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For the production of sodium sulfate, a brine is crystallized and crystals of glauber salt are generated by this process. The phase data related to the most common sodium sulfate minerals are as follows: mirabilite (Na2SO4 ⋅ 10H2O), tenardite (Na2SO4), glauberite (Na2SO4 ⋅ CaSO4), astrakanite (Na2SO4 ⋅ MgSO4 ⋅ 4H2O). The units commonly used to express the phases are moles of salt per 1000 moles of water. These latter units simplify the construction of the commonly employed four-sided Janecke phase diagrams. The cooling temperature or the speed with which the solution is cooled has an effect on the size and purity, as well as the amount of crystals produced. We seek to establish, through the population balance equations (PBE), which process variables can be modified to obtain a specific crystal size, as well as to validate the mathematical model that best predicts the amount of crystals precipitated as a function of temperature. The adjustment by least squares, cubic splines, pitzer equations and Lagrange interpolation is tested. The experimental results agree with the characteristics of the proposed models.

Intrinsic noise is inherent to many biological processes and provokes variation in gene expression in a population of isogenic cells leading to phenotypic diversity. Intrinsic noise is generated by different sources of noise such as the number of molecules, the stochastic binding and unbinding of transcription factor and/or the number and strength of transcription factor binding sites. In this work, we use numerical simulations to study the effects of the number of operators and different types of cooperativity on the Fano factor of three different molecules of the tryptophan (trp) operon of E. Coli. We analyze the Fano factor for the mRNA, anthranilate synthase and tryptophan molecules, because it represents the effects of the noise in the variation or variability of the gene expression, a larger Fano factor implies a larger variation. Our model takes into consideration the presence of intrinsic noise and all the known mechanisms of regulation. In particular, we consider hypothetical promoters in the repression mechanism with different numbers of operators and three cases of cooperativity: positive, negative, and no-cooperativity.