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A Numerical Method for Thermo-Fluid-Dynamics
Analyses of Fast Nuclear Reactors Fuel Assemblies
- Single-Phase Flow Formulation -
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March 2011
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This document is freely downloadable in Microsoft Word (doc) format
The use of this document is ABSOLUTELY UNLIMITED
Online homepage of this document is http://wp.me/p61TQ-C5
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ABSTRACT
This article describes a general numerical method for single-phase thermo-fluid-dynamics problems
applicable to the analyses of Liquid Metal-cooled Fast Nuclear Reactors (LMFR) fuel assemblies.
The governing equations are the Reynolds-averaged Navier-Stokes (RANS). The employed
discretization technique is a variant of Finite Volume Method (FVM), appropriately implemented
for triangular lattice, typical subchannel geometry of LMFBR fuel assembly. The discretized
equations, which form a large algebraic equation system, are then solved semi-implicitly.
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CONTENT
ABSTRACT.........................................................................................................................................3
CONTENT...........................................................................................................................................4
LIST OF FIGURES..............................................................................................................................5
LIST OF TABLES...............................................................................................................................6
1. FUEL ASSEMBLY OF AN LMFR.............................................................................................7
2. GOVERNING EQUATIONS......................................................................................................9
2.1. Mass Conservation Equation................................................................................................9
2.2. Momentum Conservation Equation....................................................................................11
2.2.1. Axial momentum equation.........................................................................................12
2.2.2. Transverse momentum equation................................................................................14
2.3. Energy Conservation Equation (Internal Energy)..............................................................15
3. SUMMARY OF EQUATIONS.................................................................................................19
3.1. Mass Conservation Equation..............................................................................................19
3.2. Axial Momentum Conservation Equation..........................................................................20
3.3. Transverse Momentum Conservation Equation.................................................................20
3.4. Energy Conservation Equation (Internal Energy)..............................................................20
4. SOLUTION PROCEDURE.......................................................................................................21
5. REFERENCES...........................................................................................................................27
6. APPENDIX................................................................................................................................27
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LIST OF FIGURES
Figure 1. Fuel pin arrangement in LMFR fuel assembly.....................................................................7
Figure 2. Interior, edge, and corner subchannels.................................................................................7
Figure 3. Staggered control volume system.........................................................................................8
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LIST OF TABLES
Table 1. Four equations and twelve unknowns..................................................................................21
Table 2. Four equations and four unknowns......................................................................................22
Table 3. Two equations and two unknowns.......................................................................................24
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1. FUEL ASSEMBLY OF AN LMFR
In a Liquid Metal-cooled Fast Reactor (LMFR), the way the fuel pins are arranged in a fuel
assembly forms a triangular lattice, similar to the following figure:
Figure 1. Fuel pin arrangement in LMFR fuel assembly
Therefore, as we can see that based on geometrical shape, there are three types of subchannel:
interior, edge, and corner (Figure 2).
Figure 2. Interior, edge, and corner subchannels
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In this article, there are three types of control volumes which will be used when discretizing the
governing equations, one control volume for mass and energy equations, one control volume for
axial momentum equation, and one control volume for transverse momentum equation.
Figure 3. Staggered control volume system
Figure 3 shows the three types of control volume. The axial momentum control volume is axially
staggered by half cell relative to mass and energy control volume, and the transverse momentum
control volume is laterally staggered relative to mass and energy control volume.
The subscript notations used throughout this article are explained as follow:
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2. GOVERNING EQUATIONS
Time average differential transport equations for incompressible fluid, or also known as the
Reynolds-averaged Navier-Stokes (RANS) equations are expressed as follow:
Eq. 1
Mass
Linear momentum
Internal energy
where overbars and primes in the last term denote time average and temporal fluctuation,
respectively. For simplicity of notation, the overbars will be dropped, as it is understood that here
we deal with the transport of time average quantities. The degree of implicitness chosen for the
discretized equations follows that described in detail on the papers listed in the appendix section. In
the following sections, the algebraic equation system to be solved numerically will be derived.
2.1. Mass Conservation Equation
Differential form:
Eq. 2
Finite volume form:
Eq. 3
Discretized form:
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Eq. 4
where is the advection of mass:
Eq. 5
where represents the direction of the coordinate system at gap k, expressed as:
Eq. 6
A variable is composed of spatial average term and spatial variation term:
Eq. 7
So that the variables inside the bracket in Eq. 5 can be decomposed as follows:
Eq. 8
We will use a symbol to represent the spatial variation term, , so that:
Eq. 9
We substitute Eq. 9 into mass advection term (Eq. 5):
Eq. 10
Therefore, the transport of mass due to local spatial variation is represented by the following terms:
for axial direction: and
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and for lateral direction:
For single-phase flow, the spatial variation terms are negligible for most cases ( ), which
means that we can approximate the spatial average of product as the product of spatial average:
Eq. 11
Applying this conclusion, the discretized mass conservation equation is expressed as:
Eq. 12
Eq. 13
Unknowns: , , , and .
2.2. Momentum Conservation Equation
Differential form:
Eq. 14
Here we assume that the fluid is incompressible (but thermally expandable), and the problem will
be treated and solved as a quasi-steady problem. Therefore, in each time step, fluid density can be
assumed constant, so that Eq. 14 can be re-written as follow:
Eq. 15
The reason for this manipulation will become clear later.
Momentum equation in finite volume form (approximating average of product as product of
average for the first term in LHS):
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Eq. 16
2.2.1. Axial momentum equation
One dimensional axial momentum equation:
Eq. 17
Discretized form:
Eq. 18
where:
Eq. 19
Note that it is preferable to take the flow area variables out of the pressure differential term:
Eq. 20
where:
Eq. 21
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Advection term
is the axial momentum advection term:
Eq. 22
Expressing variables appearing in Eq. 22 as averaged component and local spatially varying
component, and following the method demonstrated in Eq. 7 to Eq. 9, the average of product of
density-velocity-velocity can be expressed as:
Eq. 23
where is the spatial variation term for momentum transport. Hence, the momentum advection
term, Eq. 22 can be re-expressed as:
Eq. 24
Distributed resistance term
is the distributed resistance term for axial direction:
Eq. 25
Viscous and turbulent terms
is the summation of viscous and turbulent terms:
Eq. 26
Unknowns: , , , .
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2.2.2. Transverse momentum equation
One dimensional transverse momentum equation:
Eq. 27
Discretized form:
Eq. 28
where
Eq. 29
and:
Eq. 30
Advection term
is the transverse momentum advection term:
Eq. 31
Expressing variables appearing in Eq. 31 as averaged component and local spatially varying
component, and following the method demonstrated in Eq. 7 to Eq. 9, the average of product of
density-velocity-velocity can be expressed as:
Eq. 32
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where is the spatial variation term for momentum transport. Hence, the momentum advection
term, Eq. 31 can be re-expressed as:
Eq. 33
Distributed resistance term
is the distributed resistance term for lateral direction:
Eq. 34
Viscous and turbulent terms
is the summation of viscous and turbulent terms:
Eq. 35
Unknowns: , , , .
2.3. Energy Conservation Equation (Internal Energy)
Differential form:
Eq. 36
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Finite volume form:
Eq. 37
Approximating the average of product as the product of average for the pressure work term:
Eq. 38
Hence, Eq. 37 becomes:
Eq. 39
The first term on the RHS of Eq. 39, which represents heat conduction (molecular effect), can be
recast by utilizing the Fourier conduction law:
Eq. 40
where is the fluid thermal conductivity.
And the second term on the RHS of Eq. 39 can be manipulated as follow:
Eq. 41
where and are heat flux from fluid to solid, and from solid to fluid, respectively.
Hence, Eq. 39 becomes:
Eq. 42
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The discretized internal energy equation is expressed as follow (approximating the average of
product as the product of average for the unsteady term):
Eq. 43
Note that generally the heat source term, , and the dissipation term, , can be ignored.
Advection term
is the energy advection term:
Eq. 44
Expressing variables appearing in Eq. 44 as averaged component and local spatially varying
component, and following the method demonstrated in Eq. 7 to Eq. 9, the average of product of
density-internal energy-velocity can be expressed as:
Eq. 45
where is the spatial variation term for energy transport. Hence, the energy advection term, Eq.
44 can be re-expressed as:
Eq. 46
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Conduction term
is the conduction term (assuming is uniform in a control volume):
Eq. 47
Eq. 48
Eq. 49
where and are temperature difference between two adjacent subchannels, and centroid
distance of two adjacent subchannels, respectively. follows the following rule:
Eq. 50
Wall heat flux term
is the wall heat flux term:
Eq. 51
This term should be handled appropriately, as it is understood that this term represents the total heat
input (or output) rate from (or to) the solid surfaces in a control volume. Note that in this paper, the
heat conduction equation in fuel pin is not considered, so that the wall heat flux has to be input by
the analyst, which can be time dependent.
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Turbulent term
is the turbulent term, expressed as:
Eq. 52
Unknowns: , , , , , , , .
For most cases, the spatial variation terms are negligible, so to simplify our equation system, we
will ignore all the spatial variation terms in all equations:
Eq. 53
3. SUMMARY OF EQUATIONS
A summary of all the governing equations is presented in this section. For the sake of clarity and
simplicity, all the brackets representing surface and volume averages will be removed (short hand
notations).
3.1. Mass Conservation Equation
From Eq. 12:
Eq. 54
Eq. 55
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3.2. Axial Momentum Conservation Equation
From Eq. 20:
Eq. 56
Eq. 57
3.3. Transverse Momentum Conservation Equation
From Eq. 28:
Eq. 58
Eq. 59
3.4. Energy Conservation Equation (Internal Energy)
From Eq. 43:
Eq. 60
Eq. 61
So from now on, we will use these simpler short hand notations.
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4. SOLUTION PROCEDURE
The discretized equations form a large algebraic equation system, consisting of four transport
equations, and the following new time variables as unknowns:
Table 1. Four equations and twelve unknowns
Equation Unknowns
Continuity , , ,
Axial momentum , , ,
Transverse momentum , , ,
Internal energy , , , , , , ,
As can be seen from Table 1, there are nine variables as unknowns , and
only four equations. To be able to solve the equation system, the number of unknown must be the
same as the number of equation, so we will choose as the main variables of unknowns,
and employ five additional equations relating the five remaining variables to the four main
variables.
General forms of these constitutive equations are described as follow:
1. State equation for density
Eq. 62
2. State equation for internal energy
Eq. 63
3. Treatment of wall heat flux
Wall heat flux is specified as an input, which can be time dependent.
4. Constitutive equation for distributed resistance term of the axial momentum equation
Eq. 64
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where Kwz is a general expression for wall-fluid momentum exchange term.
5. Constitutive equation for distributed resistance term of the transverse momentum equation
Eq. 65
where Kwx is a general expression for wall-fluid momentum exchange term.
Note that there are also three terms that are not well defined yet so far: the viscous-plus-turbulent
terms in the axial and transverse momentum equations, and turbulent term in the energy equation.
They can be treated in either of two ways: simply define them as constant values, or relating them
to the four main variables. The later option has been generally chosen, since it can accommodate
local conditions of each subchannel. General expressions for the three terms are as follows:
1. Empirical correlation for viscous-plus-turbulent term of the axial momentum equation
Eq. 66
2. Empirical correlation for viscous-plus-turbulent term of the axial transverse equation
Eq. 67
3. Empirical correlation for turbulent term of the internal energy equation
Eq. 68
So now we have a closed equation system, consisting of four equations and four variables as
unknowns: p, T, u, w.
Table 2. Four equations and four unknowns
Equation Unknowns
Continuity , , , ,
Axial momentum , ,
Transverse momentum , ,
Internal energy , , , ,
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A major simplification can be made by utilizing the two momentum equations. As can be seen from
Table 2, the unknowns for the momentum equations are only velocity and pressure, this means the
system can be solved for the new time (n+1) velocities as functions of the new time pressure
gradient.
The axial momentum equation, Eq. 56, can be re-arranged to become the following form:
Eq. 69
where a0 and a1 are the collections of all other terms evaluated at old time step (n):
Eq. 70
Eq. 71
Similarly, equation for can be constructed:
Eq. 72
where:
Eq. 73
Eq. 74
And the transverse momentum equation becomes as follow:
Eq. 75
where:
Eq. 76
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Eq. 77
By substituting Eq. 69, Eq. 72, and Eq. 75 into the mass (Eq. 54) and internal energy (Eq. 60)
conservation equations, we can eliminate the two momentum equations and the two velocity
variables (u and w), so that the equation system is now reduced to only two equations (mass and
internal energy) and two unknowns (p and T).
Table 3. Two equations and two unknowns
Equation Unknowns
Continuity , , , , , ,
Internal energy , , , , , ,
To solve this large algebraic equation system, the mass and energy equations can be considered as
“zero-valued” non-linear functions:
Eq. 78
Eq. 79
Following the method described in [1], this system is then solved by using the multivariable
Newton-Raphson method:
Eq. 80
The iteration is carried out until all elements of and are sufficiently small.
The matrices are as follow (for simplicity, we will drop the n+1 superscript):
Eq. 81
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Eq. 82
Eq. 83
The above matrices can be re-arranged as follow:
Eq. 84
or in simple notations:
Eq. 85
where aij are elements of the Jacobian matrix. Now we multiply both sides of Eq. 85 with the
inverse of matrix :
Eq. 86
Eq. 87
where are new matrices as a result of the above matrix operation. Now we write Eq. 87 as
follow:
Eq. 88
Eq. 88 implies that spatial coupling is accomplished only through pressures, which means that if the
pressure correction terms are known, the temperature variable is readily obtained.
Therefore, to solve Eq. 88, first we evaluate the pressure correction terms, so we need to solve the
following equation for each calculational cell:
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Eq. 89
Once the pressure correction terms are obtained, then the temperature correction is evaluated as
follow:
Eq. 90
Note that the pressure correction formula (Eq. 89) is a discrete formulation of the Poisson equation
in terms of the pressure correction :
Eq. 91
Also note that Poisson equation is of type elliptic Partial Differential Equations (PDEs), where
disturbance introduced at any interior point, influences all other point in the domain, which is
consistent with the incompressible fluid assumption we use here. The ability to influence all other
points from interior point implies that boundary conditions are required on all boundaries.
Eq. 89 is the pressure correction formula for a single computational cell, and our task now is to
solve it for the entire computational domain. If we write Eq. 89 in matrix format (for all cells in the
computational domain), we would have a banded matrix structure, representing a linear algebraic
equation system, as shown as follow:
Eq. 92
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where is total number of subchannels per axial plane, is total number of axial planes, is an
sparse matrix, is the solution vector, and is collection of remaining terms that are not
associated with the pressure correction terms. The structure of this pressure correction matrix can be
considered as Block Tridiagonal, and many efficient algorithms are available to solve it.
5. REFERENCES
1. http://wp.me/p61TQ-rR
6. APPENDIX
Some related reading materials:
http://dx.doi.org/10.1016/0029-5493(90)90386-C (Sabena: Subassembly boiling evolution
numerical analysis)
http://dx.doi.org/10.1016/0029-5493(86)90111-1 (Analysis of low-heat-flux sodium boiling
test in a 37-pin bundle by the two-fluid model computer code SABENA)
http://dx.doi.org/10.3327/jnst.37.654 (The Multi-fluid Multi-phase Subchannel Analysis
Code KAMUI for Subassembly Accident Analysis of an LMFR)
http://dx.doi.org/10.1016/0029-5493(80)90018-7 (An overview on rod-bundle thermal-
hydraulic analysis)
http://dx.doi.org/10.1016/0029-5493(84)90208-5 (Advances in two-phase flow modeling for
LMFBR applications)
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