最佳化批次結晶程序之目標函數選擇

Abstract

結晶是一個常被用來作為液態固態分離的程序。而一個結晶槽的成品可以用結晶大小分佈函數(Crystal Size Distribution Function, CSD)來描述。成品的好壞將影響下游程序的效率，例如過濾或是乾燥程序。在結晶的過程中，同時會出現結晶成核(crystal nucleation)與結晶成長(crystal growth)現象，在大多數的情況下結晶成核現象是不受歡迎的。一個控制得好的結晶槽，可以有效抑制結晶成核現象的發生。 要最佳化一個結晶程序可以從兩方面來著手，一種是改善晶種的性質，另一種是改善冷卻的方法。而不管是使用哪一種手段，都需要一個目標函數來評斷結果的好壞。在這個研究中，我們比較前人在這個領域上使用過的目標函數，希望能夠發現一個最適合的目標函數。 在使用各個目標函數後，我們發現有些目標函數的結果藉由產生大量的成核現象以達到其目標函數值的最小值。但以一個先加入晶種的結晶程序而言，其目標是將晶種長大並避免成核現象的發生，前述達到其目標函數值的方法是與其牴觸的。事實上在真實的程序裡，在結晶之後很有可能經過一個過濾的程序將過小的晶體或是新成核的晶體移除。所以對於各個不同的目標函數，我們除了直接比較用其最佳化的結果的目標函數值以外，也比較新成核晶體被移除後的目標函數值。最後我們發現使用目標函數”最小化新成核晶體的質量”來最佳化，其結果在新成核晶體被移除後，對於各個目標函數值都有較佳的值。 對於目標函數”最小化結晶分佈函數的變異係數”，我們發現與其最佳化結晶成長曲線，使用晶種性質作為控制變數有較佳的效果。較大量的晶種質量與較集中的晶種分佈，在使用”最小化結晶分佈函數的變異係數”作為目標函數時，將有效防止大量成核現象的發生。 最後，我們探討結晶成核速率式中，在晶體體積項(Third Moment)較高的幕次對於最佳化程序的影響。我們發現，較高的幕次將抑制成核現象的發生，造成最佳化的結果有較佳的表現。另外較高的幕次也將使得最佳化晶體成長曲線在程序的一開始有較高的值。

Crystallization is a widely used process for liquid solid separation. The products from this process are crystals, which can be described by a distribution function called “crystal size distribution function (CSD)”. The properties of the crystals affect the efficiency of downstream process, such as filtration or drying. During the process both crystal nucleation and crystal growth happens, and most of the time, the crystal nucleation is undesirable. A well-controlled crystallizer can produce crystals with less crystal nucleation. Researchers have optimize a crystallization processes by improving the seed properties or the cooling policy. In both cases an objective function is required. In this work we compare the objective function that researchers have used, to see which objective function is best when optimizing the cooling policy for a batch crystallizer. The result shows that some of the objective functions are minimized by producing a large amount of nuclei. However, for a seeded batch crystallizer the idea is to grow the seed crystals while suppressing the nucleation. Moreover, in industrial practice, the product crystals would probably be filtered so that fines (nucleated crystals) would be removed. Therefore, for each objective function we also determine the objective value after the nucleated crystals are removed, to see whether the result from each objective function is still the best. After the analysis we conclud that the objective function “minimizing the nucleated crystal mass” is better than others. In this work we also discuss the utility of changing seed properties when using “minimizing weight coefficient of variation” as objective function. We found that if the seed distribution is too wide, the system would be more likely to generate a narrow distribution crystal by excess nucleation. To prevent excess nucleation and achieve a narrow product CSD, a large seed loading and a narrow seed distribution helps. Finally, we also considered the effect of using different nucleation parameters. We changed the exponent on third moment term in nucleation rate equation. The result shows that for higher value of the exponent, the nucleation rate is suppressed, and the performance is better when the growth rate trajectory is optimized using the objective function “minimizing nucleated crystal mass.” The optimized result also mention that for higher value of the exponent on third moment term in nucleation rate equation, higher growth at the beginning of the batch is desirable.