# Commission Delegated Regulation (EU) 2017/654 of 19 December 2016 supplementing Regulation (EU) 2016/1628 of the European Parliament and of the Council with regard to technical and general requirements relating to emission limits and type-approval for internal combustion engines for non-road mobile machinery, Division 3.9.3: Venturi governing equations and permissible assumptions

Extent: E+W+S+N.I.

This section describes the governing equations and permissible assumptions for calibrating a venturi and calculating flow using a venturi. Because a subsonic venturi (SSV) and a critical-flow venturi (CFV) both operate similarly, their governing equations are nearly the same, except for the equation describing their pressure ratio, r (i.e., r SSV versus r CFV). These governing equations assume one-dimensional isentropic inviscid compressible flow of an ideal gas. In point 3.9.3(d), other assumptions that may be made are described. If the assumption of an ideal gas for the measured flow is not allowed, the governing equations include a first-order correction for the behaviour of a real gas; namely, the compressibility factor, Z. If good engineering judgment dictates using a value other than Z = 1, an appropriate equation of state to determine values of Z as a function of measured pressures and temperatures may be used, or specific calibration equations may be developed based on good engineering judgment. It shall be noted that the equation for the flow coefficient, C f, is based on the ideal gas assumption that the isentropic exponent, γ, is equal to the ratio of specific heats, cp /c V . If good engineering judgment dictates using a real gas isentropic exponent, an appropriate equation of state to determine values of γ as a function of measured pressures and temperatures may be used, or specific calibration equations may be developed. Molar flow rate, ṅ [mol/s], shall be calculated by means of equation (7-139):
ṅ = C d × C f × A t × p in Z × M mix × R × T in
(7-139)
Where:
C d Discharge coefficient, as determined in point 3.9.3(a) [-]
C f Flow coefficient, as determined in point 3.9.3(b) [-]
A t Venturi throat cross-sectional area [m2]
p in Venturi inlet absolute static pressure [Pa]
Z Compressibility factor [-]
M mix Molar mass of gas mixture [kg/mol]
R Molar gas constant [J/(mol · K)]
T in Venturi inlet absolute temperature [K]
(a) Using the data collected in point 8.1.8.4 of Annex VI, C d is calculated by means of equation (7-140):
C d = ṅ ref × Z × M mix × R × T in C f × A t × p in
(7-140)
Where:
ṅ ref reference molar flow rate [mol/s]
Other symbols as per equation (7-139).
(b) C f shall be determined using one of the following methods:
(i) For CFV flow meters only, C fCFV is derived from Table 7.5 based on values for β (ratio of venturi throat to inlet diameters) and γ (ratio of specific heats of the gas mixture), using linear interpolation to find intermediate values:
Table 7.5
C fCFV versus β and γ for CFV flow meters
C fCFV
β
γ exh = 1,385
γ dexh = γ air = 1,399
0,000
0,6822
0,6846
0,400
0,6857
0,6881
0,500
0,6910
0,6934
0,550
0,6953
0,6977
0,600
0,7011
0,7036
0,625
0,7047
0,7072
0,650
0,7089
0,7114
0,675
0,7137
0,7163
0,700
0,7193
0,7219
0,720
0,7245
0,7271
0,740
0,7303
0,7329
0,760
0,7368
0,7395
0,770
0,7404
0,7431
0,780
0,7442
0,7470
0,790
0,7483
0,7511
0,800
0,7527
0,7555
0,810
0,7573
0,7602
0,820
0,7624
0,7652
0,830
0,7677
0,7707
0,840
0,7735
0,7765
0,850
0,7798
0,7828
(ii) For any CFV or SSV flow meter, equation (7-141) may be used to calculate C f:
C f = 2 × γ × r γ − 1 γ − 1 γ − 1 × β 4 − r − 2 γ 1 2
(7-141)
Where:
γ isentropic exponent [-]. For an ideal gas, this is the ratio of specific heats of the gas mixture, cp /c V
r pressure ratio, as determined in paragraph (c)(3) of this point
β ratio of venturi throat to inlet diameters
(c) The pressure ratio r shall be calculated as follows:
(i) For SSV systems only, r SSV shall be calculated by means of equation (7-142):
r SSV = 1 − Δ p SSV p in
(7-142)
Where:
Δp ssv differential static pressure; venturi inlet minus venturi throat [Pa]
(ii) For CFV systems only, r CFV shall be calculated iteratively by means of equation (7-143):
r 1 − γ γ CFV + γ − 1 2 × β 4 × r 2 γ CFV = γ + 1 2
(7-143)
(d) Any of the following simplifying assumptions of the governing equations may be made, or good engineering judgment may be used to develop more appropriate values for testing:
(i) For emission testing over the full ranges of raw exhaust gas, diluted exhaust gas and dilution air, the gas mixture may be assumed to behave as an ideal gas: Z = 1;
(ii) For the full range of raw exhaust gas a constant ratio of specific heats of γ = 1,385 may be assumed;
(iii) For the full range of diluted exhaust gas and air (e.g., calibration air or dilution air), a constant ratio of specific heats of γ = 1,399 may be assumed;
(iv) For the full range of diluted exhaust gas and air, the molar mass of the mixture, M mix [g/mol], may be considered as a function only of the amount of water in the dilution air or calibration air, x H2O, determined as described in point 3.3.2 and shall be calculated by means of equation (7-144):
M mix = M air· (1 –x H2O) +M H2O· (x H2O)
(7-144)
Where:
M air 28,96559 g/mol
M H2O 18,01528 g/mol
x H2O amount of water in the dilution or calibration air [mol/mol]
(v) For the full range of diluted exhaust gas and air, a constant molar mass of the mixture, M mix, may be assumed for all calibration and all testing as long as assumed molar mass differs no more than ± 1 % from the estimated minimum and maximum molar mass during calibration and testing. This assumption may be made if sufficient control of the amount of water in calibration air and in dilution air is ensured, or if sufficient water is removed from both calibration air and dilution air. Table 7.6 provides for examples of permissible ranges of dilution air dew point versus calibration air dew point:
Table 7.6
Examples of dilution air and calibration air dew points at which a constant M mix may be assumed
Range valid for all calibration and emission testing over the atmospheric pressure range (80,000 to 103,325) kPa.
If calibration T dew (°C) is ...
the following constant M mix (g/mol) is assumed
for the following ranges of T dew (°C) during emission tests
dry
28,96559
dry to 18
0
28,89263
dry to 21
5
28,86148
dry to 22
10
28,81911
dry to 24
15
28,76224
dry to 26
20
28,68685
– 8 to 28
25
28,58806
12 to 31
30
28,46005
23 to 34

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Source: https://www.legislation.gov.uk/eur/2017/654/annex/VII/division/3/division/3.9/division/3.9.3 (the official version)
Page: https://openlegislation.uk/eu-regulation-2017-654/annex-VII/division-3/division-3.9/division-3.9.3
Contains public sector information licensed under the Open Government Licence v3.0. Source: legislation.gov.uk, The National Archives.