals. In addition, supersaturation profiles (i.e., activity products) were determined by solution calcium analysis. In all cases the transformation was completed within approximately 80-100 h. 

The reactions produce nitrogen gas and the kinetics can be followed by measuring its volume at different times. Since the rate is relatively fast, it is convenient to record the volume as a function of time. 

The nucleation rate during shear induced crystaUization has been measured under the assumption that each morphological structure, such as spheruUtes or axialites, comes from one nucleus. Thus, by measuring the number of such structures per unit volume as a function of time the nucleation rate can be obtained. For example, by this measure there is a thirty-fold increase in the nucleation density in poly(e-caprolactone) with the application of a stress of500 Pa.(70) The nucleation density obtained by this method is not usually linear withtime.(64,68,73) However, there is a linear portion that allows for a representative nucleation rate to be obtained. Another example of the increase in nucleation rate with shear rate is illustrated in Fig. 12.18... 

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