Metal Foams: A Design Guide

Cost estimation and viability 209
the performance metric P2 is heat transfer q per unit area, ˛2 is the change in
value V associated with unit increase in q. The best solution is the one with
the smallest value of V, which, with properly chosen values of ˛1 and ˛2,
now correctly balances the conflicting objectives.
With given values of V and exchange constants ˛i, equation (16.5) defines
a relationship between the performance metrics, Pi. In two dimensions, this
plots as a family of parallel lines, as shown in Figure 16.7. The slope of the
lines is fixed by the ratio of the exchange constants, ˛1/˛2. The best solution
is that at the point along a value-line that is tangent to the trade-off surface,
because this is the one with the smallest value of V.
The method can be used to optimize the choice of material to fill a multifunctional
role, provided that the exchange constants ˛i for each performance
metric are known.
Minimizing cost as an objective
Frequently one of the objectives is that of minimizing cost, C, sothatP1DC. Since we have chosen to measure value in units of currency, unit change in
C gives unit change in V, with the result that
� �
∂V
˛1 D
D 1 ⊲16.6c⊳
∂P1 P2,...Pi
and equation (16.5) becomes
V D C C ˛1P2 C ...˛iPi .... ⊲16.7⊳
As a simple example, consider the substitution of a metal foam, M, for
an incumbent (non-foamed) material, M0, based on cost, C, and one other
performance metric, P. The value of P in the application is ˛. Substitution is
potentially possible if the value V of M is less than that, V0, oftheincumbent
M0. Thus substitution becomes a possibility when
or
V V0 D ⊲C C0⊳ C ˛ ⊲P P0⊳ 0 ⊲16.8⊳
1V D 1C C ˛1P 0
from which
1P
1C
1
˛
⊲16.9⊳
defining a set of potential applications for which M is a better choice than M0.
To visualize this, think of a plot of the performance metric, P, against
cost, C, as shown in Figure 16.8 The incumbent M0 is centered at fP0,C0g;