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An Uncertainty Analysis for a Positive Displacement Liquid Flow Calibrator Using the Water Draw TechniqueWesley B. EnglandWhere:K = Calibrator Constant [pulses/L]p = pulse count [counts]V = Volume [L]C tm= Correction for Thermal Expansion of Water Draw Vessel [-]C pl= Correction for Compressibility of Fluid Medium [-]C ts= Correction for Thermal Expansion of Flow Tube [-]C td= Correction for Thermal Expansion of Encoder [-]C ps= Correction for Pressure Expansion of the Flow Tube [-]C vs= Correction for Thermal Expansion of Fluid Vessel [-]α V= Volume Thermal Expansion Coefficient of Draw Vessel [°F -1 ]T V= Temperature of Draw Vessel [°F]T S= Reference Temperature (68 °F) [°F]P S= Reference Pressure (0 psig) [psi]P CAL= Calibrator Pressure during Draw [psi]Z 7024= Compressibility of Fluid Media [psi -1 ]α A= Area Thermal Expansion Coefficient of Flow Tube [°F -1 ]T W= Calibrator Temperature [°F]α ENC= Linear Thermal Expansion of Encoder [°F -1 ]T d= Detector Temperature [°F]d = Flow Tube Inside Diameter [in]T = Flow Tube Wall Thickness [in]γ FT= Modulus of Elasticity of Flow Tube [psi]α 7024= Thermal Expansion Coefficient of Fluid Media [°F -1 ]M E= Error in Mencius Reading [%]Calculation of Sensitivity Coefficients UsingPartial DifferentiationMeniscus Reading Error SensitivityCoefficient for Flow Tube M E[L]The Meniscus Reading Error is applied to the calibrationconstant by simply multiplying it with the CalibrationConstant equation. Therefore, the sensitivity coefficientis determined by partial differential calculus as are all thesensitivity coefficients shown below.1__K · ​ ____ ∂K ​ = ​∂M ___ 1(13)EM EPulse Count Sensitivity Coefficient p [counts]​ 1__ K · ​ ___ ∂K∂p ​ = ​ 1__ p (14)Volume Sensitivity Coefficient V [L]​ 1__ K · ​ ___ ∂K∂V ​ = ​ ___ 1V ​ (15)Calibrator Temperature SensitivityCoefficient T W[°F]​ 1__ K · ​ ____ ∂K= ​ α · ​ A [ 1−α · ​( T V−T W ) ]​−α · ​[ 1+α A· ​( T W−T__________________________________ S ) ]​∂T W​[ 1−α · ​( T V−T W ) ]​ · ​ [ 1+α A· ​( T W−T S ) ]​​ (16)Detector Sensitivity Coefficient T d[°F]α ENC​ 1__ K · ___ ​ ∂K ​ = ​ _______________∂T d1 + α ENC· ​( T d−T S )​ ​ (17)Vessel Temperature SensitivityCoefficient T V[°F]___________________________________________​ = ​ [ 1−α · ​( T V−T W ) ]​· ​[ −α · ​( 1+α V· ​( T V−T S ) )​−α V· ​( 1−α · ​( T V−T W ) ) ]​​ (18)∂T V​[ 1+α V· ​( T V−T S ) ]​1__K · ___ ∂KReference Temperature SensitivityCoefficient T S[°F]The sensitivity coefficient for the reference pressure isnot calculated here because the gauge pressure of 0 psi isfixed defined constant.Reference Pressure SensitivityCoefficient P [psi]The sensitivity coefficient for the reference pressure isnot calculated here because the gauge pressure of 0 psi isfixed defined constant.Calibrator Pressure SensitivityCoefficient P CAL[psi]​ 1__ K · _____ ​ ∂K =______________________________d + γ · T · Z​∂P CALγ · T · ​( 1 − P CAL· Z ) · ​( 1 + ​( P CAL· d ⁄ γ · T ) )​ ​ (19)Compressibility Factor Sensitivity Coefficientfor Liquid Media Z [psi -1 ]P CAL1__K · ​ ___ ∂K∂Z = ​ __________1 − P CAL· Z ​ (20)Volume Thermal Expansion SensitivityCoefficient for Liquid Media α [°F -1 ]1__K · ___ ​ ∂K ​(∂α ​ = ​ T V−T_____________ W )​1 + α · ​( T V−T W )​ ​ (21)Flow Tube inside Diameter SensitivityCoefficient d [in]P CAL1__K · ___ ​ ∂K∂d ​ = ​ ____________γ · T + P CAL· d ​ (22)Flow Tube Wall Thickness SensitivityCoefficient T [in]1__K · ___ ​ ∂K∂T ​ = ​ _________________P CAL· dT · ​( T · γ + P CAL· d )​ ​ (23)Jan • Feb • Mar 201333Cal Lab: The International Journal of Metrology