Testseitethermal energy
chemical energy
solar energy, geothermal oceanic heat chemical and nuclear combustible wind, hydraulic gradient lightning, light
process heat
geothermal energy
chemical energy of combustible
effective work
effective work
of exhaust fumes
heat power plant
combustion power plant
Fluid machines
Process of energy conversion For the generation of electric energy FM-2.2
Method of conversion
D ire
ct c
on ve
rs io
Thermoelctr., thermoionic conversion
Hydropower machines (Windpower machines)
combustion engine
open gasturbine
closed gasturbine conven- tional
steam turbine conven- tional
H ea
energy exhaust fumes
energy air
generatorgas turbine
mechanical energyenergy
cooling gas
FM-2.3
statically acting dynamically acting
decrease of fluid energy increase of fluid energy decrease of fluid energy increase of fluid energy
Mechanically acting fluid energy machine
addition of heat
hydrostatic converter
hydrostatic converter
gasturbine plant
steamturbine plant
Fluid machines Organising principle of mechanically acting fluid energy in combination with
fluid energy plants machines FM-2.4
power machine primary energy mechanical work
work machine secondary energydriving work
combustion engine gas turbine piston steam engine steam turbine water turbine wind turbine
conversion of thermal,chemical energy into mechanical energy
conversion of mechanical energy into fluid energy (kinetic energy), potential energy
piston pump centrifugal pump piston compressor turbo compressor
Driving fluid energy machine: A machine, a fluid passes trough, which drives another machine (consumer) mechanically. By this delivery of mechanical energy via the shaft (to the consumer) energy is taken out of the fluid. So, related to the shaft: energy delivering machine.
Purpose: Increase of mechanical energy (via decrease of fluid energy).
Driven fluid energy machine: A machine, a fluid passes trough, which is mechanically driven by another machine (dealer). By the supply with this mechanical energy via the shaft, energy is transmitted of the fluid. So, related to the shaft: energy receiving machine.
Purpose: Increase of fluid energy (via decrease of mechanical energy).
2
Fluid
A
E
hE > hA
hA > hE
displacement work machines with gaseous displacement
device
power- and work machines (fluid energy machines)
piston work machines
(turbo machines)
special design
displacement work machines with rigid or elastic displacement
device
pump)
gas-, steam- and waterturbine
planetary piston engine
pulse motor
Fluid machines
In case of mechanically acting fluid energy machines the exchange of work between fluid and machine is happening in terms of work due to change of volume, compression work and displacement work. As explained in the following, work due to change of the volume and compression work is connected by displacement work. It is
∫∫ −=−− A
A
E
VdpVpVppdV )(
The energy exchange via work due to change of volume requires a closed system with movable system boundary, which manages to fulfil the postulation dV 0 ≠ The simplest case of such a system is a cylinder with a moving piston. The movement of a piston is directly connected to the change in volume of a closed system. The work of the machine results from piston force F and piston travel. In losses condition this work is equal to the work due to change in volume or in case of an incompressible fluid equal to the displacement work. As the piston force is based upon a static pressure difference, we are talking about a static principle of work.
The energy exchange between fluid and machine in an open system primarily happens via compression work. The required postulation dp 0≠ is achieved via the flow around the blade profile. The changes in magnitude and direction of the flow velocity effect change in the momentum of the fluid particles. The result is the fluid force F on the blade, if the blades are mounted on a rotor, the circumferential component of the resulting fluid force multiplied with the distance of the point of load incidence, which results from the turning, provides the work of the rotor. In losses condition this work is equal to the compression work. As the force on the blade is dependent on dynamic changes of the pressure, we are talking about a dynamic principle of work.
Operating principle piston engine / turbo engine FM-2.7
Comparison of: turbo machine piston machine Sketch
Main component rotor blades piston Operating principle
via blade shape, different velocities and therefore different pressures on suction side and pressure side of the blades and turning of the rotor→ exchange of mechanical work.
Compression or expansion of the fluid in the cylinder enclosed by the valves. The translative piston force is converted into a torque by the crank shaft.
Work flow steady periodic Mass forces rotatory translative and rotatory: pistons,
piston rods, crank, valves Balance of masses perfect; rotor can be huge → P high
n high imperfect and very difficult →n small
Inlet- and outlet cross section AE, AA, big, no valve necessarily AE, AA, small, in consequent of mass forces and construction size.
Mass flow / rotational speed n m big small Specific transmission of work P/ m small big Power unit size Pmax big small Power to weight ratio G / P (G = weight of machine)
small big
Mechanical losses Pm friction in bearings only friction of pistons, crankpins, and bearings
Mobility of the installation Dimensions of machines small big Flow velocity big small Process open closed Torque steady periodic Durability relatively big relatively small Hydraulic machine has to be filled sucking
Fluid machines
Piston machine Turbo machine Construction Complex oscillating moment of many
components Simple rotating movement of the rotor
Operating principle Periodic, therefore control valves are necessary. Thereby additional moving components with mass inertia and risk of malfunction. Flywheel necessary.
Steady, thereby no control valves, no flywheel.
Flow Unsteady, thereby accelerating and decelerating forces in the streaming in and out working medium. Adjusting a medium loud on the material.
Steady state, thereby high local load on the material.
Forces By static energy conversion. Thereby high pressure forces, heavy design, limited package size.
By dynamic energy conversion. Thereby smaller forces, lighter design, bigger packages are possible.
Main part Back and forth moving piston. Thereby vibrations caused by mass forces, higher friction losses, lower operating speed. By the lubrication of the piston oil reaches the working medium
Turning wheel. Thereby complete balancing of masses, smaller friction losses, higher operating speed, oil free working medium.
Efficiency Nearly constant starting from minimum load. Dependent of power. Thereby higher losses at partial-, or overload.
Power Independent of rotation speed (except combustion engines).
Dependent of load. Thereby higher losses at part or overload. Dependent on rotation speed. Thereby minimum speed of effective operation.
Operational risk Risk of excessive pressures in case of work machines. Cut out control is necessary.
Risk runaway at power machines. Trip device is necessary.
Fluid machines Comparison piston machines to turbo machines FM-2.9
Fluid machines Basic types of piston machines in comparison to turbo machines FM-2.10
Fluid machines
mechanical engineering electrical engineering
chemical industry food industry
production of plastics steel- a. light metal constr.
drowing and cold rolling processing of stone and soil
printing and duplicating light engineering and optics
foundry paper and cardbord conv.
rubber conversion leather conversion
fine ceramics wood machining
shipbuilding
0 20 40 60 80 10002468100 20 40 60 80100 120
employees in thousend
drive technology agricultural machinery
common ariation priting and paper machines food and packing machines
fittings pumps and compressors
power machines precision tools
textile machines mining machinery
oil hydraulics and pneumatics wood working machinery
smeltery and steel mill equipement sewing machines
industrial furnaces, oil/gas weighing machine testing machines
foundry machinery fire brigade equipement
loundry and dry cleaning machines welding machines
locomotives
Number of eployees and turover of a most important industries
Production of machines and number of employees according to sectors
industry sector
sale s
961 power plants are providing the public power demand:
lower than 1 MW
over than 1000 MW
number of compressor in use
Mobile compressors (maneuverable plants)
Stationary compressors (stationary plants)
market share of compressors
operators of compressors
a) piston compressors b) rotary screw compressor, lubricated c) rotary screw compressor, dry d) multicell compressor e) piston compressors, oil free
m ar
nu m
future now
nu m
pr es
so r
volume flow 2 4 6 8 10 12 14 16 18 20 220 30m3/min
nu m
pr es
so r
volume flow 2 4 6 8 10 12 14 16 18 20 220 30m3/min
future now
Fluid machines
FM-2.13
driven machines (work machine)
general examples general examples
ac generator
dc generator
centrifugal pump
gear pump
piston compressor
turbo compressor
Fluid machines
Symbols for single elements in diagrams of heat power machines (exerpt DIN 2481) FM-2.15
Fluid machines
density
0,5548 0,2140 9,304
- 13,67 9,12 0,025
sonic speed a0 332 315 337 1260 419 259
kg/ms 17,17 19,28 16,25 8,35 9,00 13,70
m2/s 1,373 1,394 1,364 9,587 1,206 0,716
cp0 m/s grd 1004,8 912,7 1038,3 14235 2055,8 820,6
heat conductivity λ0 N/s grd 0,0242 0,0243 0,0238 0,1756 0,0220 0,0143
gas constant R 287,04 259,78 296,75 4124,0 488,18 188,88
Κ 1,402 1,399 1,400 1,409 1,313 1,301-
Prandtl number Pr0 0,71 0,72 0,072 0,68 0,84 0,79-
numbers obtained for p=1 at and T=273 (t =0)
for kinematic viscosity ν=f(T,p) for air and steam (see Traupel).
substance R [ J / kg K ]
ethylen C2H4 319,3
helium He 2077,1
FM-2.17
Fluid Liquid Steam Gas water methanol mercury water air oxygen nitrogen hydrogen helium carbon
dioxyd Properties:
(reference state pb=1 bar (o) 1 atm(+),tb=0°C)
and steam (reference state pb=1 bar, tb=100°C)
O H2O
sκ
999,8
0,0001
-0,085
19,945
810
0,000
1,19
11226
13596
0,000
0,181
284067
0,589
1,016
2,879
1,320
1,275
1,007
3,674
1,397
1,410
0,987
3,677
1,398
1,234
0,987
3,678
1,402
0,0888
0,999
3,666
1,412
0,176
0,987
3,657
1,668
1,951
1,007
3,746
1,300
Speed of sound c m/s 1412 1185 1455 473 331 315 337 1261 973 258
Specific gas constant
R J/kg K - - - 461,5 287,2 259,8 296,8 4124 2077 188,9
Dynamic viscosity
179,3 506 -150
1,229 - -
890
1,710 - -
122
1,924 - -
125
1,672 - -
117
0,782 - -
-10
1,871 - -
86
1,367 - -
242 Kinematic viscosity 610⋅ν m²/s 1,794 1,009 0,124 20,85 13,41 13,46 13,37 88,11 104,8 7,006
Specific heat capacity
_ 4,217
1,001
2,428
1,226
0,140
1,134
2,032
1,341
1,006
1,402
0,917
1,399
1,041
1,402
14,19
1,410
5,193
1,667
0,827
1,309
Heat conductivity 10⋅λ W/m K 5,683 2,14 77,9 0,247 0,237 0,243 0,240 1,620 1,453 0,150 Temp. conductivity 610⋅α m²/s 0,135 0,109 4,09 20,59 18,49 18,54 18,46 128,8 157,0 9,286
Prandtl number
Pr TP
Fluid machines
Thermodynamic systems
0
0
0
Enclosed system across the boundary flows:
0
0
0
Open system (e. g. turbo machine) across the boundary flows:
0
0
0
Fluid machines
FM-3.2
State variables
Thermal state variables Pressure p Density υρ /1= Temperature T Caloritic state variables Specific internal energy u Specific enthalpy h Specific entropy s
A state variable exists, if the difference of its values for two arbitrary states depends only on these states and not on a kind of change in state (way a or b) 1 st characteristic: for each state variable a complete differential is existing:
dT T pdpdp

∂ ∂
+
∂ ∂
=
2 rd characteristic: The difference between two states can be declared directly in case of a state variable: 1
2
2
1 ydp =∫ υ
All of these variables are dependent on the way of the change in state.(from 1 to 2) 1 st characteristic: For these variables no complete differential is existing (e.g. dq) Because at this these non-interable integrals we are often writing: e.g. qd ., qδ 2 rd characteristic: According to an agreement these non-state- variable differences are written as declared here (e.g. 12y ). We are calling it “work”, or “work difference” in contrast to “energy” or “energy difference” as in case of state variables.
Fluid machines
p = p(T,v) FM-3.3.1
General information: A homogeneous system has in every equilibrium condition definite values of the thermal state variables v, p, and T. These state variables are called thermodynamic coordinates of the system. But its state is already defined if two of these coordinates are given. For every equilibrium condition the third one is a function of the two other coordinates. The thermal state equation can be geometrically displayed as a face. For example we can plot pressure p=p(v,T) as ordinate over the v,T-plain (s. TM-3.5). Every substance has its own thermal state equation. This is an additional, independent information on the properties of the matter. It cannot be obtained by thermodynamics but only by experiments or the molecular theory of matter. Because no satisfying molecular theory is existing, the thermal state equation has to be defined by experiments. Thermal state equation p = p (T,v) thermal total differencial state-equation
),( Tpvv = ⇒ 1⇒
vT p
∂ ∂ 1− dp
Requirement: determinant of matrix has to be zero, otherwise the right side ≠ 0 Solving the determinant D gives:
With the definitions: 1−=
T p
v T
α = isobar coefficient of expansion β = isochore coefficient of compressibility γ = isothermal coefficient of compressibility
11 =⋅⋅=
u = u(T,v) FM-3.3.2
General information: The internal energy is a property of the system. In case of a simple system it consists of kinetic energy of the inferior molecular movement and of the potential energy of the molecules caused by attraction and rejection potential between the molecules. The work which is supplied to the adiabatic system increases the average speed of the molecular distance. In this way work is performed against the attraction and rejection forces between the molecules. This interpretation of the internal energy already exceeds the horizon of thermodynamics. According to the 1st fundamental theorem the internal energy is a state variable like volume, pressure and temperature. As the state of equilibrium is already defined by two (independent) state variables, the specific energy u has to be a function of two state variables. A correlation is existing which is called the calorific state equation of the simple system. Calorific state equation u = u(T,v) As the internal energy is a state function, it has a complete differential:
dv v udT
=
∂ ∂
leads to a special synonym due to history: vc is called the specific heat capacity in case of constant volume.
p v sT
),( Tphh = FM-3.3.3
General information: A flowing medium, which crosses the boundary of an open system, carries besides specific internal energy u and kinetic and potential energy specific flow energy vp ⋅ into the open system. In the 1st fundamental theorem the terms u and vp ⋅ can be combined to the enthalpy .pvuh +≡ The sum pvu + in the energy equation is a state variable like the specific energy, pressure and specific volume. Thus it is a function of two independent state variables. The functional correlation is called calorific state equation like the relation ).,( Tvuu = Calorific state equation ),( Tphh = The complete differential of the enthalpy is:
dp p hdT
v p sT
),( vtss = FM-3.3.4
General information: The 2nd fundamental theorem provides a general information on the direction of themodynamic process. This information is expressed by diving all processes into reversible, Irreversible and impossible processes. The question is does one state variable exist, which changes in characteristic and different kind for reversible, irreversible and impossible processes? This state variable, which allowes a mathematic expression of the 2rd fundamental theorem via equations or ineguations, has been established by R. CLAUSIUS and has been called entropy (s) in 1865. In a adiabatic system irreversible and impossible processes differ very clearly in the U,V-diagram by the position of their final state compared to their starting state.
2' 2 2''
pd v
We can think about connecting the final states 2,2’ and 2’’ to the starting state 1 with a quasi-static change in state. For this change in state the differencial expression ds can be calculated. It got three different values: > 0 for change in state 12’, which means irreversible adiabatic process;
T pdvdUds +
= = 0…for change in state 12, which means reversible adiabatic process;
< 0 for change in state.12’’, which means impossible process in an adiabatic system.
),( vTss =
By the value of the differencial expression du+pdv we can in case of an infinitesimal process of an adiabatic system decide, if the process is reversible, irreversible or impossible. For finite processes in an adiabatic system the value of the integral
( ) ∫∫ +−=+ 2
1 pdVUUdpVdu
can only be defined, if the path of the change in state is known, because in case of finite quasi-static change in state it is not only dependent on starting and final state, but also on the kind of change in state. Mathematical pdVdU + is not a complete differencial; there is no function ),,( VUZZ = for which
pdVdUdV V ZdU

∂ ∂
+⋅
∂ ∂
=
holds. But we can transform every incomplete differencial like pdVdU + into a complete one by diving it by an integrating denominator (in this case ),( VUT :
T VdpdH
T pdVdUdS −
= +
= .
So there is a state variable S, the entropy. Its differencial
dV V SdU
∂ ∂
+
∂ ∂
=
Exist. With the specific entropy mSs /= we get the following differencial (connection of thermal and calorific state variables): (Gibbs equation)
T vdpdh
T pdvduds −
FM-3.4
Exponent n
Change in state
T=const. p=const.
FM-3.5
Fluid machines
For an ideal gas the differential alteration of the specific entropy is:
.)( p
dpR T dTTcds p −=
If his relation is integrated from any reference temperature to the temperature T and oT an average value is assumed for , it is: pc
.lnln),(
+−=
−= ∫∫
(A) In the -diagram the isobars are exponential curves. sT , The gradient of an isobar in the -diagram can be determined using the equation above, sT , if is set: 0=dp
. pp c
.)(
.)(
,
+=
+=
+=
(B) Integration of the relation shows that isochors are exponential functions in the -diagram, sT , too, if an average value for is assumed. pc
.lnln),( 0 00
s v vR
T TcvTs v ++= (C)
The gradient of an isochor can be determined by inserting 0=vd in equation (B):
. vvs c


∂ ∂
For an ideal gas: ,Rcc vp =− is and therefore vp cc >
vvpp s
∂ ∂ in the -diagram. sT ,
This means that the gradient of the isobar is lower then the gradient of the isochor at the same temperature. The functions of the isobars and isochors show that under the set conditions these lines are running equidistant in s -direction in the -diagram. sT , That means they can be harmonized by parallel translation in this direction.
Isobar and isochor in the T,s-diagram
FM-3.6.1
Δ T'
Δ T
Isobar and isochor in the -diagram Re temperature difference of two isobars sT , From the difference of the entropy of the two isobars and at constant temperature results using Ep Ap ET equation (A):
( ) ( ) ,lnlnlnln,, 0 00
pRpTspTs E EEEAEA =−
Analogical for the two isochors and at constant temperature : Ev Av ET
( ) ( ) .ln,, E
vRvTspTs =−
This entropy is independent of the temperature. Consequently it got a constant value for two isobars or two isochors. The distance of two isobars = const., =const. in T -direction can be calculated from the entropy Ep Ap difference :0=− EA ss
( ) ( ) .0lnlnlnln,, 0000
=+−−=− p pR
T Tc
p pR
pEEEAEA
Assuming that the specific heat capacity is the same for both temperature and , it is: ET AT
.lnln E
=
As the right side of the equation is a constant value for the assumptions that have been made, it is:
EA TconstT ⋅=
To define the distance of the isobars in T -direction, the temperature difference is EA TTT −=Δ Implemented with equation (D):
( ).1−=−⋅=Δ constTTTconstT EEA
The equation indicates, that the temperature difference between two isobars inT -direction is proportional to the temperature . That means the more proceeding in positive ET s -direction, the more distance between the isobars. Meaning:
.1 '' <=
FM-3.6.2
0dv .constv
n=1
s=const. ↑(Δs>0)v=
Fluid machines
The specific heat capacity of or 0 pc 0
vc of an ideal gas is a complex function of the temperature. Exception: single atom gases like He, Ne, Ar, Kr, Xe. In this case:
0
1
2
3
4
5
6
SO2
CO2
O2
N2
H2
H2S
H2O
K Temperature
related to the temperature T
Consequences out of the energy theorem:
dudWpdvdq diss =+−
12
2
for p =const.: dTcdwdq pdiss =+
12
2
mWQ diss =+ 1212 −
c 2 1
pcm/ ⋅2 1
The average value of the specific heat capacity between
and can be interpreted as the height of a rectangle 1t 2t with the width of , which is coextensive to the 12 tt − integral over . dtc ⋅
4.20
4.15
c 1
c 2
ve rit
ab le
s pe
ci fic
c ap
ac ity
kJ /k
specific heat capacity of water at p=1,101325 bar
Determination of the average value of pc for ideal gases FM-3.6b.1
Fluid machines
The averaged specific heat capacity is listed in tables for different substances between 0°C and t: −
pc
0 00 1
Out of this the averaged specific heat capacity can be calculated for any temperature, assuming that the heat, which is necessary to heat up from to , is equal to the difference between a heating of the body from 0°C to .
1t 2t
specific heat )t(c p
010212 QQQ −=
−
⋅−⋅ =
−− −
Example of air in a temperature range =40°C, =120°C pc −
1t 2t
°
−
Determination of the average value of pc for ideal gases FM-3.6b.2
Averaged specific heat capacity of ideal gases in kJ/kgK 0−
pc as a function of temperature in Celsius-scale
t in °C Air
-60 -40
Fluid machines
FM-3.6b.3
0= =
2p
isochor
2h
2
2p
t
Δv/v
5%
p
0 0%
Related deviation of the specific volume of air in comparison to the behaviour of ideal gas
Thermal state equation: Z RT pv
=
Benedict-Webb-Rubin a.o.
0 50 100 150 200 250 300 [bar] 400
p
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Z
300
100
150
c p =100bar
t []
200
isentropic exponent k(p,t) = -v p
.
of nitrogene
Fluid machines
R
l
l
l
l
Fluid machines
Saturation pressure In a saturated mixture of gas and steam the saturation pressure ps of the steam arises. It is dependent on the temperature and the overall pressure of the mixture: )p,T(pp ss = With the increasing of the overall pressure the saturation pressure of the steam is rising a little. But it is not dependent on the kind of gas. At minor overall pressures (p < 10 bar) the saturation pressure ps is only about less than 1% higher than the steam pressure pDs of pure steam. Therefore the pressure dependency of the saturation pressure can be neglected: )T(p)p,T(p Dss = Dew point If a non-saturated mixture of gas and steam is cooled down at constant pressure, the partial pressure pD of the steam stays constant. At a special temperature it gets pD=ps. The mixture is now saturated and the first condensate is formed. The dew point is reached: DTDss p)T(p)p,T(p ==
1000
100
10
t []
),]C[t ,,exp(]mbar[ps 25236 95406401619
+° −=




−⋅⋅=
16273150922657611
t [°C]
ps [mbar]
ρWs [g/m3]
xs [g/kg]
t [°C]
ps [mbar]
ρWs [g/m3]
xs [g/kg]
-40 0.1285 0.119 0.079 20 23.385 17.28 14.89 -30 0.3802 0.339 0.237 30 42.452 30.34 27.57 -20 1.0328 0.884 0.643 40 73.813 51.07 49.57 -10 2.5992 2.140 1.621 50 123.448 82.77 87.59 0 6.1115 4.848 3.825 60 199.330 129.64 154.84 10 12.2790 9.396 7.732 70 311.770 196.86 281.76
saturation pressure ps of steam, absolute humidity ρWs and water load xs of saturated wet air at an overall
pressure of p = 1000 mbar
Mixtures of steam and gas (wet air) FM-3.9.2
Fluid machines
The state of non-saturated wet air is defined by three state variables: and the steam content. p,T Partial density or absolute humidity: V/mww ≡ρ The mass of steam is related to the volume V of wet air. According to Dalton it is: wm
TR p
w w == ρ
At a defined temperature partial pressure of the steam and absolute humidity reach a maximum, if the wet air is saturated.
wp
The ratio of absolute humidity to its maximum at the actual air temperature is called relative humidity.
)T(p/por)T(/ swwsw =≡ ρρ If non-saturated wet air is cooled down, and wp wρ stay constant until the dew point is reached. At the dew point the wet air is just saturated and it is :)T(pp Tsw =
)T(p/)T(p sTs=
Relative humidity in %
Temp. in °C
Absolute humidity in g/m3
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1.0 1.1 1.3 1.5 1.7 1.9 2.1 2.4 2.7 3.1 3.5 3.9 4.4 4.9 5.5 6.1 6.7 7.5 8.3 9.2 10.2
1.5 1.7 1.9 2.2 2.5 2.9 3.2 3.6 4.1 4.6 5.2 5.9 6.5 7.4 8.2 9.2 10.1 11.3 12.5 13.9 15.3
1.9 2.2 2.5 2.9 3.3 3.8 4.3 4.8 5.5 6.2 6.9 7.8 8.7 9.8 10.9 12.1 13.5 15.0 16.7 18.6 20.4
2.4 2.8 3.2 3.6 4.1 4.7 5.3 6.0 6.8 7.7 8.6 9.7 10.9 12.2 13.6 15.2 16.9 18.8 20.9 23.2 25.6
2.9 3.3 3.8 4.4 5.0 5.6 6.4 7.3 8.2 9.2 10.4 11.7 13.1 14.6 16.3 18.2 20.3 22.6 25.0 27.8 30.7
3.4 3.9 4.5 5.1 5.8 6.5 7.4 8.5 9.5 10.7 12.1 13.6 15.3 17.1 19.1 21.3 23.7 26.3 29.2 32.4 35.8
3.9 4.4 5.1 5.8 6.6 7.5 8.5 9.7 10.9 12.3 13.8 15.5 17.4 19.6 21.8 24.3 27.1 30.1 33.4 37.0 40.9
4.3 5.0 5.7 6.5 7.4 8.5 9.6 10.9 12.3 13.8 15.6 17.6 19.6 22.0 24.5 27.4 30.5 33.9 37.6 41.6 46.0
4.8 5.6 6.4 7.3 8.3 9.4 10.7 12.1 13.6 15.4 17.3 19.4 21.8 24.4 27.3 30.4 33.8 37.7 41.7 46.2 51.1
pw
T
Fluid machines
The technically most important two-phase area of a pure substance is the area of wet steam. Wet steam is a mixture of boiling liquid and saturated steam, which are in the state of thermodynamic equilibrium. That means they got the same temperature and the same pressure. A boiling liquid is a liquid whose state is located on the boiling curve. Saturated steam is a gas whose state is located on the dew curve. Example: Change in state during heating and evaporation of water at constant pressure p = 1 bar. The illustration is not in scale; the specific volume of saturated steam at 1 bar is 1625 times higher than the specific volume of boiling liquid.
Schematic illustration of evaporation at constant pressure. States 1 and 5 are equal to states 1 and 5 in the t,v- diagram above.
Wet steam
Steam table fort he area of wet steam of H2O
Steam table of H2O (excerpt from Baer) FM-3.9.5
Fluid machines
mΔ = mass
system at time t
tEAEAEA hqa Δ=+
t/mm;T/QQ;T/WP .
gz/chh;pvuh t ++≡+≡ 22
. EA
q EAa
specific heat ) equals the change of total enthalpy of the fluid. EA
Fluid machines
Reversible processes Idealized border case Example: Carnot-Prozess
The Carnot-Process consists of 4 reversible sub-processes 1 to 2 adiabatic, reversible (isentropic) compression → )TT(c p 1212 −=α 2 to 3 Isothermal expansion in the turbine )p/pln(TRq 2322323 ⋅=−=→ α 3 to 4 Adiabatic, reversible (isentropic) expansion )TT(c)TT(c pp 213434 −=−=→ α 4 to 1 Isothermal compression in the compressor )p/pln(TR 41441 ⋅=→ α
just cancel out each other. Both machines are running useless against each other
3412 αα −=
Thermal efficiency (Carnot)
2
1
2
12 1 The efficiency is always ηC < 1, because T1 cannot sink below TU (ambience temperature). Although the Carnot-Process is reversible, the work supplied to the process cannot completely be transformed into mechanical work. So the supplied heat is released into the ambience as effective energy and heat. Irreversible processes: (all natural processes) 2nd fundamental theorem – the principle of irreversibility
012 2
1 ≥=∫ jTdsirr1212
sealed off (adiabatic)
0<adS
adSdS >
Reversible process: ai dSdS,dS == 0 Irreversible process: ai dSdS,dS => 0 Impossible process: ai dSdS,dS << 0
Change of entropy of thermodynamic process
FM-3.12
of energy
Exergy Anergy Comments Definitions
completely transformable only exergy -
Technically and economically most important and most valuable energy (high-quality energy = “Precious energy” = exergy)
Exergy is energy, which can be transformed into any other sort of energy at defined ambience.
II Thermal energy Internal energy Displacement work (for p < pu)
limited transformable exergy fraction anergy fraction
Transformability is limited by the 2nd fundamental theorem. Ratio of exergy and anergy depends on the sort of energy, the state of energy source (system) and the state of the ambience.
)anergy p,T(
→ ⇒→→
Every kind of energy consists of exergy and anergy. But one of these can be zero.
III e.g. the “internal energy” stored in the ambience
not transformable - only anergy
Perpetuum mobile of 2nd order (impossible to transform anergy into exergy)
Anergy is energy, which cannot be transformed into exergy.
Transformability of different sorts of energy FM-3.13
Art of energy
Internal energy
Heat difference
Static enthalpy
1 b
uuu ssTh −+
Exemplary illustration of the energy fractions of static enthalpy in the T,s-diagram
1
h1
s
T
=const. 1
=const.
=const.
exergy "e" anergy "b"
Art of energy
( )12
Turbo machine
fluid enthalpy
rotation
UEnergy transformation (indirectly in 2 steps) (The flow hits plane, rotary mounted single face) 1P
st P step: high enthalpy level kinetic energy →
in the fluid of the fluid ← 2P
nd P step: kinetic energy mechanical energy →
of the fluid at the shaft ←
Development of turbo machine
Flow hits double bent,
rotary mounted single plate
Considering only one half of a double bent plate (blade of a Pelton-Wheel), the flow can be realized in an elbow, too. Flow- and force ratio in case of an elbow: The direction change generates a force F. This force F could be used in a turbo-machine
Tr an
sf er
to te
ch ni
ca lly
p ra
ct ic
ab le
tu rb
o- m
ac hi
ne s
Multiplication of the plate-impact effect per rotary mounted spoke wheel
With the elbow force at the lever r: torque M = force F x lever r Torque is increased by the multi-spoke wheel. The elbow principle is replaced with continuously passed through, bladed wheels: elbow spoke wheel bladed wheel
torque = force x lever
FM-4.1
Driving turbo-machine (compressor wheel)
Fluid machines
Fluid machines
Turbo-machines are mechanically acting fluid energy machines, which are acting towards the dynamic operating principle and are conveying fluids from one to another pressure level. In this process energy is supplied to (pressure rise) or withdrawn (pressure drop) from the fluid. UClassification:
Criterion
Pressure rise:
Pressure progression in the (rotor)- blade channel
Equal pressure machine; over pressure machine
Fluid flow direction through the machine
Axial machine Diagonal machine Radial machine
UExample: “Turbo compressor” (Compressor, fan, blower, ventilator). Driven, thermal, axial/radial over pressure turbo-machine. e.g. for overcoming the friction in pipes (natural gas pipes, …) or for pressure rise in the consumer space
Typing of turbo-machines
System boundary III stage
c = average velocity z = geodetic height u = specific internal energy p = static pressure hBtB = total enthalpy
QB12B = heat supplied to (withdrawn from) the system PBWB = mechanical work supplied to (withdrawn from) the system via the shaft
Definition of system boudaries
mΔ = mass
system at time t
FM-4.5.1
t/mm;T/QQ;T/WP .
gz/chh;pvuh t ++≡+≡ 22
EA .
EA m/Qq = The sum of supplied energy (specific technical work ,
q EAa
specific heat ) equals the change of total enthalpy of the fluid. EA
Fluid machines
FM-4.5.2
.. ===
UEquation of continuity The energy equation is valid for friction-afflicted processes, too. Using Gibbs-Equation the Uloss of energy (dissipation j)U is implemented in the energy equation.
EAEAEAEA
zgcy ΔΔ ++
2
2 = reversible fraction of technical work transmitted to the fluid EAa
0>EAj = irreversible fraction UMechanical losses
mP = mechanical loss of power (bearing friction)
KP = Power at the coupler
tEAEAEA hqa Δ=+
+= 2
22
mK PPP += 0>mP Compressor, Pump: PP;P,P KK >>0 Turbine: PP;P,P KK <<0
Fluid machines
0≠q diabatic system
permeable to heat heat supply 0>q heat removal. 0<q
0=a flow process no exchange of technical work between fluid and machine parts
0≠a work process exchange of technical work between fluid and machine parts taken up work (compressor) 0>a delivery work (turbine) 0<a Via combinations of special cases the performance of turbo-machines can be characterized in good approximation:
00 =≠ q,a adiabatic work process Assumption for calculations of turbo-machines without heat exchanger. Because the heat exchanged with the ambience is due to the small surface very small in relation to the specific technical work ( ),a/q 1<<
using: tEtA hh a qaqa −=
+=+ 1
follows in good approximation: tEtA hha −≈ .
00 ≠≠ q,a diabatic work process Assumption for calculations of turbo-machines with heat supply or heat removal in the machine, e.g. gas turbines with internally cooled hollow blades or compressors with coat cooling
tEtA hhqa −=+
00 == q,a adiabatic flow process Assumption for calculations of all non-moving parts of an adiabatic turbo- machine, e.g. entrance lug, guide wheels, volutes, exit lug
0=− tEtA hh
00 ≠= q,a diabatic flow process Assumption for calculation of non-moving heat transferring parts, e.g. coolers, heat exchangers
tEtA hhq −=
FM-4.6.1
USpecial cases of the energy theorem
00 ≠≠ q;a diabatic work process cooled gt-blades → 00 =≠ q;a adiabatic work process turbo-machine → 00 ≠= q;a diabatic flow process → heat exchanger 00 == q;a adiabatic flow process → nozzle, guide wheel
adiabatic work process
000 =≠= z;a;q Δ
adiabatic flow process
000 === z;a;q Δ
0=zgΔ e.g. very small compared to ∫= vdpy (therm. fm)
=ρ constant incompressible fluid (pump)
Special cases of the energy theorem
FM-4.6.2
11 =
UPolytrop ratioU ν
UIsentropic exponent “k”
UPolytropic exponent “n”U
polytrop
)T(c )T(c
)T()T(k v
p== κ
UDefinition:U sv
UIsentropicU UPolytropicU
if ratio (exit pressure/entrance pressure) is consequently set for pressure ratio. EA p/p →
1>= EA p/pvπ1>= EA p/pvπ
Compressor
Tubine
s
Fluid machines
Changes in state in the h,s-diagram for compressor and turbine FM-4.9
Fluid machines
Enthalpy h
( ) aAAA TT
UInternal energyU u
E EAEA jhvdpy −== ∫ Δ
Driven TM (compressor, pump)
effective power employed power loss power employed power employed power
η − = =
P P P P m V P P Pη − , , = = = − P
coupler powerˆ mechanical powerˆ internal blade powerˆ
K
m
i
iP
EA
EA tA tE EA
h h q ja j a h h q
j h h q
tA tE EAEA
j h h q
h h qa a j h h q j
In te
rn al
e ffi
ci en
y g z z y g z z j
y g z z h q g z z
2 21 2 ( )
EA A E
a c c y g z z y g z z j
y g z z h q g z z
y g z z
With the definition of polytrop and ( )A Eg z zν ≡ dh dy 0⋅ − ≈
1 1 1
EA EA
For an adiabatic change in state it is:
1
1
Definition of efficiency in build of fluid machines FM-4.11
( )
EA EA A E
( ) s s s V s
y y h y j h q h
η q
= = = ΔT s
s s
Adiabatic change in state 1
1
( ) 1 ( )
( ) 1
h h y h
EA
EA K
h q
κ κ
κ κ
the pressure ratio for κ = 1,4.
The diagram shows that the isentropic efficiency of a compression is decreasing with the rising of the pressure ratio, if the polytropic efficiency stays constant. That means in case of a polytropic compression the isentropic efficiency, which has been calculated for the whole change in state, is always lower than the one calculated for any segment of the whole change in state, because the pressure ratio of the partial compression is in any case smaller than the one of the complete compression.
Correlation between isentropic and polytropic efficiency FM-4.12
a) UCompression
b) UExpansionU
Fluid machines
Efficiency factors of thermal TM displayed via area ratios in the T,s-diagram FM-4.13
1
polV
1
polV
ol itr
op ic
e xp
on en
t n
Fluid machines
Compressor Turbine
Efficiency factors of TM ( ),s polfη η π= for 0q = FM-4.14
V pol V
h E A h E A
Δ − = = Δ −
η Δ − = = Δ −
R c
η = =
Tds −=
p Tds =
( ) ln A
( ) ( )
FM-4.15
T T pds R= − = −
c lns
streamline=streampath streamtube=space, illustrated by streamlines which form the curve K streamsting=abstraction of the stramtube onto dA
E A mAA mA A AAc dA Am m cρ ρ= = =∫ UAxial machine cB1B=cons
number of blades z → ∞
cB1B=const. cB2B=const. Meridional cut (along axis of rotation)
Cut A – A
Unwinded cylindric cut
FM-4.16
Fluid machines
UAbsolute flow
URelative flow
UAbsolute flow
Absolute streamline in La Relative streamline in La (Watcher (1) and (3) (Watcher (2))
Course of an absolute streamline Streamline by the view of a fixed watcher Example: Compressor
Fluid machines
Fluid machines
Velocity vectors For a rotating impeller (or a moving system) the following kinematic constraint counts:
UVelocity vectors →
design. Relative velocity ω →
→
c (right-angled coordinate system ) , ,a u r
ac = Axial component
rc = Radial component
mc =Meridional component
uc = Circumferential component
a axial r radial ≡ ≡
u u
a a
Velocity triangles of a turbo-machine stage (axial design)
→ →
⇒ In this way the change in flow is completely described by two velocity triangles. For the guide wheel the velocity triangles are degenerated to straight lines because of the guide wheel condition u = 0: Meaning that relative and absolute velocities are identical for guide wheels. These velocity triangles are located in the gap planes 1 and 2 tangential to the Umedial plane ofU UflowU generated by the adjacent representative streamstrings. Although the velocity triangles like the tangential planes to the bent rotationally symmetric plane of flow are not located in one plane, the velocity triangles for in- and outflow are displayed in one single drawing plane! E.g. for a diagonal compressor stage:
Conceming the definition of velocity triangles FM-4.19.2
Notation of velocity triangles of one stage
Notation predominantly in compressor literature. Here the starting points of uB1B and uB2B coincide.
Notation predominantly in turbine literature. Here the triangle tips 1 and 2 coincide.
UCompressor
UTurbine
Axial component (parallel to machine axis) ac →
Radial component (normal to machine axis) rc →
Meridional component (resulting from cBaB and cBrB) mc →
Circumferential component uc →
Circumferential velocity of the rotor at the considered point u →
ω →
Fluid machines
FM-4.19.3
Medial streamline
Gap plane (1) (before the impeller)
1 1c u w= + 1 (inflow triangle)
Gap plane (2) (after the impeller)
2 2c u w= + 2 (outflow triangle)
Gap plane (3) (after the guide wheel)
3 3c w=
Fluid machines
Stage kBmaxB-1
Stage kBmaxB
FM-4.19.5
Theorem of angular momentum: = =Σ + + = +dD
res s i a R adt M M M M M M For circumferential direction:
( )= × =ud D dm u udt dt r dmc rc
⋅
u m u mA outflow A inflow
M D D r c c dA r c c dA
The inner power of a turbo-machine is:
( ) ( )
( )
u m ai u u
P M M M P u c c dA u c c dA M
u c c P u c u c Mm m
1 2and =a resM M m m
1
2 2 1 1⋅ = = −u u p a u c u c m
Fluid machines
Impeller outflow Impeller inflow
Fluid machines
Deriving Euler’s main equation from the theorem of angular momentum FM-4.20.2
12 2 1 12u u um c r c rm= ⋅ ⋅ − ⋅ ⋅ UWheel power:
Inflow angular momentum DB1B=mB1BrB1BcBu1
Outflow angular momentum DB2B=mB2BrB2BcBu2
u uP M=ω ⋅ UCircumferential work with: U 1 2 mm m= =
2 2 1 uP
Radial machine
UFirst form of the kinematic main equation:
2 2 1 1 ( )M t u umh a u c u c u cω⋅Δ = = = − = Δ u
UThe equation counts for: 1. Along the streamline “s” (theory of streamstrings). In case of rotational symmetry valid for parts of a stage. 2. For arbitrary fluids (oil, water, gas...) 3. For all kinds of turbo-machines 4. For flow involving friction, adiabatic processes.
1 1c u w= + 1
Inflow triangle
USecond form of the kinematic main equation: From the velocity triangles:
2 2 2 2 2( )u m uc c c w c u c= + ; = − + 2 m
1
2
According to the law of cosines:
2 2 2 2 2 1 1 1 1 1 1 1 1 1
2 2 2 22 2 2 2 2 2 2 2 22
2 cos 2 2 cos 2
u
u
w c u c u c u u w c u c u c u u
α α
Out of it:
2 2 2 2 2 2 2 1 2 1 2 1
2 2 21 1 1 1 1 1 1 1 12
2 2 21 2 2 2 22 2 2 2
2 2 2
c c u u w w t
u c c u w u c u c c u w u c
h a
α α
Outflow triangle
The velocity differences are divided to: 1. Change of kinetic energy in the impeller 2 1( )c c≠ 2. Deceleration or acceleration in the impeller 2 1(w w )≠ creates a static change in pressure 3. Fraction of change in energy due to the centrifugal- respectively the centripetal-potential due to a change in the circumferential velocity ( axial machine; radial machine) 1 2u u= → 1 2u u≠ → UExpansion
Absolute system
Relative system
Transformation of the kinematic main equation FM-4.21
UConversion of the state variables in the relative system Labeling in the relative system: (‘) or index rel. In the absolute and relative system it is: UEqual staticU state values
T T p p h h′ ′ ′= = =
UUnequal totalU state values
Fluid machines
Correlation between absolute and relative system FM-4.22.1
ph T ′≠ ≠ ≠′ ′ UThe conversion of state values at a given pointU between absolute and relative system is done using the isentropic afflux of the absolute respectively relative velocities:
0 2
RT c RT
p c p
κ κ
R w RT
T
2 2
2 2
2 2
2 2
κ κ
− / = − /′
− / = − /′ − −
′ ′/ − / = / − / − −
+ The above correlations are counting only for the defined point. For the calculation of state values Ualong a streamstringU the Uconstancy of rothalpyU has to be used (indes rel*):
2 22 2trelh h w u g z∗ ≡ + / − / + ⋅ = 2 2th h c g z const. ; = + / + ⋅
2 2 22 2 1 1trel t 2RT RT c wκ κ
κ κ ∗ = − / + / − /
− − u
2 2 21 ( 2 2 2)trel tT T w u c
R κ κ
⋅ const.
These correlations are counting for UadiabaticU flow involving friction. UIn a given pointU it can be converted like:
1 11
t t t t
p p T Tp T p p p T T T
κ κκ κ κκ κ
ρ ρ
Fluid machines
FM-4.22.2
abs 2 / 2c rel 2 / 2ω Kinetic
energy 2 / 2u
h
h
h
FM-4.23.1
D
( )23
0ucΔ <
UAccelerated flow
2 1
2 1
< Δ < >
Δ <
< Δ < =
Δ =
2 1
2 1
< Δ < <
Δ >
2 1
2 1
> Δ < <
Δ >
Decelerated flow
Expansion
Velocity triangles standardized to uB2BBB
Conversion of energy in a compressor and turbine stage (q=0) FM-4.25.1
Fluid machines
1 "y vdp= = Fläche 2fgc12
∫ ∫ ( )
UGuide wheel
2 'y vdp= = Fläche 3lmd23 ∫ ∫ ( )
3
2 '
UStage
13 ' "y y y≈ − = Fläche 3abc13
13 ' "j j j≈ − = Fläche 31ce3 in contrasUt
' " s s sh h hΔ + Δ >Δ
as can be seen from
' shΔ = Fläche 3BsB lmd 3BsB
" shΔ =Fläche 2BsB fgc 2Bs
shΔ =Fläche *3s abc *3s
UGuide wheel ' (h h h 1 0 0!)Δ = − < = Fläche 0lmc0
1
irr j Tds 0!)( )
2
irr j Tds 0!)( )
2
2
UCompressor
Indexing impeller 1-2(“) guide device 2-3(‘) UTheorem of energy for adiabatic compression
Fluid machines
FM-4.26
2 2 1 1
2 2 2 2 2 2 2 1 2 1 1 2
(1 Form)
Null
= − .
− − − = + + .
>
UAdiabatic stage (with 13zΔ = 0) UFirst main theorem U(energy conservation for an open system) UStage
13 12 2 2 3 1
2 2 2 1
−′′= Δ +
UImpeller
2 2 2 2 2 2 2 1 2 1 1 2
2 2 2 c c u u w wh a − − −′′Δ = − = +
UGuide device
2 2 2 3
2 c ch −′Δ =
USecond main equationU (combined with Gibbs’s equation Tds = dh – vdp
3 3
1 1 3 3
= = = Δ − >
′′ ′′= = = Δ − >
′ ′ ′= = = Δ − >
∫ ∫ ∫ ∫ ∫ ∫
UNotice:U η η′ ≠ ′′ 13y y y′′ ′≈ + 13j j j′′ ′≈ + UIt is always:
13 13h y j y y j j h h ′′ ′ ′′ ′Δ = + = + + + ′′ ′= Δ + Δ
UGrades of reaction Ukinematic g.o.r. (g.o.r. the enthalpy diff.)
h h h h h hρ ′′ ′′Δ Δ
′′ ′Δ Δ +Δ≡ = Polytropic g.o.r. (g.o.r. of the specific work of flow)
13
′′ ′+≡ ≈ y when η η′ ′′≠
UTurbine
Indexing guide wheel 0-1(‘) impeller 1-2(“) UTheorem of energy for adiabatic compression
02ta q h+ −= Δ UEuler’s main equation
2 2 1 1 2 2 2 2 2 2 2 1 2 1 1 2
(1 Form)
Null
= − .
− − − = + + .
<
UAdiabatic stage (with 02zΔ = 0) UFirst main theorem U(energy conservation for an open system) UStage
02 12 2 2 2 0
2 2 2 1
UImpeller
2 2 2 2 2 2 2 1 2 1 1 2
2 2 2 c c u u w wh a − − −′′Δ = − = +
USecond main equation U(combined with Gibbs’s equation Tds = dh – vdp
2 2
1 1 1 1
= = = Δ − >
′′ ′′= = = Δ − >
′ ′ ′= = = Δ − >
∫ ∫ ∫ ∫ ∫ ∫
UNotice:U η η′ ′′≠ 02y y y′′ ′≈ + 02j j j′′ ′≈ + UIt is always:
02 02h y j y y j j h h ′′ ′ ′′ ′Δ = + = + + + ′′ ′= Δ + Δ
UGrades of reaction Ukinematic g.o.r. (g.o.r. the enthalpy diff.)
h h h h h h≡ρ ′′ ′′Δ Δ= ′′ ′Δ Δ +Δ
Polytropic g.o.r. (g.o.r. of the specific work of flow)
02
′′ ′′ ′′ ′+≡ ≈ when η η′ ′≠ ′
l l Dδ=
The geometry of the channels course and the cross section of the flow in the gap area are described by the Ucross section of the flowU
Fluid machines
1
2
3
lA D l D D D
lA D l D D D lA D l D D
D
0
1
2
lA D l D D D
lA D l D D D
lA D l D D D
π π π δ
π π π δ
π π π δ
mD +=
D Dν ≡ (geometric stage-characteristic)
For Uaxial machinesU is A = 2 2 4 ( )m m l aD l D D Dππ π δ⋅ ⋅ = ⋅ ⋅ = − 2
i 2 2
4 ( )m m l aD l D D Dππ π δ= ⋅ ⋅ = ⋅ ⋅ = − 2 i
Consequently for Uaxial machinesU the correlation is: 1 1l ν νδ − += because
2 22 2 4 4(1 ) (1 )(1 )a aD D
m l lD π ππ δ ν δ ν ν⋅ = + ⋅ = − + (1 ) 1lν δ ν+ = − For Uradial machinesU and v 1δ are normally independent of each other:
2 21 1 1 1 1 1 1
1
2
3
0
le g A D l D D D
lA D l D D D lA D l D D
D lA D l D D
D
ν
ν
ν
ν
UCompressor stage
UTurbine StageU
1 1 1 2 2 2 3m u 3m uc c c, , ,m uu c c u c, , , , Characteristic of guide wheel 1 1 2 2 3 31
2 2 2 2 2 2 2
m u m u m uc c c c c cu u u u u u u u, , , , , ,
Characteristic of impeller
0 0 1 1 1 2 2m u m u m uc c c c u c c u2, , , , , , , Characteristic of impeller 0 0 1 1 2 21
2 2 2 2 2 2 2
m u m u m uc c c c c cu u u u u u u u, , , , , ,
Characteristic of guide wheel
h u
3
ψ
h u
0
ψ
2 2 2 2
h h
ψρ ψ
′′ ′′Δ = = Δ
h h
h h
ψρ ψ
′′ ′′Δ = = Δ
FM-4.28.1
Fluid machines
Dissipation (hence the efficiency) is UnotU defined by the velocity triangles! UThe stage- and blade row-efficiencyU are besides the velocity ratios dependent on the change of the velocities between the reference cuts. In addition the ratio of inertial- to viscous-force, shown by the Reynolds number (Re), the compressibility, shown by the mach number (Ma), and the characteristic of the fluid, expressed by the isentropic exponent (k), are relevant.
fη = [velocity ratio (amount of diversion a.s.o.); Geometry (channel course, profile); Reynolds number; mach number; characteristic of the fluid]
fη = vel. ratio; geom.; Re; Ma; k The definition of the efficiency for a turbine stage is the reciprocal of the definition for a compressor stage. The efficiency is always defined in a way that Tη < 1 and Vη <1. Due to this the Upolytropic stage-efficiencyU is:
UCompressor stageU ( )hif …η ρ= UTurbine StageU
Compressor | y h| |< | Δ
Guide wheel: y hη ′ ′Δ′ =
Impeller: y
hη ′′ ′′Δ′′ =
Stage: y y y y yh h
St h h h h h hη ′ ′′ ′ ′′+ ′ ′′Δ Δ ′ ′′Δ Δ Δ Δ Δ Δ = ≈ = +
(Equal sign stringently counts for ' "η η= ) (1 )V St h hη η ρ η ρ′ ′′= − +
Turbine y h| |>| Δ |
Impeller: h
yη ′′Δ ′′′′ =
T St
y y y y yh h h h h h h hη
′ ′′ ′ ′′+ ′ ′′Δ Δ ′ ′′Δ Δ Δ Δ Δ Δ = ≈ =
(Equal sign stringently counts for ' "η η= ) 1 1 1(1 )
T St T Th hη η ηρ ρ′ ′′= − +
Polytropic stage-efficiency as a stage-characteristic
Polytropic stage–efficiency as a stage-characteristic FM-4.28.2
Fluid machines
2 2 2 1 2 2 2 2
2 2 1 1 2 1 2 12 2 2 2 2 2 2 3 1 2 2 1 1 3 1 3 1
2
2 22 22 1 2 11 2 12 2 2 2 2
22 22 1 31 3 12 2 2
2 2 ( ) 2 2 ( )
c c u u m m u u
c c u u m m u u
c c c cuu u u u u u u u u
c c cuu u u u u u
a u c u c c c c ch h h u c u c c c c ca


2 1
2 2
2 1 3 11 2 2 2 2 2 22 2
2 22 2 3 12
cu u u
u
−
Repetition-stage 3 1
3 1 13

= =
=
( ) ( ) ( )
( )
2 22 22 1 2 11 2 12 2 2 2 2
2 11 2 2 2
2
2
c c c cuu u u u u u u u u
c cuu u u u u
h
h u u uψ = −
2 1
2 1
c c c cu u u u u u u
c cu u u
2 12 2u u uc c c h u u
−ψ Δ= =
in addition:
2 2 2 2 2
2 11 2 2 2
2
2
c c c cuu u u u u u u u u
c cuu u u u u
hρ − − −
h u u uψ = −
c c c cu u u u u u u
u c cu u
−ψ Δ= =
in addition:
( )2
2 2 2
cu u
cu u
ψρ ∞ − −= = − = − =
Radial compressor
Axial compressor
R ep
et iti
on s
ta ge
w ith
R ep
et iti
on s
ta ge
w ith
φ =c
on st
UInflow triangle
2 2
ψ ρ= = /
UOutflow triangle
UW → Blade without effect (ρUBUhVUBU = 0) The impeller and with it the stage performs no work, because the absolute velocities do not change. If the inflow-condition cBu1B = 0 is given up, a reasonable grade of reaction ρBhVB = 1,0 results. This is a popular concept in compressor design. UR → backwards-bent blade (0,5 < ρUBUhVUBU <1) The main part of the work transmitted from impeller to fluid results in an increasing of the potential energy of the fluid. A smaller part is converted from kinetic to potential energy in the guide wheel. This kind of layout is often used in the design of stationary compressors or pumps. UA → Radial- respectively axial ending blades (ρUBUhVUBU = 0,5) The supplied energy is half by half converted to potential and kinetic energy of the fluid. A typical application is compressor design, especially for half-open impellers with high circumferential velocities (high stage pressure-ratios). UV → Forward bent-blade (0 < ρUBUhVUBU < 0,5) The impeller transforms more than half of the energy into kinetic energy. Forward-bent blades are common in ventilator design.
Fluid machines
Correlation between ψh and ρh for compressors and pumps FM-4.30.1
Fluid machines
Velocity triangles and blade grids for axial compressor repetition-stages
( 3 1 1 2 1 0 = , = , =ucc c )
Fluid machines
Fluid machines
Axial compressor repetition stages ( )1 2, 1 3 1 2,u u c c = = =
for different characteristics ,h h and ( )/ 1s t =
FM-4.31 ρ
Fluid machines
2 2
and
2 are expressed in the following diagram.
P = Pumps K = Compressors Vt = Ventilators For particular groups of machines certain angles of flow (≈ blade-angels) are common, especially for the impeller outflow-angle βB2B. The screened areas give the characteristics (ρBhV, BψBhVB) and combinations of angles (αB2B, βB2B), which meet common designs and are leading to an adequate operating behavior.
Distinktive compressor stages for different characteristics FM-4.32
Fluid machines
2 11
2 2 2
2 2 2 2 2 2 1 1 2 1 2 12
2 2 2 2 2 2 1 1 2 0 2 02
2 22 2 2 1
22 2 2 0
2 2 ( ) 2 2 ( )
h c c u u m m u u
c c c cu u u u u u
c c cu u u u
a u c u c c c c ch h u c u c c c c ca
ρ
2 2 2 2 22
2 0
c c c cuh u u u u u
ψ
− Δ : ≡ = − − − + − <
Enthalpycharacteristic
Usual postulation Radial- or diagonal-compressor stage ( )1 2u u≠ Axial compressor stage ( )1 2u u u= =
Repetition-stage 2 0
2 0 02

= =
=
2 2 2 2 2
2 11
c cu u u u

2 u u h
2 2 2 2 2 12 1 2 1
2 1
c c u
c c c cu uψ − Δ = = Δ <
in addition:
1
ucu u u u u
ψρ
h c cu u
u u u uψ = − ⋅ = − ⋅
2 1
c c u
c c c cu uψ − Δ = = Δ <
in addition: 2 0uc =
2
1
ucu u u u u
ψρ
h c cu u
u u u u= − ⋅ = − ⋅
h c w u u
ψρ ∞= − ⋅ = + = − 12 u
h c uψ = −
Fluid machines
Radial compressor
Axial compressor
R ep
et iti
on s
ta ge
w ith
B 2B
R ep
et iti
on s
ta ge
w ith
Fluid machines
UAxial turbine repetition-stage U ( 2 0=c c ) It is reasonable to design multi-stage axial turbines preferably out of repetition-stages, as it is for axial compressors, too. For the axial turbine repetition-stage (uB1B = uB2B, cB2B = cB0B) all general relations for ψBhVB and ρBhVB are simplified, as they do for compressors.
Enthalpy-characteristic
u
2 2
u u
2 01 2 1 2 2
1 2
c w u u
i.e. : and
Dependent on the size of the kinematic grade of reaction special cases are distinguished in case of axial turbine stages. Those got notations according to their practical meanings.
Adiabatic axial turbine stages
c u u u η η η
= / ; = = = / ′ ′′= = = , ;
0 5 2 0
= = = /
= , ;
= − , ;
;
FM-4.34.2 ρ =
Fluid machines
0 865 0 0 8
c u u u
ρ ψ
FM-4.34.3
0 86 0 79
c u u u
4 36 h y
ψ