Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 171

s equation for diluted suspensions (Eq. 5.2 ).
According to Odenbach [ 18 ], for ferro

Let us focus now on Einstein '

ϕ V should account
not only for the particles including their surfactant, but also the excess surfactant,
which is often used to stabilize the

fl uids, the volume fraction

fl uid. However, as he states further, it is suf-

cient to use the volume fraction of the particles including their surfactant layer for
all practical applications. For this case, the volume fraction ϕ VFF can be calculated

as follows [ 18 ]:

3

þ

d 2s

/ VFF ¼ / V ð 5 : 21 Þ

d

where d denotes the diameter of the magnetic particles, and s represents the
thickness of the surfactant layer. As reported by Odenbach [ 18 ], only few efforts
have been made to determine the real value of s in given samples of ferro uids with

higher precision. The error resulting from a lack of knowledge about the thickness
of the surfactant layer can therefore be larger than that from neglecting the excess
surfactant. Since for higher concentrations one also has to consider the interparticle
relation, Einstein

fl

s equation (Eq. 5.2 ) will not be an appropriate choice any more
for volume fractions of solid particles above 10 %.
For the application of ferro

'

fl uids, Rosensweig [ 7 ] proposed a modi ed Batch-

' '

elor s equation (Eq. 5.11). For this he assumed that the suspension s viscosity

ϕ max of suspended material:

should diverge for a certain critical volume fraction

!

2

/ VFF
/ max

g eff ¼ g L 1 2 : 5 / VFF þ ð 2 : 5 / max 1 Þ ð 5 : 22 Þ

where ϕ max represents the maximum volume (packing) fraction, which Rosensweig
[ 7 ] estimated to be 0.74. Note that this is a rather high value, especially if the
packing is random or if clustering of the magnetic particles appears. On the other
hand, suspended particles with different sizes can lead to high packing fraction.
Table 5.3 shows some examples of packing fractions for monodispersed particles.
When the ferro ow, the magnetic particles

will tend to rotate. This is due to the torque produced by the viscous forces. Under

fl uid is under the in fl uence of a shear fl

Table 5.3 The maximum packing fraction of various arrangements of monodispersed particles
[ 188 ]
Arrangement Maximum packing fraction ϕ max

Simple cube 0.52

Maximum thermodynamically stable con guration 0.548

Hexagonally packed sheets just touching 0.605

Random close packing 0.637

Face-centred cubic/hexagonal close packed 0.68

Body-centred cubic/hexagonal close packed 0.74