Terms as force on blades, torque, work (power) can be explained easily and intuitively through a description function of a water wheel:
d [m] 4,8 Fu [N] 1253,72 P [W] 2507,5 a [m] 0,25 Mk [N·m] 2993 n [min-1] 8Results of Problem 1.
The information about the force on blades, the torque, the schaft speed, and the power are significant for all types of turbomachines. For calculation of these quantities can be derived general equations, which may be used for all type of turbomachines (history of the derivation Essential equations of turbomachines are shown in the chapter 1. Water wheels and water turbines).
The velocity and the direction of the working fluid stream are changed inside the turbomachine stage. It means that some force acts to the working fluid. This force is function of mass flow rate and shape of volume through which working fluid flows (a control volume):
The control volume for the case of Equation 1b is defined so way: working fluid has the inlet to the control volume through the boundary AB (flow area A1) and outlets from the control volume through the boundary CD (flow area A2). The boundaries of the control volumes AD and BC are streamlines. The velocity is perpendicular on normal of boundaries of the control volumes AD and BC, therefore the momentum of the working fluid is zero here. The boundaries AD and BC are the same, therefore the pressure force here are the same but opposite between himsel. The control volume is not moved.
For calculation of force on blades, according Euler equation, are sufficient parameters on boundary of the control volume.
The force F→ has three spatial components as absolute velocity: at axial direction Fa (this components of the force causes a stress of rotor shaft at axial direction, this force acts to axial bearing of shaft), at radial direction Fr, at tangential direction Fu (this force does torque on rotor shaft):
Some components of force F→ can be negligible (e.g.: for case of the axial stage of turbomachine there is not radial component – flow on surface of a cylinder surface; for case of radial stage of a fan there is not axial component etc.).
The boundaries of control volume AD and BC inside blade row may not copy streamlines of absolute velocity, but suffice it to define so that flow parameters at these boundaries were the same. Boundaries of the control volume are usually defined by streamlines of relative velocities for cases of turbomachines. For this type the control volume is momentum of the stream on boundary AD the same as on boundary BC but opposite direction, therefore these momentum are eliminated between themselves:
Fx [N] -52,0064 Fy [N] 51,9596Problem 3: results.
The force F→ can be computed from relative flow through Bernoulli equation for a rotating passage [1, p. 41], but this caculation is complicated – (in this case is computed with Coriolis force which has influence force to tangential direction, and centrifugal force which has influence force to radial direction):
The force from pressure F→p acts to the control volume. If the inlet flow area (to blade passages) A1 is the same as exit flow area A2, then for a case of ρ=const., is true w1=w2 and p1=p2. This turbomachine stage is called impulse stage. The force F is the same as tangential force Fu1 for cases of impulse stages with negligible weight forces Fh and losses:
Radial stages can be designed such as an impulse stage2:
If the inlet flow area A1 is not the same as A2 (A1≠A2), then the equality of relative velocity on inlet and exit is ruled out (w1≠w2 therefore can be derived inequality p1≠p2 from Bernoulli equation). This is truth for ρ=const. or ρ≠const. This type of stage is called reaction stage:
The axial force on rotor Fa of the axial reaction stages is higher (under overpressure of the working fluid), than for cases of the action stages. The working machine blade passage is usually made as reaction stage (can be used impulse stage theoretically also, but it is not used [1, p. 38-39]). The reaction stages has less power than the impulse stages under same conditions (tangential velocity; parameters of the working fluid), see article 19. Design of axials turbomachine stages.
The radial stages can be designed as the reaction stage also:
The Euler equation can be use for "sparse" blade passages of wind turbines. The change of the velocity vector of the wind at flow through rotor is function the ratio u·c-1, which is called tip-speed ratio. The number of rotor blades is smaller with higher this ratio. It is means, the wind turbines with one blade have higher speed than three blades turbines (cheaper but higher noise), because one blade must transform the same energy as three blades:
A mean aerodynamic velocity wst is mean velocity between the inlet vector and the exit vector of the relative velocities w1, w2 of the blade passage. The Force on the blades from fluid stream F is perpendicular on the mean aerodynamic velocity for case of the incompressible flow:
The force on blade inside an axial blade row can be calculated without pressures through circulation velocity on boundary of the control volume of the blade:
From Equation 10(a) is evident, if be knew circulation in front of the rotor and behind the rotor then can be calculated the circulation around one the blade.
The forces on blades of the rotor from fluid stream develops a torque on its shaft. This torque is made up of tangential components these forces. The equation for torque of turbomachine stage is called Euler turbomachinery equation:
The torque is not function of the weight forces Fh (homogeneous accelerations field) neither the pressure forces Fp, because these forces have not any the sum forces components on tangential direction (respectively they annihilate between).
The power of the rotor can be calculated from the torque:
For cases of a turbine stages is true dP>0, for cases of a working machine stages is true dP<0.
For cases of the axial stages (flow on a cylindrical surface, r1=r2=r) the Euler turbomachinery equation can be rearranged in this form:
For case of short straight blades there is a change of the form of the velocity triangle along length the blade, therefore is approximately true:
|14.587 The power transferred between flow and the rotor with short the blades.|
At higher speed rotation is transferred smaller the torque and opposite (for the same the power) according the Euler turbomachinery equation. Therefore the diameter of shafts of high-speed rotor of turbomachines are smaller than the diameter of shafts of low-speed rotor.
The specific shaft work is a ratio between the power on rotor and mass flow rate of the working fluid (work of 1 kilogram of the working fluid during flow through the rotor blade row on tested radius r, which is transformed on torque of the machine shaft):
The quantities of the Equation 15 has different influence on the specific shaft work:
The absolute velocity is varied by all type of the turbomachine stages (there are the turbomachine stages, where c1=c2 e.g. [1, p. 40], but they do not used).
The relative velocity is not varied for a case of the impulse stages.
An influence of tangential velocity is significant for radial stages.
Working fluid flows on the spiral curve inside a spiral passage. The spiral passages can be classified to two types. The first type of the spiral passages are spiral casings5, which are used as the enter/exit from/to the turbomachine of the working fluid. The next type of the spiral passages are vaneless diffuser6, which are used on the turbocompressors and vaneless confusors7, which are used on turbines. The spiral casings are used on radial turbomachines always, except special cases.
Inside the spiral passages is presumed potential flow8 at calculation or rot c→=0.
The spiral passage has properties of a passage with constant flow area at incompressible flow ρ≈konst. – it means, flow area of the spiral casing is changed with mass flow rate on tangential direction:
ν [°] rν [mm] ν [°] rν [mm] --------------- --------------- 0 44,6 180 63,29 90 53,13 270 75,39Problem 5: results.
For case of simple shape of the spiral casing (rectangular passage section, circular passage section) there is easy solving, but for case of intricate shape of the spiral casing is usually needed CFD computation.
For case of a vaneless diffuser or a confusor is the inlet flow area different from the outlet flow area:
Pressure distribution in the working fluid flow is influenced by change of flow area and curved of flow. The centrifugal force at curved flow develops increasing of pressure on outer perimeter so called transverse pressure gradient:
|18.673 Development of transverse pressure gradient inside curved passage.|
n normal of streamline; ρ' [m] radius of curvature of flow in tested point; dp/dn [Pa·m-1] pressure gradient. Derived for assumption of potential flow, 2D flow at negligible weight forces. This equation is called Euler n-equation. The derivation of Euler n-equation is shown in the Appendix 673.
The transverse pressure gradient is developed almost everywhere inside turbomachines:
|19.771 Examples development of the transverse pressure gradient inside turbomachines.|
(a) development of transverse pressure gradient in front of Kaplan turbine; (b) development of transverse pressure gradient inside spiral casing – exit pressure of working fluid pe is higher on outer perimeter, but its velocity is lower, therefore its total energy is the same through whole exit flow area.
The pressure gradient and kinetic energy gradient in front of and behind of a blade row for a calculation of velocity triangles are need to know only. Equation of the first law of thermodynamics for open system and Euler equation of fluid dynamics at potential flow be can use for calculation these gradients as well as gradients of other quantities of flow. The advantage this equation versus the Euler n-equation is that the gradients are in relation to direction of coordinate system and not to normals:
(1) Describe equation for force on blade from fluid stream (Euler equation). (2) What is connection between inlet and exit flow area of blade-to-blade passage for case of action stage and reaction stage? (3) What is direction of pressure forces on control volume of axial reaction stage (e.g. Kaplan turbine)? Draw scheme and the force indicate. (4) Make triangle of force on blade of radial fan stage. (5) Define the mean aerodynamic velocity in the blade row. (6) What is the angle between the force on blade from flow and the mean aerodynamic velocity in the blade row? (7) What is torque on shaft of one-stage turbomachine influence through elementary flow mass of the working fluid dm (Euler turbomachinery equation)? (8) Write the equation for the specific shaft work of the turbomachine stage. (9) Describe the specific shaft work of the turbomachine stage through words. (10) Kutta–Joukowski theorem (description one quantities of equation). (11) What is the circulation around the blade in blade row? (12) What are the assumptions for the solving of the spiral passage?
ŠKORPÍK, Jiří. Základní rovnice lopatkových strojů, Transformační technologie, 2009-09, [last updated 2016-10-10]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/12.html. English version: Essential equations of turbomachines. Web: http://www.transformacni-technologie.cz/en_12.html.