Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 185

2
4

3
5 ¼ l 0 H DM
Dt

rate of magnetic work
done on fluid per unit
volume

ð 5 : 44 Þ

Then the energy equation Eq. ( 5.43 ) can be de ned as

h i

Du
Dt ¼ r _

þ l 0 H DM
Dt

¼ viscous : r~ q þ v ð 5 : 45 Þ

By expressing the internal energy u as a function of enthalpy h and the magnetic
energy (HM) (see the Chap. 1 on thermodynamics):

u ¼ h þ l 0 MH ð 5 : 46 Þ

and substituting Eq. ( 5.46 ) into Eq. ( 5.45 ), we can obtain the following relation for
the energy equation:

h i

Dh
Dt þ l 0

D
Dt MH

þ l 0 H DM ¼ viscous : r~ Þ ¼ r _ ð þ ð 5 : 47 Þ

q v

Dt

From the relation:

dMH ð Þ ¼ MdH þ HdM ð 5 : 48 Þ

Equation ( 5.47 ) can be rearranged into:

h i

Dh

Dt þ l 0 M DH

¼ viscous : r~

Dt ¼ r _

q þ v ð 5 : 49 Þ

The total derivative of the enthalpy equals (see the Chap. 1 on thermodynamics):

dh ð s ; H Þ ¼ o

h
o

o h

ds þ dH ¼ Tds l 0 MdH ð 5 : 50 Þ

s H

o

H s

Since the total derivate of the entropy equals (see the Chap. 1 on thermodynamics):

Þ ¼ o

s o s c H

T dT þ

o s

ds T ; H ð dT þ dH ¼ dH ð 5 : 51 Þ

T H H

o o o

H T T

where Eq. ( 5.51 ) can be rewritten using the Maxwell relation:

T ¼ l 0 o

s M

o

o

ð 5 : 52 Þ

H T

o

H