- Email: [email protected]

415

A CFDFinite V o l u m e M e t h o d to Generate a D e t e r m i n i s t i c Model: A p p l i c a t i o n to Stirred T a n k Reactors ~Maciel Filho, R.. ; 2Bezerra, V. M. F. 1 Laboratory of Optimization, Design and Advanced Process Control (LOPCA), College of Chemical Engineering, State University of Campinas, Email: [email protected] 2 Universidade Federal do Rio Grande do Norte - UFRN - CT -Departamento de Engenharia Quimica - Programa de p6s-gradua~o em Engenharia Q u i m i c a - PPGEQ -CEP: 59072-970 Natal - RN - Brazil- Email: [email protected]

Abstract-The objective of the present work is to analyze a deterministic model related to stirred tanks, starting from its set of partial differential equations, going forward the discretization of such set, through the Finite Volume Method and also applying a simplified procedure for obtaintion of the temperature profile of the case study considered. Comments on the discretization of the system of equations show that this particular method partitions the computational domain into a finite set of volumes or cells, assuming that the main variables are constant over each cell and this fact requires the conservation of equations being satisfied for each cell. Computational Fluid Dynamics (CFD) represents the scientific alternative to preprocess, process and post-process the fluid flow inside stirred tanks. In the core of commercial CFD packages, Finite Volume Method based discretizations for different case studies are used and the user can count with a feasible output

Keywords: stirred tanks, computational fluid dynamics, finite volume method, fluid flow.

1. I N T R O D U C T I O N Stirred tanks constitute commonly used equipments inside chemical industry. In order to study the characteristics of fluid flow inside stirred tanks, the fundamental equations of conservation (mass, momentum and energy) are used. Such approach results in deterministic models for the equipment studied. It starts from an analysis of the case study proposed. The set of fundamental relationships shows the equation of continuity (mass conservation law), the energy conservation formulae and the Navier-Stokes equations that represent the fluid flow inside the tank. In this path, the conservation laws are coupled with auxiliary equations becoming possible the subsequent numerical simulation of the system studied. It is necessary that the degree of freedom of the system be zero for possibility of solution, i.e., the number of equations is, at least, equal to the number of unknowns of the mathematical set. At this point, numerical solutions are necessary for the final output for the problem. There are diverse ways in

416 which one can choose her/his route for problem solving. Some routes can be stated as: deterministic, non-deterministic, experimental and/or non-invasive ones. In the first case, the set of partial differential equations together with initial and/or boundary conditions is discretized through the Finite Volume Method [Patankar, 1980; Maliska, 1995], p.e.. After discretization, the set of partial differential equations turns into a set of algebraic ones, to be solved through tridiagonal matrix algorithms [Patankar, 1980]. For non-deterministic models proposed for the case study, the equipment itself is considered a black box, in which the final assessment does not show conservation laws, but mathematical expressions calculating errors between inputs and outputs. At present, experimental and non-invasive techniques, as particle image velocimetry data [Fox, 1998] are used in order to compare and elucidate temperature and velocity profiles, as well as to validate data used in Computational Fluid Dynamics (CFD) simulations, p.e.. Computational Fluid Dynamics consists in studying diverse phenomena with a strong ability of post-processing the fluid flow studied in an understandable and accurate output. Examples can include stirred tank reactors and the final displays show characteristic velocity, temperature, concentartion profiles, among other, depending on the proposed problem.

2. CASE S T U D Y ar = 0 c3z exceto Ur = 0

t a_, = o

"

Parede do Tanque

&

r

li I ~j-~ Eixo (s

/

{l

4

"lamina(blade)"

i i i

l

00O=0 &

l

l

a_,= 0 Or

exceto U~,U~

l L*=0 0Z

exceto Ur

r

exceto U~ -- 0

r

Figure 1.0-Boundary conditions to a stirred tank [PIKE, 1990] Simplification of a model in mathematical terms, intends to reduce one of the among characteristics [TUCKER, 1989]: number of equations, number of terms inside the equations, degree of non-linearities, degree of coupling among the equations (mainly, stiffness degree).Taking into account our case study, it is observed that cylindrical coordinates are the pertinent representations to the deterministic model to be shaped. The main conservation equations for the system are:

417 Continuity:

1~

Op Ot

10(prUr) + (PUo)+ r Or r~--O

~z(OUz)

(1.o)

0

Simplifications:

~=0 ~0

(2.0)

ap

(3.0)

Ot

= o

that is, incompressible fluid flow is considered. In this way, Equation (1.0) has the following shape' 1 Cg(prUr)+ (pUz)=O r Or ~zz

(4.0)

Momentum equations: Component- r: OUr + Ur OUr + U 0 OU r P

&

[(

Or

63p 63 1 63 (rUt - ~Or + Ix & -r ~ r

r

-U~) -+

c~O

U z -OU ~z r ) =

r

)) q '~Ur 2~Uo~_~U]r r 2 50 2

r 2 630

CDZ2

(5.0) Disregarding angular momentum,O OU r OU r +U r 9 +Uz-~z J--OT+ltl P Ot Or

Component- z:

~- r~rr

2'rl

+ 63z2

(6.0)

418

P

c3Uz c3Uz UO c3Uz c3Uz / +U r + +U z & Or r 690 --~z/

@

[1 ~( O~z/

0z + g r ~ r r

r 0r/

(7.o)

1 632Uz 632Uz 1 -t 2 F r c~O2 C3Z2

J

Another simplification:

P

0t

z +Ur

&

z +U

Z&-zJ

op [lOIrOZO2z]

=_m+g

c~z

r&

+

&-J

c~z2

(8.0)

According to the a general representation of the system above, all the conservation laws can be represented through Equation [9.0] [Maliska, 1995]:

P r

(rUr~)+-(U~ r

rFr

(Uz~b) Ozz r Dr

+ -~ + r c~O

+

Ozz

Fo

(9.0)

+

3

DISCRETIZATION OF THE EQUATIONS, NUMERICAL METHOD OF FINITE VOLUMES:

USING

THE

The sequence of discretization follows: tomando-se a eq. (4.0) and multipling it by r, we have 0 c3 ( 9 r U r ) + r (9U)Or ~zz z

0

(10.0)

From eq. (10.0): enc9

~--(prU r )drdz - [(prU r) - ( p r U ) ](z e - z w ) ws & n r s

(11.0)

](2 r2 / rnZw 2

(12.0)

e

G3

i lr-~rr(PUz)dzdr-[(pU sw

) -(pU)

Ze

Integrating eq. (10.0) takes in the present 2D problem to:

IIr~(ouz ) d z d r - [ ( P U z ) n - ( 9 U z ) s ws

2/

re-r w 2

(13.0)

419

2 2

re - r w : ( r e _ r w 2

/re+rw/ - ( r e - r w ) r p

(14.0)

2

where rp is the medium radio of the control volume. Adding up Eqs.(13.0) and (14.0): (prUr)e (Zn - Z s ) - ( prur )w ( zn - Zs) + (PUz)n ( re - rw )rp

(15.0)

-(pUz)s(r e -rw)r p - 0 And eq. (16.0) represents discretization of the continuity equation (for the obtaintion of a deterministic 2D model) Analogously, these procedures can apply to the energy and momentum equations.

4. R E S U L T S

Figure 2 . 0 - Temperature distribution for axial and radial directions for a stirred tank [Excel v. 5.0, BEZERRA, 1997]

5. C O N C L U S I O N S : Stirred tanks are common and important equipments for chemical industry. In this way, their characterization becomes useful and demanding. Numerical simulations involving Finite Volume Method discretizations are present either in struturated algorithms or in commercial packages. The present work showed the main details for discretization of conservation equations to shape a deterministic model for a stirred tank. Moreover, it has shown a CFD-

420 Finite Volume procedure implicitly used in commercial packages, representing, p.e., a tank with simple movement of polymeric fluid inside it CFD tends to influence strongly in the scenario for representation of complex and/or simple problems.It is a low cost and relatively short time consuming to generate final displays for tanks or stirred tanks. It is concluded that many routes can be chosen to represent a same problem, but the non-invasive (CFD) ones emerge as important tools for designing and predicting na equipment and/or specific case studies. In the present case, it was shown the sequence of shaping a deterministic model in two ways: structured algorithm and afterwards numerical simulation and finally, a typical CFD post-processing for a 2D fluid flow problem. Both alternatives are important and feasible.

BIBLIOGRAPHY

BIRD, R. Byron et al., "Transport Phenomena", John Wiley & Sons, New York, 1960. BEZERRA, V. M. F. Metodologia de Obteng6o de Resultados em Fluido- Din6mica Computacional-Aplica96o a Reatores Tanques Agitados, Tese de Doutorado, UNICAMP, SP, 1997 Excel - Microsoft Excel Vers~.o 5.0 FOX, Rodney O . ; MENG, Hui, SHENG, Jian, Validation of CFD Simulations of a Stirred Tank Using Particle Image Velocimetry Data, The Canadian Journal of Chemical Engineering, Vol. 76, 611-625, June, 1998. MALISKA, C. R., TransferOncia de Calor e Mecdnica dos Fluidos ComputacionalFundamentos, Coordenadas Generalizadas, LTC, RJ, 1995. PATANKAR ,S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington DC, 1980.

Phoenics v 2.1.1, Cham, UK. PIKE, R.W., 1980, Ju, S.Y., Mulvahill, T.M., "Tridimensional Turbulent Flow in Agitated Vessels with a Nonisotropic Viscosity Turbulence Model", The Canadian Journal of Chemical Engineering, vol. 68, 3-16, 1990. TUCKER, Charles L. Fundamentals of Computer Modeling for Polymer Processing, Hanser Publishers, New York, 1989.