2. Fixed-bed reactor modelling

Shape 2ptimization of a )ixed-bed 5eactor using

$dditive 0anufacturing

Alexis Courtais^a,*, François Lesage^a, Yannick Privat^b, Cyril Pelaingre^c, Abderrazak M. Latifi^a

aLaboratoire Réactions et Génie des procédés, CNRS, Université de Lorraine, Nancy, France

bInstitut de Recherche Mathématique Avancée, CNRS, Université de Strasbourg, France

cCentre Européen de Prototypage et Outillage Rapide, Saint-Dié des Vosges, France alexis.courtais@univ-lorraine.fr

Abstract

This paper deals with a geometric shape optimization of a fixed-bed reactor. The objective is to determine the shape of the packing that maximizes the reaction conversion rate sub- jected to the process model equations, operating constraints (iso-volume, energy dissi- pated by the fluid) and manufacturing constraints. The process model is described by the mass balance equations and Navier-Stokes equations. Incompressible fluid, a homogene- ous first order reaction and steady-state conditions in the reactor are the main assumptions considered. The free software OpenFOAM is used as CFD solver. The optimization ap- proach developed is based on the adjoint system method and the resulting algorithm is tested on a two-dimensional fixed-bed reactor in laminar flow regime. The results show a significant improvement of the conversion rate and the optimal shape obtained with the manufacturing constraints can be easily printed by means of an additive manufacturing technique.

Keywords: Shape optimization, CFD, Fixed-bed reactor, Additive manufacturing.

1. Introduction

The objective of shape optimization is to deform the outer boundary of an object in order to minimize or maximize a cost function, such as the performances of a process, while satisfying given constraints. Historically, the shape optimization methods have been used in cutting edge technologies mainly in advanced areas such as aerodynamics. However, they have recently been extended to other engineering areas where the shape greatly in- fluences the performances. For example, in hydrodynamics, the shape of a pipe that min- imizes the energy dissipated by the fluid due to viscous friction was analyzed (Henrot and Privat, 2010, Tonomura et al., 2010, Courtais et al., 2019).

In chemical engineering however where the shape of unit operations (e.g. reactors, tanks, stirrers, pipes…) is an important design parameter, the shape optimization has not been extensively investigated. This important issue deserves therefore to be addressed and will probably result in a paradigm shift in optimal design and operation of processes.

Basically, there are three types of shape optimization: parametric, geometric and topo- logic. In this paper, only geometric optimization is considered. The objective is to develop an optimization approach based on Hadamard method using the adjoint system equations.

The case study is a 2D fixed-bed reactor with a laminar single phase liquid flow where a homogenous first order chemical reaction takes place. The objective is to determine the shape of the packing that maximizes the reactor conversion rate.

http://dx.doi.org/10.1016/B978-0-12-823377-1.50324-4

2. Fixed-bed reactor modelling

The optimization method developed in this work is based on the process model equations that describe the flow through the fixed-bed reactor. A two-dimensional model is there- fore developed and involves mass and momentum balance equations. Figure 1 shows the schematic representation of the fixed-bed considered.

Figure 1: Schematic representation of the fixed-bed reactor

The studied domain consists of the free volume Ω and its boundaries given by the union of the inlet (Γin), outlet (Γout), lateral wall (Γlat) and free (Γ) limits.

The momentum transport is described by the following Navier-Stokes equations along with the associated boundary conditions:

out lat

in

p

p

*

*

*

* : :

°°

°

¯

°°

°

®

­

˜

’

’

’

˜ '

on on

on in in

0 ) , (

0 0

0

QQ

U U

U U

U

U U U

in

V Q

(1)

where ν is the kinematic viscosity of the fluid, V(U,p) 2QH(U)p,, the viscous stress tensor, p the fluid absolute pressure, I the identity matrix and ε(U) the strain tensor.

The mass balance equations and their associated boundary conditions are given by Eqs.

(2). It is important to point out that the reaction takes place only in the bulk of the reactor, i.e. in Ω and not on the walls Γ and Γlat.

*

*

*

* :

°°

¯

°°

®

­

w w

’

˜ '

^lat

out in in

n C

C C

kC C C

D

on on

in 0

0 U

(2)

where C is the reactant concentration and D the constant diffusion coefficient of the reactant.

3. Optimization problem formulation

The formulation of the shape optimization problem requires the definition of a perfor- mance index, decisions variables and constraints. These ingredients are detailed below.

Ω

3.1.Performance index

The aim of this work is to determine the packing shape (i.e. the position and the shape of the free boundary Γ) of the fixed-bed reactor that maximizes the reaction conversion rate or minimizes the average outlet concentration of the reactant. The performance index is therefore defined by Eq. (3) as:

³

:

out

Cd

J( ) V ₍₃₎

3.2.Decision variable

The decision variable is defined by the free boundary Γ that will evolve with the iterations of the optimization algorithm. The other boundaries are fixed.

3.3.Equality and inequality constraints

The optimization problem is subjected to different constraints. The most obvious ones are given by the process model equations (Eqs.(1-2)). The other constraints consist of : - An iso-volume constraint introduced in order to guarantee the same residence time between initial and optimized shapes (Eq.(4)).

- An inequality constraint on the energy dissipation by the fluid due to viscous friction and is given by Eq. and (5). Such a constraint is relevant since the energy dissipation and the pressure drops are directly correlated.

1

( ) ( ) 0

C V : : V

0

2 2

2

2 ( ) 2 ( ) 0

C Q H U dx Q H U dx

: :

³ ³

- Since the resulting optimal shape will be fabricated by means of a 3D printing technique, manufacturing constraints should be accounted for in the optimization problem. They are of inequality type and impose minimum values on the pores width (Ω domain) and on the packing thickness.

4. Optimization method

The shape optimization approach developed is a gradient-based method and uses the Hadamard’s boundary variation method (Henrot and Pierre, 2006). The gradient, called shape gradient, is computed by means of adjoint system method. The shape gradient G is defined on the free boundary (Γ) and depends on the different state variables (U,p and C) and their associated adjoint states introduced by the method (Ua,pa and Ca). The gradient will allow us to compute the mesh displacement vector V that decreases the Lagrangian of the constrained optimization problem given by (Eq.(6)).

) (

) (

0 2 2

0) 2 ( ) 2 ( )

( ) ( )

(

³ ³ ³

:

* : : :

: Cd V V U dx U dx

L _{V E}

out

H Q H

Q O O

V ₍₆₎

The vector field V is the solution of Eqs. (7).

in out

in on

on

lat

G

' :

­ ° * * *

® ° ’ *

¯

out lat

*

* V V 0

V 0 Vn n

(7)

Once the vector field V is computed, the next step in the method is to move all meshpoints according to the following relation:

) )(

1 ( i

i t :

: X V (8)

where t is the method step and must be chosen small, X is the vector of each meshpoint coordinates. More details on the shape gradient calculations can be found in (Courtais et al., 2020).

5. Optimization algorithm

The optimization algorithm developed has been implemented using C++ language within the free and open source OpenFOAM software (Weller et al., 1998). The latter solves PDEs, i.e. Navier -Stokes, mass balance system and their adjoint system equations, using finite volume method. The python library “pyFoam” is used to link the iterations to each other using its utility “pyFoamMeshUtilityRunner.py”. The algorithm proceeds as fol- lows:

1. Formulation of an initial shape and generation of the associated mesh using cfMesh and snappyHexMesh, two mesh utilities supplied by OpenFOAM.

2. Resolution of the four systems of PDEs. The pressure-velocity coupling equations are solved using SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) al- gorithm.

3. Computation of the shape gradient G and the mesh displacement V. The manufactur- ing constraints are considered at this stage. The pore constraint is considered using the OpenFOAM function “wallDist”. This function computes the distance between the cells and boundaries. The thickness constraint is taken into account by computing the local distance between the obstacle and its skeleton. The skeleton of an obstacle is the set of equidistant points from the obstacle on each side (see (Feppon et al., 2018) for more details).

4. The Lagrange multipliers associated to the volume and energy constraints are up- dated by means of the following relation :

ik ki

ik₁ O HC

O (9)

where ε is a parameter with a small value, k and i refer to the constraint and iteration respectively.

5. At the end of iterations, a test on the mesh quality is carried out through three criteria:

i. the maximum value of the mesh aspect ratio which is defined as the ratio of the longer side to the shorter side of a mesh.

ii. the mesh non-orthogonality defined by the angle between the vector linking two adjacent cell centers and the normal of the face connecting cells.

iii. the face skewness defined by the distance between the face center and the intersection of the vector linking adjacent cells with their common face con- sidered (see (Holzinger, 2015) for more details).

The upper bounds of these three criteria are 10, 65 and 3.8 respectively. If the mesh quality fails, remeshing process takes place.

6. A test on the convergence is done through the maximum displacement of the mesh.

If the latter is higher than 10^-5 m, then the algorithm goes back to step 2.

6. Results and discussion

Figure 2 presents the reactant concentration profiles in the initial configuration (a) of the fixed-bed reactor and in the optimized ones without (b) and with (c) the manufacturing constraints. It can be seen that in the initial configuration, a dead zone in the reactor inlet area (i.e. light zone on the figure) appears where the reaction conversion rate is very low, thus leading to lower reactor performances. In the optimized fixed-bed without manufac- turing constraints, the dead zone has disappeared, but the reactor exhibits very narrow channels (pores) which are not easy to manufacture. In the optimized shape with con- straints however, the dead zone is no longer there and all the channels can easily be printed. On the other hand, Figure 3 presents the residence time distribution (RTD) of initial and optimized (with manufacturing constraints) configurations and shows that the fluid flow is more homogeneous in the optimized configuration. Indeed, the standard de- viation of the RTD is three times lower in the optimized reactor (i.e. 75 vs 25s). Finally, the disappearance of the dead zone and the better homogeneity of the flow the improve- ment of the conversion rate and therefore of the performance index by is about 10%.

Figure 2: Initial configuration of the fixed-bed reactor (a), optimized shape without manufacturing constraint (b), optimized shape with manufacturing constraint (c)

(a)

(b)

(c)

Figure 3: Residence time distribution of initial and optimized shapes

7. Conclusions

In this work, a geometrical shape optimization based on the adjoint system method has been developed, implemented within OpenFOAM and used in order to optimize the con- figuration of a fixed-bed reactor where a first order homogeneous reaction occurs. The objective was to determine the shape that minimizes the average concentration of the reactant at the reactor outlet. The optimization was subjected to volume, energy and man- ufacturing constraints and to momentum and mass balance equations. A significant de- crease of the performance index is obtained which results in a substantial improvement of the reactor conversion rate.

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