Abstract. Introduction

Transcription

1 UNCERTAINTY ASSOCIATED WITH PARTICULATE MATTER CONCENTRATION MEASUREMENTS FROM AGRICULTURAL OPERATIONS Jcqueline E. Price, Ron E. Lcey, Bryn W. Shw, Clvin B. Prnell, Jr. Center for Agriculturl Air Qulity Engineering nd Science Texs A&M University College Sttion, TX Abstrct Grvimetric mesurement of prticulte mtter (PM) concentrtions in mbient environments is the bsis for regultion of PM frctions (i.e. PM 1 nd PM.5 ) under the Federl Clen Air Act. While the mesurement is stright forwrd, inherent elements of uncertinty enter the nlysis, resulting in much lrger uncertinty in the concentrtion clcultion. This pper discusses the importnce of uncertinty pproximtion nd nlyzes the uncertinty inherent in grvimetric PM concentrtion mesurement. Utilizing first order Tylor Series pproximtion nd nlyticl derivtives, the overll system uncertinty is computed. Additionlly, this pper uses sensitivity nlysis of the contributing uncertinty elements in order to identify the most criticl mesurements nd their implictions on the clibrtion, opertion, nd design of PM smplers. Introduction Grvimetric mesurement of prticulte mtter (PM) concentrtion in mbient environments is the bsis for regultion of PM frctions (i.e. PM 1 nd PM.5 ) under the Federl Clen Air Act. In cotton ginning, prticulte mtter (PM) is considered the primry emitted ir pollutnt. In generl, PM emissions from gins processing picker-hrvested cotton re typiclly less thn those of gins processing stripper-hrvested cotton, nd the PM emissions from the ginning of the first hrvest of cotton re generlly less thn the PM emissions from lter hrvests (U.S. EPA, 1995). Additionlly, dt shows tht pproximtely 37% of the totl PM emitted from cotton ginning (following PM control systems) is PM 1, which describes prticulte mtter with n erodynmic equivlent dimeter less thn or equl to 1 µm (U.S. EPA, 1995). However, we know tht ssuming lognorml prticle size distribution of the PM in the ir with typicl cotton gin dust mss medin dimeter of µm nd GSD of., the mss of PM 1 on the filter equls pproximtely 16% of the totl suspended prticulte mtter mesured (Wng, ). While these PM mesurements re stright forwrd, numerous elements of uncertinty cn enter the nlysis, resulting in much lrger uncertinty in the concentrtion clcultion. This discussion covers the incorportion of uncertinty nlysis in grvimetric mesurement of prticulte mtter. A mesurement of vrible cn only provide deterministic estimte of the quntity being mesured; thus, it cn only be considered complete when supplemented by quntittive sttement of the inccurcies surrounding the mesurement. Therefore, proper experimentl plnning nd design requires n understnding of the errors inherent in these mesurements so tht the experimenter cn hve some degree of certinty in the finl mesurements nd clcultions. Uncertinty cn be defined s the sttisticl representtion of the relibility ssocited with specific set of mesurements (Yegnn et l., ). Uncertinty cn lso be described s the possible set of vlues on given mesurement nd cn be considered sttisticl vrible (Kline, 1985). The term error tkes on slightly different definition. The totl error, δ, is the difference between the mesured vlue nd the true vlue of the quntity being mesured. It cn lso be thought of s the sum of the systemtic error nd the rndom error, δ = β + ε, where β is the systemtic error nd ε is the rndom error (ANSI/ASME, 1998). This is illustrted by Figure 1. Systemtic error, β, lso known s fixed error or bis, is defined s the constnt element of the totl error, δ; therefore, this error vlue remins constnt for ech mesurement. Rndom error, ε, lso known s repetbility error or precision error, is the rndom error element of the totl error, thus ech

2 mesurement tkes on different vlue for this prt of the totl error mesurement (ANSI/ASME, 1998). Thus, the term error refers to fixed quntity, nd it cnnot be considered sttisticl vrible. Mny of the current methods of estimting the uncertinty surrounding experimentl results re bsed upon n nlysis by Kline nd McClintock (1953). With the gol in mind of determining the effects of ech potentil mesurement error, they proposed process which considers the impct of these individul uncertinties, commonly referred to s the propgtion of uncertinty (Kline nd McClintock, 1953). This process involves Tylor series pproximtion to estimte the uncertinty in vrious circumstnces. Objectives The objectives of this uncertinty nlysis re: 1. To determine the uncertinty surrounding the grvimetric prticulte mtter (PM) concentrtion using first-order Tylor series pproximtion method.. To identify the most criticl mesurements nd their implictions on the clibrtion, opertion, nd design of PM smplers using sensitivity nlysis. Methodology The impct of the individul uncertinties of ech primry mesurement in n experiment on the totl systemtic uncertinty of the experiment must be pproximted. This ide is commonly referred to s the lw of propgtion of uncertinty (ISO, 1995). The uncertinties from the individul independent vribles propgte through dt reduction eqution into resulting overll mesurement of uncertinty s demonstrted in Figure (Colemn & Steele, 1999). Primry Systemtic Uncertinty Determintion Mnufcturers specify the ccurcy of their respective mesurement instrument, nd this informtion is used in this nlysis s the vlue for the systemtic uncertinty of the mesuring device. This ccurcy specifiction tkes into ccount vrious fctors such s linerity, gin, nd zero errors (Colemn & Steele, 1999). All of the uncertinty vlues used in this discussion except for tht of the pressure drop cross the orifice meter ( P ) were obtined from the specifictions on the mnufcturers dt sheets. The uncertinty vlue given by the mnufcturer must include ny sensor or trnsducer bis in the system. In the cse of the P reding from the Hobo instrument, the bis in both the pressure trnsducer nd the Hobo dt logger must be ccounted for. Uncertinty Propgtion Clcultion With the individul systemtic uncertinties now determined, the propgted systemtic uncertinty cn be clculted. Assuming tht ll individul uncertinties re t the sme confidence level (95% confidence intervl or :1 odds in this instnce), let Y be function of independent vribles x 1, x, x 3,, x n. Therefore, the dt reduction eqution for determining Y from ech x i is ( x x ) Y = Y,..., 1, x n Furthermore, let ω be defined s the systemtic uncertinty in the result nd ω 1, ω,..., ω n s the systemtic uncertinties in ech of the bove independent vribles. Given the sme confidence intervl on ech of the independent (uncorrelted) vribles, the resulting systemtic uncertinty of Y, ω Y, cn be clculted s the positive squre root of the estimted vrince, ω y, from the following eqution (Holmn, 1) [1], ω Y = + ω Y [],

3 where the vrince, ω y, is clculted by ω δy δy δy = Y ω ω ωn δx1 δx δxn [3], or Y ( θ ω ) + ( θ ω ) + ( θ ω ) ω =... + n n 1 1 [4], where θ, the sensitivity coefficient, is defined s δy θ i = δx i [5]. Grvimetric Smpling Governing Equtions The concentrtion of prticulte mtter (PM) in the ir cn be mesure by grvimetric mens, where the PM in the ir is cptured on filter nd then weighed. The prticulte mtter concentrtion is function of the mss of PM collected in known volume of ir s indicted in eqution 6 below. W C = V [6], where C is the concentrtion, W is the mss of PM 1 collected on the filter, nd V is the totl volume of ir through the system during the entire time of smpling. Both W nd V re clculted quntities from other mesurements. Therefore, these quntities must be reduced to bsic mesurements s seen in Figure 3. First, the mss on the filter, W, is necessry. Assuming lognorml prticle size distribution of the PM in the ir with typicl cotton gin dust mss medin dimeter of µm nd GSD of., the mss of PM 1 on the filter equls pproximtely 16% of the totl suspended prticulte mtter mesured (Wng, ). Therefore, the mss of PM 1 on the filter is clculted by eqution 7. W =.16 * (W f W i ) [7], where W f is the weight of the filter nd PM fter the smpling period nd W i is the weight of the bre filter before the smpling period. These filters re weighed three times before nd fter smpling under controlled environmentl conditions (reltive humidity nd temperture hs n impct on the ccurcy), nd the men of ech of these three mesurements is used. Both W f nd W i re primry mesured quntities, so no further reduction is necessry. The totl volume of ir in ft 3, V, used during the smpling time is determined by eqution 8. V = Q *Θ [8], where Q is the volumetric flow rte in cfm nd θ is the elpsed time of the test in minutes. The elpsed time of the test, θ, is mesured quntity; however, Q is not. So, Q must be evluted further. Ech grvimetric smpler uses fn or pump to drw ir downwrd through the filter. The fn/pump setup includes n orifice meter in the line to the smpler in order to clculte the volumetric flow rte of ir through the tube. The volumetric flow rte in cfm, Q, is clculted from the pressure drop cross n

4 orifice meter s in the following eqution, which is derived from Bernoulli s eqution (Sorenson nd Prnell, 1991). ( D ) Q = 5.976* k * * P [9], where k is clibrtion constnt for the orifice meter, P is the mesured pressure drop cross the orifice meter in inches of wter using trnsducer output to dt logger to record the instntneous pressure drop cross the orifice meter, is the men ir density in lbs*ft -3, nd D is the dimeter of the orifice in inches determined by the end mill specifictions. For field smpling mesurements, the gs used is ir where the ir density in lbs*ft -3 cn be estimted by eqution 1 (Cooper nd Alley, 1994). P RH * Ps =.37 * + RH * P ( 46 + T ).596* ( 46 + T ) s [1], where P s is the sturted vpor pressure in lbs*in - t T (Engineering Toolbox, 3), T is the dry bulb temperture of the ir in degrees Fhrenheit, nd RH is the reltive humidity frction of the ir. In three of the four exmples tht follow, the vlue of k is determined ginst lminr flow element (LFE) of greter precision nd ccurcy thn the orifice meter, where the vlue of k is given by eqution 11. ` k = 5.976* Q LFE ( D ) * Pc c [11], where Q LFE is the flow given by the LFE (ft 3 *min -1 ), c is the density of the ir during clibrtion (lbs*ft -3 ), nd P c is the pressure drop cross the orifice meter during clibrtion in inches of wter. In the low volume exmple, the reding from mss flow meter (Q mssflowmeter ) is used in lieu of Q LFE in eqution 11 (to determine the k vlue). The density of the ir during clibrtion, c, is clculted using the sme eqution s, (refer to eqution 1). Sensitivity Coefficient Determintion Results nd Discussion In order to evlute the effect of ech primry mesurement on the finl concentrtion mesurement, the sensitivity must be clculted with respect to ech of these primry mesurements. The sensitivity coefficient for ech element of grvimetric smpling system is bsed on eqution 5. In order to determine the sensitivity coefficients, the systemtic uncertinty of ech instrument is necessry. Tble 1 specifies the instruments used for ech mesurement s well s the relted systemtic uncertinty s provided in the mnufcturer s specifictions. These uncertinty vlues re ssumed to be t 95% confidence intervl ( stndrd devitions from the men, lso referred to s :1 odds). Literture identifies this s Type B nlysis in which the evlution of systemtic uncertinty is bsed upon scientific judgment nd mnufcturers specifictions (NIST, 1994). With this systemtic uncertinty informtion, the sensitivity coefficient for ech vrible in equtions 6-11 is determined using prtil differentil equtions (s described by eqution 5). These prtil differentils cn be found in Appendix A. Sensitivity & Uncertinty Anlysis To determine the most sensitive input prmeters with respect to the output prticulte mtter concentrtion, sensitivity nlysis must be performed on the uncorrelted primry mesurements

5 (Yegnn et l, ). The informtion obtined from the sensitivity nlysis is used to obtin the uncertinty in the prticulte mtter concentrtion clcultion. Additionlly, this informtion helps the experimenter identify the most influentil sources of uncertinty. This proves to be importnt when the mount of uncertinty in the finl computtion needs to be reduced by identifying these influentil sources of uncertinty. This nlysis evlutes the PM 1 concentrtions in four situtions: the high volume smpling technique (Q ~ 5 cfm, which is the midpoint of the U.S. EPA defined pproprite operting flow rtes; Q ~ 39 cfm nd Q ~ 6 cfm, which re the upper nd lower limit flow rtes s defined by the U.S. EPA) nd low volume smpling technique (Q ~.6 cfm ~ 1 m 3 /min) used by the Texs A&M Center for Agriculturl Air Qulity Engineering & Science (CAAQES). It is importnt to note tht the smpling instrumenttion used by CAAQES hs less uncertinty nd vribility ssocited with ech piece of instrumenttion thn the pproved EPA smpling instrumenttion. Ech portion of Tble is summry of the sensitivity of ech independent prmeter contributing to the finl prticulte mtter concentrtion. This informtion is derived from model in Microsoft Excel s provided in Appendix B. Using the process defined in the methods section, the sensitivities of ech of the prmeters re clculted bsed on eqution 5. The uncertinty of ech secondry mesurement (the propgtion of the primry mesurements) is determined by the process s described in equtions 3 nd 4. These secondry uncertinties include not only the uncertinty in the concentrtion mesurement (ω C ) but lso the uncertinty in the mss on the filter (ω W ), the volume of ir (ω V ), the volumetric flow rte of ir (ω Q ), the density of the ir during the smpling period (ω ), the density of the ir during the orifice meter clibrtion (ω c ) nd the k vlue cross the orifice meter (ω k ). Ultimtely, the model clcultes the mount of impct of ech prmeter on the totl uncertinty in the finl concentrtion clcultion. It is importnt to note tht simply dding up the impct of ech prmeter on the finl uncertinty will yield vlue much lrger thn 1%. However, if the prmeters representing the primry mesurements re summed ( P, T, P, RH, P st, Q LFE, D, P c, T c, P c, RH c, P stc ), then the Percentge of Totl Uncertinty results in 1% of the totl uncertinty. The following scenrio evlutions re included in Tbles nd 3 (with the clcultions included in Figures 4 7): TAMU Grvimetric Smpling Q ~.6 cfm (1 m 3 /hr) TAMU Grvimetric Smpling Q ~ 39 cfm TAMU Grvimetric Smpling Q ~ 5 cfm TAMU Grvimetric Smpling Q ~ 6 cfm Tble 3 displys the overll concentrtion uncertinty for ech of the scenrios, while Tble breks down the uncertinty into the contribution of ech mesurement to the totl uncertinty. In ll four scenrios, it s importnt to note tht the leding contributor to the uncertinty in the finl concentrtion clcultion is the pressure drop cross the orifice meter. If we re to seek higher degree of certinty in our finl concentrtion clcultion, then the optiml decision would be to decrese the uncertinty in the pressure drop cross the orifice meter mesurement. Conclusions A mesurement of vrible cn only provide deterministic estimte of the quntity being mesured; thus, it cn only be considered complete when supplemented by quntittive sttement of the inccurcies surrounding the mesurement. Thus, it is extremely importnt tht ll scientific mesurements nd clcultions include sttement of uncertinty. This nlysis uses first order Tylor Series pproximtion to determine the totl uncertinty surrounding the PM concentrtion for four grvimetric smpling scenrios. In ddition to determining the totl uncertinty, the most criticl mesurements in grvimetric smpling of PM re identified using sensitivity nlysis. In evluting the uncertinty surrounding ech mesurement nd the impct on the totl uncertinty in the finl clcultion, it is notble tht the pressure

6 drop cross the orifice meter during the test s well s during clibrtion ccounts for pproximtely 6% - 8% of the totl uncertinty in ech of the four exmples. With this knowledge, the experimenter hs identified the optiml prt of the mesurement process to focus on to effectively reduce the totl uncertinty in the experiment, if desired. Thus, this nlysis hs provided systemtic method of determining which instruments in the process need to be improved on in terms of reducing overll uncertinty by using Tylor Series pproximtion pproch bsed off of the pioneering reserch by Kline nd McClintock in An uncertinty nlysis should be included in every single experimentl procedure!

7 References Americn Ntionl Stndrds Institute/Americn Society of Mechnicl Engineers (ANSI/ASME). Test Uncertinty, Performnce Test Code New York, NY: ASME. Colemn, Hugh W., nd W. Glenn Steele Experimenttion nd Uncertinty Anlysis for Engineers. nd ed. New York, NY: John Wiley & Sons. Cooper, C. Dvid, nd Alley, F.C Air Pollution Control: A Design Approch. nd ed. Prospect Heights, Illinois: Wvelnd Press, Inc. Devore, Jy L Probbility nd Sttistics for Engineering nd the Sciences. 4 th ed. Pcific Grove, CA: Brooks/Cole. The Engineering Toolbox. 3. Sturted Stem Tble in SI Units. < Lst ccessed December 18, 3. Holmn, J.P. 1. Experimentl Methods for Engineers. 7 th ed. Boston, MA: McGrw Hill. Interntionl Stndrds Orgniztion (ISO) Guide to the Expression of Uncertinty in Mesurement. Genev: ISO. Kline, S.J., nd McClintock, F.A Describing Uncertinties in Single-Smple Experiments. Mechnicl Engineering. 75: 3-8. Kline, S.J The Purposes of Uncertinty Anlysis. Journl of Fluids Engineering. 17: Ntionl Institute of Stndrds nd Technology (NIST) Guidelines for Evluting nd Expressing the Uncertinty of NIST Mesurement Results. NIST Technicl Note 197. United Sttes Deprtment of Commerce. Wshington, DC: US GPO. Sorenson, J.W., nd Prnell, Clvin B Agriculturl Processing Technology. College Sttion, TX: Texs A&M University. U.S. EPA Compiltion of Air Pollutnt Emission Fctors, AP-4, 5 th Edition, Volume I: Sttionry Point nd Are Sources. Reserch Tringle Prk, NC: U.S. GPO. USEPA CFR Prt 5. Ntionl Primry nd Secondry Ambient Air Qulity Stndrds. Reserch Tringle Prk, NC: U.S. GPO. USEPA CFR Prt 7. Ntionl Stte Operting Permit Progrms. Reserch Tringle Prk, NC: U.S. GPO. Wng, L.. A New Engineering Approch to Cyclone Design for Cotton Gins. M. S. thesis, Agriculturl Engineering Dept., Texs A&M University, College Sttion. Yegnn, A., D.G. Willimson, A.J. Grettinger.. Uncertinty Anlysis in Air Dispersion Modeling. Environmentl Modeling & Softwre. 17:

8 Tbles Tble 1. Instrument Specifiction Prmeter Instrument Systemtic Uncertinty W i, W f Srtortius SC (low volume) 1 * 1-7 g Mettler Toledo AG blnce (high volume) * 1-4 g Θ (Time) HOBO dt logger. min P Omeg PX74 Pressure Trnsducer.75 + HOBO cord.1 ma + 3 % D o End Mill Specs.5 in T HOBO Wether Sttion Temperture/RH Smrt Sensor.8 F P HOBO Wether Sttion Brometric Pressure Smrt Sensor 1 % RH HOBO Wether Sttion Temperture/RH Smrt Sensor 3 % P st Stem Tbles.1 psi Q mssflowmeter Alborg GFC17 Mss Flowmeter 1.5 % FS Q LFE Merim Instruments Model 5MC-.344 cfm P c Digitl Mnometer Dwyer Series 475 Mrk III.5 % FS T c Dvis Perception II 1 F P c Dvis Perception II 1 % RH c Dvis Perception II 5% P stc Stem Tbles.1 psi

9 Tble. Grvimetric Smpler Sensitivity Anlysis for Uncertinty Propgtion Q Volume Mss K Prmeter Units Nominl Vlue TAMU High Volume Uncertinty % of Totl Uncertinty Nominl Vlue TAMU Low Volume Uncertinty % of Totl Uncertinty Nominl Vlue EPA Lower Limit High Volume % of Totl Uncertinty Uncertinty Nominl Vlue EPA Upper Limit High Volume Uncertinty % of Totl Uncertinty W f G 9.1.E % E-7.16% E % 9.83.E % W i G 9.7.E % E-7.16% 9.7.E % 9.7.E % θ(time) Min % % % 18..3% Q Cfm % % % % P in of H O % % % % Lbs/ft % % % % k % % % % T F % % % % P Psi % % % % RH % % % % P st Psi % % % % Q LFE / Q mssflow Cfm % % % % P c in of H O % % % % D o inches % % % % c Lbs/ft % % % % c T c F % % 7 1.6% % P c Psi % % % % RH c.5.5.%.5.5.1%.5.5.1%.5.5.3% P stc psi % % % %

10 Tble 3. Totl Uncertinty for Grvimetric Smpling Under Norml Conditions Concentrtion (µg/m 3 ) Uncertinty (µg/m 3 ) Uncertinty (%) TAMU 1 m 3 /hr TAMU 39 cfm TAMU 5 cfm TAMU 6 cfm

11 Figures β δ ε x true µ k x m Figure 1. Illustrtion of Totl Error, δ β 1 β... β n Systemtic uncertinty of individul mesurements Uncertinty Anlysis Expression Y = Y (x 1, x,..., x n ) β Y Systemtic uncertinty of experimentl result Figure. Determining the systemtic uncertinty for n experiment (from Colemn & Steele, 1999)

12 Figure 3. Brekdown of Equtions

13 OR Figure 4. TAMU 5 cfm Uncertinty Anlysis

14 Figure 5. TAMU 1 m 3 /hr Uncertinty Anlysis

15 Figure 6. TAMU 39 cfm Uncertinty Anlysis

16 Figure 7. TAMU 6 cfm Uncertinty Anlysis

17 W C = (refer to eqution 6) V δc 1 = δw V δc W = δv V W = W f W i (refer to eqution 7) δw δ W f = 1 δw = 1 δw i V = Q *Θ (refer to eqution 8) δv = Θ δq δv = Q δθ P Q = 5.976* k *( D ) * (refer to eqution 9) Q = 5.976* δk P δ ( D ) * Q δd Appendix A Sensitivity Coefficient Determintion P δ = 11.95* k *( D ) * δq δ P ( D ) =.988* k * * δq =.988* k * δ ( D ) * 1 P * P ( ) 3 P RH * Ps RH * Ps ( ) ( ) = + (refer to eqution 1).37* 46 + T.596* 46 + T δ Ps 1 1 = * δrh T δ RH 1 1 = * δp s T δ 1 = δp.37* 46 + T ( )

18 k = δ δt = 5.976* δk δq δk δd 1 ( 46 + T ) Q LFE ( D ) = * P * + RH.37 * P s Pc c ( D ) LFE P 5.976* * c c δk δ P c δk δ P c = = 5.976* 5.976* ( D ) 1 * Q = 5.976* LFE 3 1 * ( D ) 1 * Q LFE * * Q (refer to eqution 11) Pc LFE c P 3 c ( D ) * Pc * c c 1 *