## Microsoft word - deltaflow berechnungsgrundlagen_e.doc

**Calculation Basics**
Telephone 089 - 80906-0, Fax 089 - 80906-200

**Welcome to systec Controls**
The deltaflow dynamic pressure probe you have
purchased is a highly precise measurement tool of
superior quality. You can count on it for the very best
performance even under the most adverse conditions.

In this booklet, the pressure unit should always be
Accurate evaluation of all measurements and precise
interpreted as absolute pressure (index abs). If gauge
calculation of flow are integral factors contributing to
pressure values are the only values which are available
(index ü or g), then the current or average ambient
pressure at the sampling site must be added. The
This booklet contains all information you will need in
average ambient pressure at sea level is 101,325Pa.

order to achieve optimal measurement results with

**Table of Contents**
Converting the Mass Flow to Other Units.6

**Basics / Units of Measurement**
All units of measurement used in this booklet are SI
units. If you use a different unit system, we
recommend that you convert your values to SI units
before performing calculations and then afterwards
convert them back to the unit system you are using.

Instructions for converting the mass flow into volume
flow, standard volume flow, or speed are described
deltaflow Basic Calculation Information 07/03

In contrast to many other primary elements, the
deltaflow’s resistance coefficient in turbulent flow is
not dependent on the flow itself or on the medium

**General Basic Calculation Information**
If for some reason you do not receive a calculation
The way flow is calculated with the deltaflow is similar
sheet with your deltaflow, you can order one from
to the way flow is calculated according to EN ISO
systec Controls by providing the serial number from
5167-1 (formerly DIN 1952). The following formula
your unit. Or you can use the free deltacalc software,
which can be downloaded from the systec Controls

**Expansion Number **ε

The expansion number ε (Epsilon) defines the effect of
the pressure loss and the resulting change in the
density of the medium on the flow measurement.

In the case of incompressible media (liquids), pressure
loss at the primary element does not result in any
change in density, so the expansion number is 1.

In the case of compressible media (gasses, steam), the
expansion number varies proportionally from 1 as the
amount of pressure loss at the primary element
increases and as the static pressure within the conduit
Because the deltaflow causes only a very minor
pressure loss, the expansion number is usually very
close to 1. You will find the expansion number at the
design point ε listed on your deltaflow calculation
When calculating flow, the expansion number is often
assumed to be a constant. To do this, the expansion
number is generally calculated as 2/3 of the maximum
flow qmax ( ε 2 / 3 ). For deltaflow purposes it is very

**Resistance Coefficient**
You will find the resistance coefficient ζ (Zeta) on
For applications in which the expansion number varies
your deltaflow calculation sheet. In some documents
significantly from 1, calculating the actual current
the probe factor K (“K number”) is used instead of ζ .

expansion number can increase the accuracy of the
There is a simple correspondence between ζ and the K
quantity measurement. The actual current expansion
number, and conversion between the two is easy:
number ε can be calculated from the design
deltaflow Basic Calculation Information 07/03

performed within the transducer when evaluating the
measurement signals from differential pressure

**Conduit Interior Diameter Conduit d**
**Current Operational Density **ρ

*B*
The interior diameter of the conduit has a great impact
on the accuracy of the flow calculation. For this
composition, the temperature, and the pressure which
reason we recommend that you determine the exact
interior diameter measurement at the sampling point
when conducting measurements which require a high
The density of liquids is only affected in a very slight
degree of accuracy. The mathematical average from
way by the pressure. For this reason, liquids are also
multiple measurements can be used when dealing with
referred to as “incompressible media.” Likewise, the
effect that temperature has on liquids is substantially
less than the effect it has on gasses or steams.

The interior diameter can change at high temperaturesdue to the thermal expansion of the material. For this
There are basically two principal approaches to
reason, the calculation sheet of your deltaflow specifies
calculating the density of media: calculation based on
the “warm” interior diameter of your conduit. This
tables and calculation based on equations.

means that it is only necessary—and practical—tocalculate the actual interior diameter at a given
Calculating density based on tables is very simple and
exact, but it does require that a density table exist and

*d *) if the process temperatures change
dramatically during the course of operation. The warm
be available. Density tables for many common media
interior diameter is calculated as follows:
can be found in the appendix of this booklet. It is
permissible to construct a linear interpolation between
sampling points when there is no phase transition (i.e.

boiling or melting point) located between the
In this formula, α represents the vertical (length)
expansion coefficient of the conduit material. For most
There are many formulas for calculating density based
steels, the value of α falls between 10*10e-6 and
on equations. These formulas vary in their ease of use
16*10e-6. A temperature increase of 100K, for
and in their accuracy. A few common formulas are
example, would result in an increased diameter
measurement of 0.13% based on the material, and the
Calculating the Density of Liquids Using a VolumeExpansion Coefficient
A table containing the vertical expansion coefficient
values of various materials is located in the Appendix.

One simple method of calculating the medium density
of liquids is to use constant volume expansion

**Measured Differential Pressure dp**
As you can see in Equation (1), the differential pressure
is located below the radical when calculating flow.

Many differential pressure transducers can
automatically extract the root given the measured
differential pressure signal. The output signal from this
type of transducer is then no longer proportional to
⎜1+ 20.7 *10

*e *− 5 (30

*K *)⎟
the differential pressure, but is instead proportional to
the root of the differential pressure.

Consult a volume of tables, and you will find an actual
It is therefore extremely important to be aware of
density of 988.0 kg/m³ at 1 bar and 50°C. The error in
whether or not the root extraction is automatically
the density calculation in this example equals 0.4%,
deltaflow Basic Calculation Information 07/03

and the resulting error in the flow calculation will
Calculating the density of gasses by means of tables is
also a simple and precise procedure.

You will find the volume expansion coefficients of
If no tables are available, or if the medium in question
several liquids in the appendix of this booklet.

is a mixture of gasses, then various calculation
equations are available, among them van der Waals,
Other Formulas for Calculating the Density of
Redlich Kwong, and many others. Various calculation
models are described in detail in the VDI-Wärmeatlas.

You will also find extensive tables with necessary
The densities of supercooled liquids and overheated
calculation constants for many different types of media
gasses, hydrocarbons, and mixtures of materials are
frequently calculated by using the model equations of
Lee and Kessler (Lee, B.I. and M.G. Kessler: AICHE J.21
The ideal gas equation is a very simple equation which
(1975) pg.510). The calculation model is described in
often provides sufficient accuracy when calculating
detail in the VDI-Wärmeatlas (Association of Engineers
density within short distances from the design point.

Thermal Atlas). You will find extensive tables with
calculation constants necessary for many different
types of media in the same reference source.

Calculating the Density of Steam and Water
The more operational pressure and temperature
The IAPWS Equation is used for the exact calculation of
deviate from the design point, the more unreliable the
the density (and other state variables) of water and
calculation. This is especially true when the operational
water steam. (IAPWS is the International Assosiation
level approaches the boiling point of the gas. When
this happens, the pressure increases and the
temperature decreases. At greater distances from the
boiling point, the ideal gas equation generally provides
The IAPWS Equation requires a considerable amount ofnumerical and mathematical process. The precise
The ideal gas equation can also be used for overheated
definition has been documented in an extensive
steam. The same conditions apply as for other gasses.

publication (W. Wagner and A. Pruss, "The IAPWSFormulation 1995 for the Thermodynamic Properties of
Ordinary Water Substance for General and ScientificUse," J. Phys. Chem. Ref. Data, 31, 387-535 (2002)).

Various pre-programmed source codes or libraries are
available for purchase (see www.ruhr-uni-
If the operational position is not too far from the
design point, the density of water can be roughly
calculated by using a constant volume expansion
coefficient (see description above). The density of
overheated steam can be calculated by using the ideal
gas equation (see description below) at small distances
According to the IAPWS 95, the actual density at 2.15
Mpa and 543.15K is 9.221 kg/m³. The error in density
calculation, then, is 0.92%, and the resulting error in
It is simple matter to calculate density by using tables
the flow measurement equals approximately 0.46%.

(refer to the Water/Steam Table in the Appendix).

Interpolating beyond the boiling point, however, could
Density Correction for Water-laden Gasses
result in large discrepancies and is not recommended.

Gases can absorb water (humidity). Mixtures of gas
and water have a different density than “pure,” or dry,
gasses. The amount of water a gas is able to absorb
increases with the temperature of that gas. Very hot
deltaflow Basic Calculation Information 07/03

gasses that have been run through a wash can
sometimes absorb a considerable amount of water and
according to the calculation sheet of your deltaflow.

their fluid data can then be significantly different from
The radical term corresponds to the output signal of a
root extracting differential pressure transducer.

The process for calculating the density of water-laden
If the final value of the transducer is set to correspond
gasses and making the necessary corrections to the
with the calculation sheet of your deltaflow, then the
flowmetering is detailed in the VDI / VDE 2040, Part 4.

root extracted output signal of the dp-transducer is
proportional to the flow measurement.

Natural gas is actually a mixture of various gasses in
Prerequisite conditions for simplification:
varying combinations. The primary elements are usually
Constant expansion number, ideal gas behavior,
methane and nitrogen as well as other hydrocarbons
There are various calculation models available for
determining gas density. The ideal gas equation is
often used for basic control purposes.

Again in this equation, the last radical term
Custody transfer applications and measurements which
corresponds to the output signal of a root-extracting
require a high degree of precision are usually
differential pressure transducer. The flow
calculated using the AGA or the GERG Equation.

measurement, therefore, results from the designed
mass flow multiplied by the output signal of a root-
The AGA NX family of equations the standard in most
extracting transmitter. The density correction is
non-European countries. This equation is described in
handled by the middle radical term, which
incorporates the operational pressure and temperature
as well as the design pressure and temperature.

The GERG88 Equation was developed by European gassuppliers and is widely-used within the European
You will find the design data (Index D) on the
region as a standard for calculating gas quantities. The
calculation sheet for your deltaflow.

GERG88 is described in the DVGW Worksheet G486.

(DVGW is he German Technical and Scientific
The simplified flow equation for gasses can also be
Association for Gas and Water and is headquartered in
used to make a rough calculation of the volume

**Simplified Calculation Equations**
**Converting the Mass Flow to Other Units**
The following simplified calculation equations can be
used for applications with limited demand for precision
The standard volume flow is primarily used in
such as in-house volume measurements and for
calculating gas volumes. The standard state of a gas is
usually based on 273.15K (0°C) and 10135.5 Pa
(1.01325 bar). The standard volume flow is calculated
Prerequisite conditions for simplification:
Constant density, incompressible liquid, constant
A table containing various standard densities is located
deltaflow Basic Calculation Information 07/03

The volume flow is used quite often for liquids. The
influences of pressure and temperature on virtually any
volume flow is calculated using the following
Because it uses a freely definable unit system, a user
can input and output any and all data in whatever
The process of determining operational density is
flowcom is easy to set up under Windows.

described above in the section “Current Operational
Detailed information about the flowcom product can
be found at: www.systec-controls.de.

The process for identifying the average velocity within
conduits is often used to determine pressure loss. The
velocity of a medium can be calculated using the

**Questions?**
No one knows the deltaflow better than we do! Please
we are more than pleased to be of assistance. Within
Germany we have a network of field representatives,
and our distributors in other countries are also pleased
to help you. You can find out who is located in your

**flowcom Flow Calculator**
The medium data for a whole series of gasses are

**http://www.systec-controls.de**
Calculator. The flowcom performs the calculations as
Or simply place a telephone call to our headquarters in
described in this information booklet and does so in a

**systec Controls Hotline:**

089 - 80906 – 108
In particular, the flowcom uses extensive tables to
calculate the medium densities and, for natural gas, it
comes pre-equipped with the GERG88 functional
The flowcom allows for the effects of the expansion
number, the thermal conduit expansion, as well as
other nonlinear characteristics of flowmeters.

Even when used independently from the deltaflow, the
flowcom can calculate and compensate for thedeltaflow Basic Calculation Information 07/03

**Appendix**
**Densities **ρ

** of Various Liquids at 20°C in kg/l**
**Densities **ρ

** of Various Gasses at 20°C and 1013.25 bar**
**Length Expansion Coefficients **α

** of Solid Materials at**
**20°C in 10e-6/K**
deltaflow Basic Calculation Information 07/03

**Volume Expansion Coefficients **γ

** of Liquids at 20°C in**
**Density **ρ

** of Air in kg/m³**
**Relative to Pressure and Temperature**
**Density **ρ

** of Water in kg/m³**
**Relative to Pressure and Temperature**
**Pressure**
**Temperature in **°

**C**
**5.00 **1000.0

**10.00 **1000.3

**20.00 **1000.8

**30.00 **1001.3

**40.00 **1001.8

**50.00 **1002.3

**60.00 **1002.8

**70.00 **1003.3

**80.00 **1003.8

**90.00 **1004.3

**100.00 **1004.8

**Volume Expansion Coefficients **γ

** of Gasses at 0.100°C**
**150.00 **1007.3

**in 10e-5/K**
**200.00 **1009.7

**250.00 **1012.1

**300.00 **1014.5

**350.00 **1016.9

**400.00 **1019.2

**450.00 **1021.5

**500.00 **1023.8

**600.00 **1028.3

**700.00 **1031.7

**800.00 **1037.0

**900.00 **1041.2

**1000.00 **1045.3

**Temperature in °C**
**Pressure**
deltaflow Basic Calculation Information 07/03

Source: http://www.systec-controls.de/files/deltaflow_calculation_basics.pdf

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