ERROR ANALYSIS A.J. Pintar and D. Caspary Department of Chemical Engineering Michigan Technological University Houghton, MI September, 2012

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1 ERROR AALYSIS AJ Pinar and D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr, 0 OVERVIEW Exprimnaion involvs h masurmn of raw daa in h laboraory or fild I is assumd ha hr is always a diffrnc bwn h masurd valu and ru valu for any masurmn Our sima of h diffrnc bwn h masurd valu and h ru valu is rpord as Uncrainy or Exprimnal Error Furhrmor, xprimnal daa will vnually b rpord dircly or hy will b usd in subsqun calculaions Thrfor, whn masurd daa ar rpord, or rsuls calculad from masurd daa ar rpord, h rpord valus mus includ an simaion of h associad uncrainy Th uncrainy in h rpord valu is simad using a ool calld Error Analysis Whn a masurd valu is rpord dircly, h rror analysis is compl whn h rror associad wih ha valu is simad and rpord If h valu is o b combind mahmaically wih ohr masurd valus and h calculad rsul is o b rpord, an addiional sp calld Propagaion of Error mus b prformd bfor rporing h rsul DATA CALCULATED FROM EMPIRICAL CORRELATIOS Ofnims you will b comparing masurd valus wih valus simad from mpirical rlaionships Ths 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 or diagrams and hav an ofn ovrlookd rror rfrrd o as nginring accuracy Unlss h spcific rfrnc sas ohrwis, nginring accuracy is assumd o b in h rang of 0-0% rror; hrfor, using a ±5% uncrainy is rcommndd TWO TYPES OF ERROR I DATA Sysmaic Error: a Has h sam sign and magniud for idnical condiions; dfind by insrumn s prcision Sysmaic Error is prdicabl b Sourcs: Mis-calibraion of Insrumns aural Phnomna, i ha ransfr in a hrmowll or in a hrmomr sm Consisn Opraor Error, i parallax c Ofn can b rmovd or compnsaion mad: Rcalibraion, adusing zro and span Corrcion Facors or Calibraion Curvs Improvd procdurs Comparison o ohr mhods d Mus b corrcd bfor daa ar rpord or usd in subsqun calculaions Random Error: a Can b posiiv or ngaiv b Can no diffrnia sourc of rror bwn h insrumn and h procss islf c Sourcs: Random Procss Flucuaions Random Insrumn Flucuaions (Insrumn Accuracy) Dgr of Subdivision of Insrumn s Scal

2 Equipmn goblins, Phas of h Moon, Miscllanous d Mus b dal wih using saisics STATISTICAL AALYSIS OF REPLICATED DATA Suppos ha hr ar masurmns of h quaniy x, (i: x, x, x 3, x 4,, x ) Whn rporing h rsuls of rplicad daa for UO Lab you will ypically rpor h man valu along wih h simad sandard rror Th following is h sp-by-sp procdur for simaing rror: Calcula h Man Valu (Esimaion of h Tru Valu) Th man valu ( x ) is dfind saisically by: xi i x (Eq ) Calcula h Varianc Th varianc is h sum of h squars of h diffrnc bwn ach masurd valu and h man valu, dividd by h numbr of rplicas minus on Varianc (σ ) is dfind saisically by: σ i x i i ( xi x) xi i ( ) ( ) (Eq ) 3 Calcula Avrag Sandard Dviaion (Masur of h Variabiliy of h daa) Whn a daa s is small w us h 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 Iniial Esimaion of Sandard Error (Masur of h dviaion of x from h ru valu) Calld h Sandard Error of h Mans ( s ) is dfind saisically by: S σ (Eq 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

3 5 Drmin Rading Error Evn hough h daa sampling shows no scar (sandard dviaion of zro) hr may sill b a random rror associad wih h daa du o h rading rror Sourcs of rading rror ( R ) ar: Snsiiviy of h insrumn (h maximum chang rquird for h insrumn o rspond) Dgr of subdivision of h scal of h insrumn (on-half h smalls subdivision) Random flucuaions in h insrumn rading in bwn sampling ims (onhalf h diffrnc bwn h maximum and minimum valus) Th valu usd for h rading rror ( R ) is h largs of h possibl valus Gnrally, som udgmn and familiariy wih h insrumn ar ndd o com up wih a good sima of h rading rror Sudns hav a ndncy o undrsima his quaniy Som considraions for rading rror in UO Lab: How much dos h roamr or prssur gaug flucua bwn radings vs h scal subdivisions? How snsiiv ar h plaform scals? How prcisly can h nd poin b drmind in iraing an organic phas (+/- how many ml)? Wha is h manufacurr s publishd accuracy for h insrumn? 6 Adus h Sandard Error for 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: σ S ( as bfor) (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 3

4 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 UL,y S,u + S,v + S, w + (Eq 8) u v,w v u,w 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) 4

5 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, ) 5

6 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 m ar of h sam ordr of magniud hn 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 D h F h 4 ( ) (,,3,,) π 0 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 6

7 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 7

8 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 ( Th ] 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 ( Th SLMTD [ + ] Th Tc [( Th ( Th ] Th Tc ln ( + ) ( Th ( Th Th Tc [ T h T c ]}/ ln ( + ) ( Th If (T h -T c ) and (T h -T c ) ar approximaly qual hn: LMTD [( Th + ( Th ] SLMTD, [ ] Th Tc Th Tc 8

9 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) 9