PROCESS DYNAMICS & CONTROL

PROCESS DYNAMICS & CONTROLBy:RALP JAYSON L. ALDAYChE - 4202THE DEVELOPMENT OF A MATHEMATICAL MODEL2OUTLINEAdditional Examples of Mathematical ModelingModeling Difficulties

3ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGExample 1. Mathematical model of a CSTRConsider a continuous stirred tank reactor system. A simple reaction A B takes place in the reactor, which is in turn cooled by a coolant that flows through a jacket around the reactor. 4

ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGThe fundamental dependent quantities for the reactor are:The total mass of the reacting mixture in the tank,The mass of chemical A in the reacting mixture in tank, andThe total energy of the reacting mixture in the tank.5ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGApply the conservation principle on the three fundamental quantities:Total mass balanceMass balance on component ATotal energy balance6ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGTotal Mass Balance

Mass Balance on Component A

7ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGTotal Energy Balance

The total energy of the reacting mixture is E = U + K + Pwhere U is the internal energy, K the kinetic energy and P the potential energy of the reacting mixture.

8ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGAssume the reactor is not moving: E = U + K + P [ kinetic & potential energies are zero ]

For liquid systems:

9ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGCharacterize Total MassWe need the density of the reacting mixture and its volume, V. The density will be a function of the concentrations a and b and of the temperature T. Quite often the dependence of density on concentration and T is weak and the density can be considered constant as the reaction proceeds.10ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGCharacterize the Mass of Component AIn this process, concentration of A and volume are considered state variables.Characterize the Total EnergyFrom thermodynamics, the enthalpy of a liquid system is a function of the temperature and the composition of the liquid system.11ADDITIONAL EXAMPLES OF MATHEMATICAL MODELING

12ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGSummary:Output variables: V, Ca, TInput variables: Cai, Fi, Ti, Q, F (when feedback control is used)Disturbances: Cai, Fi, TiManipulated variables: Q, F (occasionally Fi or Ti)Constant parameters: density, Cp, -Hr, ko, E (activation energy), R 13ADDITIONAL EXAMPLES OF MATHEMATICAL MODELINGExample 2. Mathematical model of a mixing processTwo streams 1 and 2 are being mixed in a well stirred tank, producing a product stream 3. Each of the two feed streams is composed of two components A and B, with molar concentrations ca1, cb1 and ca2, cb2, respectively. Let also F1 and F2 be the volumetric flowrates of the two streams (ft3/min, m3/min) and T1, T2 their corresponding temperatures. Finally, let ca3, cb3, F3 and T3 be the concentrations, flowrate and temperature of the product stream. A coil is also immersed in the liquid of the tank and it is used to supply or remove heat from the svstem with steam or cooling water.14MODELING DIFFICULTIESThe modeling examples discussed in the previous sections should have alerted the reader to a series of difficulties that one encounters in his efforts to develop a meaningful and realistic mathematical description of a chemical process.15MODELING DIFFICULTIESExample 1. Considering the mathematical modeling of the CSTRThe following difficulties arise:Determine with the desired accuracy the values of various parameters such as preexponential kinetic constant, ko, the activation energy, E, and the overall heat transfer coefficient, U.Although the specific heat capacities, cp and cpi have been considered constant, they are in general functions of the temperature, T, and the concentration, ca.16MODELING DIFFICULTIESExample 1. Considering the mathematical modeling of the CSTRDuring the operation of the CSTR, scaling, fouling, etc. will alter the value of the overall heat transfer coefficient.

17MODELING DIFFICULTIESThree Categories of the Difficulties Encountered During Mathematical Modeling:Those arising from poorly understood chemical or physical phenomena;Those caused by inaccurate values of various parameters; andThose caused by the size and the complexity of the resulting model.18MODELING DIFFICULTIESPoorly Understood ProcessesTypical examples:Multicomponent reaction systems with poorly known interactions among the various components and imprecisely known kinetics.Vapor-liquid or liquid-liquid thermodynamic equilibria for multicomponent systems.Heat and mass interactions in distillation columns with nonideal multicomponent mixtures, azeotropic mixtures, etc.

19MODELING DIFFICULTIESExample 2. Consider the fluidized catalytic reactor shown

20MODELING DIFFICULTIESExample 2An oil feed composed of heavy hydrocarbon molecules is mixed with catalyst and enters a fluidized bed reactor. The long molecules react on the surface of the catalyst and they are cracked into lighter product molecules (like gasoline) which leave the reactor from the top. While cracking is taking place , carbon and other heavy uncracked organic materials are deposited on the surface of the catalyst leading to its deactivation. The catalyst is then taken into a regenerator where the material deposited on its surface is burned with air. The regenerated catalyst return then to the reactor after it is mixed with fresh feed.21MODELING DIFFICULTIESIn order to model the two units, the following information must be available:The reaction rate of the cracking process;The rate with which carbon and heavy material are deposited on the catalyst (this will determine the rate of catalyst deactivation);The dependence of the above two rates on the temperature of the reactor and the quality of the feed (light or heavy); andThe rate of combustion of the carbonaceous material deposited on the catalyst, in the regenerator, and its dependence on temperature.22MODELING DIFFICULTIESImprecisely Known ParametersThe availability of accurate values for the parameters of a model is indispensable for any quantitative analysis of the behavior of a process. Unfortunately, this is not always possible. It should be pointed out that the values of the parameters do not remain constant over long periods of time. 23MODELING DIFFICULTIESThe Size and Complexity of a ModelIn an effort to develop as accurate and precise a mathematical model as possible, the size and complexity of the model increases significantly.24MODELING DIFFICULTIESConsider a distillation column with 20 trays, a reboiler and a condenser:2N + 4 = 2(20) + 4 = 44 differential equations2N + 1 = 2(20) + 1 = 41 algebraic equationsThe size of the model for such simple systems is already prohibitive. It is clear that such an extensive modeling would lead to cumbersome and hard to use models.25END OF REPORT26