SIST EN ISO 80000-11:2020
(Main)Quantities and units - Part 11: Characteristic numbers (ISO 80000-11:2019)
Quantities and units - Part 11: Characteristic numbers (ISO 80000-11:2019)
This document gives names, symbols and definitions for characteristic numbers used in the description
of transport and transfer phenomena.
Größen und Einheiten - Teil 11: Kenngrößen der Dimension Zahl (ISO 80000-11:2019)
Dieses Dokument enthält Benennungen, Formelzeichen und Definitionen für Kenngrößen der Dimension Zahl, die zur Beschreibung von Transport- und Übertragungsphänomenen verwendet werden.
Grandeurs et unités - Partie 11: Nombres caractéristiques (ISO 80000-11:2019)
Le présent document donne noms, les symboles et les définitions des nombres caractéristiques utilisés dans la description des phénomènes de transfert.
Veličine in enote - 11. del: Značilna števila (ISO 80000-11:2019)
General Information
Relations
Related Packages
Standards Content (Sample)
SLOVENSKI STANDARD
SIST EN ISO 80000-11:2020
01-december-2020
Nadomešča:
SIST EN ISO 80000-11:2013
Veličine in enote - 11. del: Značilna števila (ISO 80000-11:2019)
Quantities and units - Part 11: Characteristic numbers (ISO 80000-11:2019)
Größen und Einheiten - Teil 11: Kenngrößen der Dimension Zahl (ISO 80000-11:2019)
Grandeurs et unités - Partie 11: Nombres caractéristiques (ISO 80000-11:2019)
Ta slovenski standard je istoveten z: EN ISO 80000-11:2020
ICS:
01.060 Veličine in enote Quantities and units
SIST EN ISO 80000-11:2020 en,fr,de
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST EN ISO 80000-11:2020
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SIST EN ISO 80000-11:2020
EN ISO 80000-11
EUROPEAN STANDARD
NORME EUROPÉENNE
October 2020
EUROPÄISCHE NORM
ICS 01.060 Supersedes EN ISO 80000-11:2013
English Version
Quantities and units - Part 11: Characteristic numbers (ISO
80000-11:2019)
Grandeurs et unités - Partie 11: Nombres Größen und Einheiten - Teil 11: Kenngrößen der
caractéristiques (ISO 80000-11:2019) Dimension Zahl (ISO 80000-11:2019)
This European Standard was approved by CEN on 21 October 2020.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2020 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 80000-11:2020 E
worldwide for CEN national Members.
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SIST EN ISO 80000-11:2020
EN ISO 80000-11:2020 (E)
Contents Page
European foreword . 3
2
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SIST EN ISO 80000-11:2020
EN ISO 80000-11:2020 (E)
European foreword
The text of ISO 80000-11:2019 has been prepared by Technical Committee ISO/TC 12 "Quantities and
units” of the International Organization for Standardization (ISO) and has been taken over as
EN ISO 80000-11:2020 by Technical Committee CEN/SS F02 “Units and symbols” the secretariat of
which is held by CCMC.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by April 2021, and conflicting national standards shall be
withdrawn at the latest by April 2021.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 80000-11:2013.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the
United Kingdom.
Endorsement notice
The text of ISO 80000-11:2019 has been approved by CEN as EN ISO 80000-11:2020 without any
modification.
3
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SIST EN ISO 80000-11:2020
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SIST EN ISO 80000-11:2020
INTERNATIONAL ISO
STANDARD 80000-11
Second edition
2019-10
Quantities and units —
Part 11:
Characteristic numbers
Grandeurs et unités —
Partie 11: Nombres caractéristiques
Reference number
ISO 80000-11:2019(E)
©
ISO 2019
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SIST EN ISO 80000-11:2020
ISO 80000-11:2019(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2019
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
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Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2019 – All rights reserved
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SIST EN ISO 80000-11:2020
ISO 80000-11:2019(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Momentum transfer . 1
5 Transfer of heat .16
6 Transfer of matter in a binary mixture .24
7 Constants of matter.33
8 Magnetohydrodynamics.37
9 Miscellaneous .46
Bibliography .48
Alphabetical index .49
© ISO 2019 – All rights reserved iii
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SIST EN ISO 80000-11:2020
ISO 80000-11:2019(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 of the voluntary nature of standards, 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: www .iso
.org/ iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 12, Quantities and units, in collaboration
with Technical Committee IEC/TC 25, Quantities and units.
This second edition cancels and replaces the first edition (ISO 80000-11:2008), which has been
technically revised.
The main changes compared to the previous edition are as follows:
— the table giving the quantities and units has been simplified;
— all items have been revised in terms of the layout of the definitions, and a worded definition has
been added to each item;
— the number of items has been increased from 25 to 108 (concerns all Clauses);
— item 11-9.2 (Landau-Ginzburg number) has been transferred in this document from
ISO 80000-12:2009 (revised as ISO 80000-12:2019).
A list of all parts in the ISO 80000 and IEC 80000 series can be found on the ISO and IEC websites.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2019 – All rights reserved
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SIST EN ISO 80000-11:2020
ISO 80000-11:2019(E)
Introduction
Characteristic numbers are physical quantities of unit one, although commonly and erroneously
called “dimensionless” quantities. They are used in the studies of natural and technical processes, and
(can) present information about the behaviour of the process, or reveal similarities between different
processes.
Characteristic numbers often are described as ratios of forces in equilibrium; in some cases, however,
they are ratios of energy or work, although noted as forces in the literature; sometimes they are the
ratio of characteristic times.
Characteristic numbers can be defined by the same equation but carry different names if they are
concerned with different kinds of processes.
Characteristic numbers can be expressed as products or fractions of other characteristic numbers if
these are valid for the same kind of process. So, the clauses in this document are arranged according to
some groups of processes.
As the amount of characteristic numbers is tremendous, and their use in technology and science is not
uniform, only a small amount of them is given in this document, where their inclusion depends on their
common use. Besides, a restriction is made on the kind of processes, which are given by the Clause
headings. Nevertheless, several characteristic numbers are found in different representations of the
same physical information, e.g. multiplied by a numerical factor, as the square, the square root, or the
inverse of another representation. Only one of these have been included, the other ones are declared as
deprecated or are mentioned in the remarks column.
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SIST EN ISO 80000-11:2020
INTERNATIONAL STANDARD ISO 80000-11:2019(E)
Quantities and units —
Part 11:
Characteristic numbers
1 Scope
This document gives names, symbols and definitions for characteristic numbers used in the description
of transport and transfer phenomena.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
Names, symbols and definitions for characteristic numbers are given in Clauses 4 to 9.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Momentum transfer
Table 1 gives the names, symbols and definitions of characteristic numbers used to characterize
processes in which momentum transfer plays a predominant role. The transfer of momentum
(ISO 80000-4) basically occurs during a collision of 2 bodies, and is governed by the law of momentum
conservation. Energy dissipation can occur. In a more generalized meaning momentum transfer occurs
during the interaction of 2 subsystems moving with velocity v relative to each other. Typically, one of
the subsystems is solid and possibly rigid, with a characteristic length, which can be a length, width,
radius, etc. of a solid object, often the effective length is given by the ratio of a body’s volume to the area
of its surface.
The other subsystem is a fluid, in general liquid or gaseous, with the following properties amongst others:
— mass density ρ (ISO 80000-4);
— dynamic viscosity η (ISO 80000-4);
— kinematic viscosity ν=ηρ/ (ISO 80000-4), or
— pressure drop Δp (ISO 80000-4).
The field of science is mainly fluid dynamics (mechanics). Characteristic numbers of this kind allow
the comparison of objects of different sizes. They also can give some estimation about the change of
laminar flow to turbulent flow.
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ISO 80000-11:2019(E)
2 © ISO 2019 – All rights reserved
Table 1 — Characteristic numbers for momentum transfer
No. Name Symbol Definition Remarks
11-4.1 Reynolds num- Re quotient of inertial forces and viscous forces in a fluid flow, ex- The value of the Reynolds number gives an estimate
ber pressed by on the flow state: laminar flow or turbulent flow.
In rotating movement, the speed v = ωl, where l is the
ρvvll
Re== ; where
distance from the rotation axis and ω is the angular
ην
velocity.
ρ is mass density (ISO 80000-4),
v is speed (ISO 80000-3),
l is characteristic length (ISO 80000-3),
η is dynamic viscosity (ISO 80000-4), and
ν is kinematic viscosity (ISO 80000-4)
11-4.2 Euler number Eu relationship between pressure drop in a flow and the kinetic energy The Euler number is used to characterize losses in
per volume for flow of fluids in a pipe, expressed by the flow.
Δp A modification of the Euler number is considering the
Eu= ; where
dimensions of the containment (pipe):
2
ρv
d
Δp is drop of pressure (ISO 80000-4),
Eu′= Eu ; where
l
ρ is mass density (ISO 80000-4), and
d is inner diameter (ISO 80000-3) of the pipe, and
v is speed (ISO 80000-3)
l is length (ISO 80000-3).
11-4.3 Froude number Fr quotient of a body’s inertial forces and its gravitational forces for The Froude number can be modified by buoyancy.
flow of fluids, expressed by
Sometimes the square and sometimes the inverse of
v the Froude number as defined here is wrongly used.
Fr= ; where
lg
v is speed (ISO 80000-3) of flow,
l is characteristic length (ISO 80000-3), and
g is acceleration of free fall (ISO 80000-3)
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© ISO 2019 – All rights reserved 3
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.4 Grashof number Gr quotient of buoyancy forces due to thermal expansion which results Heating can occur near hot vertical walls, in pipes, or
in a change of mass density and viscous forces for free convection by a bluff body.
due to temperature differences, expressed by
The characteristic length can be the vertical height
32
of a hot plate, the diameter of a pipe, or the effective
Gr=ΔlgανT/ ; where
V
length of a body.
l is characteristic length (ISO 80000-3),
See also Rayleigh number (item 11-5.3).
g is acceleration of free fall (ISO 80000-3),
α is thermal cubic expansion coefficient (ISO 80000-5),
V
ΔT is difference of thermodynamic temperature T (ISO 80000-5)
between surface of the body and the fluid far away from the
body, and
ν is kinematic viscosity (ISO 80000-4)
11-4.5 Weber number We relation between inertial forces and capillary forces due to surface The fluids can be gases or liquids.
tension at the interface between two different fluids, expressed by
The different fluids often are drops moving in a gas or
2
bubbles in a liquid.
We=ργv l/ ; where
The characteristic length is commonly the diameter of
ρ is mass density (ISO 80000-4),
bubbles or drops.
v is speed (ISO 80000-3),
The square root of the Weber number is called Ray-
l is characteristic length (ISO 80000-3), and
leigh number.
γ is surface tension (ISO 80000-4)
Sometimes the square root of the Weber number as
defined here is called the Weber number. That defini-
tion is deprecated.
Interfaces only exist between two fluids which are not
miscible.
11-4.6 Mach number Ma quotient of the speed of flow and the speed of sound, expressed by The Mach number represents the relationship of iner-
tial forces compared to compression forces.
Ma=v/c ; where
For an ideal gas
v is speed (ISO 80000-3) of the body, and
p RT kT
c is speed of sound (ISO 80000-8) in the fluid
c==γ γγ= ; where γ is ratio of the
ρ M m
specific heat capacity (ISO 80000-5).
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4 © ISO 2019 – All rights reserved
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.7 Knudsen number Kn quotient of free path length of a particle and a characteristic length, The Knudsen number is a measure to estimate wheth-
expressed by er the gas in flow behaves like a continuum.
Kn=λ /l ; where The characteristic length, l, can be a characteristic
size of the gas flow region like a pipe diameter.
λ is mean free path (ISO 80000-9), and
l is characteristic length (ISO 80000-3)
11-4.8 Strouhal num- Sr, relation between a characteristic frequency and a characteristic The characteristic length, l, can be the diameter of an
ber; speed for unsteady flow with periodic behaviour, expressed by obstacle in the flow which can cause vortex shedding,
Sh
or the length of it.
Thomson num- Sr= fl/v ; where
ber
f is frequency (ISO 80000-3) of vortex shedding,
l is characteristic length (ISO 80000-3), and
v is speed (ISO 80000-3) of flow
11-4.9 drag coefficient c relation between the effective drag force and inertial forces for a The drag coefficient is strongly dependant on the
D
body moving in a fluid, expressed by shape of the body.
2F
D
c = ; where
D
2
ρv A
F is drag force (ISO 80000-4) on the body,
D
ρ is mass density (ISO 80000-4) of the fluid,
v is speed (ISO 80000-3) of the body, and
A is cross-sectional area (ISO 80000-3)
11-4.10 Bagnold number Bg quotient of drag force and gravitational force for a body moving in a The characteristic length, l, is the body’s volume di-
fluid, expressed by vided by its cross-sectional area.
2
c ρv
D
Bg= ; where
lgρ
b
c is drag coefficient (item 11-4.9) of the body,
D
ρ is mass density (ISO 80000-4) of the fluid,
v is speed (ISO 80000-3) of the body,
l is characteristic length (ISO 80000-3),
g is acceleration of free fall (ISO 80000-3), and
ρ is mass density (ISO 80000-4) of the body
b
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© ISO 2019 – All rights reserved 5
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.11 Bagnold number Ba quotient of drag force and viscous force in a fluid transferring solid
2
particles, expressed by
2
ργd
s 12/
Ba =−11/ f ; where
()
2 s
η
ρ is mass density (ISO 80000-4) of particles,
s
d is diameter (ISO 80000-3) of particles,
γ=v/d is shear rate time-derivative of shear strain
(ISO 80000-4),
η is dynamic viscosity (ISO 80000-4) of fluid, and
f is volumic fraction of solid particles
s
11-4.12 lift coefficient c , quotient of the lift force available from a wing at a given angle The lift coefficient is dependant on the shape of the
l
and the inertial force for a wing shaped body moving in a fluid, wing.
c
A
expressed by
2F F
ll
c == ; where
l
2
qS
ρv S
F is lift force (ISO 80000-4) on the wing,
l
ρ is mass density (ISO 80000-4) of the fluid,
v is speed (ISO 80000-3) of the body,
S = A cos α is effective area (ISO 80000-3) when α is the angle
of attack and A is area of the wing, and
2
q=ρv /2 is dynamic pressure.
11-4.13 thrust coeffi- c quotient of the effective thrust force available from a propeller and The thrust coefficient is dependant on the shape of the
t
cient the inertial force in a fluid, expressed by propeller.
24
cF= / ρnd ; where
()
tT
F is thrust force (ISO 80000-4) of the propeller,
T
ρ is mass density (ISO 80000-4) of the fluid,
n is rotational frequency (ISO 80000-3), and
d is tip diameter (ISO 80000-3) of the propeller
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6 © ISO 2019 – All rights reserved
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.14 Dean number Dn relation between centrifugal force and inertial force, for flows of
fluids in curved pipes, expressed by
2vrr
Dn= ; where
ν R
v is (axial) speed (ISO 80000-3),
r is radius (ISO 80000-3) of the pipe,
ν is kinematic viscosity (ISO 80000-4) of the fluid, and
R is radius of curvature (ISO 80000-3) of the path of the pipe
11-4.15 Bejan number Be quotient of mechanical work and frictional energy loss in fluid dy- A similar number exists for heat transfer (item 11-5.9).
namics in a pipe, expressed by
The kinematic viscosity is also called momentum
22
diffusivity.
Δpl ρΔpl
Be= = ; where
2
ην
η
Δp is drop of pressure (ISO 80000-4) along the pipe,
l is characteristic length (ISO 80000-3),
η is dynamic viscosity (ISO 80000-4),
ν is kinematic viscosity (ISO 80000-4), and
ρ is mass density (ISO 80000-4).
11-4.16 Lagrange num- Lg quotient of mechanical work and frictional energy loss in fluid dy- The Lagrange number is also given by
ber namics in a pipe, expressed by
La=⋅Re Eu ; where
lpΔ
Re is the Reynolds number (item 11-4.1), and
Lg= ; where
ηv
Eu is the Euler number (item 11-4.2).
l is length (ISO 80000-3) of the pipe,
Δp is drop of pressure (ISO 80000-4) along the pipe,
η is dynamic viscosity (ISO 80000-4), and
v is speed (ISO 80000-3)
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ISO 80000-11:2019(E)
© ISO 2019 – All rights reserved 7
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.17 Bingham num- Bm, quotient of yield stress and viscous stress in a viscous material for
ber; flow of viscoplastic material in channels, expressed by
Bn
plasticity num- τd
Bm= ; where
ber
ηv
τ is shear stress (ISO 80000-4),
d is characteristic diameter (ISO 80000-3), e.g. effective
channel width,
η is dynamic viscosity (ISO 80000-4), and
v is speed (ISO 80000-3)
11-4.18 Hedström num- He, quotient of yield stress and viscous stress of a viscous material at
ber flow limit for visco-plastic material in a channel, expressed by
Hd
2
τρd
0
He= ; where
2
η
τ is shear stress (ISO 80000-4) at flow limit,
0
d is characteristic diameter (ISO 80000-3), e.g. effective
channel width,
ρ is mass density (ISO 80000-4), and
η is dynamic viscosity (ISO 80000-4)
11-4.19 Bodenstein Bd mathematical expression of the transfer of matter by convection in The Bodenstein number is also given by
number reactors with respect to diffusion,
*
Bd==Pe Re⋅Sc ; where
Bd=vlD/ ; where
*
Pe is the Péclet number for mass transfer (item
v is speed (ISO 80000-3),
11-6.2),
l is length (ISO 80000-3) of the reactor, and
Re is the Reynolds number (item 11-4.1), and
D is diffusion coefficient (ISO 80000-9)
Sc=ηρ//()DD=ν is Schmidt number (item
11-7.2).
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8 © ISO 2019 – All rights reserved
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.20 Rossby number; Ro quotient of inertial forces and Coriolis forces in the context of trans- The Rossby number represents the effect of Earth's
fer of matter in geophysics, expressed by rotation on flow in pipes, rivers, ocean currents, tor-
Kiebel number
nadoes, etc.
Ro=v/2()lωϕsin ; where
E
The quantity ωϕsin is called Coriolis frequency.
E
v is speed (ISO 80000-3) of motion,
l is characteristic length (ISO 80000-3), the scale of the
phenomenon,
ω is angular velocity (ISO 80000-3) of the Earth's rotation, and
E
φ is angle (ISO 80000-3) of latitude
11-4.21 Ekman number Ek quotient of viscous forces and Coriolis forces in the context of trans- In plasma physics, the square root of this number is
fer of matter for the flow of a rotating fluid, expressed by used.
2
The Ekman number is also given by
Ek=νω/2l sinϕ ; where
()
E
Ek=Ro/Re ; where
ν is kinematic viscosity (ISO 80000-4),
Ro is the Rossby number (item 11-4.20), and
l is characteristic length (ISO 80000-3), the scale of the
phenomenon,
Re is the Reynolds number (item 11-4.1).
ω is angular frequency (ISO 80000-3) of the Earth’s rotation, and
E
φ is angle of latitude
11-4.22 elasticity num- El relation between relaxation time and diffusion time in viscoelastic See also Deborah number (item 11-7.8).
ber flows, expressed by
2
El=trν/ ; where
r
t is relaxation time (ISO 80000-12),
r
ν is kinematic viscosity (ISO 80000-4), and
r is radius (ISO 80000-3) of pipe
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© ISO 2019 – All rights reserved 9
Table 1 (continued)
No. Name Symbol Definition Remarks
11-4.23 Darcy friction f representation of pressure loss in a pipe due to friction within a
D
factor; laminar or turbulent flow of a fluid in a pipe, expressed by
Moody friction 2Δp d
f = ; where
D
factor
2
l
ρv
Δp is drop of pressure (ISO 80000-4) due to friction,
ρ is mass density (ISO 80000-4) of the fluid,
v is (average) speed (ISO 80000-3) of the fluid in the pipe,
d is diameter (ISO 80000-3) of the pipe, and
l is length (ISO 80000-3) of the pipe
11-4.24 Fanning number f , relation between shear stress and dynamic pressure in the flow of a The Fanning number describes the flow of fluids in
n
fluid in a containment, expressed by a pipe with friction at the walls represented by its
f
shear stress.
2τ
f = ; where
n Symbol f may be used where no conflicts are possible.
2
ρv
τ is shear stress (ISO 80000-4) at the wall,
ρ is mass density (ISO 80000-4) of the fluid, and
v is speed (ISO 80000-3) of the fluid in the pipe
11-4.25 Goertler num- Go characterization of the stability of laminar boundary layer flows in The Goertler number represents the ratio of centrifu-
ber; transfer of matter in a boundary layer on curved surfaces, ex- gal effects to viscous effects.
pressed by
Goertler param-
eter
vll
bb
Go= ; where
ν r
c
v is speed (ISO 80000-3),
l is boundary layer thickne
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Quantities and units - Part 11: Characteristic numbers (ISO/DIS 80000-11:2017)
Größen und Einheiten - Teil 11: Kenngrößen der Dimension Zahl (ISO/DIS 80000-
11:2017)
Grandeurs et unités - Partie 11: Nombres caractéristiques (ISO/DIS 80000-11:2017)
Ta slovenski standard je istoveten z: prEN ISO 80000-11
ICS:
01.060 9HOLþLQHLQHQRWH Quantities and units
oSIST prEN ISO 80000-11:2017 en,fr,de
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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oSIST prEN ISO 80000-11:2017
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oSIST prEN ISO 80000-11:2017
DRAFT INTERNATIONAL STANDARD
ISO/DIS 80000-11
ISO/TC 12 Secretariat: SIS
Voting begins on: Voting terminates on:
2017-01-03 2017-03-27
Quantities and units —
Part 11:
Characteristic numbers
Grandeurs et unités —
Partie 11: Nombres caractéristiques
ICS: 01.060
This document is circulated as received from the committee secretariat.
THIS DOCUMENT IS A DRAFT CIRCULATED
This draft is submitted to a parallel vote in ISO and in IEC.
FOR COMMENT AND APPROVAL. IT IS
THEREFORE SUBJECT TO CHANGE AND MAY
NOT BE REFERRED TO AS AN INTERNATIONAL
STANDARD UNTIL PUBLISHED AS SUCH.
IN ADDITION TO THEIR EVALUATION AS
ISO/CEN PARALLEL PROCESSING
BEING ACCEPTABLE FOR INDUSTRIAL,
TECHNOLOGICAL, COMMERCIAL AND
USER PURPOSES, DRAFT INTERNATIONAL
STANDARDS MAY ON OCCASION HAVE TO
BE CONSIDERED IN THE LIGHT OF THEIR
POTENTIAL TO BECOME STANDARDS TO
WHICH REFERENCE MAY BE MADE IN
Reference number
NATIONAL REGULATIONS.
ISO/DIS 80000-11:2017(E)
RECIPIENTS OF THIS DRAFT ARE INVITED
TO SUBMIT, WITH THEIR COMMENTS,
NOTIFICATION OF ANY RELEVANT PATENT
RIGHTS OF WHICH THEY ARE AWARE AND TO
©
PROVIDE SUPPORTING DOCUMENTATION. ISO 2017
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oSIST prEN ISO 80000-11:2017
ISO/DIS 80000-11:2017(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2017, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
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ISO copyright office
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ii © ISO 2017 – All rights reserved
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ISO/DIS 80000-11:2017(E)
Contents Page
Foreword . iv
1 Scope . 1
2 Normative references . 1
3 Names, symbols, and definitions . 1
4 Momentum transfer. 2
5 Transfer of heat. 15
6 Transport of matter in a binary mixture . 22
7 Constants of matter. 29
8 Magnetohydrodynamics . 31
9 Miscellaneous . 38
Bibliography. 39
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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 12, Quantities and units.
This second edition cancels and replaces the first edition (ISO 80000-11:2008).
ISO 80000 consists of the following parts, under the general title Quantities and units:
Part 1: General
Part 2: Mathematics
Part 3: Space and time
Part 4: Mechanics
Part 5: Thermodynamics
Part 7: Light and Radiation
Part 8: Acoustics
Part 9: Physical chemistry and molecular physics
Part 10: Atomic and nuclear physics
Part 11: Characteristic numbers
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ISO/DIS 80000-11:2017(E)
Part 12: Condensed matter physics
IEC 80000 consists of the following parts (in collaboration with IEC/TC 25), under the general title Quantities
and units:
Part 6: Electromagnetism
Part 13: Information science and technology
Part 14: Telebiometrics related to human physiology
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DRAFT INTERNATIONAL STANDARD ISO/DIS 80000-11:2017(E)
Quantities and units — Part 11: Characteristic numbers
1 Scope
ISO 80000-11 gives the names, symbols and definitions for characteristic numbers used in the description of
transport and transfer phenomena.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content constitutes
requirements 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.
ISO 80000-3:2006, Quantities and units — Part 3: Space and time
ISO 80000-4:2006, Quantities and units — Part 4: Mechanics
ISO 80000-5:2007, Quantities and units — Part 5: Thermodynamics
IEC 80000-6:2008, Quantities and units — Part 6: Electromagnetism
ISO 80000-8:2007, Quantities and units — Part 8: Acoustics
ISO 80000-9:2009, Quantities and units — Part 9: Physical chemistry and molecular physics
ISO 80000-9:2009, Quantities and units — Part 12: Condensed matter physics
3 Names, symbols, and definitions
The names, symbols, and definitions for characteristic numbers are given on the following pages.
Characteristic numbers are physical quantities of dimension number 1, although commonly and falsely
called dimensionless quantities. They are used in the studies of natural and technical processes, and [may]
present information about the behaviour of the process, or reveal similarities between different processes.
Characteristic numbers often are described as ratios of forces; in some cases however they are ratios of
energy or work, although noted as forces in the literature; sometimes it is the ratio of characteristic times.
Characteristic numbers may be defined by the same equation, but carry different names if they are
concerned with different kinds of processes.
Characteristic numbers may be expressed as products or fractions of other characteristic numbers if these
are valid for the same kind of process. So the following tables are arranged according to some groups of
processes.
As the amount of characteristic numbers is tremendous, and their use in technology and science is not
uniform, only a small amount of them is given here. The choice largely was depending of their common use.
Besides there was made a restriction on the kind of processes, which are displayed by the section headings.
Nevertheless several characteristic numbers are found in different representations of the same physical
information, e.g. multiplied by a numerical factor, as the square, the square root, or the inverse of other
representation. Only one of these have been chosen, the other ones declared as deprecated or mentioned in
the remarks column.
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4 Momentum transfer
The transfer of momentum (ISO 80000-4:2006, item 4-8) basically occurs during a collision of 2 bodies, and is governed by the law of momentum conservation. Energy
dissipation may occur. In a more generalized meaning momentum transfer occurs during the interaction of 2 subsystems moving with velocity 𝑣 relative to each other.
Typically one of the subsystems is solid and possibly rigid, with a characteristic length, which may be a length, width, radi us, etc. of a solid object, often the effective length is
given by the ratio of a body’s volume to the area of its surface.
The other subsystem is a fluid, in general liquid or gaseous, with the following properties amongst others:
— mass density 𝜌 (ISO 80000-4:2006, item 4-2);
— dynamic viscosity 𝜂 (ISO 80000-4:2006, item 4-23);
⁄
— kinematic viscosity 𝜈 = 𝜂 𝜌 (ISO 80000-4:2006, item 4-24), or
— pressure drop 𝛥𝑝 (ISO 80000-4:2006, item 4-15.1).
The field of science is mainly fluid dynamics (mechanics). Characteristic numbers of this kind allow the comparison of objects of different sizes. It also may give some
estimation about the change of laminar flow to turbulent flow.
No. Name Symbol Definition Remarks
11-4.1 Reynolds ratio of inertial forces to viscous forces in a fluid flow
The value of the Reynolds number gives an estimate on
𝑅𝑒
𝜌 𝑣 𝑙 𝑣 𝑙
number
(11-4.1)
the flow state: laminar flow or turbulent flow.
𝑅𝑒 = = ; where
𝜂 𝜈
𝜌 is mass density (ISO 80000-4:2006, item 4-2),
In rotating movement the speed 𝑣 is 𝑙, where 𝑙 is the
𝑣 is speed (ISO 80000-3:2006, item 3-8.1),
distance from the rotation axis and 𝜔 is the angular
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1),
velocity.
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23), and
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24)
Euler number
11-4.2 relationship between pressure drop in a flow to kinetic energy per The Euler number is used to characterize losses in the
𝐸𝑢
flow.
(11-4.2) volume for flow of fluids in a pipe
𝛥𝑝
𝐸𝑢 = ; where
2 A modification of the Euler number is considering the
𝜌 𝑣
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dimensions of the containment (pipe):
𝛥𝑝 is drop of pressure (ISO 80000-4:2006, item 4-15.1),
𝜌 is mass density (ISO 80000-4:2006, item 4-2), and
𝛥𝑝 𝑑 𝑑
′
𝐸𝑢 = = 𝐸𝑢; where
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) 2
𝑙 𝜌 𝑣 𝑙
𝑙 is length (ISO 80000-3:2006, item 3-1.1),and
𝑑 is inner diameter (ISO 80000-3:2006, item 3-1.7) of
the pipe.
The Froude number may be modified by buoyancy.
11-4.3 Froude ratio of a body’s inertial forces to its gravitational forces for flow of
𝐹𝑟
number
(11-4.3) fluids
Sometimes the square and sometimes the inverse of the
𝑣
𝐹𝑟 = ; where
Froude number as defined here is called the Froude
√𝑙 𝑔
number.
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of flow,
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1), and
The definition given here reflects that of the existing
𝑔 is acceleration of free fall (ISO 80000-3:2006, item 3-9.2)
standard. However In the majority of references the
squared value is used.
11-4.4 Grashof ratio of buoyancy forces due to thermal expansion which results in a Heating can occur near hot vertical walls, in pipes, or by a
𝐺𝑟
number bluff body.
(11-4.4) change of mass density to viscous forces for free convection due to
temperature differences
The characteristic length can be the vertical height of a
3 2
𝐺𝑟 = 𝑙 𝑔 𝛼 𝛥𝑇/𝜈 ; where
𝑉
hot plate, the diameter of a pipe, or the effective length of
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1),
a body.
𝑔 is acceleration of free fall (ISO 80000-3:2006, item 3-9.2),
See also Rayleigh number (item 11-5.3).
𝛼 is thermal cubic expansion coefficient (ISO 80000-5:2007, item 5-
𝑉
3.2),
𝛥𝑇 is difference of thermodynamic temperature 𝑇 (ISO 80000-5:2007,
item 5-1) between surface of the body and the fluid far away from
the body, and
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24)
11-4.5 Weber relationship of inertial forces compared to capillary forces for bubbles The characteristic length is commonly the diameter of
𝑊𝑒
number bubbles or drops.
(11-4.5) or drops in a fluid
2
⁄
𝑊𝑒 = 𝜌 𝑣 𝑙 𝛾; where
The square root of the Weber number is called Rayleigh
𝜌 is mass density (ISO 80000-4:2006, item 4-2),
number.
𝑣 is speed (ISO 80000-3:2006, item 3-8.1),
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𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1), and Sometimes the square root of the Weber number as
𝛾 is surface tension (ISO 80000-4:2006, item 4-25) defined here is called the Weber number. That definition
is deprecated.
Mach number
11-4.6 ratio of the speed of flow to the speed of sound The Mach number represents the relationship of inertial
𝑀𝑎
⁄
forces compared to compression forces.
(11-4.6) 𝑀𝑎 = 𝑣 𝑐; where
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of the body and
For an ideal gas
𝑐 is speed of sound (ISO 80000-8:2007, item 8-14.1) of the fluid
𝑝 𝑅𝑇 𝑘𝑇
𝑐 = 𝛾 = √𝛾 = √𝛾 ; where 𝛾 is ratio of the specific
√
𝜌 𝑀 𝑚
heat capacities (ISO 80000-5:2007, item 5-17.1).
11-4.7 Knudsen ratio of mean free path of a particle to characteristic length for gas flow The Knudsen number is a measure to estimate whether
𝐾𝑛
⁄
(11-4.7) number 𝐾𝑛 = 𝜆 𝑙; where the gas in flow behaves like a continuum.
𝜆 is mean free path (ISO 80000-9:2009, item 9-44), and
The length 𝑙 can be a characteristic size of the gas flow
𝑙 is length (ISO 80000-3:2006, item 3-1.1)
region like a pipe diameter.
11-4.8 Strouhal ratio of characteristic frequency to characteristic speed for unsteady The characteristic length 𝑙 can be the diameter of an
𝑆𝑟, 𝑆ℎ
(11-4.8) number flow with periodic behaviour obstacle in the flow which can cause vortex shedding, or
the length of it.
𝑆𝑟 = 𝑓 𝑙/𝑣; where
(Thomson
𝑓 is frequency (ISO 80000-3:2006, item 3-15.1) of vortex shedding,
number)
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1), and
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of flow
11-4.9 drag ratio of the effective drag force to inertial forces for a body moving in a The drag coefficient is strongly dependant on the shape of
𝑐
D
coefficient the body.
- fluid
2𝐹
D
𝑐 = ; where:
D
2
𝜌 𝑣 𝐴
𝐹 is drag force (ISO 80000-4:2006, item 4-9.1) on the body,
D
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of the fluid,
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of the body, and
𝐴 is the cross sectional area (ISO 80000-3:2006, item 3-3)
11-4.10 Bagnold ratio of drag force to gravitational force for a body moving in a fluid The characteristic length 𝑙 is the body’s volume divided by
𝐵𝑔
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2
number 𝑐 𝜌 𝑣 its cross sectional area.
-
d
𝐵𝑔 =
𝑙 𝑔𝜌
b
𝑐 is drag coefficient (item 11-4.9) on the body,
d
𝜌 is mass density (ISO 80000-4:2006, item 4-2), of fluid,
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of the body,
𝜌 is mass density (ISO 80000-4:2006, item 4-2), of the body,
b
𝑔 is acceleration of free fall (ISO 80000-3:2006, item 3-9.2), and
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1)
11-4.11 Bagnold ratio of drag force to viscous force in a fluid transporting solid particles
𝐵𝑎
2
2
- number
1 2
s
⁄( )
𝐵𝑎 = √1 𝑓 − 1 ; where
2
s
solid 𝜂
particles>
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of particles,
s
𝑑 is diameter (ISO 80000-3:2006, item 3-1.7) of particles,
𝛾̇= 𝑣/𝑑 is shear rate, time derivative of shear strain (ISO 80000-
4:2006, item 4-16.2),
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23) of fluid, and
𝑓 is volumic fraction of solid particles
s
lift coefficient The lift coefficient is dependent on the shape of the wing.
11-4.12 ratio of the lift force available from a wing at a given angle of attack to
𝑐 , 𝑐
l A
- the inertial force for a wing shaped body moving in a fluid
2𝐹 𝐹
𝑙 𝑙
𝑐 = = ; where
𝑙 2
𝜌 𝑣 𝑆 𝑞𝑆
𝐹 is lift force (ISO 80000-4:2006, item 4-9.1) on the wing,
𝑙
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of the fluid,
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of the body,
2
𝑞 = 𝜌𝑣 /2 is dynamic pressure, and
𝑆 = 𝐴cos𝛼 is effective area (ISO 80000-3:2006, item 3-3) when 𝛼 is the
angle of attack and A is area of the wing
11-4.13 thrust ratio of the effective thrust force available from a propeller to the The thrust coefficient is dependent on the shape of the
𝑐
t
coefficient propeller.
- inertial force in a fluid
2 4
⁄( )
𝑐 = 𝐹 𝜌 𝑛 𝑑 ; where
t T
𝐹 is thrust force (ISO 80000-4:2006, item 4-9.1) of the propeller;
T
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𝑛 is rotational frequency (ISO 80000-3:2006, item 3-15.2),
𝑑 is tip diameter (ISO 80000-3:2006, item 3-1.7) of the propeller, and
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of the fluid
11-4.14 Dean number ratio of centrifugal force to inertial force for flow of fluids in curved
𝐷𝑛
- pipes
2𝑣𝑟 𝑟
1⁄2
( )( )
𝐷𝑛 = 𝑣 𝑑 𝜌/𝜂 𝑑 /2𝑅 = ; where
√
𝜈 𝑅
𝑣 is (axial) speed (ISO 80000-3:2006, item 3-8.1),
𝑑 is diameter (ISO 80000-3:2006, item 3-1.7) of pipe,
𝜌 is mass density (ISO 80000-4:2006, item 4-2),
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23),
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24)
𝑟 = 𝑑/2 is radius of pipe
𝑅 is the radius of curvature (ISO 80000-3:2006, item 3-1.13) of the
path of the pipe
Bejan number A similar number exists for heat transfer (item 11-5.9).
11-4.15 ratio of mechanical work to frictional energy loss in fluid dynamics in a
𝐵𝑒
- pipe
2 2
Δ𝑝 𝑙 𝜌 Δ𝑝 𝑙
𝐵𝑒 = = ; where
2
𝜂 𝜈 𝜂
Δ𝑝 is drop of pressure (ISO 80000-4:2006, item 4-15.1) along a pipe,
𝑙 is length (ISO 80000-3:2006, item 3-1.1),
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23),
𝜌 is mass density (ISO 80000-4:2006, item 4-2), and
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24), momentum
diffusivity
The Lagrange number is also given by
11-4.16 Lagrange ratio of mechanical work to frictional energy loss in fluid dynamics in a
𝐿
𝑔
number
- pipe
𝐿𝑎 = 𝑅𝑒 ⋅ 𝐸𝑢; where
𝑙 Δ𝑝
𝐿𝑔 = ; where
𝜂 𝑣
𝑅𝑒 is the Reynolds number (item 11-4.1) and 𝐸𝑢 is the
𝑙 is length (ISO 80000-3:2006, item 3-1.1) of a pipe,
Euler number (item 11-4.2)
Δ𝑝 is drop of pressure (ISO 80000-4:2006, item 4-15.1) along a pipe,
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23), and
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𝑣 is speed (ISO 80000-3:2006, item 3-8.1)
11-4.17 Bingham ratio of yield stress to viscous stress in a viscous material for flow of
𝐵𝑚, 𝐵𝑛
number;
- viscoplastic material in channels
𝜏 𝑑
𝐵𝑚 = ; where
plasticity
𝜂 𝑣
number
𝜏 is shear stress (ISO 80000-4:2006, item 4-15.3),
𝑑 is characteristic diameter (ISO 80000-3:2006, item 3-1.7), e.g.
effective channel width,
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23), and
𝑣 is speed (ISO 80000-3:2006, item 3-8.1)
11-4.18 Hedström ratio of yield stress to viscous stress of a viscous material at flow
𝐻𝑒, 𝐻𝑑
- number limit.for visco-plastic material in channels at flow limit
2
𝜏 𝑑 𝜌
0
𝐻𝑒 = ; where
2
𝜂
𝜏 is shear stress (ISO 80000-4:2006, item 4-15.3) at flow limit,
0
𝑑 is characteristic diameter (ISO 80000-3:2006, item 3-1.7), e.g.
effective channel width,
𝜌 is mass density (ISO 80000-4:2006, item 4-2), and
𝜂 is dynamic viscosity (ISO 80000-4:2006, item 4-23)
11-4.19 Bodenstein representation of the transfer of matter by convection in reactors with The Bodenstein number is also given by
𝐵𝑑
number ∗
- respect to diffusion
𝐵𝑑 = 𝑃𝑒 = 𝑅𝑒 ⋅ 𝑆𝑐; where
𝐵𝑑 = 𝑣 𝑙/𝐷; where
∗
𝑃𝑒 is Péclet number for mass transfer (11-6.2),
𝑣 is speed (ISO 80000-3:2006, item 3-8.1),
𝑅𝑒 is Reynolds number (11-4.1), and
𝑙 is length (ISO 80000-3:2006, item 3-1.1) of reactor, and
⁄( ) ⁄
𝑆𝑐 = 𝜂 𝜌𝐷 = 𝜈 𝐷 is Schmidt number (11-7.2).
𝐷 is diffusion coefficient (ISO 80000-9:2009, item 9-45)
11-4.20 Rossby ratio of inertial forces to Coriolis forces in the context of transfer of The Rossby number represents the effect of earth's
𝑅𝑜
- number; matter for the flow of a rotating fluid rotation on flow in pipes, rivers, ocean currents,
⁄( )
tornadoes, etc.
Kiebel 𝑅𝑜 = 𝑣 2 𝑙 𝜔 sin𝜙 ; where
E
number
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of motion,
The quantity 𝜔 sin𝜙 is called Coriolis frequency.
E
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1), the scale
of the phenomenon;
𝜔 is angular velocity (ISO 80000-3:2006, item 3-10) of earth's
E
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rotation, and
𝜙 is angel (ISO 80000-3:2006, item 3-5) of latitude
11-4.21 Ekman ratio of viscous forces to Coriolis forces in the context of transfer of In plasma physics the square root of this number is used.
𝐸𝑘
- number matter for the flow of a rotating fluid
The Ekman number is also given by
2
⁄( )
𝐸𝑘 = 𝜈 2 𝑙 𝜔 sin𝜙 ; where
E
⁄
𝐸𝑘 = 𝑅𝑜 𝑅𝑒; where
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24),
𝑅𝑜 is the Rossby number and 𝑅𝑒 is the Reynolds number.
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1), the scale
of the phenomenon,
𝜔 is angular frequency 𝜔 (ISO 80000-3:2006, item 3-10) of earth’s
E
rotation, and
𝜙 is angel (ISO 80000-3:2006, item 3-5) of latitude
See also Deborah Number (item 11-7.8).
11-4.22 Elasticity ratio of relaxation time to diffusion time in viscoelastic flows
𝐸𝑙
2
⁄
number
- 𝐸𝑙 = 𝑡 𝜈 𝑟 ; where
𝑟
𝑡 is relaxation time (ISO 80000-12:2009, item 12-33.1),
𝑟
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24), and
𝑟 is radius (ISO 80000-3:2006, item 3-1.5) of pipe
𝑓 = 4𝑓 (Fanning friction factor)
11-4.23 Darcy friction representation of pressure loss in a pipe due to friction within the fluid
𝐷 𝑓
𝑓
D
- factor; in a laminar or turbulent flow of a fluid in a pipe
2Δ𝑝 𝑑
Moody
𝑓 = ; where
D
2
𝜌 𝑣 𝑙
friction factor
Δ𝑝 is drop of pressure (ISO 80000-4:2006, item 4-15.1) due to friction,
𝑙 is length (ISO 80000-3:2006, item 3-1.1) of the pipe,
𝑑 is diameter (ISO 80000-3:2006, item 3-1.7) of the pipe,
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of the fluid, and
𝑣 is (average) speed (ISO 80000-3:2006, item 3-8.1) of the fluid in
the pipe
11-4.25 Fanning ratio of shear stress to dynamic pressure in the flow of a fluid in a The Fanning number describes the flow of fluids in a pipe
𝑓 𝑓
n
- number , containment with friction at the walls represented by its shear stress.
2𝜏
𝑓 = ; where
n Symbol 𝑓 can be used where no conflicts are possible.
2
𝜌 𝑣
𝜏 is shear stress (ISO 80000-4:2006, item 4-15.3) at the wall,
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No. Name Symbol Definition Remarks
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of the fluid, and
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of the fluid in the pipe
11-4.26 Goertler characterization of the stability of laminar boundary layer flows in The Goertler parameter represents the ratio of centrifugal
𝐺𝑜
effects to viscous effects.
- parameter transfer of matter in a boundary layer on a curved surface
1⁄2
𝑣 𝑙 𝑙
𝑏 𝑏
( )
𝐺𝑜 = ; where
𝜈 𝑟
𝑐
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) m/s
𝑙 is boundary layer thickness (ISO 80000-3:2006, item 3-1.4),
𝑏
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24), and
𝑟 is radius of curvature (ISO 80000-3:2006, item 3-1.13)
𝑐
𝑑𝑝
11-4.27 Hagen generalization of Grashof number (11.-4.4) for forced or free convection
For free thermal convection with = 𝜌 𝑔𝛼 𝛥𝑇 the Hagen
V
𝐻𝑔, 𝐻𝑎
𝑑𝑥
- number in laminar flow
number then coincides with the Grashof number (11-4.4).
3
1 𝑑𝑝 𝑙
𝐻𝑔 = − ; where
See also Poisseuille number (item 11-4.29).
2
𝜌 𝑑𝑥 𝜈
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of fluid kg/m³,
𝑙 is a characteristic length (ISO 80000-3:2006, item 3-1.1),
𝜈 is kinematic viscosity (ISO 80000-4:2006, item 4-24), and
𝑑𝑝
is gradient of pressure (ISO 80000-4:2006, item 4-15.1)
𝑑𝑥
11-4.28 Laval number ratio of speed to the (critical) sound speed in the throat of the nozzle The Laval number is a specific kind of Mach number (item
𝐿𝑎
11-4.6).
- ⁄ ( ) ⁄( )
𝐿𝑎 = 𝑣 √ 𝑅 𝑇2𝛾 𝛾 + 1 ; where
𝑠
𝑣 is speed (ISO 80000-3:2006, item 3-8.1),
𝛾 is ratio of the specific heat capacities (ISO 80000-5:2007, item 5-
17.1),
𝑇 thermodynamic temperature (ISO 80000-5:2007, item 5-1),
𝑅
𝑅 = is specific gas constant; with
𝑠
𝑀
𝑅 = molar gas constant (ISO 80000-9:2009, item 9-42), and
𝑀 = molar mass (ISO 80000-9:2009, item 9-5)
𝑃𝑜𝑖 = 32 for laminar flow in a round pipe.
11-4.29 Poiseuille ratio of propulsive force by pressure to viscous force for a flow of fluids
𝑃𝑜𝑖
- number in a pipe
See also Hagen number (item 11-4.27).
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No. Name Symbol Definition Remarks
2
Δ𝑝 𝑑
𝑃𝑜𝑖 = − where
𝑙 𝜂 𝑣
Δ𝑝 is drop of pressure (ISO 80000-4:2006, item 4-15.1)
Pa = N/m² = kg/ms²
𝑙 is length (ISO 80000-3:2006, item 3-1.1),
𝑑 is diameter (ISO 80000-3:2006, item 3-1.7),
𝜂 is dynamic viscosity (ISO 80000-4: 2006, item 4-23), and
𝑣 is speed (ISO 80000-3:2006, item 3-8.1)
11-4.30 power ratio of power consumption by agitators due to drag (on agitator,
𝑃𝑛
- number impeller) to rotational inertial power in fluids
3 5
⁄( )
𝑃𝑛 = 𝑃 𝜌 𝑛 𝑑 ; where
𝑃 is active power (IEC 80000-6:2008, item 6-56) consumed by a
stirrer,
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of fluid,
𝑛 is rotational frequency (ISO 80000-3:2006, item 3-15.2), and
𝑑 is diameter (ISO 80000-3:2006, item 3-1.7) of stirrer
11-4.31 Richardson ratio of potential energy to kinetic energy for a falling body In geophysics differences of these quantities are of
𝑅𝑖
2
⁄
- number 𝑅𝑖 = 𝑔ℎ 𝑣 ; where interest.
𝑔 is acceleration of free fall (ISO 80000-3:2006, item 3-9.2),
ℎ is a representative height (ISO 80000-3:2006, item 3-1.3), and
𝑣 is a representative speed (ISO 80000-3:2006, item 3-8.1)
11-4.32 Reech ratio of a speed of an object submerged in water to water wave The Reech number can be used to determine the
𝑅𝑒𝑒
- number propagation speed resistance of a partially submerged object (e.g. a ship) of
2
⁄
𝑅𝑒𝑒 = 𝑔𝑙 𝑣 ; wh
...
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