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LETTER TO THE EDITOR
To the Editor:
I am writing to you to correct our mis-take in our article (‘‘A Submerged Mem-brane Reactor for Continuous PhenolHydroxylation Over TS-1’’) published inthe AIChE Journal, DOI 11514, 2008;54(7):1842-1849.
Recently we have identified an error inour article, which is ‘‘TS-1 catalyst (aver-
age particle size, 200 nm; BET surfacearea, 95 m2�g1; the Si/Ti molar ratio, 9)was provided by Baling PetrochemicalCo., SINOPEC’’. (p 1843) should bereplaced by ‘‘TS-1 catalyst (average parti-cle size, 200 nm; BET surface area, 408m2�g1; the Si/Ti molar ratio, 59) was pro-vided by Baling Petrochemical Co.,SINOPEC’’.I apologize for any inconvenience.
Wanqin JinState Key Laboratory of Materials-oriented
Chemical EngineeringNanjing University of Technology
5 Xinmofan RoadNanjing 210009
P.R. ChinaE-mail: [email protected]
To the Editor:
In a recent article on the operation ofseeded batch crystallizers,1 my co-authorsand I used the method of moments to modelcrystallizer operation and also reported thefinal crystal-size distribution function at theend of the batch. It was subsequentlybrought to my attention that we did notexplain how the final crystal-size distribu-tions were calculated, and that our methodmay be of interest to other researchers sinceresearchers working with moment modelsseldom report final crystal-size distributionfunctions, and it is well-known that thecrystal-size distribution function cannot berecovered exactly from a finite number ofmoments. Therefore, I would like to brieflyexplain how the final crystal-size distribu-tion function was calculated.From the integration of the equations in
the method of moments model, the nuclea-tion and growth rates as a function of timecan be determined. In principle, since thecalculation must only be performed once(not repeatedly for the optimization) todetermine the final crystal-size distributionone could simply solve the partial differ-ential equation
qfqt
þ GðtÞ qfqL
¼ 0
subject to the initial condition that thecrystal-size distribution is equal to the
seed crystal size distribution and the leftboundary condition
f ð0; tÞ ¼ BðtÞGðtÞ
However, there is a simpler way for theassumptions that were made in the article.The final crystal-size distribution will bethe sum of contributions from the seedsand the nuclei. The final seed crystal-sizedistribution will have the same shape asthe initial seed crystal-size distribution—itwill simply be shifted to the right by anamount equal to
R tf0GðtÞdt. For the
nuclei, at some time t* during the batch,the value of the crystal size distributionat the left boundary (L = 0) will begiven by f ð0; t�Þ ¼ Bðt�Þ=Gðt�Þ. At theend of the batch, this point in the crys-tal-size distribution will have shifted tothe right by an amount equal toR tft� GðtÞdt. So if B(t) and G(t) areknown, one can plot B(t)/G(t) vs.R tft Gðt0Þdt0 to determine the crystal-sizedistribution function of the nuclei, andthen add in the contribution from theseeds. (The smallest seed-grown crystalwill always be larger than the largestnucleated crystal under the assumptionsthat we used.) Of course, this only
works for the very restricted set ofassumptions (no breakage or agglomera-tion, size-independent growth, etc.) thatwere considered in the article. If theseassumptions were not valid, it would benecessary to solve the full partial differ-ential equation. However, then of coursethe simple method of moments couldnot be applied either.
AcknowledgmentsI thank Dr. Noriaki Kubota, Professor Emeri-
tus at Iwate University, Japan for pointing outthat we did not explain in the article the methodby which the final crystal-size distributions werecalculated, and for suggesting that I composethis letter.
Literature Cited
1. Ward JD, Yu CC, Doherty MF. A newframework and a simpler method for the de-velopment of batch crystallization recipes.AIChE J. 2011;57:606–617.
Jeffrey WardDept. of Chemical Engineering,
National Taiwan University,#1 sec. 4 Roosevelt Rd.,
Taipei 10617, TaiwanE-mail: [email protected]
� 2011 American Institute of ChemicalEngineersDOI 10.1002/aic.12661Published online May 23, 2011 in Wiley OnlineLibrary (wileyonlinelibrary.com).
� 2011 American Institute of ChemicalEngineersDOI 10.1002/aic.12662Published online May 18, 2011 in Wiley OnlineLibrary (wileyonlinelibrary.com).
AIChE JournalAIChE Journal August 2011 Vol. 57, No. 8 Published on behalf of the AIChE DOI 10.1002/aic 2289
