The numerical simulation of chemical engineering processes constitutes one of the most ... Indeed, chemical processing requires careful consideration of modelling ... materials processing, control theory, as well as other areas can be evoked.
Numerical Simulation of Chemical Engineering Processes M.O. Deville, O. Byrde, M.L. Sawley and G. Mompean IMHEF-DGM, Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland
1 Intoduction The numerical simulation of chemical engineering processes constitutes one of the most challenging problem addressed by the Computational Fluid Dynamics (CFD) community. Indeed, chemical processing requires careful consideration of modelling aspects, some of these still being open questions. It is by nature an interdisciplinary eld, where chemistry, materials processing, control theory, as well as other areas can be evoked. Furthermore, chemical engineering problems are often very sti problems from geometrical and temporal points of view. In space, chemical reactions arise at the molecular level while the
uid mechanics deals with complex uid ow phenomena at the macroscopic scale. Even the Kolmogorov scales which are connected to the dissipation rate of the turbulent kinetic energy by viscous damping are still two to three orders of magnitude above the molecular scale. In time, chemical reactions evolve on short characteristic times close to the microsecond while the uid ow involves dynamics covering several seconds to a few minutes. Both space and time sti ness render any numerical simulation a complex and almost untractable problem. Returning to the uid mechanics viewpoint, multi-phase
ows are very often present and non-Newtonian and viscoelastic e ects a ect deeply the physical behaviour of such processes. The industrial applications are numerous: food processing, glues, paints, polymer solutions, injection molding, biomedical engineering, glass making, etc. The elaboration of improved technologies, quality requirements for the products, and constraints in a global economy call for a design cycle avoiding whenever possible all empirical concepts and building upon enhanced advanced tools originating from the computational sciences. In this paper, we will concentrate on single phase ows avoiding any attempt to handle chemical reactions, giving some examples that indicate the state-of-the-art capabilities of CFD applied to particular relevant chemical engineering applications. The paper is organized as follows. In Section 2, the modelling of glass ow in large melting furnaces is surveyed. New links to control theory are mentioned, sheddding some light on interdisciplinary design approaches. Section 3 reports on new developments about three-dimensional transient simulation of viscoelastic uids. Finally, Section 4 presents the results obtained by parallel computation of an incompressible uid in an in-line static mixer. 1
2 Glass ow modelling Glass is melted in very large furnaces where the heat is supplied by oil combustion in burners placed laterally on both sides of the production unit. One key issue to run eciently this process is the energy saving made through an automated control of the heat supply. On-line regulation is taking place in order that the heat budget remains in a range established a priori by operating and economical considerations. Hereafter, we review the modelling steps that have been carried out over the last fteen years bringing major improvements and opening new routes towards a system analysis on the numerical furnace. Most of the recent developments were introduced by N. Vanandruel [1] and embodied in a nite element programme. In the early eighties, two-dimensional ows in the longitudinal symmetry plane were computed using the incompressible Navier-Stokes equations coupled with the temperature equation via the Boussinesq approximation. Molten glass at high temperatures around 1500 C behaves as a Newtonian uid with a viscosity law depending strongly on the temperature. The Boussinesq approximation takes the buoyancy e ects into account in the gravity term of the equations. The volumic mass varies linearly with the temperature such that the uid is heavier in cold conditions and lighter when it is hot. Everywhere else in the equations, the volumic mass is considered as a constant. The heat transfer is made by conduction (this is a marginal process as glass is an insulating material), convection and radiation. This last physical transfer occurs in the glass volume and may be described by the Rosseland approximation (a model used in astrophysics for the stars) which represents the radiative heat transfer by an equivalent radiative conductivity depending on the third power of the temperature. In the late eighties, modi ed thermal conditions were introduced into the model by the radiative slip condition. This model allows a temperature correction along the furnace walls where the physics changes completely as the radiation heat transfer is no longer inside the volume, but becomes two-dimensional when one approaches the surfaces. In 1990, electrical boosting performed by carbon electrodes placed on the bottom wall of the furnace was handled by the solution of the Poisson equation for the potential of the electrical eld. The Joule e ect was also incorporated in the temperature equation. In 1991, bubblers were modelled by a forcing term in the momentum equations. The bubbles are produced as an air curtain inside the glass volume in order to enhance the glass mixing in the (upstream) fusion section of the furnace. Due to the hardware development in terms of memory size increase and computing time speed-up, the early nineties have seen the rst three-dimensional simulation with the full geometrical complexity of the furnace: neck, steps, etc. In 1993, the rst twodimensional transient simulations were performed in the symmetry plane. Now, threedimensional transient studies are in the realm of feasibility. Modelling improvements are tackling the batch composition from the thermal point of view, the radiative exchange between the glass bath and the ceiling, etc. The numerical "furnace" is an investigation tool where all the predictive control is performed [2]. The predictive algorithms, control theory and estimation procedures are tested and designed on the numerical simulation instead of the real units. This approach results in savings in the design cycle, the needed experimental quantities and tests. 2
3 Non-Newtonian uid ows In several industrial applications, viscoelastic uids are owing through three-dimensional complex geometries and the time-dependent e ects are very important. To deal with these problems, an ecient nite-volume algorithm was designed to treat complex geometries. This choice is based on the requirements of memory space and time consumption to solve 3D unsteady ows. The complete representation of the three-dimensional characteristics and the time dependent behaviour have a great advantage over the standard tools. This section of the paper describes brie y the numerical simulation of viscoelastic uids. The equations for viscoelastic uids, of an upper-convected Maxwell and Oldroyd-B type, are solved using a nite volume technique derived from the SOLA algorithm [4]. This technique was developed for three-dimensional unsteady ows using a non-uniform staggered grid. The primitive variables, velocity, pressure and extra-stresses are used in the formulation. On the staggered mesh, the pressure and the normal extra-stress components are treated at the centre of the control volume. The velocities are computed in the centre of the faces and the cross components of the extra-stress tensor are attached to nodes located at the mid-edges. The non-linear terms in the momentum and constitutive equations are taken into account and are discretized in space using a third-order quadratic upstream scheme proposed by Leonard [5]. The decoupling procedure for the pressure is derived from a SOLA type algorithm. This procedure yields a Poisson pressure equation. The matrix of the pressure system is positive de nite and symmetric. It can be solved either by a Cholesky factorization or a preconditioned conjugate gradient method. The velocities are then calculated from the momentum equation. The solution results in a velocity and pressure eld that enforce the conservation equations. The details of this procedure can be found in reference [6]. To solve the viscoelastic problem, the extra-stress components are treated by an explicit Euler scheme. The time step is chosen using the Courant-Friedrichs-Lewy (CFL) condition. 3.1
Transport equations
The equations governing the velocity components (Ui), the pressure (P ) and the extrastresses (Tij ) elds are obtained from the mass, momentum and constitutive equations [7]: (i) mass conservation @Ui @xi
(ii) momentum conservation @Ui @t
+
@ (Uj Ui) @xj
!
= 0;
(1)
@P i = ? @x + @x@ (o @U + Tij ); @x
i
j
j
(2)
where o is the Newtonian viscosity, the uid volumic mass and Tij the viscoelastic extra-stress components. 3
(iii) constitutive equation for an Oldroyd-B uid: Tij
@ @Ui @Uj @Ui @Uj ij + ( @T + ( Um Tij ) ? Tkj ? Tki ) = 1 ( + @x ): @t @x @x @x @x
m
k
k
j
i
(3)
Here, is the relaxation time and 1 is the constant viscosity of the viscoelastic extrastress tensor. A dimensionless number called the Deborah number compares the internal material time to the time characteristic of the ow based, for example, on the shear rate. The higher the Deborah number, the sti er the numerical problem. This is due to the fact that the hyperbolic nature of the constitutive relation (3) is more prominent. 3.2
Studied cases
Case studies have been conducted for the start-up Couette ow and the four-to-one planar contraction, in both two- and three-dimensional geometries. The numerical solutions are in very good agreement with analytical solutions for the start-up Couette ow. One known diculty in the numerical simulations of viscoelastic ows is how to treat the geometrical singularities. The existence of a stress singularity, in geometries such as contractions, has proved to be a hard test case for numerical methods. The geometrical singularities are sources of steep gradients in the ow, which can cause the loss of convergence for several numerical steady state algorithms. The code was tested in the 4:1 contraction giving excellent results. The size of the corner vortex for the four-to-one contraction, in the 2D case, is in good agreement with previous computations by Marchal and Crochet [8]. These two cases, start-up Couette ow and the 2D 4:1 contractions, were used with the aim of validating the numerical code in the three spatial directions and testing the convergence of the code. New results showing the vector eld, streamlines and extra-stresses were obtained for the 3D 4:1 contraction at high Deborah numbers. The time-dependent nature of these
ows was analyzed taking the solution at various times, from the initial condition to the steady-state, as frames in a motion picture. Results for vortex formation and growth were obtained, then the di erences between the simulated 2D and 3D planar contractions were discussed. To our knowledge, this is the rst time-dependent numerical simulation of a viscoelastic uid through a three-dimensional 4:1 contraction. Grid independence has always been veri ed; for all cases, the calculations demonstrated convergence to the steady state for a Deborah number up to 157 in the 3D 4:1 contraction. These results are the subject of two papers, [9] and [10]. Figure 1 shows the normal stress di erence for an Oldroyd-B uid through a 4:1 contraction. The positive peak near the singularity and negative values along the symmetry axis can be observed. This behaviour is in agreement with the measurements of the normal stress di erence in viscoelastic ows [11]
4 Flow in a static mixer Static mixers are in-line mixing devices which consist of mixing elements inserted in a length of pipe [12]. Mixers of this type are used in continuous operation, with the energy for mixing being derived from the pressure loss incurred as the process uids ow 4
through the mixing elements. A variety of element designs are available from the various manufacturers, with the number of elements required for a particular application being dependent on the diculty of the mixing task (more elements are necessary for dicult tasks). The range of applications for static mixers is very wide, and includes both laminar and turbulent processes. Laminar ow mixing proceeds by a combination of ow division and ow re-orientation while, for turbulent ows, the elements cause a higher level of turbulence than in the corresponding empty pipe. The optimization of chemical mixers is traditionally performed by trial and error, with much depending on previous experience and wide safety margins. Some of the diculties inherent to this approach can be overcome by numerical ow simulation. The ability to numerically simulate the ow through a mixer can contribute signi cantly to the understanding of the mixing process and provide for better, faster and cheaper design optimization. The geometry considered in the present study corresponds to the Kenics KM static mixer [13, 14, 15]. This mixer design has been employed in the process industry since the mid-1960s, mainly for the in-line blending of liquids under laminar ow conditions. The Kenics mixer is comprised of a series of mixing elements aligned at 90o , each element consisting of a short helix of one and a half pipe diameters in length. Each helix has a twist of 180o with right-hand and left-hand elements being arranged alternately in the pipe. Various con gurations have been considered for the Kenics mixer, di ering in the number of mixing elements. Results are presented in this paper for a con guration having 6 elements. Computations of the steady ow in the mixer has been performed for a range of Reynolds number, 15 Re 100. To resolve the detailed ow features present in the Kenics mixer, a high-performance parallel computer system (the 256-processor Cray T3D system installed at the EPFLausanne) has been employed. This has enabled numerical simulations to be performed using computational meshes signi cantly ner than are currently employed by industry. 4.1
Flow solver
The parallel ow solver used in this study is based on a multi-block code that was developed for the numerical simulation of 3D unsteady, turbulent, incompressible ows [17]. This code solves the Reynolds-averaged Navier-Stokes equations on block-structured computational meshes. The numerical method employed is based on a cell-centered nite volume discretization with an arti cial compressibility method to couple the pressure and velocity elds. A high-order upwind spatial discretization scheme based on the approximate Riemann solver of Roe is employed for the advection terms, while the di usion terms are discretized using a central approximation. An ADI method is used to solve for the steady ow considered in the present study. Parallelism is achieved by dividing the computational domain into a number of blocks (sub-domains), with the ow equations being resolved in all blocks in parallel by assigning one block to each processor [16]. Communication between processors is necessary to exchange data at the interface of neighbouring blocks. The communication overhead is minimized by data localization using two layers of ghost cells surrounding each block. 5
Data are exchanged between blocks using message passing via the PVM library. The parallel code has demonstrated good performance scalability on the Cray T3D for numbers of processors varying from 1 to 256. Communication overhead has been measured to account for less than 10% of the total resolution time. The code has been employed for the computation of a number of di erent applications of industrial interest [16]. To determine the eciency of a chemical mixer, it is necessary to establish means by which the extent of the uid mixing can be gauged both qualitatively and quantitatively. In the present study, this is achieved through the calculation of the trajectories of particles through the mixer. Such an approach can be applied directly to a single species uid considered in the present study. To quantify the mixing, it is necessary to study the trajectories of large number of particles. This is achieved by replicating the computed
ow elds on each processor of the Cray T3D, enabling up to 1 million particle trajectories to be calculated in parallel. Such calculations will allow the determination of statistical averages that are signi cantly more accurate than has been achieved to date. 4.2
Results
Figure 2 shows, for a ow with Re = 100, the spatial evolution along the mixer of 130,000 particles divided into two initially distinct groups of "red" and "blue" particles. These plots illustrate the randomization of the particles via a process of stretching and folding, indicating a high level of mixing of the uid due to its passage through the mixer. The in uence of the Reynolds number on the extent of mixing is qualitatively shown in Fig. 3. From these plots it can be observed that for low Reynolds number, several "islands" are visible in the particle trace pattern at the mixer exit. These islands, which contain particles of only one colour, inhibit a complete mixing of the uid. Similar behaviour has been observed in the results of computations of Stokes ow (Re = 0) in the Kenics mixer [15].
References [1] N. Vanandruel, Simulation Numerique de la Thermoconvection dans les Fours a Verre, Ph.D. Thesis, Universite Catholique de Louvain, Louvain-la-Neuve, 1994. [2] N. Vanandruel, Thermoconvection Simulation and Control in Melting Glass Furnaces, Proceedings of the IMACS-COST Conference, Lausanne, September 1995, Vieweg Verlag, Braunschweig, 1996. [3] B. Caswell, Report on the IXth International Workshop on Numerical Methods in Non-Newtonian Flows, Journal of Non-Newtonian Fluid Mechanics, 62, 99 (1996). [4] C.W. Hirt, B.D. Nichols and N.C. Romero, SOLA - Numerical solution algorithm for transient uid ow, Los Alamos Laboratory, Report LA-5852 (1975). [5] B.P. Leonard, A stable accurate convective modelling procedure based on quadratic upstream interpolation, Computer methods in Applied Mechanics and Engineering, 19, 59-88 (1979). 6
[6] G. Mompean, Three-equation turbulence model for prediction of the mean square temperature variance in grid-generated ows and round jets, International Journal of Heat and Mass Transfer, 37, 1165 (1994). [7] D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, (Springer-Verlag, New York, 1984). [8] J.M. Marchal and M.J. Crochet, Hermitian nite elements for calculating viscoelastic
ow, Journal of Non-Newtonian Fluid Mechanics 20, 187 (1986). [9] G. Mompean and M. Deville, Time-dependent 3D numerical simulation of OldroydB uid using nite volume method, Proceedings of the IMACS-COST Conference, Lausanne, September 1995, Vieweg Verlag, Braunschweig, pp. 171-179, 1996. [10] G. Mompean and M. Deville, Recent developments in three-dimensional unsteady
ows of non-Newtonian Fluids, Revue de l'Institut Francais du Petrole, 51, 261-267, 1996. [11] L.M. Quinzani, R.C. Amstrong and R.A. Brown, Birefringence and laser-Doppler velocimetry (LDV) studies of viscoelastic ow through a planar contraction, Journal of Non-Newtonian Fluid Mechanics 52, 1-36 (1994). [12] N. Harnby, M.F. Edwards and A.W. Nienow, Mixing in the Process Industries, (Butterworth-Heinemann, Oxford, 2nd ed., 1992). [13] J. Arimond and L. Erwin, A simulation of a motionless mixer, Chemical Engineering Communications, 37, 105-126 (1985). [14] A. Bakker and E.M. Marshall, Laminar mixing with Kenics in-line mixers, Fluent Inc. Users' Group Meeting, Burlington VT, Oct. 13-15, 1992. [15] F.H. Ling and X. Zang, A numerical study on mixing in the Kenics static mixer, Chemical Engineering Communications, 136, 119-141 (1995). [16] O. Byrde, D. Cobut, J.-D. Reymond and M.L. Sawley, Parallel multi-block computation of incompressible ow for industrial applications, in Parallel Computational Fluid Dynamics: Implementations and Results using Parallel Computers, A. Ecer et al. (eds), (North-Holland, Amsterdam, 1996), pp. 447-454. [17] Y.P. Marx, A numerical method for the solution of the incompressible Navier-Stokes equations, IMHEF Report T-91-3 (1991).
5 Figure captions Figure 1. Normal stress di erence for the ow of an Oldroyd-B uid through a 4:1 contraction. 7
Figure 2. Fluid particle locations computed for di erent positions along the mixer (from top left to bottom right: at inlet, near the end of each of the six mixing elements, and at the outlet) for ow with Re = 100. Figure 3. Fluid particle locations computed at the mixer exit for di erent values of Reynolds number (from top left to bottom right: Re = 15, 25, 50 and 100).
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