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.