13
1.4 Basic Laws for Stationary Duct Parts
on the duct wall stand still. No work is associated with them for a control volume
positioned with its envelope upon the duct wall. So we considered, when drawing
up the energy balance, the pressure force work and the friction force work on the
envelope as being zero. Neither is there any work of the friction force onto the inlet
and outlet sections (velocity perpendicular to these sections). The total work of the
friction force is thus zero. The total work consists of displacement work and deformation work. The latter is the sum of volume change work and form change work.
The volume change work of the friction force is zero, however (principally: friction
force tangential to the surface). With a stationary duct part, the displacement work
of the friction force on the moving fluid equals the form change work in value. The
latter is the source of friction heat (see fluid mechanics and see Sect. 2.1.5).
Energy dissipation is reduction of the work capacity of the sum of the energies in
a system. For a steady flow, the sum of the energies is the sum of enthalpy, kinetic
energy and gravitational potential energy (Eq. 1.7). The work capacity is expressed
by the mechanical energy, i.e. the part of the energy that can be used for work generation. The mechanism that reduces the mechanical energy is the deformation work
associated to friction forces, either friction internally in the fluid or friction between
the fluid and walls. The deformation work converts mechanical energy into heat.
Technically, it is mostly said that there is head loss. The dissipation mechanism with
the friction force will be further analysed in Chap. 2, Sect. 2.1.5. The effect of dissipation may be grasped already now by taking the difference between the energy
equation (1.7) and the work equation (1.5):
(1.9)
We thus obtain the expression of the second law of thermodynamics on the production of entropy ( s) for an infinitesimal duct part. Eq. (1.9) already gives confidence
that the term dq irr represents energy dissipation. The equation may also be written as
This expression shows that friction causes heating of the fluid. The third term is
heating by compression (volume change work of pressure). This term is zero for a
constant density fluid.
When applying the energy theorem, we put the envelope of the control volume
onto the duct wall. Since the choice of a control volume is arbitrary, we may also opt
to position its envelope just inside the fluid. Then, the envelope has velocity v and
-dq irr represents the total work of the friction force on the control volume. The new
choice does not change the formulation of the momentum equation, but changes the
formulation of the energy equation. Within the one-dimensional flow representation, velocity is uniform over a cross-section. This implies that we should consider
the entire flow retardation due to wall friction as concentrated between the envelope
of the control volume and the duct wall. Within this zone, dissipation by friction
or
irr
irr
1
1
dh
dp dq
dq Tds dh
dp dq
dq.
r
r
−
−
=
= −
=
+
.
irr
1
de dq
dq p d ( )
r
=
+ −
1.4 Basic Laws for Stationary Duct Parts
on the duct wall stand still. No work is associated with them for a control volume
positioned with its envelope upon the duct wall. So we considered, when drawing
up the energy balance, the pressure force work and the friction force work on the
envelope as being zero. Neither is there any work of the friction force onto the inlet
and outlet sections (velocity perpendicular to these sections). The total work of the
friction force is thus zero. The total work consists of displacement work and deformation work. The latter is the sum of volume change work and form change work.
The volume change work of the friction force is zero, however (principally: friction
force tangential to the surface). With a stationary duct part, the displacement work
of the friction force on the moving fluid equals the form change work in value. The
latter is the source of friction heat (see fluid mechanics and see Sect. 2.1.5).
Energy dissipation is reduction of the work capacity of the sum of the energies in
a system. For a steady flow, the sum of the energies is the sum of enthalpy, kinetic
energy and gravitational potential energy (Eq. 1.7). The work capacity is expressed
by the mechanical energy, i.e. the part of the energy that can be used for work generation. The mechanism that reduces the mechanical energy is the deformation work
associated to friction forces, either friction internally in the fluid or friction between
the fluid and walls. The deformation work converts mechanical energy into heat.
Technically, it is mostly said that there is head loss. The dissipation mechanism with
the friction force will be further analysed in Chap. 2, Sect. 2.1.5. The effect of dissipation may be grasped already now by taking the difference between the energy
equation (1.7) and the work equation (1.5):
(1.9)
We thus obtain the expression of the second law of thermodynamics on the production of entropy ( s) for an infinitesimal duct part. Eq. (1.9) already gives confidence
that the term dq irr represents energy dissipation. The equation may also be written as
This expression shows that friction causes heating of the fluid. The third term is
heating by compression (volume change work of pressure). This term is zero for a
constant density fluid.
When applying the energy theorem, we put the envelope of the control volume
onto the duct wall. Since the choice of a control volume is arbitrary, we may also opt
to position its envelope just inside the fluid. Then, the envelope has velocity v and
-dq irr represents the total work of the friction force on the control volume. The new
choice does not change the formulation of the momentum equation, but changes the
formulation of the energy equation. Within the one-dimensional flow representation, velocity is uniform over a cross-section. This implies that we should consider
the entire flow retardation due to wall friction as concentrated between the envelope
of the control volume and the duct wall. Within this zone, dissipation by friction
or
irr
irr
1
1
dh
dp dq
dq Tds dh
dp dq
dq.
r
r
−
−
=
= −
=
+
.
irr
1
de dq
dq p d ( )
r
=
+ −