Self-study notes - GAUSSIAN PLUMES

Transcription

1 Slf-stud nots - GASSIAN PLMES You should rad appropriat txt books in ordr to undrstand th maning of thos words which ar givn hr in italics. onsidr a point sourc somwhr in th air whr a pollutant is rlasd at a constant rat (kg/s). Th wind is blowing continuousl in a dirction x (masurd in mtrs from th sourc) with a spd (m/s). Th plum sprads as it movs in th x dirction such that th local concntrations (x,,) (kg/m 3 ) at an point in spac form distributions which hav shaps that ar Gaussian or normal in plans normal to th x dirction. In th latral, or, dirction th profil shap is givn b whilst in th vrtical, or, dirction it is givn b π π

2 Th paramtrs and (m) ar th standard dviations of ths Gaussian distributions, which indicat th sprad of th plum in th and dirctions, rspctivl. Th incras with th distanc x from th sourc. Th ara undr th distribution, dtrmind b intgration of th functions givn abov btwn plus and minus infinit, is qual to unit. ombining ths two-dimnsional shap distributions b multipling th functions togthr givs us th function for th shap of th distribution in thr-dimnsions (a kind of hill of pollutant). π Th concntration at an point is givn b ( x,, ) π Hnc, th concntration is qual to th rat of mission from th sourc dividd b th wind spd and thn multiplid b th shaping function. This distribution masurs and normall from th x-axis (th x-axis ma also b considrd to b th dirction of th cntr-lin of th plum. In practic, th sourc will usuall b raisd abov th ground (for xampl th xit of a chimn). Hnc w nd to modif th coordinat so that it is masurd from th ground. H ffctiv hight of plum cntr-lin (m) h s hight of sourc abov ground (m) h initial plum ris (m) coordinat masurd vrticall from th ground to a point in th plum (m)

3 Hnc, rlativ to th plum axis th nw vrtical coordinat is (-H) giving ( ) π H) (, x, Thr is on furthr modification to b mad to this quation. nlik a plum sprading in fr air, most plums will b mittd clos to th ground, lik th cas of th chimn shown abov. Hnc, as th plum sprads downwards (as wll as upwards) as it movs downwind from th sourc, it will vntuall hit th ground. larl, th plum cannot continu to sprad into th ground! Instad, it is rflctd back into th air abov th ground. Th ffct of th ground boundar is includd in th concntration quation mathmaticall b using a fictitious mirror-imag sourc (S i S) of th sam strngth ( i ) placd at th sam distanc from th ground (h s ) but on th othr sid of th boundar. Hnc, at an point P thr is a contribution to th concntration (x,,) from both th ral sourc (S) and th imaginar sourc (S i ). Th vrtical distanc to P from th cntr-lin of th ral plum is (-H). Th vrtical distanc to P from th cntr-lin of th imaginar sourc is (+H). Th latral distanc () into th pag is th sam for both sourcs. Hnc, th total concntration at P is ( ) + π + H) ( H) (, x, You should rmmbr this basic quation!

4 If onl concntrations at ground lvl ar rquird (for xampl in assssing th xposur of crops or humans to th pollutant) thn w can simplif th quation b stting 0. This givs ( x,, 0) π H It should b notd that th maximum concntration occurs whn H It is usful to rmmbr this littl rul. At larg distancs from th sourc, whr is much largr than H, th concntration varis in proportion to /(. ). If onl concntrations at ground lvl on th cntr-lin of th plum (along th x-axis dirction) ar rquird thn th quation is simplifid furthr sinc both 0 and 0. This givs ( x, 0, 0) π H sing th basic quation, if w know th rat of mission from th sourc (), th prvailing wind spd () and dirction (x) and th hight of th cntr-lin of th plum abov ground (H), w can dtrmin th concntration () at an point (x,,). Howvr, to do this w nd information about th plum sprad b obtaining valus for and. Thr ar man formula and smi-mpirical xprssions availabl for dtrmining and undr diffrnt conditions of atmosphric stabilit. A rasonabl approximation in rgions nar to th sourc whn th sourc is lvatd abov th ground (such as at th top of a chimn) is I. x and I. x whr I and I ar th turbulnt wind spd fluctuations (turbulnc intnsitis) in th and dirctions, rspctivl.

5 ndr nutral atmosphric conditions it has bn found that, ovr a rang of hights corrsponding to th vrtical plum sprad, cntrd at approximatl h s, I and I ma b stimatd as 0.88 I hs ln o and 0.50 I hs ln o whr ln is th natural log, h s is th rlas hight and o is th arodnamic roughnss rprsnting diffrnt topographic ground conditions (s nots on atmosphric boundar lars). This simpl modl for th turbulnc intnsitis bars som similarit with th approximations w usd arlir in th cours for I x, I and I and ithr ma b usd in practic. For gnral cass of diffrnt atmosphric conditions th following tpical valus appl Thrmal stratification Latral intnsit (I ) Vrtical intnsit (I ) Extrml unstabl Modratl unstabl Nar nutral Modratl stabl Extrml stabl It ma b sn that th turbulnc intnsitis, spciall th vrtical wind spd fluctuations, incras as atmosphric conditions bcom mor unstabl. On othr factor that nds to b considrd in practic is th plum ris ( h). This is th path or trajctor of th plum cntr-lin aftr it lavs th sourc. Its cours dpnds upon atmosphric conditions and th amount of buoanc and vrtical momntum in th initial plum at th sourc. Buoanc forcs causs th plum ris to var with x /3 Momntum forcs caus th plum ris to var with x /3 Hnc, th shap of th trajctor will dpnd on which forcs dominat th plum. If th chimn plum is buoant th plum riss to a maximum lvl of h 0.h s to h 0.6h s abov th sourc, dpnding on atmosphric conditions. If th initial momntum of th plum dominats thn an approximat xprssion for th final plum ris is h 3 D W whr D diamtr of th sourc,.g. chimn xit diamtr (m) W initial vrtical vlocit of th plum (m/s) In practic, th initial vrtical vlocit is oftn similar to th prvailing wind spd. So, for a tpical chimn diamtr of, sa, 3m th final plum ris is of th ordr of 9m. In man cass th valu of th plum ris is so small, rlativ to th magnitud of th rlas hight and th si of th plum, that th plum cntr-lin ma b considrd to b horiontal for simplicit.

6 Gross Scrning Analsis a simpl initial assssmnt It is oftn usful to prform a simpl scrning analsis bfor appling a mor rfind analsis. A gross scrning analsis will quickl idntif th ordr of magnitud of th xpctd concntrations and ma vn show that no problm xists, in which cas mor advancd modlling is unncssar. A usful formula for stimating worst cas man concntrations downwind of a point sourc is: whr: 0 H 9 W sourc strngth or mission rat of gas or particulat [kg/s] worst cas concntration [µg/m 3 ] worst cas wind spd at hight 0 m, usuall m/s W worst cas cloud width [m] (usuall w assum W 0.x, whr x is distanc from th sourc) H worst cas cloud dpth (usuall w assum H 50 m in worst cas) This quation is ssntiall a statmnt of th consrvation of pollutant mass, but it illustrats man of th basic paramtr dpndncis in disprsion modling. Rfrring to Figur, w assum a uniform concntration in th plum passing through th downwind plan HW. () Equation () follows from th fact that th flux of pollutant through an plan must qual th sourc rat. Equation () illustrats svral important dpndncis that should b satisfid b all plum modls:. Th man concntration is invrsl proportional to man wind spd.. Th man concntration is dirctl proportional to th rlas rat. 3. Th man concntration is invrsl proportional to th plum cross-sctional ara. As an xampl of th abov, suppos a small amount (-kg) of ammonia is rlasd ovr a priod of 30 minuts in an accidntal rlas. Assuming a light wind of m/s, dos this rlas pos an risk to th occupants of a hospital locatd 5 km downwind? For this xampl th stimat of th plum width is W m, thus, 9 ( 0 µ g / kg) x ( kg /800 s) µ g / m H W m / s x 50m x 500m This concntration turns out to b quivalnt to 0.03 ppm (and is, in fact, 500 tims blow th prsonal xposur limit (PEL) associatd with ngativ halth ffcts du to prolongd xposur to ammonia). Thrfor, w can safl sa that thr is no risk. In such a cas thr is also no nd to prform an advancd modlling to assss th risk.

7 ustions () Sulphur dioxid is mittd at a rat of kg/s from th top of a chimn that is 0m high. Th plum initiall riss vrticall a furthr 0m abov th chimn xit, bfor bing convctd horiontall b a wind spd of 5m/s undr conditions of nutral stabilit. Th surrounding trrain is flat with a roughnss lngth o of 0.0m. alculat; (a) Th concntration (kg/m 3 ) on th plum cntr-lin at a distanc of 800m downwind of th chimn.[answr: 53 µg/m 3 ] (b) Th ground lvl concntration at a distanc of 800m downwind of th chimn (that is, along th x-axis). [Answr: 57 µg/m 3 ] (c) Th location (x) whr th maximum ground lvl concntration occurs downwind of th chimn on th x- axis. [Answr: 54.3 m] (d) Th concntration at this location. [Answr: 05 µg/m 3 ] () A 00m tall chimn stack mits hdrogn chlorid (dnsit.64kg/m 3 ) at a rat of m 3 /s. Th plum initiall riss a furthr 5m dirctl abov th xit bfor bing convctd horiontall b a wind blowing at a spd of 0m/s undr nutral atmosphric conditions. Th trrain has a roughnss lngth of o 0.03m. A small housing dvlopmnt commncs at a location which is a distanc of x 500m downwind of th stack and 500m from th cntr-lin of th plum. (a) (b) What is th pollution concntration (kg/m 3 ) at ground lvl at th start of th housing dvlopmnt? [Answr: 43 µg/m 3 ] If th wind dirction changd so that th plum axis pointd dirctl towards th housing dvlopmnt, what would b th nw ground lvl concntration at th sam location as bfor? (Assum that th wind spd and th rat of mission rmain unchangd). [Answr: 537 µg/m 3 ] (3) A 00m tall chimn mits sulphur dioxid at a rat of.5kg/s in a highl buoant plum. Th plum ris ( h) aftr xiting th chimn follows a trajctor which is givn b th quation: h 0.3 x /3 (all dimnsions in mtrs). Th wind spd is 0m/s and th local ground roughnss lngth is o 0.0m. What is th ground lvl concntration along th x-axis at a distanc of.5km down wind of th chimn? [Answr: 400 µg/m 3 ] (4) A ground lvl accidntal rlas of 0.5 g/s occurs at a 4 m hight in trrain with a roughnss lngth of o 0.0m. Assuming a wind spd of m/s what is th maximum ground lvl concntration 4 km downwind stimatd from th gross scrning mthod and from th gaussian plum quation? [Answrs: 5 µg/m 3, 0.56 µg/m 3 ]