Metal Foams: A Design Guide

Thermal management and heat transfer 183
the cellular metal as a geometric variant on a staggered bank of cylinders
(BOC) oriented normal to the fluid flow. The modified BOC solution has
proportionality coefficients that reflect the geometric differences between the
foam and the cylinder. This approach has been validated experimentally and
the unknown coefficients calibrated. Only the results are given here.
The heat transfer coefficient, Hc, for the cellular metal is (Lu et al., 1998):
Hc D 2Q
d keff
� �
� 2b�
Bieff tanh Bieff
⊲13.2⊳
d
Here Q / s, keff is an effective thermal conductivity related to the actual
thermal conductivity of the constituent metal, ks, by:
keff D 0.28 ks
⊲13.3⊳
and b is the thickness of the medium (Figure 13.1). The coefficient of 0.28
has been determined by experimental calibration, using infrared imaging of
the cellular medium (Bastawros and Evans, 1997; Bastawros et al., in press).
The heat transfer that occurs from the metal ligaments into the fluid can be
expressed through a non-dimensional quantity referred to as the Biot number:
Bi D h
⊲13.4⊳
dks
where h is the local heat transfer coefficient. The Biot number is governed by
the dynamics of fluid flow in the cellular medium. The established solutions
for a staggered bank of cylinders (e.g. Holman, 1989) are:
Bi D 0.91Pr 0.36 Re 0.4 ⊲ka/ks⊳ ⊲Re � 40⊳ ⊲13.5⊳
D 0.62Pr 0.36 Re 0.5 ⊲ka/ks⊳ ⊲Re > 40⊳
where Re, the Reynolds number, is
Re D vfd
a
⊲13.6⊳
with vf the free stream velocity of the fluid, a its kinematic viscosity, ka
its thermal conductivity and Pr is the Prandtl number (of order unity). For
the cellular metal, Bi will differ from equation (13.5) by a proportionality
coefficient (analogous to that for the thermal conductivity) resulting in an
effective value:
Bieff D 1.2 Bi ⊲13.7⊳
where the coefficient 1.2 has been determined by experimental calibration
(Bastawros et al., in press).
This set of equations provides a complete characterization of the heat
transfer coefficient. The trends are found upon introducing the properties of