Active cooling by metallic sandwich structures with periodic cores T.J. Lu a,b, * , L. Valdevit c , A.G. Evans c a Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK b State Key Laboratory for Mechanical Behavior of Materials, Xian Jiaotong University, Xian 710049, P.R. China c Materials Department, University of California, Santa Barbara, CA 93106-5050, USA Abstract We review the thermal characteristics of all-metallic sandwich structures with two dimen- sional prismatic and truss cores. Results are presented based on measurements in conjunction with analytical modeling and numerical simulation. The periodic nature of these core struc- tures allows derivation of the macroscopic quantities of interest—namely, the overall Nusselt number and friction factor—by means of correlations derived at the unit cell level. A fin anal- ogy model is used to bridge length scales. Various measurements and simulations are used to examine the robustness of this approach and the limitations discussed. Topological prefer- ences are addressed in terms scaling relations obtained with three dimensionless parame- ters—friction factor, Nusselt number and Reynolds number—expressed both at the panel and the cell levels. Countervailing influences of topology on the Nusselt number and friction factor are found. Case studies are presented to illustrate that the topology preference is highly application dependent. Ó 2005 Elsevier Ltd. All rights reserved. 0079-6425/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2005.03.001 * Corresponding author. Tel.: +44 1223 766316; fax: +44 1223 332662. E-mail address: [email protected](T.J. Lu). Progress in Materials Science 50 (2005) 789–815 www.elsevier.com/locate/pmatsci
Transcript
Progress in Materials Science 50 (2005) 789–815
www.elsevier.com/locate/pmatsci
Active cooling by metallic sandwich structureswith periodic cores
T.J. Lu a,b,*, L. Valdevit c, A.G. Evans c
a Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UKb State Key Laboratory for Mechanical Behavior of Materials, Xian Jiaotong University,
Xian 710049, P.R. Chinac Materials Department, University of California, Santa Barbara, CA 93106-5050, USA
Abstract
We review the thermal characteristics of all-metallic sandwich structures with two dimen-
sional prismatic and truss cores. Results are presented based on measurements in conjunction
with analytical modeling and numerical simulation. The periodic nature of these core struc-
tures allows derivation of the macroscopic quantities of interest—namely, the overall Nusselt
number and friction factor—by means of correlations derived at the unit cell level. A fin anal-
ogy model is used to bridge length scales. Various measurements and simulations are used to
examine the robustness of this approach and the limitations discussed. Topological prefer-
ences are addressed in terms scaling relations obtained with three dimensionless parame-
ters—friction factor, Nusselt number and Reynolds number—expressed both at the panel
and the cell levels. Countervailing influences of topology on the Nusselt number and friction
factor are found. Case studies are presented to illustrate that the topology preference is highly
application dependent.
� 2005 Elsevier Ltd. All rights reserved.
0079-6425/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.
All-metallic lightweight sandwich structures with periodic truss and prismaticcores (Fig. 1) have the potential for simultaneous load bearing and active cooling.
The load bearing characteristics have been well-documented in the recent literature,
demonstrating the structural efficiency benefits of various topologies [1–7]. Con-
tinuous channels of these sandwiches allow internal fluid transport, enabling simul-
taneous active cooling [8–15]. A comprehensive assessment of these cooling
capabilities, involving comparisons between different core topologies, has yet to be
assembled. The present review addresses this deficiency.
The attributes of open cell sandwich structures for active cooling are comparedusing three non-dimensional groups: (a) the Nusselt number, Nu, characterizing
the heat transfer, (b) the friction factor, f, (or the loss coefficient, Kcell), for the pres-
sure drop and (c) the Reynolds number, Re, for the fluid flow rate. When appropri-
ately defined, results presented in terms of these parameters allow comparison with
louvered fins [16], corrugated ducts [17], sintered beds [18], open-celled metal foams
[19–24] and other competing concepts. Unless stated otherwise, air-cooling is to be
considered: although many of the formulae can be used to examine trends when
other fluids are used.The approaches used to analyze truss and prismatic cores are distinguished by
the shapes of the representative unit volumes (Figs. 2 and 3). Due to periodicity,
prismatic systems [8,9] are analyzed by using a narrow through-thickness corruga-
tion (Fig. 2), with characteristic dimension dictated by the shape and size of each
cell along the corrugation. For truss structures [also referred to as lattice-frame
materials (LFMs)] [10–15], the core members and the faces are embodied within a
common unit cell (see Fig. 3 for tetragonal cells). The characteristic dimension is
the cell edge length (or cell size). Such deterministic approaches differ from the
Nomenclature
A area (m2)cp specific heat at constant pressure (J/kgK)
Cp pressure coefficient
d Strut diameter (m)
dp unit cell length (m)
dc core wall thickness for prismatic panels (m)
Dh hydraulic diameter (m)
h local heat transfer coefficient (W/m2K)�h overall heat transfer coefficient (W/m2K)H, W, L height, width and length of sandwich (m)
k thermal conductivity (W/mK)
lc core member length (m)_m mass flow rate (kg/s)
n unit cell index
N total number of cells in flow direction
Ns number of corrugated cells per unit width
p static pressure (Pa)P pumping power (W)
q heat flux (W/m2)
Q heat flow (W)
Ropen ratio of open area to total area
Sx, Sy pitch lengths (m)
t time (s)
T temperature (K)
U flow velocity (m/s)x, y, z cartesian coordinates
Greek symbols
a, as non-dimensional parameters
g, n non-dimensional coordinates
j thermal diffusivity (m2/s)
l viscosity (kg/ms)
q density (kg/m3)�q relative density
s time constant (s)
Non-dimensional groups
Bi Biot number
f friction factor
j Colburn factor
Kcell pressure loss coefficient
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 791
Nu Nusselt number
Pr Prandtl number
Re Reynolds number
Subscripts
dp non-dimensional groups based on unit cell length
f fluid
h non-dimensional groups based on hydraulic diameter
H non-dimensional groups based on core height
s solid
x evaluated at location x along the panel0 at time t = 0
ss steady-state
792 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
effective porous medium approach widely used for stochastic foams [20,22] or sin-
tered beds [18,25].
2. Thermal definitions
The heat transfer coefficient at any location x (Fig. 4) along a sandwich panel
heated on one face is:
�hx ¼qðxÞ
T wðxÞ � T fðxÞð2:1Þ
where q(x) and Tw(x) are the imposed heat flux and the wall temperature of the
heated face (averaged in y) and Tf(x) is the bulk fluid temperature.1 The bar distin-
guishes this quantity from the local heat transfer coefficient at the core element level.
Integration along the length of the panel gives:
�h ¼ 1
L
Z L
0
�hx dx ¼ QWLDT
ð2:2Þ
with Q the total heat transferred to the fluid and DT the temperature difference:
DT ¼ DT lm ¼ ðT w � T inÞ � ðT w � T outÞlnfðT w � T inÞ=ðT w � T outÞg
for iso-temperature b:c:’s
T wðxÞ � T fðxÞ for iso-flux b:c:’s
8<:
ð2:3Þ
1 The bulk fluid temperature, or mixing-cup temperature, is defined as: T f ðxÞ ¼R
cross�sectionuT dAR
cross�sectionu dA
.
Fig. 1. Typical prismatic and truss topologies.
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 793
Note that for a fully developed thermal profile, �h ¼ �hx, and that, for iso-flux
boundary conditions, the difference between the wall and the fluid temperature is
independent of x. Ascertaining �h is of primary importance for heat exchanger design,since it is a direct measure of the heat that can be exchanged for a given temperature
difference.
To specify the thermal efficiency of a panel, the three non-dimensional para-
meters of interest are the Nusselt number, Reynolds number and friction factor,
given by:
NuH ¼�hHkf
; ReH ¼ qfUHlf
; f H ¼ ðDp=LÞHqfU
2=2ð2:4Þ
where Dp is the overall pressure drop (over the length L), U the mean velocity at the
inlet, qf the fluid density, lf its viscosity and kf its thermal conductivity (qfU2/2 is the
dynamic pressure). The length scale H is arbitrary, but selected to facilitate
Fig. 2. Unit cells used to model the thermal performance of prismatic structures.
794 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
comparisons among different technologies.2 The value of NuH depends on the core
topology and relative density, as well as the solid and fluid properties (namely, the
ratio of the solid and fluid conductivities and the Prandtl number of the fluid). It
embodies three different heat transfer mechanisms: conduction through the core,convection core-fluid and convection face-fluid. The challenge is to ascertain the heat
transfer coefficient and the pressure drop for each of the core topologies over a wide
range of Reynolds number. The protocol relies on knowledge of the point-wise heat
transfer characteristics of the constituent elements of the panel. This information is
typically available in the literature, subject to some simplifications. Two correlations
are especially useful: one for flow in prismatic ducts and the other for flow over
banks of tubes. In the next section, we present these correlations, and in Section 4
apply them to the calculation of NuH for the panels of interest.
2 Another possible choice would be the hydraulic diameter of the overall panel, which for an infinitely
large panel is equal to 2H.
Fig. 3. The tetrahedral truss unit.
Fig. 4. The panel used for analysis identifying the dimensions, the heat source and the flow.
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 795
On scaling grounds, for fully developed flow in ducts, NuH = f (ReH,Pr), where
Pr = m/a is the Prandtl number (roughly 0.7 for most gases, including air). For most
geometries and flows,3 the heat transfer performance of sandwich panels that use dif-ferent working fluids may be compared using the Colburn factor j ¼ NuH
ReHPr Pr2=3 in lieu
of NuH. In the following section, we present results for the Nusselt number and fric-
tion factor widely used in the heat transfer community.
3. Basic scaling
3.1. Prismatic structures
For prismatic structures, the characteristic dimension of the unit cell is its hydrau-
lic diameter, Dh, related to the cell edge length lc and topology by:
3 Incorporating Re in the Colburn factor is an attempt to derive an analogy between heat transfer and
fluid friction: j and NuH should scale with Re in the same way, and j should not depend on Pr.
Table 1
Non-dimensional coefficients and Nusselt number for cells having different shapes
Cell type ca cH gfin Nuh
Triangular-4 6.93 1.73 2.0 3.0
Triangular-6 6.94 1.16 3.0 3.0
Square-3 4.0 1.5 1.78 3.61
Square-4 4.0 1.0 2.0 3.61
Hexagon 2.31 1.16 1.5 4.02
796 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
Dh ¼ 4lc
ffiffiffiffiffiffiffiffiffiffiffi1 � �q
p=cA ð3:1Þ
where �q is the relative density of the cellular material in the core and the non-dimen-
sional coefficient cA depends on cell shape (Table 1, Fig. 2). The resulting non-dimen-
sional parameters are:
Nuh ¼ hDh
kf
; Reh ¼ qfUDh
lf
; f h ¼ ðDp=LÞDh
qfU2=2
ð3:2Þ
where h is the local heat transfer coefficient averaged over the perimeter of the cell.4
When sufficient cells exist through the thickness and when the cell wall is a good con-
ductor (such that the solid temperature over a cell is nearly uniform in z), the fully
developed value of Nuh depends on the shape of the cell, but not on its size.
As a preamble, correlations for prismatic ducts subject to longitudinally uniform
heat flux and peripherally uniform temperature (the so-called H1 boundary condi-
tions) are recalled. All correlations apply to fully developed thermal and velocity
profiles. The most authoritative source for the friction factor of ducts is Moody�sdiagram [26]; although derived for circular ducts, it can be applied to nearly any
cross-section, provided that the diameter is replaced by the hydraulic diameter. In
the laminar regime, fh � Re�1h , where the proportionality constant depends only
on the duct geometry (typically close to 64).5 In the turbulent regime, it is a strong
function of the duct wall roughness j [27]. For smooth ducts:
fh ¼ 0.316Re�0.25h ð3:3Þ
whereas for rough walls:
fh ¼ 4½1.74 logðDh=jÞ þ 2.78�2 ð3:4ÞThe Nusselt number is independent on the Reynolds number in the laminar regime
(Nuh � 4, depending slightly on the cross-section geometry), whereas for turbulent
flow [27]:
Nuh ¼ ðfh=8ÞðReh � 1000ÞPr
1 þ 12.7ffiffiffiffiffiffiffiffiffifh=8
pðPr2=3 � 1Þ
ð3:5Þ
4 h ¼ qðxÞT sðxÞ�T f ðxÞ
, where q(x) and T sðxÞ are peripherally averaged quantities at location x.
5 Note that many authors define fh ¼ ðDp=LÞDh
2qf U2 [27]. With this definition, fhReh � 16, and Eqs. (3.3)–(3.5)
change accordingly.
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 797
Similar correlations have been suggested for sandwich panels with prismatic cores,
with friction factor [14,26,28]:
fh � ðDp=LÞDh
qfU2=2
� 1
ð1 � �qÞ2Að1 � �qÞ
Reh
þ B�
ð3:6Þ
For smooth channels, measurements give: A � 140, B � 0.02 (Fig. 5a). The first term
in parenthesis characterizes the laminar range and the second applies when the flow
is form–dominant. The transition occurs when Reh � 2000. Since the Reynolds num-
ber and the friction factor both contain the hydraulic diameter, the shape and size of
the channels have a strong effect: in the laminar range Dp scales as (cA/lc)2 (see [9])
and in the form dominant range as (cA/lc).The Nusselt number in the laminar regime is fixed by the cell shape (Table 1),
whereas in the turbulent range, it increases with increase in Reynolds number. A
power law correlation can be attempted [28]:
Nuh � ðRehÞaPrb ð3:7Þ
Fig. 5. (a) Friction factor and (b) Nusselt number as a function of Reynolds number for prismatic cells.
798 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
where the coefficient a and b are determined by data fitting (a � 0.8,b � 0.4). Mea-
surements of the type presented in Fig. 5b reveal the experimental challenges. The
apparent dependence of Nuh upon the Reynolds number in the laminar regime is
attributed to the dominance of entry effects, which must be eliminated before
embarking on a thermal assessment.Perturbations on the channel surface increase both fh and Nuh but not proportion-
ally. Designs that maximize Nuh/fh include louvered fins (Fig. 6) and corrugated
ducts (Fig. 7), as well as lithography-based micro-channels that promote flow mixing
at low Reynolds number [29].
3.2. Truss structures
For truss structures, since the pressure loss per unit cell is well-defined [10–12,30](and largely caused by form drag), Kcell replaces the friction factor. The non-dimen-
sional parameters at the unit cell level are thus:
Nudp ¼hcelldp
kf
; Redp ¼qfUdp
lf
; Kcell ¼Dpcell
qfU2=2
ð3:8Þ
where dp is the cell dimension, and Dpcell is the static pressure drop over the cell. Note
that the usual definition of fH is still valid, and will be used later to compare resultsfor panels with different core topology. (For prismatic structures, a similar set of
parameters can be defined for each unit cell, if dp in (3.5) is replaced by Dh.) Pressure
Fig. 6. Compact heat exchanger with triangular channel and louvered fins [16].
Fig. 7. Corrugated ducts [17].
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 799
drop measurements (Fig. 8a) reveal that, when Redp < 2000, the fluid-flow is laminar
(viscous effects dominate) and Kcell � A=Redp , with A � 1000. (Note that A � 64 for
flow in a smooth pipe.) When Redp > 3000, form-drag predominates and Kcell
depends on the direction of flow relative to the unit cell (Fig. 3). For both prismatic
and truss structures, geometrical similarity dictates that [11,14]:
fh ¼ 1 � Ropen
Ropen
�2
; Kcell ¼1 � Ropen
Ropen
�2
ð3:9Þ
where Ropen is the ratio of open area to the cross-sectional area normal to the flow
direction [11,14,31]. Note that (3.9) compares favorably with the test data for textiles
(Fig. 8b).
The heat transfer coefficient at the strut level can be evaluated by envisaging the
trusses as staggered arrays of inclined cylinders, resulting in the following correla-
tions [33]:
Nud ¼ hstrutdkf
¼1.04Re0.4
d Pr0.36 Red ¼ 1–500
0.71Re0.5d Pr0.36 Red ¼ 500–1000
0.35Re0.6d Pr0.36 Red ¼ 103–105
8><>: ð3:10Þ
where d is the strut diameter. Note that the transition to turbulent flow occurs atRed � 1000. The effect of cylinder inclination with respect to the flow direction is
small [33]. For tetrahedral trusses (angle relative to the face sheet about 55�), the
Nusselt number is diminished by a factor 0.9 [33].
Fig. 8. Measurements for square and diamond textile cores converted into cell level parameters: (a)
friction factor and (b) Nusselt number.
800 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
4. Thermal performance
The procedure for evaluating the Nusselt number involves the following five
steps.
(i) A repetitive unit cell, thickness H, is identified.
(ii) The fluid temperature Tf(x) is calculated by imposing energy conservation.
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 801
(iii) Energy balance at the face–core interface is invoked to determine the heat flux
entering the core.
(iv) The temperature of the solid within the unit cell is calculated by using a 1D fin
analogy, as a function of Tf(x).
(v) The local heat transfer coefficient,
h ¼ qðx; y; zÞT sðx; y; zÞ � T fðxÞ
is either found in the literature, or derived using either experimental techniques
or numerical methods.
4.1. Prismatic cores
In order to illustrate the method, we report the analytical derivation of NuH for a
corrugated core panel with one face subject to constant heat flux and the other ther-
mally insulated. Analogous expressions can be obtained with different boundary con-
ditions [9,32]. With the effect of the horizontal fins neglected (they will be included
later), the corrugated wall behaves as a one-dimensional corrugated fin (Fig. 2) sub-
jected to a heat flux on one end and insulated at the other. Its temperature satisfies
the fin equation:
o2T sðnÞon2
� 2hksdc
½T sðnÞ � T f ¼ 0 ð4:1Þ
where n = (z/H)cH is the non-dimensional location, with cH a tortuosity coefficient
(Table 1). We solve (4.1), with boundary conditions:
�ks
oT s
on
n¼0
¼ q1
oT s
on
n¼cHH
¼ 0
8>>><>>>:
ð4:2Þ
where q1 is the heat flux entering the corrugated wall from one face (yet to be calcu-
lated). The resulting temperature in the solid is:
T sðx; nÞ ¼ T fðxÞ þq1
ksmcoshðmðcHH � nÞÞ
sinhðmcHHÞ ð4:3Þ
with m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2h=ksdc
p.
The temperatures averaged over the cross-section of the cellular array are gener-
ally deemed invariant with the vertical location in the channel, z (Fig. 4). In such
cases, for a panel with Ns core elements per unit width (the y-direction), the First
Law requires that [9]:
_mcpdT f
dx¼ qW ð4:4Þ
802 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
Here q is the imposed heat flux on the top face sheet, _m ¼ qfUWH is the mass flow
rate, and cp is the specific heat of the fluid. The solution is:
T fðxÞ ¼ T in þq
HqfcpUx. ð4:5Þ
In order to calculate q1, we impose the energy balance:
qWL ¼ fwqw þ gfinN s
dc
Wq1
� WL ð4:6Þ
where qw is the heat flux into the fluid from the faces, fw = 1 � Nscwdc/W representsthe area fraction of the face sheet in direct contact with the fluid, and gfin is a correc-
tion factor that accounts for the contribution to heat transfer from the horizontal
fins (Table 1). The number of cells per unit width, Ns = cNW/lc, with cN = 2/3 for hex-
agonal cells and cN = 1 for all others; similarly, cw = 1 for square cells, and
cw ¼ffiffiffi3
p=2 for all others. By noting that the heat flux from the faces is qw(x) =
h[Tw(x) � Tf(x)] (where Tw(x) = Ts(n = 0,x)), we can express q1 as a function of q.
By inserting the definition of the overall heat transfer coefficient (2.1), the ensuing
ð4:7ÞThis result allows the overall thermal characteristics to be related to the Nusselt
number evaluated at the cell level, Nuh. Moreover, the scaling with cell size, lc, is
included (though intimately linked with the other geometric and thermal terms).Note that the same expression would be obtained under iso-thermal boundary
conditions at both faces [9].
4.2. Truss cores
Trends are ascertained by considering that the fluid and wall temperatures are
constant within a unit cell; but increasing cell-by-cell along the stream direction.
The solid temperature variation along each strut in the nth cell from the entranceis governed by:
o2T sðg; nÞog2
� hstrutPksAstrut
ðT sðg; nÞ � T fðnÞÞ ¼ 0 ð4:8Þ
where g is a local coordinate along the strut axis, Ts the strut temperature, hstrut the
heat transfer coefficient averaged over the strut surface (assumed constant in the unit
cell), P the strut perimeter and Astrut its cross-sectional area. Solving with one face
subject to constant heat flux and the other thermally insulated gives the heat flow rate
entering each strut:
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 803
Qstrut ¼SxSyq
SxSy �pd2
4
�hstrut
ksmcoshðmLÞsinhðmLÞ þ
3
4pd2
ð4:9Þ
where Sx and Sy are the longitudinal and transverse pitches of the unit cell (Fig. 3)
and m2 = hstrutP/(ksAstrut). An energy balance leads to the mean fluid temperature:
T fðnÞ ¼ T fð0Þ þqSxn
qfHUcpð4:10Þ
When there is only one cell across the core height (Fig. 1), the heat flow entering
through the vertex is equal to the sum of the heat flows in all of the struts forming
the unit cell (Qvertex = 3Qstrut for a tetrahedral truss). The total heat flow is thus:
Q ¼ Qvertex þ Qwall ð4:11Þwhere Qwall is heat removed from the face. The unit-cell averaged Nusselt number is:
Nudp ¼hcelldp
kf
ð4:12Þ
where
hcell ¼ ksm tanhðmLÞSxSy �
pd2
4
�hstrut
ksmcoshðmLÞsinhðmLÞ þ
3
4pd2
SxSyð4:13Þ
Finally, the overall heat transfer coefficient at the panel level is:
NuH ¼ Nudp
Hdp
¼ Hmks
kf
tanhðmLÞSxSy �
pd2
4
�hstrut
ksmcoshðmLÞsinhðmLÞ þ
3
4pd2
SxSyð4:14Þ
By recalling that the core density of a truss core panel is �q ¼ 3pðd=LÞ2 [3], we can
rewrite (4.14) as:
NuH¼ 2
ffiffiffiffiffiffi2p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiNud
�qks
kf
stanh 2
ffiffiffiffiffiffi3p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiNud
�qkf
ks
s !
� 1
21 � �q
6ffiffiffi3
p � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Nudkf
ks
tanh�1
s2ffiffiffiffiffiffi3p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiNud
�qks
kf
s !þ �q
2ffiffiffi3
p( )
ð4:15Þ
Since the above analysis is based on fully developed flow, it cannot capture entry or
exit effects. In practice, the first few cells contribute less significantly than the innercells, due to a different flow pattern. Entry and exit effects are discussed in the fol-
lowing subsection.
4.2.1. Entry and exit effects for truss core panels
The static pressure drops when the flow enters the panel and rises when it exits.
The pressure changes include reversible (flow-area change) as well as irreversible
804 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
(boundary layer separation) processes. Upon entering, the flow undergoes rapid
contraction, expressed (Fig. 9a) as the distribution of pressure coefficient, Cp(x,y) =
[p(x,y) � pin]/(qfU2/2), with the inlet pressure, pin, obtained from a steady CFD (com-
putational fluid dynamics) calculation. Along the flow direction, the static pressure
drops linearly from cell-to-cell (Fig. 9b). The entry and exit effects on the pressuredrop may be estimated as:
Kentry ¼DP 1
qU 2=2� Kcell; Kexit ¼
DP 2
qU 2=2� Kcell ð4:16Þ
where DP1 and DP2 are, respectively, the pressure drop and rise at the entry and exit
regions. The results (Fig. 9) indicate that there is almost no entry effect, but that the
pressure increases by �0.3Kcell at the exit.
Fig. 9. Entry and exit regions observed from the steady CFD calculation at Redp ¼ 2.0 � 104 for a
pyramidal truss core panel: (a) contour map of static pressure coefficient on face and (b) static pressure
distribution along A–A.
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 805
To obtain the corresponding effect on the Nusselt number [13], the wall temper-
ature has been monitored at every cell, as well as the fluid temperature at both the
inlet and outlet (Fig. 10). The Nusselt number may then be ascertained using:
NuðnÞ ¼ dp
kf
qT wðnÞ � T fðnÞ
ð4:17Þ
Fig. 10. (a) Variation of the cell averaged Nusselt numbers along the flow direction for two orientations
(Redp ¼ 4600 for orientation A and Redp ¼ 6000 for orientation B) and (b) distribution of Nusselt number
on facesheet at Redp ¼ 4600 for orientation A. Data were obtained from the thermochromic liquid crystal
measurements on a panel with pyramidal truss core; the flow is from left to right.
806 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
where
T fðnÞ ffi T f ;in þ ðT f;out � T f;inÞn=N ð4:18ÞThe variation of the cell-averaged Nusselt number, normalized by that averaged over
cells in the fully developed flow regime (Fig. 10) indicates that the first cell of orien-
tation A has NuH about 65% of the average, whilst that in orientation B is 85% of the
average. In orientation A, the entry region decreases the overall heat transfer by
about 6% when seven unit cells are used in the flow direction. For orientation B,
a decrease of 2.5% occurs when eight unit cells were used.
4.3. Overall trends
The countervailing influences of topology on the heat transfer and on the pressure
drop are evident from (3.6) and (4.7) or (4.15). Most evident is the effect of cell size.
At a fixed relative density, small cells increase the surface area density (total surface
area per unit volume), and hence the heat transfer, but also increase the pressure
drop. Accordingly, there will generally be an optimum governed by the system
requirements [9,11]. Generally this optimum is at the meso-scale [(lc)opt of order
mm]. The relative density introduces similar countervailing influences resulting inan optimum [9,11]: typically, �qopt � 20%, but the preferred density varies somewhat
with Reynolds numbers.
All other terms affect the performance in a monotonic manner. Namely, a high
thermal conductivity material is always beneficial [11,12] and hexagonal cross-sec-
tions are always preferable to square or triangular shapes [9]. Deeper channels (lar-
ger H) are also superior from a thermal perspective, but the associated weight
penalties must be addressed, often resulting in an optimum. Specific design choices
incorporating these trends are dependent on detailed knowledge about the local val-ues of the heat transfer coefficients, examined next.
5. Cell parameters
Beyond the scaling relationships described in Section 3 additional precision and
insight can be gained by conducting a combination of experimental measurements
and simulations at the cell level. The protocols are illustrated by recent results ob-tained for truss cores.
For measurement purposes, the following methods are used.
(i) Particle image velocimetry (PIV) is used to determine the velocity field by mea-
suring the displacement of illuminated particles with a double-pulsed Nd:YAG
laser, and an oil-fluorescent dye mixture is used for flow visualisation.
(ii) Pressure distributions on the surface are measured with purpose-built pressure
tapping rings.(iii) Surface temperatures are measured using transient liquid crystal (TLC) ther-
mography: by monitoring color changes with a digital camera connected to
an image acquisition system.
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 807
To prevent flow disturbances caused by the insertion of a lighting source and an
image capturing system into the flow, enlarged versions can be built using hollow
Perspex tubes, and a borescope used to monitor a TLC color display from withinthe tube.
The heat transfer coefficient is ascertained by inserting a heating mesh before the
test section, and measuring changes in the temperature at the surface with time, Ts(t),
which varies as:
T sðtÞ � T 0
T ss � T 0
¼ 1 � 1
1 þ a2s
ea2
erfcðaÞ
þ �e�t=s a2s
1 þ a2s
1 þ 1
as
1
p
ffiffiffits
rþ 2
p
X1n¼1
1
ne�n2=4 sinh n
ffiffiffits
r !" #( )
ð5:1Þ
with a ¼ hffiffiffiffiffijt
p=ks and as ¼ h
ffiffiffiffiffijs
p=ks : T ss is the steady state wall temperature, T0 is
the initial temperature of the solid, j is the thermal diffusivity of the solid, and s is atime constant. Since h is the only unknown parameter, it can be ascertained by fitting
(5.1) to the measured temperatures, Ts(t). The local heat transfer coefficient for the
truss is:
hcell ¼Z Z
Acell
hðx; y; zÞdA ð5:2Þ
Computational fluid dynamics (CFD) codes, such as FLUENT, can be used to obtain
complementary information. The codes allow the pressure gradient and the heat
transfer coefficient to be determined with the assumption that the flow is incompress-
ible and that the thermodynamic properties (heat capacity, thermal conductivity, vis-
cosity and density) are invariant with temperature. A periodic boundary condition is
imposed at the inflow and outflow faces of the unit cell. The velocity profiles andmagnitudes at the inflow and the outflow faces are set to be identical to conserve
the mass flow crossing the faces.
While the pressure and temperature at these faces differ, the shapes of the
profiles are identical; only the mean changes. Generally, the mass flow rate is speci-
fied and the pressure gradient obtained from the calculations. The mean flow velocity
is obtained by integrating over the periodic surfaces. For numerical accuracy, the
non-dimensional wall distance, y+, must not exceed 8 (y+ = qfushc/lf with the friction
velocity us ¼ ðsw=qfÞ1=2
, sw the wall shear stress, and hc the distance between thefirst grid cell and the walls). The consequence is that the CFD simulation is compu-
tationally intensive, especially for the calculation of surface heat transfer coeffi-
cient: for flow in the form–dominant range, a typical mesh contains several million
cells.
A heat transfer map of the surface of the lower end-wall (Fig. 11) reveals the for-
mation of horseshoe and arch-shaped vortices: both by measurement (via thermo-
chromic liquid crystals) and simulation; the accompanying flow field is presented
Fig. 11. Nusselt number variations in a tetrahedral truss system showing the vortices in the vicinity of the
attachment to the faces: (a) measurements and (b) CFD simulations.
808 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
in Fig. 12. These vortices increase the local heat transfer by promoting flow mixing.
Even though the spacing between the vertices is large, the staggered vertices and thewakes originating from the struts contribute to end-wall heat transfer. Details listed
in Table 2 indicate the following contributions to the heat transfer:
(i) The truss region before flow separation provides 37%;
(ii) The truss region after the flow separation contributes 20%;
(iii) The end-wall encompassing the vortex structure contributes 18%;
(iv) The remnant end-wall contributes 25%.
While the above experimental measurements and CFD simulations are very useful
for identifying and understanding the individual mechanisms of heat transport, the
procedure is usually complicated and extremely time consuming.
Fig. 12. Simulated local flow patterns in a tetrahedral truss system at Redp ¼ 2.0 � 104 showing the
formation of vortex structures.
Table 2
Estimated contribution of local flow features. All of the data were measured at Redp = 2.0 · 104 for both
the facesheets and the struts
Sandwich
component
Location Fraction of
total area
Area–averaged
Nudp
(measured)
Fraction
of averaged
Nudp
Total
area–averaged
Nudp
Fraction
of total
Nudp
Strut Before
separation
0.23 293 0.37 260 0.57
After
separation
0.17 215 0.20
Facesheet Influence of
largest vortex
0.22 181 0.18 163 0.43
Least influence
of vortex
0.38 152 0.25
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 809
6. Topology preferences
By converting the preceding results into relationships based on the core thickness
H, plots of NuH(ReH) and fH(ReH) encompassing all of the competing core topolo-
gies can be constructed. The results are summarized on Fig. 13. Each topology cat-
egory is encompassed by an ellipse, because both NuH(ReH) and fH(ReH) are affected
by the cell size, the material, the relative density and the thermal boundary condi-
tions. While the Reynolds number can in principle vary from arbitrarily small toarbitrarily large for each topology, the limits on Re shown in Fig. 13 are set by exper-
imental conditions: the lower limit by the accuracy of the pressure tappings and the
upper by the pump capacity. These vary as the core topology is varied. Upon
comparison of the two plots, note the countervailing influences of topology on the
Fig. 13. A compilation of results for: (a) friction factor as a function of the Reynolds number at the panel
level and (b) the Nusselt number as a function of the Reynolds number.
810 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
Nusselt number and friction factor. The implication is that the topology preference
depends on the application. In most cases, the choice is dictated by a maximum allow-able temperature in the metal and/or the fluid, subject to either a specified pressure
drop or pumping power. A natural goal would be to maximize the total heat trans-
ferred to the fluid.
To facilitate use of the maps (Fig. 13), we recall some scaling relations. For aes-
thetic reasons, the subscript H was omitted from the quantities Nu, f and Re, but all
these quantities are defined at the panel level. The total heat flux scales as:
Q � NuDTkfðWL=HÞ ð6:1Þ
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 811
with DT representing the relevant fluid-metal temperature difference: DT = Tw � Tf
for iso-flux and DT = D Tlogmean for iso-temperature boundary conditions. The pres-
sure drop and pumping power scale as:
Dp � ðf Re2Þ l2
q
�ðLH�3Þ
P � DpWHU � ðf Re3Þ l3
q2
�ðLWH�3Þ
ð6:2Þ
Other quantities of interest are:
QDT
� ðNuÞðkfÞðWL=HÞ ð6:3Þ
QDTDp
� Nu
f Re2
�kfqf
l2f
�ðWH 2Þ ð6:4Þ
and
QDTP
� Nu
f Re3
�kfq2
f
l3f
�ðH 2Þ ð6:5Þ
All quantities are the product of three terms: the first depends on the heat sink design
and the Reynolds number, the second depends only on the fluid and the third is a func-tion of the external geometry.6 For the ensuing assessments, the fluid and the external
dimensions of the panel are fixed, such that only the first term need be considered.
Case I. When there is no constraint on either the pressure or the pumping power,
and the goal is to maximize the heat transferred per unit temperature dif-
ference, clearly Nu must be as large as possible. Then, Fig. 13a immedi-
ately reveals the best technologies. They include foams, prismatic cores
and trusses.Case II. If the quantity to be maximized is the heat transferred per unit tempera-
ture difference and pressure drop, then Nu/fRe2 must be as large as pos-
sible. The preferred system can be visualized using Fig. 14, since
contours of constant Nu/fRe2 become lines of slope 2. Inspection reveals
that the best candidates are laminar empty channels, louvered fins and
corrugated ducts. Note, however, that when the panel must support bend-
ing loads, all of these systems are excluded, because of their extremely low
core shear strength and stiffness. In this case, truss and prismatic coresand textiles outperform all other options.
Case III. The same argument can be repeated for conditions where the pumping
power is to be controlled, resulting in lines of slope 3. In this case,
corrugated ducts are the best option, whereas all the other concepts are
all roughly equivalent.
6 While it may be tempting to use the materials index to optimize the properties of the fluid; this would
be erroneous, since the conductivities and the Prandtl number are embedded in Nu.
Fig. 14. Thermal efficiency parameters.
812 T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815
Table 3
Constrained optimization for varying topology selections
Q/(DTDp) Q/DT > (Q/DT)min Nu/fRe2 Nu > Numin High heat flux: truss cores,
prismatic cores
Low heat flux: empty
channels,
corrugated ducts
1/Dp Q/DT > (Q/DT)min 1/fRe2 Nu > Numin High heat flux: truss cores
Low heat flux: corrugated
ducts
T.J. Lu et al. / Progress in Materials Science 50 (2005) 789–815 813
For each of these cases, the designs were unconstrained. In typical applications,
either the pressure drop or the pumping power might be specified. The ensuing con-
strained optimization can be analyzed as follows (Table 3). Suppose the goal is to
maximize Q/DT subject to the constraint Dp < (Dp)max. For fixed fluid properties
and external dimensions, this is equivalent to maximizing Nu subject to fRe2 <( fRe2)max. The preferred systems can be ascertained as follows: (a) Superpose on
Fig. 13a a line of slope �2 with intercept log[(fRe2)max]: all technologies that lie
above the line are excluded. (b) Use Fig. 13b to maximize Nu by considering only
the technologies retained from (a). At very low (Dp)max, the preference would be
the corrugated duct: whereas at high (Dp)max, foams, textiles and truss cores are
superior. Similarly, Q/(DTDp), or Q/(DTP), can be optimized subject to the con-
straint Dp < (Dp)max. The preferred technology emerges as the corrugated duct,
regardless of the pressure drop. Note that, for any given problem, the specific choicedepends on a finer resolution than offered by Figs. 13 and 14, based on details of size,
material and thermal boundary conditions.
The constraint on the maximum temperature in the metal is more difficult to ex-
press graphically. For iso-flux conditions:
T metalmax � T fluid
in ¼ DT þ QqfcpUHW
ð6:6Þ
the maximum temperature depends not only on DT, but also on the velocity. When-
ever such a constraint is to be met, the maps can only provide initial screening; after
identifying a few good options, the designer needs to revert to the formulae.
Acknowledgement
TJL wishes to thank the U.S. Office of Naval Research for partial financial sup-
port through ONR/ONRIFO grant number N000140110271, the National Basic Re-
search Program of China through Grant No. 2004CB619303, and the National
Science Foundation of China through Grant No. 10328203. LV is particularly grate-
ful to H.A. Stone, Harvard University, for innumerable helpful discussions. Theauthors would also like to thank T. Kim, C.Y. Zhao, J. Tian and H.P. Hodson
for their contributions.
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