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Jfaltigroap IJlffuflion Theory DE 8 6 011* 41 for Hteutral Atom Transport in Plasaas PDF

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Preview Jfaltigroap IJlffuflion Theory DE 8 6 011* 41 for Hteutral Atom Transport in Plasaas

MtiTO DISCLAIMER This report was prepared as an account of work sponsored by an agency or the United Stales Government. Neither the United Statu Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liabiJity or responsi­ bility for the accuracy, completeness, or usefulness of any in form at ion, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer­ ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, reconv mcndalioD, or favoring by Ihc United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. UCLA/PPG—-916 A Two-Di«ensional Finite Element n l U i1 Jfaltigroap IJlffuflion Theory D E 86 011*41 for Hteutral Atom Transport in Plasaas Mohammad Z. Hasan and Robert W, Conn Mechanical, Aerospace and Nuclear Engineering Department Center for Plasma Physics and Fusion Engineering University of California, Los Angeles Los Angeles, California 90024 February, 1986 UCLA/PPG-916 Submitted to "Journal of Computational Physics" fjr publication MTHBUTiW GF TW3 B0Sd-.KT IS UKLiaO tP ABSTRACT Solution of the energy dependent diffusion equation in two dimensions is formulated by multigroup approximation of the energy variable and general triangular nesh, finite element discretization of the spatial domain. Finite element formulation is done by Galerkin's method. Based on this formulation, a two-dimensional multigroup finite element diffusion theory code, FENAT, has been developed for the transport of neutral atoms in fusion plasmas. FENAT solves the multigroup diffu£"rin equation in X-Y cartesian and R-Z cylindrical/ toroidal geometries. Use of the finite element method allows solution of problems in which the plasma cross-section has an arbitrary shape. The accuracy of FENAT has been verified by comparing results to those obtained using the two-dimensional discrete ordinate transport theory code, DOT-4.3. Results of application of FENAT to the transport of limiter-originated neutral atoms in a tokamak fusion machine are presented. 1 I, INTRODUCTION The importance of the neutral atom distribution in a fusion plasma has long been recognized [l,2j. Neutral atoms influence the plasma particle and energy balance by undergoing ionization and charge exchange reactions in the plasma and by introducing hlgh-Z impurity atoms through processes such as sputtering and impact desorption. Neutral atom sputtering of limiter/divertor components and vacuum vessel walls in a fusion device affect the lifetime of these components, in addition to the problem of impurity production and plasma contamination. Also, knowledge of the energy spectrum of neutral atoms emerging from the plasma can be used for diagnostic purposes, such as in the determination of the plasma ion temperature. Fluid plasma codes used to describe plasma behavior in tokamaks, such as WHIST [31 or BALDUR [4], use the results of neutral calculation as input to plasma simulation. For all these reasons, the accurate and efficient description of neutral atom transport in complex geometries is a key problem in fusion research. In general, neutral atom transport ha3 to be solved numerically because of the non-homogeneity or spatial variation of the plasma temperature and density. Several numerical solutions in one-dimensional geometry are available. Burrell [5] has developed the code, NEUCG, based on Che solution of an integral equation for a one-dimensional cylindrical plasma. Audenaerde, Emmert, and Gordiner [6] have developed the one-dimensional slab geometry code, SPUDNUT, based on the numerical solacion ai an integral equation for the charge-exchange collision density. Hughes and Post [7] have written a one- dimensional cylindrical geometry code for neutral atom transport using the Monte Carlo method. Duchs et al. [8] solved for the neutral atom distribucion in one-dimensional cylindrical plasma using a charge-exchange generation method, allowing up to 10 successive generations of charge-exchange 2 neutrals. Garcia et al. [9J numerically solved for the neutral atom distribution in a one-dimensional slab plasma and in a half-space using the F„-method. Neutral atom distribution in multidimensional plasma geometries have been obtained using the Monte Carlo methods. Heifetz et al. (10] and Reiter and Nicolai [111 have reported multidimensional Monte Carlo codes for neutral atom transport. Codes written specifically for neutron transport, such as ANISN [12], DOT-4.3 [13] and TRIDENT" [14], can be used to calculate the distribution of neutral atoms in plasmas. Application of ANISN is reported in reference [15]. Often, a one-dimensional approximation of the fusion plasma is not realistic because there are localized sources of neutrals or because the geometry is not readily simplified. He have shown elsewhere [16,171 that energy dependent diffusion theory with Pj-Marshak boundary conditions provides very good accuracy for the transport of neutral atoms in fusion plasmas. A two-dimensional diffusion theory code for the transport of neutral atoms will be much faster then transport theory and Monte Carlo codes. Therefore, the code FENAT has been developed based on the finite element solution of the energy dependent diffusion equation in X-Y cartesian and R-Z cylindrical/toroidal geometries. The energy dependence is treated by the multigroup method. In section II, the multigroup diffusion equation is written in (x,y,f) toroidal coordinates. Finite element formulation of the solution by Galerkin's method is treated in section III, and the solution of the resulting system of algebraic equations and the chavacteristics of the code, FENAT, are presented in section TV. Section v contains verification of the accuracy of FENAT and the results of sample calculations for neutral atom transport in the TEXTOR tokamak [18]. 3 II. DIFFUSION EQUATION IN TOROIDAL GEOMETRY The steady state energy and space dependent diffusion equation for neutral atom transport is -7'Dfr,E)j'iXr,E)+a(r,E>iXr,E> - / <£' ^I.E' *E>#r,£' )+S(r,E) (!) t: v where <Kr,E) is the total neutral atom flux, D(r,E) is the diffusion coefficient, or (r,E) is the total plasma-neutral interaction cross-section (Ionization plus charge exchange), o (r,E'*E) is the charge exchange cross- section where a neutral atom with energy E' charge exchanges with a plasma ion of energy E, and S (r,E) is the volumetric source of neutral atoms. A common volumetric source would be recombination source which Is important in high density plasma. Treating the energy dependence by the multigraup method, the group diffusion equation corresponding to Eq. (1) becomes [19,20J G g=l,2,....G where g denotes a particular energy group. For P,-diffusion theory with isotropic charge exchange (scattering) cross-section, the diffusion coefficient D is given by l/3o . In Eq. (2), or , is the group to group ^ t *fi Re charge exchange cross-section. Evaluation of these group cross-sections is discussed in ref. [2]J. The multlgroup cross-sections used here are calculated by the code PLASMX [21]. Equation (2) represents a set of G partial differential equations coupled by the first term in the right hand 4 side. Equation (2) is valid for any coordinate system provided proper expressions are used for the gradient and divergence terms. This equation will now be written in the toroidal coordinate system (x.y,1^ shown in Fig. I. The metric (hj.h^h^j) of this system Is given by hj=h=l and h=(R+x). 2 3 Gradient and divergence terms in this system are [22] -A hj 3x * + h 3y y + h 3* * 2 3 and ,.„ - L~ ri_(!ln2 JB 2, Vi JB _LVi JB + + ( hlh2h3 ^ hI ^ h2 ^ ^ h3 where x,y and f are the unit vectors in the respective directions of the toroidal coordinates (x,y.*). Using these expressions in Eq. (2), with A=f and B=D 7$ , the flroup diffusion equation in toroidal coordinates (x,y, 1) becomes: is ^* i a ^ s ^ - OHXT VR+X>D-K " aso ^R+X)D -|" 77--2 -^R+x) V« + at.** B R R R G S a , + , + S (3) When written in the toroidal coordinates (r.S,1?), where x = rcosS and y = rsinf), Eq. (3) becomes the same as that obtained by Pomraning and Stevens [23] by first writing the transport equation in (r,8,19 and then taking the P. approximation of the angular flux to derive the Pj diffusion equation. With toroidal symmetry (i.e. Y-symmetry), Eq. (3) becomes 5 3<Ji _ 34 G (R+XT ax' s Sx sy g 3y t,g g g'-"gV v,g gl=1 — in domain fl . Without the factor (R+x) Eq. (4) is the group diffusion equation in X-Y Cartesian coordinates, and with (R+x) replaced by R and y by Z, it becomes the group diffusion equation in R-Z cylindrical coordinates. The boundary conditions considered for Eq. (4) can be written as: J* = S"' + £ Js (5) in a out where J and J t are the scalar current of neutral atoms at the surface, S in ou Q is the surface source and a is the reflection coefficient for the neutral atoms from the wall material. Using the Pi-Marshak expressions for J< and J in Eq. (5) [20] , the boundary conditions can be written after some out algebra, as; 9+ . * 2S — on 9S2, the boundary of the domain S2 . where -=-— is the normal derivative of neutral atom flux at the surface. on Special cases included in Eq. (6) are free surface boundary conditions (os=0), reflection conditions (o=l), and general albedo boundary conditions (a neither 0 nor 1). If there is no boundary source, S =0. 6 III. GALERKIN FORMULATION OF SOLUTION Finite element solution of Eqs. (4) and (6) is formulated by Galerkin's method. Since the diffusion equation is self-adjoint, formulation by the variational method will result in the same equations as by Galerkin formulation. The group diffusion equation can be written in operator notation L* - q (7) g g where, L = -V • D V + o S- t >g and q = E ^ • ^_ • , + S For clarity, the subscript or supercript 'g' will be dropped from now on except where ambiguity may arise. Let the neutral flux, 4>, be approximated as: * = [N] {*} (3) where, [N] = row vector of suitably choosen basis functions (functions of the spatial coordinates) {*} = column vector of unknown flux (nodal values) In Galerkin formulation, the weak form of Eq. (7) is [24] / [BJ (L+ - q) dV = 0 a 7 or, Q -/[N]TV • (DV<j.)dV + /[N]To (f dV - ! [N]T{ E a *, +S }dV = 0 fl a r n '=.is "*g v g where IN]™ is the transpose of [N]. The first terra of the above equation can be simplified by using vector identity and Gauss' divergence theorem [17]. When this is done, the above equation becomes 3* t J [H]T • D7« dV - / (Nj D ~ iS + I [NjT<J i dV - a an * n t T G / fNJ (I ".*,.*«,. + S > dV = 0 (9) In two dimensions, the basis functions, N, are functions of x and y. Therefore, from Eq. (8). ff«^fT<« OO) Let the domain ft be divided into a finite number of subdoraains or elements (triangular) as shown in. Fig. 2. Then the integrals over the domain ft and its boundary ail in Eq. (9) can be replaced by Che sura ox Che integrals over the domain, ft , of each element and over the boundary, 311 , of each boundary elements. Sow using Eqs. (10) and (6) for D -^~ In F.q. (9) and writing out m the gradient terms, the finite element equations for a single element, e, for the solution of the group diffusion equation, Eq. (4), in R-Z cylindrical/ toroidal coordinates is obtained. 8 2* /r^T-^'e-T <«**)". + 2* ^ V^ <*X**>". e + 2,r / 7 <TrS> [«]T<t> (R+x)dL + 2n / [N]T[Njo. {*}(R+x)d4 = e e p 2ir E a / fN]TlNj f*Jg'(R+x)dA + 2ir j [N]TSe(R+x) dA *• '=lg "ft e ft ff * e e 2S e 2ir /T^A- [N]T(R+X) dL (11) 3fiu ' e In Eq. (11), expressions for differential volume and surface elements (with toroidal symmetry), dV = 2<R + x)dzdy = 2<R + x) dAand dS = 2n(R + x)dL e e e e respectively, have been used. The integrations in E^. (11) can be evaluated either analytically or numerically depending <-xi the complexity of the known btsis functions, N(x,y). When this is done, we obtain an element matrix equation as ASC*}e = tq}8 (12) where A is a square matrix of size (n x n), n being the number of unknown nodal values in the element, e. Matrix representation for the whole domain £2 and its boundary 32 is obtained by properly adding Eq. (12) for all the elements in the domain, ft. To obtain explicit expressions for A and {qJ , we need basis functions, N(x,y). Let the neutral flux $ be represented by a linear polynomial in an element, e, = a + <*x + ay (13) 2 9

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Submitted to "Journal of Computational Physics" fjr publication .. C Arfken, Mathematical Methods for Physicists, Acad. Press, 1970 M.Z. Bason & R.H. Conn.
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