Moment-correction algorithms

Exercise 3.4 Use the moment-correction algorithm to modify the following moment set ... 

In this case the moments that have been changed, which are responsible for the invalidity of the set, are mi and mo. When this moment set is fed to the correction algorithm, the following answer is obtained ... 

This second correction algorithm consists of calculating the new moment set from those corresponding to log-normal distributions. A log-normal distribution has the following functional form ... 

Typically two of the lower-order moments m, m2, and m3 are selected. However, in order to accommodate more moments of the original set, sometimes the final moments are calculated as the arithmetic average of the moments of two log-normal distributions. A typical choice could be to use for the first log-normal distribution mo, mi, and m3, and for the second mo, m2, and m3. In this way, after the correction only mo and m3 will be identical to the original ones, but a certain degree of control is achieved on both mi and m2. A Matlab script implementing this correction algorithm can be found in Section A.2.2 of Appendix A. 

Below a Matlab script with the implementation of the algorithm for the correction of an unrealizable moment set is reported. The script first analyzes the moment set and then, if it is unrealizable, the script identifies the moment that has to be changed the least to make the set realizable. The procedure is iterated until the moment set becomes realizable or the maximum iteration number is reached. Realizability is verified by ensuring positiveness of the second-order differences however, other more stringent conditions could more effectively be used. 

However, the particle motion depends on the droplet shape and the number of electrodes that the droplet overlays at any given moment. Since this is not known a priori, we use local estimation and control at each time step of our simulation to compute the pressure boundary conditions needed to realize the desired flow field. At each instant in time, the control algorithm is provided with the droplet shape and particle locations, as would be available through a vision sensing system. Any deviation of the particles from their desired trajectories that may arise from thermal fluctuatimis, external disturbances, and actuation errors is corrected using feedback of the particle positions. We now give an overview of our algorithm ... 

The closed equation (12.383) does not necessary conserve the moments of the distribution due to the macroscopic or finite grid resolution employed in the size domain, thus some sort of ad hoc numerical correction must be induced to enforce the conservative moment properties. It is noted that it is mainly at this point in the formulation of the numerical algorithms that the class method of Hounslow et al. [88], the discrete fixed pivot method of Kumar and Ramkrishna [112] and the multi-group approach used by Carrica et al. [30], among others, differs to some extent. The problem in question is related to the birth terms only. Following the discrete fixed pivot method of Kumar and Ramkrishna [112], the formation of a particle of size in size range... 

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