Skip to main content
SpringerLink
Log in

Two-dimensional cellular metals as multifunctional structures: topology optimization

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

Highly porous two-dimensional (2D) cellular metals have multifunctional attributes, with tailorable structures to achieve multifunctional performance. The focus of this study is to explore the optimal cellular topology of 2D cellular metals for heat dissipation, and to investigate the eligibility of different heat enhancement techniques for more efficient heat dissipation. An analytical approach for the optimal design of metallic 2D cellular materials, cooled by single-phase laminar forced convection in various flow configurations, is proposed and validated by comparison with full numerical simulations. The optimal design is characterized by two subsidiary dimensionless parameters: one reflecting the trade-off between convection and fluid friction, and the other reflecting the optimal balance between conduction and convection. A heat transfer enhancement technique––boundary layer redevelopment––is subsequently introduced and its feasibility examined experimentally. Future research directions in specific areas are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Abbreviations

A :

area

c :

coefficient

c p :

specific heat at constant pressure

C:

perimeter

D h :

hydraulic diameter

e :

roughness

E :

Young’s modulus

\( \tilde{E} \) :

Young’s modulus of a cellular solid

f :

friction factor

\( f_{D_{h}} \) :

friction factor based on the size of a single duct

f H :

friction factor based on the height of the 2D cellular core

F :

function

g :

acceleration due to Earth’s gravity

Gr :

Grashof number

h :

heat transfer coefficient

\( \bar{h} \) :

overall heat transfer coefficient

h(x):

local heat transfer coefficient

H :

overall height of a 2D cellular solid

Inf:

infinity

k :

thermal conductivity

k eff :

effective thermal conductivity of a 2D cellular solid

k e,ax :

effective thermal conductivity of a 2D cellular solid in axial direction

k e,r :

effective thermal conductivity of a 2D cellular solid in lateral direction

K :

pressure loss coefficient

l bottom :

length of the bottom parallel side in trapezoidal cells

l top :

length of the top parallel side in trapezoidal cells

L :

overall length of a 2D cellular solid

L ch :

length of the inlet channel defined in the computation domain in Fig. 3.1(b)

L hy :

hydrodynamic developing length

M :

mass flow rate

M :

dimensionless mass flow rate

n :

normal direction

N H :

the cell number along the height of a 2D cellular solid

N L :

block number

N W :

the cell number along the width of a 2D cellular solid

Nu :

Nusselt number

Nu H :

Nusselt number based on the height of the 2D cellular core

p :

pressure

p d :

dynamic pressure measured by the Pitot tube

\( \Updelta p \) :

pressure loss

\( {\mathbf{\Updelta p}} \) :

dimensionless pressure loss

\( \Updelta p_{\text{l}} \) :

local pressure loss

\( \Updelta p_{\text{f}} \) :

frictional pressure loss

P :

pumping power

P :

dimensionless pumping power

Pr :

Prandtl number

\( q^{\prime\prime} \) :

heat flux

Q :

total heat transfer

Q :

dimensionless heat transfer

R :

thermal resistance

Ra :

Rayleigh number

Re :

Reynolds number

\( Re_{{D_{\text{h}} }} \) :

Reynolds number based on the size of a single duct

\( Re_{H} \) :

Reynolds number based on the height of the 2D cellular core

t :

cell-wall thickness

\( t_{\text{face}} \) :

face-sheet thickness

T :

temperature

u, v, w:

x, y, z components of a velocity vector in Cartesian coordinate

U :

velocity vector

U c :

centre line velocity in the inlet channel upstream of the specimen

U m :

mean velocity in the inlet channel upstream of the specimen

\( U^{\prime}_{\text{m}} \) :

mean velocity averaged over the specimen’s porosity

W :

overall width of a 2D cellular solid

x, y, z :

Cartesian coordinate (shown in Fig. 1)

x + :

dimensionless axial distance in hydrodynamic developing region

x*:

dimensionless axial distance in thermal developing region

α:

thermal diffusivity

β:

coefficient of thermal expansion

ε :

porosity

ϕ :

surface area density

μ :

dynamic viscosity

ν :

kinematic viscosity

θ :

base angle in triangular, trapezoidal, and hexagonal cells

ρ :

relative density

\( \tilde{\rho } \) :

density of a cellular solid

σ:

strength

\( \tilde{\sigma } \) :

strength of a cellular solid

\( \tau_{\text{w}} \) :

wall shear stress

Ψ:

shape factor

M :

flow configuration of fixed mass flow rate

Δp :

flow configuration of fixed pressure drop

P :

flow configuration of fixed pumping power

Crt:

critical

Eff:

effective

F:

fluid

In:

inlet

Max:

maximum

Min:

minimum

N:

normal direction

Opt:

optimal

Out:

outlet

Ref:

reference

S:

solid

W:

wall

SC:

sudden contraction

SE:

sudden expansion

References

  1. Seepersad CC, Fernandez MG, Panchal JH, Choi HJ, Allen JK, McDowell DL, Misstree F (2004) Foundations for a systems-based approach for materials design. In: 10th AIAA/SSMo multidisciplinary analysis and optimization conference, Albany

  2. Gu S, Lu TJ, Evans A (2001) On the design of two-dimensional cellular metals for combined heat dissipation and structural load capacity. Int J Heat Mass Transf 44:2163–2175

    Article  MATH  Google Scholar 

  3. Hayes AM AJW, Dempsey BM, McDowell DL (2004) Mechanics of linear cellular alloys. Mech Mater 36:691–713

    Article  Google Scholar 

  4. Seepersad CC, Dempsey BM, Allen JK, Mistree F, McDowell DL (2004) Design of multifunctional honeycomb materials. AIAA J 42:1025–1033

    Article  Google Scholar 

  5. Wen T, Xu F, Lu TJ (2008) 2D cellular metals as a multifunctional structures––I. review

  6. Lu TJ (1999) Heat transfer efficiency of metal honeycombs. Int J Heat Mass Transf 42:2031–2040

    Article  MATH  Google Scholar 

  7. Yilmaz A, Buyukalaca O, Yilmaz T (2000) Optimum shape and dimensions of ducts for convective heat transfer in laminar flow at constant wall temperature. Int J Heat Mass Transf 43:767–775

    Article  MATH  Google Scholar 

  8. Muzychka YS (2005) Constructal design of forced convection cooled microchannel heat sinks and heat exchangers. Int J Heat Mass Transf 48:3119–3127

    Article  MATH  Google Scholar 

  9. Bejan A (1995) Convection heat transfer. Wiley, New York

    MATH  Google Scholar 

  10. Wen T, Xu F, Lu TJ (2007) Structural optimization of two-dimensional cellular metals cooled by forced convection. Int J Heat Mass Transf 50:2590–2604

    Article  MATH  Google Scholar 

  11. Mereu S, Sciubba E, Bejan A (1993) The optimal cooling of a stack of heat generating boards with fixed pressure drop, flowrate or pumping power. Int J Heat Mass Transf 36:3677–3686

    Article  Google Scholar 

  12. Bejan A, Sciubba E (1992) The optimal spacing of parallel plates cooled by forced convection. Int J Heat Mass Transf 35:3259–3264

    Article  Google Scholar 

  13. Bergles AE (2003) High-flux processes through enhanced heat transfer. In: 5th international conference on boiling heat transfer, Montego Bay

  14. Xu JL, Gan YH, Zhang DC, Li XH (2005) Microscale heat transfer enhancement using thermal boundary layer redeveloping concept. Int J Heat Mass Transf 48:1662–1674

    Article  Google Scholar 

  15. Wen T (2007) Thermo-fluid characteristics of metallic 2D cellular materials, In: Department of Engineering. vol PhD Thesis, University of Cambridge, Cambridge

  16. Wen T, Tian J, Lu TJ, Queheillalt DT, Wadley HNG (2006) Forced convection in metallic honeycomb structures. Int J Heat Mass Transf 49:3313–3324

    Article  Google Scholar 

  17. Kumar RS, McDowell DL (2004) Rapid preliminary design of rectangular linear cellular alloys for maximum heat transfer. AIAA J 42:1652–1661

    Article  Google Scholar 

  18. Papautsky I, Ameel T, Frazier AB (2001) A review of laminar single-phase flow in microchannels. In: Proceedings of 2001 ASME international mechanical engineering congress and exposition, New York

  19. Rostami AA, Mujumdar AS, Saniei N (2001) Flow and heat transfer for gas flowing in microchannel: a review. Heat Mass Transf 38:359–367

    Article  Google Scholar 

  20. Morini GL (2004) Single-phase convective heat transfer in microchannel: a review of experimental results. Int J Therm Sci 43:631–651

    Article  Google Scholar 

  21. Kandlikar SG (2005) High flux heat removal with microchannels––a roadmap of challenges and opportunities. Heat Transf Eng 26:5–14

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the Overseas Research Studentship (ORS) and Overseas Trust Scholarship of Cambridge University, the National Natural Science Foundation of China (10572111, 10632060), the National Outstanding Youth Foundation, the National 111 Project of China (B06024), and the National Basic Research Program of China (2006CB601202).

Author information

Authors and Affiliations

  1. Engineering Department, Cambridge University, Cambridge, CB2 1PZ, UK

    F. Xu & T. Wen

  2. MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, 710049, Xi’an, People’s Republic of China

    T. J. Lu

Authors
  1. F. Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. J. Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. J. Lu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, F., Wen, T. & Lu, T.J. Two-dimensional cellular metals as multifunctional structures: topology optimization. Heat Mass Transfer 45, 485–501 (2009). https://doi.org/10.1007/s00231-008-0450-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-008-0450-0

Keywords

Search

Navigation