Comparison of Efflux Time between Cylindrical and Conical Tanks Through an Exit Pipe

Transcription

1 International Journal of Applie Science an Enineerin 0. 9, : - Comparison of Efflux Time between Cylinrical an Conical Tanks Trou an Exit Pipe C. V. Subbarao * epartment of Cemical Enineerin,MVGR Collee of Enineerin, Cintalavalasa,Vizianaaram-5005, Anra Praes, Inia Abstract: Matematical equations for efflux time urin ravity rainin of a Newtonian liqui (below its bubble point) from lare open storae tanks of cylinrical an conical sapes (Were te flow in te respective tanks is essentially laminar) trou an exit pipe of same lent an cross sectional area (te flow in te exit pipe line bein turbulent) locate at te bottom of te respective storae tanks are evelope. Te equations are ultimately simplifie an written in imensionless forms. Tese equations will be of use in arrivin at te minimum time require for rainin te contents of te respective eometries of storae vessels. To rain te same volume of liqui, te efflux time equations so evelope are compare to fin out wic of te tanks consiere rain faster. Keywors: Efflux time; Newtonian liqui; open storae tank; exit pipe; minimum time.. Introuction Processin equipment an storae vessels use in te cemical an allie inustries appear in a lare variety of sapes. Te reason for use of ifferent eometrical sapes of vessels may inclue convenience, insulation requirements, floor space, material costs, corrosion, safety consierations etc. art an Sommerfel (995) mentione tat te time require to rain tese vessels off teir liqui contents is known as efflux time an tis is of crucial importance uner many emerency situations besies prouctivity consierations []. Tey evelope expressions for efflux time for annular an toriaial containers trou restricte orifice. Jouse (00) reporte moelin an experimental work for efflux time urin ravity rainin of a Newtonian liqui trou restricte orifice []. Vanoen an Roce.jr (995) presente teir experimental work on efflux time for a cylinrical tank trou an exit pipe for te case of turbulent flow in te exit pipe[]. Te Reynols number rane consiere was between 0,000 to 60,000 an exit pipe lent was meter. owever, tere is a possibility of formation of pockets of air or vapor wen intense turbulent conitions prevail as in te case cite. Morison (00) carrie out moelin an simulation work for efflux time for a cylinrical tank trou an exit pipe at aroun Reynols number of 6,000 usin computational tools []. Te autor use a rouness value of 0 in is calculations. Te autor also reporte tat te friction factor equation use is vali for Reynols number >5000. Joye et al (00) reporte work on efflux time trou an exit pipe for bot laminar an turbulent flow conitions in te exit * Corresponin autor; subbaraocv@reiffmail.com Accepte for Publication: April, 0 0 Caoyan University of Tecnoloy, ISSN -9 Int. J. Appl. Sci. En., 0. 9,

2 C. V. Subbarao pipe [5]. Tey mentione tat turbulent flow (in te exit pipe) solutions are useful in many plant situations. Tey mae an assumption of constant friction factor in te exit pipe line. Subbarao et al (008 a, b) moele te efflux time equation for rainin te contents of cylinrical storae vessel trou exit pipe(s) for turbulent flow in te exit pipe [6, ]. Tey name te simplifie form of efflux time equation as moifie form of Toricelli equation an introuce a term calle moifie from of acceleration ue to ravity m. Tey also reporte tat urin rainin of a liqui from a lare cylinrical storae vessel trou exit pipe, Froue number remains constant an is influence by iameter an lent of te exit pipe. Tey furter mentione tat polymer aitions influence te efflux time. Efflux time can also be influence by te eometry of te vessel. In te present work, matematical equations for efflux time wile rainin a Newtonian liqui from a lare open storae tanks of two ifferent eometries trou exit pipe are erive. Te exit pipe iameter for bot te eometries consiere is assume to be same an te matematical analysis is base on macroscopic balances. Bir et al (006) mentione tat macroscopic balances are useful for initial appraisal of an enineerin problem [8]. Tey are sometimes use to erive approximate relations wic will ten be verifie wit experimental ata for terms wic ave been omitte or about wic tere is insufficient information. Te eometries consiere for evelopment of efflux time equation are Cylinrical tank wit a flat bottom Conical tank Te expressions evelope are compare to fin out wic of te tanks consiere rains faster.. evelopment of matematical equation for efflux time for ifferent eometries Suppose an open tank of iven eometry (Fiure, ) provie wit an exit pipe is plue an initially fille wit a Newtonian an incompressible liqui. Te liqui leaves te sation- wen te exit pipe is unplue. It is esire to fin te time require to rain te contents of te respective storae vessels, not te exit pipe. I GV () Fiure. Tank alon wit Exit Pipe. Int. J. Appl. Sci. En., 0. 9,

3 Comparison of Efflux Time between Cylinrical an Conical Tanks Trou an Exit Pipe R r GV Fiure. Conical tank alon wit exit pipe Writin te mass balance equation, Rate of mass in Rate of mass out Rate of mass accumulation W W ( V ) () t For te present system W 0 an W V π ence W ( V ) () t Te mecanical enery balance equation between station- an station- can be written as P V + + Z P V + + Z + f V () P For te present system, (since top an bottom are open to atmospere), At any eit, Z Z + Subbarao et al [6, ] use macroscopic balances an accounte V for bot f V an wile simplifyin te matematical equation for efflux time. Tey assume a constant friction factor wile simplifyin te equation. Teir experiments sueste tat f >>. ence nelectin allows evelopment of an alternative equa- V V V tion for efflux time. Wit te above, Eq. can be written as P Int. J. Appl. Sci. En., 0. 9, 5

4 C. V. Subbarao V fv + ( + ) () Furter, liqui rains very slowly (Since te iameter of tank is very lare compare to te iameter of te exit pipe trou wic te liqui rains), V 0& for turbulent flow in te exit pipe, te friction factor is iven by Eqn. becomes 0.09 f an 0.5 Re 0.09 f V 0.5 V ( + ) ( ) /.8 * (5) /.. evelopment of Matematical equation for efflux time for a cylinrical tank For incompressible flui, Substitutin te value of V from Eq.5 an V π in Eqn. /.8 * ( + ) ( ) (6) / t Separatin te variables an interatin between (t0) an to complete rainin 0 an (tt ) / / t 0.8* * + () / t 0.8* * / + (8) / 0.8* * + (9) Were t (0) /.. evelopment of Matematical equation for efflux time for a conical tank As sown in te Fiure, a conical tank as to be raine by means of an exit pipe (wen te flow in te exit pipe is turbulent). Applyin mass balance equation for incompressible liquis 6 Int. J. Appl. Sci. En., 0. 9, Te imensions are sown in te Fiure. Te tank is fille wit a Newtonian liqui an te liqui is raine from. It is esire to fin te efflux time require to rain te contents of te storae vessel.

5 Comparison of Efflux Time between Cylinrical an Conical Tanks Trou an Exit Pipe Int. J. Appl. Sci. En., 0. 9, ( r t π ( ) ( ) *.8 * / / π + () From Fiure for te conical tank R r () R r () Substitute r value in above equation R t π ( ) ( ) *.8 * / / π + t ( ) ( ) / / *.8 * + Te above equation upon interation between te limits ( at t0 an 0 at tt ) ives te followin equation for efflux time t * 0/ / / / () efinin in terms of imensionless roups / t (5) *.5 0 X (6) Were / / / X (). Results an iscussion Wile erivin te above equations, te contraction coefficient term is nelecte an te flui motion aroun te respective storae tanks is also nelecte. Te efflux time equations for cylinrical an conical tanks erive above are influence only by imensions of te tanks an pysical properties of te liqui, not on te Reynols

6 C. V. Subbarao number. Tey are also expecte to be muc more useful for te case of variable friction factor as well. In orer to rain te same volume of liqui, te matematical equations erive are now compare to fin out wic of te tanks consiere rain faster (i.e, wose efflux time is te lowest). wit tat of Cone Te efflux time equation for cylinrical tank an tat of conical tank are compare by takin ratio of. Tis is obtaine by iviin Eq.9 wit Eq.6.. Comparison of Efflux time of Cyliner *( + / +. * / / ) + *( + ) *( + ) 5 0 / (8) Wen tis ratio is >, it can be conclue tat conical tank rains faster tan a cyliner. Wen same volume of liqui is to be raine, i.e π πr (9) Te equation suests tat te volume of liqui in bot te tanks is influence by bot te iameter an eit of liqui in te tank. Tis ives rise to te followin two cases. (a) Wen te iameter of cyliner is same as maximum iameter of cone i.e ( R ) In tis case, Eq9 becomes an eq.8 can be written in terms of an as *(( + ) / / +.* / ) + *(( + ) ) *(( + 5 ) 0 / ) (0) Since te ratio is a function of / an / is always >0 for all values of &, a plot of vs is sown in fiure. Te plot suests tat te ratio is > for all values of / suestin te efflux time for cylinrical tank is reater tan conical tank. ence conical tank rains faster tan a cylinrical tank wen te maximum iameter of cone is same as tat of cyliner. Te plot also suests tat te ratio increases as / increases. / can be increase by keepin constant (Fiure 5) an increasin or by keepin constant an ecreasin (fiure 6) 8 Int. J. Appl. Sci. En., 0. 9,

7 Comparison of Efflux Time between Cylinrical an Conical Tanks Trou an Exit Pipe Fiure. Efflux time ratio for ifferent values of ( /)>0)) ( ) Fiure. Efflux time ratio for ifferent values of ( /)>0, fixe)) ( ) Fiure 5. Efflux time ratio for ifferent values of ( /)>0, fixe)) ( ) Int. J. Appl. Sci. En., 0. 9, 9

8 C. V. Subbarao Fiure 6. Efflux time ratio for ifferent values of ( /)>0)) ( ) (b) Wen te eit of liqui in bot te tanks is same i.e In tis case, Eq.0 becomes () Eq.8 can be written as / * / / () 0 / * ( + ) + * ( + ) * ( + ) 5 A plot of vs is sown in fiure 6. From te plot, it can be observe tat te ratio is > suestin rainin time for cyliner is reater tan tat of a cone uner turbu- lent flow conitions in te exit pipe. (c) Wile erivin te above equations for efflux time, (eq. an ), te contraction coefficient term is nelecte. Te value of contraction coefficient for cyliner is iven as.5 by Joye et al (00) were as Mott [9] reporte a contraction coefficient of 0 for cone. Tis will furter increase te efflux time ratio. ence cone is expecte to rain muc rapily tan cyliner as preicte by te above equations.. Conclusions Some of te conclusions of te present work are (a) For rainin te same volume of liqui trou exit pipe of same iameter, efflux time for cone < efflux time for cyliner. Te efflux time ratio is influence only by /. (b) Te efflux time ratio is influence by bot iameter as well as eit of liqui in te storae vessels. (c) Te plots (Fi. & 6) also suest tat efflux time ratio is more for te case of equal iameters of tanks tan equal eits of liqui in te tanks. Te teoretical values obtaine are to be verifie experimentally to fin out te exact ratio of efflux times for te eometries consiere. 0 Int. J. Appl. Sci. En., 0. 9,

9 Comparison of Efflux Time between Cylinrical an Conical Tanks Trou an Exit Pipe Nomenclature iameter of cyliner, m iameter of cone corresponin to eit iameter of exit pipe, m Acceleration ue to ravity, m/sec eit of liqui in te tank at any time, m, eit of liqui at any time t for cyliner an cone, m, eit of liqui at t0 for cyliner an cone, m f Friction factor, imensionless ent of exit pipe, m Re Reynols number, imensionless t,t Efflux time for cyliner an cone, sec V Volume of liqui, m V &V Velocities at station- an, m/sec P & P Pressures at station an station, N/m R Raius of cyliner, m R Raius of cyliner corresponin to eit, m r Raius of cone corresponin to eit, m W & W Mass flow rate at station an, k/sec Z & Z µ Elevations at &, Z Z +, m ensity of liqui, k/m Viscosity of liqui, k/m.sec imensionless time for cylinrical tank, sec imensionless time for conical tank, sec References [ ] art, W. Peter an Sommerfel, T Expression for ravity rainae of annular an Toroial containers. Process Safety Proress,, : 8-. [ ] ibii, J. N. 00. Mecanics of slow rainin of lare tank uner ravity, American Journal of Pyscis,, : 0-0. [ ] Vonoen,. B. an Roce, Jr. E. C Efflux time from tanks wit exit pipes an fittins. International Journal of Enineerin Eucation, 5: 06-. [ ] Morrison, K. R. 00. Moelin an computation tecniques for flui mecanics experiments. International Journal of Enineerin Eucation, : [ 5] Joye,.. an Barret, B. C. 00. Te tank rainae problem revisite: o tese equations really work? Canaian journal of Cemical Enineerin, 6: [ 6] Subbarao, C. V., Kin, P., an Prasa, V. S. R. K, 008a. Effect of polymer aitives on te mecanics of slow rainin of lare tank uner ravity. ARPN Journal of Enineerin an Applie Science, : [ ] Subbarao, C. V., Kin, P., an Prasa, V. S. R. K, 008b. Effect of polymer aitives on te ynamics of a Flui for once trou system, International Journal of Flui Mecanics Researc, 5: -9. [ 8] Bir, B. R., Stewart, W. E., an itfoot, E. N Transport Penomena. secon E., Wiley Inia. [ 9] Mott, R Applie flui mecanics. ixt eition. Int. J. Appl. Sci. En., 0. 9,