Copyright©Jiří Škorpík, 2016-2022
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The forces acting from the fluid flow on the machine surfaces can be determined from the momentum change theorem. Its special form is also used to calculate the forces acting on the blades in the blade row from the fluid flow, a classical problem in turbomachinery.
The forces acting from the fluid flow on the machine surfaces can be determined from the momentum change theorem (Newton's second law). According to the momentum change theorem, the change in fluid momentum over time is equal to the sum of the external forces acting on the fluid in the control volume. In the case of applying this law to a fluid enclosed in a control volume V_{C} (Figure 1), the external forces considered are: the pressure forces from the surrounding fluid at the boundaries of the control volume F_{p}, the weight of the fluid in the control volume F_{h}, and the forces exerted by the bodies inside and at the boundaries of the control volume F_{b}. The change in momentum of the fluid inside the control volume is also equal to the difference of the product of velocity and mass flow between the inlet and outlet of the control volume [Bathie, 1984], [Kadrnožka 2003].
When calculating turbomachine blade forces, define control volume boundaries where parameters for momentum change theorem are known. Therefore, the control volume in Figure 2 is defined to pass through the center of the blade channel, or the boundaries AD and BC are spaced apart by the pitch of blades^{1.}. The boundaries AD and BC are the expected streamlines of the relative velocities W of the velocity triangle^{1.} of this blade row. The blade passages are equal in a single blade row, so that the action of the pressure forces at the AB boundaries cancel with the action of the pressure forces at the DC boundaries. The integration of the product of the absolute velocity V and the mass flow is also cancelled at these boundaries, see Equation 2.
Typical for the investigation of forces in the turbomachine stage is the use of cylindrical coordinate system^{1.}. The force F in the cylindrical coordinate system has three spatial components, namely a component in the radial direction F_{r}, in the tangential direction F_{θ} (this force produces torque) and in the axial direction F_{a} (it causes stress on the rotor in the axial direction and is collected by the thrust bearing) - these force components are abbreviated as radial, tangential and axial forces.
The force acting on the blade is approximately perpendicular to the mean aerodynamic velocity in the blade row W_{m}, which is the mean of the relative velocities at the inlet W_{1} and outlet W_{2}. Respectively, it can be shown (see Equation 3) that the resulting force on the blade from the incompressible fluid flow F is perpendicular to the mean aerodynamic velocity w_{m} in lossless flow.
The design of the energy distribution or transformation in the turbomachine stage is based on two directions. In the direction perpendicular to the meridian direction^{1.}, the Euler work distribution, which is the local value of the internal work, is designed. In the meridional direction, the design of the reaction stage, which describes the distribution of energy transformations between the stator and the rotor of the stage, decides the characteristics of the stage.
The Euler work is the fluid work transferred to the blades in the surroundings of the streamline under investigation, see Figure 4(b). The difference with the internal work^{1.} of the stage w_{i} is that the internal work of the stage is the average work of all the working fluid flowing through the stage (including gaps) and can be determined from the complete energy balance of the stage, see Figure 4(a). So some of the fluid will do more Euler work than others, but their average is w_{i}. For real stages, the largest Euler work is in the core of the flow (at the mean diameter of the blades), where the losses are smallest. Conversely, at the edges of the blades, or near their hubs and tips, the Euler work is smallest due to high frictional losses and internal leakage. The Euler work can be determined from the velocity triangles on the streamline under investigation, see Equation 4(c) - Euler turbomachinery equation.
Similarly, the stage efficiency can be determined in two ways, either to the Euler work (Euler efficiency) or to the internal work (internal stage efficiency), as done in Problem 4.
Reaction is the ratio between the change in the static enthalpy in the rotor blade row and the change in the stagnation enthalpy of the stage (Formula 5), or the change in the static enthalpy of the stage - depending on convention [Kadrnožka, 2004], [Japikse, 1997], [Bathie 1984], [Ingram, 2009]. Thus, it describes the distribution of the energy transformation between the stator and rotor blade rows of the stage.
Reaction is determined to a specific streamline (radius) similar to Euler work. To calculate the reaction, it is important to know the construction of the h-s diagram, from which the differences in specific enthalpies Δh_{s} and Δh_{R} can be determined (see Problem 5, Problem 6). h-s diagrams and a description of their construction are given in Figure 6. In the case of hydraulic machines, the required pressure differences Δp_{s} and Δp_{R} can also be determined from Bernoulli's equation for relative flow, see Problem 7 and Problem 6.
Most stages are designed with a variable reaction over the height of the blades, while a common requirement in stage design is a reaction at a mean radius of about 0.5 (even a little higher for radial stages due to the difference in blade speeds on the rotor), since at maximum Euler work the absolute velocities in the stator passages are about the same as the relative velocities in the rotor passages, and hence the distribution of losses between stator and rotor is uniform, see Figure 7.
Laval turbines^{1.} and Pelton turbines show minimal the reaction. Among working machines, low reactions are used in certain fans. At low reaction, compressive force on the rotor is small, calling these stages equal pressure or impulse stages. Conversely, stages with significant reaction yield greater compressive force, termed overpressure or reaction stages.
In axial stages, as reaction increases, blade camber decreases (required momentum change drops), reducing sensitivity to flow separation from the profile decreases.
Ideal flow equations are derived for ideal fluid flow without internal losses. Although they are ideal flow equations, they are crucial for basic design of turbomachinery flow components, prediction of properties, analysis of the effect of flow component shape on internal losses of the machine, and understanding the causes of failures or problem operation of turbomachines.
The basic equations describing ideal flow velocities are the potential flow equations. The flow is considered to be potential (meaning that the velocity can only be calculated using the coordinates of a point according to a potential function V=f(x, y, z) in the case of a orthogonal coordinate system, or according to a function V=f(r, θ, a) in the case of a cylindrical coordinate system), where such a flow is referred to as an axisymmetric potential flow. Other quantities of ideal flow can be calculated from the energy equations and the Euler equation of hydrodynamics.
For potential flow, the curl velocity vector must be zero throughout the volume (Equation 8a). For axisymmetric flow, gradients of velocity components in the tangential direction (Equation 8b) must be zero in a cylindrical coordinate system, because the tangential coordinates are closed and the velocity at the origin of the tangential axis must be identical to that at the end of the coordinates. These conditions yield special velocity Equations (8c-h), applicable to other fluid quantities.
The product r·V_{θ} is called the circulation of the tangential component of the velocity, which is constitutive, so it has the same properties as the irrotational vortex [Škorpík, 2023, p. 1.40]. If the circulation is constant, then the difference of the circulations in front of and behind the rotor is also constant and then also according to the Euler work equation (Equation 4) the Euler work of the potential flow will be constant along the length of the blades, see Problem 8.
The equations of axisymmetric potential flow can also be applied to spiral paths, for example in spiral passages (Problem 9) or in bladeless diffusers and confusers (Problem 10).
The Euler equation of hydrodynamics for potential flow can also be used to calculate other state variables, for example, the article Internal fluid friction and boundary layer development [Škorpík, 2023b]. From this equation one can read, among other things, that the pressure gradient of a potential flow without the effect of gravitational acceleration cannot have a tangential component, because the gradient of velocity or kinetic energy does not have one either, see Problem 8.
1: | entry: | d; z; z_{H2O}; V | 4: | calculation: | F_{h,x}; A; m; p_{1}; p_{2}; F_{p,x}; F_{x} | |||||||
2: | derivation: | eq. for F_{x}; F_{y}; F_{z} | 5: | calculation: | V_{C}; m; F_{h,z}; F_{pz}; F_{z} | |||||||
3: | read off: | ρ; g; p_{at} | 6: | calculation: | F_{y} |
1: | entry: | m; p_{1}; z; r_{1}; r_{2}; V1; V_{2}; α_{2}; b | 2: | calculation: | V_{2r}; F_{r,p}; F_{r} | |||||||
3: | calculation: | V_{2θ}; F_{θ} | 4: | calculation: | F |
1: | entry: | r_{t}; r_{h}; V_{1θ}; V_{a}; V_{2}; N; z | 4: | calculation: | r_{m}; U; -W_{2θ}; W_{1θ}; W_{mθ}; W_{m}; β_{m}; ε; F_{a}; F | |||||||
2: | read off: | ρ | 5: | read off: | p_{at}; g | |||||||
3: | calculation: | A_{1}; A_{2}; Q; m; F_{θ} | 6: | calculation: | V_{1}; p_{1}; p_{2} |
1: | entry: | Δh_{is}; L_{w}; V_{1}; W_{2}; V_{0}; V_{2}; W_{1}; U_{1}; U_{2} | 3: | calculation: | w_{i} | |||||||
2: | calculation: | w_{E} | 4: | calculation: | w_{is}; η_{E}; η_{i} |
1: | entry: | V_{1}; V_{2}; W_{1}; W_{2} | 3: | calculation: | w_{E}; Δh_{s}; R | |||||||
2: | calculation: | Δh_{R} |
1: | entry: | Δp_{s}; ρ; U_{2}; U_{1} | 3: | calculation: | Δh_{R}; R | |||||||
2: | calculation: | Δh_{s} |
1: | entry: | r_{1}; V_{1}; V_{2}; N; α_{1}; Δz | 4: | derivation: | equation for Δp_{R} | |||||||
2: | derivation: | equation for Δp_{s} | 5: | read off: | g; L_{w,0-2}; L_{w,1-2} | |||||||
3: | calculation: | U_{1}; V_{1θ}; w_{E} | 6: | calculation: | R |
1: | entry: | w_{E}; r_{t}; r_{h}; N; V_{a}; V_{2}; p_{2}; ρ | 6: | calculation: | p_{1h}; p_{1m}; p_{1t} | |||||||
2: | calculation: | r_{m}; U_{h}; U_{m}; U_{t}; V_{1θh}; V_{1θm}; V_{1θt} | 7: | calculation: | Δp_{s}; Δp_{Rh}; Δp_{Rm}; Δp_{Rt}; R_{h}; R_{m}; R_{t} | |||||||
3: | calculation: | V_{1h}; V_{1m}; V_{1t}; α_{1h}; α_{1m}; α_{1t} | 8: | derivation: | equation for grad p_{1} | |||||||
4: | calculation: | W_{1θh}; W_{1θm}; W_{1θt}; W_{1h}; W_{1m}; W_{1t}; β_{1h}; β_{1m}; β_{1t} | 9: | derivation: | equation for Δp_{1} | |||||||
5: | calculation: | W_{2h}; W_{2m}; W_{2t}; β_{2h}; β_{2m}; β2t |
1: | entry: | R; r_{3}; r_{2}; r_{1}; b_{2}; b_{1}; Δp_{s}; N; Q; p_{1s}; ρ | 5: | calculation: | p3s; V_{3θ}; V_{3r}; V_{3}; p_{3} | |||||||
2: | derivation: | equation for r_{θ} | 6: | proof: | α=const. | |||||||
3: | calculation: | w_{E}; U_{2}; V_{2θ}; C | 7: | discussion: | on effect of friction | |||||||
4: | calculation: | r_{θ} for selected θ | 8: | discussion: | velocity and pressure distribution at casing outlet |
1: | entry: | V_{2θ}; V_{2r}; r_{2}; p_{2}; Δp_{S}; t_{2} | 4: | solution: | r_{3} z m_{2}=m_{3} | |||||||
2: | read off: | h_{3}; t_{3} from h-s diagram | 5: | calculation: | α_{2}; α_{3} | |||||||
3: | calculation: | V_{3}; ρ_{3} | 6: | comapare: | α_{2} vs α_{3} |