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Calculation Basics
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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 ρB
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 *10e − 5 (30K )⎟ 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 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:
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
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
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
deltaflow Basic Calculation Information 07/03


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