ISO 8426-1:2025

Hydraulic fluid power - Determination of derived displacement of positive displacement pumps and motors - Part 1: Two-step Toet method

This document specifies the two-step Toet method for the determination of the derived displacement of hydraulic fluid power positive displacement pumps and motors under steady-state conditions. A value for the derived displacement is determined from measurements at multiple shaft speeds.

Transmissions hydrauliques — Détermination de la cylindrée calculée des pompes et moteurs volumétriques — Partie 1: Méthode Toet à deux étapes

General Information

• Status: Published
• Publication Date: 13-Aug-2025
• ICS: 23.100.10 - Pumps and motors
• Technical Committee: ISO/TC 131/SC 8 - Product testing
• Drafting Committee: ISO/TC 131/SC 8 - Product testing
• Current Stage: 6060 - International Standard published
• Start Date: 14-Aug-2025
• Due Date: 21-Sep-2025
• Completion Date: 14-Aug-2025

Relations

• Revises: ISO 8426:2008 - Hydraulic fluid power - Positive displacement pumps and motors - Determination of derived capacity
• Effective Date: 02-Jul-2022

Overview

ISO 8426-1:2025 - "Hydraulic fluid power - Determination of derived displacement of positive displacement pumps and motors - Part 1: Two-step Toet method" specifies a standardized laboratory method to determine the derived displacement (derived capacity) of positive-displacement hydraulic pumps and motors under steady-state conditions. The standard defines how to collect flow and speed data across multiple shaft speeds and differential pressures and prescribes the two-step Toet regression procedure to compute the derived displacement value.

Key technical topics and requirements

• Scope and definitions: Applies to hydraulic fluid power positive-displacement pumps and motors. Terminology follows ISO 5598.
• Test planning:
  • Use five or more shaft speeds and five or more differential pressures (≥25 measurement points total).
  • Shaft speeds and differential pressures should be increased in approximately equal increments; targeted values must meet a ±5% tolerance to avoid clustering.
  • Guidance for maximum targeted differential pressure is provided relative to rated pressure (different rules for ≤50 bar and >50 bar).
• Measurement requirements:
  • Follow ISO 4409:2019 for basic steady-state testing, flowmeter placement, fluid properties, temperature, and measurement accuracy classes.
  • Special considerations for variable displacement units, reverse-flow operation, and whether the unit is tested as a pump or motor (results can differ).
• Computation - Two-step Toet method:
  • Step 1: For each targeted differential pressure, perform ordinary least squares (OLS) regression of high-pressure flow rate versus shaft speed to obtain slopes (α_i).
  • Step 2: Regress the obtained slopes α_i versus differential pressure Δp; the intercept at Δp = 0 is the derived displacement V (slope of flow vs speed at zero Δp).
  • Formulae and fitting guidance are provided in Annex B; test report examples are in Annex A.
• Reporting:
  • Required test report information includes unit ID, test date/location, speed and pressure ranges, fluid properties, raw measured values (p_in, p_out, Δp, n, flow, temperature), derived displacement, coefficients and correlation statistics.

Practical applications and users

ISO 8426-1:2025 is intended for:

• Test laboratories and certification bodies performing hydraulic component characterization.
• Manufacturers of hydraulic pumps and motors for product development, quality control and datasheet validation.
• R&D and engineering teams assessing volumetric performance and design changes.
• Specifiers and procurement engineers who need comparable, traceable derived displacement data for system design and component selection.

Using this standard ensures repeatable, comparable derived-displacement measurements that support performance verification, troubleshooting, and compliance with industry best practices.

Related standards

• ISO 4409:2019 - Methods of testing and presenting basic steady-state performance for positive-displacement pumps and motors (referenced for test setup and measurement accuracy).
• ISO 5598 - Fluid power systems and components - Vocabulary (terms and definitions used).

Standards Content (Sample)

International
Standard
ISO 8426-1
First edition
Hydraulic fluid power —
2025-08
Determination of derived
displacement of positive
displacement pumps and motors —
Part 1:
Two-step Toet method
Transmissions hydrauliques — Détermination de la cylindrée
calculée des pompes et moteurs volumétriques —
Partie 1: Méthode Toet à deux étapes
Reference number
© ISO 2025
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]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and units. 1
5 Test procedure . 2
5.1 General requirements .2
5.2 Number of steady-state shaft speeds and differential pressures .3
5.3 Variable displacement unit .3
5.4 Reverse flow unit .3
5.5 Unit used as pump or motor .3
6 Calculation of derived displacement . 4
6.1 General .4
6.2 The two-step Toet method .4
7 Test report . 5
Annex A (informative) Test report examples . 7
Annex B (informative) Formulae for fitting a line .15
Bibliography .16

iii
Foreword
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This document was prepared by Technical Committee ISO/TC 131, Fluid power systems, Subcommittee SC 8,
Product testing.
A list of all parts in the ISO 8426 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
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iv
Introduction
In hydraulic fluid power systems, power is transmitted and controlled through a liquid under pressure
within an enclosed circuit. Two types of components of such systems are the positive displacement pumps
and motors. One of the technical parameters of these components is the derived displacement, also known
as derived capacity. The term "derived displacement" is preferred over "derived capacity". This document
describes the test procedure and analytical approach of the two-step Toet method for the determination of
the derived displacement of hydraulic fluid power positive displacement pumps and motors.

v
International Standard ISO 8426-1:2025(en)
Hydraulic fluid power — Determination of derived
displacement of positive displacement pumps and motors —
Part 1:
Two-step Toet method
1 Scope
This document specifies the two-step Toet method for the determination of the derived displacement of
hydraulic fluid power positive displacement pumps and motors under steady-state conditions. A value for
the derived displacement is determined from measurements at multiple shaft speeds.
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 4409:2019, Hydraulic fluid power — Positive-displacement pumps, motors and integral transmissions —
Methods of testing and presenting basic steady state performance
ISO 5598, Fluid power systems and components — Vocabulary
3 Terms and definitions
For the purposes of this document, the terms and definitions are given in ISO 5598 and the following apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
unit
hydraulic positive displacement pump or motor
4 Symbols and units
The symbols used throughout this document are based on ISO 4391 and shown in Table 1. Graphical symbols
are depicted in accordance with ISO 1219 series.
A diagram showing the use of symbols and graphical symbols is shown in Figure 1.

Table 1 — Symbols, terms and units
Symbol Term Units
α slope of high pressure flow rate vs shaft speed cm /r
i
j number of shaft speeds —
k number of measurement points —

number of differential pressures —
n
shaft speed r/min
a
Δp differential pressure bar (MPa)
a
p rated max pressure bar (MPa)
rated
a
maximum targeted differential pressure bar (MPa)
Δp
maxt, arget
a
minimum targeted differential pressure bar (MPa)
Δp
mint, arget
a
p inlet pressure bar (MPa)
a
target inlet pressure bar (MPa)
p
1,target
a
p outlet pressure bar (MPa)
a
target outlet pressure bar (MPa)
p
2,target
a
case pressure bar (MPa)
p
d
q high pressure flow rate l/min
ve
inlet temperature °C
θ
V derived displacement cm /r
i
a 5 2
1 bar = 10 Pa = 0,1 MPa; 1 Pa = 1 N/m .
a)  Pump b)  Motor
Figure 1 — Example use of symbols, subscripts and graphical symbols for pumps and motors
5 Test procedure
5.1 General requirements
Measurements for positive displacement pumps shall be carried out in accordance with ISO 4409:2019, 5.1
and 5.2.
Measurements for positive displacement motors shall be carried out in accordance with ISO 4409:2019, 5.1
and 5.3.
Suspect values shall be re-evaluated at the appropriate conditions to determine if the data were in error or
an anomaly in expected performance exists at these specified test conditions.
5.2 Number of steady-state shaft speeds and differential pressures
Five or more shaft speeds and five or more differential pressures shall be used. This implies 25 or more
measurement points shall be used.
The maximum targeted differential pressure, Δp , of the test is defined as follows:
maxt, arget
— If p ≤50 bar ,
rated
— for a pump use Δpp=− p ;
maxt,,argetrated 1 target
— for a motor use Δpp=− p .
maxt,,argetrated 2 target
— If p >50 bar ,
rated
p −50
rated
— for a pump use Δp = +−50 p ;
maxt,,arget 1 target
p −50
rated
— for a motor use Δp = +−50 p .
maxt,,arget 2 target
The differential pressure shall be increased in approximately equal increments from minimum targeted
differential pressure, Δp , to maximum targeted differential pressure, Δp , of the unit to
mint, arget maxt, arget
avoid clustering some of the points around one target value. A tolerance of ±5 % of Δp for the
maxt, arget
targeted differential pressures shall be met.
The shaft speed shall be increased in approximately equal increments from minimum targeted shaft speed
for constant rotation (not zero rev/min) to maximum rated shaft speed of the unit to avoid clustering some
of the points around one value. A tolerance of ±5 % of the maximum rated shaft speed for the targeted shaft
speeds shall be met.
5.3 Variable displacement unit
The data for performing the estimation of the derived displacement of variable displacement pump or motor
shall be obtained following ISO 4409:2019, 5.2.4 (pump) and respectively 5.3.4 (motor).
5.4 Reverse flow unit
If the unit can be operated in both flow directions and the derived displacement depends on the flow
direction, the derived displacement shall be determined for each.
5.5 Unit used as pump or motor
A unit intended to be used as pump shall be tested as pump and a unit intended to be used as motor shall be
tested as motor. The derived displacement determined when a given unit is tested as a pump can differ from
that determined when the unit is tested as a motor. If both operation modes are considered necessary, the
test shall be carried out for both operation modes.

6 Calculation of derived displacement
6.1 General
The mathematical definition of the derived displacement is shown in Formula 1. Expressing Formula 1 in
words, the derived displacement is the slope of the high pressure flow rate with respect to shaft speed
evaluated at zero differential pressure. One of the fundamental implications o
...