Hybrid Mixed Discretization Methods for Combustion Problems in

Hybrid Mixed Discretization Methods for Combustion Problems in Porous Media Peter Knabner and Gerhard Summ Friedrich-Alexander-University Erlangen-Nurnberg, Institute of Applied Mathematics, Martensstrae 3, D-91058 Erlangen, Germany. E-mail: fknabner,[email protected]

Abstract. This paper presents numerical methods for the simulation of combus-

tion processes in porous media. The equations governing the ow in the porous medium are discretized by the mixed nite element method on Raviart{Thomas elements of lowest order. Hybridization is applied to transform the resulting equations into a form more convenient for the solution by multigrid methods. Special emphasis is put on the elimination of the mass ux values which requires the solution of local nonlinear subsystems. For the discretization of the species and energy conservation equations a cell-centered nite volume scheme is used. Some results of numerical simulations demonstrate the in uence, which layers of dierent porosity of the solid matrix can have on the localization of the combustion zone.

1 Introduction In recent years, the request for environment-friendly combustion systems has led to the development of a new burner concept: combustion in porous media. In the developed burners a premixed gas-air mixture is constrained to

ow through and combust within the pores of a porous medium, typically a ceramic foam. Several authors performed numerical calculations to study the behaviour of dierent types of porous media combusters. In [6] a burner is considered without any additional cooling, in which radiation is the only ame stabilization mechanism. The authors perform transient simulations of a onedimensional model with given ow rate using nite dierence expressions. By contrast, the authors in [7] consider a combuster-heater system with coolant tubes embedded in the porous medium. They develop a mathematical model for the two-dimensional stationary case and simulate uid ow, heat transfer and combustion in the porous matrix using a control volume nite dierence approach. In both cases the porous medium is assumed to be homogeneous. We will consider another type of porous medium burner, which was developed by Trimis and Durst [9] and is characterized by the following facts:  The porous medium consists of several layers with dierent porosity and thermo-physical properties. Typically we have low porosity near the inlet and high porosity in the combustion zone.  The outer wall of the combuster is cooled by water.

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Peter Knabner and Gerhard Summ

We deal with a two-dimensional time-dependent model for this type of burner. After introducing this model we want to present numerical methods for the discretization of the governing equations and the solution of the resulting system of nonlinear equations. Finally we present results of numerical simulations that show how layers of dierent porosity contribute to the localization and stabilization of the ame in the porous medium combuster.

2 The Mathematical Model We restrict our considerations here to the case of an exothermic one-step reaction mechanism. Then the mathematical model consists of the following equations which have been derived by the method of volume averaging [8]. The ow in the porous medium is governed by the continuity equation  @t  + div (m) = 0 ; (1) the Darcy-Forchheimer equation, a nonlinear extension of Darcy's law,   c 1 +p F jmj m + rp = 0 (2)

 k

k

and an equation of state; we use the ideal gas law p = R0 T W : (3) The unknowns in this system are the density , the pressure p and the mass

ux density m. The porosity  and permeability k of the porous medium, the viscosity  and molecular weight W of the gas mixture are known as well as the Forchheimer constant cF and the universal gas constant R0 . The species conservation equation  @t (y) + div (m y ? D ry) = ?r_ (4) describes the transport of the reactant in the pores. Here y is the mass fraction of the reactant, D the mass diusion coecient and r_ the rate of consumption of the reactant given by the Arrhenius model r_ = By exp(?E=R0T ) ; where B is the frequency factor and E the activation energy. Assuming thermal equilibrium between the gas mixture and the solid the heat transport is described by the energy conservation equation ?
@t (cp  + (1 ? )cs s ) T ?  @tp +div (cp m T ? e rT ) = Qr_ + FQ ; (5) where T denotes the temperature and e = g +(1 ? )s the eective heat conductivity. The speci c heat at constant pressure cp of the gas mixture, the speci c heat cs and density s of the solid are assumed to be constant. They are known as well as the heat conductivities g and s of the gas and the solid, resp., the heat of reaction Q and the power density of an arti cial heat source FQ .

Mixed Discretization Methods for Combustion in Porous Media

3

3 Numerical Solution of the Model Equations For the numerical simulation of the processes inside a porous combuster, we implemented the following algorithm in the frame of the toolbox UG [2], which provides numerical procedures for computations on unstructured grids. Using the method of lines we rst discretize the governing equations in space { details will be explained in the sequel { and integrate the resulting system of ordinary dierential equations using the implicit Euler method. This way we get in each time step a nonlinear equation system. For the solution of this system we decouple the ow problem from the transport equations and solve the resulting subsystems alternately in a xpoint iteration, which terminates, when the solution processes of both subsystems are converged (cf. Fig. 1). Each subsystem is linearized by Newton's method and the resulting linear problems are solved by appropriate multigrid methods.

m



ow problem solver



; p

transport problem solver





6

?

T ; y

Fig. 1. Solution of the problem by xpoint iteration of the two subsystems

3.1 Spatial Discretization and Solution of the Flow Problem

To de ne the discretization in space we rst have to introduce a decomposition Th of the computational domain (the combuster) into triangular elements K . For every K 2 Th we denote by Pk (K ), k  0, the set of polynomials of degree  k on K . For every edge e 2 Eh , being the set of edges of Th , we de ne a unique unit normal vector ne . To preserve the form given by (1) and (2) and to get a good approximation for the mass ux density m, which appears also in equations (4) and (5), we use the mixed nite element method on Raviart{Thomas elements of lowest order (see e.g. [4]) for the spatial discretization of (1) and (2), as proposed in [5]. Thus m is approximated by mh 2 Vh , where



n

Vh := RT0( ; Th ) := vh 2 H (div; ) vh jK 2 RT0(K ) 8 K 2 Th

and RT0 (K ) is de ned by

x ?
2 RT (K ) := P (K ) + P (K ) : 0

0

y

0

o

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Peter Knabner and Gerhard Summ

Note that any vh jK 2 RT0 (K ) is uniquely de ned by the ux across the edges of K Z ve := vh jK
ne ds ; e  @K ; e

and that the property vh 2 H (div; ) requires the continuity of these ux values. Using the basis fwe ge2Eh of Vh satisfying

 we
nf ds = 10 ifif ee =6= ff f

Z

the corresponding degrees of freedom are given by the uxes ve , e 2 Eh . Corresponding to the choice of Vh the pressure p is approximated by a piecewise constant function ph 2 Qh, where Qh is de ned by







Qh := qh 2 L2 ( ) qh jK 2 P0 (K ) 8 K 2 Th :

For simplicity we consider the case of Dirichlet boundary conditions p = g on @ . Then the discrete mixed formulation reads as follows: Find (mh ; ph) 2 Vh  Qh such that for every (vh ; qh ) 2 Vh  Qh

Z



  cF  Z p + j m j ( m
v ) dx ? div(vh ) ph dx h h h (ph ) k k

Z + g (vh
n) ds = 0 Z Z @

 @t ((ph )) qh dx + div (mh ) qh dx = 0 : 1





(6a)

(6b)

To be able to use the multigrid algorithm proposed in [3] for the solution of the linearized resulting equations, we apply the following implementational technique, called hybridization, to our discrete mixed formulation. We eliminate the continuity constraints in the de nition of Vh and enforce the required continuity instead through additional equations involving Lagrange multipliers de ned on the edges e 2 Eh . Thus we replace Vh by

n

?

Vh := RT?1( ; Th ) := vh 2 L2 ( )


2 v j 2 RT (K ) 8 K 2 T o hK 0 h

and de ne the space of Lagrange multipliers by





Z

gh := h 2 L2 (Eh ) h je 2 P0 (e) 8e 2 Eh ; (h ? g) ds = 0 8 e  @

e

where Eh = [e2Eh e. Then the hybridized mixed formulation reads as:



;

Mixed Discretization Methods for Combustion in Porous Media

5

Find (m  h; ph; h) 2 Vh  Qh  gh, such that for every (vh; qh; h) 2

Vh  Qh  0h Z 1   cF  XZ p + j m  j ( m 
v  ) dx ? div (vh ) ph dx h h h (ph ) k k K 2T hK

Z

+

X Z

K 2Th @K

 @t ((ph )) qh dx +

XZ

K 2Th K

X Z

K 2Th @K

(7a)

h (vh
nK ) ds = 0 div (m  h) qh dx = 0 (7b)

h (m  h
nK ) ds = 0 : (7c)

The solutions m  h and ph of (7a)-(7c) coincide with the solutions mh and ph of (6a)-(6b), in particular m  h 2 Vh . Therefore we can drop the bar in the

following considerations. Furthermore the additionally computed Lagrange multipliers can be used in the reconstruction of a more accurate approximation for p (for more details, see e.g. [4] and the references cited there). Note that every vh 2 Vh is given by the degrees of freedom vK;e , which are de ned by

Z

vK;e = vh jK
nK ds ; K 2 Th ; e  @K ; e

where nK denotes the unit outer normal of K , and that the corresponding basis functions w K;e vanish in n K . Thus the unknown functions mh , ph and h have the representation

mh =

X X

K 2Th e2@K

mK;e w K;e ; ph =

X

K 2Th

pK K and h =

X

e2Eh

e e ;

where we denote by K and e the characteristic function of the element K and the edge e, resp.. Using the basis vectors wK;e, K 2 Th , e  @K , as test functions we obtain from (7a) nonlinear equations of the form

0 Z @ X mK;e (ph ) 1

e@K

K

1 0 1 X c  @ + pF mK;f wK;f A (wK;e
wK;e) dxA k k f @K

(8)

+ pK + e =: FK;e = 0 :

Due to the discontinuity in the de nition of Vh the unknowns mK;e , e  @K , appear only in the equations FK;f , f  @K , corresponding to the same element K . This fact can be exploited to compensate the main drawback of this procedure, the introduction of additional degrees of freedom, by solving

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Peter Knabner and Gerhard Summ

the local systems (FK;f )f @K and eliminating the approximate mass ux values mK;e from the global system. Unfortunately, owing to the nonlinearity of (8) it is not possible to nd a closed form solution for the mK;e . Nevertheless by monotonicity arguments it can be proven, that the local subsystems are uniquely solvable and their jacobian is invertible. Thus it is possible to compute the approximate mass

ux values mK;e , e  @K , by a local Newton iteration during the assembling procedure. The elimination of the mK;e introduces a nonlinearity into the originally linear equations arising from (7b) and (7c). Using Newton's method to solve these equations we have to compute the jacobian of the global system. By the implicit function theorem this also requires the invertibility of the local jacobians (@FK;f =@mK;e )f;e@K . The linearized system after time discretization is solved by the multigrid method using (block-)Gau-Seidel iteration as smoothing operator. As the resulting global system for the approximate pressure degrees of freedom pK , K 2 Th , and the Lagrange multipliers e , e 2 Eh, is equivalent to a certain nonconforming nite element method (cf. [3]), we can apply the intergrid transfer operators developed for this nonconforming method in the computation of the coarse grid correction. For the de nition of these intergrid transfer operators we refer to [3].

3.2 Spatial Discretization of the Transport Problem Equations (4) and (5) are discretized by a cell-centered nite volume scheme. Thus Th and yh , the approximate solutions for temperature and mass fraction of the reactant, resp., are also functions in Qh . In contrast to the discretization of the ow problem, the corresponding uxes are not computed directly, but are approximated using an upwind scheme for the convective part and a dierence scheme for the diusive part. This saves computational cost by reducing the number of degrees of freedom, but restricts the construction of the triangulation Th , as it inhibits the use of triangles with obtuse angles [1]. As (4) and (5) are coupled by the nonlinear reaction term r_ , the discrete equations arising from their nite volume discretization are linearized by Newton's method. The resulting linear equation system is solved by a multigrid algorithm. Here the smoothing operator is given by a block-ILU decomposition and the intergrid transfer operators are chosen according to the intergrid transfer operators for the element values given in [3].

4 A Case Study: Comparison of Homogeneous and Layered Porous Media We consider a simple rectangular geometry, which represents the upper half of a porous combuster and may consist of two layers with dierent porosity (cf. Fig. 2). The lower boundary (?S ) is assumed to be a line of symmetry,

Mixed Discretization Methods for Combustion in Porous Media

7

while the upper boundary (?C ) corresponds to the cooled outer wall. The cooling at the boundary part ?C1 is stronger than at ?C2 . ΓC 1

ΓC 2

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

ΓI A

ΓO B

ΓS

ignition zone

Fig. 2. Upper half of a rectangular burner geometry Beginning with a stationary ow eld, the combustion process is started by an arti cial heat source in the ignition zone. After the ignition of the fuel, the arti cial heat source is switched o and the simulation is continued until a stationary solution is reached. Only the values for porosity and permeability vary between the computations, all other coecients remain unchanged. For a con guration with dierent values for porosity and permeability in the two layers ( = 0:3; k = 10?8 m2 in layer A and  = 0:8; k = 10?7 m2 in layer B) the steady state combustion zone is located near the interface between the two layers, as can be seen from Fig. 3. It cannot move towards the inlet, because the heat loss in layer A is too strong due to the high eective heat conductivity e . 5

.0 0.06 0

60

0

1200

0

00

0.04

00

10

0

10

80

0.04

80

0.06

0.02

0.02

−0.02

0

−0.04

80

1200

0

80 60

0

10

00

−0.06

00

10

−0.04

0.01

0

−0.02

0.03

0

−0.06 0 .0

5

0.05

0.1

0.15

0.2

0.25

0.3

0.05

0.1

0.15

0.2

0.25

0.3

(a) (b) Fig. 3. Steady state solution for layers of dierent porosity: (a) Temperature distribution and (b) mass fraction of reactant

Simulations for homogeneous porous media lead to totally dierent results. For values of  = 0:3 and k = 10?8 m2 the combustion process dies out after switching o the heat source. In this case e is high in the whole combuster, such that more heat is lost through conduction towards the cooled wall than produced by the combustion process. On the other hand, if we consider the values  = 0:8 and k = 10?7 m2 , the eective heat conductivity is

8

Peter Knabner and Gerhard Summ

low all over the combuster. Thus the produced heat cannot be carried away fast enough, the combustion zone moves towards the inlet and nally reaches the inlet. Further simulations for homogeneous porous media with dierent, but constant porosity and permeability values show either extinction of the ame or movement of the combustion zone towards the inlet. This indicates clearly that layers with dierent porosity contribute signi cantly to the localization of the combustion zone inside the porous medium combuster.

5 Conclusion We presented an algorithm for the two-dimensional transient numerical simulation of combustion processes in porous media. The discretization of the governing equations is based on the mixed nite element method on Raviart{ Thomas elements of lowest order. We discussed the hybridization of the equations arising from the ow problem, that requires the elementwise solution of nonlinear systems, and mentioned the main features of the cell-centered nite volume scheme used for the discretization of the transport equations. The presented results of simulations indicate clearly, that layers with dierent porosity of the solid matrix contribute signi cantly to the localization of the combustion zone.

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