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Journal of Nuclear Science and Technology

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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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990 K.-J. PARK et al.

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

JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY


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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

VOL. 40, NO. 12, DECEMBER 2003


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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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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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design of a spacer grid using axiomatic design

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