CN106126931B - The accelerated method that a kind of nuclear reactor subchannel calculates - Google Patents

The accelerated method that a kind of nuclear reactor subchannel calculates Download PDF

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CN106126931B
CN106126931B CN201610472929.3A CN201610472929A CN106126931B CN 106126931 B CN106126931 B CN 106126931B CN 201610472929 A CN201610472929 A CN 201610472929A CN 106126931 B CN106126931 B CN 106126931B
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CN106126931A (en
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曹良志
陈军
刘宙宇
吴宏春
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Xian Jiaotong University
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Abstract

The invention discloses the accelerated method that a kind of nuclear reactor subchannel calculates, 1, linearize subchannel quality, energy and conservation of momentum Nonlinear System of Equations；2nd, the subchannel equation after linearisation is solved using Krylov subspace method, equation coefficient matrix using compressed line non-zero memory technology is stored and is used and restarts the GMRES Algorithm for Solving system of linear equations；The present invention is a kind of method that reactor core subchannel is accelerated to calculate while reduce memory consumption, this method solves the Large Unsymmetric system of linear equations during Whole core subchannel calculates by using Krylov subspace method, including energy equation, axial momentum equation, pressure correction equation, there is higher computational efficiency.

Description

The accelerated method that a kind of nuclear reactor subchannel calculates
Technical field
The present invention relates to nuclear reactor design and security fields, are that a kind of nuclear reactor subchannel calculating adds Fast method.
Background technology
Traditional presurized water reactor Whole core thermal hydraulic analysis generally adopts one channel model, which thinks each passage of reactor core Between be isolated, closing, without considering between adjacency channel quality, momentum and energy exchange, this model is simple, Calculation amount is small, but is only applicable to calculate enclosed passage.Therefore Sub-channel mode has been developed, which thinks adjacency channel cooling agent Between there are quality, momentum and energy exchange.Effect is mixed due to considering this hand over, the enthalpy and temperature of cooling agent in the passage of heat Than without considering low when handing over mixed, corresponding fuel temperature can also be declined, this security for being conducive to improve reactor and Economy.
But a small number of passage of heats cannot be only calculated as one channel model in the calculating of Sub-channel mode, but it is right Substantial amounts of flux is analyzed, therefore when the subchannel for carrying out Whole core calculates, calculation amount and the calculator memory consumed All it is insufferable.Therefore, it is necessary to a kind of accelerator path computations and the method for reducing memory computer consumption.
The content of the invention
In order to overcome the above-mentioned problems of the prior art, it is an object of the invention to provide a kind of nuclear reactor The accelerated method of path computation, is a kind of method that reactor core subchannel is accelerated to calculate while reduce memory consumption, and this method passes through The Large Unsymmetric system of linear equations during Whole core subchannel calculates is solved using Krylov subspace method, including energy equation, Axial momentum equation, pressure correction equation have higher computational efficiency.
In order to achieve the above object, the present invention adopts the following technical scheme that：
The accelerated method that a kind of nuclear reactor subchannel calculates, including following two big steps：
Step 1：Subchannel quality, energy and conservation of momentum Nonlinear System of Equations are linearized, definition is three following Transfer function fij、βkjAnd βkj' to consider the influence of alms giver's member：
In formula,
mij--- mixture axial direction mass flow；
wkjIt is positive value, with e to pass through the crossflow velocity of interval kikDirection, e are represented togetherikwkjIt is represented more than 0 from l passages L ' passages are flowed to, otherwise on the contrary；
--- for the algebraic mean of the effective speed, i.e. adjacency channel effective speed of interval k；
Three transfer functions in formula (1) are updated to subchannel energy, axial momentum, transverse momentum and mass equation In, just obtain the subchannel equation after following four linearisations：
1) energy equation
In formula,
fij、fij-1--- the transfer function of passage (i, j) and (i, j-1)；
mij、mij-1--- the mixture axial direction mass flow of passage (i, j) and (i, j-1)；
△Xj--- the height of axial locking nub j；
w'kj--- it is the turbulent velocity through interval k；
βkj--- the transfer function of interval k
wkjIt is positive value, with e to pass through the crossflow velocity of interval kikDirection, e are represented togetherikwkjIt is represented more than 0 from l passages L ' passages are flowed to, otherwise on the contrary；
hij、hnj、hij+1And hij-1--- passage (i, j), (n, j), the mixture enthalpy of (i, j+1) and (i, j-1)；
Pr--- the heat week of fuel rod r, fuel rod r are adjacent with passage i；
Φir--- all shares adjacent with passage i of heat of fuel rod r；
q”rj--- the heat flow density of fuel rod to fluid；
Ck--- horizontal thermal conductivity factor, as following formula calculates：
--- using the algebraic mean value of the thermal conductivity factor of adjacent as border k two passages i and n；
GT--- the geometrical factor of thermal conductivity factor, for considering the energy transmission caused by heat conduction between subchannel；
lk--- the centre-to-centre spacing of two neighboring subchannel
Tij、Tnj--- passage (i, j) and (n, j) temperature；
2) axial momentum equation
In formula：
U'ij、U'ij-1--- the effective speed of passage (i, j) and (i, j-1) represent to control in vivo average speed；
fT--- turbulent flow factor of momentum；
v'*--- effective ratio volume；
Aij、Anj--- the cross-sectional area of passage (i, j) and (n, j)
K'ij--- Pressure drop factor；
gc--- unit conversion factor；
G --- acceleration of gravity；
Pij、Pij-1--- the pressure at layer (i, j) and (i, j-1)；
ρij--- the density of passage (i, j)
The angle of θ --- passage and vertical plane；
3) transverse momentum equation
In formula：
--- for the algebraically of the effective speed, i.e. adjacency channel effective speed of the interval k on layer j and j-1 It is average；
sk--- the length of subchannel dividing line k；
lk--- the centre-to-centre spacing of two neighboring subchannel；
KG--- the crossing current friction pressure drop factor；
Pkj-1--- the pressure difference of the interval two neighboring passages of k；4) mass-conservation equation
In formula：
ECij--- mass-conservation equation residual error
δPij、δPij-1、δPij-2--- the pressure variety at layer (i, j), (i, j-1) and (i, j-2)；
Step 2：Subchannel equation (2)~(5) after linearisation are solved using Krylov subspace method, to equation coefficient square Battle array is stored using compressed line non-zero memory technology CSR, and CSR technologies are using three one-dimension arrays a, ia and ja come storage size size Nonzero element and corresponding position for the coefficient matrices A of m × n.The nonzero element number in A matrixes is remembered for nnz, then：
1) one-dimension array a, array length nnz, array element be respectively in coefficient matrices A from left to right, from top to bottom The nonzero element value of arrangement；
2) one-dimension array ia, array length m+1, per first nonzero element of a line total non-in packing coefficient matrix A Order in neutral element, the last one element are nnz+1；
3) one-dimension array ja, array length nnz, per office of a line nonzero element in the row in packing coefficient matrix A Portion's order.
Compared with prior art, the present invention has following outstanding advantages：
(1) non-zero memory technology is used to coefficient matrix, avoids generating big coefficient matrix, it is direct with Gaussian reduction etc. Method is compared and greatlys save calculator memory.
(2) energy, momentum and the pressure correction equation during subchannel calculates are solved using Krylov subspace, is matched with Gauss The fixed point iterations methods such as Dare iteration, overrelaxation iteration, which are compared, has higher computational efficiency.
Description of the drawings
Fig. 1 is axial control volume and variable-definition schematic diagram.
Fig. 2 is crosswise joint volume and variable-definition schematic diagram.
Fig. 3 is typical presurized water reactor a quarter component diagram.
Fig. 4 is direct method and GMRES Algorithm for Solving Comparative result schematic diagrames.
Specific embodiment
The present invention is described in further details with reference to the accompanying drawings and detailed description：
For the control volume in Fig. 1 and Fig. 2, stable state subchannel analysis solves the stable state conservation side of following difference form Journey：
(1) mass-conservation equation
The mass-conservation equation of control volume (i, j) stable state difference form is as follows：
(2) energy conservation equation
The energy conservation equation of stable state difference form is as follows：
(3) axial momentum conservation equation
The axial momentum conservation equation of stable state difference form is as follows：
(4) transverse momentum conservation equation
The transverse momentum conservation equation of stable state difference form is as follows：
Four conservation equation (6)~(9) constitute Nonlinear System of Equations, and solution procedure is by Nonlinear System of Equations linear line Property, and iterative solution.In each linearization calculation, a variable is only solved, other variables are solved as known quantity.
The subchannel equation of four linearisations in (2)~(5) is obtained by three transfer functions in definition (1).
Observation type (2) is it can be found that this is the system of linear equations that a coefficient matrix is five-diagonal matrix, by solving this Equation group can obtain enthalpy hijDistribution.
Equally it can be found that formula (3) is the system of linear equations that a coefficient matrix is five-diagonal matrix, by solving the party Journey group can obtain enthalpy mijDistribution.
Pressure field is axially and transversely assumed in the solution of the equation of momentum, a correct pressure field should make to be calculated Velocity field meet continuity equation.But according to the so given calculated velocity field of pressure field, continuity may not necessarily be met Equation, therefore given pressure field is made improvements, that is, carry out pressure correction.
According to mass-conservation equation, for control volume (i, j), conservation of mass residual error can be write as following form：
In order to which residual error is made to be equal to zero, according to Newton iteration method, pressure field is modified：
In formula (11),
Formula (12) is substituted into formula (11) and arranges the mass equation formula (5) after being linearized.
It is the system of linear equations that coefficient matrix is seven diagonal matrix by observing visible (5).
It can be found by the derivation abbreviation of subchannel conservation equation, energy equation, axial momentum equation, transverse momentum equation What is finally solved with pressure correction equation is all the system of linear equations shaped like formula (13), when carrying out Whole core or a quarter heap When the subchannel of core calculates, the scale of the coefficient matrices A of the system of linear equations is often very big, up to million magnitudes, therefore reasonable section Non-zero memory technology and choose the Solving Linear method of a stability and high efficiency for save calculator memory and Improving computational efficiency is just particularly important.
Ax=b (13)
(1) the non-zero memory technology of reasonable science
It is big come storage size using three one-dimension arrays a, ia and ja using compressed line non-zero memory technology CSR, CSR technology The nonzero element of the small coefficient matrices A for m × n and corresponding position.The nonzero element number in A matrixes is remembered for nnz, then：1) one Dimension group a, array length nnz, array element be respectively in coefficient matrices A from left to right, the non-zero entry arranged from top to bottom Element value；
2) one-dimension array ia, array length m+1, per first nonzero element of a line total non-in packing coefficient matrix A Order in neutral element, the last one element are nnz+1
3) one-dimension array ja, array length nnz, per office of a line nonzero element in the row in packing coefficient matrix A Portion's order；
Such as size be 4 × 4 coefficient matrices A：
Storing corresponding three one-dimension arrays using CSR technologies is respectively：
A=[1 563 6]
IA=[1 245 6]
JA=[1 123 2]
It, can be logical for large-scale son to avoid the neutral element in packing coefficient matrix A by non-zero memory technology CSR Road problem, such as Whole core subchannel problem, coefficient matrix scale reach million magnitudes, and CSR technologies can greatly save storage Space.
(2) the Solving Linear method of stability and high efficiency
The method for solving of system of linear equations mainly includes direct method, primary iteration method and sciagraphy.Common direct method has Gaussian reduction and triangle decomposition method although their theories are simply easy to implement, are handling extensive fabric problem luck Calculation amount and to consume memory too big, can not practicality.In contrast, primary iteration method is just more suitable for the sparse diagonal dominance square of high-order The solution of battle array, this method mainly include Jacobi iterative method, Gauss-Seidel iteration method and over-relaxation iterative method three classes.But Equation group of the primary iteration method after subchannel linearisation is solved is inefficient, the equation being primarily due to after subchannel linearisation The coefficient matrix of group is weak diagonally dominant matrix.The iterative method for solving such system of linear equations most practicality and high efficiency at present surely belongs to project Method.It is not directly to be found in n ties up Euclidean space when solving system of linear equations such as according to the thought of sciagraphy Solution vector, but a certain subspace in find its approximate solution：And this sub-spaces is at present mainly using Krylov Space, corresponding system of linear equations sciagraphy are also thus referred to as Krylov subspace solution.
For shaped like linear algebraic equation systems, define initial residual vectorFor：
In formula：
--- solution vectorArbitrary iterative initial value.
Then RnM dimension Krylov subspace can be denoted as：
Krylov subspace solution is exactly by using Dinko Petrov-gal the Liao Dynasty gold bar part：
So as to tie up affine subspace in mIn search out the approximate solution of equation group.Wherein LmIt is another m Subspace.Different LmSelection mode generates different classes of Krylov subspace solution.Most typical two kinds of selections mode Have：First, it choosesOrThis kind of method is mainly based upon Arnoldi orthogonalization procedures 's；Second, it choosesThis kind of method is mainly based upon Lanczos biorthogonalization processes, wherein ATFor matrix A Transposition.Anyway, Krylov subspace solution is actually to have done such a approximation：
In formula：
qm-1--- m-1 rank multinomials.
Obviously, Krylov subspace solution is exactly using a polynomial expansion approximation in itself.
Linear algebraic equation systems solution huge number based on above-mentioned Krylov subspace theory, wherein it is more famous as Conjugate gradient method (CG), Lanczos methods and the minimum remaining method (GMRES) of broad sense etc..They respectively have feature, specific to choose Any method will be depending on treating the property of solving equations coefficient matrix.Such as CG methods and Lanzos methods often can be used only in pair In the solution for claiming the coefficient matrix of positive definite；And GMRES methods there almost are not matrix properties in addition to requiring coefficient matrix nonsingular There are other tightened up requirements.Therefore, GMRES methods also just become the large size that current scientific algorithm field is most widely used One of sparse matrix linear algebraic equation systems Krylov subspace solution.It will be introduced briefly GMRES rudimentary algorithms below.
GMRES algorithms are the Krolov subspaces solutions based on Arnoldi orthogonalization procedures.It has chosenResidual vector can so be made for subspaceIn institute's directed quantity all reach minimization, The typical algorithm of GMRES methods is as follows：
(1) calculate
(2) (m+1) × m rank matrixes are definedJuxtaposition
(3) j=1,2 ..., m are done
(4) calculate
(5) i=1,2 ..., j are done：
(6)
(7)
(8) i Xun Huans are terminated；
(9)If hJ+1, j=0, then m=j is made, and jumps to 12 steps；
(10)
(11) j Xun Huans are terminated；
(12) calculateAnd(wherein, VmFor withFor n × m of the i-th row Rank matrix,For the first row of (m+1) × (m+1) rank unit matrix)；
(13) terminate.
It can be proved that it is with the increase of subspace exponent number m to complete calculation amount needed for above-mentioned GMRES methods and amount of storage
And at least it is in respectively O (m2) n and O (mn) increase.Fortunately, theoretically it is proved the subspace needed for above-mentioned algorithm When exponent number at most gets n, iteration can restrain.And in common practical application, subspace exponent number m is often much small In n, this also just makes GMRES methods have more attraction.But for some problems, especially when matrix exponent number is very high When, GMRES methods may need higher subspace dimension that could obtain the solution for meeting required precision, this can undoubtedly bring bigger Computation burden.There are mainly two types of approach at present to solve the problems, such as this：Periodically restart GMRES algorithms and block Arnoldi orthogonalization procedures.Wherein opposite execution is simple, widely used to surely belong to the former.Restart GMRES calculations below to be typical Method：
(1) calculate
(2) by vectorStart, using the Arnoldi orthogonalization procedure identical with foregoing GMRES algorithms, generate Arnoldi orthogonal basis VmAnd matrix
(3) calculateAnd
(4) if solution vectorMeet required precision then to terminate, otherwise putAnd jump to the 1st step.
It is that the GMRES algorithms that iteration realization is once restarted are walked per m above.
Gaussian reduction is respectively adopted and GMRES methods solve typical presurized water reactor a quarter component shown in Fig. 3 and complete group Part, comparison is as shown in fig. 4, it can be seen that with the increase of computational problem scale, the acceleration effect of GMRES is got over the time required to calculating Significantly.

Claims (1)

1. a kind of accelerated method that nuclear reactor subchannel calculates, it is characterised in that：Step is as follows：
Step 1：Subchannel quality, energy and conservation of momentum Nonlinear System of Equations are linearized, following three conversions of definition Function fij、βkjAnd βkj' to consider the influence of alms giver's member：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>&gt;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;le;</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&gt;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;le;</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mover> <msup> <mi>U</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&gt;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mover> <msup> <mi>U</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;le;</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
In formula：
mij--- passage (i, j) mixture axial direction mass flow；
wkjIt is positive value, with e to pass through the crossflow velocity of interval kikDirection, e are represented togetherikwkjIt represents to flow to from l passages more than 0 L ' passages, otherwise on the contrary；
--- for the algebraic mean of the effective speed, i.e. adjacency channel effective speed of interval k；
Three transfer functions in formula (1) are updated in subchannel energy, axial momentum, transverse momentum and mass equation, just Obtain the subchannel equation after following four linearisations：
1) energy equation
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <mo>&amp;lsqb;</mo> <msubsup> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>}</mo> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <mo>&amp;lsqb;</mo> <msubsup> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>r</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>P</mi> <mi>r</mi> </msub> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>q</mi> <mrow> <mi>r</mi> <mi>j</mi> </mrow> <mrow> <mo>&amp;prime;</mo> <mo>&amp;prime;</mo> </mrow> </msubsup> <mo>-</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>C</mi> <mi>k</mi> </msub> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>T</mi> <mrow> <mi>n</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
In formula：
fij、fij-1--- the transfer function of passage (i, j) and (i, j-1)；
mij、mij-1--- the mixture axial direction mass flow of passage (i, j) and (i, j-1)；
△Xj--- the height of axial locking nub j；
w'kj--- it is the turbulent velocity through interval k；
βkj--- the transfer function of interval k；
wkjIt is positive value, with e to pass through the crossflow velocity of interval kikDirection, e are represented togetherikwkjIt represents to flow to from l passages more than 0 L ' passages, otherwise on the contrary；
hij、hnj、hij+1And hij-1--- passage (i, j), (n, j), the mixture enthalpy of (i, j+1) and (i, j-1)；
Pr--- the heat week of fuel rod r, fuel rod r are adjacent with passage i；
Φir--- all shares adjacent with passage i of heat of fuel rod r；
q″rj--- the heat flow density of fuel rod to fluid；
Ck--- horizontal thermal conductivity factor, as following formula calculates：
<mrow> <msub> <mi>C</mi> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>G</mi> <mi>T</mi> </msub> <mover> <mi>k</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> <msub> <mi>l</mi> <mi>k</mi> </msub> </mfrac> </mrow>
--- using the algebraic mean value of the thermal conductivity factor of adjacent as border k two passages i and n；
GT--- the geometrical factor of thermal conductivity factor, for considering the energy transmission caused by heat conduction between subchannel；
lk--- the centre-to-centre spacing of two neighboring subchannel；
Tij、Tnj--- passage (i, j) and (n, j) temperature；
2) axial momentum equation
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>&amp;lsqb;</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>+</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>T</mi> </msub> <msubsup> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <msub> <mrow> <mo>(</mo> <mfrac> <msup> <mi>v</mi> <mrow> <mo>&amp;prime;</mo> <mo>*</mo> </mrow> </msup> <mi>A</mi> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>K</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>|</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>|</mo> <mo>&amp;rsqb;</mo> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>T</mi> </msub> <msubsup> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>&amp;rsqb;</mo> <mfrac> <msup> <mi>v</mi> <mrow> <mo>&amp;prime;</mo> <mo>*</mo> </mrow> </msup> <msub> <mi>A</mi> <mrow> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mfrac> <msub> <mi>m</mi> <mrow> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>g</mi> <mi>c</mi> </msub> <msub> <mi>A</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>g&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <msub> <mi>A</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>cos</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
In formula：
U′ij、U′ij-1--- the effective speed of passage (i, j) and (i, j-1) represent to control in vivo average speed；
fT--- turbulent flow factor of momentum；
v'*--- effective ratio volume；
Aij、Anj--- the cross-sectional area of passage (i, j) and (n, j)；
K′ij--- Pressure drop factor；
gc--- unit conversion factor；
G --- acceleration of gravity；
Pij、Pij-1--- the pressure at layer (i, j) and (i, j-1)；
ρij--- the density of passage (i, j)；
The angle of θ --- passage and vertical plane；
3) transverse momentum equation
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>&amp;lsqb;</mo> <msubsup> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <msub> <mover> <msup> <mi>U</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <msub> <mover> <msup> <mi>U</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>k</mi> </msub> <msub> <mi>l</mi> <mi>k</mi> </msub> </mrow> </mfrac> <msub> <mi>K</mi> <mi>G</mi> </msub> <msup> <mi>v</mi> <mrow> <mo>&amp;prime;</mo> <mo>*</mo> </mrow> </msup> <mo>|</mo> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <mo>&amp;rsqb;</mo> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi>s</mi> <mi>k</mi> </msub> <msub> <mi>l</mi> <mi>k</mi> </msub> </mfrac> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <msub> <mi>g</mi> <mi>c</mi> </msub> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <msub> <mover> <msup> <mi>U</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;beta;</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> <msub> <mover> <msup> <mi>U</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
In formula：
--- for the algebraic mean of the effective speed, i.e. adjacency channel effective speed of the interval k on layer j and j-1；
sk--- the length of subchannel dividing line k；
lk--- the centre-to-centre spacing of two neighboring subchannel；
KG--- the crossing current friction pressure drop factor；
Pkj-1--- the pressure difference of the interval two neighboring passages of k；
4) mass-conservation equation
<mrow> <mo>-</mo> <msub> <mi>EC</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mi>&amp;delta;P</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mi>&amp;delta;P</mi> <mrow> <mi>i</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> <msub> <mi>&amp;delta;P</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;X</mi> <mi>j</mi> </msub> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>i</mi> </mrow> </munder> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>w</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mi>&amp;delta;P</mi> <mrow> <mi>k</mi> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
In formula：
ECij--- mass-conservation equation residual error；
δPij、δPij-1、δPij-2--- the pressure variety at layer (i, j), (i, j-1) and (i, j-2)；
Step 2：Subchannel equation (2)~(5) after linearisation are solved using Krylov subspace method, equation coefficient matrix is adopted It is stored with compressed line non-zero memory technology CSR, CSR technologies use three one-dimension arrays a, ia and ja come storage size size for m The nonzero element of the coefficient matrices A of × n and corresponding position；The nonzero element number in A matrixes is remembered for nnz, then：
1) one-dimension array a, array length nnz, array element be respectively in coefficient matrices A from left to right, arrange from top to bottom Nonzero element value；
2) one-dimension array ia, array length m+1, per first nonzero element of a line in total non-zero entry in packing coefficient matrix A Order in element, the last one element are nnz+1；
3) one-dimension array ja, array length nnz, it is suitable per part of a line nonzero element in the row in packing coefficient matrix A Sequence.
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