ISO 12167-1:2016

Plain bearings — Hydrostatic plain journal bearings with drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings with drainage grooves

ISO 12167-1:2016 applies to hydrostatic plain journal bearings under steady-state conditions. In ISO 12167-1:2016, only bearings with oil drainage grooves between the recesses are taken into account. As compared to bearings without oil drainage grooves, this type needs higher power with the same stiffness behaviour.

Paliers lisses — Paliers lisses radiaux hydrostatiques avec rainures d'écoulement fonctionnant en régime stationnaire — Partie 1: Calcul pour la lubrification des paliers lisses radiaux avec rainures d'écoulement

General Information

• Status: Withdrawn
• Publication Date: 18-Sep-2016
• ICS: 21.100.10 - Plain bearings
• Technical Committee: ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications
• Drafting Committee: ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications
• Current Stage: 9599 - Withdrawal of International Standard
• Completion Date: 14-Jun-2023

Relations

• Revises: ISO 12167-1:2001 - Plain bearings — Hydrostatic plain journal bearings with drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings with drainage grooves
• Effective Date: 12-Dec-2015
• Revised: ISO 12167-1:2023 - Plain bearings — Hydrostatic plain journal bearings with drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings with drainage grooves
• Effective Date: 06-Jun-2022

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 12167-1
Second edition
2016-09-15
Plain bearings — Hydrostatic plain
journal bearings with drainage
grooves under steady-state
conditions —
Part 1:
Calculation of oil-lubricated plain
journal bearings with drainage grooves
Paliers lisses — Paliers lisses radiaux hydrostatiques avec rainures
d’écoulement fonctionnant en régime stationnaire —
Partie 1: Calcul pour la lubrification des paliers lisses radiaux avec
rainures d’écoulement
Reference number
ISO 12167-1:2016(E)
©
ISO 2016

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ISO 12167-1:2016(E)

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© ISO 2016, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
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ISO 12167-1:2016(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Bases of calculation and boundary conditions . 1
4 Symbols, terms and units . 3
5 Method of calculation . 5
5.1 General . 5
5.2 Load-carrying capacity . 6
5.3 Lubricant flow rate and pumping power . 8
5.4 Frictional power. 9
5.5 Optimization .10
5.6 Temperatures and viscosities .11
5.7 Minimum pressure in recesses .11
Annex A (normative) Description of the approximation method for the calculation of
hydrostatic plain journal bearings .13
Annex B (informative) Example of calculation according to the method given in Annex A.23
Bibliography .32
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ISO 12167-1:2016(E)

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment,
as well as information about ISO’s adherence to the World Trade Organization (WTO) principles in the
Technical Barriers to Trade (TBT) see the following URL: www.iso.org/iso/foreword.html.
The committee responsible for this document is ISO/TC 123, Plain bearings.
This second edition cancels and replaces the first edition (ISO 12167-1:2001), of which it constitutes a
minor revision.
ISO 12167 consists of the following parts, under the general title Plain bearings — Hydrostatic plain
journal bearings with drainage grooves under steady-state conditions:
— Part 1: Calculation of oil-lubricated plain journal bearings with drainage grooves
— Part 2: Characteristic values for the calculation of oil-lubricated plain journal bearings with
drainage grooves
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ISO 12167-1:2016(E)

Introduction
Hydrostatic bearings use external lubrication to support pressure on the bearings; thus, are less prone
to wear and tear, run quietly, and have wide useable speed, as well as high stiffness and damping
capacity. These properties also demonstrate the special importance of plain journal bearings in
different fields of application such as in machine tools.
Basic calculations described in this part of ISO 12167 may be applied to bearings with different numbers
of recesses and different width/diameter ratios for identical recess geometry.
Oil is fed to each bearing recess by means of a common pump with constant pumping pressure (system
p = constant) and through preceding linear restrictors, e.g. capillaries.
en
The calculation procedures listed in this part of ISO 12167 enable the user to calculate and assess a given
bearing design, as well as to design a bearing as a function of some optional parameters. Furthermore,
this part of ISO 12167 contains the design of the required lubrication system including the calculation
of the restrictor data.
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INTERNATIONAL STANDARD ISO 12167-1:2016(E)
Plain bearings — Hydrostatic plain journal bearings with
drainage grooves under steady-state conditions —
Part 1:
Calculation of oil-lubricated plain journal bearings with
drainage grooves
1 Scope
This part of ISO 12167 applies to hydrostatic plain journal bearings under steady-state conditions.
In this part of ISO 12167, only bearings with oil drainage grooves between the recesses are taken into
account. As compared to bearings without oil drainage grooves, this type needs higher power with the
same stiffness behaviour.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 3448, Industrial liquid lubricants — ISO viscosity classification
ISO 12167-2:2001, Plain bearings — Hydrostatic plain journal bearings with drainage grooves under
steady-state conditions — Part 2: Characteristic values for the calculation of oil-lubricated plain journal
bearings with drainage grooves
3 Bases of calculation and boundary conditions
Calculation in accordance with this part of ISO 12167 is the mathematical determination of the
operational parameters of hydrostatic plain journal bearings as a function of operating conditions,
bearing geometry and lubrication data. This means the determination of eccentricities, load-
carrying capacity, stiffness, required feed pressure, oil flow rate, frictional and pumping power, and
temperature rise. Besides the hydrostatic pressure build up, the influence of hydrodynamic effects is
also approximated.
Reynolds’ differential formula furnishes the theoretical basis for the calculation of hydrostatic bearings.
In most practical cases of application, it is, however, possible to arrive at sufficiently exact results by
approximation.
The approximation used in this part of ISO 12167 is based on two basic formulae intended to describe
the flow through the bearing lands, which can be derived from Reynolds’ differential formula when
special boundary conditions are observed. The Hagen-Poiseuille law describes the pressure flow in a
parallel clearance gap and the Couette formula the drag flow in the bearing clearance gap caused by
shaft rotation. A detailed presentation of the theoretical background of the calculation procedure is
included in Annex A.
The following important premises are applicable to the calculation procedures described in this part of
ISO 12167:
a) all lubricant flows in the lubrication clearance gap are laminar;
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ISO 12167-1:2016(E)

b) the lubricant adheres completely to the sliding surfaces;
c) the lubricant is an incompressible Newtonian fluid;
d) in the whole lubrication clearance gap, as well as in the preceding restrictors, the lubricant is
partially isoviscous;
e) a lubrication clearance gap completely filled with lubricant is the basis of frictional behaviour;
f) fluctuations of pressure in the lubricant film normal to the sliding surfaces do not take place;
g) bearing and journal have completely rigid surfaces;
h) the radii of curvature of the surfaces in relative motion to each other are large in comparison to the
lubricant film thickness;
i) the clearance gap height in the axial direction is constant (axial parallel clearance gap);
j) the pressure over the recess area is constant;
k) there is no motion normal to the sliding surfaces.
The bearing consists of Z cylindrical segments and rectangular recess of the same size and is
supplied with oil through restrictors of the same flow characteristics. Each segment consists of a
circumferential part between two centre lines of axial drainage grooves. With the aid of the above-
mentioned approximation formulae, all parameters required for the design or calculation of bearings
can be determined. The application of the similarity principle results in dimensionless similarity values
for load-carrying capacity, stiffness, oil flow rate, friction, recess pressures, etc.
The results indicated in this part of ISO 12167 in the form of tables and diagrams are restricted to
operating ranges common in practice for hydrostatic bearings. Thus, the range of the bearing
eccentricity (displacement under load) is limited to ε = 0 to 0,5.
Limitation to this eccentricity range means a considerable simplification of the calculation procedure
as the load-carrying capacity is a nearly linear function of the eccentricity. However, the applicability
of this procedure is hardly restricted as in practice eccentricities ε > 0,5 are mostly undesirable for
reasons of operational safety. A further assumption for the calculations is the approximated optimum
[1]
restrictor ratio ξ = 1 for the stiffness behaviour.
As for the outside dimensions of the bearing, this part of ISO 12167 is restricted to the range bearing
width/bearing diameter B/D = 0,3 to 1, which is common in practical cases of application. The recess
depth is larger than the clearance gap height by a factor of 10 to 100. When calculating the friction
losses, the friction loss over the recess in relation to the friction over the bearing lands can generally
be neglected on account of the above premises. However, this does not apply when the bearing shall be
optimized with regard to its total power losses.
To take into account the load direction of a bearing, it is necessary to distinguish between the two
extreme cases, load in the direction of recess centre and load in the direction of land centre.
Apart from the aforementioned boundary conditions, some other requirements are to be mentioned for
the design of hydrostatic bearings in order to ensure their functioning under all operating conditions.
In general, a bearing shall be designed in such a manner that a clearance gap height of at least 50 % to
60 % of the initial clearance gap height is ensured when the maximum possible load is applied. With
this in mind, particular attention shall be paid to misalignments of the shaft in the bearing due to shaft
deflection which may result in contact between shaft and bearing edge and thus in damage of the bearing.
In addition, the parallel clearance gap required for the calculation is no longer present in such a case.
In the case where the shaft is in contact with the bearing lands when the hydrostatic pressure is
switched off, it might be necessary to check the contact zones with regard to rising surface pressures.
It shall be ensured that the heat originating in the bearing does not lead to a non-permissible rise in the
temperature of the oil.
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ISO 12167-1:2016(E)

If necessary, a means of cooling the oil shall be provided. Furthermore, the oil shall be filtered in order
to avoid choking of the capillaries and damage to the sliding surfaces.
Low pressure in the relieved recess shall also be avoided, as this leads to air being drawn in from the
environment and this would lead to a decrease in stiffness (see 5.7).
4 Symbols, terms and units
Table 1 — Symbols, terms and units
Symbol Term Unit
a Inertia factor 1
2
A Land area m
lan
 
A
* lan
* 1
Relative land area A = 
lan
A
lan  
π××BD
 
2
A Recess area m
p
b Width perpendicular to the direction of flow m
 
π×D
b m
ax Width of axial outlet b = −+lb
()
 
ax cG
Z
 
b m
c
Width of circumferential outlet bB=− l
()
cax
b Width of drainage groove m
G
B Bearing width m
c Stiffness coefficient N/m
c Specific heat capacity of the lubricant (p = constant) J/kg·K
p
 
C Radial clearance CD=−D /2 m
R
()
RB J
 
 
d Diameter of capillaries m
cp
D Bearing diameter (D : shaft; D : bearing; D ≈ D ≈ D ) m
J B J B
e Eccentricity (shaft displacement) m
f Relative film thickness [ f = h/C ] 1
R
f 1
en,i
Relative film thickness at ϕϕ= ′
1,i
f 1
ex,i
′
Relative film thickness at ϕϕ=
2,i
F Load-carrying capacity (load) N
F* Characteristic value of load-carrying capacity [F* = F/(B × D × p )] 1
en
*
Characteristic value of effective load-carrying capacity 1
F
eff
*
Characteristic value of effective load-carrying capacity for N = 0 1
F
eff,0
h Local lubricant film thickness (clearance gap height) m
h Minimum lubricant film thickness (minimum clearance gap height) m
min
h Depth of recess m
p
K Speed-dependent parameter 1
rot
l Length in the direction of flow m
l Axial land length m
ax
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ISO 12167-1:2016(E)

Table 1 (continued)
Symbol Term Unit
l Circumferential land length m
c
l Length of capillaries m
cp
-1
N Rotational frequency (speed) s
p Recess pressure, general Pa
Pa
p
Specific bearing load [ p = F/(B × D)]
p Feed pressure (pump pressure) Pa
en
p Pressure in recess i Pa
i
P * Pressure ratio [P * = P /P ] 1
i i i en
p Pressure in recess i, when ε = 0 Pa
i, 0
P* Power ratio (P* = P /P ) 1
f p
P Frictional power W
f
P Pumping power W
p
P Total power (P = P + P ) W
tot tot p f
*
Characteristic value of total power 1
P
tot
3
Q Lubricant flow rate (for complete bearing) m /s
Q* Lubricant flow rate parameter 1
3
Q Lubricant flow rate from capillary into recess 1 m /s
cp, i
3
R Flow resistance of capillaries Pa⋅s/m
cp
 
12××η l
ax
  3
R Flow resistance of one axial land R = Pa⋅s/m
lan, ax
lan,ax
 3 
bC×
 ax R 
3
R Pa⋅s/m
lan, c
 
12××η l
c
 
Flow resistance of one circumferential land R =
lan,c
3
 
bC×
cR
 
3
R Pa⋅s/m
P, 0
 
R
lan,ax
 
Flow resistance of one recess, when ε = 0, R =
p,0
 
21×+κ
()
 
Re Reynolds number 1
So Sommerfeld number 1
T Temperature °C
T Mean temperature in the bearings; see Formula (15) °C
B
ΔT Temperature difference °C
u Flow velocity m/s
U Circumferential speed m/s
Average velocity in restrictor m/s
w
Z Number of recesses 1
α Position of first recess related to recess centre measured from load direction; rad
see Figure A.3
β Attitude angle of shaft °
g Exponent in viscosity formula 1
ε Relative eccentricity (ε = e/C ) 1
R
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ISO 12167-1:2016(E)

Table 1 (continued)
Symbol Term Unit
η Dynamic viscosity Pa⋅s
η Pa⋅s
Dynamic viscosity for TT=
B
B
κ 1
 
R
lan, ax lb×
ax c
 
Resistance ratio κ ==
 
R lb×
 
lan, c cax
 
ξ 1
 
R
cp
 
Restrictor ratio ξ =
 
R
 
P, 0
 
π 1
f
 
ηω×
B
 
Relative frictional pressure π =
f
 2 
P ×ψ
 en 
3
ρ Density kg/m
2
τ Shearing stress N/m
φ Angular coordinate measured from radius opposite to eccentricity, e; rad
see Figure A.3
 
2×C
R
 
ψ Relative bearing clearance ψ = 1
 
D
 
-1
ω Angular velocity (ω = 2 × π × Ν) s
5 Method of calculation
5.1 General
This part of ISO 12167 covers the calculation, as well as the design, of hydrostatic plain journal
bearings. In this case, calculation is understood to be the verification of the operational parameters of
a hydrostatic bearing with known geometrical and lubrication data. In the case of a design calculation,
with the given methods of calculation, it is possible to determine the missing data for the required
bearing geometry, the lubrication data and the operational parameters on the basis of a few initial data
(e.g. required load-carrying capacity, stiffness, rotational frequency).
In both cases, the calculations are carried out according to an approximation method based on the
Hagen-Poiseuille and the Couette formulae, mentioned in Clause 3. The bearing parameters calculated
according to this method are given as relative values in the form of tables and diagrams as a function of
different parameters. The procedure for the calculation or design of bearings is described in 5.2 to 5.7.
This includes the determination of different bearing parameters with the aid of the given calculation
formulae or the tables and diagrams. The following calculation items are explained in detail:
a) determination of load-carrying capacity with and without taking into account shaft rotation;
b) calculation of lubricant flow rate and pumping power;
c) determination of frictional power with and without consideration of losses in the bearing recesses;
d) procedure for bearing optimization with regard to minimum total power loss.
For all calculations, it is necessary to check whether the important premise of laminar flow in the
bearing clearance gap, in the bearing recess and in the capillary is met. This is checked by determining
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ISO 12167-1:2016(E)

the Reynolds numbers. Furthermore, the portion of the inertia factor in the pressure differences shall
be kept low at the capillary (see A.3.1).
If the boundary conditions defined in Clause 3 are observed, this method of calculation yields results
with deviations which can be neglected for the requirements of practice, in comparison with an exact
calculation by solving the Reynolds differential formula.
5.2 Load-carrying capacity
Unless indicated otherwise, it is assumed in the following that capillaries with a linear characteristic
are used as restrictors and that the restrictor ratio is ξ = 1. Furthermore, the difference is only made
between the two cases, “load in direction of recess centre” and “load in direction of land centre”. For this
reason, it is no longer mentioned in each individual case that the characteristic values are a function of
the three parameters, “restrictor type”, “restrictor ratio” and “load direction relative to the bearing”.
Even under the abovementioned premises, the characteristic value of load carrying capacity [Formula (1)]
F p
*
F = = (1)
BD××p p
en en
still depends on the following parameters:
— number of recesses, Z;
— width/diameter ratio, B/D;
— relative axial land width, l /B;
ax
— relative land width in circumferential direction, l /D;
c
— relative groove width, b /D;
G
— relative journal eccentricity, ε;
— relative frictional pressure when the difference is only made between the two cases, “load on recess
centre” and “load on land centre”:
ηω×
B
π = (2)
f
2
p ×ψ
en
NOTE The Sommerfeld number, So, common with hydrodynamic plain journal bearings can be set up as
follows:
2 *
pF×ψ
So = =
ηω× π
Bf
In ISO 12167-2:2001, Figures 1 and 2, the functions F*(ε, π ) and β(ε, π ) are represented for Z = 4, ξ = 1,
f f
B/D = 1, l /B = 0,1, l /D = 0,1, b /D = 0,05, i.e. restriction by means of capillaries, load in direction of
ax c G
centre of bearing recess.
These figures show the influence of rotation on the characteristic value of load-carrying capacity and
the attitude angle.
For the calculation of a geometrically similar bearing, it is possible to determine the minimum lubricant
film thickness when values are given, e.g. for F, B, D, p , ω, ψ, and η (determination of η according to
en B B
5.6, if applicable).
All parameters are given for the determination of F* according to Formula (1) and π according to
f
Formula (2). For this geometry, the relevant values for ε and β can be taken from ISO 12167-2:2001,
Figures 1 and 2 and thus, h = C (1 − ε).
min R
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ISO 12167-1:2016(E)

According to the approximation method described in Annex A, it transpires that the characteristic
value of effective load-carrying capacity is no longer a function of the ratio B/D.
*
F F
*
F = = (3)
eff
bZ××bP× b Zb×
caxenc ax
×
D π×B
If the resistance ratio
R
lb×
lan, ax
ax c
κ == (4)
R lb×
lan, c cax
and the speed dependent parameter
ξκ××π ×l
fc
K =
rot
D
(5)
K
rot
K =
rot,nom
1+κ
is introduced, there remains a dependence on the following parameters:
*
FZ,,ϕκ,,K ε
()
effG rot
*
If, in addition, advantage is taken of the fact that the function (ε) is nearly linear for ε ≤ 0,5, then it
F
eff
*
is practically sufficient to know that the function Ffεϕ= 04,,= ZK,,κ for the calculation
() ()
effG rot
of the load carrying capacity.
For K = 0, i.e. for the stationary shaft, the characteristic value of effective load-carrying capacity for
rot
ε = 0,4 only depends on three parameters:
*
Ffεϕ= 04,,= Z ,κ
()
()
effG
*
Thus, in ISO 12167-2:2001, Figure 3, F ε = 04, for Z = 4 and 6 can be given via κ for different
()
eff,0
φ values.
G
The influence of the rotational movement on the characteristic value of load-carrying capacity is taken
*
F
eff
into account by the ratio = fZ,,ϕκ, K .
()
Grot
*
F
eff,0
**
For Z = 4, the ratio FF/ is shown in ISO 12167-2:2001, Figure 4. The hydrodynamically conditioned
effeff, 0
increase of the load-carrying capacity can be easily recognized when presented in such a manner.
*
If, e.g. Z and all parameters are given for the determination of F according to Formula (3), κ according
eff
to Formula (4) and K according to Formula (5), then the minimum lubricant film thickness developing
rot
during operation can be determined.
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ISO 12167-1:2016(E)

*
After having calculated φ , κ and K , the value for F ε = 04, is taken from ISO 12167-2:2001,
G rot, nom ()
eff,0
** *
Figure 3 and the value for FF/,ε = 04 from ISO 12167-2:2001, Figure 4, F is calculated
()
effeff,0 eff
according to Formula (3) and then the eccentricity is obtained as follows:
*
04, × F
eff
*
F
eff *
εε= 04,,×=F 04
() ()
eff, 0
*
F
eff, 0
and the minimum lubricant film thickness is h = C × (1 – ε).
min R
5.3 Lubricant flow rate and pumping power
The characteristic value for the lubricant flow rate is given by
Q ×η
B
*
Q = (6)
3
Cp×
Ren
It depends only slightly on the relative journal eccentricity ε, the load direction relative to the bearing
and the relative frictional pressure π , or the speed dependent parameter K .
f rot
By approximation, the lubricant flow rate can be calculated as follows (see also A.3.4):
l
ax
1−
ZB κ +1
B
QQ**εε≤ 0,5 ≈= 0 = ×× × (7)
() ()
D l κ
61+ξ
()
c
D
R
P 6××η l
1
cp
P Bax
==ε 0 with ξ = and R =
()
P,0
3
1+ξ p R
bC×+1 κ
en P,0
()
ax R
The flow resistance of the capillaries according to A.3.2.2 is given by
128××η l
cp cp
R ×+1 a
()
cp =
4
π×d
cp
with the non-linear portion (inertia factor)
10, 8 4××Q ρ
a =×
32 η ××lZ
cp cp
By converting Formula (6), the lubricant flow rate can be calculated when the parameters η , C , p , ξ,
B R en
B/D, and l /B are given.
ax
For optimized bearings, Q* shall be taken from ISO 12167-2:2001, Table 1. The pumping power, without
considering the pump efficiency, is given by
23
pC×
* en R
PQ=×pQ=× (8)
pen
η
B
According to the approximation method, Q* is again determined according to Formula (7), thus it is the
characteristic value of both flow rate and pumping power.
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ISO 12167-1:2016(E)

5.4 Frictional power
The characteristic value for the frictional power is given by
 ×
C
P
* f R
 = (9)
P
f
2
 ××BD×
η
U
B
Friction generates in the lands as well as in the recess area. The land area related to the total surface of
the bearing π × B × D is given by
 
 
l l l l b
2
*
ax caxaxG
A =×  ×+π Z ××12−×  −×Z × 
lan
 
π B D B B D
 
 
 
According to the approximation method, the characteristic value for the frictional power in the land
area is given by
π
**
PA=×
f,lanlan
2
1 -ε
and in the recess area
C
**R
P =×π 41×× − A .
( )
f,P lan
h
p
Thus, the characteristic value for the total amount of friction is given by
 
 
4×C
1 1
* * R
 
 
PA=×π × + ×−1 (10)
f lan
 
 * 
2 h
A
p
1 −ε
 
 lan 
 
The actual frictional power is obtained by converting Formula (9):
2
η ××UB×D
*
B
PP=×
ff
C
R
© ISO 2016 – All rights reserved 9

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ISO 12167-1:2016(E)

5.5 Optimization
When optimizing according to the power consumption, the total power loss, i.e. the sum of pumping
and frictional power, is minimized. According to 5.3 and 5.4, the total power is given by
2
2 3
η ××UB × D
pC×
B
* *
en R
PP=+PQ=× +×P
totp f f
η C
B R
With Formulae (8) and (9), this can be written as follows:
*
 
P
Q
f
 
PF=×ω ××C ×+1 (11)
totR
B  P 
*
p
4 ××F × π  
f
D
[2]
Following a proposal of Vermeulen, the ratio of frictional power to pumping power is introduced as
an optional parameter and designated with P*. Thus, the characteristic value for the total power loss is
given by
**
QP×+1
( )
P
* tot
P = = (12)
tot
FC××ω B
*
R
4××F × π
f
D
Serial calculations have shown that the power minimum which can be obtained in the relatively wide
range, P* = 1 to 3, depends only slightly on the chosen power ratio, P*. An approximated optimization
with the mean value P* = 2 may be carried out.
The relative frictional pressure in Formula (12) cannot be chosen freely, as it is linked to the chosen
power ratio, P*:
*
* *
×
B 1 Q
P P
* 2 f
 =× 4××  or =×  (13)
P ππ
f f
*
D 2 B
*
Q
×
P
f
D
When P*, B/D, ε, η /C and ξ are given, the characteristic value of total power according to Formula (12)
P R
to be minimized remains only a function of Z, l /B, l /D and b /D.
ax c G
*
In ISO 12167-2:2001, Figures 5 to 12, for P* =
...