1065.640—Flow meter calibration calculations.
This section describes the calculations for calibrating various flow meters. After you calibrate a flow meter using these calculations, use the calculations described in § 1065.642 to calculate flow during an emission test. Paragraph (a) of this section first describes how to convert reference flow meter outputs for use in the calibration equations, which are presented on a molar basis. The remaining paragraphs describe the calibration calculations that are specific to certain types of flow meters.
(a) Reference meter conversions.
The calibration equations in this section use molar flow rate, n
ref, as a reference quantity. If your reference meter outputs a flow rate in a different quantity, such as standard volume rate, V
stdref, actual volume rate, V
actref, or mass rate, m
ref, convert your reference meter output to a molar flow rate using the following equations, noting that while values for volume rate, mass rate, pressure, temperature, and molar mass may change during an emission test, you should ensure that they are as constant as practical for each individual set point during a flow meter calibration:
Eq. 1065.640-1
Where:
nref = reference molar flow rate.
Vstdref = reference volume flow rate, corrected to a standard pressure and a standard temperature.
Vactref = reference volume flow rate at the actual pressure and temperature of the flow rate.
mref = reference mass flow.
Pstd = standard pressure.
Pact = actual pressure of the flow rate.
Tstd = standard temperature.
Tact = actual temperature of the flow rate.
R = molar gas constant.
Mmix = molar mass of the flow rate.
Example 1:
Vstdref = 1000.00 ft3/min = 0.471948 m3/s
P = 29.9213 in Hg @ 32 °F = 101325 Pa
T = 68.0 °F = 293.15 K
R = 8.314472 J/(mol·K)
nref = 19.619 mol/s
Example 2:
mref = 17.2683 kg/min = 287.805 g/s
Mmix = 28.7805 g/mol
nref = 10.0000 mol/s
(b) PDP calibration calculations.
For each restrictor position, calculate the following values from the mean values determined in § 1065.340, as follows:
Example:
nref = 25.096 mol/s
R = 8.314472 J/(mol·K)
Tin = 299.5 K
Pin = 98290 Pa
fnPDP = 1205.1 rev/min = 20.085 rev/s
Vrev = 0.03166 m3/rev
Example:
fnPDP = 1205.1 rev/min = 20.085 rev/s
Pout = 100.103 kPa
Pin= 98.290 kPa
Ks = 0.006700 s/rev
(3)
Perform a least-squares regression of PDP volume pumped per revolution, V
rev, versus PDP slip correction factor, K
s, by calculating slope, a
1, and intercept, a
0, as described in § 1065.602.
(4)
Repeat the procedure in paragraphs (b)(1) through (3) of this section for every speed that you run your PDP.
| f nPDP (rev/min) | a 1 (m3/min) | a 0 (m3/rev) |
| 755.0 | 50.43 | 0.056 |
| 987.6 | 49.86 | −0.013 |
| 1254.5 | 48.54 | 0.028 |
| 1401.3 | 47.30 | −0.061 |
(6)
For each speed at which you operate the PDP, use the corresponding slope, a
1, and intercept, a
0, to calculate flow rate during emission testing as described in § 1065.642.
(c) Venturi governing equations and permissible assumptions.
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 paragraph (c)(4) of this section, we describe other assumptions that you may make, depending upon how you conduct your emission tests. If we do not allow you to assume that the measured flow is an ideal gas, the governing equations include a first-order correction for the behavior of a real gas; namely, the compressibility factor, Z. If good engineering judgment dictates using a value other than Z =1, you may either use an appropriate equation of state to determine values of Z as a function of measured pressures and temperatures, or you may develop your own calibration equations based on good engineering judgment. Note 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, C
p /C
v. If good engineering judgment dictates using a real gas isentropic exponent, you may either use an appropriate equation of state to determine values of γ as a function of measured pressures and temperatures, or you may develop your own calibration equations based on good engineering judgment. Calculate molar flow rate, n, as follows:
Where:
Cd = Discharge coefficient, as determined in paragraph (c)(1) of this section.
Cf = Flow coefficient, as determined in paragraph (c)(2) of this section.
At = Venturi throat cross-sectional area.
pin = Venturi inlet absolute static pressure.
Z = Compressibility factor.
Mmix = Molar mass of gas mixture.
R = Molar gas constant.
Tin = Venturi inlet absolute temperature.
Where:
nref = A reference molar flow rate.
(i)
For CFV flow meters only, determine C
fCFV from the following table based on your values for β and γ, using linear interpolation to find intermediate values:
| CfCFV | ||
|---|---|---|
| β | γexh =1.385 | γdexh =γair =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 |
Where:
γ = isentropic exponent. For an ideal gas, this is the ratio of specific heats of the gas mixture, Cp/Cv.
r = Pressure ratio, as determined in paragraph (c)(3) of this section.
β = Ratio of venturi throat to inlet diameters.
Where:
ΔpSSV = Differential static pressure; venturi inlet minus venturi throat.
(4)
You may make any of the following simplifying assumptions of the governing equations, or you may use good engineering judgment to develop more appropriate values for your testing:
(i)
For emission testing over the full ranges of raw exhaust, diluted exhaust and dilution air, you may assume that the gas mixture behaves as an ideal gas: Z = 1.
(ii)
For the full range of raw exhaust you may assume a constant ratio of specific heats of γ =1.385.
(iii)
For the full range of diluted exhaust and air (e.g., calibration air or dilution air), you may assume a constant ratio of specific heats of γ = 1.399.
(iv)
For the full range of diluted exhaust and air, you may assume the molar mass of the mixture is a function only of the amount of water in the dilution air or calibration air, x
H2O, determined as described in § 1065.645, as follows:
Example:
Mair = 28.96559 g/mol
xH2O = 0.0169 mol/mol
MH2O = 18.01528 g/mol
Mmix = 28.96559 · (1 0.0169) 18.01528 · 0.0169
Mmix = 28.7805 g/mol
(v)
For the full range of diluted exhaust and air, you may assume a constant molar mass of the mixture, M
mix, for all calibration and all testing as long as your assumed molar mass differs no more than ±1% from the estimated minimum and maximum molar mass during calibration and testing. You may assume this, using good engineering judgment, if you sufficiently control the amount of water in calibration air and in dilution air or if you remove sufficient water from both calibration air and dilution air. The following table gives examples of permissible ranges of dilution air dewpoint versus calibration air dewpoint:
| If calibration Tdew (°C) is... | assume the following constant Mmix (g/mol)... | for the following ranges of Tdew (°C) during emission tests a |
|---|---|---|
| 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. |
| a Range valid for all calibration and emission testing over the atmospheric pressure range (80.000 to 103.325) kPa. | ||
(5)
The following example illustrates the use of the governing equations to calculate the discharge coefficient, C
d, of an SSV flow meter at one reference flow meter value. Note that calculating C
d for a CFV flow meter would be similar, except that C
f would be determined from Table 2 of this section or calculated iteratively using values of β and γ as described in paragraph (c)(2) of this section.
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A
t = 0.01824 m 2
p
in = 99132.0 Pa
γ = 1.399
β = 0.8
Δp = 2.312 kPa
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(1)
Calculate the Reynolds number, Re
#, for each reference molar flow rate, using the throat diameter of the venturi, d
t. Because the dynamic viscosity, µ, is needed to compute Re
#, you may use your own fluid viscosity model to determine µ for your calibration gas (usually air), using good engineering judgment. Alternatively, you may use the Sutherland three-coefficient viscosity model to approximate µ, as shown in the following sample calculation for Re
# :
Where, using the Sutherland three-coefficient viscosity model:
Where:
µ = Dynamic viscosity of calibration gas.
µ0 = Sutherland reference viscosity.
T0 = Sutherland reference temperature.
S = Sutherland constant.
| Gas a | µ0 | T0 | S | Temp range within ± 2% error | Pressure limit |
|---|---|---|---|---|---|
| kg/(m·s) | K | K | K | kPa | |
| Air | 1.716·10−5 | 273 | 111 | 170 to 1,900 | ≤ 1,800 |
| CO2 | 1.370·10−5 | 273 | 222 | 190 to 1,700 | ≤ 3,600 |
| H2O | 1.12·10−5 | 350 | 1,064 | 360 to 1,500 | ≤ 10,000 |
| O2 | 1.919·10−5 | 273 | 139 | 190 to 2,000 | ≤ 2,500 |
| N2 | 1.663·10−5 | 273 | 107 | 100 to 1,500 | ≤ 1,600 |
| a Use tabulated parameters only for the pure gases, as listed. Do not combine parameters in calculations to calculate viscosities of gas mixtures. | |||||
Example:
µ0 = 1.716 · 10−5 kg/(m·s)
T0 = 273.11 K
S = 110.56 K
µ = 1.837·10−5 kg/(m·s)
Mmix = 28.7805 g/mol
nref = 57.625 mol/s
dt = 152.4 mm
Tin = 298.15 K
Re# = 7.541·105
(2)
Create an equation for C
d versus Re
#, using paired values of (Re
#, C
d ). For the equation, you may use any mathematical expression, including a polynomial or a power series. The following equation is an example of a commonly used mathematical expression for relating C
d and Re
# :
(3)
Perform a least-squares regression analysis to determine the best-fit coefficients to the equation and calculate the equation's regression statistics, SEE and r 2, according to § 1065.602.
(4)
If the equation meets the criteria of SEE ≤ 0.5% · n
refmax and r 2 ≥ 0.995, you may use the equation to determine C
d for emission tests, as described in § 1065.642.
(5)
If the SEE and r 2 criteria are not met, you may use good engineering judgment to omit calibration data points to meet the regression statistics. You must use at least seven calibration data points to meet the criteria.
(6)
If omitting points does not resolve outliers, take corrective action. For example, select another mathematical expression for the C
d versus Re
# equation, check for leaks, or repeat the calibration process. If you must repeat the process, we recommend applying tighter tolerances to measurements and allowing more time for flows to stabilize.
(7)
Once you have an equation that meets the regression criteria, you may use the equation only to determine flow rates that are within the range of the reference flow rates used to meet the C
d versus Re
# equation's regression criteria.
(e) CFV calibration.
Some CFV flow meters consist of a single venturi and some consist of multiple venturis, where different combinations of venturis are used to meter different flow rates. For CFV flow meters that consist of multiple venturis, either calibrate each venturi independently to determine a separate discharge coefficient, C
[=E T=8052]d, for each venturi, or calibrate each combination of venturis as one venturi. In the case where you calibrate a combination of venturis, use the sum of the active venturi throat areas as A
[=E T=8052]t, the square root of the sum of the squares of the active venturi throat diameters as d
[=E T=8052]t, and the ratio of the venturi throat to inlet diameters as the ratio of the square root of the sum of the active venturi throat diameters (d
[=E T=8052]t) to the diameter of the common entrance to all of the venturis (D ). To determine the C
[=E T=8052]d for a single venturi or a single combination of venturis, perform the following steps:
(1)
Use the data collected at each calibration set point to calculate an individual C
[=E T=8052]d for each point using Eq. 1065.640-4.
(2)
Calculate the mean and standard deviation of all the C
[=E T=8052]d values according to Eqs. 1065.602-1 and 1065.602-2.
(3)
If the standard deviation of all the C
d values is less than or equal to 0.3% of the mean C
d, use the mean C
d in Eq 1065.642-6, and use the CFV only down to the lowest r measured during calibration using the following equation:
Where:
ΔpCFV = Differential static pressure; venturi inlet minus venturi outlet.
(4)
If the standard deviation of all the C
[=E T=8052]d values exceeds 0.3% of the mean C
[=E T=8052]d, omit the C
[=E T=8052]d values corresponding to the data point collected at the lowest r measured during calibration.
(5)
If the number of remaining data points is less than seven, take corrective action by checking your calibration data or repeating the calibration process. If you repeat the calibration process, we recommend checking for leaks, applying tighter tolerances to measurements and allowing more time for flows to stabilize.
(6)
If the number of remaining C
[=E T=8052]d values is seven or greater, recalculate the mean and standard deviation of the remaining C
[=E T=8052]d values.
(7)
If the standard deviation of the remaining C
[=E T=8052]d values is less than or equal to 0.3% of the mean of the remaining C
[=E T=8052]d, use that mean C
[=E T=8052]d in Eq. 1065.642-6, and use the CFV values only down to the lowest r associated with the remaining C
[=E T=8052]d.
(8)
If the standard deviation of the remaining C
[=E T=8052]d still exceeds 0.3% of the mean of the remaining C
[=E T=8052]d values, repeat the steps in paragraph (e)(4) through (8) of this section.