16
1 Working Principles
The work dW is zero on a rotor, since forces affecting the rotor perform no work
within the frame that is turning with the rotor. Variation of gravitational potential
energy within the rotor is negligible, with as a consequence:
(1.13)
(1.14)
From now on, we will use the term stagnation enthalpy or total enthalpy. This is
the enthalpy adopted by a fluid particle, if flow locally would be brought to a halt
adiabatically. With the subscript 0 denoting the stagnation, the total enthalpy in the
absolute frame and the total enthalpy in the relative frame are
Similarly, we will use the term total pressure or stagnation pressure as the pressure
obtained if the flow locally would be brought to zero velocity in absence of heat
exchange (adiabatic) and without losses (reversible). If necessary to make the difference with total quantities, the basic quantities will be called the static ones; so:
static pressure and static enthalpy.
The energy equation in the relative frame is sometimes expressed using a quantity I h
u
r
=
−
0
1 2
2
, called rothalpy ( rotating total enthalpy). The centrifugal force is
2
2 2
2
1
1
2
2
Cf
r
(
r )
(
u )
W
W
=
= ∇
= − ∇ −
. Thus, the term − 1 2
2
u is the associated
potential energy. The energy equation then becomes dI dq
= . For an axial machine,
u = constant, and the energy equation simplifies to dh
dq
0r =
. We will not often use
the term rothalpy. This means that the work term associated to the centrifugal force
will mostly be written explicitly as in (1.13) and (1.14).
1.5.2 Moment of Momentum in the Absolute Frame: Rotor Work
Figure 1.7 sketches the meridional section of a streamtube enclosing the entire
blade height of a pump rotor. The meridional section of a rotating object is a section
containing the axis of rotation. This term is borrowed from geography. v m denotes
the meridional component of the absolute velocity. The velocity component in the
circumferential (or tangential) direction is denoted by v u . According to the mean
line flow representation, there is no velocity component perpendicular to the average circumferential streamsurface. The meridional velocity may be decomposed in
an axial component v a and a radial component v r .
Figure 1.7 shows two possible rotor forms. The left-hand rotor has a shroud. This
type is called closed or shrouded. The blades of the right-hand rotor fit directly in
the casing with a clearance. This type is termed open or unshrouded. The part of the
rotor that fits on the shaft is called the hub. The inner part of the rotor that carries
Work :
2
2
1
1
irr
2
2
1
d w
dp dq
d u ,
r
+
+
=
Energy :
2
2
1
1
2
2
dh d w
d u dq.
+
=
+
h h
v
0
1 2
2
= +
, h
h
w
r
0
1 2
2
= +
.
1 Working Principles
The work dW is zero on a rotor, since forces affecting the rotor perform no work
within the frame that is turning with the rotor. Variation of gravitational potential
energy within the rotor is negligible, with as a consequence:
(1.13)
(1.14)
From now on, we will use the term stagnation enthalpy or total enthalpy. This is
the enthalpy adopted by a fluid particle, if flow locally would be brought to a halt
adiabatically. With the subscript 0 denoting the stagnation, the total enthalpy in the
absolute frame and the total enthalpy in the relative frame are
Similarly, we will use the term total pressure or stagnation pressure as the pressure
obtained if the flow locally would be brought to zero velocity in absence of heat
exchange (adiabatic) and without losses (reversible). If necessary to make the difference with total quantities, the basic quantities will be called the static ones; so:
static pressure and static enthalpy.
The energy equation in the relative frame is sometimes expressed using a quantity I h
u
r
=
−
0
1 2
2
, called rothalpy ( rotating total enthalpy). The centrifugal force is
2
2 2
2
1
1
2
2
Cf
r
(
r )
(
u )
W
W
=
= ∇
= − ∇ −
. Thus, the term − 1 2
2
u is the associated
potential energy. The energy equation then becomes dI dq
= . For an axial machine,
u = constant, and the energy equation simplifies to dh
dq
0r =
. We will not often use
the term rothalpy. This means that the work term associated to the centrifugal force
will mostly be written explicitly as in (1.13) and (1.14).
1.5.2 Moment of Momentum in the Absolute Frame: Rotor Work
Figure 1.7 sketches the meridional section of a streamtube enclosing the entire
blade height of a pump rotor. The meridional section of a rotating object is a section
containing the axis of rotation. This term is borrowed from geography. v m denotes
the meridional component of the absolute velocity. The velocity component in the
circumferential (or tangential) direction is denoted by v u . According to the mean
line flow representation, there is no velocity component perpendicular to the average circumferential streamsurface. The meridional velocity may be decomposed in
an axial component v a and a radial component v r .
Figure 1.7 shows two possible rotor forms. The left-hand rotor has a shroud. This
type is called closed or shrouded. The blades of the right-hand rotor fit directly in
the casing with a clearance. This type is termed open or unshrouded. The part of the
rotor that fits on the shaft is called the hub. The inner part of the rotor that carries
Work :
2
2
1
1
irr
2
2
1
d w
dp dq
d u ,
r
+
+
=
Energy :
2
2
1
1
2
2
dh d w
d u dq.
+
=
+
h h
v
0
1 2
2
= +
, h
h
w
r
0
1 2
2
= +
.