RP-02 Field: Determination of Mass Calibration Values and Related Uncertainties
N.Dupuis-Désormeaux, Senior Engineer - Gravimetry
January 2003
Amendment, in PDF format, 156KB
Table of Contents
1. Calibration And Tolerance Testing Differences
1.1 Calibration
1.2 Tolerance Testing
1.3 Calculated Difference Dt
1.3.1 Dt When Using An Electronic Mass Comparator
1.3.2 Dt When Using An Equal Arm Balance
2.1 Absolute Mass Equation
2.2 Conventional Mass Equation
2.3.Conventional Mass Approximate Equation And Tolerance Testing
3.1 Combined Uncertainty Equation For Absolute Mass
3.2 Combined Uncertainty Equation For Conventional Mass
4.1 Uncertainty Of The Reference Weight
4.2 Uncertainty Of Air Density
4.3 Uncertainty Of Material Density Of Reference And Uncertainty Of Material Density Of Weight Being Calibrated
4.4 Uncertainty Of The Process
5. Uncertainty Multipliers
5.1 Density Of Reference And Density Of Weight Being Calibrated
5.2 Air Density
5.3 Conventional Air Density
5.4 Conventional Mass Density
6. Summary - At A Glance
7. References
Background
The importance of quantifying measurement uncertainty has grown steadily with the pace of industrialization that followed the second World War. However, most companies and even most governments developed their own methods for calculating and expressing uncertainty without much consideration to international harmonization. In 1977, the Comité International de Poids et Mesures (CIPM) requested that the Bureau International de Poids et Mesures (BIPM) develop a document to harmonize methods of expression of uncertainty. In 1980, a BIPM work group developed the International Recommendation INC-1: Expression of Experimental Uncertainties. It was then decided that the responsibility of developing a detailed guide on how to express uncertainties according to INC-1 should be transferred to the International Standardization Organization (ISO). The BIPM/ISO Guide to the Expression of Uncertainty in Measurement was first published in 1995 as ISBN 92-67-10188-9.
The methods and calculations proposed herein are in accordance with the Guide to the Expression of Uncertainty in Measurement, 1995. Please refer to the above document for any background information.
Definitions
Below, you will find definitions and the mathematical relationships between types of calibrations.
In the regions, all calibrations and verifications (tolerance testing) are performed in terms of conventional mass.
Definitions
Absolute mass, Xabs
Absolute mass is obtained when weighing in vacuum. It is an intrinsic property of an artifact and is independent of gravity or buoyancy.
Effective mass, Xe = Xabs (1 - (ρat/ρX))
Effective mass is the absolute mass times the buoyancy factor (in parenthesis above). In other words, effective mass is less than the absolute mass by the mass of air that it displaces. (Note: ρat is the air density and ρx is the material density of the mass).
Conventional mass, Xc = Xabs (1 - (ρac/ρX)) / (1 - (ρac/ρc))
The conventional mass is by definition: "the mass Xc that a standard of density ρc equal to 8000 kg/m3 would require to have to balance a given mass Xabs of density ρx when this measurement is performed in an air density ρac of 1.2 kg/m3 and a temperature of 20ºC." In other words:
Xc - ρac Vc = Xabs - ρac Vx
Xc (1-(ρac/ρc)) = Xabs (1-( ρac/ρx))Xc = Xabs (1-(ρac/ρx)) / (1-( ρac/ρc))
OIML R111(2001) definition 2.6 - Conventional Mass: For a weight taken at 20ºC, the conventional mass is the mass of a reference weight of a density of 8 000 kg/m3 which it balances in air of a density of 1.2 kg/m3.
Please note that in the past, the term “apparent mass” was used to describe “conventional mass”.
Calibration
Any measurement performed in order to determine the mass of an object, and must necessarily include a statement of the uncertainty attributable to this measurement.
Tolerance Testing
Any measurement performed in order to determine if the conventional mass of an object being verified rests within pre-determined tolerances. When the deviation from the nominal value is beyond pre-determined limits, the object being verified is adjusted. A statement that the mass is within pre-determined tolerances is issued and the uncertainty attributable to this measurement is set equivalent to the applicable tolerance.
District Standard
The standard calibrated at the Calibration Services Laboratory in Ottawa, and used as the reference for the calibration or tolerance test of field working standard(s).
Field Working Standard
Designates the standard that is used in the day-to-day verification of devices.
Test Weight
Designates any unknown object being calibrated.
Nominal Value
The value which a standard is intended to have, or the designation by which it is commonly known. For example, "a 1 kg mass" is a statement of nominal value; the corresponding actual mass may be somewhat higher or lower, say 1,000 350 kg.
Single Comparison
Single comparison calibration refers to a method where the test mass and the reference are only compared once. This comparison is effected by using the substitution method.
Substitution Weighing
Substitution weighing is also called differential weighing or comparison weighing and refers to the procedure where a standard and a test mass are weighed consecutively on the same load receiving element and the difference in their indicated value forms the basis for the calibration.
Value of a Quantity
The expression of a quantity in terms of a number and an appropriate unit or measurement.
Example: 12 kg
True Value (of a quantity)
The value which characterizes a perfectly defined quantity, in the conditions which exist when that quantity is considered.
Note: The true value of a quantity is an ideal concept and, in general, cannot be known exactly.
Uncertainty of Measurement
An estimate characterizing the range of values within which the true value of a measurand lies.
Note: Uncertainty of measurement comprises, in general, many components. Some of these components may be estimated on the basis of the statistical distribution of the results of a series of measurements and can be characterized by experimental standard deviations.
Accuracy Of Measurement
The closeness of the agreement between the result of a measurement and the expected true value of the measurand. In other words, accuracy refers to how close a measurement is to its true value.
Repeatability Of Measurements
The closeness of the agreement between the results of successive measurements of the same measurand carried out over a short period of time and subject to all of the following conditions being constant:
- the method of measurement
- the observer
- the location
- the condition of use
Good repeatability is represented by small short-term process uncertainties (short-term process standard deviations)
Note: Repeatability may be expressed quantitatively in terms of the dispersion of the results over a short period of time , i.e., the standard deviation of one set of readings.
Repeatability is a measure of the ability of a measuring instrument and operator to yield, for the same applied load and the same conditions, indications which agree amongst themselves.
Reproducibility Of Measurements
The closeness of the agreement between the results of measurements of the same measurand where the individual measurements are carried out over a long period of time and under changing conditions such as:
- the method of measurement
- the observer
- the location
- the conditions of use
Good reproducibility is represented by small long-term process uncertainties (long-term process standard deviations)
Note: Reproducibility may be expressed quantitatively in terms of the dispersion of the results over a long period of time, i.e., the standard deviation of a process obtained with a number of sets of points.
Reproducibility is a measure of the ability of a measuring instrument being used by different operators in various conditions, to yield, for the same applied load, indications which agree amongst themselves. It is established by determining the standard deviation of the process over a certain period of time; which can also be described as the long-term repeatability of the device, operators and conditions.
Sensitivity
The change in the response of a measuring instrument corresponding to a change in load.
Calibration and Tolerance Testing Differences
It is easy to understand the concept of buoyancy when speaking of a liquid. A boat rests on the surface of the water because of its material density, its shape and the density of water. The same is true when speaking of buoyancy of objects in air. In accordance with Archimedes' principles of buoyancy, the weight of a body immersed in a fluid diminishes by an amount equal to the fluid it displaces. Air, just like water, is a fluid.
The calibrated mass of a test weight is obtained by comparing the weight of the test weight to the weight of a standard. If, however, these do not have the same material density, the effect of the upward push exerted by air will be different. For example, when comparing a cast iron weight to a stainless steel standard, the effect of buoyancy in air must be taken into consideration unless certain conditions are met.
As will be seen in the following section, the calibration equation when comparing a test weight to a standard in terms of conventional mass is:
mct = ((ρt - ρac) / (ρt - ρat)) { (mcR (ρR - ρat)/(ρR - ρac)) + (Dt ρc / ( ρc - ρac)) }
All terms are expressed in the same unit of mass
mct = conventional mass of unknown mass t
mcR = conventional mass of reference R
Dt = calculated difference between unknown mass t and reference R
ρt = density of the unknown mass t (kg/m3) - usually provided by the manufacturer; otherwise, an estimate can be made by knowing the material and class
ρR = density of the reference R (kg/m3) - usually found on calibration certificate for the reference; otherwise, an estimate can be made by knowing the material and class
ρat = air density at the time of the calibration (kg/m3) - see formula provided in section 1.2
ρa = conventional air density; equal by definition to 1.2 kg/m3
ρc = conventional mass density; equal by definition to 8000 kg/m3
When tolerance testing, the conventional mass of the weight being verified is NOT obtained explicitly but, rather, a value is obtained that determines if the "conventional mass" is within pre-determined tolerances. For example, a tolerance testing result will be stated as: 10 kg and will be said to be within the tolerances of Schedule IV, Part III of the Weights and Measure Regulations (local standard used to inspect devices for weighing other than precious metals; having a tolerance of ± 0.005%).
From the calibration equation,
mct = ((ρt - ρac) / (ρt - ρat)) { (mcR (ρR - ρat)/(ρR - ρac)) + (Dt ρc / ( ρc - ρac)) }
it can be seen that only when the actual air density (ρat) is close to the "conventional" value of the air density (ρac), i.e., at ρac 1.2 kg/m3, can we assume that:
mct = mcR + (Dt ρc) / (ρc - ρac)
mct = mcR + Dt
mc test weight = mc district standard + Dt
This will be the case if the temperature is close to 20ºC, the relative humidity close to 50%, and the barometric pressure close to 101.4 kPa.
Performing tolerance testing is in fact the same as using the last equation above without stating specifically the value of Dt (the difference in mass between the reference and the test weight)
The actual air density (ρat) can be calculated with the following equation (provided by the Canada National Research Council), where ‘P' is the pressure, ‘T' is the temperature and ‘Hrel' is the relative humidity:
ρat = (0.092695) + (1.18 x 10-5) P (in Pa) - (4.15 x 10-3) T (in C) - (1.22 x10-4) Hrel (in %)
When the temperature is close to 20ºC, the relative humidity close to 50%, and the barometric pressure close to 101.4 kPa, the actual air density (ρat) will be close to the "conventional" value of the air density (ρac) of 1.2 kg/m3.
Please note that due to the smaller tolerances involved, calibration of M1 weights shall only be performed in the above conditions - in other words: indoors.
For calibration of weights of class M1-2, M2, M2-3 or M3 taking place at temperatures outside the range stated above buoyancy can be significant. However, if the material densities of section 2.4.5 of RP01-Field (repeated below) are respected, this effect will be small enough to be negligible.
Material Density Limits:
- F2: 6400 kg/m3 to 8400 kg/m3
- M1: 6000 kg/m3 to 8400 kg/m3
- M1-2: 6000 kg/m3 to 8400 kg/m3
- M2: 6000 kg/m3 to 8400 kg/m3
- M2-3: 4400 kg/m3 to 8400 kg/m3
- M3: 4400 kg/m3 to 8400 kg/m3
Note however that when calibrating outdoors, it is preferable to calibrate when the temperature is between 15ºC and 25ºC. This can be explained by the fact that, since the air pressure normally varies between 90kPa and 110kPa, when the temperature is between 15ºC and 25ºC the air density will be lower than 1.32 kg/m3 and can be deemed "close enough" to the conventional air density value.
Calculated Difference Dt
1.3.1 Dt when using an Electronic Mass Comparator
As stated in the section above, when calibrating or tolerance testing a test mass against a standard, we have the following approximate equation:
mc test weight = mc district standard + Dt
In the above, Dt is the calculated difference in mass between the test weight and the standard. This calculated difference is as follows:
Dt = SR (I test weight - I district standard)
Where
I test weight = indicated reading on the comparator for the test weight
I district standard = indicated reading on the comparator for the district standard
SR = Sensitivity Reciprocal at that nominal value
1.3.2 Dt when using an Equal Arm Balance
Again, when calibrating or tolerance testing a test mass against a standard, we have the following approximate equation:
mc test weight mc district standard + Dt
In the above, Dt is the calculated difference in mass between the test weight and the standard. This calculated difference is as follows:
Dt = SR (CRPtest weight - CRPdistrict standard)
Where
CRPtest weight = calculated rest point for the test weight
CRP standard = calculated rest point for the district standard
SR = Sensitivity Reciprocal at that nominal value
Calibration Equations
In the regions, all calibrations are performed in terms of conventional mass. Section 2.1 is provided for information only and the reader can therefore skip to section 2.2
When performing comparison weighing in air, we must account for buoyancy effects and the effective mass is what is compared. In other words, we are comparing the absolute mass of an unknown test weight minus the air displaced by this object to the absolute mass of a reference minus the air that it displaces; the following calibration equation holds:
mt - ρat vt = mR - ρat vR + Dt
vR = volume of reference (standard mass) R
vt = volume of unknown mass t
Rewriting this equation, we have:
(mt - ρat vt ) - (mR - ρat vR ) = Dt
(mt - mR) - ( ρat vt - ρat vR ) = Dt
In other words, the calculated difference (Dt) between the mass and the reference is equal to: the difference between the absolute mass of the unknown and that of the reference, minus the difference in the flotation of the two masses. In practice, this difference Dt is obtained by looking at the indicated values for the masses and multiplying the result by the sensitivity reciprocal (see section 1.3).
Since vt = mt / ρt and vR = mR / ρR , the first equation above can be expressed as:
(2.1)
mt - ρat mt / ρt = mR - ρat mR / ρR + Dt
mt(1 - ( ρat /ρ t)) = mR(1 - ( ρat/ ρR)) + Dt
mt = [ mR (1 - ( ρat/ ρR)) + Dt ] / ( 1 - ( ρat/ ρt))
mt = absolute mass of unknown t
mR = absolute mass of reference R
Dt = calculated difference between unknown mass t and reference R (see section 1.3)
ρt = density of the unknown mass t (kg/m3) - usually provided by the manufacturer; otherwise, an estimate can be made by knowing the material and class
ρR = density of the reference R (kg/m3) - usually found on calibration certificate for the reference; otherwise, an estimate can be made by knowing the material and class
ρat = air density at the time of calibration (kg/m3); the formula provided in section 1.2 can be used
2.2 Conventional Mass Equation
Conventional mass is a mass value established by, as the name implies, a set of conventions. Using the definition, we can convert absolute mass values into conventional mass values.
Recall that the conventional mass is by definition: "the mass Xc that a standard of density ρc equal to 8000 kg/m3 would require to have to balance a given mass Xabs of density ρx when this measurement is performed in an air density ρac of 1.2 kg/m3 and a temperature of 20ºC." In other words:
Xc - ρac vc Xabs - ρac vx
Xc (1-( ρac/ ρc)) Xabs (1-( ρac/ ρx))
Xc Xabs (1-( ρac/ ρx)) / (1-(ρac/ ρc))
OIML R111(2001) definition 2.6 - Conventional Mass: For a weight taken at 20ºC, the conventional mass is the mass of a reference weight of a density of 8 000 kg/m3 which it balances in air of a density of 1.2 kg/m3.
Using notation similar to that used in section 2.1, we have that for a given mass mt, its conventional mass value mct is:
mct - ρac vc mt - ρac vt
mct (1-( ρac/ ρc)) mt (1-( ρac/ ρt))
mct mt (1-( ρac/ρt)) / (1-( ρac/ ρc))
Expressed for mt, we have:
mt mct (1- ( ρac/ ρc)) / (1-( ρac/ ρt))
where
mct = conventional mass value of unknown mass t
ρac = conventional value of the air density, set equal by convention to 1.2 kg/m3
vc = volume of unknown mass t if it had a density set equal by convention to 8000 kg/m3
mt = absolute mass of unknown mass t
vt = volume of unknown mass t
Note that vt is equal to mt/ ρt and vc is equal to mct/ ρc. The second equation in the above definition is obtained by inserting mct/ ρc for vc, and mt/ ρt for vt , in the first equation as follows:
mct - ρac vc mt - ρac vt
mct (1-( ρac/ ρc)) mt (1-( ρac/ ρt))
The third and fourth equations in the definition are obtained by rearranging the above equation. Therefore, expressing mt in terms of its conventional mass mct, we have:
mct (1-(ρac/ρc)) / (1-( ρac/ρt)) = mt
Similarly, looking at mR in terms of its conventional mass mcR, we have:
mcR (1-(ρac/ρc)) / (1-(ρac/ ρR)) mR
Now using the general mass equation of section 2.1
mt - ρat vρ = mR - ρat vR + Dt
and replacing mt and mR by their equivalence in conventional mass, we have:
mt - ρat vt = mR - ρat vR + Dt
[mt] (1 - ( ρat/ ρt)) = mR (1 - ( ρat/ ρR)) + Dt
[mct (1- ( ρac/ ρc)) / (1-( ρac/ ρt))] (1 - ( ρat/ ρt)) = [mcR (1 - ( ρac/ ρc)) / (1- ( ρac/ ρR))] (1 - ( ρat/ ρR)) + Dt
(mct ( ρc - ρac) ( ρt - ρat) / ρc ( ρt - ρac)) = (mcR ( ρc - ρac) ( ρR - ρat) / ρc ( ρR - ρac)) + Dt
mct = (( ρc - ρac) ( ρt - ρat) / ρc ( ρt - ρac)) -1{ [mcR ( ρc - ρac) ( ρR - ρat) / ρc ( ρR - ρac)] + Dt }
mct = mcR ( ρR - ρat) ( ρt - ρac) / ( ρt - ρat)( ρR - ρac) + Dt ρc ( ρt - ρac) / ( ρt - ρat)( ρc - ρac)
mct = ((1 - ρc/ ρt ) / (1 - ρat/ ρt )) {mcR (1-( ρat/ ρR ))/(1-( ρac/ ρR )) + Dt / (1 - ( ρac/ ρc))}
Therefore:
(2.2.A)
mct = (( ρt - ρac) / (ρt - ρat)) { (mcR ( ρR - ρat)/( ρR - ρac)) + (Dt ρcρc - ρac)) }
where mct , mcR and Dt are expressed in the same unit of mass
where
mct = conventional mass of unknown mass t
ρt = density of the unknown mass t (kg/m3) - usually provided by the manufacturer; otherwise, an estimate can be made by knowing the material and the class
ρac= conventional air density; equal by definition to 1.2 kg/m3
ρat = air density at the time of calibration (kg/m3); the formula provided in section 1.2 can be used
mcR = conventional mass of reference R
ρR = density of the reference R (kg/m3) - usually found on calibration certificate for the reference; otherwise, an estimate can be made by knowing the material and the class
Dt = calculated difference between unknown mass t and reference R. See section 1.3
ρc = conventional mass density; equal by definition to 8000 kg/m3.
Please note that writing the above expression (2.2 A) in terms of mR instead mcR , we get:
mt - ρat (vt) = mR - ρat (vR) + Dt
[mt] (1 - ( ρat / ρa)) = mR (1 - ( ρat / ρatR)) + Dt
[mct (1 - ( ρat/ ρa)) / (1- ( ρat/ ρa))] (1 - ( ρat / ρa)) = mR (1 - ( ρat / ρR)) + Dt
mct (( ρC - ρat) ( ρt - ρat)) / ( ρc ( ρt - ρac)) = mR (1 - ( ρat / R)) + Dt
mct = [( ρc ( ρt - ρac)) / (( ρc - ρac) ( ρt - ρat)) ] {mR (1 - ( ρat / ρR)) + Dt }
(2.2.B)
mct = [(ρc (ρt - ρac)) / (( ρc - ρac) ( ρt - ρat)) ] {mR (1 - ( ρat / ρR)) + Dt }
where mct, mR and Dt are expressed in the same unit of mass
2.3 Conventional Mass Approximate Equation and ToleranceTesting
As was seen in section 1.2, when certain conditions are met, equation 2.2.A (below)
mct = (( ρt - ρac) / ( ρt - ρat)) { (mcR ( ρR - ρat)/( ρR - ρac)) + (Dt ρc / ( ρc - ρac)) }
can be approximated with the following:
mct mcR + (Dt ρc) / ( ρc - ρac)
mct mcR + Dt
(2.3)
mc test weight = mc district standard + Dt
where mct, mR and Dt are expressed in the same unit of mass
where
mct = conventional mass of unknown mass t
mcR = conventional mass of reference R
Dt = calculated difference between unknown mass t and reference R. See section 1.3
The above equation holds for both calibration in terms of conventional mass and tolerance testing when the conditions listed in section 1.2 are met.
Uncertainty Equations
In the regions, all calibrations are performed in terms of conventional mass. Section 3.1 is provided for information only and the reader can therefore skip to section 3.2
3.1 Combined Uncertainty Equation for Absolute Mass
The equation below is the equation for the combined uncertainty associated with a calibration in terms of absolute mass.
u2 (mt) u2(MrR) + (mt/ t-mR/ρR)2u2(ρat) + ρat2(-mt/ρt2)2u2(ρt) + u2(Dt)
where all terms are expressed in the same unit of mass
Note that u2(mR) used in the above equation should already contain the term ρat2(-mR/ ρR2)2 u2(ρR) .
where
u(mt) = combined standard uncertainty associated with the evaluation of the absolute mass of t
u(mR) = combined standard uncertainty associated with the absolute mass of reference R - usually found on the calibration certificate for reference R. See section 4.1
u(ρat) = standard uncertainty associated with the estimation of the air density at the time of the calibration (kg/m3). See section 4.2
u(ρt) = standard uncertainty associated with the density of the unknown mass t. See section 4.3.
u(ρR) = standard uncertainty associated with the density of the reference. See section 4.3.
u(Dt) = standard process uncertainty associated with the calculated difference - the long-term process uncertainty of the procedure and equipment used in determining the difference between mass t and reference R. See section 4.4
mt = absolute mass of unknown mass t
mR = absolute mass of reference (standard mass) R
ρt = density of the unknown mass t (kg/m3) - usually provided by manufacturer; otherwise, an estimate can be made by knowing the material and the class. See section 5.1
ρR = density of the reference R (kg/m3) - usually found on calibration certificate for reference R; otherwise, an estimate can be made by knowing the material and the class. See section 5.1
ρat = air density at the time of the calibration (kg/m3) - the formula provided in section 5.2 can be used.
≈ = approximately equal
NOTE: The terms involved in the uncertainty equation 3.1 are described in detail in the following sections 4 and 5.
The above is obtained through mathematical derivation of the weighing equation. The reader can find the necessary information regarding how this equation is obtained by consulting the following box.
Rearranging the first equation in 2.1 above yields:
mt = mR - ρat vR + ρat vt + Dt
Pose f = mt = mR + ρat (vt - vR) + Dt
The function f is f(mR, ρat, vt, vR, Dt). In other words: f is a function of mR , ρat , vt , vR and Dt. By the Law of Propagation of Uncertainty (see section 5.1.2 of the ISO Guide to the Expression of Uncertainty in Measurement) and assuming no covariance, we have:
(ucombined(mt))2 = (∂f/∂mR(u(mR)))2 + (∂f/∂ρat (u(ρat)))2 + (∂f/∂vt (u(vt)))2 + (∂f/∂vR (u(vR)))2 + (∂f/∂Dt(u(Dt)))2
Taking the partial derivatives of the function f(mR, ρat, vt, vR, Dt), we have:
∂f/∂mR ≈ 1
∂f/∂vR ≈ ρat
∂f/∂vt ≈ ρat
∂f/∂ρat ≈ (vt - vR)
∂f/∂Dt ≈ 1
Please note that using the equation: mt(1-(ρat/ρt)) = (mR - mR ρat/ρR + Dt), we get:
mt = {1/(1-(ρat/ρt))} {mR - mRρat/ρR + Dt} instead of equation "f" above, we have the following derivatives without approximation:
∂f/∂mR = {ρt/(ρt - ρat)} χ {(ρR -ρat)/ρR}
∂f/∂Dt = {ρt /(ρt - ρat) χ1}
∂f/∂R = {ρt /(ρt - ρat)χ - ρat mR/ R2}
∂f/ ∂ρt χ {ρt /(ρt - ρat)}{- ρat mt/ρt2}
∂f/∂ρat = {ρt /(ρt - ρat)χ(mR/ ρR)((ρR - ρt)/(ρt - ρat) + Dt /(ρt - ρat)}
Hence:
(ucombined(mt))2 ≈ [1(u(mR))]2+[(vt-vR)(u( at))]2+ [ρat(u(vt))]2 + [- ρat(u(vR))]2 + [1(u(Dt))]2
(ucombined(mt))2 ≈ [u(mR)]2+[(mt/ρt - mR/ρR)(u(ρat))]2+[ρat(u(vt))]2+[-ρat(u(vR))]2+[u(Dt)]2
Note that, again by the Law of Propagation of Uncertainty, we obtain
(u(vt))2 = u2(mt/ρt) = ( ∂(vt)/ ∂mt)2 u2(mt) + ( (∂vt)/ ∂tρ)2 u2(ρt) = (1/ρt)2 u2(mt) +(-mt/ t2)2 u2(ρt).
However, if the density determination is performed independently from the mass determination, we have:
(u(vt))2 ≈ ( ∂(vt)/ρt)2 u2(ρt) (-mt/ρt2)2 u2(ρt) and similarly (u(vR))2 ≈ (-mR/ R2)2 u2(ρR).
Therefore:
(ucombined(mt))2 ≈ [1(u(mR))]2+[(vt-vR)(u(ρat))]2+ [ρat(u(vt))]2 + [-ρat(u(vR))]2 + [1(u(Dt))]2
≈ [u(mR)]2 + [(mt/ρt - mR/ρR)× u(ρat)]2 + [ρat(-mt/ρt2)× u(ρt)]2 + [-ρat(-mR/ρR2)× u(ρR)]2+[u(Dt)]2
u2 (mt) ≈ u2(mR) + (mt/ρt-mR/ρR)2u2(ρat) + ρat2(-mt/ρt2)2u2(ρt) + ρat2 (-mR/ρR2)2 u2(ρR) + u2(Dt)
Since ρat2 (-mR/ρR2)2 u2(ρR) should already be included in u2(mR), the combined standard uncertainty for the absolute mass mt is the square root of the following:
u2 (mt) ≈ u2(mR) + (mt/ρt-mR/ρR)2u2(ρat) + ρat2(-mt/ρt2)2u2(ρt) + u2(Dt)
where all terms are expressed in the same unit of mass
Please note that the above equation without approximation is:
u2(mt) = [ρt /(ρt-ρat)]2 { u2(Dt) + [(ρR-ρat)/ρR]2 u2(mR) + [mRρat/(ρR)2]2 u2(ρR) + [-mtρat/(ρt)2]2 u2(ρt) + [(mR/ρR)(ρR-ρt)/(ρt-ρat) + Dt
where all terms are expressed in the same unit of mass
3.2 Combined Uncertainty Equation for Conventional Mass
The equation below is the equation for the combined uncertainty associated with a calibration or tolerance test in terms of conventional mass.
(3.2.A)
u2(mct) ≈ u2(mcR)+ ρc2/(ρc - ρac)2 u2(Dt)+ [mcR/(ρR - ρac) - mct/(ρt - ρac)] 2u2(ρat) + (ρac -ρat)2 [(mct2 u2(ρt)) / ρt2 (ρt - ρac)2]
where all terms are expressed in the same unit of mass
The u2(mcR) term used in the above equation should already contain the term
(ρac -ρat)2 [ (mcR 2u2(ρR))/ρR2 (ρR - ρac)2] .
If the unknown mass is of the same nominal value ‘N’ as the reference weight, the above equation can be further approximated with:
(3.2.A approximate)
u2(mct) ≈ u2(mcR) + u2(Dt) + N2 [1/ρR - 1/ρt]2 u2(ρat) + N2 [(ρac -ρat)2 / ρt4 ] u2(ρt)
where all terms are expressed in the same unit of mass
where
u(mct) = combined standard uncertainty associated with the evaluation of the conventional mass of t
u(mcR) = combined standard uncertainty associated with the conventional mass of reference R - usually found on the calibration certificate for reference R. If only u(mR ) is known, formula 3.2.B should be used.
u(mR) = combined standard uncertainty associated with the absolute mass of reference R - usually found on the calibration certificate for reference R. See section 4.1
u(ρat) = standard uncertainty associated with the estimation of the air density at the time of the calibration (kg/m3). See section 4.2
u(ρt) = standard uncertainty associated with the density of the unknown mass t (kg/m3) - See section 4.3.
u(ρR) = standard uncertainty associated with the density of the reference..See section 4.3.
u(Dt) = standard process uncertainty associated with the calculated difference - the long-term process uncertainty of the procedure and equipment used in determining the difference between mass t and reference R. See section 4.4
mt = absolute mass of unknown mass t
mR = absolute mass of reference (standard mass) R
ρt = density of the unknown mass t (kg/m3) - usually provided by manufacturer; otherwise, an estimate can be made by knowing the material and the class. See section 5.1
ρR = density of the reference R (kg/m3) - usually found on calibration certificate for reference R; otherwise, an estimate can be made by knowing the material and the class. See section 5.1
ρat = air density at the time of the calibration (kg/m3) - the formula provided in section 5.2.can be used.
ρac = conventional air density; equal by definition to 1.2 kg/m3 .
ρc = conventional mass density; equal by definition to 8000 kg/m3.
N = nominal value of BOTH the unknown mass and the reference mass
≈ = approximately equal
NOTE: The terms involved in the uncertainty equations 3.2.A and 3.2.B(below) are described in detail in the following sections 4 and 5.
If ever there is a case when, for a calibration or tolerance test in conventional mass, the value of the district standard (or reference) is only known in terms of absolute mass, the equation below is used for the combined uncertainty:
(3.2.B)
u2(mct) ≈ (1-(ρac/ρc))-2 {u2(mR) + (mR/ρR - mt/ρt)2u2(ρat) + (ρac -ρat)2 mt2 u2(ρ)/ρt4 + u2(Dt)}
where all terms are expressed in the same unit of mass
The u2(mR) term used in equation above should contain the term ρat2 mR2 u2(ρR)/ρR4.
The above equations are obtained through mathematical derivation of their respective weighing equations. The reader can find the necessary information regarding how these equations are obtained by consulting the following boxes.
Equation 3.2.A:
First, let us look at equation (3.2.A). It was obtained with equation 2.2.A as follows:
Replacing mt and mR by their equivalence in conventional mass and inserting into the equation
f = mt = mR + ρat (vt - vR) + Dt (see section 3.1), we get:
[mt] = [mR] + ρat (vt - vR) + Dt
[mct - mct (ρac/ρc) + ρacvt] = [mcR - mcR(ρac/ρc) + ρacvR] + ρat (vt - vR) + Dt
mct (1- (ρac/ρc)) = mcR (1- (ρac/ρc)) + ρac (vR - vt) - ρat (vR - vt) + Dt
Pose g = mct = mcR + (1- (ρac /ρc))-1 {(ρac - ρat )(vR - vt) + Dt }
Using the Law of Propagation of Uncertainty, the combined variance for mct assuming no covariance between the parameters is:
(ucombined(mct))2 = (∂g/∂mcR (u(mcR)))2 + (∂g/∂ρac(u(ρac)))2 + (∂g/∂ρc (u(ρc)))2 + (∂g/∂ρat (u(ρat)))2 + (∂g/∂vt (u(vt)))2+ (∂g/∂vR (u(vR)))2 + (∂g/∂Dt (u(Dt)))2
By definition, u(ρc) and u(ρac) are equal to zero; thus the above equation reduces to:
(ucombined(mct))2 =
(∂g/∂mcR (u(mcR)))2 + (∂g/∂ρat (u(ρat)))2 + (∂g/∂vt (u(vt)))2
+ (∂g/∂vR (u(vR)))2 + (∂g/∂Dt (u(Dt)))2
Recall that (u(vt))2 ≈ (-mt/ρt2)2 u2(ρt) and (u(vR))2 ≈ (-mR/ρR2)2 u2(ρR). See section 3.1 for details.
where ∂g/∂mcR≈ 1
∂g/∂ρat ≈ -(vR - vt)(1-(ρac/ρc))-1 = - (mR/ρR - mt/ρt)(1-(ρac/ρc))-1
∂g/∂vt ≈ -(ρac-ρat)(1-(ρac/ρc))-1
∂g/∂vR ≈ (ρac -ρat)(1-(ρac/ρc))-1
∂g/∂Dt ≈ (1-(ρac/ρc))-2
Please note that using the full equation 2.2.A. instead of “g”, we have the following derivatives without approximation:
∂g/∂Dt = [(ρt-ρac)/(ρt-ρat)] [ρc/(ρc-ρac)]
∂g/∂mcR = [(ρt-ρac)/(ρt-ρat)] [(ρM’-ρat)/(ρM’-ρac)]
∂g/∂ρat = [(ρt-ρac)/(ρt-ρat)] [(mct (ρt-ρat)/(ρt-ρac)2) - (mcR/(ρR-ρac))]
∂g/∂ρR = [(ρt-ρac)/(ρt-ρat)] [mcR(ρat-ρ
∂g/∂ρt = [(ρt-ρac)/(ρt-ρat)] [mct (ρac-ρat)/(ρt-ρac)2]
Thus
(ucombined(mct))2 ≈
[ 1 × u(mcR) ]2 + [ - (mR/ρR - mt/ρt) × (1-(ρac/ρc))-1 × u(ρat) ]2
+ [ - (ρac-ρat) × (1-(ρac/ρc))-1 × u(vt) ]2 + [ (ρac -ρat) × (1-(ρac/ρc))-1 × u(vR) ]2
+ [ (1-(ρac/ρc))-1 × u(Dt) ]2
(ucombined(mct))2 ≈
u(mcR)2 + (1-(ρac/ρc))-2{[-(mR/ρR - mt/ρt) u(ρat)]2 + [-(ρac-ρat)u(vt)]2 + [(ρac -ρat)u(vR)]2 + u(Dt)2}
Since (u(vt))2 ≈ (Mvt/Mρt)2 u2(ρt) ≈ (-mt/ρt2)2 u2(ρt) and
(u(vR))2 ≈ (MvR/MρR)2 u2(ρR) ≈ (-mR/ρR2)2 u2(ρR)
(ucombined(mct))2 ≈
u(mcR)2 + (1-(ρac/ρc))-2 {[-(mR/ρR - mt/ρt) u(ρat)]2 + [-(ρac-ρat)(-mt/ρt2) u(ρt) ]2 + [ (ρac -ρat)(-mR/ρR2) u(ρR) ]2 + u(Dt)2 }
Again, replacing mt and mR by their equivalence in conventional mass
(ucombined(mct))2 ≈
u(mcR)2
+ (1-(ρac/ρc))-2 {u(Dt)2}
+ (1-(ρac/ρc))-2 {- ([mcR ρR (1-(ρac/ρc))/(ρR-ρac)] /ρR - [mct ρt (1-(ρac/ρc))/(ρt-ρac)] /ρt) u(ρat) }2
+ (1-(ρac/ρc))-2 {-(ρac-ρat)(-[(mct ρt (1-(ρac/ρc)) /(ρt-ρac)] /ρt2) u(ρt) }2
+ (1-(ρac/ρc))-2 { (ρac -ρat)(-[(mcR ρR (1-(ρac/ρc)) /(ρR-ρac)]/ρR2) u(ρR)}2
(ucombined(mct))2 ≈
u(mcR)2 + (1-(ρac/ρc))-2 {u(Dt)2}
+ (1-(ρac/ρc))-2 { (- (1-(ρac/ρc)))2 ( mcR/(ρR-ρac) - mct /(ρt-ρac) )2 u2 (ρat)
+ (1-(ρac/ρc))-2 {(-(1-(ρac/ρc)))2 (ρac-ρat)2 (-[(mct /(ρt-ρac)] /ρt)2 u2(ρt)
+ (1-(ρac/ρc))-2 { (1-(ρac/ρc))2 (ρac -ρat)2 (-[(mcR/(ρR-ρac)]/ρR)2 u2 (ρR)
u2(mct) ≈ u2(mcR)+ ρc2/(ρc - ρac)2 u2(Dt)+ [mcR/(ρR - ρac) - mct/(ρt - ρac)] 2u2(ρat)
+ (ρac -ρat)2 [ (mcR 2u2(ρR))/ρR2 (ρR - ρac)2 + (mct2 u2(ρt)) / ρt2(ρt - ρac)2]
Since (ρac -ρat)2 [ (mcR 2u2(ρR))/ρR2 (ρR - ρac)2] should already be included in u2(mR), the combined standard uncertainty for mct expressed in terms of mcR is the square root of the following:
(3.2.A)
u2(mct) ≈ u2(mcR)+ ρc2/(ρc - ρac)2 u2(Dt)+ [mcR/(ρR - ρac) - mct/(ρt - ρac)] 2u2(ρat) + (ρac -ρat)2 [(mct2 u2(ρt)) / ρt2 (ρt - ρac)2]
where all terms are expressed in the same unit of mass
Please note that equation 3.2.A without approximation is:
u2(mct) = [(ρt-ρac)/(ρt-ρat)]2 {[(ρM’-ρat)/(ρM’-ρac)]2 u2(mcR)+ ρc2/(ρc-ρac)2 u2(Dt) + [mcR(ρat-ρac)/(ρR-ρac)2]2 u2(ρR) + [(mct (ρt-ρat)/(ρt-ρac)2) - (mcR/(ρR-ρac))]2 u2(ρat) + [mct (ρac-ρat)/(ρt-ρac)2]2 u2(ρt)}
Please also note that u2(Dt) is the base value of the process uncertainty for the nominal value being calibrated or tolerance tested, as obtained with RP-03Field and implicitly includes both variations in the value of “SR” and variations in the indicated values.
EQUATION 3.2.B:
Now looking at equation (3.2.B). It was obtained with equation 2.2.B as follows:
Replacing only mt by its equivalence in conventional mass and inserting into the equation
f = mt = mR + ρat (vt - vR) + Dt, we get:
[mct - mct (ρac/ρc) +ρacvt] = [mR] + ρat (vt - vR) + Dt
mct (1-(ρac/ρc)) = [mR] + ρat (vt - vR) + Dt - ρacvt
mct = (1-(ρac/ρc)) -1{mR + ρat (vt - vR) + Dt - ρacvt}
Pose h = mct = (1-(ρac /ρc))-1 { mR + ρat (vt - vR) - ρac vt + Dt }
Using the Law of Propagation of Uncertainty as in the previous section, and remembering that u(ρc) and u(ρac) are equal to zero, the combined variance for mct, assuming no covariance between the parameters, is:
(ucombined(mct))2 =
(∂h/∂mR (u(mR)))2+ (∂h/∂ρat (u(ρat)))2+ (∂h/∂vt (u(vt)))2+ (∂h/∂vR (u(vR)))2 + (∂h/∂Dt (u(Dt)))2
where
∂h/∂mR = (1-(ρac/ρc))-1
∂h/∂ρat = -(vR - vt)(1-(ρac/ρc))-1
∂h/∂vt = -(ρac-ρat)(1-(ρac/ρc))-1
∂h/∂vR = (ρat)(1-(ρac/ρc))-1
∂h/∂Dt = (1-(ρac/ρc))-2
(ucombined(mct))2 ≈
((1-(ρac/ρc))-1 u(mR))2 + (-(vR - vt)(1-(ρac/ρc))-1 u(ρat)) 2 + (-(ρac-ρat)(1-(ρac/ρc))-1 u(vt))2 + (-(ρat)(1-(ρac/ρc))-1 u(vR))2 + ((1-(ρac/ρc))-1 u(Dt))2
Recall that (u(vt))2 . (-mt/ρt2)2 u2(ρt) and (u(vR))2 ≈ (-mR/ρR2)2 u2(ρR). See section 3.1 for details.
(ucombined(mct))2 ≈
(1-(ρac/ρc))-2 {u(mR))2 + (-(mR/ρR - mt/ρt)u(ρat))2 + (-(ρac-ρat)(-mt/ρt2)u(ρt))2
+ (-(ρat)(-mR/ρR2)u(ρR))2 + (u(Dt))2}
u2(mct) ≈
(1-(ρac/ρc))-2 {u2(mR) + (mR/ρR - mt/ρt)2u2(ρat) + (ρac -ρat)2 mt2 u2(ρt)/ρt4
+ ρat2 mR2 u2(ρR)/ρR4 + u2(Dt)}
Since (1-(ρac/ρc))-2 {ρat2 mR2 u2(ρR)/ρR4} should already be included in u2(mR), the combined standard uncertainty for mct expressed in terms of mR is the square root of the following:
(3.2.B)
u2(mct) ≈ (1-(ρac/ρc))-2 {u2(mR) + (mR/ρR - mt/ρt)2u2(ρat) + (ρac -ρat)2 mt2 u2(ρt)/ρt4 + u2(Dt)}
where all terms are expressed in the same unit of mass
The u2(mR) term used in equation above should contain the term ρat2 mR2 u2(ρR)/ρR4.
Note: Some prefer to assume that the density of the object being measured is unimportant in conventional weighing. However, as can be seen by looking at the above expression, both the density of the test mass (ρt) and the uncertainty of this density (u(ρt)) are included in the equation for u(mct) and can only be neglected if certain conditions are met.
Components Of Uncertainty
4.1 Uncertainty of the Reference Weight: u(mR) or u(mcR)
The uncertainties u(mR) or u(mcR) are the combined standard uncertainties associated with, respectively, the absolute mass and the conventional mass of the reference R. These values are usually found on the calibration certificate for the reference weight.
As noted previously, in order to perform tolerance testing, the reference weight shall be of a class at least one level above that of the mass being tolerance tested. In other words the tolerance of the reference weight must be smaller than one third that of the weight being tolerance tested.
As a condition for the reference weight to be acceptable during its calibration, we had that:
U(mcR) ≤ 1/3 [mpe (tolerance) of mcR]
- where U(mcR) is the expanded uncertainty associated with the determination of the conventional mass of the reference. When calculating the expanded uncertainty, we use a coverage factor of k=2, which means that we multiply the combined uncertainty by two to get the expanded uncertainty. In other words: U(mcR) = 2 u(mcR)
Now, since the reference weight must be of at least one class higher than the weight being tolerance tested, we have that:
[mpe (tolerance) of mcR] ≤ 1/3 [mpe (tolerance) of mct]
Therefore:
U(mcR) ≤ 1/3 [mpe (tolerance) of mcR] or
U(mcR) ≤ 1/3 [1/3 mpe (tolerance) of mct]
Since U(mcR) = 2 u(mcR), then
2u(mcR) ≤ 1/9 [mpe (tolerance) of mct]
u(mcR) ≤ 1/18 [mpe (tolerance) of mct]
Hence, in order to perform tolerance testing, the expanded uncertainty of the reference (standard mass) used must first be known to be of at least one class above that of the mass under test; in other words U(mcR) = 2 u(mcR) shall be less than 1/9 of the tolerance of the mass under test, which implies that u(mcR) ≤ 1/18 [mpe (tolerance) of mct] .
Although this section only addresses uncertainty of the reference, please recall that, as mentioned in RP-01FIELD: Laboratory Calibration Procedures for Standards of Mass and in RP-03FIELD: Calibration Performance Monitoring, we know that for any mass being calibrated, we must also have 3u(Dt) ≤ 1/3 mpe (tolerance) of that mass; thus, u(Dt) ≤ 1/9 tolerance of mct.
4.2 Uncertainty of Air Density: u(ρat)
To calculate the air density at the time of calibration, we can use the approximate equation provided by the Canada National Research Council (CNRC) as follows:
ρat . [(0.092695) + (1.18 x 10-5) P (in Pa) - (4.15 x 10-3) T (in°C) - (1.22 x10-4) Hrel (in %) ] kg/m3
Inside:
The air density near sea level, at a temperature close to 20°C and humidity close to 50%RH, varies by no more than approximately ± 10% ρac . In this case (ρac - ρat) = 10%(1.2 kg/m3) = ± 0.12 kg/m3. If we assume a rectangular distribution, we have u(ρat) = ± (0.12 kg/m3) ÷ √3 = ± 0.07 kg/m3. This scenario corresponds to calibrations performed indoors. Therefore, if the air density indoors is not measured at the time of calibration, it can be assumed to be (1.2 ± 0.07) kg/m3 , and u(ρat) is ± 0.07 kg/m3.
Outside
For calibrations performed outdoors, the air density for outside temperature and humidity may vary by as much as ± 24% ρac with the different seasons. This fact can be observed by looking at the following expected variations for different locations across the country:
- temperature: -40°C to +40 °C
- relative humidity: 20% to 100%
- pressure: 90 kPa to 110 kPa
we get a maximum air density when H=20%, T= -40°C and P=110 kPa; ρat = 1.554 kg/m3
we get a minimum air density when H=100%, T= 40°C and P= 90 kPa ; ρat = 0.976 kg/m3
Hence, the maximum variation of ρat for an outdoor environment can be set at: ± (1.554 - 0.976) ÷ 2 = ± 0.29 kg/m3. If we treat this value as a rectangular distribution, we get u(ρat) = ± 0.29 kg/m3 ÷ √3 = ± 0.17 kg/m3. Therefore, if the air density outdoors is not measured when the calibration is performed outdoors, it can be assumed to be (1.2 ± 0.17) kg/m3 and u(ρat) is ± 0.17 kg/m3.
Inside: The value of the uncertainty of the air density to be used in all the uncertainty equations for calibrations performed inside under stable conditions is:
u(ρat) = ± 0.07 kg/m3.
Outside: The value of the uncertainty of the air density to be used in all the uncertainty equations for calibrations performed outside, for example railway test car calibrations, is:
u(ρat) = ± 0.17 kg/m3.
4.3 Uncertainty of Material Density of the Reference u(ρR) and
Uncertainty of Material Density of Weight Being Calibrated u(ρt)
When calibrating or tolerance testing a field working standard (or industry weight) using a district standard as a reference, we have that ρR is the density of the district standard, and u(ρR) is the uncertainty associated with this density determination, ρt is the density of the field working standard, and u(ρt) is the associated uncertainty of its density determination.
The value (or an estimate) of the density and its associated uncertainty for a district standard will be indicated on its calibration certificate issued by the CSL.
Unless provided by the manufacturer of the field working standard, a rectangular distribution estimate based on the limits of the material densities can be used for u(ρt). In other words, if the material densities of section 1.2 are respected, we have that, for class M1, M1-2 and M2 weights, the value of u(ρt) for the field working standard can be approximated as: ± (8400 - 6000) kg/m3 ÷ 2√ 3 = ± 693 (kg/m3) or approximately ± 700 kg/m3. For classes M2-3 and M3 the value of u(ρt) can be approximated with:
± (8400 - 4400) kg/m3 ÷ 2√ 3 = ± 1155 (kg/m3) or approximately ± 1200 kg/m3.
When calibrating or tolerance testing using district standards, we have
u(ρR) = the value provided by the CSL
u(ρt) M1, M1-2 and M2 weights * = ± 700 kg/m3 or the value provided by the manufacturer of the field inspection standard, whichever is greater
u(ρt) M2-3 and M3 weights = ± 1200 kg/m3 or the value provided by the manufacturer of the field inspection standard, whichever is greater
Please recall that u(ρR) should not be accounted for twice as it should already be included in u(mR) or u(mcR).
Note *: if unsure that the material densities of section 1.2 have been respected, it is best to use the estimate of ± 1200 kg/m3.
4.4 Uncertainty of the Process u(Dt) (Amendment)
The process uncertainty u(Dt), which is the long-term performance (repeatability and reproducibility) of the device used in calibrating or tolerance testing must be
smaller than 1/9 of the tolerance that will be applied to the standard being calibrated or tolerance tested.
The process uncertainty u(Dt) is a function of two main components: short-term variance (S2i) and long-term variance u2(LTi).
u2(Dt) = (St)2 + u2(LTt)
The short-term variance is based on the individual (daily) calibrations and the long-term variance is calculated over many days of data by using the base value of the Process Variance u2(Di) for the Check Standards of the same (or similar) value as the mass being calibrated.
The base value of the Process Variance of the Check Standards u2(Di) is continuously monitored and adjusted by using RP-03FIELD: Mass Calibration Process Monitoring. It is downloaded directly into the calculation of u(Dt).
When performing tolerance testing or calibration with a single comparison, only one value is obtained for this type of verification, hence a variance (S2i) cannot be calculated and we use the readability of the device (its sensitivity or half the graduation) divided by /3 instead of Si. The division by /3 is performed to treat the data as a rectangular distribution.
As mentioned above, the base value of the Process Variance u2(Di) for the Check Standard of the same (or similar) nominal value as the weight being calibrated is used as an estimate of u2(LTi).
When performing a calibration with more than one comparative weighing, we use the range (the difference between the maximum and minimum values obtained) as an estimate of the short-term variance. Please note that the range is divided by 2/3 to treat it as a rectangular distribution.
• Tolerance Testing and Single Comparative Weighing (“r” =1):
Readability = the sensitivity or half the graduation (division) of the device used, expressed in the same units as u2(Dt)
u2(Di) = the base value of the Process Variance for the Check Standard of the same value as the mass ‘t’ being calibrated
• Comparative Weighing with “r” Comparisons (r>1):
u2(Dt) = (Sζ)2 + u2(Di)
where Sζ is:
Readability = the sensitivity or half the graduation (division) of the device used, expressed in the same units as u2(Dt)
Sζ = the standard deviation of the “r” comparisons, approximated with the range
Maximum (x) = the maximum value obtained over the “r” comparisons
minimum (x) = the minimum value obtained over the “r” comparisons
u2(Di) = the base value of the Process Variance for the Check Standard of the same value as the mass ‘t’ being calibrated
Please note that since u(Dt) must be smaller than 1/9 of the tolerance, then the readability of the device as expressed by either the sensitivity or by half the graduation must also be smaller than 1/9 of the tolerance
Uncertainty Multipliers
5.1 Density ρR of the reference and density ρt of weight being calibrated
When calibrating in absolute mass or in conventional mass, an estimate of the density of the test mass and of the reference must be known. This estimate can be obtained from the manufacturer of the weight or by knowing the material.
When tolerance testing field working standards, it is sufficient to ensure that the material density is within the limits listed in section 1.2 above. An estimate of the material density can be obtained from the manufacturer of the weight, by knowing the material or by taking the average value for the limits provided in section 1.2.
For the purposes of tolerance testing or calibration of the field working standards using a District Standard as the reference, we have:
ρR = value provided by the CSL on the certificate for the District Standard
ρt = value obtained from the manufacturer of the field working standard, by knowing the material or by taking the average of the limits listed in section 1.2.
As discussed in 4.2, the air density near sea level, at a temperature close to 20°C and humidity close to 50%RH, varies by no more than approximately ± 10% from the value 1.2 kg/m3.
Since we can’t really look at a worst case scenario, the value ρat = 1.2 kg/m3 is used when approximating. Should we wish to calculate the actual air density, the following equation (provided by the Canada National Research Council) can be used:
ρat ≈ (0.092695) + (1.18 x 10-5) P (in Pa) - (4.15 x 10-3) T (in°C) - (1.22 x10-4) Hrel (in %)
In other words, the value of air density to be used in all the calibration and uncertainty equations is either the true value calculated by using the NRC equation provided above or is approximated with ρat = 1.2 kg/m3.
The conventional air density is a theoretical value that has been set to ρac = 1.2 kg/m3.
The value of conventional air density to be used in all the calibration and uncertainty equations is: ρac = 1.2 kg/m3.
The conventional mass density is a theoretical value that has been set to ρc = 8000 kg/m3 .
The value of conventional mass density to be used in all the calibration and uncertainty equations is: ρc = 8000 kg/m3.
Summary - At a Glance
NOTE: All terms are expressed in the same unit of mass
Calibration with Single Comparison
Absolute Mass - Calibration Equation
(2.1) mt = [ mR (1 - (ρat/ρR)) + Dt ] / ( 1 - (ρat/ρt))
Absolute Mass - Uncertainty Equation
(3.1) u2 (mt). u2(mR) + (mt/ρt-mR/ρR)2u2(ρat) + ρat2(-mt/ρt2)2u2(ρt) + u2(Dt)
Conventional Mass - Calibration Equations
(2.2.A) mct = ((ρt - ρac) / (ρt - ρat)) { (mcR (ρR - ρat)/(ρR - ρac)) + (Dt ρc / ( ρc - ρac)) }
(2.2.B) mct = [(ρc (ρt - ρac)) / ((ρc - ρac) (ρt - ρat)) ] {mR (1 - (ρat /ρR)) + Dt }
Conventional Mass - Uncertainty Equations
(3.2.A) u2(mct) ≈ u2(mcR)+ ρc2/(ρc - ρac)2 u2(Dt)+ [mcR/(ρR - ρac) - mct/(ρt - ρac)] 2u2(ρat) + (ρac -ρat)2 [(mct2 u2(ρt)) / ρt2 (ρt - ρac)2]
(3.2.A_approx) u2(mct) ≈ u2(mcR) + u2(Dt) + N2 [1/ρR - 1/ρt]2 u2(ρat) + N2 [(ρac -ρat)2 / ρt4 ] u2(ρt)
Tolerance Testing
Tolerance Testing Equation (in Conventional Mass)
(2.3) mct ≈ mcR + Dt
Tolerance Testing - Uncertainty Equations (in Conventional Mass)
(3.2.A) u2(mct) ≈ u2(mcR)+ ρc2/(ρc - ρac)2 u2(Dt)+ [mcR/(ρR - ρac) - mct/(ρt - ρac)] 2u2(ρat) + (ρac -ρat)2 [(mct2 u2(ρt)) / ρt2 (ρt - ρac)2]
(3.2.A_approx) u2(mct) ≈ u2(mcR) + u2(Dt) + N2 [1/ρR - 1/ρt]2 u2(ρat) + N2 [(ρac -ρat)2 / ρt4 ] u2(ρt)
where
u(mct)= combined standard uncertainty associated with the evaluation of the conventional mass of t
u(mcR)= combined standard uncertainty associated with the conventional mass of reference R - usually found on the calibration certificate for reference R. If only u(mR ) is known, formula 3.2.B should be used.
u(mR)= combined standard uncertainty associated with the absolute mass of reference R - usually found on the calibration certificate for reference R. See section 4.1
u(ρat) = standard uncertainty associated with the estimation of the air density at the time of the calibration (kg/m3). See section 4.2
u(ρt) = standard uncertainty associated with the density of the unknown mass t (kg/m3) - See section 4.3.
u(ρR) = standard uncertainty associated with the density of the reference..See section 4.3.
u(Dt) = standard process uncertainty associated with the calculated difference - the long-term process uncertainty of the procedure and equipment used in determining the difference between mass t and reference R. See section 4.4
mt = absolute mass of unknown mass t
mR = absolute mass of reference (standard mass) R
ρt = density of the unknown mass t (kg/m3) - usually provided by manufacturer; otherwise, an estimate can be made by knowing the material and class. See section 5.1
ρR = density of the reference R (kg/m3) - usually found on calibration certificate for reference R; otherwise, an estimate can be made by knowing the material and class. See section 5.1
ρat = air density at the time of the calibration (kg/m3) - the formula provided in section 5.2.can be used.
ρac = conventional air density; equal by definition to 1.2 kg/m3 .
ρc = conventional mass density; equal by definition to 8000 kg/m3.
N = nominal value of BOTH the unknown mass and the reference mass
≈ = approximately equal
References
1. Chapman, George D., Orthogonal Designs for the Calibration of Kilogram Submultiples, National Research Council of Canada, Report No. AMS-002 NRCC 25819, November 1993.
2. Debler, E., A Theoretical Approach to Measurement - Uncertainty of Mass, Conventional Mass and Force, OIML Bulletin, Volume XXXV, Number 4, October 1994.
3. Dupuis-Désormeaux, N., Procedure for Estimating Uncertainties in Mass Calibrations, Industry Canada, third edition, November 1994.
4. Joint International Committee ISO/IEC/OIML/BIPM- TAG-4, Guide to the Expression of Uncertainty in Measurement, first edition, June 1992.
5. OIML Recommendation R111 - Weights of classes E1, E2, F1, F2, M1, M2, M3, Edition 1994.
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