Metal Foams: A Design Guide

184 Metal Foams: A Design Guide
the foam (d, / s and ks), its thickness b, and the fluid properties ( a, ka and
Pr), as well as its velocity vf. The caveat is that the proportionality constants
in equations (13.3) and (13.7) have been calibrated for only one category of
open cell foam: the DUOCEL range of materials. Open-cell foams having
different morphology are expected to have different coefficients. Moreover, if
/ s and d are vastly different from the values used in the calibration, new
domains of fluid dynamics may arise, resulting again in deviations from the
predictions.
The substrate attached to the cellular medium also contributes to the heat
transfer. In the absence of a significant thermal constriction, this contribution
may be added to Hc. Additional interface effects can reduce Hc, but we shall
ignore these.
13.3 Heat fluxes
The heat, Q, flowing into the fluid through the cellular medium per unit width
is related to the heat transfer coefficient by:
Q D LHc1Tℓm
⊲13.8⊳
where L is the length of the foam layer in Figure 13.1. Here 1Tℓm is the
logarithmic mean temperature. It is related to the temperature of the heat
source T1 as well as fluid temperature at the inlet, T0, and that at the outlet,
Te by:
Te T0
1Tℓm D
⊲13.9⊳
ℓn[⊲T1 T0⊳/⊲T1 Te⊳]
Usually, T1 and T0 are specified by the application. Accordingly, Te must be
assessed in order to determine Q. For preliminary estimates, the approximation
1Tℓm ³ T1 T0 ⊲13.10⊳
may be used. Explicit determination requires either experimental measurements
or application of the following expressions governing the fluid flows.
The temperature in the fluid along the x-direction varies as
Tf D T1 ⊲T1 T0⊳ exp⊲ x/ℓ⊳ ⊲13.11⊳
where ℓ is a transfer length governed by the properties of the cellular metal,
the fluid and the substrate. In the absence of a thermal resistance at the attachments,
this length is:
ℓ D
2 keff
acpbvf
�
Bieff
�
1 C Q
1.5
�
2b � 1
tanh Bieff
d
⊲13.12⊳