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Discussion about water turbine energy equations

Discussion about pump energy equations

Adiabatic expansion inside heat turbine ● Polytropic expansion inside heat turbine

Adiabatic compression inside compressor ● Polytropic compression inside compressor

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*Appendices pages are only in Czech language; tip: You can use https://translate.google.com for translate of the appendices.

*Appendices pages are only in Czech language; tip: You can use https://translate.google.com for translate of the appendices.

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

www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2019. All rights reserved. This work was published without any linguistic and editorial revisions.

www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2019. All rights reserved. This work was published without any linguistic and editorial revisions.

● 13. Energy balances of turbomachines ●

Inside a turbomachine can be changed of __enthalpy__, kinetic energy and potential energy of a working gas according the equation for __First law of thermodynamics for open system__ and it is can use for all types of turbomachines. In next chapters there is description of transformation energy for various types of the turbomachines (include losses), so as it is usually used under practice.

In chapter 11. Internal power output/input of turbomachine *P _{i}* is defined specific internal work

For cases of hydraulic machines is usually used a term

The internal power and efficiency is not influenced by the internal losses only, but even a leakage of machine see chapter

The water turbines can transform potential, kinetic and __pressure energy__ of a working liquid to work and heat-internal losses (increase of internal energy of water):

1.303 Specific internal work of a water turbine.a [J·kg_{i}^{-1}] specific internal work; p [Pa] pressure of working liquid; c [m·s^{-1}] absolute velocity of flow; ρ [kg·m^{-3}] density of working liquid; g [m·s^{-2}] gravitational acceleration; h [m] height of levels; y [J·kg^{-1}] specific total energy of working liquid; z [J·kg^{-1}] specific internal losses. Derived from Bernoulli equation. |

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● 13. Energy balances of turbomachines ●

An internal efficiency of the water turbine is ratio between *a _{i}* and the change of specific total energy of the working liquid between the inlet flange and the exit flange:

The water turbine are one of the most efficiency turbomachines with *η _{i}* up to

The water turbine usually feeds an electric generator, which has adequate power on its contacts.

Work is done only inside the impeller of turbine and the change of specific energy of water inside the impeller is function of a difference of kinetic energy and pressure __gradient__ (at negligible influence of the change of potential energy). Hence is advantageous used a reaction stage *p*_{1}>p_{2}, then is velocity less and therefore also the internal losses, which are function of the velocity.

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● 13. Energy balances of turbomachines ●

For technical reasons and safety of the water turbines are not placed tightly above the level of lower tank (a risk of flooding of machine room etc.). Hence, for case turbine with the reaction stage is inserted a **draft tube** between the turbine and the lower tank:

The end of the draft tube is tightly below the surface of the lower tank and is filled water. Its height is function of difference of pressure between the turbine exit and the surface in lower tank. It evident, the pressure on the turbine exit must be less then the pressure on the surface of the lower tank. The pressure on the turbine exit *p _{2}* must not be less then saturation pressure of water in this point. In the opposite case the water column would be crashed under development of steam. Because pressure

Calculate the internal power output and the maximum length of a draft tube of a water turbine. Difference of the levels is *136 m*, the flow rate *46 m*^{3}·s^{-1}. Do not take into account any losses.The solution of this problem is shown in the Appendix 597.

This case is similar such as the water turbines, but in rotodynamic pumps is transformed work (internal power input of the rotodynamic pump) to energy of the working liquid (potential, kinetic, pressure energy). The purpose of the pumps is to increase of specific total energy of the working liquid from a state *y _{0}* to required state

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● 13. Energy balances of turbomachines ●

The letter **S** denotes the suction side of pump (here working liquid enters to the pump), the letter **V** denotes the delivery side of pump (here working liquid leaves from the pump). For the internal work of the pump is turth *0>a*_{i}.

Internal efficiency of the pump is ratio between the change of the total energy of the liquid during flow through the pump and the absolute value of internal work *a _{i}*:

Maximum *η _{i}* of pumps can be higher then

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● 13. Energy balances of turbomachines ●

During pumping of the working liquid can be neglected change of kinetic energy between the surface inside the suction and the delivery tank:

7.302 A practical calculation of an increasing of the specific total energy of the working liquid inside the pump.The losses are calculated from a characteristic of piping system. The end of the tube is on a flanges of the pump. The derivation of this equation is shown in the Appendix 302. |

Inside a heat turbine is transformed heat and ethalpy to work during expansion of the working gas. The working substance (liquid, gas, steam ...) is usually heated outside turbine, e.g. in case of steam turbines is made steam inside steam boiler (steam cycle), in case of a combustion turbine is made hot gas inside a combustion chamber in front of turbine (Brayton cycle).

The expansion of the working gas can be __adiabaticaly__^{(3)} inside heat turbine, therefore *q=0* nebo *q≈0*. There are cases where heat transfer with surroundings has influence on expansion (so called __polytropic__ expansion *q≠0*). In two next chapters is descripted the cases adiabatic and polytropic expansion inside heat turbines.

Inside the heat turbines is higher temperature of gas than is temperature of the surroundings, but are thermally isolated therefore heat transfer with surroundings is not actual.

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● 13. Energy balances of turbomachines ●

The working gas inside turbine expands from pressure *p _{i}* on pressure

For the difference specific enthalpy *i _{i}-i_{e}* can be use the equation also:

From previous equations is evident, maximum *a _{i}* at adiabatic expansion is reached for case isentropic process, therefore the isentropic expansion is used as comparative for assessment of internal efficiency of the heat turbine:

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● 13. Energy balances of turbomachines ●

11.604 The internal efficiency of the heat turbine in relation to isentropic process. |

Maximum *η _{i}* of heat turbines can be higher than

Calculate *P*_{i} of a steam turbine and the quality of the vapor on the exit turbine. Through the turbine flows *33 t·h*^{-1} of steam, *η*_{i}=75 %, specific isentropic work of turbine is *1259,59 kJ·kg*^{-1}, exit pressure of turbine is *3 kPa*, inlet pressure and temperature of turbine are *3,5 MPa*, *450 °C*.The solution of this problem is shown in the Appendix 871.

In cases heat transfer with surroundings influences expansion inside turbine. For example a cooling of temperature stressed of parts of the turbine etc. For these cases the expansion is more similar a polytropic expansion than adiabatic expansion.

^{(4)}Design of area in T-s diagram which is equivalent to work at expansion of gas- An equivalent area to heat
*q*is subtracted from the area which equivalent to work*a*, if it has negative sign and vice versa. An equivalent area to enthalpy change at constant pressure (_{pol}*i*) is subtracted from the area_{e}-i_{e,q}*a*if it has positive sign and vice versa._{pol}

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● 13. Energy balances of turbomachines ●

For case polytropic equations is maximum *a _{i}* reached for case reversible polytropic expansion, therefore the reversible polytropic expansion is used as comparative for assessment of internal efficiency of heat turbine at polytropic expansion:

14.908 Internal efficiency of turbine in relation to reversible polytropic expansion.a [J·kg_{p}^{-1}] specific work at comparative (proposed) ideal polytropic expansion (usually isothermal expansion). |

The above knowledge can be also applied to the description of expansion at heat transfer with surroundings in one stage of heat turbine.

Inside the compressor is transformed work to the pressure energy of working gas.

It is evident *a*_{i}<0. The difference of kinetic energy on input and exit usually are negligible in relation the difference of enthalpy. The Derivation of this equation is from the general equation for internal work of the turbomachine at negligible change of potential energy.

The compression of the working gas can be presumed adiabaticaly inside compressor, therefore *q=0* nebo *q≈0*. There are cases where heat transfer with surroundings has influence on the compression (so called polytropic compression *q≠0*). In two next chapters is descripted the cases adiabatic and polytropic compression inside heat compressors.

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● 13. Energy balances of turbomachines ●

The working gas is compressed from pressure *p _{i}* on pressure

If is not known i-s diagram of the working gas, is can used for a calculation of enthalpy difference *i _{i}-i_{e}* also the

At adiabatic compression can be maximum *a _{i}* reached for case isentropic compression, therefore the isentropic compression is used as comparative for assessment of the internal efficiency of the compressor at adiabatic compression:

17.609 Internal efficiency of the compressor at adiabatic compression. |

A median of *η _{i}* is about

A significant parameter of the compressors is also the **compression ratio** of compressor of the static or stagnation pressures:

18.610 The compression ratio of the compressor.ε [-] compression ratio. |

The turbocompressor is driven by an electrical engine, by combustion turbine or turbo-expander. The turbocompressor is driven by a steam turbine in industrial plants with high consumption of pressurized air.

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● 13. Energy balances of turbomachines ●

In cases heat transfer with surroundings influences compression inside compressors. For example a __turbocompressor with cooling__ etc. For these cases the compression is more similar a polytropic compression than adiabatic compression.

^{(5)}Design of area in T-s diagram which is equivalent to work at compression of gas- The internal ideal polytropic work of compressor
*a*has negative sign, therefore for an image area, which equivalent to the internal work_{pol}*a*is true: the area, which equivalent to heat_{i}*q*is subtracted from the area*a*if it has positive sign and vice versa. An equivalent area to an enthalpy change at constant pressure (_{pol}*i*) is subtracted from the area_{e}-i_{e,q}*a*if it has negative sign and vice versa._{pol}

In this case, temperature of cooling fluid must be smaller than the temperature of working gas on input of the turbocompressor

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● 13. Energy balances of turbomachines ●

At polytropic compression can be maximum *a _{i}* reached for case reversible polytropic compression, therefore the reversible polytropic compression is used as comparative for assessment of the internal efficiency of the compressor at polytropic compression:

21.1003 Internal efficiency of compressor in relation to reversible polytropic compression.a [J·kg_{p}^{-1}] specific work at comparative (proposed) ideal polytropic compression (a, because _{p}≠a_{pol}a is influenced by losses at real compression see problem [26.612])._{pol} |

The above knowledge can be also applied to the description of compression at heat transfer with surroundings in one stage of turbomachine.

The fans are machines, which provide force of gas flow (compensate of pressure losses) with a small increase of its pressure. For calculation of the fans are usually used assumptions *ρ≈const.*, *t≈const.*-incompressible flow. The work for the working gas is transformed on the pressure energy and the kinetic energy, a change of potential energy is negligible inside the fan. Because the compression ratio *ε _{c}* in the fan is very small is better to describe the transformation energy based on the increase stagnation pressure

22.309 Specific internal work of the fan shown in i-s digram.Δp [Pa] increase of stagnation pressure in fan. The assumption: potential energy is negligible. The derivation of this equation is shown in the Appendix 309._{c} |

The internal efficiency of the fan is defined as ratio between specific internal work of the fan without losses and the internal work of the fan:

23.581 Internal efficiency and power input of the fan.V [m^{•}^{3}·s^{-1}] volume flow; m˙ [kg·s^{-1}] mass flow rate through fan. |

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● 13. Energy balances of turbomachines ●

A portion of the kinetic energy, which flows through the rotor of the wind turbine is transformed to work (in case is negligible influence potential energy, internal heat energy and pressure energy). The flow area of the stream-tube of the wind rotor is increases under decreasing of the wind velocity inside. The slow-motion air flow of air behind the turbine is a barrier for surrounding flow (the flow on outside stream-tube), which must gradually wrap it. Therefore the stream-tube is developed (separated air flow, which flows through rotor from other air flow) far upstream of the turbine. Specific work of wind turbine is specific work of air inside stream-tube of rotor:

The axial wind turbine is a reaction stage. The stream-tube in front of turbine works as of a diffuser (pressure increasing) where the kinetic energy is transformed to pressure energy. The stream-tube behind turbine works as of a draft tube, because closely behind of turbine is developed a underpressure:

**Efficiency of wind turbine** *η _{i}* is defined as the ratio between specific work

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● 13. Energy balances of turbomachines ●

The propellers, marine propellers are machines without a housing. The transformation energy by these machines is similar as for the case the wind turbines. The kinetic energy of the working fluid during flow through the rotor is increased-the velocity is increasing:

The propeller is used to make a **thrust** – a force, which causes a motion of an airplane or it is used for overcome the drag forces and weight forces during uniform flight. The most significant function of the propeller is to make the thrust:

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● 13. Energy balances of turbomachines ●

The most significant quantity of the propeller drive is the thrust *T*, which allows the airplane an flight with velocity *v*, therefore **the efficiency of the propeller** is function these both quantities. The efficiency of the propeller shows qualitation of transformation of power input to the thrust. The efficiency transformation kinetic energy of realtive velocities of flow to the thrust is called **propulsion efficiency**:

For optimal condition of the flight is efficiency of propeller *η _{i}* more than

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● 13. Energy balances of turbomachines ●

1. *Unified Propulsion*, 2011. [Online] kurz v rámci projektu MIT OpenCourseWare Massachusetts Institute of Technology. Adresa: http://ocw.mit.edu.

2. MATTINGLY, Jack, HEISER, William, PRATT, David. *Aircraft Engine Design*, 2002. Second edition. Reston: American Institute of Aeronautics and Astronautics, ISBN 1-56347-538-3.

3. KADRNOŽKA, Jaroslav. *Teorie lopatkových strojů*, 1991. 3. vydání, přepracované. Brno: Vysoké učení technické v Brně, ISBN 80-214-0275-X.

4. KRBEK, Jaroslav. *Tepelné turbíny a turbokompresory*, 1990. 3. vydání. Brno: Vysoké učení technické v Brně, ISBN 80-214-0236-9.

This document is English version of the original in Czech language: ŠKORPÍK, Jiří. Energetické bilance lopatkových strojů, *Transformační technologie*, 2009-10, [last updated 2017-11-19]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z https://www.transformacni-technologie.cz/13.html. English version: Energy balances of turbomachines. Web: https://www.transformacni-technologie.cz\en_13.html.

©Jiří Škorpík, LICENCE

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*Appendices pages are only in Czech language; tip: You can use https://translate.google.com for translate of the appendices.