Turbo Machines
report
Momentum and Moment of Momentum Theories
student's name: Abdollah Mohsen Mohamed
ID: 302018259
Energy Engineering Program Helwan University
SUPERVISOR: Dr. Mohammad Nawar
Momentum and Moment of Momentum Theories
Momentum theories are based on the application of the conservation laws of fluid-mechanics
under some simplifying assumptions. In this chapter, the results are derived directly from the
Navier-Stokes equations and the derivation steps are given in details. The standard axial
momentum theory without wake rotation is presented and the underlying assumptions and
inconsistencies are discussed. The chapter proceed with the development of the general
momentum theory which leads to a system of equations which is unfortunately difficult to
close. Simplifying assumptions are introduced to close the system, leading to: the general axial
momentum theory without wake rotation, and the streamtube theory with wake rotation. The
latter forms the basis of the blade-element-momentum method.
- Introduction
Assumptions The momentum theory is based on the application of the conservation laws of
fluidmechanics 1 under the assumption that the flow is steady, incompressible and
axisymmetric, that the fluid is homogeneous and inviscid and that the rotor loads are
axisymmetric and concentrated onto an actuator disk. The actuator disk assumption tacitly
implies that the rotor consists of an infinite number of blades since the actuator disk can be
thought of as an infinite number of lifting lines distributed over the azimuth. A pressure drop
is present at the actuator disk but the longitudinal and radial velocities are assumed
continuous across it. No molecular or turbulent mixing between the free-stream flow and the
wake flow occurs and a shear vorticity layer (slipstream) is present between the wake and the
free-flow. The total aerodynamic force, moment and power that the fluid apply on the actuator
disk are the thrust T, the torque Q and power P. The actuator disk applies on the fluid the
opposite quantities. The scalar quantities T, Q and P are chosen such as they are positive for
a usual wind turbine operating condition. No assumption is made on the nature of the forces
and the actuator disk forces may be non-conservative [14]. Notations The notations used in
this chapter are introduced in Figure 9.1. The actuator disk is centered on the z-axis and normal
to it. Polar coordinates (r,θ,z) are adopted. Since the problem is assumed axisymmetric, the
geometry is invariant in the azimuthal coordinate and the parameters will have no
dependency in the variable θ. Four planes are defined with the notation 0, +, − and w.
The quantities far-upstream and far-downstream are noted with the subscript 0 and w
respectively.
The quantities exactly upstream and downstream of the actuator disk are noted with the +
and − subscript respectively.
Upper-case letters are used to design the value of the velocity in these specific planes, e.g. the
longitudinal velocity in these planes is written U0, Uz+ , Uz− and Uzw . Across the actuator
disk, the longitudinal velocity is continuous and no distinction will be made between the + and
− values: Uz+ ≡ Uz− ≡ Uz . When the velocities are evaluated outside of these four planes, a
lower-case is used, e.g. (ur ,uθ ,uz).
Figure 9.1: Notations for the momentum analyses of the actuator disk. Definitions of the planes “0”, “+”, “−” and
“w”. An annular streamtube is illustrated where the gray-areas represent its cross section in the different planes.
he application of the conservation laws (Equation 2.19-Equation 2.22) to a fixed control
volume (CV) of closed surface boundary ∂CV surrounding the entire rotor gives:
Since the thrust and the torque are directed along ez , the z-component of the equations of
conservation of momentum are of interest. The control volumes used in the momentum
theory analysis consist of streamtubes delimited by two planes orthogonal to the z-axis. An
example of such a control volume is shown in Figure 9.1. Assuming that the streamtube control
volume is delimited by the coordinates z1 and z2, with z1 < z2, then Equation 9.1 and the zcomponents of Equation 9.2 and Equation 9.3 become:
where Sz is the surface of the streamtube at z and dT and dQ are the elementary loads from
the actuator disk that are comprised within the control volume. The integral over the lateral
surface has no contribution since u · n = 0 along a streamline. The pressure integral from
Equation 9.3 is not present since it only has a component along eθ . If the control volume
does not enclose the actuator disk, the thrust and torque should be set to zero in the above
equations.
Actuator disk The actuator disk assumption implies that a pressure jump ∆p(r) is created over
the rotor. In the axial momentum theory of section 9.2, this pressure drop is introduced
artificially, while in section 9.3 this pressure drop can be attributed to the change of azimuthal
velocity imparted by the disk on the flow. The elementary thrust dT(r) over an annulus of
section dA(r) = 2πrdr is equal to the pressure force normal to the disk:
Despite the pressure jump over the actuator disk, the velocity is assumed to be continuous
across
it. The elementary power associated with this local force is then dP(r) = dT(r)Uz(r). A wind
turbine extracts energy from the flow while propellers provide energy to the flow. The
extraction or addition of energy by the rotor implies a progressive change in kinetic energy
(i.e. velocity) as the flows passes the rotordT(r) over an annulus of section dA(r) = 2πrdr is
equal to the pressure force normal to the disk: dT(r) = ∆p(r)dA(r) = 2πrdr∆p(r) (9.8) Despite
the pressure jump over the actuator disk, the velocity is assumed to be continuous across it.
The elementary power associated with this local force is then dP(r) = dT(r)Uz(r). A wind turbine
extracts energy from the flow while propellers provide energy to the flow. The extraction or
addition of energy by the rotor implies a progressive change in kinetic energy (i.e. velocity) as
the flows passes the rotor Simplified axial momentum theory (no wake rotation) Notations
and assumptions The simplified 1D momentum theory is attributed to the work of Rankine
[10] and R.E. Froude [4]. It is developed in details in chapter II of the work of Glauert [5, p. 182190]. The assumptions of section 9.1 are supplemented with the following: the pressure drop
across the rotor is uniform over the rotor area, the axial velocity is uniform over the rotor area,
there is no rotational velocity in the wake, and the static pressure far upstream and
downstream is equal to the undisturbed ambient static pressure. The notations adopted and
the expected evolution of the velocity and pressure are given in Figure 9.2 using a wind turbine
convention. Since the flow is assumed to be axial, the subscript z is dropped in this section:
the axial velocity evolves from U0 upstream to U at the rotor and Uw in the far-wake. The
actuator disk of constant loading generates a tubular vortex sheet at the rim of the disk which
stretches from the disk to infinity. The vortex sheet forms a tubular surface of revolution of
circular cross-section where the revolution axis is the wake axis. This model is described in
subsection 5.2.3. The radius of the vortex sheet is determined by the fact that the vortex sheet
is force-free and hence the vortex sheet follows the streamline of the flow that passes through
the edge of the disk. The intensity of the vortex sheet progressively increases towards the farwake in order to satisfy the fact that the vortex sheet convects with the mean of the velocity
on both its sides. The vortex sheet surface reaches an equilibrium radius in the far-wake and
from this point on the wake corresponds to a semi-infinite vortex cylinder of constant radius.
The flow is irrotational everywhere outside of this vorticity sheet. The vorticity view of the
actuator disk system is convenient for some of the treatments done in this chapter.
Pressure and velocity evolution through the actuator disk for the 1D momentum theory. A wind turbine
convention is adopted. The thrust force T is represented as the force exerted by the fluid on the actuator disk.
The assumptions of axial momentum theory are further discussed in subsection 9.2.4 since
they lead to inconsistencies: a radial flow is present, the pressure forces on the streamtube
are not properly accounted for and the axial velocity is not constant over the rotor. Despite
these inconsistencies, the 1D momentum theory presents the advantage to provide a
satisfactory account of the performance of a rotor and the flow around it using only few
parameters.
-Determination of power, thrust and rotor velocity
Axial and uniform velocities By assumption, the velocity far-upstream, U0, and the rotor
velocity U are uniform and purely axial. The far-wake velocity Uw is also uniform and purely
axial as a consequence of the assumption of the constant pressure jump and the equilibrium
of pressure in the far-wake. This can be shown by consideration of the vortex system
generated by the actuator disk which tends to an infinite vortex cylinder in the far wake, within
which the velocity is constant (see e.g. subsection 36.1.3). The result can also be shown by
applying Bernoulli’s equation as follows. The assumptions of steady state homogeneous,
inviscid, incompressible fluid2 allows the application of the form of Bernoulli’s equation given
in Equation 2.120 on both side of the actuator disk. The application gives:
Using the assumption of pressure equilibrium in the far-wake, i.e. p0 = pw, the pressure jump
across the disk is found to be:
Since the pressure drop and U0 are independent of r it is seen that Uw is also independent
of r. Mass and momentum conservation The continuity equation given in Equation 9.1 is
applied successively to the control volumes CV1 and CV2 (or CV0) shown in Figure 9.3. The
term u · n is
Control volumes used for the axial momentum theory analyses. The control surface follows
the streamlines of the flow. The upstream and downstream planes are assumed to be at
infinity. The cross sections of the volumes form a full disk. The control volume CV0 is such
that CV0=CV1∪CV2∪CV3.
zero along the streamtube surface since the surface follows the streamlines. Only the integral
over the surfaces normal to the rotor plane remains. Using the fact that all velocities are
uniform and
axial, the application of Equation 9.1 leads to
The conservation of mass implies that the mass flow rate ˙m is constant in every cross section
of the streamtube. The application of the conservation of axial (z) momentum, Equation 9.2,
to the control volume CV0 leads to:
The pressure integral can be shown to vanish as discussed in subsection 9.2.4. Using the
definition of the mass flow given in Equation 9.12, the thrust force applied by the fluid on the
disk can then be expressed indifferently:
Two methods can be used to determine the velocity at the rotor.
Method 1: Equating powers The conservation of enthalpy given in Equation 9.4 is applied to
the control volume CV0:
After introducing the definition of the mass flow from Equation 9.12 and simplifying it, the
power extracted by the actuator disk is:
The extracted power is the work rate of the thrust (given in Equation 9.14
Equating the two expressions of the power from Equation 9.17 and Equation 9.16 leads to the
expression of the velocity at the rotor
The induced velocity at the rotor and in the far-wake are written Ui and Ui,w respectively such
that U = U0 +Ui and Uw = U0 +Ui,w. The result from Equation 9.18 implies that Ui,w = 2Ui
Method 2: Equating thrust The expression of the thrust force as function of the pressure drop
(Equation 9.8) is simplified since the velocity, and hence the pressure, is assumed constant
along the disk
Using the pressure drop ∆p given in Equation 9.11, the thrust from Equation 9.19 becomes
) Equating the expressions of the thrust obtained using the pressure jump (Equation 9.20) and
using the momentum change (Equation 9.14) giv
9.2.3 Induction factors and rotor performance
The velocity in the far-wake is expressed as function of a using Equation 9.21:
Based on the results of subsection 9.2.2, the power and thrust can easily be expressed as
function of the induction factor a by using U = U0(1−a) and Uw = U0(1−2a):
The dimensionless power and thrust coefficients defined in Equation 4.9 are then:
Equation 9.27 is inverted to give a:
The power coefficient is expressed as function of the velocity ratio Uw/U0 as:
The radius of the far-wake is determined by using the conservation of mass Equation 9.12.
Results are given in section 15.1 and Equation 15.1. The variations of Cp with respect to a or
Uw/U0 are found on Figure 9.4. To obtain the maximum power that can be harvested,
Equation 9.26 is
Figure 9.4: Variation of Cp with respect to the induction factor a and the velocity ratio Uw/U0
. The maximum value of Cp is 0.593, known as the Betz limit and found at 1/3 for the two
parameters due to their relation from Equation 9.23.
differentiated with respect to a and a local maximum is found. The result gives a = 1/3 which
corresponds to:
The value of maximum power coefficient Cpmax = 16/27 was found by Betz [1] and Joukowski
[9] and is now commonly referred to as the Betz limit, the Betz factor, or the Betz-Joukowski
limit. In the absence of a rotor, the kinetic energy of the air passing through the area A during
the time dt is 1 2 ρU 3 0 dt. Betz result implies that a maximum of 59.3% of this energy can
be extracted by a rotor. Many sources of losses are neglected in this theory, e.g. losses due to
viscosity, wake rotation, finite number of blades. The Betz-limit is thus conservative and it
states an upper-limit that can likely not be exceeded. The presence of a diffuser (or shroud)
can increase the power extraction at the rotor (see e.g. [3, 7, 12]). Yet, depending on the area
used for the normalization of the power coefficient, it is not obvious that the Betz-limit can be
exceeded. The vortex model of Joukowski predicts values of CP above the Betz-limit for small
tip-speed ratios. This has led to discussions in the literature and it was suggested that this
result was unphysical and due to the omission of the pressure force acting on the control
volume [12]. In practice, typical modern 3-bladed wind turbines do not exceed a power
coefficient of 0.49 and will operate at power coefficients below 0.4 above their rated wind
speed.