IEC 62097:2009

Hydraulic machines, radial and axial - Performance conversion method from model to prototype

IEC 62097:2009 is applicable to the assessment of the efficiency and performance of prototype hydraulic machine from model test results, with consideration of scale effect including the effect of surface roughness. This document is intended to be used for the assessment of the results of contractual model tests of hydraulic machines.

Machines hydrauliques, radiales et axiales - Méthode de conversion des performances du modèle au prototype

La CEI 62097:2009 est applicable à l'évaluation de l'efficacité et de la performance de machines hydrauliques prototypes à partir d'essais sur modèle d'essais avec la considération des effets d'échelle incluant l'effet de rugosité de surface. Ce document est prévue pour être employée pour l'évaluation des résultats des essais contractuels sur modèle réduit de machines hydrauliques.

General Information

• Status: Published
• Publication Date: 10-Feb-2009
• ICS: 27.140 - Hydraulic energy engineering
• Technical Committee: TC 4 - Hydraulic turbines
• Drafting Committee: WG 18 - TC 4/WG 18
• Current Stage: DELPUB - Deleted Publication
• Start Date: 08-Jan-2019
• Completion Date: 26-Oct-2025

Standards Content (Sample)

IEC 62097 ®
Edition 1.0 2009-02
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Hydraulic machines, radial and axial – Performance conversion method from
model to prototype
Machines hydrauliques, radiales et axiales – Méthode de conversion des
performances du modèle au prototype

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IEC 62097 ®
Edition 1.0 2009-02
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Hydraulic machines, radial and axial – Performance conversion method from
model to prototype
Machines hydrauliques, radiales et axiales – Méthode de conversion des
performances du modèle au prototype

INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
XC
CODE PRIX
ICS 27.140 ISBN 978-2-88910-619-6
– 2 – 62097 © IEC:2009
CONTENTS
FOREWORD.5

INTRODUCTION.7

1 Scope.9

2 Normative references .9

3 Terms, definitions, symbols and units .9

3.1 System of units .9

3.2 List of terms .9

3.2.1 Subscripts’ list .9
3.2.2 Terms, definitions, symbols and units .10
4 Scale-effect formula .13
4.1 General .13
4.1.1 Scalable losses .13
4.1.2 Basic formulae of the scale effect on hydrodynamic friction losses .15
4.2 Specific hydraulic energy efficiency.17
4.2.1 Step-up formula.17
4.2.2 Roughness of model and prototype.19
4.2.3 Direct step-up for a whole turbine .22
4.3 Power efficiency (disc friction).23
4.3.1 Step-up formula.23
4.3.2 Roughness of model and prototype.23
4.4 Volumetric efficiency .24
5 Standardized values of scalable losses and pertinent parameters .24
5.1 General .24
5.2 Specific speed.25
5.3 Parameters for specific hydraulic energy efficiency step-up.25
5.4 Parameters for power efficiency (disc friction) step-up.26
6 Calculation of prototype performance .27
6.1 General .27
6.2 Hydraulic efficiency .27
6.3 Specific hydraulic energy .28
6.4 Discharge.28
6.5 Torque .29

6.6 Power.29
6.7 Required input data .30
7 Calculation procedure.31
Annex A (informative) Basic formulae and their approximation.33
Annex B (informative) Scale effect on specific hydraulic energy losses of radial flow
machines .43
Annex C (informative) Scale effect on specific hydraulic energy losses of axial flow
machines [10] .63
Annex D (informative) Scale effect on disc friction loss .70
Annex E (informative) Leakage loss evaluation for non homologous seals .76
Bibliography.83

Figure 1 – Basic concept for step-up considering surface roughness .16

62097 © IEC:2009 – 3 –
Figure 2 – IEC criteria of surface roughness given in Tables 1 and 2 .20

Figure 3 – Francis Runner blade and fillets .21

Figure 4 – Runner blade axial flow.22

Figure 5 – Guide vanes.22

Figure 6 – Calculation steps of step-up values.32

Figure A.1 – Flux diagram for a turbine .34

Figure A.2 – Flux diagram for a pump .35

Figure B.1 – Loss coefficient versus Reynolds number and surface roughness .44

Figure B.2 – Different characteristics of λ in transition zone.45
Figure B.3 – Representative dimensions of component passages .48
Figure B.4 – Relative scalable hydraulic energy loss in each component of Francis
turbine .54
Figure B.5 – Relative scalable hydraulic energy loss in each component of pump-
turbine in turbine operation .55
Figure B.6 – Relative scalable hydraulic energy loss in each component of pump-
turbine in pump operation .56
Figure B.7 – κ and κ in each component of Francis turbine.57
uCO dCO
Figure B.8 – κ and κ in each component of pump-turbine in turbine operation.58

uCO dCO
Figure B.9 – κ and κ in each component of pump-turbine in pump operation .59
uCO dCO
Figure B.10 – d and d for Francis turbine .60
ECOref Eref
Figure B.11 – d and d for pump-turbine in turbine operation .61

ECOref Eref
Figure B.12 – d and d for pump-turbine in pump-operation .62

ECOref Eref
Figure C.1 – δ for Kaplan turbines .66
Eref
Figure D.1 – Disc friction loss ratio δ .72
Tref
Figure D.2 – Dimension factor κ .74
T
Figure D.3 – Disc friction loss index d .75
Tref
Figure E.1 – Examples of typical design of runner seals (crown side) .78
Figure E.2 – Examples of typical design of runner seals (band side) .79

Table 1 – Maximum recommended prototype runner roughness for new turbines (μm).21
Table 2 – Maximum recommended prototype guide vane roughness for new turbines
(μm).22

Table 3 – Permissible deviation of the geometry of model seals from the prototype .24
Table 4 – Scalable loss index d and velocity factor κ for Francis turbines.25

ECOref uCO
Table 5 – Scalable loss index d and velocity index κ for pump-turbines in
ECOref uCO
turbine operation.26
Table 6 – Scalable loss index d and velocity index κ for pump-turbines in
ECOref uCO
pump operation.26
Table 7 – Scalable loss index d and velocity factor κ for axial flow machines .26

ECOref uCO
Table 8 – Required input data for the calculation of the prototype performance .30
Table B.1 – d and κ for step-up calculation of whole turbine .51
Eref u0
Table B.2 – Criteria for the surface roughness for the application of the direct step-up
formula .52

– 4 – 62097 © IEC:2009
d
EST
Table C.1 – Ratio of for Francis turbines and pump-turbines .68
δ
EST
Table C.2 – Parameters to obtain Δ for axial flow machines .68
ECO
62097 © IEC:2009 – 5 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION

_____________
HYDRAULIC MACHINES, RADIAL AND AXIAL –

PERFORMANCE CONVERSION METHOD
FROM MODEL TO PROTOTYPE
FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
international co-operation on all questions concerning standardization in the electrical and electronic fields. To
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6) All users should ensure that they have the latest edition of this publication.
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Publications.
8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
International Standard IEC 62097 has been prepared by technical committee 4: Hydraulic
turbines.
The text of this standard is based on the following documents:
FDIS Report of voting
4/242A/FDIS 4/243/RVD
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
This publication contains attached files in the form of Excel file. These files are intended to be
used as a complement and do not form an integral part of this publication.

– 6 – 62097 © IEC:2009
The committee has decided that the contents of this publication will remain unchanged until

the maintenance result data indicated on the IEC web site under “http://webstore.iec.ch” in the

data related to the specific publication. At this date, the publication will be

• recommended;
• withdrawn;
• replaced by a revised edition;

• or amended.
62097 © IEC:2009 – 7 –
INTRODUCTION
0.1 General remarks
This International Standard establishes the prototype hydraulic machine efficiency from model

test results, with consideration of scale effect including the effect of surface roughness.

Advances in the technology of hydraulic turbo-machines used for hydroelectric power plants

indicate the necessity of revising the scale effect formula given in 3.8 of IEC 60193. [1] The

advance in knowledge of scale effects originates from work done by research institutes,

manufacturers and relevant working groups within the organizations of IEC and IAHR. [1 - 7]

The method of calculating prototype efficiencies, as given in this standard, is supported by
experimental work and theoretical research on flow analysis and has been simplified for
practical reasons and agreed as a convention. [8 – 10] The method is representing the
present state of knowledge of the scale-up of performance from model to a homologous
prototype.
Homology is not limited to the geometric similarity of the machine components, it also calls for
homologous velocity triangles at the inlet and outlet of the runner/impeller. [2] Therefore,
compared to IEC 60193, a higher attention has to be paid to the geometry of guide vanes.
According to the present state of knowledge, it is certain that, in most cases, the formula for
the efficiency step-up calculation given in the IEC 60193 and earlier standards, overstated the
step-up increment of the efficiency for the prototype. Therefore, in the case where a user
wants to restudy a project for which a calculation of efficiency step-up was done based on any
previous method, the user shall re-calculate the efficiency step-up with the new method given
in this standard, before restudying the project of concern.
This standard is intended to be used mainly for the assessment of the results of contractual
model tests of hydraulic machines. If it is used for other purposes such as evaluation of
refurbishment of machines having very rough surfaces, special care should be taken as
described in Annex B.
Due to the lack of sufficient knowledge about the loss distribution in Deriaz turbines and
storage pumps, this standard does not provide the scale effect formula for them.
An excel work sheet concerning the step-up procedures of hydraulic machine performance
from model to prototype is indicated at the end of this Standard to facilitate the calculation of
the step-up value.
0.2 Basic features
A fundamental difference compared to the IEC 60193 formula is the standardization of
scalable losses. In a previous standard (see 3.8 of IEC 60193:1999 [1]), a loss distribution
factor V has been defined and standardized, with the disadvantage that turbine designs which
are not optimized benefit from their lower technological level.
This is certainly not correct, since a low efficiency design has high non-scalable losses, like
incidence losses, whereby the amount of scalable losses is about constant for all
manufacturers, for a given type and a given specific speed of a hydraulic machine.
This standard avoids all the inconsistencies connected with IEC 60193:1999. (see 3.8 of [1])
A new basic feature of this standard is the separate consideration of losses in specific
hydraulic energy, disc friction losses and leakage losses. [5], [8 – 10]
—————————
Numbers in square brackets refer to the bibliography.

– 8 – 62097 © IEC:2009
Above all, in this standard, the scale-up of the hydraulic performance is not only driven by the

dependence of friction losses on Reynolds number Re, but also the effect of surface

roughness Ra has been implemented.

Since the roughness of the actual machine component differs from part to part, scale effect is

evaluated for each individual part separately and then is finally summed up to obtain the

overall step-up for a complete turbine. [10] For radial flow machines, the evaluation of scale

effect is conducted on five separate parts; spiral case, stay vanes, guide vanes, runner and

draft tube. For axial flow machines, the scalable losses in individual parts are not fully

clarified yet and are dealt with in two parts; runner blades and all the other stationary parts
inclusive.
The calculation procedures according to this standard are summarized in Clause 7 and Excel
sheets are provided as an Attachment to this standard to facilitate the step-up calculation.
In case that the Excel sheets are used for evaluation of the results of a contractual model
test, each concerned party shall execute the calculation individually for cross-check using
common input data agreed on in advance.

62097 © IEC:2009 – 9 –
HYDRAULIC MACHINES, RADIAL AND AXIAL –

PERFORMANCE CONVERSION METHOD
FROM MODEL TO PROTOTYPE
1 Scope
This International Standard is applicable to the assessment of the efficiency and performance

of prototype hydraulic machine from model test results, with consideration of scale effect
including the effect of surface roughness.
This standard is intended to be used for the assessment of the results of contractual model
tests of hydraulic machines.
2 Normative references
The following referenced documents are indispensable for the application of this document.
For dated references, only the edition cited applies. For undated references, the latest edition
of the referenced document (including any amendments) applies.
IEC 60193:1999, Hydraulic turbines, storage pumps and pump-turbines – Model acceptance
tests
3 Terms, definitions, symbols and units
3.1 System of units
The International System of Units (SI) is used throughout this standard. All terms are given in
SI Base Units or derived coherent units. Any other system of units may be used after written
agreement of the contracting parties.
3.2 List of terms
For the purposes of this document, the terms and definitions of IEC 60193 apply, as well as
the following terms, definitions, symbols and units.
3.2.1 Subscripts’ list
Term Symbol Term Symbol
model M component CO
prototype P
specific energy E spiral case SP
volumetric Q stay vane SV
torque or disc friction T guide vane GV
in general term
represented by
reference ref runner RU
CO
hydraulic diameter d draft tube DT
velocity u stationary part ST
hydraulic h
optimum point opt
off design point off
– 10 – 62097 © IEC:2009
3.2.2 Terms, definitions, symbols and units

Term Definition Symbol Unit
Radial flow machines Francis turbines and Francis type reversible pump-turbines - -

Axial flow machines Kaplan turbines, bulb turbines and fixed blade propeller - -

turbines
Reference diameter Reference diameter of the hydraulic machine D m

(see Figure 3 of IEC 60193)
Hydraulic diameter 4 times sectional area divided by the circumference of the m
d
h
section
Sand roughness Equivalent sand roughness [11] k m
S
Arithmetical mean Deviation of the surface profile represented by the Ra m
roughness arithmetical mean value
–2
Acceleration due to Local value of gravitational acceleration at the place of g m s
gravity testing as a function of altitude and latitude (see
IEC 60193)
–3
Density of water Mass per unit volume of water (see IEC 60193) kg m
ρ
Dynamic viscosity A quantity characterizing the mechanical behaviour of a μ Pa s
fluid
2 –1
Kinematic viscosity Ratio of the dynamic viscosity to the density of the fluid. ν m s
Values are given as a function of temperature. (see
IEC 60193)
3 –1
Discharge Volume of water per unit time flowing through any section Q m s
in the system
–1
Mass flow rate Mass of water flowing through any section of the system kg s
(ρ Q)
per unit time
3 –1
Discharge of machine Discharge flowing through the high pressure reference Q m s
section
3 –1
Leakage flow rate Volume of water per unit time flowing through the runner q m s
seal clearances
3 –1
Net discharge Volume of water per unit time flowing through Q m s
m
runner/impeller. It corresponds to Q -q in case of turbine
and Q +q in case of pump.
–1
Mean velocity Discharge divided by the sectional area of water passage v m s
–1
Peripheral velocity Peripheral velocity at the reference diameter u m s
–1
Rotational speed Number of revolutions per unit time n S
–1
Specific hydraulic Specific energy of water available between the high and E J kg
energy of machine low pressure reference sections 1 and 2 of the machine

taking into account the influence of compressibility (see
IEC 60193)
–1
Specific hydraulic Turbine: Net specific hydraulic energy working on the E J kg
m
energy of runner
runner/impeller
Pump: Specific hydraulic energy produced by the impeller
–1
E J kg
m
–1
Specific hydraulic Specific hydraulic energy loss in stationary part which E J kg
Ls
energy loss in includes both friction loss and kinetic loss

stationary part
–1
Specific hydraulic Specific hydraulic energy loss in runner/impeller which E J kg
Lm
energy loss in includes both friction loss and kinetic loss

runner/impeller
–1
Friction loss of Specific hydraulic energy loss caused by the friction on the E J kg
Lf
specific hydraulic surface of water passages
energy
62097 © IEC:2009 – 11 –
Term Definition Symbol Unit
–1
Kinetic loss of Specific hydraulic energy loss caused by the hydraulic E J kg
lk
specific hydraulic phenomena other than surface friction, such as turbulence,

energy separation of flow, abrupt change of water passage, etc.

Turbine net head or H = E / g H m

pump delivery head
Turbine output or The mechanical power delivered by the turbine shaft or to P W

pump input the pump shaft, assigning to the hydraulic machine the

mechanical losses of the relevant bearings and shaft seals
(see Figures A.1 and A.2)
Hydraulic power The power available for producing power (turbine) or P W
h
imparted to the water (pump)
P = E (ρQ )
h 1
Mechanical power of The power transmitted through the coupling between shaft P W
m
runner/ impeller and runner (impeller).
Power of Turbine: Power produced by the runner corresponding to P W
r
runner/impeller
E (ρQ ) or P +P
m m m Ld
Pump: Power produced by the impeller represented by
P W
r
E (ρQ ) or P -P
m m m Ld
Disc friction loss Loss power caused by the friction on the outer surface of P W
Ld
the runner/impeller
Bearing loss power Loss power caused by the friction of the shaft bearing and P W
Lm
shaft seal
Runner/impeller Torque transmitted through the coupling of the T N m
m
torque runner/impeller and the shaft corresponding to the
mechanical power of runner/impeller, P .
m
Hydraulic efficiency -
Turbine: η =P /P       Pump: η =P /P η
h m h h h m h
Specific hydraulic Turbine: η = E /E       Pump: η = E /E η -
E m h E h m E
energy efficiency
(see Figures A.1 and A.2)
Volumetric efficiency -
Turbine: η = Q /Q      Pump: η = Q /Q η
Q m 1 Q 1 m Q
(see Figures A.1 and A.2)
Power efficiency (disc -
Turbine : η = P /P       Pump : η = P /P η
T m r T r m T
friction efficiency)
(see Figures A.1 and A.2)
Mechanical efficiency Turbine: η = P/P        Pump: η = P /P η -
m m m m m
(see Figures A.1 and A.2)
Efficiency step-up Difference between efficiencies at two hydraulically similar Δη -
operating conditions
Efficiency step-up Ratio of efficiency step-up against model efficiency
Δ
ratio
Δη
Δ =
η
M
Reynolds number Reynolds number of the machine Re -
Re = D u / ν
Reynolds number of Re -
Re = d v / ν
d
d h
component passage
Friction loss Friction loss coefficient for a pipe. λ -
coefficient for pipe
E
flow
Lf
λ =
L v
d 2
where d pipe diameter (m)
L pipe length (m)
– 12 – 62097 © IEC:2009
Term Definition Symbol Unit
Friction loss Friction loss coefficient for a flat plate. C -

f
coefficient for a flat
E
plate
Lf
C =
f
BL w
Q 2
where B width of a flat plate (m)

L length of a flat plate (m)
Q discharge passing by the plate (m /s)

w relative flow velocity (m/s)

Disc friction loss Friction loss coefficient for a rotating disc C -
m
coefficient
P
Ld
C =
m
π
ρ n D
d
where
D diameter of the rotating disc (m)
d
Relative scalable Scalable specific hydraulic energy loss divided by E, which δ -
E
hydraulic energy loss is dependent on Reynolds number and roughness (in most
cases, it is represented in %)
δ = E /E
E lf
Relative non-scalable Non-scalable specific hydraulic energy loss divided by E, -
δ
ns
hydraulic energy loss which remains constant regardless of Reynolds number
and roughness
δ = E /E
ns lk
Reference scalable δ value for a model with smooth surface operating at a δ -
E Eref
hydraulic energy loss
reference Reynolds number Re = 7 ×10
Reference scalable δ for each component passage δ -
Eref ECOref
hydraulic energy loss
in component
passage
Relative disc friction -
δ
Disc friction loss P divided by P
T
Ld m
loss
P
Ld
δ =
T
P
m
Reference disc -
δ
δ value for a model with fairly smooth surface operating at
Tref
T
friction loss
a reference Reynolds number Re = 7 × 10

Flow velocity factor Ratio of the maximum relative flow velocity in each κ -
uCO
for each component component passage against the peripheral velocity u
passage
v
CO
κ =
uCO
u
Dimension factor for Ratio of the hydraulic diameter of each component κ -
dCO
each component passage against the reference diameter
passage
d
hCO
κ =
dCO
D
62097 © IEC:2009 – 13 –
Term Definition Symbol Unit
Dimension factor for Ratio of the diameter of the runner crown or runner band -
κ
T
disc friction loss against the reference diameter

D
d
κ =
T
D
D diameter of the runner crown or the runner band,
:
d
whichever larger
Scalable hydraulic d -
δ
ECOref ECOref
d =
energy loss index for ECOref
0,2
1 + 0,351()κ × κ
uCO dCO
each component
passage
Scalable disc friction d -
δ Tref
Tref
loss index
d =
Tref
0,4
1+ 0,154 κ
T
Loss distribution Ratio of scalable loss to total loss V -
factor
δ
V =
1− η
h
Specific speed N -
0,5
QE
nQ
N =
QE
0,75
E
Speed factor n -
ED
nD
n =
ED
0,5
E
Discharge factor Q -
Q ED
Q =
ED
2 0,5
D E
Power factor P -
P ED
m
P =
ED
2 1,5
ρ D E
Energy coefficient E -
nD
E
E =
nD
2 2
n D
Discharge coefficient Q -
Q nD
Q =
nD
nD
Power coefficient P -
P nD
m
P =
nD
3 5
ρ n D
4 Scale-effect formula
4.1 General
4.1.1 Scalable losses
The energy flux through hydraulic machines and the various losses produced in the energy
conversion process in a hydraulic machine can be typically illustrated by the flux diagrams
shown in A.1. [4]
As a consequence, one of the main features of the new scale up formula as stated in this
standard is the separate consideration on three efficiency components. They are specific

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hydraulic energy efficiency η , volumetric efficiency η and power efficiency η . In this
E Q T
standard, scale effect on each of these efficiency components is considered.

Among the losses corresponding to these efficiency components, the following losses are

subject to scale effect by the difference of Reynolds number and the relative roughness. Then

these losses are referred to as “scalable losses” in this standard.

• Specific hydraulic energy loss due to friction: E
Lf
• Leakage loss: q
• Disc friction loss: P
Ld
It is considered in this standard that the relative magnitude of each scalable loss to each
corresponding performance parameter, except for discharge, ( δ = E E and δ = P P )
E Lf T Ld m
is given as a function of the specific speed for each type of machine.
E is the sum of the friction loss in various parts of the machine and it is expressed as the
Lf
sum of the friction loss in each component as E = E . The scale effect on this loss is
∑
Lf LfCO
caused by the difference of Reynolds number and relative roughness between model and
prototype and assessed by the formula shown as Equation 1.
The rest of the specific hydraulic energy loss is called “kinetic loss" or “non-scalable loss" and
expressed as E = E . It is considered that the ratio of E against E remains the
Lk ∑ LkCO Lk m
same through the model and the prototype.
The scale effect on the leakage loss, q, is caused by the change of the friction loss coefficient
of the seal clearance of the runner/impeller. In most cases, the leakage loss itself is minor
and the scale effect on this loss is relatively very small.
Therefore, in case that the geometry of the seal is maintained homologous between the model
and the prototype within the criteria given in Table 3, the scale effect on the leakage loss is
disregarded and η of the prototype is considered to be the same as that of the model. (See
Q
E.3)
In case that the geometry of the model is not homologous to the prototype, this standard
recommends to use the correction formula for η as set out in E.2.
Q
Similarly to E , the scale effect on the disc friction P is caused by the difference in
Lf Ld
Reynolds number and the relative roughness of the outer surface of the runner/impeller

between the model and the prototype. Due to the presence of the radial flow and the distortion
of the boundary layer in the limited space between the runner/impeller and the stationary
parts, the scale effect on P appears in a slightly different manner than on E . It is
Ld Lf
considered in this standard that the scale effect on the disc friction may be assessed by a
scale effect formula shown as Equation 7. (See Annex D)
In case of axial flow machines, the friction loss of the surface of runner hub is negligibly small
and its scale effect is disregarded.
Therefore, in this standard, only the scale effect on the losses corresponding to the efficiency
components; η and η , are considered for radial flow machines and only η is considered for
E T E
axial flow machines.
62097 © IEC:2009 – 15 –
4.1.2 Basic formulae of the scale effect on hydrodynamic friction losses

Another new feature of the new scale effect formula is the consideration of surface

roughness. The basic physical background for consideration of surface quality is the

Colebrook diagram. By some manipulation and simplification, the implicit Colebrook formula

can be converted into as expression as shown below. [4, 6]

0,2
⎡ ⎤
⎛ k Re ⎞
4 S 0
⎢ ⎥
⎜ ⎟
λ = λ 0,74 8 ×10 × + + 0,26 (1)
⎜ ⎟
⎢ ⎥
d Re
h d
⎝ ⎠
⎣ ⎦
where
Re = 7 ×10                   λ = 0,008 5
0 0
k  sand roughness
S
d hydraulic diameter of the water passage
h
d × v d × v
h h
Re Reynolds number of the water passage Re = = Re
d
d
ν D × u
Practically, the surface roughness of model and prototype are represented by the arithmetical
mean roughness Ra as stated in 4.2.2. Regarding the relationship between the sand
roughness k and Ra, wide spread results have been reported so far. In this standard,
S
however, it is considered that the relationship can be expressed by the following formula:
k Ra
S
≅ 5 (2)
d d
h h
NOTE For very rough surfaces, considerations as described in (2) and in Note 2 of B.1 should be taken into
account.
Then, Equation 1 is rewritten as follows;
0,2
⎡ ⎤
⎛ Re ⎞
Ra D × u
5 0
⎢ ⎥
⎜ ⎟
λ = λ 0,74 4 ×10 × + × + 0,26 (3)
⎜ ⎟
⎢ ⎥
d d × v Re
⎝ h h ⎠
⎣ ⎦
Figure 1 sketches the basic concept for the step-up from model to prototype conditions
including surface roughness. Example P shows the case of a smooth prototype machine. P
3 2
shows the case of a prototype machine of reasonable roughness, whereby P shows the
example of a very rough surface where even a decrease of efficiency compared to the model

will occur.
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Rough surface
Smooth surface
P
M
λ
P
P
Re Re
M P
IEC  201/09
Re
Figure 1 – Basic concept for step-up considering surface roughness
In order to calculate the difference of hydraulic efficiency between two hydraulically similar
operating points M and P at different Reynolds numbers and different surface roughness
conditions, the following formulae can be derived by using Equation 3 (see A.2 (2)).
⎛ ⎞
Δη λ − λ
E M P
⎜ ⎟
Δ = = δ (4)
E Eref
⎜ ⎟
η λ
EM ref
⎝ ⎠
The Colebrook diagram is valid for pipe flow, but it can be demonstrated that also friction loss
coefficients of flat plate flow can be approximated with sufficient accuracy by similar
equations as shown below.
0,2
⎡ ⎤
⎛ k Re ⎞
5 S 0
⎢ ⎥
C = C 0,80⎜10 + ⎟ + 0,20
f f0
⎜ ⎟
⎢ L Re ⎥
⎝ f ⎠
⎣ ⎦
(5)
0,2
⎡ ⎤
⎛ Re ⎞
Ra D × u
5 0
⎢ ⎥
= C 0,80⎜5 ×10 + × ⎟ + 0,20
f0
⎜ ⎟
⎢ L L × w Re ⎥
⎝ ⎠
⎣ ⎦
where
Re = 7 ×10            C = 0,003 2
0 f0
L × w L × w
Re Reynolds number of the plate Re = = Re
f f
ν D × u
L length of the plate
w relative flow velocity on the plate
By replacing λ in Equation 4 by C given by Equation 5, Equation 4 is used to calculate the
f
scale effect of the friction loss of runner blades of axial flow machines.

62097 © IEC:2009 – 17 –
Similar formula of friction loss coefficient for disc friction is established as follows [9];

(See Annex D).
0,2
⎡ ⎤
⎛ k Re ⎞
ST 0
⎢ ⎥
⎜ ⎟
C = C 0,85 1,5 ×10 × + + 0,15
m m0
⎜ ⎟
⎢ a Re ⎥
⎝ T⎠
⎣ ⎦
(6)
0,2
⎡ ⎤
⎛ ⎞
Ra D Re
⎢ 4 T 0 ⎥
⎜ ⎟
= C 0,85 7,5 × 10 + × + 0,15
m0
⎢ ⎜ ⎟ ⎥
a Re
2a
⎝ ⎠
⎢ ⎥
⎣ ⎦
where
Re = 7 ×10          C = 0,001 9
0 m0
k  equivalent sand roughness corresponding to Ra k =5Ra

ST T ST T
Ra weighted average of the arithmetical mean roughness of the outer surface of the
T
runner and the surface of the stationary part facing to the runner as given by
Equation 13
Re Reynolds number of the disc
T
2 2 2
D
a ω a ω 2a
d
Re = = Re = Re = Re
T
2 2
ν Du
D 2D
a radius of runner crown or band, whichever larger (m)
D
d
a =
ω angular velocity of the disc (rad/s)
By using Equation 6, step-up formula for power efficiency (disc friction) is obtained as follows
(see A.2 (4)):
Δη ⎛ C − C ⎞
T mM mP
⎜ ⎟
Δ = = δ (7)
T Tref
⎜ ⎟
η C
TM ⎝ mref ⎠
4.2 Specific hydraulic energy efficiency
4.2.1 Step-up formula
The scalable losses δ as appeared in Equation 4 are referred to those of a model with
Eref
smooth surface operating at a reference Reynolds number Re = 7 × 10 and have been

ref
established as a function of type and specific speed of a hydraulic machine. They are
standardized and set out in Annex B for radial flow machines and Annex C for axial flow
machines.
By putting the new scale effect formula Equation 3 into Equation 4, the following formula for
the individual step-up for a machine component is derived (see B.2).

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⎛ ⎞
Δη λ − λ
ECO COM COP
⎜ ⎟
Δ = = δ
ECO ECOref
⎜ ⎟
η λ
EM COref
⎝ ⎠
0,2 0,2
⎡ ⎤
6 6
⎛ ⎞ ⎛ ⎞
Ra Ra
7 × 10 7 × 10
⎢ 5 COM 5 COP ⎥
⎜ ⎟ ⎜ ⎟
4 × 10 κ + − 4 × 10 κ +
uCO uCO
⎢⎜ ⎟ ⎜ ⎟ ⎥
D Re D Re
M M P P (8)
⎝ ⎠ ⎝ ⎠
⎢ ⎥
= δ
ECOref
0,2
⎢ ⎥
1 + 0,35()κ × κ
uCO dCO
⎢ ⎥
⎢ ⎥
⎣ ⎦
0,2 0,2
⎡ ⎤
6 6
⎛ ⎞ ⎛ ⎞
Ra 7 × 10 Ra 7 × 10
5 COM 5 COP
⎢ ⎥
⎜ ⎟ ⎜ ⎟
= d 4 × 10 κ + − 4 × 10 κ +
ECOref uCO uCO
⎢⎜ ⎟ ⎜ ⎟ ⎥
D Re D Re
M M P P
⎝ ⎠ ⎝ ⎠
⎢ ⎥
⎣ ⎦
where
δ standardized reference scalable loss for each component passage when the
ECOref
machine Reynolds number Re is equal to the reference Reynolds number
M
( 7 ×10 )  (see A.2 (2) and B.2 (2))
κ standardized flow velocity factor for each component passage (see B.2 (1))
uCO
κ standardized dimension factor for each component passage (see B.2 (1))
dCO
δ scalable loss index for each component passage (see B.2 (2))

ECOre
...