Determination of the Flow Rate of Different Fluids by a Rotameter*

Transcription

1 БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ. BULGARIAN ACADEMY OF SCIENCES ПРОБЛЕМИ НА ТЕХНИЧЕСКАТА КИБЕРНЕТИКА И РОБОТИКАТА, 55 PROBLEMS OF ENGINEERING CYBERNETICS AND ROBOTICS, 55 София Soia Determination o the Flow Rate o Dierent Fluids by a Rotameter* Bogdan Stoyanov 1, Jordan Beyazov 1 Institute o Metal Science Acad. A. Balevski, 1574 Soia bogsto@abv.bg Institute o Inormation Technologies, 1113 Soia yorbe@abv.bg 1. Introduction Rotameters are devices or direct measuring o the low o moving luid. An accounting element o the value measured is an elux rotational loat, moving vertically in a conic transparent pipe as a result o the action o the luid running through the pipe. The value o the quantity measured is deined by the height o loat going up. The rotameters have ound wide application in practice due to their simple construction, distinct indications along a linear scale, possibility to measure small lows o luids and gases, including aggressive ones, suiciently wide range o measuring, etc. The principal diagram o a classic rotameter is shown in Fig. 1. The main elements o its construction are a transparent conic pipe 1 and a loat, made o a conic tip, a cylindrical body 3 and a board 4. There are some screw-shaped channels cut along the periphery o the board and due to them the loat is constantly rotating around its axis, positioning in the middle o the luid low, not rubbing the pipe walls. The loat is an elux body, which takes on the dynamic pressure o the luid low and supports constant drop o the static pressure in ront o and behind it. The luid low ascending in the conic pipe o the device lits the loat at height, at that the annular oriice S between the loat board and the inner surace o the conic pipe reaches a value, at which the orces acting on the loat are balanced. The height at which the loat is raised corresponds to a deinite value o the low rate o the passing luid. * The investigation are realized in accordance with Project No Fluid systems or technological processes control IIT BAS, and Project No Investigations in the optimization o the control o the processes o costing and cooling o metal melts, Institute o Metal Science BAS. 7 1

2 a b Fig. 1. Deining o a luid low by a rotameter The ormula or volumetric low Q (m 3 /s) o the luid running through the conic pipe o the rotameter, based on the equation or continuity o the luid medium and on Bernoulli s equation, applied or sections a a and b b, has the orm given in Fig. 1a, (1) Q S gl, m 3 /s where: is a discharge coeicient o the luid low through the conic pipe; S the least surace o the annular oriice o the luid low, corresponding to the height o the loat with respect to the initial section o the conic pipe, m ; g acceleration o the loat mass, m/s ; L loat length, m; () 1,, is loat thickness, kg/m 3 ; g luid thickness, kg/m 3. The discharge coeicient in (1) is analogical to the discharge coeicient o a luid low with a narrowing. It depends on the loat shape and on the resistance the luid low meets against the loat: (3) 7, where is a coeicient o the loat shape, determined by the relation

3 V (4), S L where V is the loat volume, m 3 ; S = D /4 surace area, (m ) o the cross section o the board o a loat with a diameter D (Fig.1), equal to the inner diameter o the conic pipe at the starting position o the loat at =0; drag coeicient o the loat, depending on the loat shape and on Reynolds criterion: (5) gv, S where is the rate o the luid low. Replacing (4) in (5) and expressing the speed o the luid low by = Q/S, and the ormula is obtained o the basic unctional relation o the parameters in a rotameter with ree movement o a loat and a pipe o an arbitrary proile (m ): (6) S Q. gl This ormula indicates that at established equilibrium position o the loat, the surace area o the smallest annular oriice between it and the pipe wall is directly proportional to the volumetric low o the passing luid. The geometric dependence o the surace area o the throttling section S on the height o loat liting is deined according to the scheme in Fig. 1 (m ), (7) S = (Dtg + tg ), where is the angle o the cone-orming line with relation to the pipe axis. A generalized static characteristic is obtained rom equations (1) and (7) or determination o the volumetric low with the help o a rotameter with a conic pipe and a loat is (m 3 /s) (8) Q K a b, where: (9) K gl, (10) a = Dtg, (11) b = tg. As seen rom equation (8), the value o the low rate o a given luid running through a rotameter with known parameters o the pipe and the loat, is a unction o the discharge coeicient, the thickness o the luid and o the loat and the height o loat liting. The discharge coeicient in this equation is analogical to the discharge 7 3

4 coeicient or throttling devices measuring the low rate in a vertical pipeline. It depends on the loat shape and on the critical Reynolds number Re. For the loat, shown in Fig. 1, that is most requently used in practice, at Re = 8000 up to Re =10000, is approximately equal to The relation o the loat rate Q and the loat travel, as seen in equation (8), is not proportional. The rotameter scale division accomplished according to this relation, is non-linear. The measuring o the low rate in this case will be diicult and imprecise due to the possibility or coarse errors in its reading. For pipes with small conicity (rom the order o 1:100) and not long loat travel, the expression (b ) in equation (8) gets insigniicant values and it can be ignored, at that the error will be within the limits o the admissible in practice values up to %. Then the generalized characteristic o the rotameter or measuring a volumetric low, will obtain the orm (m 3 /s): (1) Q Ka.. Multiplying the two sides o this equality by the thickness o the luid (kg/m 3 ), we will get the generalized characteristic o the rotameter or measuring the mass low G (kg/s): (13) G Ka ( ). For the purpose o comparison Fig. shows in a graphic orm the relations (8) and (1) o a rotameter oten used in practice or measuring the volumetric low o air. The measuring range is rom up to 0 m 3 /h, at temperature 0 о С and nominal operating pressure o kра. The rotameter pipe is with conicity o 1:100, and the loat is made o X10CrNi18.9 alloy with thickness = kg/m 3. The shape and the size o the loat are given in Fig. 1b. As seen in the diagram, at scale division o the rotameter with respect to linear equation (1), the ma ximal error o the measuring range is 0.6%, which is considerably smaller than the admissible error in low rate measuring. This gives reason or the practical use o this equation in the scale division o rotameters o the type discussed in measuring low rates o dierent luids. 0 Q, m 3 /h , mm 7 4 by equation (8) by equation (1) Fig.. Relations o the rotameter gauge according to equations (8) and (1)

5 3. Gauge transormation o a rotameter or low rate o dierent luids The rotameters oered by the companies are scaled only or measuring the low rate o water or air. Their direct application or other luids, without any corrections, leads to measuring errors. ence, the problem o gauge transormation o the rotameters proposed or measuring low rates o other luids, has got great practical signiicance. The gauge transormation o a given rotameter rom the type discussed or another luid (the low rate o which is to be measured) may be realized according to equation (1) or equation (13). Fig. 3 shows the characteristics o equation (1) or measuring the volumetric low o several dierent luids with the help o the rotameter considered. The calculations are made or nominal pressure and temperature o the luids. Using these characteristics, the scale o a rotameter or air can be transormed into a scale measuring the volumetric low o another luid. The maximum relative error in measuring the low rate o other luids can be directly determined rom the diagram, i this is accomplished without gauge transormation o the rotameter or air. It can be seen, that even or oxygen, which is with thickness close to that o air, the error would be greater than 5 %. In measuring gases, the greater the dierence is between the thickness o the gas measured and the air thickness, the greater the inaccuracy is. The gauge transormation o a rotameter or another luid may be realized also directly [7], multiplying the low rate values according to the existing scale, adjusted or a given luid (air or water), by a correction coeicient C. This coeicient may be deined using equation (1) or (13). The correction coeicient C Q o a scale or volumetric low is obtained rom (1): Q, m 3 /h , mm Ammonia Air Oxygen Argon Carbon dioxide Chlorine Fig. 3. Characteristics o the rotameter or dierent luids 7 5

6 (14), α'' ' '' C C C Q Q Q where: α' '' ' '' (15) C Q ' is a correction coeicient o luids viscosities; ' '' (16) C Q '' ' is a correction coeicient o luids thickness; ''' and ', '' are the coeicients o the low rate and thickness respectively o the luid, or which the scale is divided and the luid, or which gauge transormation is to be made. For most cases in practice, the coeicients or luids low rates the one or which the rotameter is designed or and the one subject to gauge transormation, are o very close values and in these cases the coeicient C Q may be accepted as 1 and then C = C Q. For gases, due to their smaller thickness compared to that o the loat, ormula (16) is simpliied, the expression in the brackets receiving also the value 1 and then or the correction coeicient C Q it is obtained: (17) C Q ' ''. The correction coeicient C G o a scale or mass low is obtained rom (13): '' '' ( '' ) (18) C C C G G G ' ' ( ' ), where: (19) C G '', ' '' ( '' ) (0) C G ' ( ' ). Accepting the same assumptions, as or the scale or volumetric low above mentioned, the simpliied expression or the correction coeicient C G o the scale or mass low will take the orm: (1) C G '' '. 7 6

7 I the absolute pressure P'' and the absolute temperature T'' o the luid, which low rate will be measured, dier rom the nominal values P' and T', or which the rotameter scale is divided, the correction coeicients C Q and C G will be respectively: ' P' T'' () C Q, '' P'' T' '' P'' T' (3) C G. ' P' T'' Fig. 4 is an illustration o the graphic relation o the gauge transormation by the correction coeicient () o the rotameter scale or air into a scale measuring the volumetric low o Argon at parameters dierent than the nominal ones pressure o bar and temperature o 0 o С. Q, m 3 /h , mm Air: 1bar, 15 C Argon: bar, 0 C Fig. 4. An example o the gauge transormation o a rotameter or air into a scale or measuring Argon o o low with the help o a correction coeicient. 4. Determination o the error in low measuring by a rotameter The general mean square relative error in measuring the volumetric low by a rotameter, derived or equation (1), has the orm l (4) Q S, 4 4( ) where:,,,, S l are the mean square relative errors o the low coeicient, o the annular oriice, o the loat length, o the material thickness, and o the thickness 7 7

8 o the luid being measured. The ormula determining the general mean square relative error in measuring mass low by a rotameter, is analogical. In rotameter scale division, i the values o the loat length and its thickness are careully determined, the errors rom these values can be avoided. The error S is obtained rom inaccurate positioning o the loat in its rising due to its riction with the pipe walls. For the type o rotameters considered, with a ree moving loat in a conic pipe, there is not any considerable riction practically and this error is also ignored. The error is a result o the alteration o the turbulence degree o the luid low in the rotameter. In many reerences [1, 3, 9] the value o 0.5% is accepted or this error. The error is a result o the alteration in the luid thickness due to alteration in its pressure and temperature. The value o this error is accepted to be 0.5% or luids and 0.75% or gases. Taking into consideration the measuring error, it can be accepted that in measuring a low by a rotameter o the type discussed, the general relative error will be smaller than 1.5%. R e e r e n c e s 1. B o s h n i a k, L. Measurem en ts in therm otechn ical investigation s. St. Peterburg, Machinostroenie, 1974 (in Russian).. K r e m l e v s k i i, P. Division and design o lowmeters. St. Peterburg, Machinostroenie, 1978 (in Russian). 3. P r e o b r a j e n s k i i, V. Thermotechnical measurements and devices. Moskow, Energy, 1978 (in Russian). 4. S t o y a n o v, B., V. I l i e v, J. B e y a z o v. Control and measuring o a low by a rotameter with a cone loat. Technicheska misul, ХХХІІ, 1995, No 1, 18-4 (in Bulgarian). 5. S t o y a n o v, B., J. B e y a z o v. Flow control and measurement. Problems o Engineering Cybernetics and Robotics, 43, 1995, portolio/cholera_ treatment/data_iles/ rotameter_calcs.pd Определение дебита любых флюидов ротаметром Богдан Стоянов*, Йордан Беязов** * Институт металловедения Акад. А. Балевски, 1574 София bogsto@abv.bg **Институт информационных технологий, 1113 София yorbe@abv.bg (Р е з ю м е) Выведены теоретические зависимости для определения объемного и массового дебита любых флюидов, измеряемых ротаметром из конусной трубки и поплавка. Предложен метод преградуирования скалы ротаметра, используя упрощенные зависимости дебита от поднимания поплавка в трубе и метод прямого перечисления скалы ротаметра посредством коррекционного коэффициента. Анализированны относительные ошибки при измерении дебита ротаметром. 7 8