EN ISO 80000-11:2020

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

• Status: Published
• Publication Date: 30-Jan-2021
• Withdrawal Date: 29-Apr-2021
• ICS: 01.060 - Quantities and units
• Technical Committee: CEN/SS F02 - Units and symbols
• Drafting Committee: CEN/SS F02 - Units and symbols
• Current Stage: 6060 - Definitive text made available (DAV) - Publishing
• Start Date: 28-Oct-2020
• Completion Date: 28-Oct-2020
• Ref Project: SIST EN ISO 80000-11:2020 - Quantities and units - Part 11: Characteristic numbers (ISO 80000-11:2019)

Relations

• Replaces: EN ISO 80000-11:2013 - Quantities and units - Part 11: Characteristic numbers (ISO 80000-11:2008)
• Effective Date: 08-Jun-2022

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

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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.
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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.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Fax: [phone]
Email: [e-mail]
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.
© ISO 2019 – All rights reserved v
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SIST EN ISO 80000-11:2020
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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.
© ISO 2019 – All rights reserved 1
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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
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):
Δp is drop of pressure (ISO 80000-4),
Eu′= Eu ; where
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
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
of a hot plate, the diameter of a pipe, or the effective
Gr=ΔlgανT/ ; where
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),
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
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,

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

body moving in a fluid, expressed by shape of the body.
c = ; where
ρv A
F is drag force (ISO 80000-4) on the body,
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.
c ρv
Bg= ; where
lgρ
c is drag coefficient (item 11-4.9) of the body,
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
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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

particles, expressed by
ργd
s 12/
Ba =−11/ f ; where
2 s
ρ is mass density (ISO 80000-4) of particles,
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

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

and the inertial force for a wing shaped body moving in a fluid, wing.
expressed by
2F F
c == ; where
ρv S
F is lift force (ISO 80000-4) on the wing,
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
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

cient the inertial force in a fluid, expressed by propeller.
cF= / ρnd ; where
F is thrust force (ISO 80000-4) of the propeller,
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
diffusivity.
Δpl ρΔpl
Be= = ; where
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
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
plasticity num- τd
Bm= ; where
ber
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
τρd
He= ; where
τ is shear stress (ISO 80000-4) at flow limit,
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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SIST EN ISO 80000-11:2020
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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
The quantity ωϕsin is called Coriolis frequency.
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
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.
The Ekman number is also given by
Ek=νω/2l sinϕ ; where
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
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
El=trν/ ; where
t is relaxation time (ISO 80000-12),
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

factor; laminar or turbulent flow of a fluid in a pipe, expressed by
Moody friction 2Δp d
f = ; where
factor
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

fluid in a containment, expressed by a pipe with friction at the walls represented by its

shear stress.
f = ; where
n Symbol f may be used where no conflicts are possible.
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
 
Go= ; where
 
ν r
 
v is speed (ISO 80000-3),
l is boundary layer thickne
...

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01-marec-2017
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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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ii © ISO 2017 – All rights reserved
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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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 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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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
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
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-
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
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

coefficient the body.
- fluid
2𝐹
𝑐 = ; where:
𝜌 𝑣 𝐴
𝐹 is drag force (ISO 80000-4:2006, item 4-9.1) on the body,
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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number 𝑐 𝜌 𝑣 its cross sectional area.
𝐵𝑔 =
𝑙 𝑔𝜌
𝑐 is drag coefficient (item 11-4.9) on the body,
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,
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
- number 1 2
⁄( )
𝐵𝑎 = √1 𝑓 − 1 ; where
solid 𝜂
particles>
𝜌 is mass density (ISO 80000-4:2006, item 4-2) of particles,
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
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,
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,
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

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;
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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
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

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
𝜏 𝑑 𝜌
𝐻𝑒 = ; where
𝜏 is shear stress (ISO 80000-4:2006, item 4-15.3) at flow limit,
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
number
𝑣 is speed (ISO 80000-3:2006, item 3-8.1) of motion,
The quantity 𝜔 sin𝜙 is called Coriolis frequency.
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
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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
⁄( )
𝐸𝑘 = 𝜈 2 𝑙 𝜔 sin𝜙 ; where
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
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
number
- 𝐸𝑙 = 𝑡 𝜈 𝑟 ; where
𝑡 is relaxation time (ISO 80000-12:2009, item 12-33.1),
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
- factor; in a laminar or turbulent flow of a fluid in a pipe
2Δ𝑝  𝑑
Moody
𝑓 = ; where
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

- 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.
 𝜌 𝑣
𝜏 is shear stress (ISO 80000-4:2006, item 4-15.3) at the wall,
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oSIST prEN ISO 80000-11:2017
ISO/DIS 80000-11:2017(E)
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
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),
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)
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
𝐻𝑔, 𝐻𝑎
𝑑𝑥
- number in laminar flow
number then coincides with the Grashof number (11-4.4).
1 𝑑𝑝 𝑙
𝐻𝑔 = − ; where
See also Poisseuille number (item 11-4.29).
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
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 = is specific gas constant; with
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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oSIST prEN ISO 80000-11:2017
ISO/DIS 80000-11:2017(E)
No. Name Symbol Definition Remarks
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
- 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

𝑅𝑒𝑒 = 𝑔𝑙 𝑣 ; wh
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