Uncertainty Analysis. P=pRT

Transcription

1 Uncertanty Analyss In many, f not most, exermental stuatons, the fnal desred result s not measured drectly. Instead, measurements of several varables are substtuted nto a data reducton equaton to obtan the desred quantty. As an examle, suose the densty of a flowng gas stream s desred. rect measurements of gas densty are dffcult, so nstead the ressure and temerature T are measured and, assumng the gas can be treated as deal, the densty can be found from the deal gas equaton of state, T The queston that naturally arses s: How do the uncertantes n the ndvdual measured varables and T roagate through the data reducton equaton nto the fnal result for? The urose of uncertanty analyss s to answer ths mortant queston. As dscussed n revous sectons, bas s a fxed error that can be reduced by calbraton. On the other hand, recson error s a random error that can be reduced by obtanng multle measurements. Because of the dfferng nature of these two comonents of uncertanty, t s desrable to nvestgate ther roagaton nto the exermental result searately. Ths aroach s consstent wth that recommended n the ANSI/ASME Standard on Measurement Uncertanty. A. roagaton of Bas Errors The stes nvolved n determnng the bas lmt for the exermental result are sketched below. Each of the ndvdual measurement varables (X, X,,X s subject to several, say M, elemental bas errors. The bas lmts for each of these elemental sources are combned n some manner to obtan the overall bas lmt (,,, for each varable. In turn, the technques of uncertanty analyss are used to combne the bas lmts for the measured varables nto a bas lmt for the exermental result. The followng materal wll descrbe the detals of the method requred to carry out ths rocedure.

2 erhas the most dffcult ste n ths rocess s the dentfcaton and quantfcaton of the elemental bas lmts that affect each of the measurement varables. As dscussed n an earler secton, elemental bas error sources can generally be laced n three categores: calbraton, data acquston, and data reducton. However, assgnng magntudes to these sources s not a straghtforward task. Unlke estmatng the recson lmt from comutaton of the recson ndex S, there s no statstcal calculaton that can be done to estmate the bas lmt. Snce t s a fxed error, the bas s the same for each measurement. However, ts magntude, beng the dfference between the samle mean and the true value, s unknown because the true value s always unknown n any realstc exerment. Thus, the elemental bas lmts must always be estmated. In addton, the bas lmt estmates are made at a 95% confdence level for consstency wth the recson lmt determnatons. Ths can be nterreted to mean that the magntude of the bas β s less than or equal to the bas lmt at a 95% level of confdence. Informaton on bas errors can be nferred from comarson of ndeendent measurements that deend on dfferent hyscal rncles or that have been ndeendently calbrated. Bas lmt estmates can also be made based on revous exerence of the exermenter or other ndvduals, nstrument manufacturer s nformaton and secfcatons, and comarson of measurements wth known values. Once the bas lmts for the elemental error sources are estmated, they must be combned n some manner to obtan the bas lmt for each measured varable. The referred method

3 for dong ths s the root-sum-square (SS technque. or measurement varable X ths s gven by: [( (... ( ] where s the 95% confdence estmate of the bas lmt for the measurement. The next ste n the rocedure s to aly uncertanty analyss to determne how the bas lmts (,,, for the ndvdual varables roagate through the data reducton equaton to form the bas lmt for the exermental result. The data reducton equaton s taken to be of the form (X, X,, X where t s assumed that ths relaton s contnuous and has contnuous dervatves n the doman of nterest and that the bas lmts for the measurement varables are ndeendent of one another. Under these condtons, the bas lmt for the result s gven by the uncertanty analyss exresson (see oleman and Steele, Exermentaton and Uncertanty Analyss for Engneers, ley, Aendx B. X X M... X Use of ths exresson wll be demonstrated n an examle at the end of ths secton. B. roagaton of recson Errors The rocedure for determnng the recson lmt for an exermental result s generally smlar to that for determnng the bas lmt. In fact, a sketch of the rocedure would look dentcal to the one gven revously for the bas lmt wth the substtuton of for and recson for bas. The measurement of each varable (X, X,, X s nfluenced by recson errors from a number of elemental error sources. These random errors combne to cause the recson error n the measurement of each varable; the latter s quantfed by determnng the recson lmt (,,, of each measured varable. These ndvdual measurement recson lmts are then roagated through the uncertanty analyss to obtan the recson lmt for the result. The way n whch the recson lmts of the ndvdual measurement varables are determned deends both on the tye of exerment and the hase under consderaton. or examle, n the desgn hase of a new exerment, before any equment has been secfed or data obtaned, estmates for the recson lmts are made based on all avalable nformaton: the exermenter s exerence, that of others, manufacturer s secfcatons, etc. At ths stage n the exerment, the recson lmt assocated wth the measurement system may be the only recson error source consdered. As a general rule-of-thumb, the recson lmt resultng from the readablty of an analog nstrument can be taken as one-half of the smallest scale dvson. Lkewse, for a dgtal outut, the recson lmt assocated wth the readablty s

4 one-half of the least dgt n the outut. or cases n whch the recson lmt s estmated, the estmate should be that band whch wll contan the mean value of the varable wth 95% confdence. urng the executon hase of an exerment, revous measurements may be avalable wth whch to determne each of the values. In other cases, multle measurements of the varable may be made durng the actual exerment, from whch the recson ndex S and recson lmt can be calculated from a samle of N readngs. Several comments are requred here to clarfy ths rocedure. rom the dscusson n recedng sectons, recall that the arorate recson lmt to use wth a varable X that s determned from a sngle readng s the recson ndex of the samle oulaton tmes the t factor taken from the t-dstrbuton table for N < or t.0 for N, X ts X Of course, for a sngle readng X must be estmated or must be avalable from revous measurements. As dscussed revously, the ± X band around the measurement X contans the mean value of the measure varable wth 95% confdence. Therefore, n the uncertanty analyss equatons gven below, X and should be nterreted as X and X when the value of X used n the data reducton equaton s determned from a sngle readng. hen the value of the measurement s determned as the mean X of N searate readngs, then the recson lmt of the samle mean ts X X ts X N should be used. In ths case, the ± X band around the samle mean X contans the mean value of the measured varable wth 95% confdence. Therefore, f the value of the varable that s used n the data reducton equaton s determned as the mean of N searate measurements, the values X and X should be used n the uncertanty analyss equatons. hen several searate factors can be dentfed as causng the recson error n a measured varable, t may sometmes be desrable to determne the recson lmt by consderng the contrbutons of the elemental error sources. The rocedure s smlar to that dscussed revously for the determnaton of the bas lmt. or the X measurement varable, suose that M elemental recson error sources are dentfed and ther 95% confdence recson lmts

5 are determned as (, (,, ( M. Then, the overall recson lmt s gven by the SS exresson at the 95% confdence level. [( (... ( ] Another factor that must be consdered n estmatng recson lmts s the tme erod over whch the samle oulaton s obtaned. Ths s summarzed n the followng rule: ata sets for determnng estmates of recson ndces should be acqured over a tme erod that s long relatve to the tme scales of the factors that have a sgnfcant nfluence on the data and that contrbute to the recson errors. If ths rule s not followed, the recson lmt estmatons would not nclude long tme (low frequency varatons that affect the measurement, and therefore these lmt estmates would be naccurate. Once the 95% recson lmt for each of the measured varables X n the data reducton equaton (X, X,, X Is determned, the 95% recson lmt for the exermental result s found from the uncertanty analyss exresson M X X... X (oleman and Steele, Exermentaton and Uncertanty Analyss for Engneers, ley, aendx B. Note that ths exresson s dentcal n form to the one used for determnaton of the bas lmt of the result. It s assumed that the data reducton equaton s contnuous and has contnuous dervatves n the doman of nterest and that the recson lmts for the measured varables are ndeendent of one another. In addton, t s assumed that the result s determned from the reducton equaton only once at a gven exermental condton usng ether a sngle measurement X or the mean value X of N reeated measurements.. Uncertanty of the Exermental esult In order to determne the overall uncertanty U of the exermental result, the bas and recson lmts, and, must be combned. Ths s accomlshed usng the root-sumsquare (SS method U (

6 thereby rovdng 95% coverage of the true value. Examle The drag coeffcent,, s to be reorted for the flow of water over a strut-mounted shere. The drag force s measured drectly wth a force transducer, the freestream velocty s measured wth a tot-statc robe, and the shere dameter s measured wth a mcrometer. The table below gves nomnal values of the measurement varables and the water densty, as well as estmates for the bas and recson lmts of each varable at the 95% confdence level. The bas lmts have been estmated based on manufacturer s secfcatons and revous exerence wth the nstruments durng ndeendent calbratons. The recson lmts, on the other hand, have been determned from multle measurements of each varable, together wth comutaton of the recson ndces, tsx S X, and. Estmate the overall uncertanty n the reorted drag coeffcent X N U at a confdence level of 95%. Measured arable, X Nomnal alue Bas Lmt, recson Lmt, rag force, 0.5N 0.0N 0.0N ater densty, 99 kg/m 0.% reestream velocty, 5m/s 0.m/s 0.m/s Shere dameter, 0mm 0.mm 0.05mm SOLUTION Before begnnng the detals of the soluton, dscusson of the ercentage uncertanty lsted for the water densty s n order. hen usng tabular or curve-ft reference values for quanttes such as materal roertes, t s mortant to remember that these are not true values. ather, they are best estmates based on exermental data that have uncertantes assocated wth them. However, once a table or curve-ft equaton has been chosen to determne a roerty, the same value wll be obtaned for a gven exermental condton no matter how many tmes the table or equaton s used. Thus, the recson lmt assocated wth a roerty value determned from a table or equaton s zero. All of the uncertanty n exermental roerty data s combned nto a bas lmt that s the best estmate of the overall uncertanty n the data used to generate the table or equaton. In the current case, the 95% bas lmt estmate for (exressed here as a ercentage s qute low, 0.%, snce the densty of water s well known and s also relatvely nsenstve to envronmental factors such as temerature varatons. rtng the exresson for the data reducton equaton and the uncertanty exresson for the bas lmt :

7 omutng the dervatves: 6 6 Substtutng nto the uncertanty exresson for : Thus, the bas lmt deends most strongly on and,.e., the factors n the data reducton equaton wth the largest exonents, whch s always the case for reducton exressons of ower law form.

8 Substtutng n numercal values: ( x0 (x0 (.6x0 (x0 (.6x0 6.0% (6.00x0 Note that the bas lmt contrbutons from the drag force and velocty measurements are equal and domnate the bas uncertanty for and that the bas lmt on densty contrbutes neglgbly. Now consderng the recson lmts, the uncertanty exresson for s: The dervatves have already been carred out n consderng the bas lmt, so the followng result can be wrtten mmedately: Substtutng numercal values: x0 (x0 (5.x0 (.096x0 9.7% (9.65x0

9 ombnng the bas and recson lmts by the SS method: U ( ( [ ] % x0.60x0 U Thus, under these condtons the total uncertanty n the drag coeffcent s.% at a 95% confdence level. Snce the nomnal value of s 0.50 (0.0 (99(5 0.5 Ths result can also be wrtten as 0.50 ± 0.05 at 95% confdence.