Page 336 - Materials Science and Engineering An Introduction

308 • Chapter 9 / Phase Diagrams

Simultaneous solution of these two equations leads to the lever rule expressions for this par-
ticular situation,
C a - C 0
W L = (9.1b)
C a - C L
W a = C 0 - C L (9.2b)
C a - C L

For multiphase alloys, it is often more convenient to specify relative phase amount
in terms of volume fraction rather than mass fraction. Phase volume fractions are pre-
ferred because they (rather than mass fractions) may be determined from examination
of the microstructure; furthermore, the properties of a multiphase alloy may be esti-
mated on the basis of volume fractions.
For an alloy consisting of a and b phases, the volume fraction of the a phase, V a ,
is defined as

a phase volume v a (9.5)
fraction—dependence V a =
on volumes of a and v a + v b
b phases
where v a and v b denote the volumes of the respective phases in the alloy. An analogous
expression exists for V b , and, for an alloy consisting of just two phases, it is the case that
V a + V b = 1.
On occasion conversion from mass fraction to volume fraction (or vice versa) is
desired. Equations that facilitate these conversions are as follows:

W a
V a = r a (9.6a)
W a W b
+
Conversion of mass r a r b
fractions of a and
b phases to volume
fractions W b
r b
V b = (9.6b)
W a W b
+
r a r b
and

W a = V a r a (9.7a)
Conversion of volume V a r a + V b r b
fractions of a and
b phases to mass
fractions W b = V b r b (9.7b)
V a r a + V b r b