09.Jan.2014
The population balance equation (PBE) is an integro-partial differential equation with nonlinear source term. The PBE is known to admit analytical solutions only for a few cases with restricted forms of interaction kernels. We propose for the first time a novel converging sequence of continuous approximations to the number concentration function as a solution to the population balance equation (PBE). These approximations are internally consistent with respect to any finite number of desired moments. The uniqueness and convergence of such a sequence are assured by being an optimal solution to the constrained NLP, which maximizes the constrained Shannon entropy function. The solution is an optimal functional containing the maximum missed information about the distribution. This entropy maximization problem is a convex program and is solved by converting the constrained NLP into a set of transport equations in terms of the optimal Lagrange multipliers. Since differential form of the Lagrange multipliers is used, the method is given the name the Differential Maximum Entropy (DMaxEnt) method. The DMaxEnt method is tested using many standard and even complex liquid–liquid extraction processes, where the population balance modeling is needed.https://www.researchgate.net/publication/260043652_Solution_of_the_Population_Balance_Equation_using_the_Differential_Maximum_Entropy_Method_DMaxEntM_An_application_to_liquid_extraction_columns
The population balance equation (PBE) is an integro-partial differential equation with nonlinear source term. The PBE is known to admit analytical solutions only for a few cases with restricted forms of interaction kernels. We propose for the first time a novel converging sequence of continuous approximations to the number concentration function as a solution to the population balance equation (PBE). These approximations are internally consistent with respect to any finite number of desired moments. The uniqueness and convergence of such a sequence are assured by being an optimal solution to the constrained NLP, which maximizes the constrained Shannon entropy function. The solution is an optimal functional containing the maximum missed information about the distribution. This entropy maximization problem is a convex program and is solved by converting the constrained NLP into a set of transport equations in terms of the optimal Lagrange multipliers. Since differential form of the Lagrange multipliers is used, the method is given the name the Differential Maximum Entropy (DMaxEnt) method. The DMaxEnt method is tested using many standard and even complex liquid–liquid extraction processes, where the population balance modeling is needed.
https://www.researchgate.net/publication/260043652_Solution_of_the_Population_Balance_Equation_using_the_Differential_Maximum_Entropy_Method_DMaxEntM_An_application_to_liquid_extraction_columns
أنا طالبة في الهندسة الصناعية, اختياري لهذا التخصص كان بناءً على أهميته الكبيرة كمهنة في الحاضر والمستقبل ... رغد بركات
الهندسة الصناعية تساعدك على اتخاذ قرارات أفضل، وتعطي أشكالا أخرى من مبادئ الهندسة بشكل عملي وعلمي في آن. ... محمود صلاح
قسم الهندسة الكيميائية قسم جميل جدا تعلمت فيه الكثير ومما تعلمته فيه جدية العمل وروح الفريق الواحد .. ... رغد الشويكي