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

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Outline Description of mathematical models Strategy in mathematical modeling Summary

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Description

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Mathematical model mathematical expressions that represents some phenomena

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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

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Translation into math expression Verbal statement: The concentration of reactant A is decreasing with increasing time Mathematical expression: C A = C Ao – k 1 t 3 – k 2 t 2 – k 3 t or ?? or

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Categories of mathematical models Empirical model Deterministic model Stochastic model

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Empirical models C A = C Ao – k 1 t 3 – k 2 t 2 – k 3 t Any mathematical expressions with some adjustable parameters No particular theoretical background Example: polynomials

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Deterministic models Based on theoretically accepted laws Example: From kinetics:

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

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

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Comparison EmpiricalDeterministicStochastic Difficulty <>>>> Accuracy <>>>>>

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Focus for undergraduate level DETERMINISTIC MODELS

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Strategy

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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

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

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Example: Batch reactor data Time Concentration of reactant, C A T1T1 T2T2 T3T3 T 1 < T 2 < T 3 What can you infer from the data?

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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?

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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 C A and time temperature is incidental factor

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

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

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Example to find model constants Experimental data: For T 1 : t C A (-dC A /dt) t 1 C A1 y 1 t 2 C A2 y 2... t n C An y 3

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Trial and error procedure Try n=1 -(dC A /dt) C a n (with n=1) Plot of experimental data should be linear if it is true that n = 1 If the plot of experimental data is not linear, try another value of n

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Determination of k 2 Assume it is correct that n = 1 The value of k 2 is then the slope of the plot -(dC A /dt) C a n (with n=1) Slope = k 2

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Incorporating the incidentals From the example, you may get 3 different values of k 2 for three different temperatures: Tk 2 T 1 (k 2 ) 1 T 2 (k 2 ) 2 T 3 (k 2 ) 3

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Correlating k 2 and T Back to basic: use Arrhenius correlation: k = A exp (- E/R/T)

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Correlating k 2 and T Plot of ln k 2 vs. (1/T)

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Correlating k 2 and T ln k 2 1/T Slope = -E/R Intercept = ln A

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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!

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Utilizing kinetics model in CSTR design Back to basic: use mass balance (assuming isothermal reactor) F vin C Ain F vout C Aout Rate of mass in + Rate of mass formed by reaction - Rate of mass reacted - Rate of mass out = Rate of accumulation

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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 F vin.C Ain + 0- (-r A ).V - F vout.C A =0 USEFUL FOR SCALE-UP

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

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Example: Scale-up mixing Power number Reynold number

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Summary

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

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

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

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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’

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Have a nice journey to be chemical engineers!

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