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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. Principle of transformation of energy inside a hydraulic or a heat turbomachine is the same, but is useful use special version of this equation for hydraulic machines (for *ρ≐constant*) and heat machines (for *ρ≠constant*). 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 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):

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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.

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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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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 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.

**Problem 1.**id597

Pi [MW] 61,371 l [m] 8,73 |
Problem 1: results.l [m] length of draft tube. |

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

7.id302 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__* 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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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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11.id604 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%. Other parameters are shown in Problem 1 [6.].

**Problem 2.**id871

Pi [MW] 8,6597 ie [kJ·kg-1] 2393,3575 x [%] 93,8016 |
Problem 2: results.x [%] quality of vapor on exit turbine. |

The *Equation 9* describes the energy state of working gas on the input and on the exit of the turbine regardless of number of the stages. This equation can also be applicable on one stage. The sum of the energy balance all stages and the energy balance of other section of the turbine must be equal the energy balance of the stage section of the turbine as one entire:

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It is evident, a portion of the heat from the loss processes inside previous stage is used inside next stage during expansion. Only the heat from the loss processes inside last stage of the heat turbine are not used.

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.

- 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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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:

15.id908 Internal efficiency of turbine in relation to reversible polytropic expansion. |

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

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Inside the compressor is transformed work to the pressure energy of working gas during compression for ideal case.

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.

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

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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:

18.id609 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:

19.id610 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.

In case multi-stage turbocompressors, the resulting compression is composed of several sub-compression, their number is equal number of the stages inside turbocompressor:

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The internal losses *z* of one turbocompressor stage reduces the internal efficiency *η _{i}* of next stage.

The air with temperature *15 °C* and pressure *0,1013 MPa* enters to a turbocompressor, on exit of the turbocompressor the air has temperature *293 °C* and pressure *0,802 MPa*. Calculate work *a*_{i}, *a*_{iz}, internal efficiency *η*_{i} and factor *1+f*. Number of stages of the turbocompressor is *12*. Use simplification *c*_{p}=const..

**Problem 3.**id122

This problem is published in [4].

This problem is published in [4].

ai [kJ·kg-1] -279,251 aiz [kJ·kg-1] -233,2122 ηi [-] 0,8355 1+f [-] 1,0437 |
Problem 3: results. |

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.

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- 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

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:

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23.id1003 Internal efficiency of compressor in relation to reversible polytropic compression. |

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 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, then from equation of First law of thermodynamic for open system:

24.id308 Specific internal work of the fan.The assumption: q=0, potential energy is negligible. |

The profile of static and dynamic pressure change is shown on i-s diagram as like for the case turbocompressors. 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

25.id309 Specific internal work of the fan shown in i-s digram.Δp [Pa] increase of stagnation pressure in fan; _{c}Δp [Pa] _{z}pressure drop (difference of stagnation pressure between isentropic and real process). The assumption: q=0, potential energy is negligible. The derivation of this equation is shown in the Appendix 309. |

The internal efficiency of the fan is defined as ratio between specific internal work of the fan and the internal work of the fan at isentropic increase of pressure:

26.id581 Internal efficiency and internal 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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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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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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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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*Unified Propulsion*, 2011. [Online] kurz v rámci projektu MIT OpenCourseWare Massachusetts Institute of Technology. Adresa: http://ocw.mit.edu.- MATTINGLY, Jack, HEISER, William, PRATT, David,
*Aircraft Engine Design*, 2002. Second edition. Reston: American Institute of Aeronautics and Astronautics, ISBN 1-56347-538-3. - KADRNOŽKA, Jaroslav.
*Teorie lopatkových strojů*, 1991. 3. vydání, přepracované. Brno: Vysoké učení technické v Brně, ISBN 80-214-0275-X. - 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 2015-02]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/energeticke-bilance-lopatkovych-stroju.html. English version: Energy balances of turbomachines. Web: http://www.transformacni-technologie.cz\en_energeticke-bilance-lopatkovych-stroju.html.

©Jiří Škorpík, LICENCE

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