Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 191

dp

ds þ q v dv ds þ q g dh ds l 0 M dH ds ¼

0 ð 5 : 66 Þ

If Eq. ( 5.66 ) is now integrated from the section, e.g. denoted by 1 to the section
denoted by 2, then the following form will be obtained:

Z

2

Þ l 0 Z

2

dp
q þ

v 2 v 1 M

q dH ¼ 2 þ gh 2 h 1

ð 0 ð 5 : 67 Þ

1 1

uid with a constant
density, Eq. ( 5.67 ) takes the following form, with dimensions per unit of volume:

Since ferro fl uid can be considered as an incompressible fl

2 þ q gh 1 l 0 Z 2 þ q gh 2 l 0 Z

H 1 H 2

p 1 þ q v 1 p 2 þ q v 2

MdH ¼ MdH ð 5 : 68 Þ

0 0

uid is placed in a
tube, which is cooled at one end (cold part with temperature of T 1 = T 2 ) and heated
at the other end (hot part with temperature of T 3 = T 4 ). In the middle of the tube, the
constant magnetic

Let us consider now the example shown in Fig. 5.5 b. The ferro fl

eld region is given by the magnetic eld source, by the

direction of the magnetic eld along the tube axis (direction from cold to hot part).

Because of the higher magnetization in the cold part, the ferro fl uid will be attracted

to the magnetic uid is heated, its magnetization

will be reduced. If one now applies the ferrohydrodynamic Bernoulli equation
(Eq. 5.68 ), by neglecting gravitational forces and keeping in mind that the kinetic
energy for the constant cross-section of the tube will remain constant, the following
expression can be made between the stations 1 and 2, which are placed in the free
space:

eld. In the hot part, since the ferro fl

p 2 l 0 MH 2

p 1 ¼ ð 5 : 69 Þ

Analogously to Eq. ( 5.69 ), the pressure difference can be shown for stations
3 and 4:

p 3 l 0 MH 3 ¼

p 4 ð 5 : 70 Þ

where:

Z

H

1
H

M ¼ MdH ð 5 : 71 Þ

0