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Journal of Nuclear Science and Technology
ISSN: 0022-3131 (Print) 1881-1248 (Online) Journal homepage: http://www.tandfonline.com/loi/tnst20
Design of a Spacer Grid Using Axiomatic Design
Ki-Jong PARK , Byung-Soo KANG , Kee-Nam SONG & Gyung-Jin PARK
To cite this article: Ki-Jong PARK , Byung-Soo KANG , Kee-Nam SONG & Gyung-Jin PARK (2003)Design of a Spacer Grid Using Axiomatic Design, Journal of Nuclear Science and Technology,
40:12, 989-997
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Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 40, No. 12, p. 989–997 (December 2003)
ORIGINAL PAPER
Design of a Spacer Grid Using Axiomatic Design
Ki-Jong PARK1, Byung-Soo KANG2, Kee-Nam SONG3 and Gyung-Jin PARK4,*
1
Department of Machine Design and Production Engineering, Hanyang University, 1271 Sa-1 dong, Ansan, Kyunggi-do, 425-791, Korea2Center of Innovative Design Optimization Technology, 17 Haengdang-dong, Seongdong-gu, Seoul, 133-791, Korea
3Korea Atomic Energy Research Institute, 150 Dukjin-dong, Yusong-gu, Taejon, 305-353, Korea4 Division of Mechanical and Information Management Engineering,
Hanyang University, 1271 Sa-1 dong, Ansan, Kyunggi-do, 425-791, Korea
(Received March 3, 2003 and accepted in revised form July 30, 2003)
Recently, much attention is focused on the design of fuel assemblies in the Pressurized Light Water Reactor (PWR).
The spacer grid is one of the main structural components in a fuel assembly. It supports fuel rods, guides cooling
water, and maintains geometry from external impact loads. In this research, a new shape of the spacer grid is designed
by axiomatic approach. The Independence Axiom is utilized for the design. For the conceptual design, functional
requirements (FRs) are defined and corresponding design parameters (DPs) are found to satisfy corresponding FRs in
sequence. Overall configuration and shapes are determined in this process. Detailed design is carried out based on the
sequence from axiomatic design. For the detailed design, the system performances are evaluated by using linear andnonlinear finite element analyses. The dimensions are determined by optimization. Some commercial codes are utilized
for the analysis and design.
KEYWORDS: axiomatic design, independence axiom, decoupled design, design parameter, functional require-
ment, impact load, shape optimization, contact pressure, PWR type reactors, nuclear fuel rod support grid
I. Introduction
The nuclear fuel assembly in Fig. 1 is used in a PWR. It is
composed of a top end piece, a bottom end piece, guide thim-
bles, fuel rods, and spacer grids. The slenderness ratio of the
fuel rod is so high that several spacer grids need to support
the rod in order to prevent its unstable structural behavior.
The actual supporting parts in the spacer grid are the springs
and dimples as illustrated in Fig. 2. Structural performance
of these supporting parts is critical for stable support of the
fuel rod.1–7) Moreover, in an abnormal operating environment
such as in an earthquake, most of the external impact loads
are mainly applied to the spacer grids supporting the fuel
rods. Control rods normally reside outside of the spacer grids.
Under an abnormal operating environment or when control-
ling the core power, control rods must be quickly inserted in
the nuclear reaction zone through the guide thimbles that are
fixed to spacer grids via sleeves or welding. Therefore, defor-
mation of spacers needs to be limited to safely maintain theguide thimbles.8–11)
The spacer design has to consider many complex engineer-
ing aspects such as structural aspects, metallurgy, thermal-
hydraulics, manufacturing limitations, etc. In this research,
the design of a spacer grid is conducted from structural view-
points. Other complex aspects are indirectly considered with
the results of previous researches4–7) in the design process.
The conceptual design process is proposed by the axiomatic
approach.12–17) The proposed process is reasonable and sys-
tematic compared to the conventional design process based
on experience and sophisticated analysis methods.
Two axioms exist in axiomatic design. One is the Indepen-
∗Corresponding author, Tel. +82-31-400-5246, Fax. +82-31-407-
0755, E-mail: [email protected]
Fig. 1 Nuclear fuel assembly
dence Axiom and the other is the Information Axiom. The
Independence Axiom is employed to design the spacer grid.
After the conceptual design, detailed design is performed to
solve each problem indicated by the design matrix of the ax-
iomatic approach. The detailed design includes three struc-
tural analysis problems. They are evaluations of the inner
strap strength using non-linear dynamic analysis, the con-
tact behavior between the fuel rod and the grid spring using
non-linear static analysis, and the strength of the grid spring
arms using linear static analysis. The finite element method is
adopted for the analyses.18) Some commercial software sys-
tems are utilized. LS-DYNA3D,19) an explicit dynamic anal-
ysis software system and ABAQUS/Standard20) an implicit
analysis software system, are used for nonlinear dynamic
analysis and nonlinear static contact analysis, respectively. Todetermine the final shape of the grid spring arms, numerical
structural optimization is employed by using a commercial
989
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Fig. 2 Unit spacer grid structure
software system called GENESIS21,22) which is capable of
structural optimization with linear static analysis.
II. Axiomatic Design15)
Design is a continuous interplay between what to achieve
and how to achieve it. The designer determines what to
obtain by defining the design objectives from the Customer
Attributes (CAs). The “what to achieve” items are the func-
tional requirements (FRs). The answer to the question, “how
to achieve them” is obtained by the definition of design pa-
rameters (DPs) in the physical domain. In the axiomatic ap-
proach, a design is the creation of the solutions that can obtain
stated objectives through mapping from FRs to DPs through
the proper selection of DPs that satisfy the FRs. Then, it is
the obligation of the designer to select the appropriate FRs
and their corresponding DPs. If a completed design does not
satisfy perceived needs or prescribed FRs, the designer must
create a new idea to change the DPs or FRs. This process
is repeated until the designer gets reasonable FRs and corre-
sponding DPs.
The Independence Axiom in axiomatic design suggests that
one DP influences only a corresponding FR in the mapping
process between the functional domain and the physical do-
main; i.e. it suggests one-to-one mapping of FRs and DPs.
Figure 3 is a graphical interpretation of the general map-
ping process between the functional domains and physical do-
mains. Consider the following design equation to understand
the implications of the Independence Axiom:
{FRs} = [ A]{DPs}, (1)
where {FRs} is the vector of the functional requirements,{DPs} is the vector of the design parameter, and [ A] is the de-
sign matrix that identifies the relationship between {FRs} and
{DPs}. The design matrix consists of three types as follows:
–Coupled design
FR1
FR2
FR3
=
X X X X X X
X X X
DP1
DP2
DP3
, (2)
–Decoupled design
FR1
FR2
FR3
=
X O O
X X O
X X X
DP1
DP2
DP3
, (3)
–Uncoupled design
FR1
FR2
FR3
=
X O OO X O
O O X
DP1
DP2
DP3
, (4)
where X means an FR and a DP have a certain relationship
and O means they have no relationship.
In Eq. (2), DP1, DP2, and DP3 are to be determined si-
multaneously to satisfy FR1. However, even though they sat-
isfy FR1, they cannot be guaranteed to satisfy FR2 or FR3.
Therefore, many trials and errors are needed to find the cor-
rect values of all DPs. It is possible to conduct a sequential
design in the case of Eq. (3). That is, we can determine thedesign parameters in the sequence of DP1, DP2, and DP3. In
Eq. (4), an FR corresponds exclusively to only one DP. A
designer can treat one FR–DP set regardless of the remaining
two FR–DP sets. Therefore, when a variation exists in a cer-
tain FR–DP set, there is no influence from the variation over
the other FR–DP sets. That is, each FR is independent of the
other FRs.
The Independence Axiom recommends the uncoupled de-
sign as shown in Eq. (4). In this design, the relationship be-
(a) General process (b) The hierarchy and zigzag mappingprocess for this research
Fig. 3 Concept of domain, mapping and decomposition
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Design of a Spacer Grid Using Axiomatic Design 991
tween the FR and the DP is one-to-one. If it is impossible
to achieve this design, the next best is a decoupled design as
shown in Eq. (3). The coupled design in Eq. (2) is not desir-
able from the viewpoint of axiomatic design. In this case, the
designer must create new ideas for FRs or DPs to achieve a
decoupled design or an uncoupled design.
The Information Axiom is used to select the best designout of several designs that satisfy the Independence Axiom.
According to the Information Axiom, the best design among
those satisfying the Independence Axiom is the one with the
least information content. The axiomatic design research
group says that the information content may be inversely pro-
portional to the probability of achieving design goals. In this
work, the design of a spacer grid is carried out by using only
the Independence Axiom because multiple design candidates
are not extracted.
The FRs and DPs are decomposed to make a hierarchy
and a zigzag mapping process is used during the decompo-
sition.12–17) The process is illustrated in Fig. 3. The FRs for a
design problem are decomposed into a hierarchy, and the to-
tal design description at any level of the hierarchy consists of
the engineering aspects needed to satisfy the stated objectives.
Thus, the FR–DP mapping process takes place over a number
of levels of abstraction, but a given set of FRs must be suc-
cessfully mapped to a set of DPs in the physical domain prior
to the decomposition of the FRs. Iterations between FR-to-
DP mapping and functional decomposition need zigzagging
processes between the functional and physical domains. Ac-
tually, when the hierarchy arrives at the bottom leaves, the
design is completed.
III. Design of a Spacer Grid Using the IndependenceAxiom
1. Description of the Problem
As mentioned earlier, the spacer grid in Fig. 2 is a part of
the fuel assembly that supports the fuel rod. Here are the
general features that a spacer grid must have.4–7) First, it must
make the fuel rod stationary. Second, it must supply a cooling
flow path that encourages heat transfer from the hot fuel rod to
the coolant. Third, it must protect the control rod guide path
in any abnormal operating environment. A designer needs to
consider the above features in designing a spacer grid; the
third feature, the design for consistent safe operation such as
safe shutdown of the reactor in an emergency is especially
important.
The above general features are related to various complex
engineering fields such as structural mechanics, metallurgy,
thermal-hydraulics and manufacturing. In reality, it is ex-
tremely difficult to consider all the disciplines simultaneously
even in modern engineering. Therefore, when a component is
designed by using a certain discipline, data from other dis-
ciplines are generally assigned fixed values. By the same
token, some values are fixed as constants from other disci-
plines4–7) or some aspects are ignored due to simplification of
the model. For example, the finite element (FE) model uti-
lized in this research is illustrated in Fig. 4 and it has 5 by 5grids. Actually, the real full model has 16 by 16 grids. The
simplified spacer assembly model without the thimble sleeve
Fig. 4 Applied boundary condition for impact FE analysis model
of 5 by 5 cell grid
and fuel rod is enough to grasp the design trend at the con-
ceptual design stage because their effects are not large.4–7)
Welding spots are modeled by merging the nodes and they are
tuned by experiments. And several parameters such as strap
thickness, material property, etc. are also not considered in the
design process. These parameters have been determined from
other disciplines such as structural mechanics and thermal-hydraulics. Thus, this work is limited to those boundaries.
In an abnormal operating environment such as in an earth-
quake, lateral impact loads are applied to the spacer grids
as illustrated in Fig. 5. The spacer gird must protect the
guide path of the control rod against these lateral impact
loads. Therefore, the spacer grid must have sufficient strength
against this load.
A fuel rod contacts with the spring because the spring sup-
Fig. 5 Unit spacer grid under impact load
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Fig. 6 Contact between a nuclear fuel rod and a spring
ports the fuel rod as illustrated in Fig. 6. Thus, there must be
contact pressure on the contact surface. Flow induced vibra-
tion from the coolant also causes the fretting phenomenon,
which results in wearing down of fuel rods and leakage of
radioactive products. Therefore, wearing down of the fuel
rod should be prevented. When a fuel rod is inserted into the
spacer grid, most deformation occurs not in the dimple, but
in the spring because the dimple is stiffer. Generally, the de-
formation behavior is elasto-plastic. The grid spring loses its
strength due to irradiation-induced relaxation and creep-down
of fuel rod diameter. Consideration must be made at the de-
sign stage to keep the supporting reaction force throughout
the lifetime of the reactor. Therefore, plastic deformation in
the spring must be minimized and the grid spring must sup-
port the fuel rod with the proper initial force. Based on these
observations, FRs and DPs are defined from a zigzagging pro-
cess and decomposition.
2. Conceptual Design by Definition of the Design Equa-
tion
A spacer grid must have sufficient strength to protect the
guide path of the control rod against lateral impact loads. The
spacer grid assembly is composed of inner and outer straps,
fuel rods, thimble sleeves, etc. As mentioned earlier, thimble
sleeves and fuel rods have little influence on strength of the
structure. Thus, they are not considered in the analysis and
design process. Therefore, the main components responsi-
ble for the strength of the spacer grid are the outer and inner
straps as illustrated in Fig. 5. The strength of the outer strapis mainly dependent on its thickness. But the thickness of the
outer strap is fixed to 0.664 mm because the simplified model
of 5 by 5 grid cells is used in this work. Actually, the outer
strap is located around the edges of the 16 by 16 full model.
Thus, we can only control the strength of the inner strap to
resist the impact load. The main part of the inner strap which
resists the impact load is illustrated in Fig. 7. Therefore, FR1
and DP1 are defined as follows:
FR1: Control strength of the inner strap
DP1: Dimension l in Fig. 7.
The spacer grid must safely support the fuel rod. Thus, FR2
and DP2 are as follows:
FR2: Support fuel rod safelyDP2: Supporting part for the fuel rod (spring).
The type of the design matrix is determined by the relation-
Fig. 7 Unit inner strap in a spacer grid without the spring
ship between FRs and DPs. In this work, the supporting part
for the fuel rod is placed at the center of the inner strap as
illustrated in Fig. 7. Therefore, after the strength of the inner
strap (FR1) is determined, the space for the supporting partcan be fixed. The design is a decoupled design as
FR1
FR2
=
X O
X X
DP1
DP2
. (5)
The supporting part needs to prevent leaking of radioactiv-
ity caused by wearing due to contact pressure and the fretting
phenomenon. Also, it needs to reduce the plastic deformation
to maintain the required spring force. Thus FR2 is decom-
posed into the following two functional requirements:
FR21: Reduce the contact pressure on the contact surface
FR22: Reduce the maximum stress of spring under a spe-
cific spring force.
Once the contact force is introduced, the contact pressure
is determined by the contact area. Thus the contact area can
control the contact pressure. The spring force and the strength
of the spring are the result of the shape and thickness of the
spring. The thickness of the spring cannot be designed due
to restrictions in the manufacturing process. Thus the shape
of the spring controls the spring force and the stress in the
spring. Therefore, the above second level FRs are mapped to
the following second level DPs:
DP21: The shape of the contact part of the spring
DP22: The shape of the spring arms.
The fixed shape of the contact area of the spring imposes
a small restriction on the shape change of the spring so thatFR21 is hardly related to DP22. The design equation for the
second level is as follows:FR21
FR22
=
X O
X X
DP21
DP22
. (6)
As design matrices in Eqs. (5) and (6) indicate, the spacer
grid is designed in the sequence of DP1, DP21 and DP22.
3. Detailed Design of the Design Parameter Dimensionl
Prior to decision of DP1, it is necessary to define the criti-
cal impact load of a spacer grid from experiments as follows.
In an experiment, the magnitude of impact load is gradually
increased and applied to a spacer grid. If the maximum re-action force of the spacer grid does not increase any more at
the i th step, then the impact load applied at the (i −1)th step
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Fig. 8 Schematic diagram for free fall shock machine
Table 1 Material properties of Zircaloy-4
Elastic Plastic
E σ y ρν
σ ε
(GPa) (MPa) (kg/m3) (MPa)
328.0 0.0105.15 328.0 6,550 0.294
443.0 0.340
is considered as the critical impact load.10)
First the dimension of l is determined so that the critical im-
pact load of the designed spacer grid is higher than or equal
to a nominal value. The nominal value can be obtained from
an experiment with the existing spacer grids that are currently
used in reactor. Experiments with existing ones have beencarried out a few times to get the mean value of the critical
impact load using the free-fall tester as illustrated in Fig. 8.18)
Since the mean value of the critical impact load from the ex-
periments is 4,500N, the nominal value of the critical impact
load is set to 4,500N.
For numerical simulation, the FE model in Fig. 4 is devel-
oped.19) The model has a rigid sphere with mass and initial
velocity, a rigid plate and the spacer grid with shell elements
for impact analysis. The impact load is obtained from the
contact force among them. For higher velocity, larger impact
load is developed. The target for this simulation is to find a
value of l for the critical impact load with the nominal valueof 4,500N. Material properties in Table 1 are used in the sim-
ulation. As a result of several nonlinear analyses with sev-
eral candidate values of l , the critical impact load of a spacer
grid with l of 4.374 mm is slightly higher than the nominal
value. Therefore, the dimension of DP1 is determined to be
4.374 mm.
4. Detailed Design of the Shape of the Contact Area
(DP21) and the Shape of the Spring (DP22)
Due to the coolant flow, the fuel rod vibrates with the sup-
ports from the springs.5) The relative infinitesimal motion be-
tween two bodies causes fretting wear on the contact surfaces
of the fuel rod and spring. A design should be performedto minimize the wear and it can be achieved through mini-
Fig. 9 Contact pressure contour and contact area
Table 2 Contact pressure and area for DP21
Original design Improved design
Contact pressure (N/mm2) 2,190.0 323.0
Contact area (mm2) 0.1126 0.8528
mization and uniform distribution of the local contact pres-
sure (DP21). Improved shapes are searched through many
trials and errors in simulation. The original shape and the
final result are illustrated in Fig. 9 and Table 2. In the new
design, the contact pressure is considerably reduced and the
contact area is larger than that of the original design.
Next, the shape of the spring (DP22) should be determined
to minimize the maximum stress. The spring is deformed by
the manufacturing tolerance of the fuel rod assembly, exces-
sive shipping loads, and the loading condition in the nuclear
reactor. The load applied to the spring in a spacer grid can beexpressed by Eq. (7). Due to the radiation by neutrons in the
reactor, only about 8% of the initial spring force remains af-
ter long-term operation. A load with about 2N from the fluid
induced vibration is applied to the spring.9,11) Therefore, to
support the fuel rod throughout the operating period, the ini-
tial spring force of a spacer grid must be greater than 25N as
shown in Eq. (8) which is a brief expression of Eq. (7):
F spring × 0.08 > 2N (7)
F spring > 25N after manufacturing of fuel assemblies. (8)
After manufacturing of fuel assemblies, it has been found
that the spring of the spacer grid was deformed by about
0.2 mm due to the insertion of the fuel rod through the spacer
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994 K.-J. PARK et al.
grid.9,11) At that very moment, the supporting load of the
spring is equivalent to Eq. (8). It is the initial supporting
condition which the spring must have before operating the
reactor. However, the fuel rod is pulled into each spacer grid
cell and the maximum height difference among grid cells is
0.4 mm. The grid spring has the same deflection during the
manufacturing process as illustrated in Fig. 10. The heightdifference is caused by manufacturing tolerances. After in-
sertion of the rod, the performance of the grid spring can be
deteriorated if the excessively deformed spring is not able to
be recovered to the initial displacement of 0.2 mm with 25N.
Generally speaking, considering the above deformation and
height difference, a load of 50N is applied to a linear spring
and the linear spring is deformed by 0.4 mm. That means the
best spring in a spacer grid has the ideal behavior characteris-
tics as illustrated in Fig. 11. However, most of the springs ac-
tually exhibit partial plastic deformation at the displacement
of 0.4 mm. Thus, the spring can be nearly linear if the plastic
Fig. 10 Manufacturing process of fuel assembly and schematic di-
agram
Fig. 11 Ideal force–displacement curve for a grid spring
deformation is minimized in the above force-deflection range.
It is noted that the design is a decoupled design as shown
in Eq. (5). Thus DP1 fixes a space for the spring. Moreover,
the shape of the contact part somewhat reduces the room for
designing the shape of the spring as shown in Eq. (6). Under
these circumstances, the problem is defined with the maxi-
mum stress as the objective function. The optimization prob-lem is formulated for the spring shape DP22 as follows:
Find DP22
to minimize maximum stress
subject to [K ]{δ} = { f }
δmax = 0.4 mm, (9)
where [K ] is the stiffness matrix, {δ} is the displacement vec-
tor, { f } is the external force of the finite element analysis
equation, and δmax is the deflection at the center of the spring.
When a maximum property is included in the optimization
formulation, the problem can be solved by using an artificialvariable as follows:23)
Find DP22
to minimize β
subject to [K ]{δ} = { f }
σ < β (at all the elements of FE analysis)
δmax = 0.4 mm, (10)
where β is the artificial variable. The artificial variable β
is used for the objective function of a min-max problem.
Thus, the artificial variable β is minimized while the con-
straints including all the stresses are satisfied. The shape of
the spring obtained from this formulation minimizes the max-
imum stress subject to the displacement of 0.4 mm under the
given constant load in the elastic range. For shape optimiza-
tion,24,25) a quarter FE model of the spring is utilized as illus-
trated in Fig. 12 and design variables are indicated in Fig. 13.
DV x is the x th design variable in Fig. 13 and Table 3. That is,
DP22 is a vector which consists of eleven design variables in
Fig. 13. They are the changes of the coordinates on selected
nodal points. Overall shape changes of the FE model can be
obtained by interpolation or extrapolation between them.22)
Fig. 12 Initial unit inner strap and its quarter model
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Fig. 16 Force–velocity curve of the designed spacer grid
vspace*-3pt
FE analysis are illustrated in Fig. 16. To save analysis time,
two spacer grids are selected, without the spring illustrated in
Fig. 7, and with the spring illustrated in Fig. 15. Compared
to the spacer grid without the spring, the critical load of the
product with the spring is slightly higher by 100N. Thus, it is
not exactly true that the supporting part (DP2) does not affect
the strength of the inner strap (FR1) as mentioned in Eq. (5).
However, the influence of DP2 upon FR1 is sufficiently small.
Therefore, the spring effect is considered to be negligible for
the strength of the inner strap, and the design matrix is a de-
coupled one as shown Eq. (5). This consideration is backed upby a theorem which implies that if the amount of the effect by
a DP on an FR is less than the designer specified tolerance in
an element of design matrix, that element can be neglected.15)
The design matrix of the spacer grid in this work is a de-
coupled one. Thus, if change is required for the strength of
the inner strap, the shape of the spring should be redesigned.
And if the loading condition of the spring is to be changed,
only the shape of the spring can be changed, not the strength
of the inner strap.
V. Conclusions
A conceptual design process was proposed for a spacergrid using the Independence Axiom. Functional requirements
were defined and mapped onto appropriate design parameters.
A functional requirement of the first level was decomposed
into two functional requirements of the second level. The de-
sign was found to be decoupled and detailed designs were
carried out based on the sequences that the design equations
indicated. In the detailed design, finite element analyses and
numerical optimizations were employed. The performance of
the new design was significantly improved. The research was
conducted for a simplified model with 5 by 5 grids while the
full model has 16 by 16 grids. Currently, design work with 16
by 16 grids is being performed with a larger number of design
variables and the same method explained in this paper.
The functional requirements in this work were defined from
a structural viewpoint. But the real working environment of
spacer grids should be analyzed from various viewpoints such
as thermodynamics, fluid dynamics, structural dynamics, and
nuclear engineering. If these are considered, the functional re-
quirements might be changed or even conflict with those from
non-structural dynamics considerations. These days, multi-
disciplinary design optimization (MDO) is being developedto consider multiple disciplines in the optimization process.
Therefore, it is necessary to employ an MDO method in fu-
ture studies.
Acknowledgments
This research was supported by the high performance
spacer grid structure program of the Korea Atomic Energy
Research Institute. This research was also supported by the
Center of Innovative Design Optimization Technology, Korea
Science and Engineering Foundation. The authors are thank-
ful to Mrs. MiSun Park for her correction of the manuscript.
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VOL. 40, NO. 12, DECEMBER 2003