# Title 40

## SECTION 1065.640

### 1065.640 Flow meter calibration calculations.

§ 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:

*n*ref = reference molar flow rate.

*v*stdref = reference volume flow rate, corrected to a standard pressure and a standard temperature.

*v*actref = reference volume flow rate at the actual pressure and temperature of the flow rate.

*m*ref = reference mass flow.

*p*std = standard pressure.

*p*act = actual pressure of the flow rate.

*T*std = standard temperature.

*T*act = actual temperature of the flow rate.

*R*= molar gas constant.

*M*mix = molar mass of the flow rate. Example 1:

*v*stdref = 1000.00 ft 3/min = 0.471948 m 3/s

*p*std = 29.9213 in Hg @32 °F = 101.325 kPa = 101325 Pa = 101325 kg/(m·s 2)

*T*std = 68.0 °F = 293.15 K

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

*n*ref = 19.619 mol/s Example 2:

*m*ref = 17.2683 kg/min = 287.805 g/s

*M*mix = 28.7805 g/mol

*n*ref = 10.0000 mol/s

(b) *PDP calibration calculations.* 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 §
1065.340 as follows:

*n*ref = mean reference molar flow rate.

*R*= molar gas constant.

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

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

*f*nPDP = mean PDP speed. Example:

*n*ref = 25.096 mol/s

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

*T*in = 299.5 K

*P*in = 98.290 kPa = 98290 Pa = 98290 kg/(m·s 2)

*f*nPDP = 1205.1 r/min = 20.085 r/s

*V*rev = 0.03166 m 3/r

(2) Calculate a PDP slip correction factor, *K*s, for each
restrictor position from the mean values determined in § 1065.340
as follows:

*f*nPDP = mean PDP speed.

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

*P*in = mean static absolute pressure at the PDP inlet. Example:

*f*nPDP = 1205.1 r/min = 20.085 r/s

*P*out = 100.103 kPa

*P*in = 98.290 kPa

*K*s = 0.006700 s/r

(3) Perform a least-squares regression of *V*rev, versus
*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.

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

Table 1 of § 1065.640 - 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 (b) 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 flow of an ideal gas. Paragraph (c)(5) 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 pressures and temperatures, or
you may develop your own calibration equations based on good
engineering judgment.

(1) Calculate molar flow rate, *n* , as follows:

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

*C*f = flow coefficient, as determined in paragraph (c)(3)
of this section.

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

*p*in = venturi inlet absolute static pressure.

*Z* = compressibility factor.

*M*mix = molar mass of gas mixture.

*R* = molar gas constant.

*T*in = venturi inlet absolute temperature.

(2) Using the data collected in § 1065.340, calculate *C*d
for each flow rate using the following equation:

*n*ref = a reference molar flow rate.

(3) Determine *C*f using one of the following methods:

(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:

Table 2 of § 1065.640-CfCFV Versus b and g for CFV Flow Meters

CfCFV | ||
---|---|---|

b | gexh = 385 |
gdexh = gair = 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, you may use the following
equation to calculate *C*f for each flow rate:

*C*p/

*C*v.

*r*= pressure ratio, as determined in paragraph (c)(4) of this section. b = ratio of venturi throat to inlet diameters.

(4) Calculate *r* as follows:

(i) For SSV systems only, calculate *r*SSV using the
following equation:

*Δp*SSV = Differential static pressure; venturi inlet minus venturi throat.

(ii) For CFV systems only, calculate *r*CFV iteratively
using the following equation:

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

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

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

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

(iv) 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 § 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

(v) For diluted exhaust and dilution 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:

Table 3 of § 1065.640 - 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} |
---|---|---|

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.

(6) 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. 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 b and g as described in paragraph (c)(2) of this
section.

*n*ref = 57.625 mol/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 = 99132.0 Pa = 99132 kg/(m·s 2) g = 1.399 b = 0.8

*Δp*= 2.312 kPa

*C*f = 0.274

*C*d = 0.982

(d) *SSV calibration.* Perform the following steps to
calibrate an SSV flow meter:

(1) Calculate the Reynolds number, *Re*#, for each
reference molar flow rate, *n* ref, using the throat diameter
of the venturi, *d*t. 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 4 of § 1065.640 - Sutherland Three-Coefficient Viscosity Model Parameters

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

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 |

H2 | 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 |

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

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

*T*0 = 273 K

*S*= 111 K

*µ*= 1.838 · 10−5 kg/(m·s)

*M*mix = 28.7805 g/mol

*n*ref = 57.625 mol/s

*d*t = 152.4 mm = 0.1524 m

*T*in = 298.15 K

*Re*# = 7.538·10 8

(2) 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*#:

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

(4) 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 § 1065.642.

(5) If the equation does not meet the specified statistical criterion, 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.

(6) 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.

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

(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*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*t, the square root of the sum of
the squares of the active venturi throat diameters as *d*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*t) to the diameter of the common entrance to all
the venturis. (*D*). To determine the *C*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*d for each point using Eq.
1065.640-4.

(2) Calculate the mean and standard deviation of all the
*C*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. 1065.642-4, 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.

(4) If the standard deviation of all the *C*d values
exceeds 0.3% of the mean *C*d, omit the *C*d value
corresponding to the data point collected at the highest *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*d values is seven or
greater, recalculate the mean and standard deviation of the
remaining *C*d values.

(7) If the standard deviation of the remaining *C*d values
is less than or equal to 0.3% of the mean of the remaining
*C*d, use that mean *C*d in Eq. 1065.642-4, and use the
CFV values only up to the highest *r* associated with the
remaining *C*d.

(8) If the standard deviation of the remaining *C*d still
exceeds 0.3% of the mean of the remaining *C*d values, repeat
the steps in paragraph (e)(4) through (8) of this section.