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نہ میرے دل پہ مرا کوئی اختیار رہا

نہ میرے دل پہ مرا کوئی اختیار رہا
کہ میری جان کے دشمن پہ یہ نثار رہا

وہ ہر جگہ پہ مرے نام سے ہی جانا گیا
یہ اور بات وہ اوروں کا غمگسار رہا

لیے ہیں رنج سدا ہی خلوص کے بدلے
کہ دل جلوں کا ہمیشہ یہ کاروبار رہا

تمام رات رہا ہم کلام تاروں سے
لباس جس کے مقدر کا تار تار رہا

وہ کوئی اور نہیں تھا وہ تیرا تائبؔ تھا
تمام عمر جسے تیرا انتظار رہا

From the Chief Editor's Desk

Medical journals are a credible source of disseminating research and innovations, and Launching a medical journal is a challenging task. Many medical science journals are establishing a platform to publish quality research but still the task is tough and requires perseverance and hard work. Shalamar Medical and Dental College (SMDC) Lahore, strives to promote a culture of research. As part of this initiative, SMDC had launched ‘Medical Journal of Sakina Begum Institute’, but publication of the second volume was delayed due to the pandemic. Moreover, the name of the journal had to be changed due to some administrative issues. However, the committed editorial team was successful in bringing efforts for the latest issue to fruition.

Numerical Solutions of Population Balance Models in Dispersed Systems

This work focuses on the modeling and numerical approximation of population balance models (PBMs) for simulating dispersed systems, especially the batch crystallization pro- cess. Apart from applying the existing numerical schemes, new numerical techniques are introduced for solving these models efficiently and accurately. The effects of nucleation, growth, aggregation, breakage, and fines dissolution phenomena on the crystal size distri- bution (CSD) are investigated. An alternative quadrature method of moments (QMOM) is introduced for solving the single-variate length-based PBM incorporating simultaneous nucleation, growth, aggregation and breakage phenomena. In the proposed QMOM, or- thogonal polynomials, formed by lower order moments, are used to find the quadrature points and weights. To ensure better accuracy of the scheme, a third order orthogonal polynomial, utilizing the first six moments, is selected to calculate the quadrature points (abscissas) and corresponding quadrature weights. Therefore, at least a six moment sys- tem is needed to solve. This choice of polynomial gives a three-point Gaussian quadrature rule which generally yields exact results for polynomials of degree five or less. A mathe- matical model is derived for simulating batch crystallization process incorporating crystals nucleation, size-dependent growth and dissolution of small nuclei below certain critical size in a recycling pipe. Moreover, a time delay in the dissolution unit is also incorporated in the model. The dissolution of small crystals (fines dissolution) is helpful to further improve the product CSD. It withdraws and dissolve excessive fines from the quiescent zone of crystallizer which are generated during periods of high supersaturation. This ef- fectively shifts the CSD towards right and often makes the distribution narrow. A new numerical scheme is introduced for simulating this model. The method of characteristics, the Duhamel’s principle, and the QMOM are employed together to devise the proposed numerical scheme. Several test problems are considered and the numerical results are val- idated against available analytical solutions and the finite volume scheme (FVS). It was found that the suggested numerical methods have capability to solve the given models efficiently and accurately.
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