Theory and Calibration Procedures for the Use of a Rotameter

Transcription

1 T I : T O S H E R I S L I N G O U R S E ppendix F Theoy and alibation ocedues o the Use o a Rotaete F. Noenclatue = coss-sectional aea o the loat = annula aea between the cicueence o the loat and the inside cicueence o the ete tube at that position = da coeicient = lenth which is chaacteistic o the physical syste unde study (used to calculate Reynolds Nube) d = lenth which is chaacteistic o the physical syste unde study = diaete o the loat t = diaete o the tube at the loat position = local acceleation due to avity c = diensional constant = ass o the loat = olecula weiht o the eteed as,, 3...etc.= value o olecula weiht o the eteed as at conditions,, 3...etc. Re = Reynolds Nube Re/ = diensionless acto deined by Equation F-4 = absolute pessue o the eteed as,, 3...etc. = values o absolute pessue at conditions,, 3...etc. = voluetic low ate thouh the ete at conditions o pessue ( ), tepeatue (T ), and olecula weiht ( ) R = univesal as constant T = absolute tepeatue o the eteed as T, T, T 3...etc. = values o absolute tepeatue at conditions,, 3...etc. v = aveae as velocity thouh the annula aea o the ete V = volue o the loat µ = viscosity o lowin luid (used to calculate Reynolds Nube) F-

2 T I : T O S H E R I S L I N G O U R S E µ = viscosity o the eteed as = density o lowin luid (used to calculate Reynolds Nube) = density o the loat = density o the eteed as F. esciption o a Rotaete The otaete (Fiue F-) is a vaiable aea ete which consists o a vetical, tapeed, tanspaent tube containin a loat. The loat oves upwad as the luid low inceases. vaiable in o annulus is ceated between the oute diaete o the loat and the inne wall o the tube. s the loat oves upwad in the tube, the aea o the annulus inceases. The loat will continue to ove upwad until a pessue dop acoss the loat, which is unique o each otaete, is eached. This pessue dop acoss the loat is constant eadless o the low ate. Gaduations ae etched on the side o the tube so that an instantaneous eadin ay be obseved. Fiue F-. Rotaete. F-

3 T I : T O S H E R I S L I N G O U R S E F.3 evelopent o Flow Equations Geneal Flow Rate Equations ee body diaa o the oces actin upon the otaete loat is shown in Fiue F-. The weiht o the loat is equal to the oce o avity actin on the loat. The buoyant oce is equal to the weiht o the as that is displaced by the loat. The da oce is equal to the ictional oces actin between the loat and the ovin as stea. Fiue F-. Foces actin upon a otaete loat. atheatically, these oces ae as ollows: a oce V Weiht o loat c v c Whee: V Buoyant oce = coss sectional aea o the loat = da coeicient = local acceleation due to avity = diensional constant c c F-3

4 T I : T O S H E R I S L I N G O U R S E v = aveae as velocity thouh the annula aea o the ete V = volue o the loat = density o the loat = density o the eteed as When the oces actin in an upwad diection exactly equal the oce actin in a downwad diection, the loat will eain stationay in the tube. Equatin these oces yields: c v V c V c ancellin like tes ( c ) and eaanin yields: v V - V Solvin o v and actoin out V and o the ist two tes yields: (Eq. F-) V v The aea o the loat is equal to 4, whee is the diaete o the loat. Substitutin 4 o in Equation F- yields: (Eq. F-) 8V v Let equal 8, whee is called a ete coeicient and is dependent on the da coeicient. Substitutin o 8 in Equation F- yields: (Eq. F-3) v V Because the da coeicient is dependent on Reynolds Nube, ust also be a unction o Reynolds Nube. Because the density o the as lowin in the otaete is vey sall copaed to the density o the loat, it can be F-4

5 T I : T O S H E R I S L I N G O U R S E inoed in the ( ) te. odiyin the ( ) te in Equation F-3 yields: (Eq. F-4) v V The voluetic low ate (,.) thouh the otaete is equal to the poduct o the velocity (v) and the annula aea o the ete ( ). Substitutin o v in Equation F-4 yields: V Reaanin tes and eovin o the adical yields: (Eq. F-5) V The density o the loat is equal to the ass o the loat ( ) divided by the volue o the loat. Substitutin V o in Equation F-5 and cancellin the V s yields: (Eq. F-6) The density o the as ixtue passin thouh the ete ( ) is equal to, whee is the absolute pessue at the ete, is the appaent olecula weiht o the as ixtue passin thouh the ete, R is the univesal as constant, and T is the absolute tepeatue o the as ixtue. Substitutin o in Equation F-6 yields the eneal low ate equation o a otaete: (Eq. F-7) oputation o Reynolds Nube Reynolds Nube is deined as vd, whee v is the velocity low, d is a lenth which is chaacteistic o the physical syste unde study, is the density o the lowin luid, and is the viscosity o the lowin luid. When F-5

6 T I : T O S H E R I S L I N G O U R S E calculatin Reynolds Nube o a as lowin thouh a otaete, the lenth chaacteistic o the physical syste ( d ) is the dieence between the tube diaete ( ) and the diaete o the loat ( ). Theeoe, Reynolds Nube ay be calculated by usin the ollowin equation: (Eq. F-8) v Re μ The aveae velocity o low thouh the otaete is iven by whee is the voluetic low ate thouh the ete and is the annula aea between the inside cicueence o the tube at the loat position. Substitutin o v in Equation F-8 yields: (Eq. F-9) Re The density o the lowin luid is equal to, whee is the absolute pessue o the eteed as, is the appaent olecula weiht o the eteed as, R is the univesal as constant, and T is the absolute tepeatue o the eteed as. Substitutin o in Equation F-9 yields: (Eq. F-0) Re ddin the subscipt to the viscosity te in Equation F-0 to denote the viscosity o the eteed as yields the ollowin equation, which is used to calculate Reynolds Nube o as low in a otaete. (Eq. F-) Re F.4 oon actices in the Use o a Rotaete o Gas Flow easueent It can be seen o Equation F-7 that the voluetic low ate thouh a otaete can be calculated when such physical chaacteistics as the diaete and the ass o the loat and the annula aea o the ete at each tube eadin ae known, povidin easueents ae ade o the tepeatue, pessue, and olecula weiht o the eteed as. Beoe these calculations o the F-6

7 T I : T O S H E R I S L I N G O U R S E voluetic low ate can be ade, data ust be known about the ete coeicient,. The ete coeicient bein a unction o Reynolds Nube is ultiately a unction o the conditions at which the ete is bein used. To obtain data on the ete coeicient, the ete ust be calibated. Howeve, because o the ease involved in usin calibation cuves, coon pactice is to use calibation cuves to deteine voluetic low ates instead o calculatin the low ates o aw data. ocedues o the alibation o a Rotaete coon aaneent o equipent o calibatin a otaete is shown in Fiue F-3. Fiue F-3. Test setup o calibatin a otaete. Flow thouh the calibation tain is contolled by the etein valve. t vaious settins o the otaete loat, easueents ae ade o the low ate thouh the tain and o the pessue and tepeatue o the as stea at the otaete. The tepeatue o the as stea is usually assued to be the sae as the tepeatue o the abient ai. I the test ete siniicantly aects the pessue o tepeatue o the as stea, easueents should also be ade o the actual pessue and tepeatue at the test ete. typical otaete calibation cuve is illustated in Fiue F-4. F-7

8 T I : T O S H E R I S L I N G O U R S E Fiue F-4. Rotaete calibation cuve. To ake the calibation cuve useul, the tepeatue and pessue o the voluetic low ate ust be speciied. Univesal alibation uve The noal aaneent o the coponents in a saplin tain is shown in Fiue F-5. Since the ete is usually installed downstea o the pollutant collecto, it can be expected to opeate unde widely vayin conditions o pessue, tepeatue, and olecula weiht. This equies a dieent calibation cuve o each condition o pessue, tepeatue, and olecula weiht. This can be acilitated by dawin a aily o calibation cuves, which would backet the anticipated ane o pessues, tepeatues and olecula weihts, as shown in Fiue F-6. Fiue F-5. aneent o saplin coponents. F-8

9 T I : T O S H E R I S L I N G O U R S E Fiue F-6. Faily o otaete calibation cuves. Opeation o a otaete unde extee saplin conditions, paticulaly extee tepeatues, coplicates the calibation setup. It is diicult, i not ipossible, o ost laboatoies to be able to calibate low etein devices at hih tepeatues o unusual as ixtues (especially whee toxic ases ae involved). Fo these easons, it is desiable to develop a calibation cuve which is independent o the actual expected saplin conditions. s peviously entioned, the low thouh a otaete is dependent upon the value o, the ete coeicient (see Equation F-7), which is a unction o the Reynolds Nube o the low in the otaete. Theeoe, to be independent o the saplin conditions, the calibation cuve ust be in tes o and Re. evelopent o a Univesal alibation uve Solvin Equation F-7 o ives the ollowin elationship: (Eq. F-) ividin Equation F- by Equation F- yields: Re ancellin the like tes and yields: μ F-9

10 T I : T O S H E R I S L I N G O U R S E F-0 μ Re Sipliyin: μ Re obinin the like tes,, and T yields: (Eq. F-3) μ Re Sipliyin the and elationship in Equation F-3 yields a diensionless acto which has no liitations on eithe Reynolds Nube o the ete coeicient. (Eq. F-4) Re plot o the diensionless acto Re deined by Equation F-4 vesus the ete coeicient as calculated o Equation F- on eula aph pape will yield a univesal calibation cuve which is independent o the saplin conditions. Such a plot is illustated in Fiue F-7.

11 T I : T O S H E R I S L I N G O U R S E Re Fiue F-7. univesal calibation cuve o a otaete. F.5 Use o the Univesal alibation uve o a Rotaete To eteine an Existin Flow Rate To deteine an existin low ate, easueents ust be ade o the as tepeatue and pessue as well as the loat position. ata o the anuactue o the otaete will yield inoation on the diaete o the tube at the vaious loat positions and on the diaete and ast o the loat. The appaent olecula weiht o the as bein eteed can be calculated i the coposition o the as stea is known. The viscosity o the as stea can be deteined i the tepeatue o the as stea is known (see ey s heical Eninee s Handbook). Fo this data the Re acto (see Equation F-4) can be calculated. The univesal calibation cuve is then enteed at the calculated value o Re and the coespondin is noted. is then calculated o Equation F-7. To Establish a Requied Saplin Rate To establish a equied saplin ate, estiates ae ade o the eteed as pessue ( ), the eteed as tepeatue ( T ), the appaent olecula weiht o the eteed as ( ), and the aea o the ete ( ) which will exist at the desied saplin ate. Usin these estiated values, the ete coeicient,, is calculated (see Equation F-) o the desied saplin F-

12 T I : T O S H E R I S L I N G O U R S E ate. The univesal calibation cuve (see Fiue F-7) is enteed at this value o and the coespondin acto is noted. is solved by usin the ollowin equation which is a eaaneent o Equation F-4: (Eq. F-5) Re The loat position can be deteined o the value. Fo soe otaetes the value o is the tube eadin divided by 00. I the aea o the ete coespondin to this loat position is not equal to the oiinal estiated value o the ete aea, the new value o aea is used as an estiate and the entie pocedue is epeated until the estiated aea and the calculated aea ae equal. Then upon settin the loat position at this tube eadin, T,, and, ae noted. I they ae dieent o the oiinal estiates, the pocedue is epeated usin the obseved values o T,, and as estiates. Expeience will aid in selectin oiinal estiates that ae nealy accuate so that the equied saplin ate ay be set aily apidly. To edict alibation uves The above techniques ae vey cubesoe to apply in the ield and, as a esult, the univesal calibation cuve should not be used in such a anne. The eal utility o the univesal calibation cuve is that it can be used to pedict calibation cuves at any set o conditions. This esults in a eat eduction in laboatoy wok in that the otaete need only be calibated once and not evey tie the conditions at which the ete is opeated chane. The ist step in pedictin calibation cuves o the univesal calibation cuve o a otaete is to ascetain the anticipated ete opeatin ane o the saplin application o concen. Once this opeatin ane is established, an abitay selection o a point on the univesal calibation cuve is ade (see point in Fiue F-8). The coodinates o point a, point b ) Re, and point c ( ae deteined. Values o T,, and and the value o Re calculate a value o by eans o the ollowin equation: ae used to (Eq. F-5) Re F-

13 T I : T O S H E R I S L I N G O U R S E Fiue F-8. edictin calibation cuves o the univesal calibation cuve (N Re=Re, o Reynolds Nube). The aea o the ete ( ) is calculated o this value o and is used alon with the assued values ot,, and and the value o o the univesal calibation cuve to calculate a voluetic low ate by eans o the ollowin equation: (Eq. F-7) This pocedue is epeated until enouh points ae available to plot a noal calibation cuve. The entie pocedue is epeated usin new values o tepeatue, pessue, and olecula weiht until a aily o calibation cuves is plotted. O couse, this aily o cuves should backet the anticipated ete opeatin conditions o the saplin application o concen. The voluetic low ate ( ) is plotted vesus eithe the aea o the ete ( ) o the tube eadin that coesponds to the ete aea. Field opeation is eatly sipliied i the tube eadin is used. typical aily o calibation cuves is shown in Fiue F-9. F-3

14 T I : T O S H E R I S L I N G O U R S E Fiue F-9. alibation cuves pedicted o univesal calibation cuve. Notice that these cuves ae siila to the calibation cuves illustated in Fiue F-6. The dieence between the is the anne in which they wee obtained. The cuves o Fiue F-6 wee obtained by an actual laboatoy calibation un o each set o conditions illustated, wheeas the cuves o Fiue F-9 wee obtained by atheatical anipulation o data o only one calibation un. This can, o couse, save consideable laboatoy tie. In addition, it ay not be possible to ascetain, in the laboatoy, calibation data at extee conditions, paticulaly at hih tepeatues. F-4