### Syllabus

**Marking system**

**Written part:** Two open textbook tests (Test I and Test II) focussing on practical skills of problem solving will be given at seminars during the semester. Resitting both tests will be possible during the examination period. Please note, that only one or two resits will be organized during the examination period.

Maximum score from each test is 50 points, i.e., maximum score from the written part is 100 points. In order to pass the written part and proceed to the oral exam, it is necessary to obtain at least 20 points from each test and the sum of both tests must be 50 or more points. (When resitting a particular test, its result can be only improved, meaning that test result is disregarded if it is worse than the result achieved before).

The course material is divided into Tests I and II according to the type of equations obtained by model derivation:

**Test I:**algebraic equations, initial value problem ordinary differential equations (AE/ODE)**Test II:**differential-algebraic equations, partial differential equations and boundary value problem ordinary differential equations spatially discretized using the finite volume method (DAE/PDE)

**Oral exam:** Conditions for passing the written part must be met before attempting the exam. Student is asked to answer two questions selected from the list of topics. Maximum score from the exam is 100 points (50 from each question). At least 20 points from each question must be obtained to pass the exam.

**Marking:** The total score will be the sum of both tests and oral exam; mark for the course will be allocated as follows:

Total score | Mark |

less then 100 | F |

100-119 | E |

120-139 | D |

140-159 | C |

160-179 | B |

180-200 | A |

### Exam questions

1. Systems of algebraic equations – various types of problems. Number of solutions of algebraic equations. Newton iteration method, the importance of initial guess, convergence and divergence.

2. Phase equilibria models and calculations. Gibbs phase rule. Boiling point and dew point of the mixture.

3. Transformation of bi-linear balance equations into the set of linear equations. Sequential and global methods for flow-sheets representing balance problems. Polynomial approximation of experimental data and splines.

4. Flash distillation without and with enthalpy balance. Discussion of state variables and various combinations of known (i.e., initially specified) parameters. Calculation with phase-equilibria model or with the equilibrium given by tabulated data.

5. Three-component extraction with partially miscible solvents. Description of phase equilibrium (triangle) diagram. Description of single-stage extraction with partially miscible solvents.

6. Systems of ordinary differential equations. Numerical solution of the initial value problem by explicit or implicit methods. Typical problems involving reaction kinetics in the batch reactor.

7. Control of the liquid level in the tank with feed and outlet streams, where outlet is equipped with the control valve. PI controller. Measured and manipulated variables.

8. Enthalpy balance of continuous stirred tank reactor. Heat transfer through the reactor wall to the reactor jacket.

9. Various ways of describing the cooling jacket in the reactor. PID controller of reactor temperature – configuration of the controller for cooling and for heating. Steam hammer.

10. Differential-algebraic equations. Classification of systems of DAE equations. Numerical solution of DAE equations. Consistent initial conditions.

11. Semi-batch reactor with non-constant density of liquid phase. Transformation of equations into linear-implicit form.

12. Batch distillation of 3-component mixture. The system of differential and algebraic equations for this problem.

13. Semi-batch reactor with the reaction in gas phase and with the ideal pressure controller. Index of the system of DAE equations. Solution of problems with PID controllers including the derivative term.

14. Problems in the form of partial differential equations (PDE). Example of elliptic, parabolic and hyperbolic PDE. Finite volume method.

15. Application of the finite volume method on the reaction-diffusion problem in the spherical particle. Implementation of boundary conditions.

16. Application of the finite volume method on the problem of food sterilization in the can.

17. Dynamical model of absorption column with stages. The assumption of equilibrium and non-equilibrium plates.

18. Absorption in the packed column including the kinetics of mass transfer. Introduction of simplifying assumptions. Axial dispersion (back-mixing).

19. Convection-dispersion problems. Balance of tubular reactor with axial dispersion. Method of lines. Up-wind difference formulas.