6o^P- ^OIIOI-- ^ ^^^. ^_ ^^^^^
Development of a Detailed Core Flow Analysis Code EGG-M—90249 for Prismatic Fuel Reactorst
DE90 013092
Ralph G. Bennett
Idaho National Engineering Laboratory EG&G Idaho, Inc.
Idaho Falls, ID 83415
The development of a computer code for the analysis of the detailed flow of helium in prismatic fuel reactors is reported. The code, called BYPASS, solves a finite difference control volume formulation of the compressible, steady state fluid flow in highly cross- connected flow paths typical of the Modular High-Temperature Gas Cooled Reactor (MHTGR). The discretization of the flow in a core region typically considers the main coolant flow paths, the bypass gap flow paths, and the crossflow connections between them.
Introduction
The detailed analysis of the core flow distribution in prismatic fuel reactors is of interest for Modular High Temperature Gas Cooled Reactor (MHTGR) design and safety analyses. Such analyses involve the steady state flow of helium through highly cross-connected flow paths in and around the prismatic fuel elements. Several computer codes have been previously developed for this purpose'' -2, However, since they are proprietary codes, they are not generally available for independent MHTGR design confirmation.
The previously developed codes are not known to consider the exchange or diversion of flow between individual bypass gaps with much detail. Such a capability could be important in the analysis of potential fuel block motion, such as occurred in the Fort St. Vrain reactor, or for the analysis of the conditions around a flow blockage or misloaded fuel block.
This work develops a computer code with fairly general purpose capabilities for modeling the flow in regions of prismatic fuel cores. The code, called BYPASS, solves a finite difference control volume formulation of the compressible, steady state fluid flow in highly cross-connected flow paths typical of the MHTGR. The discretization of a core region typically considers the main coolant flow paths, the bypass gap flow paths, and the crossflow connections between them. A few other special flow paths also exist and can be modeled, like the control rod and shutdown absorber material channels, or the target cooling channels in the New Production Reactor version of the MHTGR. BYPASS allows the discretization of coolant channels and bypass gaps into fluid cells. Depending on the detail required in the analysis, individual coolant channels may be treated separately or lumped together in a given fluid cell. Bypass gaps between fuel or reflector columns may similarly be lumped or treated separately. When treated separately, the bypass gaps need to be laterally connected along their edges, usually three at a time. This is accomplished by exchange flow junctions in BYPASS, which nxidel the exchange of mass, momentum, and energy in the Y-shaped junctions between bypass gaps (see Fig. 1). Up to two exchange flow junctions may be connected to a given cell. The crossflow of helium between the end faces of fuel blocks may be modeled in BYPASS with crossflow links that connect the main coolant fluid cells to the bypass gap fluid cells. Up to six crossflow links may be made to a given cell.
"fThis work was supported by the U.S. Department of Energy Assistant Secretary for Office of Nuclear Programs, ,'—S\v\
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Exchange^
Flows
Bypass Gaps Between Elements
I
Figure 1. Flow geometry in the BYPASS code.
Fluid Flow Equations
In this section, the fluid flow equations are derived for an individual cell. A more detailed treatment is found in the BYPASS Theory Manual.^ The derivation begins with ideal one-dimensional axial flow, with terms added to represent additional fluxes of mass, axial momentum, and energy. Additional equations are added to the set to conserve lateral exchanges of mass, momentum, and energy. The starting point for the derivation are the one-dimensional (axial) steady state equations of mass, energy, and momentum conservation for a compressible fluid:
j-{pVA) = 0 (1) -^{puVA) + p-^{VA) = 0 (2)
dz oz
/ . P A ^ + A ^ = 0 (3)
These equations are integrated over the appropriate fluid cell control volumes shown in Fig. 2 to yield the following control volume difference equations:
[pylA,-[pVlAs = 0 (4) [puV]^A^-[puVlA, + JPciVsA^-VsAs) = 0 (5)
Xe{Pc+P,)As{V^ + 2V,{V,-V,,)-V,]) + g,A,{p^-p,) = 0 (g)
For the mass and energy equation integrations, the control volume and fluid cell are identical. For the momentum equation integration, the control volume is staggered axially with respect to the fluid cell to assure numerical stability of the resulting difference equations. Because of the staggered grid, the integration of the top and bottom half-height momentum control volumes in the coolant channels results in obviously specialized equations that are not discussed here.
The relationship of neighbor cells to the central cell is shown in Fig. 2. The geometry illustrated in the figure is typical of a single bypass flow gap in prismatic fuel: The cell is long and wide, but relatively thin.
The front and back neighbor cells are solid graphite, which exert frictional drag forces on the flowing helium in the cell, and also heat the helium. The upper and lower neighbors are extended along the coolant channel that the cell resides in. The east and west neighbors are typically the four bypass gaps that are connected to the central cell by the Y-shaped junctions in hexagonal prismatic fuel. The crossflow neighbors are more remote cells that are connected to the cell by way of small gaps between the end faces of the fuel or reflector blocks. Crossflow connections are considered to connect the north (upper) ends of fluid cells. It should be noted that the arrangement of neighbors for a bypass gap is one of the more highly-connected in prismatic fuel. In this work a fluid cell could generally be connected to up to12 neighbors in total, i.e., 1 upper, 1 lower, 4 exchange flow, and 6 crossflow. The derivations will proceed for the most complex case, and the results may be simplified (by ignoring connections) and applied to simpler fluid cells, such as the main coolant channels, in a straighforward way. Guidance on the discretization of core regions is given in the BYPASS User's Manual.'^
Control Volume for Mass and Energy
Staggered Control Volume for
Axial Momentum
Figure 2. Control volumes for mass and energy, axial momentum, and lateral momentum. Control volumes shown are the same height as an MHTGR fuel/reflector block, but may be made arbitrarily smaller.
Additional Terms in the Cell Equations
The additional terms to be added to Eqs. 4-6 representing lateral fluxes of mass, energy, and momentum between cells, and additional physical effects are considered next. Terms are added to the cell continuity equation (Eq. 4) that reflect the lateral flux of mass to east and west exchange flow neighbors, and to crossflow neighbors. The cell continuity equation is written:
[P^L^N - [py]s^S +PCVEAX +PcywAx + Y.[P^SENSEyCR]A:R = 0 (7)
crossflows
Terms are added to the cell energy equation (Eq. 5) that reflect the lateral flux of internal energy and flow work to east and west exchange flow neighbors, lateral flux of internal energy and flow work to crossflow neighbors, convective heat flow from the front and back graphite walls, frictional heating of the fluid, and, work done against buoyant forces. The cell energy equation is written:
[P"^]^^N - [puVl^s + UPciVsAN - VSAS) + [U]PCVEAX + [u]pcV^A^
+Pc {VW + VE )AX + X {[P^^SENSEycR ]A:K + JPCPSENSEVCJIA:!! )
crossflows
^ u ^
(8)
Terms are added to the cell momentum equation (Eq. 6) that reflect the lateral flux of axial momentum to east and west exchange flow neighbors, lateral flux of axial momentum to crossflow neighbors, wall friction, buoyancy force, axial shear forces from east and west exchange flow neighbors, and loss of momentum due to sudden changes in flow area. The cell momentum equation is written:
'Ae{Pc+PLM^'N+2V,{V,-V,,)-V,]) + g,A,{p,-p,)
+[ys]Pc{ywc + yEc)ABHc + lys]Pc{ywL + ^^j^^THL + J^Pc^s max(F,^^,£Vc/('0)Ac«
cross/lows
v\v\
Additional Balance Equations
The above discussion treated the derivation of continuity, energy, and momentum balance equations for general fluid cells. This section derives the additional balance equations needed to close the set of equations.
Exchange flow is not a comnnon tenn in the MHTGR literature. It is defined in this paper as the lateral flow of mass, momentum, and energy between adjacent bypass flow cells. This commonly occurs between sets of three flow cells at a time, because of the hexagonal shape of the fuel. Such a connection of three cells along a common edge is called a 3-way junction, and is shown in Figure 2. A 2-way junction is also available for analysis with BYPASS, for laying out symmetric patterns of cells, and for refining a mesh of cells. The equation of continuity for a 3-way junction is simply the requirement that the mass flows per unit time along their shared edge sum to zero:
A A^i + Pa^^z + P34^3 = 0 (10)
The use of subscripts 1 to 3 in this equation is simply a convenience - each of these velocities is already defined as either an east or west velocity for their respective fluid cells, as are the densities and areas. The equation of continuity for a 2-way junction simply omits the third term in Eq. 10.
The momentum balance for exchange flow is derived by integrating the lateral forces on the control volume shown in Figure 2, consisting of one half of a fluid cell (either the east or west half). The momentum loss in one of these 'legs' is written:
^ o ^ x ( / ' y - A ) + /2/(A."i.^.25jAyJv;|W^c//c = 0 • (11)
Crossflow is a common term in the MHTGR literature, defined as the lateral flow of mass, momentum, and energy between neighboring main coolant and bypass flow cells. This commonly occurs where the end faces of the hexagonal fuel blocks butt together. The momentum loss in each crossflow link is just due to the friction pressure drop, plus an optional loss coefficient that accounts for form losses in the complex flow geometry. The crossflow link momentum balance between cells '1' and '2' is written:
«.(ft-P.)+4/(A±£..fl±ii,v„,D„]^(a±ft]]^y
+ ^ v c . ( ^ ) ^ = 0 <12)
Recall that crossflow is connected at the north end of the fluid cells. The normal MHTGR axial flow is downward, which would carry any crossflow of mass, momentum, and energy down into the central, acceptor fluid cell. However, should the flow reverse, the crossflow should then be donored into the upper fluid cell, not the central fluid cell. This prescription for 'double donoring' is not currently implemented in the BYPASS code, and may be a desirable modification in the future.
If the total core flowrate, WCORE. 'S specified instead of the core pressure drop, the following three equations are added to the system: A mass balance is defined over the axial flow channels:
Kor.+ Z K ] ^ ^ = 0 (13)
channels
and upper and lower mixed-mean plena internal energies are determined through two additional mixture equations:
max(W„„,0)[cyT.^^ -u^op)+ S P C ^ N ( " C " ^op)^N = 0 (14)
upflow channels
T^^{Wcor,^0)(cyT„^-Usar) + X P C ^ J ( " C - UBOT)AS = 0 (15)
dow/i/low charuuls
This allows for the calculation of natural circulation conditions in the MHTGR core, where upflow and downflow channels coexist and draw fluid from the mixed lower and upper plena, respectively.
Correlations for helium properties in the BYPASS code are adapted from several references^'6'7 care was taken to code each helium property in only one function in the BYPASS code, so that they could be modified or applied to other gases. Preference was given to more efficient polynomial correlations, rather than to tabulated correlations.
Solution Scheme
The equations to be solved (Eqs. 7-15) for a steady state flow distribution in the BYPASS code are viewed as a set of N nonlinear equations in N variables. The number of equations (or variables) is derived from the size of the problem:
N = 3(#cells)-h(#channels) + 4(#3-wayjunctions)-i-3(#2-way junctions)
-!-(# crossflow links) -I- 3 {if the flowrate is specified) (16) The system of N nonlinear equations to be solved is formally written:
J2\ 1' 2.-'- Nj- , or as a column vector: F = 0 (17)
JN\^\^^2,"' ^N) ~ "
The Newton-Raphson solution technique constructs a tangent hyperplane (i.e., first derivative of the system), and attempts to step to the root along the tangent direction:
x ' ' ^ ' = x ' ' - J - ' F (18) where the Jacobian matrix is defined as:
dF^ .. .
^ iFs.
oXi dXf]
In the BYPASS code the partial derivatives in the Jacobian matrix are all evaluated as symbolic derivatives on the N nonlinear equations. Many of the elements of the Jacobian are zero, making it quite likely that the Jacobian may be inverted by sparse and/or iterative methods. The terms in the Jacobian are derived in the Theory Manual.^ Generally the Jacobian is very economical to evaluate, but very expensive to invert as its size increases. The cunrent method of inversion is an LU decomposition with partial pivoting. Aside from the potential for sparse matrix techniques on the inversion, the evaluation of a Jacobian may even be unnecessary, given that other methods^-l^ (especially Steffensen's) are known.
The BYPASS code makes no special ordering of the equations or variables in the system arrays. BYPASS assembles the equations in the following order:
• Each fluid cell is considered in turn, with continuity, energy, and momentum equations assembled for each. If the cell is the bottom cell in a channel, a bottom cell momentum equation is assembled instead of the normal cell momentum equation;
• Each channel is considered in turn, with a top cell momentum equation assembled for the top cell in the channel;
• Each exchange flow junction is considered in turn. If the junction is a 3-way junction, then a continuity equation and three leg pressure drop equations are assembled. If the junction is a 2-way junction, then a continuity equation and two leg pressure drop equations are assembled;
• Each crossflow path is considered in turn. A pressure drop equation for the link is assembled;
• If the core flowrate is specified, then three additional equations (a mass balance, and upper and lower plenum mixture equations) are assembled.
The matrix inversion for the Jacobian in each iteration uses a version of LU decomposition found in the UNPACK subroutine libraryi 1 -12. The matrix inversion is carried out in double precision. Problems of up to about 1000 equations can be solved on a large personal computer in several hours, allowing for about 4 or 5 iterations (matrix inverses). The matrix inversion routine, DGECO, from the LINPACK library estimates the reciprocal of the condition number of the Jacobian matrix at atwut 10'^ to 10"''0 for typical systems of equations in BYPASS posed at MHTGR operating conditions. This is felt to be adequate justification for employing double precision in BYPASS.
With the Newton-Raphson solution scheme, the convergence of BYPASS seems quite good. The Newton-Raphson method is known to be a second order accurate methodic, and the residuals in BYPASS are frequently observed to decrease by two orders of magnitude each iteration after the third or fourth iteration. The residuals tend to zero somewhat more slowly in the first several iterations. Problems with a lot of important influences in the energy equation (e.g., strong heating from the walls) tend to take an iteration or two more than other problems. The convergence control is as follows: At the end of each iteration, the maximum absolute value of each residual is compared to a pre-assigned tolerance. The problem ends if all residuals are smaller than this tolerance.
Sample Calculation
The BYPASS code has not been extensively validated, especially its capabilities in exchange flow and crossflow. Comparison of single channel results have been made against the MORECA codei'*'"'^ with 1% agreement in pressure drop and temperature rise. A number of sample problems similar to the one shown here have been developed, and are documented in a User's Manual.1^
For illustration of its capabilities, the sample problem considers the flow in a fuel column, and includes main coolant, bypass, and crossflow. The flow is heated by the fuel blocks in the column, and is representative of normal reactor operating conditions. A constriction is imposed on one of the bypass gaps in order to stimulate a flow of helium around it. The column of blocks to be modeled is shown in Figure 3. The sample problem considers the flow down through this column, under the small effect of a constriction in one of the bypass gaps halfway down the column. Also shown in the figure are the boundary conditions for the problem. The inlet temperature of the helium is taken to be 283 °C at the top of the column. The inlet pressure is 6.379 MPa. The outlet pressure is 6.366 MPa. The graphite fuel block surface temperatures are approximated by the temperatures shown in the figure, representing a typical linear temperature profile in the MHTGR of 28 °C increase per fuel block. The bypass gap to be constricted will be arbitrarily taken at a point halfway down the column. The bypass gaps will have a nominal (hot) dimension of 0.305 cm; the constriction narrows the bypass gap at the midpoint to 0.152 cm.
The discretization of the fuel column is considered in Fig. 4. The detail on the left shows the coarse discretization of each block into 8 axial channels - 4 bypass gaps and 4 lumped main coolant channels.
The exchange flow and crossflow connections are schematically indicated in the detail at the lower left in the figure. The full set of connections for channel 6 (a bypass gap) are shown in the detail at the right.
The results for the sample problem are shown in Fig. 5. Average helium temperatures are shown, and average flow velocities (nVsec) are indicated for a number of locations in the bypass gaps. The solution exhibits a fair amount of flow diversion around the constriction in channel 8. The crossflow velocities are not shown in the figure, but are negligibly small in this problem. The main coolant channel velocities are not shown, but are about 25 nVsec at the inlet, and 30 m/sec at the outlet. A small unphysical 'wiggle' is evident in the exchange flow two cells downstream of the constriction. The axial flow velocity directly below the constriction seems somewhat high, this is probably due to neglecting a form loss term in the exchange flow pressure drop equation (Eq. 11).
6.379 MPa 283 °C helium
Constricted Bypass Gap
in this Area
^
A A A . A A A A A A . A A A A A A . A A A A A A . A A A A A A . A A A A A A . A A A . A A A A A A . A A A A A A . A A A A A A
A A A A A A .
310 °C graphite
338 °C graphite
366 °C graphite
394 °C graphite
421 °C graphite
449 °C graphite
6.366 MPa
Figure 3. Symmetry section of six fuel blocks arranged in a column analyzed in the sample problem.
Boundary conditions of pressure, helium inlet temperature, and graphite block temperatures are shown.
Channel Number
6 7 5 2
Detailed Cell, Crossflow Link, and Junction Numbers for Channel 6 and its Neighbors
Figure 4. Discretization of the axial flow channels in the sample problem. Cell connections for all the cells in channel 6 are illustrated at the right: bypass gap cells are schematically shown as rectangles, lumped main coolant cells are schematically shown as cylinders.
283 °C helium
301 °C helium
325 °C helium
351 °C helium
385 °C helium
409 "C helium
435 '='C helium
Figure 5. Selected flow pattern and temperature results for the sample problem Velocities shown are in m/sec. Main coolant and crossflow velocities are discussed in the text
Summary
A computer code with fairly general capabilities has been developed for nrx)deling the very detailed, highly connected flow paths in prismatic fuel cores typical of the MHTGR. Initial, limited verification of the code has been undertaken only for simple one-dimensional flow. While the initial choice of the Newton- Raphson method and full Jacobian matrix inversion is appropriate for code validation and verification efforts, recommendations aimed at performance improvements for production use of the code have been suggested.
Nomenclature Control Generic Volume
SymbQl Variable Description of Variable A Area
As Area of south face of center cell A N ...of north face
A x ...of east or west edge
A i ...of east or west edge of exchange cell 1 A2 ...of east or west edge of exchange cell 2 A3 ...of east or west edge of exchange cell 3
A B H C •Of bottom half of east or west edge
A T H L -Of top half of east or west edge of the lower cell AcR ...of the crossflow link
Cy Heat capacity at constant volume D Hydraulic diameter
DcR ...of the crossflow link f f(p,u,V,D) Friction factor
F Vector of residuals (= 0 at final solution) g Acceleration due to gravity
go Gravitational constant h h(p,u,V,D) Heat transfer coefficient H Height
He ...of center cell H L ...of lower cell
J NxN Jacobian matrix of system 3 Thermal equivalent of mechanical work K Loss coefficient
Kvs ...associated with a cell south velocity KvcR ...associated with a crossflow velocity LcR Equivalent length of the crossflow link N Number of equations in the system
p
PC PL Ri Pi P2 PsENSE T
T F T B
^ i n f l o w u
uc
U L UBOT UTOP
U l U2 V
Vs
V N
V s s VsHW VSHE V i V2 Vs
V E
Vw
V E C
Vwc V E L VWL VcR W W c
W C O R E
X z 5
8N
6s
Pressure (absolute) ...in center cell ...in lower cell
...in neighboring junction ...infirst leg of a junction ...in second leg of a junction
Factor (+1 or -1) accounting for the positive sense of the crossflow velocity vector Temperature
...of front (graphite) face of cell ...of back (graphite) face of cell
...of makeup flow entering from the heat transport system Internal energy
...in center cell ...in lower cell ...in lower plenum ...in upper plenum ...in first leg of a junction ...in second leg of a junction Velocity
Axial velocity at south face of center cell ...at north face
...at south face of lower cell Shear velocity in fluid at west face
at east face
Exchange velocity in leg 1 ...in leg 2
...in leg 3
...at east face of center cell ...at east face of center cell ...at east face of center cell ...at west face of center cell ...at east face of lower cell ...at west face of lower cell Crossflow velocity in a link Width of center cell
Desired core flowrate (+ve is downflow) Vector of fluid flow variables
Axial position
Bypass gap dimension or thickness ...of the north face of a cell
...of the south face of a cell
Pc PL PI P2 P3
Density ...in center cell ...in lower cell
...in first leg of a junction ...in second leg of a junction ...in third leg of a junction
Donor cell treatment, i.e., quantity in brackets referenced to upstream values which follow from the velocity in brackets being greater or less than zero.
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6 J. Goodman, et. al.. The Thermodynamic and Transport Properties of Helium. GA-A13400, October, 1975.
7 G. Melese and R. Katz, Thermal and Flow Design of Helium-Cooled Reactors. Chicago: American Nuclear Society, 1984, Appendix B.
8 R. Bennett, The BYPASS Computer Code. Vol. 1: Theory Manual." EGG-2604, May, 1990, Appendix A.
9 G. Dahlquist and A. Bj6rk, Numerical Methods. Englewood Cliffs: Prentice-Hall, 1974, pp. 222-237.
""^ J. Ortega, and W. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press, 1970, Chapter 7.
1 •• J. Dongarra, et. al., LINPACK User's Guide. Philadelphia: SIAM, 1979.
12 T. Coleman, and C. Van Loan, Handbook for Matrix Computations. Philadelphia: SIAM, 1988.
13 w . Press, et. al.. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge Univ.
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1^ S. Ball, ORECA-1: A Digital Computer Code for Simulating the Dvnamics of HTGR Cores for Emergency Cooling Accidents. ORNIJTM-5159, April, 1976
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