Example In the control of a chemical process temperature is measured by a sensor that according to the manufacturer has a calibration of 0 50C In 20 measurements in a test of the temperature sensor s repeatability an average value of 1500C was determined with a standard deviation of 1 50C It is estimated that there can be a spatial variation of 20C and an installation effect of 10C In a separate test of the data transmission system it was determined that for 10 measurements the standard deviation was 0 50C The control program uses a linear relationship between temperature and voltage which can introduce an uncertainty of up to 100C in the applicable range What are the elemental errors classify as B and P calibration B repeatability P spatial variation B installation B data acquisition P linearity B If we put numbers to these we have BIAS calibration 0 50C spatial variation 20C installation 10C linearity 10C Precision repeatability P 1 50C 20 measurements data acquisition P 0 50C 10 Measurements Combine elemental Bias Errors B 52 22 12 22 C 2 5C Combine elemental precision Errors USE SD NOT SDOM P 52 1 52 1 58 C ux B 2 tv P P What is n 1 2 2 ux 2 52 tv P 1 58 repeatability P 1 50C 20 measurements data acquisition P 0 50C 10 Measurements What is tn P t23 95 2 07 P 95 2 K 2 Pk n k K1 4 Pk k 1 vk 1 5 n 2 So we have 1 2 2 4 C 1 2 2 u x 2 52 2 07 1 58 0 Total uncertainty of a single T measurement 4 0 5 vk Nk 1 2 4 1 5 0 5 19 9 2 22 9 23 In the previous example we looked at the uncertainty in a single measurement of temperature based on available data We would use SDOM if we were interested in the best estimate of the true temperature R f x1 x2 x3 We know that R f x1 x2 x3 We need to figure out uR f Px1 Px2 Px3 Bx1 Bx2 Bx3 Again we can divide these elemental errors into Pk Elemental Standard Random Uncertainty Where P is given by P Sx Sx P N We propagate the systematic Single measurement error through L 2 B i Bi i 1 Measurement of the mean We propagate the standard random error through L 2 P i Pi i 1 Bk Elemental Systematic Errors 1 2 1 2 So at this point we have P and B P is the measurement Standard Random Uncertainty B is the measurement Systematic Uncertainty We have assumed Bias Errors reported at 95 We combine the measurement standard random uncertainty and the measurement systematic uncertainty through 2 K 2 k Pk k 1 n 1 4 K k Pk 2 2 2 u x B tv P P k 1 vk Finally the degrees of freedom in this case is calculated from Example The following relation is used to calculate the thermal efficiency h of a natural gas powered P internal combustion engine h m f Hv P is the output power kW Mf is the mass flow rate kg s Hv is the heating value of the natural gas kJ kg The average values of P Mf and Hv are 50 kW 0 2 kg minute and 49 180 kJ kg The systematic uncertainties have been evaluated as 0 2 kW 0 003 kg minute and 1500 kJ kg respectively Now 16 measurements of P give a standard deviation of 0 3kW And 10 measurements of Hv give a standard deviation of 167 6 kJ kg Calculate the total uncertainty in a single measurement of the efficiency of the engine at 95 confidence level How do we attack this problem We need to propagate the uncertainties into h First what is the expected value of h kJ 50 P s h 0 3050 m f Hv kg m kJ 0 2 49 180 m 60s kg Let s start with the precision error What are the elemental standard random uncertainties single measurement PP 0 3 kW PH 167 6 kJ kg How do we propagate these 0 1 2 2 h 2 h 2 h P PP Pm PH P m H OK we need the sensitivity coefficients h 1 1 1 0 0061 kW P m f H v 3 33x10 3 49 100 Pm f h 50 3 33x10 3 6 kg 6 2 x10 2 3 2 H kJ m f H v 3 33x10 49 100 Now we just need to plug in 1 2 2 h 2 h P PP PH 2 1x10 3 P H Great Now we need to attack the bias error The systematic uncertainties P Mf and Hv have been evaluated as 0 2 kW 0 003 kg minute and 1500 kJ kg respectively 1 2 2 2 h 2 h h B BP Bm BH P m H At this point we know everything except what h PH v 50 49 100 s 91 5 2 2 3 m kg m f H v 3 33x10 49 100 Great Now plug everything in careful w units and solve for B 1 2 2 2 h 2 h h B BP Bm BH 10 43x10 3 P m H We need to combine these uh B 2 tn P P 1 2 2 B 10 43x10 3 P 2 1x10 3 Now we are left to find the degrees of freedom 2 k Pk n k 1 4 K k Pk vk k 1 K t23 95 2 2 n uh 0 011 P4 h h PP P H P H np nH 4 4 23 n 30 5 1 1 95 Determine all sources of error divide into bias and precision Find an equation for the variable of interest in terms of given quantities Differentiate the equation to find standard random uncertainty systematic general equation plug in to find these Almostuncertainty the same asinthe previous example Why quantities Determine degrees of freedom t value Systematic uncertainty given at any confidence level tv P b Generally assume Bias error have high degrees of freedom n large t 2 At 95 confidence Bias errors typically listed at 95 confidence B We can convert this to a standard systematic uncertainly b b Here t z 1 B 2 Giving about a 68 confidence interval P Sx Now the standard random uncertainty is Sx P N Now we have standard b and P and combine them with the RSS with the result multiplied by the t value u x tv p b 2 P 2 Alternative book ux B 2 tv P P These are different 1 2 2 b P 2 2 Called combined standard uncertainty Our approach 2 Pk v k K1 4 Pk K n k 1 Alternate approach 2 2 2 2 Pk bk v Kk 1 4 K 4 Pk bk K k 1 2 2 ux B 2 tv P P k 1 nk k 1 nk ux tv p b2 P 2 …