Time scale factor

Fitzgerald et al. (1984) measured pressure fluctuations in an atmospheric fluidized bed combustor and a quarter-scale cold model. The full set of scaling parameters was matched between the beds. The autocorrelation function of the pressure fluctuations was similar for the two beds but not within the 95% confidence levels they had anticipated. The amplitude of the autocorrelation function for the hot combustor was significantly lower than that for the cold model. Also, the experimentally determined time-scaling factor differed from the theoretical value by 24%. They suggested that the differences could be due to electrostatic effects. Particle sphericity and size distribution were not discussed failure to match these could also have influenced the hydrodynamic similarity of the two beds. Bed pressure fluctuations were measured using a single pressure point which, as discussed previously, may not accurately represent the local hydrodynamics within the bed. Similar results were... 

Scale Up of Process. The scale up of fluidized bed coating processes has received little attention in the literature. Current practices in the pharmaceutical industry are reviewed by Mehta (1988). The basic approach described by Mehta (1988) is to scale the airflow and liquid spray rates based on the cross-sectional area for gas flow. This seems reasonable except for the fact that in the scaling of the equipment, the height of the bed increases with increasing batch size. For this reason, a time scale factor is also required. 

If the circle does contain the origin of the polar chart, the distance between origin of the chart and the center of your circle is the actual residual unbalance present on the rotor correction plane. Measure the distance in units of scale you choose in Step 1 and multiply this number by the scale factor determined in Step 6. Distance in units of scale between origin and center of the circle times scale factor equals actual residual unbalance. 

Time scaling may be used as long as the time scaling factor is the same for aU formulations. Different time scales for each fonnulation indicate absence of an IVIVC. 

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

Is there evidence that a different time-scaling factor should be used for the AEGL-2 ... 

In Fig. 9, Type 1 is the ideal case, a one to one relationship between (Y = bx) in vitro and in vivo data. Type 2 [(y = hx — a), lower line] represents the case where a dissolution exists in vitro without any absorption in vivo. This case is physiologically possible and could possibly be explained by a time lag for example. The FDA suggests in this case a time scaling on condition that the time scaling factor is the same for all formulations. Different time scales for each formulation indicate the absence of an IVIVC. Another possibility would be to modify the dissolution technique. Type 3 (y = bx + a, upper line) represents an impossible case where the drug is dissolved and absorbed in vivo before any in vitro dissolution. This case implies an inappropriate dissolution technique that must be modified. ... 

Sequence Pulse cycles (phase) Cycle time Scale factor for Iff... 

Because k = g(0.50)//o5o (at a constant temperature, 7), it follows that l/to so (times corrected by subtraction of Q is proportional to k at temperature, T. The reciprocals of the time-scaling factors, t so, (or at any other reference value of a) can thus be used as measures of the rate coefficient, and hence the temperature dependence of /q 50 can be used to calculate the activation energy, E, of reaction without the necessity of identifying the kinetic model, g(flr) [14] ... 

The parameter p is called the time scaling factor for the Laguerre functions. This parameter plays an important role in their practical application and will be discussed in detail in Section 2.3. (Note The set of Laguerre functions presented in Equations (2.4) differs by a factor of —1 for even values of i when compared with the set of Laguerre functions presented by Lee (1960). However, this does not affect the orthonormal properties of these functions.)... 

We can compute the coefficients of the Laguerre model using Equations (2.14) for a positive time scaling factor p... 

The first term on the right-hand side of Equation (2.33) or Equation (2.35) is independent of the time scaling factor p and therefore only the second term is a function of p. Hence, for a given model order N, the minimum error E with respect to p corresponds to the maximum of with respect to p. Therefore, the problem of searching for an optimal time scaling factor p is converted to finding the maximum of the loss function defined by... 

The optimal choice of the time scaling factor p described by Clowes (1965) is generalized here for any L2 stable system. 

Theorem 2.1 Given that the Laguerre coefficients cj can be obtained from Equations (2.14), and assuming that the true system G s) is L2 stable, then the derivative of the loss function V with respect to the time scaling factor p is given by... 

Optimal time scaling factor for first order plus delay systems... 

First order plus delay systems are commonly encoimtered in the process industries and therefore it is important to consider the choice of an optimal time scaling factor for this class of systems. Our intention is to derive some empirical rules based on the process time delay and time constant so that a near optimal time scaling factor can be found with little computational effort. 

Example 2.1 illustrated that the optimal value of p for a first order system is equal to the inverse of the process time constant. If the process is higher order but without time delay, satisfactory results can be obtained if p is chosen based on the dominant time constant of the process. However, the presence of delay can greatly affect the optimal choice of p. To examine this problem, we shall first derive an analytical solution for the Laguerre coefficients associated with a first order plus delay system and then find empirical rules for choosing the optimal time scaling factor p. 

For 7 values greater than 1.5, we have found that it is better to select the optimal time scaling factor by examining the zeros of Equation (2.63) for the chosen value of N. 

The first step in choosing the optimal time scaling factor p is to identify the interval in which it might be located. Using the estimate of the process settling time T, we can form an interval pmintPmax], where the lower end of the interval, Pmin — is chosen to approximately correspond to... 

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