A Wavelet-Galerkin Method applied to Separation

A Wavelet-Galerkin Method applied to Separation Processes R. v. Watzdorf1, K. Urban2 , W. Dahmen2, and W. Marquardt1

1 2

Lehrstuhl fur Prozetechnik, RWTH Aachen University of Technology Institut fur Geometrie und Praktische Mathematik, RWTH Aachen University of Technology, 52064 Aachen, Germany

1. Continuous Modeling of Fluid Mixtures

Many uid mixtures encountered in chemical and hydrocarbon processing industries are ill-de ned in the sense that they contain far too many components for a detailed compositional analysis and subsequent modeling in terms of pure component mass balances. Common examples of such mixtures are frequently related to processes of high economic relevance and include petroleum and reservoir uids as well as polymer solutions and polyreaction systems [6]. The concept of continuous thermodynamics is a well-established approach for the modeling of these mixtures [2, 6]. The compositional complexity of the mixture is represented in terms of a time-dependent continuous distribution function F(; t) of some characterizing uid property  (e.g. molecular weight, natural boiling point or Single Carbon Number). The relation between the discrete mole fraction of any component and the distribution function is given by xl (t) =

Z


l

F(; t)d :

(1.1)

Using de nition eq. (1.1), a complete framework of thermodynamic relations analogous to the discrete case can be derived [6, 13]. The most simple, generic model of a separation process is a single equilibrium ash unit. Applying the usual assumptions1 (i.e. negligible vapor holdup, equilibrium, perfect stirred liquid phase, no chemical reactions) and assuming a constant molar holdup in the vessel, the model equations for the general dynamic case relating the distribution functions F V(; t) and F L(; t) of the vapor and liquid phase read as follows: L P L V (1.2) n @F @t = PF ? LF ? V F ; 0 = FV ? K FL ; (1.3) 1

These assumptions are no prerequisites of the Wavelet-Galerkin method. Their only purpose is to furnish a simple illustrative dynamic model of a multicomponent separation process.

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0 = 1?

Z

1

0 Z

1

F Ld ;

(1.4)

0 = 1? F V d ; (1.5) 0 0 = P ?L?V ; (1.6) 0 = P ? V : (1.7) L V The phase equilibrium coecient K(T; p; F ; F ), which is a functional of the unknown distribution functions F(; t) and a function of temperature T and pressure p, is provided by some suitable thermodynamic property correlation. A simple control law based on the degree of vaporization eq. (1.7) is used instead of the energy balance. As in the discrete case, the system of equations is linearly dependent and the normalization constraint eq. (1.4) can be dropped. 2. Solution Techniques

The set of model equations (1.2) to (1.7) can be considered as an operator equation Lu = f in a certain Hilbert space H equipped with an inner product (
;
)H. Here L : H ! H is some operator, f 2 H is given and u 2 H is the unknown solution.

2.1 Method of weighted residuals { Petrov{Galerkin Method A well known solution technique is the Method of Weighted Residuals, also referred as Petrov{Galerkin method. Let Ym and Zm be nite dimensional subspaces of H spanned by y1 ; : : :; ym and z1 ; : : :; zm , respectively. The unknown u is approximated by an element um of the trial space Ym : P um = mi=1 ai yi . The unknown coecients ai are determined by testing the residual R(um ) := Lum ? f by the elements of the test space Zm , i.e.
? (2.1) R(um ); zi H = 0; 8i = 1; : : :; m: The Galerkin method uses the same spaces for both trial and test space. A detailed discussion of the application of MWR-techniques to engineering problems can be found in Villadsen and Michelsen [14].

2.2 Multiscale Methods In general, multiscale methods are approximation processes, where a sequence of nested scales is used for approximation. The initial solution can be computed eciently on a coarse scale. To obtain a more exact approximation the relationship of dierent scales is used such that ner details are added to the approximative solution, see [7].

Wavelet-Galerkin Method applied to Separation Processes

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In terms of a Petrov{Galerkin method the single trial (and test) S space is replaced by a sequence of nested subspaces V = fVj gj 0 such that j 0 Vj is dense in H. These spaces are often given in terms of their bases, i.e. Vj = span j ; j := fj;k : k 2 Ij g; (2.2) where j;k 2 H and Ij denote some set of indices. The detail spaces are given by some appropriate complements Wj of Vj in Vj +1 , i.e. Vj +1 = Vj  Wj : (2.3) Then one is interested in a nding a basis j for the complement space Wj : Wj = span j ; j := f j;k : k 2 Jj g: (2.4) With the notations W?1 := V0 , ?1;k := 0;k and J?1 := I0 we obtain an alternative expression for Vj : Vj =

j ?1 M

i=?1

Wi

(2.5)

and an approximation uj 2 Vj of an element u 2 H can either be written in

single scale representation

uj =

X

k2Ij

bj;k j;k

(2.6)

or in multiscale representation uj =

j ?1 X X

i=?1 k2Ji

ai;k i;k :

(2.7)

This approach gives rise to ecient compression techniques [8]. If Wj is the orthogonal complement and if the j are orthogonal bases (ONB) for Wj , S then := j ?1 j is an ONB of H. If furthermore j constitutes an ONB of Vj the following relations hold: bj;k = (u; j;k)H ; ai;k = (u; i;k )H : (2.8)

2.3 Wavelets

Wavelets form a tool for constructing stable bases of H = L2 (R). The basis j of the space Vj is constructed by means of dyadic dilations and translations of a single scaling function : j;k () := 2j=2(2j  ? k); k 2 Ij := Z: (2.9) Under certain conditions on the generator  the sequence V is called a multiresolution analysis [11]. Similarly we have k 2 Jj := Z: (2.10) j;k () := 2j=2 (2j  ? k);

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If Wj is the orthogonal complement we obtain orthogonal wavelets [9]. In this case the transformation Tn , which relates the single and the multiscale coef cients is the Fast Wavelet Transform [5]. If V is a multiresolution analysis, then a re nement equation is valid for : X () = k (2 ? k) ; fk gk2 2 `2 (Z) : (2.11)

Z

Z

Z

Z

k2

The fact that the space Wj is contained in Vj +1 gives rise to an equation analogous to eq. (2.11) for the wavelet : X () = k (2 ? k) ; f k gk2 2 `2 (Z) : (2.12) k2

For many problems the solution does not extend over the whole space R. Hence one is interested in a basis for bounded intervals, e. g. for the space L2 ([0; 1]). There are several papers considering this problem, e.,g. [1, 4]. However, for the simplest choice of , namely H := [0;1] , where [0;1] denotes the characteristic function for the interval [0; 1], the construction of an ONB of L2 ([0; 1]) is very easy. The corresponding orthonormal wavelets are known as the Haar wavelets H and will be considered in the following.

2.4 Trial solution formulation Using the Wavelet{Galerkin method with Ym = Zm = Vn , m = 2n, gives rise to a formulation of the trial solution ~ t) = F(;

nX ?1

X

i=?1 k2Ji

ai;k (t) i;k () =

X

k2In

bn;k (t)n;k ()

(2.13)

in terms of the multiscale or single scale basis, respectively [10].

2.5 Representation of concentration distributions The composition of the process streams is expressed in terms of a possibly large sequence of mole fractions x 2 Rm. Without loss of generality equidistant intervals can be assumed, e. g. l = 2?n l. Using de nition (1.1), a relation between the continuous distribution function F  ();  2 fP;L;Vg and the mole fractions x can be obtained using the single-scale basis Hn : xl (t) = =

Z


l

Z

F  (; t) d =

Z

X


l k2In

bn;k (t)Hn;k () d

bn;l (t)Hn;l () d = 2? 2 bn;l (t) :

R

n

(2.14) (2.15)

Wavelet-Galerkin Method applied to Separation Processes

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2.6 Spatial discretization By virtue of orthonormality the projection eq. (2.1) of the residuals onto the H : i 2 f?1; : : :; ng; k 2 Ji g leads to the following simple trial functions f i;k expressions for the mass balance eq. (1.2) and normalization equation (1.5) daL n dti;k = PaPi;k ? LaLi;k ? V aVi;k fi 2 f0; : : :; ng; k 2 Jig ; (2.16) aL?1;0 = 1 ; (2.17) V a?1;0 = 1 : (2.18) However, chemical engineering models comprise usually a large number of algebraic constraints highly nonlinear in composition. The projection of eq. (1.3) gives rise to the following quantities: Z H () H ()d ? 1  i; j  n; k 2 Ji ; s 2 Jj : (2.19) K(T; p; F~ V; F~ L) i;k j;s To avoid the evaluation of these quantities, the continuous phase equilibrium function is expressed in terms of the single-space basis Hn , e.g. K(T; p; F~ L; F~ V) 2 Vn . Using the fast Wavelet transform Tn?1 the single-scale representation of the trial solution is obtained and the residuum of (1.3) can be formulated in terms of the single-scale basis Hn . The residuum is thus expressed by means of scaling functions of common level n. A subsequent fast wavelet transform provides the residuum of (1.3) in the multiscale representation, whose inner products with the weight functions can easily be evaluated due to orthogonality. This approach exactly recovers the discrete model without any deviations due to the continuous model formulation if the Haar basis is used. However, in order to simplify the calculations, a continuous formulation of the phase equilibrium function independent of the distribution functions F  can be applied. For the given implementation, a local thermodynamic model [3] relating the continuous phase equilibria coecient to the fugacity of a reference component and the distribution variable  was tted to the rigorous thermodynamic properties: S + A2 () : ln (K()P) = A1() ln fref (2.20) Using this simple relation the inner products eq. (2.19) can be evaluated analytically giving rise to a system of dierential-algebraic equations (DAE) which can readily be implemented in the dynamic simulation environment Diva [12] and integrated by standard DAE algorithms. Figure 2.1 gives a comparison between the steady state solution for the liquid and the vapour phase mole fractions calculated by the Wavelet-Galerkin method and by a full discrete model. As expected the composition pro les are almost identical, the deviations are resulting from the interpolation of the physical property routines due to the formulation (2.20) of the phase equilibrium coecient in the wavelet ash. Note that this is no error but merely a

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dierent modeling philosophy compared to the assumption of a piecewise constant phase equilibrium coecient, which would exactly recover the discrete model. 0.14 Feed wavelet flash 0.12

Feed discrete flash Vapor wavelet flash

Mole fraction

0.1

Vapor discrete flash Liquid wavelet flash

0.08

Liquid discrete flash

0.06

0.04

0.02

0 0

10

20

30 40 Component index

50

60

70

Fig. 2.1. Composition pro les of equilibrium ash calculated with WaveletGalerkin method and full discrete model.

0.08

Molefraction

0.06 0.04 0.02 0 0 500

Step response of liquid phase composition to change in boilup ratio calculated with Wavelet-Galerkin method.

Fig. 2.2.

1000

70 60

1500

50 40 30

2000 Time

[s]

20 2500

10 0

Component Index

The implemented model can be used to perform dynamic simulation runs. Figure 2.2 shows a step response of the liquid phase composition to a change in the evaporization rate  (corresponding to an increase of heat duty) at time t = 500s from ? = 0:1 to + = 0:9 calculated by the wavelet ash model. The composition pro le is pushed toward the high boiling components and the amount of lights in the liquid phase is strongly reduced.

Wavelet-Galerkin Method applied to Separation Processes

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3. Conclusions

A new approach for the modeling of complex multicomponent mixtures has been sketched using a local, wavelet based representation of the unknown distribution functions. It incorporates the full discrete model as a special case and requires no involuntary additional assumptions on the physical model. A rst implementation of a dynamic model for a separation process highlights the practical applicability of the approach. The technique outlined can be generalized to accommodate any problem speci c wavelet family and allows for a model reduction by tresholding of the multi-scale representation of the trial solution. Further work will address these issues. References

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