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September 29, 1999 Department of Chemical Engineering University of South Florida Tampa, USA

Population Balance Techniques in Chemical Engineering. by. Richard Gilbert & Nihat M. Gürmen. September 29, 1999 Department of Chemical Engineering University of South Florida Tampa, USA. Part I -- Overview. What is the Population Balance Technique (PBT)?.

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September 29, 1999 Department of Chemical Engineering University of South Florida Tampa, USA

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  1. Population Balance Techniques in Chemical Engineering by Richard Gilbert & Nihat M. Gürmen September 29, 1999 Department of Chemical Engineering University of South Florida Tampa, USA

  2. Part I -- Overview R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  3. What is the Population Balance Technique (PBT)? PBT is a mathematical framework for an accounting procedure for particles of certain types you are interested in. The technique is very useful where identity of individual particles is modified or destroyed by coalescenceor breakage. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  4. (Dis)advantage of PBT Advantage • Analysis of complex dispersed phase system Disadvantage • Difficult integro-partial differential equations R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  5. Application Areas • colloidal systems • crystallization • fluidization • microbial growths • demographic analysis R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  6. Ni(q,t) No(q,t) Tampa Immigration Emigration n(q,t) Birth Rate Death Rate Origins of population balances: Demographic Analysis • Time = t • Age = q R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  7. A Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer Qi, Ci, ni Particle Size Distribution (PSD) Qo, Co, n R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  8. Growth Rate Growth Kinetics Growth Rate Supersaturation Mass Balance Feed Nucleation Kinetics Population Balance PSD Nucleation Rate Crystal Area (from Theory of Particulate Processes, Randolp and Larson, p. 3, 1988) Information diagram showing feedback interaction R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  9. Part II -- Mathematical Background R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  10. Population Density, n(L) Population Density, n(L) Size, L Size, L Exponential Distribution Normal Distribution Two common density distributions by particle number R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  11. Exponential density distribution by particle number N1 Cumulative Population, N(L) Population Density, n(L) N1 n1 Size, L Size, L L1 L1 N1 is the number of particles less than size L1 n1 is the number of particles per size L1 R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  12. Population Density, n(L) Ntotal Lmax Size, L Normal density distribution by particle number Ntotal = Total number of particles Cumulative Population, N(L) Lmax Size, L R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  13. 1 Normalized Population Density, f(L) 0 Lmax Size, L Normalization of a distribution One way to normalize n(L) normalized Area under the curve R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  14. Average properties of a distribution The two important parameters of a particle size distribution are * How large are the particles? mean size, * How much variation do they have with respect to the mean size? coefficient of variation, c.v. where 2(variance)is R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  15. Mean, = the first moment about zero Moments of a distribution j-th moment, mj, of a distribution f(L) about L1 Variance, 2= the second moment about the mean R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  16. Skewness, 1 = measure of the symmetry about the mean (zero for symmetric distributions) Kurtosis, 2= measure of the shape of tails of a distribution Further average properties: Skewness and Kurtosis j-th moment, j, of a distribution f(L) about mean R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  17. Part III -- Formulation of Population Balance Technique R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  18. Check these Assumptions Basic Assumptions of PBT(Population Balance Technique) • Particles are numerous enough to approximate a continuum • Each particle has identical trajectory in particle phase space S spanned by the chosen independent variables • Systems can be micro- or macrodistributed R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  19. Basic Definitions Number density function n(S,t) is defined in an (m+3)-dimensional spaceS consisted of 3 external (spatial) coordinates m internal coordinates (size, age, etc.) Total number of particles is given by SpaceS R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  20. The particle number continuity equation a subregion R1 from the Lagrangian viewpoint R1 S R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  21. Convenient variable and operator definitions where is the set of internal and external coordinates spanning the phase space R1 R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  22. Applying the product rule of differentiation to the LHS R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  23. Substituting all the terms derived earlier As the region R1 is arbitrary R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  24. In terms of m+3 coordinates Micro-distributed Population Balance Equation Averaging the equation in the external coordinates Macro-distributed Population Balance Equation R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  25. B - D terms represent the rate of coalescence conventionally collision integrals are used for B and D the rate at which a bubble of volume u coalesces with a bubble of volume v-u to make a new bubble of volume v is a death function consistent with the above birth function would be R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  26. Coagulation kernel, C(x,y) C(x,y) : the rate at which bubbles of volumes x and y collide and coalesce. in the modeling of aerosols two of the functions used for C(x,y) are where Ka is the coalescence rate constant 1) Brownian motion 2) Shear flow R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  27. Simplifications for a Solvable System • dynamic system => t • spatially distributed => x, y, z • single internal variable, size => L Growth rate G is at most linearly dependent with particle size => R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  28. Moment Transformation Defining the jth moment of the number density function as Averaging PB in in the L dimension R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  29. Microdistributed form of moment transformation j = 0,1,2,... ³ k Macrodistributed form of moment transformation j = 0,1,2,... ³ k R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  30. If Assumptions Do Not Allow Moment Transformations • You have to use other methods of solving PDEs like • method of lines • finite element methods difficult if both of your variables go to infinity R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  31. Part IV -- Examples R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  32. Example 1 : Demographic Analysis • neglect spatial variations of population • one internal coordinate, age q Ni(q,t) No(q,t) Immigration Emigration Tampa n(q,t) Set up the general population balance equation? R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  33. Qi, Ci, ni Qo, Co, n Example 2: Steady-state MSMPR Crystallizer The system is at steady-state Volume of the tank : V Outflow equals the inflow Feed stream is free of particles Growth rate of particles is independent of size There are no particles formed by agglomeration or coalescnce Derive the model equations for the system. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  34. References • BOOK • Randolph A. D. and M. A. Larson, Theory of Particulate Processes, 2nd edition, 1988, Academic Press • PAPERS • Hounslow M. J., R. L. Ryall, and V. R. Marhsall, A discretized population balance for nucleation, growth, and aggregation, AIChE Journal, 34:11, p. 1821-1832, 1988 • Hulburt H. M. and T. Akiyama, Liouville equations for agglomeration and dispersion processes, I&EC Fundamentals, 8:2, p. 319-324, 1969 • Ramkrishna D., The prospects of population balances, Chemical Engineering Education, p. 14-17,43, 1978 R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

  35. THE END R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

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