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Discussion about Euler equation ● Relation between mean aerodynamic velocity and force F ● Force on blade and circulation of velocity

Discussion about Euler turbomachinery equation

Discussion about specific shaft work

The article of online continued resource Transformační technologie; ISSN 1804-8293;

www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2020. All rights reserved. This work was published without any linguistic and editorial revisions..

www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2020. All rights reserved. This work was published without any linguistic and editorial revisions..

Terms as force on blades, torque, work (power) can be explained easily and intuitively through a description function of a water wheel:

What dimensions must have a water wheel on a weir about height *0,6 m* and at mass flow rate *0,7 m*^{3}·s^{-1}? For calculation will be used empirical knowledges of millwrights and millers [2]. The solution of this problem is shown in the Appendix 255.

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):

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Velocities and forces are vectors, but the arrow above the symbol does not usually write. **H** [N] **momentum of working fluid**; **F**_{h} [N] weight forces, which act on working fluid inside control volume (gravitational force); **F**_{p} [N] pressure forces, which act on working fluid on border of control volume; **F**_{t} [N] force acting on working fluid from bodies inside or on border of control volume (blades, casing etc.); **F** [N] force acting on bodies inside or on border of control volume (blades, casing atc.) from working fluid; **w** [m·s^{-1}] relative velocity; **u** [m·s^{-1}] tangential velocity; **m**^{•} [kg·s^{-1}] mass flow rate of working fluid which flows through control volume; **m** [m] weight of working fluid inside control volume; **p** [Pa] pressure; **A** [m^{2}] surface of control volume; **g** [m·s^{-2}] gravitational acceleration. **Ψ** streamline of absolute velocity, **K** control volume; **1** inlet to control volume; **2** exit from control volume. This equation is called **Euler equation**, because it was derived by __Leonhard Euler__ as first. This equation agrees for stationary flow through the control volume. The derivation of the Euler equation is shown in the Appendix 196.

The control volume for the case of *Figure 1b* is defined so way: working fluid has the inlet to the control volume through the boundary *AB* (flow area *A _{1}*) and outlets from the control volume through the boundary

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

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.).

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● 12. Essential equations of turbomachines ●

At right from sectional view is 3D view of the impeller.

What force on blades of water turbine (axial type)? It is known dimension of the rotor, the flow mass rate, the shaft speed and the velocities of working gas. The solution of this problem is shown in the Appendix 583.

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:

Water flows through of the pipe with velocity *4 m·s*^{-1}. What is force on the pipe between the flanges from flow of liquid? Inside diameter of the pipe is *23 mm*, difference of height between the upper and bottom flange is *1,2 m*, difference between pressure inside pipe and atmospheric pressure is *2 m* water column. Ideal fluid and flow without losses. The solution of this problem is shown in the Appendix 254.

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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):

Describe forces inside blade passage of radial impeller of fan from fluid stream, if its relative velocities are known and its geometry. Inlet and outlet pressure are same *p*_{1}=p_{2}. The solution of this problem is shown in the Appendix 584.

The force from pressure *F ^{→}_{p}* acts to the control volume. If the inlet flow area (to blade passages)

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● 12. Essential equations of turbomachines ●

This is true for flow without losses. In other way does not have to be truth if

Radial stages can be designed such as an impulse stage^{(2)}:

It is true at negligible centrifugal force inside working fluid volume between radius

If the inlet flow area *A _{1}* is not the same as

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The Kaplan turbine is reaction turbine, it means, in front of the rotor is pressure higher than behind rotor (it can be easily identified by the fact that the blades forms convergent passages and for relative velocities

The axial force on rotor *F _{a}* 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

The radial stages can be designed as the reaction stage also:

The picture shows an example of an reaction radial stage.

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

Momentum of wind is changed in all flow area formed by blades.

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The derivation of the formula is shown in the Appendix 981.

A **mean aerodynamic velocity w _{st}** is mean velocity between the inlet vector and the exit vector of the relative velocities

Assumptions: incompressible flow, lossless flow (__isentropic__ flow – subscript **iz**), axial stage (*r*_{1}=r_{2}). **w**_{st} [m·s^{-1}] mean aerodynamic velocity in blade passage; **β**_{st} [rad] angle of mean aerodynamic velocity; **ε** [rad] angle of the force *F*. For incompressible flow is true *w*_{1a}=w_{2a}=w_{a, st}. The derivation of this equation is shown in the Appendix 248.

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:

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Assumptions: blade length is elementary and the pitch of the blade *s* is constant. **ρ** [kg·m^{-3}] density; **F**_{l} [N·m^{-1}] force on airfoil about elementary length *1 m*; **Γ** [m^{2}·s^{-1}] circulation of velocity. **Γ** **circulation of velocity around the blade in blade row** (on the curve of the control volume of blade *ABCD* on *Figure 10*); **z** number of blades; **z·Γ** circulation around *z* blades; **Γ**_{R1} circulation in front of the rotor blade row; **Γ**_{R2} circulation behind the rotor blade row. The derivation of this equation is shown in the Appendix 588.

From *Equation 11(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**:

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The torque is not function of the weight forces *F _{h}* (homogeneous accelerations field) neither the pressure forces

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, *r _{1}=r_{2}=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:

15.587 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):

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The specific shaft work

The quantities of the *Equation 16* 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 *c _{1}=c_{2}* 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 casings^{(5)}, which are used as the enter/exit from/to the turbomachine of the working fluid. The next type of the spiral passages are vaneless __diffuser__^{(6)}, which are used on the turbocompressors and vaneless confusors^{(7)}, which are used on turbines.

^{(5)}Spiral casings- For case of turbocompressors is the inlet to the spiral casing in radial direction and the outlet tangential direction, for a case of turbines is these opposite.

^{(6)}Vaneless diffuser- The inlet and the outlet of the vaneless diffuser is in radial direction and working fluid is pressured inside (the pressure of the working fluid increasing here). The vaneless diffuser is usually formed by two annular wall, see bellow. On the vaneless diffuser continues a spiral casing.

^{(7)}Vaneless confuser- The vaneless confuser is similar to the vaneless diffuser. Inside the vaneless confuser working fluid is expanded (the pressure of the working fluid decreasing here). A feeder for the vaneless confuser is a spiral casing.

Inside the spiral passages is presumed __potential flow__^{(8)} 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:

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Sometimes the spiral casing is computed with condition

Design a spiral casing of low pressure radial fan with forward curved blades. Casing has rectangular passage section. Impeller perimeter is *65,82 mm*, width of casing is *26,3 mm*, tangential direction of absolute velocity at exhaust of impeller is *19,23 m·s*^{-1} and flow of air is *100 m*^{3}·h^{-1}. Discuss influence of width of casing on casing radius. The solution of this problem is shown in the Appendix 264.

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:

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Derive equation of change of pressure inside a vaneless diffuser. Is there a change angle *α* (the angle between absolute velocity and tangential direction)? Calculate for case of incompressible flow. The solution of this problem is shown in the Appendix 407.

**Problem 6.**407

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__:

19.673 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:

20.771 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 p is higher on outer perimeter, but its velocity is lower, therefore its total energy is the same through whole exit flow area._{e} |

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:

Calculate the pressure gradient in front of and behind of an axial water turbine. Assumptions: potential flow of ideal liquid. The solution of this problem is shown in the Appendix 736.

**Problem 7.**736

Calculate the pressure gradient inside the spiral casing from *Problem 5*. The solution of this problem is shown in the Appendix 725.

**Problem 8.**725

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1. KADRNOŽKA, Jaroslav. *Lopatkové stroje*, 2003. 1. vydání, upravené. Brno: Akademické nakladatelství CERM, s.r.o., ISBN 80-7204-297-1.

2. KŘIVANOVÁ, Magda, ŠTĚPÁN, Luděk. *Dílo a život mlynářů a sekerníků v Čechách*, vydalo nakladatelství ARGO 2000, ISBN- 80-7203-254-2, 2001.

ŠKORPÍK, Jiří. Základní rovnice lopatkových strojů, *Transformační technologie*, 2009-09, [last updated 2017-10-25]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z https://www.transformacni-technologie.cz/12.html. English version: Essential equations of turbomachines. Web: https://www.transformacni-technologie.cz/en_12.html.

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