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An Uncertainty Analysis for a Positive Displacement Liquid Flow Calibrator Using the Water Draw TechniqueWesley B. EnglandUncertainty Summary for Thermal Expansion ofMIL-C-7024 and Gear Oil Blend:Type of Uncertainty: Relative Type BSensitivity Coefficient: ​ 1__ K · ​ ___ ∂K ​(∂α ​ = ​ T V−T_____________ W )​1 + α · ​( T V−T W )​ ​Distribution: NormalOne Standard Uncertainty:________________________________​( 7.111769 % ⁄ 100 ) · ​( 4.90069×10 -4 °F )​= 1.742629×102-5 °FFlow Tube Inside Diameter, d [in]The Flow Tube inside diameter of 4.625 inches wastaken from the Flow Technology Test Report for MT10SN: MT02060175, Report No. 75888, dated July 31, 2002[11]. It was assumed that the nominal value of 4.625inches was measured accurately to within 0.0001 in.Therefore:Uncertainty Summary for Flow Tube Inside Diameter:Type of Uncertainty: Relative Type BP CALSensitivity Coefficient: ​ 1__ K · ​ ___ ∂K∂d ​ = ​ ____________γ · T + P CAL· d ​Distribution: Rectangular0.0001 inOne Standard Uncertainty: ​ ________​√ ​ = 5.7735×10 -5 in3 ​Flow Tube Wall Thickness, T [in]The Flow Tube wall thickness of 0.312 inches wastaken from the Flow Technology Test Report for MT10SN: MT02060175, Report No. 75888, dated July 31, 2002[11]. It was assumed that the nominal value of 0.312inches was measured accurately to within 0.0001 in.Therefore:Uncertainty Summary for Flow Tube Wall Thickness:Type of Uncertainty: Relative Type BSensitivity Coefficient: ​ 1__ K · ​ ___ ∂K∂T ​ = ________________​P CAL· dT · ​( T · γ + P CAL· d )​ ​Distribution: Rectangular0.0001 inOne Standard Uncertainty: ​ ________​√ ​ = 5.7735×10 -5 in3 ​Linear Thermal Expansion Coefficientof Encoder, α ENC[°F -1 ]This Flow Calibrator utilizes a Mitutoyo Model AT112Encoder that, according to the Mitutoyo online catalogspecifications [12], has a thermal expansion coefficient of(8±1)×10 -6 °C -1 with an uncertainty of ±1×10 -6 °C -1 whichconverts into (5 ⁄ 9) °C ⁄ °F · 8×10 -6 ⁄ °C = 4.44×10 -6 ⁄ °F withan uncertainty of (5 ⁄ 9) °C ⁄ °F · ±1×10 -6 ⁄ °C = 5.56×10 -7 ⁄ °F.Uncertainty Summary for Linear Thermal ExpansionCoefficient of the Encoder:Type of Uncertainty: Relative Type BSensitivity Coefficient: ​ 1__ K · ​ _____ ∂K ​( T d−T​ = ​_______________ S )​∂α ENC1 + α ENC· ​( T d−T S )​ ​Distribution: RectangularOne Standard Uncertainty:____________​ 5.56×10-6 °F -1​√ ​ = 3.21×10 -7 °F -13 ​Area Thermal Expansion Coefficientof Flow Tube, α A[°F -1 ]The Flow Tube is constructed from 316 Stainless Steel,which according to API Manual of Petroleum MeasurementStandards Chapter 12-Calculation of Petroleum Quantitiespage 16 Table 6 [13] has a linear thermal expansioncoefficient of 8.83x10 -6 °F -1 . Since we want an area thermalexpansion coefficient this value must be multiplied by2 therefore the area thermal expansion coefficient of thestainless steel flow tube is 2(8.83×10 -6 ) = 1.766×10 -5 °F -1 . TheNIST online Engineering Tool Box [14] States that the valuefor the linear thermal expansion of 316 stainless steel isknown within 3 to 5%. Therefore, taking 5% as the worstcase, the following is obtained for the uncertainty in thevolume area thermal expansion coefficient:U = ​√ ________________________________________________( ​( 5 ⁄ 100 ) · ​( 8.83×10 -6 °F -1 ) ) 2 ​ + ​( ​( 5 ⁄ 100 ) · ​( 8.83×10 -6 °F -1 ) ) 2= 6.2438×10 -7 °F -1 . (45)Uncertainty Summary of Area Thermal ExpansionCoefficient of 316 Stainless Steel:Type of Uncertainty: Relative Type BSensitivity Coefficient: ​ 1__ K · ​ ____ ∂K ​( T W−T​ = ​______________ S )​∂α A1 + α A· ​( T W−T S )​ ​Distribution: RectangularOne Standard Uncertainty:______________6.2438×10 -7 °F -1​√ ​ = 3.605×10 -7 °F -13 ​Jan • Feb • Mar 201339Cal Lab: The International Journal of Metrology