MODELLING AND SIMULATION OF CHEMICAL INDUSTRIAL REACTORS

Zdenka Prokopová and Roman Prokop Faculty of Applied Informatics Tomas Bata University in Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic E-mail: [email protected]

KEYWORDS subsequently: In the continuous reactor case, should it Modelling, Simulation, Chemical reactor, Steady-state. be preferred a piston flow or perfect mixing one? For producing high-volume chemicals, flow reactors are ABSTRACT usually preferred. The ideal piston flow reactor exactly The paper is focused on analysis, mathematical duplicates the kinetic behavior of the ideal batch modelling and simulation of reactors which are used in reactor. For small-volume chemicals, the economics the chemical and tanning technology. Material and frequently favor batch reactors. Batch reactors are used energy balances are the key issues of mathematical for the greater number of products, but flow reactors models of chemical reactors and processes. The produce the larger volume as measured in tons. combination with chemical kinetics and transport Flow reactors are operated continuously; that is, at effects an intellectual basis for chemical reactor design steady state with reactants continuously entering the can be obtained. The contribution brings a special vessel and with products continuously leaving. Batch tanning facility – paddle tumbler which can be reactors are operated discontinuously. A batch reaction described as a type of rotary batch reactor and cycle has periods for charging, reaction, and continuous-flow circulation reactor for the cyclohexane discharging. The continuous nature of a flow reactor production. A mathematical dynamic model is derived lends itself to larger productivities and greater and the optimal parameters were computed. economies of scale than the cyclic operation of a batch reactor. The volume productivity for batch systems is INTRODUCTION identical to that of piston flow reactors and is higher than most real flow reactors. However, this volume The contribution presents a utilization of material productivity is achieved only when the reaction is balances with kinetic expressions for elementary actually occurring and not when the reactor is being chemical reaction in the phase of the reactor design. charged or discharged, being cleaned, and so on. Within Many design engineers grope in the dark when they are the class of flow reactors, piston flow is usually desired facing to labyrinth of chemical and physical properties for reasons of productivity and selectivity. in the real world of technological plants. The paper A useful and interesting application of reactor design for gives simple techniques and an example how those chemical engineers can be seen in (Froment and Bischotf problems can be overcome. The final derived equations 1990), (King and Winterbottom 1998). The treatment of of mathematical models are solved for quite simple but biochemical and polymer reaction engineering is frequent and important chemical reactor. described very extensively in (Nauman 2002), The behavior of real chemical process is obviously too (Levenspiel 1998), (Schmidt 1998), (Smith 1956). There complex for complete mathematical analysis. We are are emphases on numerical solutions which are needed obliged to use an approximate description which is able for most practical problems in chemical reactor design. to deliver sufficiently accurate results. Cardinal Sophisticated numerical techniques are rarely necessary. simplifications of the mathematical description can be The aim is to make the techniques understandable and obtained by the following supposals: easily accessible and allow continued focus on the - the process does not change its properties during the chemistry and physics of the process. observation period, - the relation between input and output variables of THEORETICAL BACKGROUND the process is linear, - all process variables can be measured continuously Types of elementary reactions and have a continuous-time trend. In the phase of design process of many chemical, Consider the reaction of two chemical species according biochemical, tanning and other reactors there arise some to the stoichiometric equation: fundamental and relevant questions. Among them, the first is: Should the reactor be batch or continuous? And A + B → P (1)

Proceedings 23rd European Conference on Modelling and Simulation ©ECMS Javier Otamendi, Andrzej Bargiela, José Luis Montes, Luis Miguel Doncel Pedrera (Editors) ISBN: 978-0-9553018-8-9 / ISBN: 978-0-9553018-9-6 (CD)

This reaction is said to be homogeneous if it occurs within a single phase. It means that they take place only k 3A ⎯⎯→ P ℜ = ka3 in the gas phase or in a single liquid phase. These k 2 reactions are said to be elementary if they result from a 2A + B ⎯⎯→ P ℜ = ka b (8) k single interaction (i.e., a collision) between the A + B + C ⎯⎯→ P ℜ = kabc molecules appearing on the left-hand side of equation (1). The rate at which collisions occur between A and B Types of ideal reactors molecules should be proportional to their concentrations, a and b. Not all collisions cause a Generally, there are four basic kinds of ideal reactors: reaction, but at constant environmental conditions (e.g., temperature) some definite fraction should react. Thus, 1) The batch reactor we expect This is the most common type of industrial reactor. Batch reactors has no input or output of mass after the ℜ = = k[A][B] kab (2) initial charging. The amounts of individual components may change due to reaction but not due to flow into or where k is a constant of proportionality known as the out of the reactor. The component balance for rate constant. component A satisfies equation (for constant volume V Note that the rate constant k is positive so that ℜ is batch reactor): positive. ℜ is the rate of the reaction, not the rate at which a particular component reacts. Components A d(Va) da = ℜ V ⇒ = ℜ (9) and B are consumed by the reaction or equation (1) and dt A dt A thus are "formed" at a negative rate: The solution of ordinary differential equation - ODE (9) ℜ = ℜ = −kab (3) A B requires an initial condition: a=a0 at t=0. The rate of the reaction ℜA depends on concentration a while the product P is formed at a positive rate: ( ℜA = −ka ).

ℜ = + P kab (4) 2) The piston flow reactor The piston flow reactor is continuous-flow reactor with The adopted sign convention supposes that the rate of the similar behavior as a batch reactor. It is usually a reaction is always positive. The rate of formation of visualized as a long tube in which materials stay a component is positive when the component is formed together and react as they flow down the tube. The by the reaction and it is negative when the component is composition of material flowing through a piston flow consumed. reactor depends on time and also on position in the tube. The age of material at point z it t, and the composition at First-Order Reactions this point is given by the constant-volume version of the k A ⎯⎯→ P ℜ = ka (5) component balance for a batch reactor (9) with the substitution t = z / u :

The rate constant k for the first-order reaction has units -1 da of [s ]. u = ℜA (10) dt Second-Order Reactions 1) One reactant: where z is distance measured from the begin of the tube and u is the velocity of the fluid.

⎯⎯→k ℜ = 2 2A P ka (6) 3) The continuous-flow stirred tank reactor The most important assumption is that the continuous- 2) Two reactants: flow stirred tank reactor is perfectly mixed. In that case there are only two possible values for concentration – k the inlet flow has concentration a and everywhere else A + B ⎯⎯→ P ℜ = kab (7) in concentration aout. The component balance for component A in the single reactions is: in the case of second-order reaction the rate constant k 3 -1 -1 has units of [m mol s ]. Qa + ℜ V = Qa (11) in A out Third- Order Reactions In principle, there are three types of third-order 4) The completely segregated, continuous-flow stirred reactions: tank reactor This type of reactor is interesting theoretically, but has

very limited practical utilization. - Heat of the hydrolysis reaction is inconsiderable. Obviously, no real reactor can achieve ideal state. - The cylinder shape reactor has radius at least 10 However, it is often possible to design reactors that very times greater than the thickness of wall, it means closely approach ideal limits. that the temperature profile can be described as “infinite plate“. Temperature dependence of reactions - Dependence of all physical parameters of model on the temperature is negligible. Chemical reactions can be isothermal or nonisothermal. On the mentioned assumptions the quantitative In the case of isothermal reaction, the operating mathematical model can be expressed by the following temperature and the rate constants are arbitrary relations: assigned. For nonisothermal reactions, the temperature varies from the point to point in the reactor, the ∂ ∂ 2 T (x,T ) = T < < > temperature dependence directly enters the design at (c,t);0 x b;t 0 (14) ∂t ∂ 2 calculations. The rate constant k for elementary x ∂T (t) ∂T reactions is expressed by Arrhenius equation: m c 0 = Sλ (0,t) (15) 0 p0 ∂t ∂x = ⎛ − E ⎞ T (x,0) Tp (16) k = k T m exp⎜ ⎟ (12) 0 RT = ⎝ ⎠ T (b,t) Tp (17) T (0,t) = T (18) where parameter m=0 (or m=0.5, 1 for the special cases 0 = of reactions) depends on used theoretical model. E is T0 (0) T0 p (19) activation energy and the fraction E/R is called an activation temperature. A non-stationary temperature profile in the reactor wall A general energy balance (for a flow reactor) can be is described by equation (14). Equation (15) describes a written as: balance between the speed of reaction mixture temperature decrease and the heat transfer through the dT reactor wall. Initial conditions are expressed by ρ.c .V = q .ρ .c .T p dt in in pin in (13) equations (16) and (17). Equations (18) and (19) − q .ρ .c .T −VΔH kc − Fα(T − T ) represent conditions of ideal heat transfer. out out pout out R A ext The time response of the reaction mixture temperature depends on the reactor wall thickness b, the thermal dT ρ conductivity coefficient at, the weight of wall and where term .c p .V is accumulation of energy, dt reactor filling (m, m0) and the specific heat capacity of Δ V HRkcA is a heat generated by reaction and wall and reaction mixture (cp, cp0). The dimensional α − value of reaction mixture temperature T0(t) depends on F (T Text ) is heat transferred out of reactor. its initial temperature T and the temperature of 0p surrounding T which can be identical with the initial ILLUSTRATIVE EXAMPLES p temperature of the reactor wall (Kolomaznik et al.

2005). The rotary batch reactor – paddle tumbler From all mentioned parameters only the weight of

reactor filling m and its initial temperature can be Creation of the mathematical model of wood paddle 0 changed. The weight of the reaction mixture and its tumbler is one of the aims of our research. The non- initial temperature have to be chosen so that its resulting isothermal and non-adiabatic paddle tumbler for an temperature do not decrease under the desired limit enzymatic hydrolysis of chromium tannery wastes is border (the reaction speed would be too small). considered. Because of simplification we can describe For the first approximation without nonlinearities of the the paddle tumbler as a type of rotary batch reactor. temperature field in the reactor wall can be used very Application of paddle tumbler for an enzymatic simple mathematical model. It is presented as a quasi- hydrolysis in tannery has two basic reasons. Firstly, the stationary model in the form: paddle tumblers are usually used directly in the tannery wastes point of origin. Secondly, it needs only small λ − dT0 = S − investment to plant changes for its available utilization. m0c0 (T0 Tp ) (20) Both reasons support economical and environmental dt b aspects of the desired process. The presented mathematical model should respect the The solution of the model (11) is: following assumptions (Kolomaznik et al. 2006): - Reaction mixture is intimately stirred by motion of − ⎛ T0 p Tp ⎞ Sλt reactor. ln⎜ ⎟ = (21) ⎜ − ⎟ - Heat transfer on the both sides of wall is ideal. ⎝ T0 Ts ⎠ bm0c0

The continuous-flow circulation reactor The time response of the reaction mixture concentration Another presented reactor is a continuous-flow depends on the input flow of the benzene, the circulation reactor for the cyclohexane production. concentration of input stream cBv, the volume of reactor Cyclohexane, as an important base material for filling, the specific heat capacity of reaction mixture cp polyamide manufacturing, makes by catalytic and the rate constant k. The solution of ordinary hydrogenation of benzene. A large number of reaction differential equation (25) requires an initial condition: heat transfers during hydrogenation process. Therefore, cBv = cB0 at t=0. benzene and cyclohexane mixture hydrogenation From all mentioned parameters the input flow stream proceed in continuous time circulating reactor under into the reactor qv and its initial concentration cBv can be hydrogen excess. The presented mathematical model changed. should respect the following conditions: - Reaction mixture is perfectly mixed by circulation. EXPERIMENTS - The volume of reaction mixture is constant. - All technological parameters in reactor are constant. The rotary batch reactor – paddle tumbler - Heat transfer on the both sides of wall is ideal. The hydrogenation mechanism of benzene is supposed A rotary batch reactor with diameter 1,5 m and high 1 m to be a first-order reaction: was used for the measuring of the temperature characteristics. Experiments took place directly in tannery industrial conditions. In the first phase 18 k + 3H ⎯⎯→ experiments were performed. Results of some measured 2 (22) Benzene Cyclohexane variables were influenced by different factors e.g. cover rB = −k ⋅ cB leakage. From these reasons these corrupted data were not be inserted into final measurement sets. These dates The material balance for the component benzene were used for comparison of simulation characteristics satisfies the following differential equation (for constant with real values as well as for the determination of volume V of reactor): hardly measured parameters. Experiments used for comparison were carried out in a cold reactor and the dc filling was preheated by steam to 70-80°C. Initial q .c = q.c +V.k.c +V B (23) v Bv B B dt parameters of are presented in Table 1.

The temperature balance in the reactor satisfies the Table 1: Initial parameters for comparative experiments differential equation: Experiment Surrounding Initial Weight number temperature temperature of qv.ρv.c pv.Tv + (−ΔHR )V.k.cB = dT (24) [°C] of filling filling q.ρ.c .T + F.α(T − T ) +V.ρ.c p x p dt [°C] [kg] experiment 14 21 80 108 where the relation (−ΔHR )V.k.aB is the heat generated experiment 17 20 80,5 155 by the reaction and F.α(T − Tx ) represents the heat transfer into surroundings. experiment 18 20 76 155 The rate constant k for elementary reactions is expressed by Arrhenius equation (12). After some The thermal conductivity (λ) of the wood paddle mathematical manipulations the equation of the benzene tumbler (strengthened with iron tires) was drawn from concentration variation takes the form: measured experimental dates. The solution of quasi- stationary model (14) was used. The final line (see Figure 1.) was obtained from dependence of relation − E dcB qv q − = ⋅ c − ⋅ c − k e RT ⋅ c (25) ⎛ T0 p Tp ⎞ dt V Bv V B 0 B ln⎜ ⎟ on the time. The thermal conductivities ⎜ − ⎟ ⎝ T0 Ts ⎠ Similarly, variation of the reaction mixture temperature were calculated by the regression analysis from in the reactor is: measured data. Then the average value of thermal conductivity λ= 0.1031 [W/(m .K)] was calculated for dT q k.c q the later utilization. = v T + (−ΔH ). B − T dt V v R ρ.c V p (26) F.α − (T − Tx ) V.ρ.c p

0. 35 This approximation was obtained by a four-parameter 0.3 method adopted from (Astrom 1995) applied to step

0. 25 responses. measured 0.2 The continuous-flow circulation reactor

0. 15 calculated The continuous-flow circulation reactor with V=0,5 [m3] volume was used for the simulation of the ln (top-tp/t0-ts) 0.1 concentration and temperature characteristics. Initial 3 -1 0. 05 input flow into the reactor was qv=0.0067 [m s ] with 3 the initial concentration of benzene cBv=1.9 [kg/m ]. -1 0 Initial rate constant was considered k0= 1.616.e14 [s ] 0 20 40 60 80 100 120 140 160 180 ρ -3 time [min] density of mass =985 [kg m ], specific heat capacity cp = 4.050 [J/kg.K], transfer heat coefficient α = 435.00 2 2 Figure 1: Demonstration of thermal conductivity [W/m .K], surface F=5.5 [m ], activation energy -1 calculation from measured dates. For experiment 18: H=4.8e4 [Jmol ], initial mass temperature Tv=320 [K] λ=0.1036 [W/m .K] and temperature of surrounding Tx=290 [K].

The main problem consists in holding of the reaction Steady-State Analysis Results mixture temperature in the range which is sufficient for The dependence of the output variable upon the input in a protein yield. Simulations were performed with using the steady-state is shown in the Figure 3. The reactor of the averaged thermal conductivity. The minimal steady-state characteristics were obtained by solution of reactor filling was determined as m =113 [kg] upon the 0 equations (12), (25) and (26). The initial conditions dependence of the reaction mixture temperature on the were computed by a standard optimization method - reactor filling. The obtained parameters from experiments were used iterative procedure. for simulation calculations. For the first step, mathematical model (14)-(19) was used. However, the 0.03 calculated temperature profiles do not exactly follow the measured data. The difference is caused by simplifications but the main error is only in the initial phase when the heat transfer from reaction mixture into 0.02 reactor wall is very high (accumulation of energy in the wall).

76 cbs [kg/m3] 0.01

74

72 ]

C 0 70 0.04 0.06 0.08 0.1 0.12 0.14 0.16 qvs [m3/s] 68 66 Figure 3: Steady-state input-output behavior of the

temperature [ continuous-flow circulation reactor 64 62 Dynamic Analysis Results

60 For dynamic analysis purposes one input and one output 0 50 100 150 200 time [min] choices were considered. Then were defined input u and output y as deviations from their steady-state values – s s Figure 2: Comparison of measured (-*) and approximated u(t) = q(t) – q (t) and y(t) = cB(t) – cB (t). (--) temperature characteristic (experiment 18). Graphical interpretation of simulation experiments – the output y time responses to input u step changes are Described rotary reactor can be for example depicted on Figure 4. approximated by the first order system in the form:

−15 B(s) G(s) = ≈ (27) 98s + 0.8 A(s)

1.2 REFERENCES 1

1 Felder, R. M. and Rousseau, R. W. 2000. Elementary Principles of Chemical Processes, 3rd ed., Wiley, New 0.8 York Froment, F. and K. B. Bischotf. 1990. Chemical Reactor 0.6 2 Analysis and Design, 2nd ed., Wiley, New York. King, M. B. and M. B. Winterbottom. 1998. Reactor Design 0.4 For Chemical Engineers, Chapman & Hall, London. y(t) [kg/m3] 0.2 3 Nauman, E. B. 2002. Chemical reactor design, optimization, and scaleup. McGRAW-HILL, New York. ISBN: 0-07-137753- 4 0 0. 5 Levenspiel O., Chemical Reaction Engineering. 3rd ed., -0.2 Wiley, New York. 1998. 0 5 10 15 20 Schmidt, L. D. 1956.The Engineering of Chemical Reactions, t [min] Oxford University Press, New York. 1998. Figure 4: The output y time responses to input u step Smith, J. M. 1998. Chemical Engineering Kinetics. McGRAW- changes, u=-0.08(1), -0.07(2), -0.05(3), 0.05(4), 0.08(5) HILL, New York. p. 131. Kolomazník, K., Prokopová, Z., Vašek, V., Bailey, D. G. 2006. “Development of control algorithm for the Model approximation optimized soaking of cured hides”. JALCA. Vol. 101, No. 9, pp. 309-316. ISSN: 0002-9726. For control requirement, described continuous time Kolomazník, K., Janáčová, D., Prokopová, Z. 2005. circulation reactor was approximated by the second “Modeling of Raw Hide Soaking”. In: WSEAS order system in the form: Transactions on Information Science and Applications. Hellenic Naval Academy. Vol. 11(2), 5,.pp. 1994-1998, 1.9 B(s) ISSN: 1790-0832. G(s) = ≈ (28) Reklaitis, G. V., Schneider, D. R. 1983.Introduction to 2 71s + 10s + 0.5 A(s) Material and Energy Balances, Wiley, New York.

The approximation was obtained by a four-parameter method adopted from (Astrom and Hagglund 1995) AUTHOR BIOGRAPHIES applied to step responses. ZDENKA PROKOPOVÁ was born in

Rimavská Sobota, Slovak Republic. She CONCLUSION graduated from Slovak Technical University in 1988, with a master’s degree Developed mathematical models (especially quasi- in automatic control theory. Doctor’s stationary) enable to set up different changes of various degree she has received in technical parameters without any complex industrial experiments. cybernetics in 1993 from the same university. Since A deep analysis and comparisons of experimental and 1995 she is working in Tomas Bata University in Zlin, simulation (computed) data were performed for Faculty of Applied Informatics. She is now working verification of models. Some parameters and constants there as an associating professor. Research activities: of mathematical relations had to be derived previously mathematical modeling, simulation, control of from measured industrial variables. Especially, the technological systems, programming and application of thermal conductivity of wood barrel wall had to be database systems. Her e-mail address is : computed. This mathematical derivation was much [email protected]. simpler then performing experiments and analyzing data. After the establishing of constants and models, the ROMAN PROKOP was born in minimization of the reactor filling in the first case and Hodonin, Czech Republic in 1952. He initial input flow or initial concentration of benzene into graduated in Cybernetics from the Czech the reactor in the second case can be optimized. For the Technical University in Prague in 1976. control requirements we have tried to approximate He received post graduate diploma in 1983 presented models as the first or second order input- from the Slovak Technical University. output system. All computations were performed in the Since 1995 he has been at Tomas Bata University in MATLAB+SIMULINK environment. Zlín, where he presently holds the position of full

professor of the Department of Automation and Control Acknowledgments Engineering and a vice-rector of the university. His

research activities include algebraic methods in control The work has been supported by the Ministry of theory, robust and adaptive control, autotuning and Education of the Czech Republic in the range of the optimization techniques. His e-mail address is : project No. MSM 7088352102. This support is very [email protected]. gratefully acknowledged.