Empirical Prediction of Fitting Densities in Industrial Workrooms for Ray Tracing. 1 Introduction. 2 Ray Tracing using DRAYCUB

Transcription

1 Empiical Pediction of Fitting Densities in Industial Wokooms fo Ray Tacing Katina Scheebnyj, Muay Hodgson Univesity of Bitish Columbia, SOEH-MECH, Acoustics and Noise Reseach Goup, 226 East Mall, Vancouve, B.C., V6T 1Z3, Canada, The objective of this study was to develop an empiical method, based on measuable physical fitting desciptos, fom which fequency-dependent fitting densities could be calculated in a wokoom fo use in the pediction of the sound level in that wokoom by ay tacing. The study involved eleven typical wokooms of vaying dimensions and types, quantities, and distibutions of fittings, in which Hz octave-band sound popagation cuves and fitting dimensions had peviously been measued. Physical desciptos consideed included the Kuttuff fitting density and othes involving the numbe, aveage height, and total volume of the fittings. Pedictions of octave-band sound popagation cuves wee made fo vaious fitting densities using a ay tacing pogam. These wee compaed (based on the total deviation fom the pedicted cuves) with measuements of the sound popagation levels to detemine the best fit fitting density. A coelation analysis was then pefomed between those fitting densities and vaious physical desciptos to find pediction models. Regession analyses of combinations of the highest coelated paametes found the highest coefficients of detemination anged fom.39 to.61 fo octave bands between Hz. The aveage fitting to wokoom height atio, the volume atio of the fittings to the wokoom, and the numbe of fittings wee the paametes that pedicted the fitting density best. 1 Intoduction Noise is an impotant pat of the health and safety of people; in industial wokooms sevee consequences could esult when noise levels ae too high (i.e. heaing loss and miscommunication). To educe and minimize noise effectively, it must fist be modelled as accuately as possible. Modelling an industial wokoom involves quantifying the fittings (obstacles) by the fitting density. The objective of this study was to develop an empiical method, based on measuable physical fitting desciptos, fom which fequencydependent fitting densities could be calculated in a wokoom fo use in the pediction of the sound level by ay tacing. Pevious eseach has found DRAYCUB, a ay tacing appoach, to accuately pedict sound popagation cuves in wokooms [1]. These sound popagation cuves, defined as SP() = L p ()-L w [db], can be pedicted fo given fitting densities using DRAYCUB. This pape compaes DRAYCUB s sound popagation cuves with measuements in wokooms to find which fitting densities best pedict the data. Two methods of compaison wee used: a slope-fit method, based on the ate of decay of sound popagation levels with distance (the slope of the sound popagation levels); and a bestfit method, based on the total deviation fom the pedicted cuves. Results found bette ageement with the best-fit method and it will thus be the focus of this pape. Thee has been a geat deal of eseach done on both ay tacing and empiical modelling fo pedicting sound levels in industial wokooms [2,3]. Although empiical methods may be fundamentally valid, ay tacing is a moe accuate appoach [2]. A simila study to the one pesented in this pape has been done which used the empiical models in PlantNoise, a model that uses empiical data fo modelling, pedicting, visualizing and aualising noise in industial wokooms [4]. Measued esults wee compaed with the sound popagation cuves pedicted by PlantNoise fo fitting densities anging fom.25 to.15 m -1, and the best-fit fitting densities wee coelated with physical desciptos. Simple linea egession was then used with the moe highly coelated desciptos (eithe h/h, the atio of aveage fitting height to wokoom height; o N f, the numbe of fittings) to detemine an empiical elationship at octave band fequencies fom 125 to 4 Hz. The R-squaed values anged between.25 to.47 using this method. 2 Ray Tacing using DRAYCUB DRAYCUB [5] is a compute pogam developed fom the Ondet and Baby algoithm [6] which pedicts sound popagation cuves based on the geomety (size and shape of the oom), the fittings (the obstacles in the wokooms, as descibed by thei absoption coefficient and fitting density, [m -1 ]), the location and powe level of souces(s), the location of eceives, the absoption due to ai, and the absoption and diffuse-eflection coefficients of each suface at each octave band fequency. Rays emitted fom a souce in andom diections ae taced about the wokoom using Monte Calo methods and vecto geomety, based on paametes set by the use (including the numbe of taced tajectoies, the numbe of ays emitted by the souce, and the paametes mentioned above). The 2359

2 Foum Acusticum 25 Budapest Scheebnyj, Hodgson eceives ecod the numbe of ays, the enegy of each ay, and the total enegy of the ays that pass though them. This infomation is then used to calculate the sound pessue levels at the eceives. Thee assumptions wee made while modeling the fittings: they ae dimensionless; they scatte omnidiectionally; and they ae distibuted accoding to a Poisson distibution (if the distance between each pai of fittings wee plotted, the fequency distibution of the distances would fom a Poisson distibution). The invese of the mean distance ( ) between these pais, fo which the distibution eaches a maximum, epesents the fitting density of a given fitted oom, ( 1 ). The units of ae m -1. Kuttuff agued that the fitting density is elated to the suface aea of the fittings. The Kuttuff fitting density, K, is defined as the total exposed fitting aea divided by fou times the volume of the egion unde consideation [1] as descibed by A V. K f, tot 4 Although this elationship does not suggest a fequency-dependence, it is in fact a high-fequency limit. This paamete will be efeed to late when detemining the fitting densities of the wokooms. 3 Wokooms and Modelling Eleven typical wokooms involved in the poduction of food and pesonal cae poducts wee used in this poject (the tem typical hee efes to thei size, constuction, shape, and fitting distibution). The layout of these wokooms was based on poductions lines, whee machines wee often joined by conveyo belts. Plans of the wokooms wee used to calculate the dimensions with as much detail as possible (i.e. cones and sloped ceilings). Typical dimensions wee 2-3 m on both sides; one was as lage as 85 x 5 m 2. Typical constuction mateials included concete floos, masony walls and steel deck ceilings. Although the diffuse-eflection coefficient vaies with fequency, sound fields ae not highly dependant on it and a value of.75 was used fo all cases based on pevious eseach [7]. Since it was assumed that all wokooms ae of typical constuction, an aveage oom absoption coefficient was used fo each fequency. Based on pevious eseach, the following values wee used:.15/.15/.8/.7/.6/.6, fo octave bands anging fom 125 to 4 Hz, espectively [8]. Although these values can vay by appoximately ±25 % [9], it was not expected that DRAYCUB pedictions would be vey sensitive to changes in the absoption coefficients. The fittings consideed, machines and conveyos (the two main obstacles in wokooms), had complex geometies. To detemine the physical desciptos elated to fittings (i.e. thei aveage height, fitting volume, etc.), the machines and conveyos wee consideed as eithe ectangula boxes o vetical cylindes. DRAYCUB allows the effect of the wokoom fittings to be taken into account, so the aveage height of the fittings in a oom was also used as the height of the fitted volume. When including the conveyos, this aveage height was lowe than when only using the machines. As mentioned, the physical desciptos chosen fo this study wee ones that ae easy to quantify and/o measue. They included the Kuttuff fitting density, ( k [m -1 ]); the aveage fitting height, (h [m]); the numbe of fittings, (N f ); the pecent of floo aea coveed by fittings, (%FAC); the total fitting volume to wokoom volume, (V f /V); and the aveage fitting height to aveage wokoom height (h/h). These paametes wee coelated with the best-fit fitting densities to identify which wee most highly elated to. The numeical value fo a good coelation depends on the goodness of fit,, fo the final data. Coelation was consideed weak if.5 and stong if.8 1 [11]. 4 Wokoom Measuements An omnidiectional loudspeake aay was used to geneate boadband noise in the eleven wokooms. The sound pessue levels wee then measued in Hz octave bands at inceasing distances fom the souce (i.e..5, 1, 2, 5, 1 m, etc.) and soundpopagation levels wee found at each fequency. The paths of sound popagation wee chosen though aeas of typical fittings. The two fitting cases will be efeed to as Machines and Machines and Conveyos fo the emainde of this pape. Data fo the Machines case was not obtained fo one of the eleven wokooms. 5 Results and Analysis DRAYCUB was used to pedict sound popagation cuves fo each wokoom at each fequency fo vaying -values (,.1,.15,.2,.25,.3,.4,.5,.6,.7, and.8 m -1 ), fo eceives in two diections (each with a single souce), and fo the aveage fitting height of the Machines and of the Machines and Conveyos configuations. Typical esults ae illustated in Figue

3 Foum Acusticum 25 Budapest Scheebnyj, Hodgson L p - L w (db) L p - L w (db) Machines (1 Hz) Distance fom Souce (m) Machines and Conveyos (1 Hz) Distance fom Souce (m) Data Data Figue 1: Sound popagation cuves measued and pedicted by DRAYCUB fo fitting densities fom = to.8 m -1 fo a typical wokoom at 1 Hz. Data epesents the measuements taken in the wokoom. Although the two gaphs in Figue 1 appea simila, upon close examination, the Machine cuves decay slightly faste than the Machines and Conveyos cuves. It is also appaent fom the measued data points that sound-popagation levels do not follow a constant linea slope as distance inceases logaithmically, but athe thee is a beak point nea 1 m (keep in mind the x-axis has been plotted logaithmically). As a esult, logaithmic egession was used to fit thee slopes to the data plotted in Figue 1: one called the full-distance, that included all data points (fom =.5 m to the fa end of the oom); one called the nea distance (fom =.5 m up to but not including = 1 m); and one called the fa distance (fom = 1 m until the fa end of the oom). The best-fit -values wee identified by summing the diffeences between the measued data points (L p -L w ) and the sound popagation level fo each distance measuement (these wee calculated based on the equation fo a line found using linea egession in the pevious pat). In most cases, this sum passed fom negative to positive values ove the -value ange; the fitting density with a sum closest to zeo was ecoded. Fo example, the data fo the Machines and Conveyos of Figue 1 is pesented in Table 1. In this case, the best fit fitting density is.4 m -1. It also occued in a few cases that the sums wee all positive o all negative, in which case the sum closest to zeo was used (this was not as accuate a choice as the only options wee m -1 o.8 m -1 ). Fo example, Table 2 illustates the case whee all the values wee negative and the best-fit fitting density was set to m -1. At each fequency, in each oom, six best-fit fitting densities wee found. As descibed ealie, the values at each fequency wee then coelated with physical desciptos that chaacteized the coesponding wokoom fittings. Only single coelations wee possible and thus atios of some of the physical desciptos listed above wee combined and also used in the analysis. Fo all but the lowest fequency, the highest coelation values fo both the nea and fa distances wee neve both highe than that fo the full distance (fo 125 Hz, the highest coelation values wee full : -.49; nea : -.69; fa : -.52). This suggests that thee is no advantage in splitting the distance fo the best-fit data (o at least not with the tansition at 1 m). The split distances wee set aside and the full distance was used fo all futhe calculations. As shown in Table 3, the highest coefficients of detemination fo the Machines and the Machines and Conveyos cases wee vey close. The two cases howeve didn't necessaily use the same physical desciptos. Table 1: Machines and Conveyos data fom Figue 1 at 1 Hz (m) Slope (ln(x)) B-Value Total Stdev L p-l w Exp. / Table 2: Machine data fom the same wokoom as in Figue 1, but at 25 Hz (m) Slope (ln(x)) B-Value Total Stdev L p-l w Exp. /

4 Foum Acusticum 25 Budapest Scheebnyj, Hodgson Table 3: Highest adjusted coefficients of detemination ( 2 ) found fo the full distance Fequency (Hz) Machines Machines and Conveyos Fo the majoity of the fequencies, including conveyos did not impove the accuacy (the Machines had a highe coefficient of detemination). Unfotunately, it is at midange fequencies (5 1 Hz) that this is not tue. Since only 11 ooms wee used to find these data (1 fo the Machines), the diffeence at 1 Hz may not be that significant. Although using the Machines and Conveyos data is moe accuate fo the 5 Hz case, the additional time and effot equied to measue and include conveyos in calculations may not be wothwhile. Fo the puposes of this study, since 2 =.61 at 5 Hz and 2 =.6 at 1 Hz ae the two highest values fo the Machines case, it is assumed that they will pedict the fitting densities with sufficient accuacy. Geate accuacy is possible, but it comes at the pice of additional field measuements. The most highly coelated physical desciptos fo the best-fit method fo Machines using the full distance data wee h/h and N f. Using vaious combinations of the physical paametes (up to thee paametes at a time) in multipleegession analyses, equations wee found to pedict the fitting density. The equations with the best fit, o highest adjusted 2 value based on only one physical descipto, ae listed as Equations (1a 1f): h H 2.15 h H 2.43 h H 2.41 h H 2.5 h H N f (1a) (1b) (1c) (1d) (1e) (1f) Highe adjusted 2 values wee obtained with the addition of a second paamete, as listed in Equations (2a-2f): V f h V H V f h V H (2a) (2b) V f h V H V f h V H V f h V H h N f 12 H (2c) (2d) (2e) (2f) The 2 values at all fequencies ae highe with two paametes, except at 2 Hz, whee they ae equal. The addition of a thid paamete did not incease the 2 value futhe. Of the 1 wokooms used in this pat of the analysis (data fo the 11th wokoom was only available fo Machines and Conveyos), it was suspected that one o moe of the ooms may be unlike the othes, acting as outlies athe than tend data. To quantify this fo the paametes used in Equations 1 and 2 (as well as fo k, out of inteest), the best-fit fitting densities wee plotted with the value of each paamete at each fequency. All gaphs showed simila esults; the gaphs fo V f /V and fo h/h have been included as Figues 2a and 2b. Ideally, the elationship will be linea. Best-Fit Fitting Density (m -1 ) V f /V 125 Hz 25 Hz 5 Hz 1 Hz 2 Hz 4 Hz Figue 2a: Scatte plot showing the elation between V f /V and the best-fit fitting density fo 1 wokooms Best-Fit Fitting Density (m -1 ) h/h 125 Hz 25 Hz 5 Hz 1 Hz 2 Hz 4 Hz Figue 2b. Scatte plot showing the elation between h/h and the best-fit fitting density fo 1 wokooms 2362

5 Foum Acusticum 25 Budapest Scheebnyj, Hodgson Table 4: The standad deviation and some of its popeties fo each octave band (all ae in units of db) Fequency (Hz) Aveage Standad Standad of the Aveage Standad Maximum Standad Minimum Standad It does not appea fom Figues 2a and 2b that the points take the shape of a paticulaly staight line. As a esult, it is difficult to identify outlies. Pehaps with moe data (i.e. moe wokooms), the geneal tend would be moe appaent and outlies could be emoved; fo the time being all data was used fo the analysis. Thee might have been an offset due to eos in the calibation of the equipment that was used to make the measuements in the wokooms. If the standad deviation of the best-fit fitting density was simila at each fequency, it is likely that this was indeed the case. The aveage standad deviation, its standad deviation, the maximum standad deviation, as well as the minimum standad deviation fo each fequency have been tabulated in Table 4. These esults will be discussed in futhe detail in the following section. 6 Discussion of Best-Fit Results The tems found in Equations 2 wee consideably moe significant than those found using the slope-fit method: 67% of the tems wee significant as compaed to 31% fom the slope-fit method. It was expected that the physical paametes used in the equations to pedict the fitting density would epesent the density and both the vetical and hoizontal size and distibution of the fittings. If that assumption wee tue, physically ealistic conclusions could be dawn fom Equations (2a-2f), which used h/h, N f, and V f /V. Also, the epetition of two of the same paametes in five of the six equations suggests that those paametes may be stongly associated with the fitting density, unlike in the case of the slope fit model whee thee wee diffeent paametes fo each fequency. It was supising howeve that the Kuttuff fitting density did not coelate well in this wok; it does not appea in any of the equations fo eithe the best-fit method, o the slope-fit method. An inteesting esult was found at 2 Hz: it appeas as though Equations (1e) and (2e) ae equally as accuate ( 2 =.39 fo both). Fom an engineeing pespective, they ae equivalent, and Equation (1e) should be used fo simplicity. The coefficient of detemination fo Equation (2e) howeve is.5 highe, and since V f /V will aleady be calculated (it is equied fo fou othe fequency bands), the inclusion of this tem does not equie much additional wok. Because the coelation values wee aely vey high (the highest value fo the full distance fo Machines using the best-fit data was.74), thee was quite a bit of vaiance in Figues 2a 2b. As a esult, not only was it difficult to notice any outlies, it was also difficult to see stong tends. Moe data (i.e. moe wokooms) may help distinguish between the two. As it is now, thee is no justification fo emoving any wokoom fom the analysis. If thee wee a constant offset due to the calibation of the equipment, the standad deviation of the total best fit values would be the same fo all ooms at a given fequency band. In othe wods, the standad deviation of the total best-fit values standad deviation would be. Fo example, the total best fit value at 25 Hz fo one wokoom is.9 (as shown in Table 2). The standad deviation is 1.3. These two values exist fo each wokoom. The standad deviation of the standad deviation listed in the tables simila to Table 2 should be if thee is a constant offset fom oom to oom. This is not the case based on the data fom Table 4; the standad deviations of the standad deviations fo a given fequency ae all above 1 db. In addition, the distibution within each fequency band is quite wide (all ange fom ~3.5 to 4.5 db wide) suggesting that the offset, if thee is one, is not sufficiently constant oom to oom to be distinguished fom uncetainties in the actual measuements. A supising finding in this wok was that many of the coelation coefficients, paticulaly fo the highly coelated paamete h/h, wee negative. This is not intuitive; it is expected that the fitting density would incease with an incease in the physical desciptos used in this epot. Fo example, conside the paamete h/h: as the fitting height inceases elative to the height of the wokoom, the fittings take up moe oom in the wokoom and thus the fitting density should incease. In fact, the opposite took place, with the aveage coelation coefficient aound.7. With the exception of N f, this phenomenon eoccus with all the paametes in the best-fit analysis. At this time, no explanation is available fo this behavio. Sound popagation modeling is not a pecise science; studies have found ay tacing only to be accuate to ± 2 dba[12]. Assuming aveage values fo the absoption coefficients as well as one value fo each of the diffuse-eflection coefficients as explained in the Theoy section may in fact be limiting the accuacy of the ay tacing. 2363

6 Foum Acusticum 25 Budapest Scheebnyj, Hodgson 7 Conclusions The sound popagation cuves fo vaious fitting densities in 11 wokooms wee pedicted using DRAYCUB. This data was compaed with expeimental data, and linea egession was used to compae the cuves. Compaisons of the slopes and the best fits of these cuves wee made fo the full, nea, and fa distance as well as with fittings compising only machines, and both machines and conveyos. Physical desciptos wee then coelated with the fitting densities to obtain equations that could late be used to pedict the fitting density in othe wokooms. A thoough analysis found that thee was no accuacy advantage in including the conveyos in the model, no did splitting the distance into a nea and fa section with a tansition point of 1 m fom the souce, o using the slope-fit method, impove accuacy. Equations with coefficients of detemination fom.39 to.61 wee found using the best-fit appoach. The physical desciptos equied fo the pediction of the fitting density in wokooms ae V f /V, h/h, and N f. In othe wods, the effect of fittings can be found fom the numbe and dimensions of the fittings. The esults found thee to be little vaiation in the soundpopagation cuves close to the souce: within appoximately 3 m, all the cuves ae simila. At lage souce/eceive distances, the vaiation was often aound 7 db, lage enough that a pope selection fo fitting density is necessay. It may be infomative in futue wok to change the definition of what was consideed nea and fa distances in this study (i.e. vay the tansition point fom 1 m). If modeling is epeated using DRAYCUB, it might also be infomative to use fitting densities lage than.8 m -1 since the fitting densities in seveal wokooms wee pedicted to be geate than that. It is also ecommended that futue wok look futhe into why most coelations wee negative. The absolute best physical paametes to descibe the fitting density ae still unknown; this study povides good esults based on a limited numbe (and a limited numbe of combinations). Moe wok is equied to obtain data that can be compaed with all physical desciptos and all combinations theeof. Although the Kuttuff fitting density did not coelate well in this wok, it may (pehaps in combination with anothe (o new) paamete(s)) fit in anothe elationship quite well. A futhe analysis on non-linea elationships that combines multiple physical desciptos may also be useful fo a bette pediction model of the fitting density in wokooms. Futue wok may also include validation of the equations fom this wok by using them to pedict the fitting densities in othe wokooms. Fo the time being, the equations found in this study povide easonable accuacy in the pediction of fitting densities in wokooms. Refeences [1] M.R. Hodgson, ' On the accuacy of models fo pedicting the sound popagation in fitted ooms', J. Acoust. Soc. Am. 88 pp (199) [2] M.R. Hodgson, 'Sound-Popagation Cuves in Industial Wokooms: Statistical Tends and Empiical Pediction Models', Building Acoustics, Vol. 3. No. 1. pp (1996) [3] M.R. Hodgson, 'Ray-tacing evaluation of empiical models fo pedicting noise in industial wokshops', Applied Acoustics, Vol. 64. pp (23) [4] A. Matella and M. Hodgson, 'Optimization and Expeimental Validation of the PlantNoise System in Unileve Wokooms', Repot to Unileve UK, Univesity of Bitish Columbia (23). [5] M.R. Hodgson, '(D,E)RAYCUB Use s Manual', Depatment of Mechanical Engineeing, Univesity of Bitish Columbia, Vancouve, BC, Canada. [6] A.M. Ondet and J.L. Baby, 'Modeling of sound popagation in fitted wokshops using ay tacing'. J. Acoust. Soc. Am, Vol. 85 No. 2. pp (1989). [7] M.R. Hodgson, 'Evidence of diffuse suface eflections in ooms', J. Acoust. Soc. Am, Vol. 89. pp (1991) [8] M.R. Hodgson, 'Effective Densities and Absoption Coefficients of Fittings in Industial Wokooms', ACUSTICA acta acustica, Vol. 85. pp (1999) [9] M.R. Hodgson, 'Towad a poven method fo pedicting factoy sound popagation', Poceedings of Inte-Noise 86. II, pp (1986) [1] N. Heeema and M. Hodgson, 'Empiical models fo pedicting noise levels, evebeation times and fitting densities in industial wokooms', Applied Acoustics Vo. 57. pp (1999) [11] J.L. Devoe, 'Pobability and Statistics fo Engineeing and the Sciences, Sixth Edition', 24 Books/Cole, a division of Thomson Leaning, Canada. pp (24) [12] M.R. Hodgson, 'Ray-Tacing Pediction of Noise Levels in a Nuclea Powe-Geneation Station', Applied Acoustics, Vol. 52. No. 1. pp (1997) 2364