Compressible flows in turbomachinery (dimensional analysis only) Types of compressible flow turbomachinery Compressors (air and gas) Centrifugal Axial Turbines (gas and steam) Axial Inward flow radial (IFR)

Dimensional analysis of compressible flow turbomachinery Purpose To obtain compressible turbomachine characteristics and performance parameters Applicable to any turbomachine of the types listed on the previous slide Dimensional analysis of a compressor, for example Kinetic compressor characteristics and performance may be described ito the following 14 parameters:

,,,,,,,,,,,,, 02020101 RCCPTNDPTm VP

The dimensions of the above parameters can all be expressed ito the

dimensions of mass M, length L, time T and temperature . The number of parameters is large and would make interpretation of modelling- or test data impossible. The number of parameters may be

reduced, to a limited extent, by substitution of the ratio of to with , the ratio of CP to CV with , and substitution of R, T01 and T02 with RT01 and RT02. The following parameters are used:

,,,,,,,,,,02020101PRTNDPRTm

The number of parameters is reduced from 14 to 11. However, this is not the only advantage of the above substitutions. None of the above parameters have temperature as a dimension, as it has been eliminated by using RT01 and RT02, instead of R, T01 and T02. The number of dimensions is thus reduced from 4 to 3. It will be shown that the resulting number of dimensionless groups is still too large to obtain interpretable results. Further reductions are required; these will be carried out after the dimensional analysis.

Dimensionless numbers (-groups) are obtained by combination of the dimensions of the above parameters

kjihgfedcba PRTNDPRTm 02020101

(1)

gfedcba LTLTLTMLTLMTTLM 12121221000

kjih TMLTL 2122 (2)

The number of parameters is 11

The number of dimensions is 3 (i.e. M, L, T)

The number of -groups is 11 3 = 8* * The problem with interpretability of such a large number of -groups will be addressed later

Notes

For such a large number of dimensionless groups it is difficult to determine which exponents to express ito which

It is not advisable to select primary parameters

A method of elimination of parameters is followed here instead

1. and are already dimensionless; their exponents are equal to 1

121 jk (3)

11

k

(4)

The first dimensionless number is the compressor efficiency

22

j

(5)

The second dimensionless number is the ratio of specific heats of the gas

2. Parameters with the same dimensions are combined in -groups; their ratios are dimensionless (i) RT01 and RT02

3302013

hbRTRT (6)

33 222200 hb TLTLTL (7)

33330 hbhb (8)

with h3 arbitrary

333010202013

hhhTTRTRT

(9)

For h3 = 1

01023TT (10)

The third dimensionless number is the stagnation temperature ratio

(ii) P01 and P02

4402014

icPP (11)

44 2121000 ic TMLTMLTLM (12)

44440 icic (13)

with i4 arbitrary

444010202014

iiiPPPP

(14)

For i4 = 1

01024PP (15)

The fourth dimensionless number is the stagnation pressure ratio

(iii) and D

55

5

gdD (16)

550 gd LLL (17)

55550 gdgd (18)

with g5 arbitrary

5555

gggDD (19)

For g5 = 1

D 5 (20) The fifth dimensionless number is the relative surface roughness

The first five dimensionless numbers have been determined Only three dimensionless numbers are currently unknown In order to determine these dimensionless numbers, dimensions of the parameters are combined as follows

fedcba NDPRTm 0101

(21)

The dimensions of the -group in Equation (21) must be zero

fedcba TLTLTMLTLMTTLM 12121221000 (22)

3. The remaining three dimensionless numbers are formed as follows

fedcba NDPRTm

6number group- form

0101

(23)

fedcba NDPRTm

7number group- form

0101

(24)

8number group- form

0101

fedcba NDPRTm

(25)

(i) Dimensionless -group number 6

666601016

dcbaDPRTm (26)

6666 21221000 dcba LTMLTLMTTLM (27) Exponent equations

66 0 : caM (28a)

66620 : dcbL (28b)

666220 : cbaT (28c)

Solutions of exponent equations

266ab (29a)

66ac (29b)

66 2ad (29c) with a6 arbitrary

6

6666

2

01

012

01

2

016

a

aaaa

DP

RTmDPRTm

(30)

For a6 = 1

2

01

01

6DP

RTm

(31)

The sixth dimensionless number is the nondimensional mass flow

(ii) Dimensionless -group number 7

777701017

edcbNDPRT (32)

7777 12122000 edcb TLTMLTLTLM (33) Exponent equations

7 0 : cM (34a)

77720 : dcbL (34b)

777220 : ecbT (34c)

Solutions of exponent equations

07c (35a)

772bd (35b)

77 2be (35c) with b7 arbitrary

7

777

22

01220

01017

b

bbb

DN

RTNDPRT

(36)

For b7 = -

01

7RT

ND

(37)

The seventh dimensionless number is the nondimensional rotational speed

(iii) Dimensionless -group number 8

8888

018

fedcNDP (38)

8888 12121000 fedc TLTLTMLTLM (39) Exponent equations

8 0 : cM (40a)

88820 : fdcL (40b)

88820 : fecT (40c)

Solutions of exponent equations

08c (41a)

288de (41b)

288df (41c)

with d8 arbitrary

8

888 220

018

d

ddd DNNDP

(42)

For d8 = 2

2

8

ND

(43)

The eighth dimensionless number is the Reynolds number

8 dimensionless numbers have been determined for a compressor Although dimensional analysis has been applied to reduce the number of parameters, in the interest of improving interpretability of data, 8 dimensionless groups may still be considered to be too many

In order to further improve interpretability, -groups that do not significantly influence compressor performance are ignored Dimensionless groups that may be ignored are discussed next

(i) Reynolds numbers associated with flows through compressors are high, in the turbulent range. A change in Re would therefore not have a significant effect on performance, and is generally ignored in

compressor performance analyses and tests. (7 -groups left) (ii) Relative roughness has a constant effect at high Reynolds numbers. For a given machine the effect of changes in relative

roughness may therefore be ignored. (6 -groups left)

(iii) Ratio of gas specific heat constants is sensitive to changes in temperature, such as those occurring in gas turbine engines. The of the air flowing through the compressor differs from that of the combustion gas through the turbine. However, for a compressor,

OR for a turbine, a constant may be assumed with acceptable accuracy. Thermomachines 4A (TRM4A11) will address this in more detail next year (various formulations of efficiency will be

addressed as well). (5 -groups left)

The remaining 5 -groups are treated as follows (iv) Compressor efficiency couples temperature- and pressure ratios

Figure 1: T-s-diagramme of a compression process [CRSS, 2009]

The relationship among compressor isentropic efficiency C, stagnation temperature ratio T02/T01, stagnation pressure ratio P02/P01 and is

11

1

1

01

02

01

02

P

P

T

T

C (44a)

The relationship among compressor polytropic efficiency PC, stagnation temperature ratio T02/T01, stagnation pressure ratio P02/P01 and is

PC

P

P

T

T 1

01

02

01

02

(44b)

(v) Nondimensional mass flow and -rotational speed may be simplified for a given machine. With R and D fixed for a given machine, the mass flow- and speed ratios become:

010101

01

2

01

01;

T

N

RT

ND

P

Tm

DP

RTm

(45)

The above relationships are not dimensionless, but provide a logical and interpretable way to represent compressor performance, and may be graphically depicted by means of so-called compressor maps

(vi) Representation of compressor performance Stagnation pressure ratio and isentropic efficiency are given as follows ito nondimensional mass flow and -rotational speed

0101

01

1

01

02 ,T

N

P

Tmf

P

P

(46a)

0101

01

2 ,T

N

P

TmfC

(46b)

The 14 parameters initially selected have been combined into 4 groups of nondimensional parameters

For a known stagnation pressure ratio, efficiency and ratio of specific heats, stagnation temperature ratio may be obtained by means of Equations (44) Figures 1a & 1b, from CRSS [2009]*, show a typical centrifugal compressor map The figures show stagnation pressure ratio and isentropic efficiency vs nondimensional mass flow for various nondimensional rotational speeds (Normally pressure ratio and efficiency are shown on the same map, but for the sake of clarity they are shown separately here) * Saravanamuttoo, H.I.H., Rogers, G.F.C., Cohen, H., & Straznicky, P.V., Gas Turbine Theory, 6th Edition, Pearson Prentice Hall, Harlow, Essex, 2009

Figure 1a: Centrifugal compressor map (rP) [CRSS, 2009]

.

Figure 1b: Centrifugal compressor map (C) [CRSS, 2009]

Points of maximum efficiency

.

The map shown in Figure 1a is a family of stagnation pressure ratio vs nondimensional flow curves of a centrifugal compressor, similar to the pressure vs flow curves of pumps and fans, for various non-dimensional rotational speeds. The surge line is a locus of points of the lowest nondimensional flows for which compressor operation will be stable, for a range of nondimensional rotational speeds from 0,6 to 1,0. Flows to the right of the surge line, for any given rotational speed, are stable, while those to the left are unstable. (See also Figures 4.20 a & b on slide 67 of the lectures of week 4, for unstable ranges of kinetic pumps). A typical axial compressor map is shown in Figures 2a and 2b.

Figure 2a: Axial compressor map (rP) [CRSS, 2009]

.

Figure 2b: Axial compressor map (C) [CRSS, 2009]

.

Physical interpretation of nondimensional mass flow and -rotational speed Nondimensional mass flow [CRSS, 2009]

Faaa M

RT

C

RTD

RTAC

PD

RTAC

PD

RTm

0101

2

01

01

2

01

01

2

01

(47) (the axial flow Mach number) where Ca is absolute flow velocity and A is cross-sectional area

Nondimensional rotational speed [CRSS, 2009]

RM

RT

U

RT

ND

0101 (48)

(the rotational speed Mach number) where U is blade velocity

More dimensionless numbers

The flow coefficient of a compressor may be expressed as

RF

a MMU

C

(49)

The load- or temperature coefficient of a compressor may be expressed as

U

c

U

ccU

U

w

U

h

2

2

22

0 1

(50)

Compressible flow turbines Figure 3, from Hill & Peterson [1992]* shows a schematic of an axial turbine stage, e.g. that of a gas turbine Figure 4, from Hill & Peterson [1992]* shows the velocity triangles of an axial turbine stage * Hill, P.G., & Peterson, C.R., Mechanics and Thermodynamics of Propulsion, 2nd Edition, Addison Wesley, Reading, MA, 1992

Figure 3: Section through an axial turbine stage [Hill & Peterson, 1992]

Figure 4: Axial turbine stage velocity triangles [Hill & Peterson, 1992]

Turbine characteristics A turbine map, from Dixon [2005]*, is shown in Figure 5. Turbine maps are sometimes plotted on inverse axes, i.e. with nondimensional mass flow versus stagnation pressure ratio, as shown in Figure 6 [CRSS, 2009]. In Figure 6, the pressures and temperatures have the subscripts of turbines, i.e. T03 & P03 at entry and T04 & P04 at exit; P03 > P04 (more about that in TRM4A11). * Dixon, S.L., Fluid Mechanics and Thermodynamics of Turbomachinery, 5th Edition, in SI/Metric units, Elsevier Butterworth - Heinemann, Burlington, MA, USA, 2005

Figure 5: Turbine map (rP) [Dixon, 2005]

Figure 6: Turbine map (rP & ) on inverted axes [CRSS, 2009]

At low mass flows and pressure ratios, flow increases with an increase in pressure ratio. However, further increases in pressure ratio gradually saturate the flow, until it chokes. At higher rotational speeds choking occurs at higher pressure ratios. When the flow is choked, any increase in pressure ratio fails to increase the flow.

The degree of reaction R, or simply reaction of a turbine is the ratio of static enthalpy change in the rotor to the stagnation enthalpy change in the whole stage, i.e. in the rotor and stator

U

ww

hh

hhR

2

32

0301

32

(51)

It is clear from Equation (51) above that R is dimensionless

Effect of the shapes of the velocity triangles on degree of reaction The velocity triangles of the axial turbine stage shown in Figure 4 are symmetric. Symmetric velocity triangles are characteristic of a turbomachine with a 50% degree of reaction. The velocity triangles of turbomachines with degrees of reaction other than 50% are skewed to one side. Velocity triangles of axial turbine stages with zero-, 50%- and 100% reactions are shown in Figures 7a, 7b and 7c respectively. For a 0% reaction stage, the relative velocities at entry and exit, i.e. w2 and w3 are equal. For a 50% reaction stage, c3 = w2 and w3 = c2. For a 100% reaction stage, the absolute velocities at entry and exit, i.e. c2 and c3 are equal.

Figure 7a: Velocity triangles of a zero reaction axial turbine stage

(3 = 2, w3 = w2) [Dixon, 2005] (rotated through 90 relative to Figure 4)

Figure 7b: Velocity triangles of a 50% reaction axial turbine stage

(3 = 2, 2 = 3, c3 = w2, w3 = c2) [Dixon, 2005] (rotated through 90 relative to Figure 4)

Figure 7c: Velocity triangles of a 100% reaction axial turbine stage

(3 = 2) [Dixon, 2005] (rotated through 90 relative to Figure 4)

Velocity triangles of an axial compressor stage with a reaction of smaller than 50% are shown in Figure 8a, while those of an axial compressor stage with a reaction of larger than 50% are shown in Figure 8b. Velocity triangles of an axial compressor stage with a 50%

reaction are symmetric, with 1 = 2 (not shown).

Figure 8a: Velocity triangles of a an axial compressor stage with

a reaction of smaller than 50% (2 < 1) [Dixon, 2005]

c1

c2 w1

w2

Figure 8b: Velocity triangles of a an axial compressor stage with

a reaction of larger than 50% (2 > 1) [Dixon, 2005]

c1

w1 c2

w2

Effect of degree of reaction on turbine performance

Figure 9 [Hill & Peterson, 1992] shows load coefficient ( = h0/U2) versus flow coefficient ( = Ca/U) for various stator outlet angles 2 (in a typical range of 60 to 75) and reaction R ranging from 0 to 1, of an

axial turbine with zero exit swirl (c3 = 0).

It can be seen that load increases with a decrease in reaction R, for any given 2. However, this is at the cost of a higher flow coefficient , and therefore a higher axial velocity Ca for the same blade speed U, with accompanying flow losses.

Figure 9: vs for R ranging from 0 to 1 and 2 from 60 to 75

Steam turbines The field of turbomachinery is too vast to cover in a part of a semester. The MKE3B21 curriculum therefore has to be demarcated and limited to selected turbomachines. Turbomachines most used are pumps, fans, hydraulic-, gas-, steam- and wind turbines. Theory of the first three types, i.e. pumps, fans and hydraulic turbines has been covered in the lectures in this subject. Gas turbines will be covered in more detail in TRM4A11. Wind turbines are omitted. Steam turbine theory is covered in more detail in Thermomachines 4B.