Calibration of a gauge block of nominal length 50 mm
The calibration of the grade 0 gauge block (ISO 3650) of 50 mm nominal length is carried out by comparison using a comparator and a calibrated gauge block of the same nominal length and the same material as reference standard. The difference in central length is determined in vertical position of the two gauge blocks using two length indicators contacting the upper and lower measuring faces.
The actual length of the gauge block to be calibrated is related to the actual length of the reference standard by the equation
l_{X}' = l_{S}' + δl
with δl being the measured length difference. l_{X} blocks under the measurement conditions, in particular at a temperature which on account of the uncertainty in the measurement of laboratory temperature may not be identical with the reference temperature for length measurements.
Model Equation:
l_{X} = l_{S} + δl_{D} + δl + δl_{C}  L * (α_{av} * δt + δα * Δt_{av} + u_{at})  δl_{V}

List of Quantities:
Quantity  Unit  Definition 

l_{X}  mm  length of the gauge block to be calibrated 
l_{S}  mm  length of the reference gauge block at the reference temperature of t_{0}=20 °C according to its calibration certificate 
δl_{D}  mm  Change of the length of the reference gauge block since its last calibration due to drift 
δl  mm  observed difference in length between the unknown and the reference gauge block 
δl_{C}  mm  correction for nonlinearity and offset of the comparator 
L  mm  nominal length of the gauge blocks under consideration 
α_{av}  K^{1}  average of the thermal expansion coefficients of the unknown and the reference gauge block 
δt  K  difference in temperature between the unknown the reference gauge block 
δα  K^{1}  difference in the thermal expansion coefficients between the unknown and the reference gauge block 
Δt_{av}  K  deviation of the average temperature of the unknown and the standard gauge block from the reference temperature 
u_{at}  coorection for second order terms of (δα * Δt_{av})  
δl_{V}  mm  correction for noncentral contacting of the measuring faces of the unknown gauge block 
l_{S}: 
Type B normal distribution Value: 50.000020 mm Expanded Uncertainty: 30·10^{6} mm Coverage Factor: 2.0 
REFERENCE STANDARD: The length of the reference gauge block together with the associated expanded uncertainty of measurement is given in the calibration certificate of a set of gauge blocks as 50,000 02 mm ±30 nm (coverage factor k=2).
δl_{D}: 
Type B triangular distribution Value: 0 mm Halfwidth of Limits: 30·10^{6} mm 
DRIFT OF THE STANDARD: The temporal drift of the length of the reference gauge block is estimated from previous calibrations to be zero with limits ±30 nm. General experience with gauge blocks of this type suggest that the zero drift is the most probable value and that a triangular probability distribution can be assumed.
δl: 
Type A Method of observation: Direct Number of observation: 5
Arithmetic Mean: 94.00·10^{6} mm 
δl_{C}: 
Type B rectangular distribution Value: 0 mm Halfwidth of Limits: 32·10^{6} mm 
COMPARATOR: The comparator has been verified to meet the specifications stated in EALG21. From this it can be ascertained that for length differences D up to ±10 µm corrections to the indicated length difference are within the limits ±(30 nm + 0,02 * abs(D)). Taking into account the tolerances of the grade 0 gauge block to be calibrated and the grade K reference gauge block the maximum length difference will be within ±1 µm leading to the limits of ±32 nm for nonlinearity and offset corrections of the comparator used.
L: 
Constant Value: 50 mm 
Nominal length of the gauge block to be calibrated.
α_{av}: 
Type B rectangular distribution Value: 11.5·10^{6} K^{1} Halfwidth of Limits: 1·10^{6} K^{1} 
TEMPERATURE CORRECTION: Based on the calibration certificate of the reference gauge block and the manufacturer’s data for the gauge block to be calibrated the linear thermal expansion coefficient of the steel gauge blocks is assumed to be within the interval (11,5±1,0)×10^{6} °C^{1}.
δt: 
Type B rectangular distribution Value: 0 K Halfwidth of Limits: 0.05 K 
TEMPERATURE CORRECTION: Before calibration care is taken to ensure that the gauge blocks assume ambient temperature of the measuring room. The remaining difference in temperature between the standard and the gauge block to be calibrated is estimated to be within ±0,05 K.
δα: 
Type B triangular distribution Value: 0.0 K^{1} Halfwidth of Limits: 2·10^{6} K^{1} 
TEMPERATURE CORRECTION: Combining the two rectangular distributions the difference in linear thermal expansion coefficient is triangularly distributed within the limits ±2E6 °C^{1}.
Δt_{av}: 
Type B rectangular distribution Value: 0 K Halfwidth of Limits: 0.5 K 
TEMPERATURE CORRECTION: The deviation of the mean temperature of measurement from the reference temperature t_{0} = 20 °C is estimated to be within ±0,5 °C.
u_{at}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.236·10^{6} Coverage Factor: 1.0 
TEMPERATURE CORRECTION: The best estimates of the difference in linear expansion coefficients (δα) and the deviations of the mean temperature from the reference temperature (Δt_{av}) are zero. Therefore second order terms have to be taken into account in the evaluation of their uncertainty contribution resulting in the product of standard uncertainties associated with the factors of the product term (δα × Δt_{av}) in the model equation. The final standard uncertainty is u(δα × Δt_{av}) = 0,236·10^{6}.
δl_{V}: 
Type B rectangular distribution Value: 0 mm Halfwidth of Limits: 6.7·10^{6} mm 
VARATION IN LENGTH: For gauge blocks of grade 0 the variation in length determined from measurements at the centre and the four corners has to be within ±0,12 mm (ISO 3650). Assuming that this variation occurs on the measuring faces along the short edge of length 9 mm and that the central length is measured inside a circle of radius 0,5 mm the correction due to central misalignment of the contacting point is estimated to be within ±6,7 nm.
Uncertainty Budgets:
l_{X}: length of the gauge block to be calibrated
Quantity  Value 
Standard Uncertainty 
Distribution 
Sensitivity Coefficient 
Uncertainty Contribution 
Index 

l_{S}  50.0000200 mm  15.0·10^{6} mm  normal  1.0  15·10^{6} mm  19.3 % 
δl_{D}  0.0 mm  12.2·10^{6} mm  triangular  1.0  12·10^{6} mm  12.8 % 
δl  94.00·10^{6} mm  4.75·10^{6} mm  normal  1.0  4.7·10^{6} mm  1.9 % 
δl_{C}  0.0 mm  18.5·10^{6} mm  rectangular  1.0  18·10^{6} mm  29.2 % 
L  50.0 mm  
α_{av}  11.500·10^{6} K^{1}  577·10^{9} K^{1}  rectangular  0.0  0.0 mm  0.0 % 
δt  0.0 K  0.0289 K  rectangular  580·10^{6}  17·10^{6} mm  23.6 % 
δα  0.0 K^{1}  816·10^{9} K^{1}  triangular  0.0  0.0 mm  0.0 % 
Δt_{av}  0.0 K  0.289 K  rectangular  0.0  0.0 mm  0.0 % 
u_{at}  0.0  236·10^{9}  normal  50  12·10^{6} mm  11.9 % 
δl_{V}  0.0 mm  3.87·10^{6} mm  rectangular  1.0  3.9·10^{6} mm  1.3 % 
l_{X}  49.9999260 mm  34.2·10^{6} mm 
Results:
Quantity  Value 
Expanded Uncertainty 
Coverage factor 
Coverage 

l_{X}  49.999926 mm  68·10^{6} mm  2.00  95% (ttable 95.45%) 