# Title 40

## SECTION 1066.625

### 1066.625 Flow meter calibration calculations.

§ 1066.625 Flow meter calibration calculations.This section describes the calculations for calibrating various flow meters based on mass flow rates. Calibrate your flow meter according to 40 CFR 1065.640 instead if you calculate emissions based on molar flow rates.

(a) *PDP calibration.* Perform the following steps to
calibrate a PDP flow meter:

(1) Calculate PDP volume pumped per revolution, *V*rev, for
each restrictor position from the mean values determined in §
1066.140:

*V*ref = mean flow rate of the reference flow meter. Tin = mean temperature at the PDP inlet.

*p*std = standard pressure = 101.325 kPa.

*f*nPDP = mean PDP speed. Pin = mean static absolute pressure at the PDP inlet.

*T*std = standard temperature = 293.15 K. Example:

*V*ref = 0.1651 m 3/s Tin = 299.5 K

*p*std = 101.325 kPa

*f*nPDP = 1205.1 r/min = 20.085 r/s Pin = 98.290 kPa

*T*std = 293.15 K

*V*rev = 0.00866 m 3/r

(2) Calculate a PDP slip correction factor, *K*s for each
restrictor position from the mean values determined in §
1066.140:

*f*mPDP = mean PDP speed.

*p*out = mean static absolute pressure at the PDP outlet.

*p*in = mean static absolute pressure at the PDP inlet.

(3) Perform a least-squares regression of *V*rev, versus
*K*s, by calculating slope, *a*1, and intercept,
*a*0, as described in 40 CFR 1065.602.

(4) Repeat the procedure in paragraphs (a)(1) through (3) of this section for every speed that you run your PDP.

(5) The following example illustrates a range of typical values for different PDP speeds:

Table 1 of § 1066.625 - Example of PDP Calibration Data

f nPDP (revolution/s) |
a1 (m ^{3}/s) |
a0 (m ^{3}/revolution) |
---|---|---|

12.6 | 0.841 | 0.056 |

16.5 | 0.831 | −0.013 |

20.9 | 0.809 | 0.028 |

23.4 | 0.788 | −0.061 |

(6) For each speed at which you operate the PDP, use the appropriate regression equation from this paragraph (a) to calculate flow rate during emission testing as described in § 1066.630.

(b) *SSV calibration.* The equations governing SSV flow
assume one-dimensional isentropic inviscid flow of an ideal gas.
Paragraph (b)(2)(iv) of this section describes other assumptions
that may apply. If good engineering judgment dictates that you
account for gas compressibility, you may either use an appropriate
equation of state to determine values of *Z* as a function of
measured pressure and temperature, 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, g, 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 pressure and
temperature, or you may develop your own calibration equations
based on good engineering judgment.

(1) Calculate volume flow rate at standard reference conditions,
*V* std, as follows

*C*d = discharge coefficient, as determined in paragraph (b)(2)(i) of this section.

*C*f = flow coefficient, as determined in paragraph (b)(2)(ii) of this section.

*A*t = cross-sectional area at the venturi throat.

*R*= molar gas constant.

*p*in = static absolute pressure at the venturi inlet.

*T*std = standard temperature.

*p*std = standard pressure.

*Z*= compressibility factor.

*M*mix = molar mass of gas mixture.

*T*in = absolute temperature at the venturi inlet.

(2) Perform the following steps to calibrate an SSV flow meter:

(i) Using the data collected in § 1066.140, calculate *C*d
for each flow rate using the following equation:

*V*ref = measured volume flow rate from the reference flow meter.

(ii) Use the following equation to calculate *C*f for each
flow rate:

*C*p/

*C*v.

*r*= pressure ratio, as determined in paragraph (b)(2)(iii) of this section. b = ratio of venturi throat diameter to inlet diameter.

(iii) Calculate *r* using the following equation:

*p*= differential static pressure, calculated as venturi inlet pressure minus venturi throat pressure.

(iv) You may apply any of the following simplifying assumptions or develop other values as appropriate for your test configuration, consistent with good engineering judgment:

(A) For raw exhaust, diluted exhaust, and dilution air, you may
assume that the gas mixture behaves as an ideal gas (*Z* =
1).

(B) For raw exhaust, you may assume g = 1.385.

(C) For diluted exhaust and dilution air, you may assume g = 1.399.

(D) For diluted exhaust and dilution air, you may assume the
molar mass of the mixture, *M*mix, is a function only of the
amount of water in the dilution air or calibration air, as
follows:

*M*air = molar mass of dry air.

*x*H2O = amount of H2O in the dilution air or calibration air, determined as described in 40 CFR 1065.645.

*M*H2O = molar mass of water. Example:

*M*air = 28.96559 g/mol

*x*H2O = 0.0169 mol/mol

*M*H2O = 18.01528 g/mol

*M*mix = 28.96559 · (1 − 0.0169) + 18.01528 · 0.0169

*M*mix = 28.7805 g/mol

(E) For diluted exhaust and dilution air, you may assume a
constant molar mass of the mixture, *M*mix, for all
calibration and all testing if you control the amount of water in
dilution air and in calibration air, as illustrated in the
following table:

Table 2 of § 1066.625 - Examples of Dilution Air and Calibration Air Dewpoints at Which You May Assume a Constant Mmix

If calibration Tdew ( °C) is . . . | assume the following constant Mmix (g/mol) . . . |
for the following ranges of
Tdew ( °C) during emission tests ^{a} |
---|---|---|

≤0 | 28.96559 | ≤18 |

0 | 28.89263 | ≤21 |

5 | 28.86148 | ≤22 |

10 | 28.81911 | ≤24 |

15 | 28.76224 | ≤26 |

20 | 28.68685 | −8 to 28 |

25 | 28.58806 | 12 to 31 |

30 | 28.46005 | 23 to 34 |

^{a} The specified ranges are valid
for all calibration and emission testing over the atmospheric
pressure range (80.000 to 103.325) kPa.

(v) The following example illustrates the use of the governing
equations to calculate *C*d of an SSV flow meter at one
reference flow meter value:

*V*ref = 2.395 m 3/s

*Z*= 1

*M*mix = 28.7805 g/mol = 0.0287805 kg/mol

*R*= 8.314472 J/(mol·K) = 8.314472 (m 2·kg)/(s 2·mol·K)

*T*in = 298.15 K

*A*t = 0.01824 m 2

*p*in = 99.132 kPa = 99132 Pa = 99132 kg/(m·s 2) g = 1.399 b = 0.8

*Δp*= 7.653 kPa

*C*f = 0.472

*C*d = 0.985

(vi) Calculate the Reynolds number, *Re*#, for each
reference flow rate at standard conditions, *V* refstd, using
the throat diameter of the venturi, *d*t, and the air density
at standard conditions, rstd. Because the dynamic viscosity, m, is
needed to compute *Re*#, you may use your own fluid viscosity
model to determine m for your calibration gas (usually air), using
good engineering judgment. Alternatively, you may use the
Sutherland three-coefficient viscosity model to approximate m, as
shown in the following sample calculation for *Re*#:

Where, using the Sutherland three-coefficient viscosity model:

Where: m0 = Sutherland reference viscosity.*T*0 = Sutherland reference temperature.

*S*= Sutherland constant.

Table 3 of § 1066.625 - Sutherland Three-Coefficient Viscosity Model Parameters

Gas
^{1} |
m0 | T0 | S | Temperature range within ±2%
error ^{2} |
Pressure limit
^{2} |
---|---|---|---|---|---|

kg/(m·s) | K | K | K | kPa | |

Air | 1.716·10−5 | 273 | 111 | 170 to 1900 | ≤1800. |

CO2 | 1.370·10−5 | 273 | 222 | 190 to 1700 | ≤3600. |

H2O | 1.12·10−5 | 350 | 1064 | 360 to 1500 | ≤10000. |

O2 | 1.919·10−5 | 273 | 139 | 190 to 2000 | ≤2500. |

N2 | 1.663·10−5 | 273 | 107 | 100 to 1500 | ≤1600. |

^{1} Use tabulated parameters only
for the pure gases, as listed. Do not combine parameters in
calculations to calculate viscosities of gas mixtures.

^{2} The model results are valid
only for ambient conditions in the specified ranges.

*T*0 = 273 K

*S*= 111 K

*T*in = 298.15 K

*d*t = 152.4 mm = 0.1524 m rstd = 1.1509 kg/m 3

*Re*# = 1.3027·10 6

(vii) Calculate r using the following equation:

Example:*ρ*std = 1.1964 kg/m 3

(viii) Create an equation for *C*d as a function of
*Re*#, using paired values of the two quantities. The equation
may involve 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*#:

(ix) Perform a least-squares regression analysis to determine
the best-fit coefficients for the equation and calculate *SEE*
as described in 40 CFR 1065.602.

(x) If the equation meets the criterion of *SEE* ≤0.5% ⋅
*C*dmax, you may use the equation for the corresponding range
of *Re*#, as described in § 1066.630(b).

(xi) If the equation does not meet the specified statistical criteria, you may use good engineering judgment to omit calibration data points; however, you must use at least seven calibration data points to demonstrate that you meet the criterion. For example, this may involve narrowing the range of flow rates for a better curve fit.

(xii) Take corrective action if the equation does not meet the
specified statistical criterion even after omitting calibration
data points. 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 calibration
process, we recommend applying tighter tolerances to measurements
and allowing more time for flows to stabilize.

(xiii) Once you have an equation that meets the specified
statistical criterion, you may use the equation only for the
corresponding range of *Re*#.

(c) *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
calibration coefficient, *K*v, for each venturi, or calibrate
each combination of venturis as one venturi by determining
*K*v for the system.

(1) To determine *K*v for a single venturi or a combination
of venturis, perform the following steps:

(i) Calculate an individual *K*v for each calibration set
point for each restrictor position using the following
equation:

*V*refstd= mean flow rate from the reference flow meter, corrected to standard reference conditions.

*T*in= mean temperature at the venturi inlet.

*P*in= mean static absolute pressure at the venturi inlet.

(ii) Calculate the mean and standard deviation of all the
*K*v values (see 40 CFR 1065.602). Verify choked flow by
plotting *K*v as a function of *p*in. *K*v will have
a relatively constant value for choked flow; as vacuum pressure
increases, the venturi will become unchoked and *K*v will
decrease. Paragraphs (c)(1)(iii) through (viii) of this section
describe how to verify your range of choked flow.

(iii) If the standard deviation of all the *K*v values is
less than or equal to 0.3% of the mean *K*v, use the mean
*K*v in Eq. 1066.630-7, and use the CFV only up to the highest
venturi pressure ratio, *r*, measured during calibration using
the following equation:

*p*CFV = differential static pressure; venturi inlet minus venturi outlet.

*p*in = mean static absolute pressure at the venturi inlet.

(iv) If the standard deviation of all the *K*v values
exceeds 0.3% of the mean *K*v, omit the *K*v value
corresponding to the data point collected at the highest *r*
measured during calibration.

(v) 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.

(vi) If the number of remaining *K*v values is seven or
greater, recalculate the mean and standard deviation of the
remaining *K*v values.

(vii) If the standard deviation of the remaining *K*v
values is less than or equal to 0.3% of the mean of the remaining
*K*v, use that mean *K*v in Eq 1066.630-7, and use the
CFV values only up to the highest *r* associated with the
remaining *K*v.

(viii) If the standard deviation of the remaining *K*v
still exceeds 0.3% of the mean of the remaining *K*v values,
repeat the steps in paragraph (c)(1)(iv) through (vii) of this
section.

(2) During exhaust emission tests, monitor sonic flow in the CFV
by monitoring *r.* Based on the calibration data selected to
meet the standard deviation criterion in paragraphs (c)(1)(iv) and
(vii) of this section, in which *K*v is constant, select the
data values associated with the calibration point with the lowest
absolute venturi inlet pressure to determine the *r* limit.
Calculate *r* during the exhaust emission test using Eq.
1066.625-8 to demonstrate that the value of *r* during all
emission tests is less than or equal to the *r* limit derived
from the CFV calibration data.