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Mathematical Modeling

Mathematical Modeling. Overview on Mathematical Modeling in Chemical Engineering By Wiratni, PhD Chemical Engineering Gadjah Mada University Yogyakarta - Indonesia. Outline. Description of mathematical models Strategy in mathematical modeling Summary. Description. Mathematical model.

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Mathematical Modeling

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  1. Mathematical Modeling Overview on Mathematical Modeling in Chemical Engineering By Wiratni, PhD Chemical Engineering Gadjah Mada University Yogyakarta - Indonesia

  2. Outline • Description of mathematical models • Strategy in mathematical modeling • Summary

  3. Description

  4. Mathematical model mathematical expressions that represents some phenomena

  5. Why need mathematical models? • Engineering is more easily conducted by means of NUMBERS • Everything needs to be QUANTIFIED • Engineer must be able to TRANSLATE any problems into NUMBERS, by which they make sizing of process equipment

  6. Translation into math expression • Verbal statement: The concentration of reactant A is decreasing with increasing time • Mathematical expression: CA = CAo – k1t3 – k2t2 – k3t ?? or or

  7. Categories of mathematical models • Empirical model • Deterministic model • Stochastic model

  8. Empirical models CA = CAo – k1t3 – k2t2 – k3t • Any mathematical expressions with some adjustable parameters • No particular theoretical background • Example: polynomials

  9. Deterministic models • Based on theoretically accepted laws • Example: From kinetics:

  10. Chemical Engineering Tools • Mass balance • Energy balance • Rate processes • Equilibrium • Economics • Humanity Basic principles in mathematical model development

  11. Stochastic models • Take into account the uncertainty of the phenomenon • Incorporating the concept of probability into deterministic model • Example: in processes involving living organisms

  12. Comparison

  13. Focus for undergraduate level DETERMINISTIC MODELS

  14. Strategy

  15. Principles in math modeling • Think simple: separate the incidental from the essentials; focus on the essentials • Back to basic: mass balance, energy balance, rate processes, equilibrium

  16. Principles in math modeling INCIDENTAL Refining Fundamental Model FUNDAMENTAL MODEL Improved Accuracy ESSENTIAL Equilibrium Mass balance Rate process Energy balance

  17. Example: Batch reactor data T1 < T2 < T3 What can you infer from the data? Concentration of reactant, CA T1 T2 T3 Time

  18. Observation • Reactant depletion over time • Different set of data for different temperature • Reactant depletes more quickly at higher temperature Which one essential and which one incidental?

  19. Essential vs. incidental • What essential: DEPEND ON YOUR MOST FUNDAMENTAL PURPOSE • With respect to batch reaction data: your final goal is to design a reactor • The size of reactor: depend on how fast the reaction is  the essential of the model is relation between CA and time  temperature is incidental factor

  20. Focus on the essential • How to correlate CA and time? • Back to basic: rate process for reaction is governed by reaction kinetics law • In batch reactor (one of the possibilities): -rA =

  21. Model’s variables and constants • Variable: Independent Dependent • Constants: Adjustable parameters to fit the data on particular mathematical model

  22. Example to find model constants • Experimental data: For T1: t CA (-dCA/dt) t1 CA1 y1 t2 CA2 y2 . . . . . . tn CAn y3

  23. Trial and error procedure • Try n=1 Plot of experimental data should be linear if it is true that n = 1 -(dCA/dt) If the plot of experimental data is not linear, try another value of n Can (with n=1)

  24. Determination of k2 • Assume it is correct that n = 1 • The value of k2 is then the slope of the plot -(dCA/dt) Slope = k2 Can (with n=1)

  25. Incorporating the incidentals • From the example, you may get 3 different values of k2 for three different temperatures: T k2 T1 (k2)1 T2 (k2)2 T3 (k2)3

  26. Correlating k2 and T • Back to basic: use Arrhenius correlation: k = A exp (- E/R/T)

  27. Correlating k2 and T Plot of ln k2 vs. (1/T)

  28. Correlating k2 and T Intercept = ln A ln k2 Slope = -E/R 1/T

  29. Complete model • Combining the essential and incidental, you get kinetic model: • You can predict the concentration of remaining A in the reactor at any time and temperature!

  30. Utilizing kinetics model in CSTR design • Back to basic: use mass balance (assuming isothermal reactor) Fvin CAin Fvout CAout Rate of mass in Rate of mass formed by reaction Rate of mass reacted Rate of mass out Rate of accumulation + - - =

  31. Steady-state CSTR modeling Rate of mass in Rate of mass formed by reaction Rate of mass reacted Rate of mass out Rate of accumulation + - - = Fvin.CAin (-rA).V Fvout.CA + 0 - - = 0 USEFUL FOR SCALE-UP

  32. Other tools for scale up • Correlations among dimensionless groups • Empirical, but ok as long as you pick the correct dimensionless groups

  33. Example: Scale-up mixing Power number Reynold number

  34. Summary

  35. QUANTITATIVE APPROACH IN SCALE-UP (Mathematical Modeling) Math modeling Model improvement Pilot plant scale Commercial scale Lab scale Engineering design Simulation

  36. Important tools for scale-up • Reliable correlations of dimensionless groups • Reliable mathematical models • Numerical methods to solve the models

  37. Required fundamentals • Mass balance • Energy balance • Rate processes Physical: momentum transfer, mass transfer, heat transfer Chemical: reaction rate • Equilibrium: Phase equilibrium Chemical equilibrium

  38. Accuracy • Highly accurate models: time/energy consuming, costly • Moderately accurate models: quick, low cost • For engineering purpose: does not need 100% (absolute) correct answers  we can do with ‘careful estimation based on theoretical supports’

  39. Have a nice journey to be chemical engineers!

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