Introduction to Measurement, Error Analysis, Propagation of Error, and Reporting Experimental Results

Transcription

1 Inroducion o Masurmn, Error Analysis, Propagaion of Error, and Rporing Exprimnal Rsuls AJ Pinar, TD Drummr, D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr, 03 Ovrviw Exprimnaion involvs h obsrvaion and masurmn of a physical propry or a characrisic of somhing A fundamnal rul of scinific masurmn sas ha i is nvr possibl o xacly masur h ru valu of any characrisic, only approximaions of h ru valu For his rason, i is h rsponsibiliy of h nginrs and sciniss o rpor all masurd valus along wih an simaion of h uncrainy in h masurmn Th procss of simaing h ru valu of h masurmn and is associad uncrainy is calld rror analysis Furhrmor, whn a masurd valu is rpord dircly, h rror analysis is compl whn h rror associad wih ha valu is simad and rpord In ohr cass, h masurd valu is o b combind mahmaically wih ohr masurd valus and h calculad rsul is h final rpord valu In his cas, h rrors associad wih ach masurd valu mus b combind o sima h uncrainy in h rsul This addiional calculaion is calld propagaion of rror This papr prsns a procdur for rror analysis and propagaion of rror for us in Uni Opraions Laboraory rpors Sourcs of Masurmn Error Human Error: This is also rfrrd o as Gross Error Carful planning and xcuion of your xprimn should prvn misaks Rading Error: This is a combinaion of h insrumn s accuracy and prcision and can b found in h manufacurr s spcificaions for h insrumn a Accuracy rfrs o an insrumn s abiliy o masur h ru valu of a characrisic This dscribs how clos h masurmn is o h ru valu b Prcision rfrs o h randomnss of h masurd valu du o variaion in h masuring dvic This dscribs h rpaabiliy of h masurmn c Rading rror is rad undr h gnral cagoris of Sysmaic Error and Random Error 3 Sysmaic Error: This is somims calld drmina rror a Has h sam sign and magniud for idnical condiions; sysmaic rror is prdicabl b Sourcs of sysmaic rror: i Mis-calibraion of insrumns This class of sysmaic rror rfrs o h insrumn s accuracy Could b du o a zro offs or impropr insrumn span ii aural phnomna or inhrn characrisics of h insrumn Could b du o hysrsis or h linarizaion of a non-linar rspons, or could b du o h mhod usd, i masuring surfac mpraur of a pip o rprsn fluid mpraur iii Consisn opraor rror, i parallax c Ofn can b rmovd or compnsaion mad: i Rcalibraion, adusing zro and span

2 ii Corrcion facors or calibraion curvs iii Improvd procdurs iv Comparisons o ohr mhods d Mus b corrcd bfor daa ar rpord or usd in subsqun calculaions 4 Random Error: This is a combinaion of h randomnss of h masurmn procss and h randomnss of h characrisic you ar masuring I is also calld indrmina rror a Can b posiiv or ngaiv and has varying magniud, is no prdicabl b You can no diffrnia h sourc of h flucuaions causd by h masuring insrumn from hos of h procss islf c Sourcs of random rror: i Random procss flucuaions i Equipmn goblins, moon phas, miscllanous ii Random insrumn flucuaions (rfrrd o in h insrumn manufacurr s daa sh as insrumn prcision) iii Dgr of subdivision of insrumn scal and your abiliy o prcisly rad h scal d Random rror is quanifid using Saisical mhods Uncrainy in Valus Obaind from Empirical Rlaionships Ofnims you will b comparing your masurd valus wih valus calculad from mpirical rlaionships Ths mpirical valus will also hav an rror (or uncrainy) associad wih hm Thorical valus for fricion facors, ha ransfr cofficins, mass ransfr cofficins, c ar usually obaind from corrlaing quaions and diagrams and hav an ofn ovrlookd rror rfrrd o as nginring accuracy Unlss h spcific rfrnc sas ohrwis, nginring accuracy can b assumd o b in h rang of 0-0%; hrfor, using a ±5% uncrainy is rcommndd Exprimnal Planning and Daa Collcion Aciviis Wih many of h xprimns in Uni Opraions Lab you will b askd o masur a uni opraion s prformanc a svral diffrn sady sa condiions To rpor h rsul a ach sady sa, you will collc daa for wo rasons Firs, you will vrify ha h uni opraion is a sady sa Scond, you will mak a numbr of rpa masurmns a his sady sa a rgular, prdrmind im inrvals so ha you can prdic h ru valu of ach masurmn and sima is associad rror An imporan firs sp in planning xprimnal work is o idnify wha h xprimnal rsuls should look lik From hr, drmin wha nds o b masurd and how i should b masurd In many cass hr ar choics in h yp of masuring insrumn you could us Qui ofn, an insrumn or mhod ha yilds high prcision masurmns aks mor im or ffor o us Using rial calculaions, drmin h ffc of ha insrumn s prcision on h final calculad rsuls and slc an appropria insrumn Insrumn prcision can usually b obaind from h manufacurr s daa sh This valu is your rading rror and should b rcordd in h laboraory nobook along wih h modl and srial numbr of h dvic long bfor you sar any lab work Anohr sp in h xprimnal planning procss is o drmin h numbr of rplicas rquird o characriz ach masurmn and h masurmn s uncrainy An infini numbr of rplicas can b avragd oghr o rpor h ru valu of h masurmn xacly Tim, rsourcs, and ohr pracical limiaions prvn his So, drmin how many rplicas you will nd in ordr o characriz h masurmn Minimally, i aks rplicas o calcula a sandard dviaion Howvr, b awar ha a sandard dviaion calculad around or 3 rplicas

3 has lil or no maning and will rsul in a larg associad uncrainy Fiv valus should b considrd as a minimum Finally, chck ha h daa gahring aciviis fi wihin h schduld laboraory im Bfor saring any xprimnaion on lab day, i is your rsponsibiliy o vrify ha masuring insrumns ar proprly calibrad If possibl, wo-poin or hr-poin calibraions ar prformd For xampl, a mpraur dvic can b placd in an ic bah, chckd a room mpraur, and in boiling war o vrify h calibraion; or svral sandard soluions can b carfully prpard and h snsor rang chckd a hs known poins Rcord any zro or calibraion offss in your laboraory nobook Prpar a calibraion curv if ncssary Add appropria columns in your spradsh o corrc masurd valus If a wo-poin calibraion is no possibl, hn minimally chck h zro or a rs rading agains a known and rusd dvic For xampl, prssur gaugs can b chckd a amosphric prssur Whil collcing daa, chck for gross misaks and rpa xprimns if ncssary Early in h day, chck ach opraor for possibl sysmaic rror, i from parallax or impropr rading chniqu and corrc immdialy Any rmaining variaion in rplicad masurd valus is rad as random rror and mus b quanifid using saisics Quanifying Random Error Saisical Analysis of Rplicad Daa Whn rporing h rsuls of a masurd valu for Uni Opraions Lab you will ypically rpor h man valu of your masurd rplicas along wih an simad sandard rror Unforunaly, hr is no singl mhod for calculaing h ru valu of a saisic in all siuaions For xampl: in on cas, mulipl rplicas of sady sa daa can b rcordd a som s frquncy In anohr siuaion, on or mor rprsnaiv sampls of a largr bach of marial ar s asid o prform an analyical masurmn Ths siuaions ar vry diffrn and mus b rad diffrnly In ihr cas, h goal in masurmn is o drmin h ru valu of somhing If an nginr could ak all h possibl masurmns (rplicas) of h characrisic, h man valu of hs rplicas would b h ru valu of wha is bing masurd This would b calld h grand avrag or populaion man and is rprsnd by h Grk lr, µ W could also calcula a sandard dviaion around his grand avrag o quanify h disprsion of daa around h avrag Sinc im and rsourcs ar limid, i is usually no pracical o ak all possibl masurmns So, for his xampl suppos ha hr ar masurmns of a quaniy y, (i: y, y, y 3, y 4,, y ) Ths masurmns rprsn a subs of all h possibl rplicas ha could hav bn masurd and hrfor rprsn a sampl of h nir populaion Th sampl avrag is calld h sampl man and is rprsnd by h familiar symbol, x Sinc h sampl man was no calculad from h nir populaion, i can b xpcd ha h sampl man will diffr slighly from h populaion man (h ru valu of h masurmn) A saisic calld h Sandard Error of h Mans can hn b calculad o sima h diffrnc bwn h sampl man and h populaion man 3

4 A suggsd procdur for rporing h sampl man and calculaing h uncrainy follows: Calcula h Man Valu of h Daa S Th man valu ( x ) is dfind by: xi i x (Eq ) Calcula h Sampl Varianc Th sampl varianc is h sum of h squars of h diffrnc bwn ach masurd valu and h sampl s man valu, dividd by h numbr of rplicas minus on Varianc (σ ) is: x i i ( xi x) xi σ i i ( ) ( ) (Eq ) 3 Calcula Avrag Sandard Dviaion of h Sampl Whn a daa s is small calcula an avrag sandard dviaion o dscrib h magniud of h sprad in h daa Avrag Sandard Dviaion is simply calld Sandard Dviaion (σ) and is dfind as h squar roo of h varianc, i h squar roo of h xprssion labld Eq 4 Mak an Iniial Esimaion of h Sandard Error (Masur of h dviaion of x from h ru valu) Also calld h Sandard Error of h Mans (SEM) SEM is dfind saisically by: SEM σ (Eq 3) 5 Compar h SEM o h Rading Error A masurmn can b no mor prcis han h masuring insrumn Evn if h rcordd daa shows no scar (sandard dviaion of zro) hr may sill b an uncrainy in h daa du o h rading rror Sourcs of rading rror ( R ) can b: Snsiiviy of h insrumn (h maximum chang rquird for h insrumn o rspond) Dgr of subdivision of h scal of h insrumn (gnrally, on-half h smalls subdivision) or h display s rsoluion Th valu usd for h rading rror ( R ) usually can b found in h manufacurr s daa sh If non is availabl, us ±½ h smalls incrmn of h dvic Gnrally, som udgmn and familiariy wih h insrumn ar ndd o com up wih a good sima of h rading rror Som considraions for rading rror in UO Lab: Wha ar h scal subdivions of h roomr or prssur gaug? How snsiiv ar h plaform scals? How prcisly can you find h ndpoin in iraing, +/- how many ml? Wha is h manufacurr s publishd accuracy for h insrumn? 4

5 6 Adus h Sandard Error for a Combind Random Error and Rading Error Onc a valu is drmind for h rading rror ( R ) i is compard o h sandard dviaion (σ) from (Eq ) o obain h sandard rror as follows: If R << σ, hn: S SEM σ (Eq 4) Bu, if R >> σ, us: R S (Eq 5) 3 (Th origin of h 3 in Eq 5 is h Poisson Disribuion) If R and σ ar of h sam ordr of magniud hn us h avrag of h wo rrors: σ R S + (Eq 6) 3 I can b shown saisically ha, for normally disribud daa, h ru valu of x (h individual masurmn) lis somwhr bwn: x - and x + (wih 683% confidnc) S x - and x + (wih 950% confidnc) S S S x - 3 and x + 3 (wih 997% confidnc) S S 5

6 ESTIMATIO OF ERROR I A CALCULATED RESULT Whn masurd valus ar usd in calculaions, h rror associad wih ach masurd valu will affc h uncrainy in h final calculad rsul Th rror in ach rm of h quaion mus b combind wih h rror in h ohr rms This is calld Propagaion of Error An simaion of h rror in h calculad rsul mus b calculad and rpord along wih h rsul Mhod: If y is h dsird quaniy and all h individual u, v, w, ar h raw daa ndd o calcula y, w can rprsn h gnral funcion as: y f(u, v, w, ) You would ypically run a s of idnical rpad xprimns and find h individual valus of u, v, w, x, calcula h man valu of ach u, v, w, Th man valu of y can b calculad by using h man valus of in h funcional rlaionship: y f(u,v,w,) Thn, o sima h rror associad wih y, us ihr of h wo following mhods: A Roo Mans Squar Error ( RMS ) Th Roo Man Squar Error has a basis in saisics: RMS, y f f + u S,u v, w v u, w S, v f + w u, v S, w + u, v, w (Eq 7) whr h man valus ( uvw,,,) ar usd o valua h drivaivs in h abov xprssion Th RMS Error is dious o calcula by hand and is bs suid o spradshs B Uppr Esima of h Propagad Error An uppr limi o h rror can b simad as follows: f f f S,u + S,v + S, w (Eq 8) u v w UL,y + v,w u,w u,v whr h man valus ( uvw,,,) ar usd o valua h drivaivs in h abov xprssion This mhod is asir o us for hand calculaions o ha RMS < UL always Thus, using UL will giv a mor consrvaiv sima of h rror SIGIFICAT FIGURES Whn rporing a valu and is associad rror us h appropria numbr of significan figurs (SF) For masurd valus, h numbr of SF is a funcion of h prcision of h masuring dvic Whn a calculad rsul combins mor han on masurd or simad valu h corrc numbr of SF is h sam as ha of h las of all h masurd valus Th corrc numbr of SF for simad rror is ypically on lss han h numbr of SF of h calculad rsul (somims wo, bu ofnims only on) 6

7 ERROR AALYSIS OF FLOW RATE BY REPLICATED PAIL AD SCALE MEASUREMETS On common mhod of masuring flow ra is o masur h mass of liquid collcd in a barrl or pail (w F -w 0 ) ovr a im inrval () If rplicad masurmns () hav bn mad of h final and iniial mass (w F, and w 0, ) and h im inrval ( ), i would b incorrc o drmin h man, varianc, c of (w F, w 0, and ) and hn calcula h mass flow ra (m) and is rror Th corrc procdur would b as follows: Calcula h mass flow ra for ach masurmn (m ): m& ( wf, w 0, ) (,,3,,) Calcula h man valu of h flow ra ( &m ): m& m& 3 Calcula h sandard dviaion of &m : σ m ( m& m& ) ( ) 4 Drmin h rading rror associad wih ach mass flow ra ( &m ) du o propagaion of h rading rrors in w F, w 0, and : ( ) Rm, [ Rw, F + Rw, ] [( wf w ) R, ] Drmin h avrag rading rror associad wih h mass flow ra: Rm, ( Rm, ) 7

8 6 Combin h rading rror and h sandard dviaion as bfor: If R,m << σ m, hn m, σ Sm If R,m >> σ m, hn Sm, Rm, 3 If R,m and σ ar of h sam ordr of magniud hn m Sm, σ m Rm, ( + ) 3 ERROR AALYSIS OF FLOW RATE BY REPLICATED MEASUREMETS OF CHAGE I LIQUID LEVEL I A TAK On common mhod of masuring volumric flow ra (Q) is o masur h chang in liquid lvl in a ank (h F -h 0 ) ovr a im inrval () If rplicad masurmns () hav bn mad of h final and iniial liquid lvls (h F, and h 0, ) and h im inrval ( ), an rror analysis can b prformd in h sam way as for h pail and scal mhod: Calcula h volumric flow ra for ach masurmn (Q ): Q πd h F h 0 4 ( ) (,,3,,) whr D is h insid diamr of h ank (assumd o hav no rror associad wih i) Calcula h man valu of h flow ra ( Q): Q Q 8

9 3 Calcula h sandard dviaion of Q: σ Q ( Q Q) ( ) 4 Drmin h rading rror associad wih ach flow ra (Q ) du o propagaion of h rading rrors in h F, h 0, and : ( R,Q ) πd R,h + R,h (h F h 0 ) F R, 5 Drmin h avrag rading rror associad wih h flow ra: RQ, ( RQ, ) 6 Combin h rading rror and h sandard dviaion as bfor: If R,Q << σ Q, hn SQ Q, σ If R,Q >> σ Q, hn SQ, RQ, 3 If R,Q and σ Q ar of h sam ordr of magniud hn SQ, σ Q RQ, ( + ) 3 9

10 EXAMPLE -- ERROR I CALCULATED VALUE OF THE OVERALL HEAT TRASFER COEFFICIET Th ovrall ha ransfr cofficin (U) is obaind from: U Q AT ( T) h c LM whr ( T T ) LMTD h c LM [( Th Tc) ( Th Tc) ] ln[ ( Th Tc ) ( T T ) ] h c Th rror in h calculad valu of U du o rrors in Q, A, and h mpraurs (T h, T h, T c, T c ) is givn by: whr SU, SQ, QSLMTD, QSA, + + ALMTD ( ) ALMTD ( ) A ( LMTD) Th Tc Th Tc Th Tc, { [( ) ( ) ] ln ( ) ( Th Tc) ( Th Tc) SLMTD [ + ] Th Tc [( Th Tc) ( Th Tc) ] Th Tc ln ( + ) ( Th Tc) ( Th Tc) Th Tc [ T h T c ]}/ ln ( + ) ( Th Tc) If (T h -T c ) and (T h -T c ) ar approximaly qual hn: LMTD [( Th Tc) + ( Th Tc) ], [ ] SLMTD Th Tc Th Tc 0

11 TABLE OF OMECLATURE A Ha Transfr Ara D S R h F, h 0 Insid Diamr of Tank Sandard Error Rading Error Final and Iniial Liquid Lvls, rspcivly, in Volumric Flow Ra Masurmn i, Rfr o a Paricular Sampl or Daa Poin LMTD Log-Man Tmpraur Diffrnc m Q σ σ T h, T c U Mass Flow Ra umbr of Daa (Sampl) Poins Volumric Flow Ra; Ha Transfr Ra Sandard Dviaion Varianc Tmpraur of Ho and Cold Fluids, rspcivly Tim Inrval for Flow Ra Masurmn Ovrall Ha Transfr Cofficin u, v, w Indpndn Variabls Usd in a Calculaion w F, w 0 Final and Iniial Mass, rspcivly, in Pail and Scal Mhod x x i y Man Valu of x Sampld Valu of x Dpndn Variabl Drmind in a Calculaion REFERECES Bragg, GB, Principls of Exprimnaion and Masurmn, Prnic-Hall, Englwood Cliffs, J, (974) Barry, AB, Errors in Pracical Masurmn in Scinc, Enginring, and Tchnology, Wily, Y, (978) Lyon, AJ, Daling wih Daa, Prgamon Prss, Y, (970)