Mechanics of slow draining of large cylindrical tank under gravity

Transcription

1 Mechanics of slow drainin of lare cylindrical tank under ravity Ch.V. Subbarao 1, Divya 2, D. Appala Naidu 2 and P. Kin 2 1 Departent of Cheical Enineerin, MVGR Collee of Enineerin, Chintalavalasa, Vizianaara , Andhra Pradesh, India. 2 Departent of Cheical enineerin, Andhra University, Visakhapatna, A.P, India. Correspondin Author Eail:subbaraochv@rediffail.co Abstract The results of experients on efflux tie durin slow drainin of Newtonian liquid under ravity fro a lare cylindrical open storae tank throuh an exit pipe are presented in this paper. The atheatical equation developed by one of the authors for efflux tie for sinle exit pipe is used to verify the experiental values. The variables studied are dia. of storae tank, dia. of exit pipe, initial heiht of liquid in the tank. It has been observed that durin drainin, Froude nuber reains constant and is independent of initial heiht of liquid in the tank. Froude nuber is decreased with increased exit pipe lenth and is also found to be independent of dia. of storae tank. Keywords: Efflux tie, cylindrical tank, exit pipe, Froude nuber 1. Introduction Different eoetries of storae tanks are bein used by process and cheical industries. Selection of a particular eoetry of tank is based on insulation requireents, floor space, corrosion requireents, aterial costs etc. The tie required to epty the liquid content fro the storae vessel is known as efflux tie [1] and this is iportant under eerency situations as well as fro productivity point of view. Joye and Barret [2] reported atheatical equations for efflux tie durin drainin of a Newtonian liquid (below its bubble point) fro a cylindrical tank throuh an exit pipe (for the case of turbulent flow in the exit pipe). The authors assued constant friction factor while developin the atheatical equation. The authors used contraction coefficient value of 1.5 for coparin the experiental values with the odel values. Subbarao and co-researchers [3,4,5] ade the sae assuption of constant friction factor and developed equation for efflux tie. The authors stated that is proportional to (Fr) 2, where is odified for of acceleration due to ravity and Fr is the Froude nuber. The siplified equation for efflux tie is naed as odified for of Torricelli equation. The authors used Polyacrylaide solutions of different concentrations and reported that polyer additions decrease the efflux tie and hence increase the Froude nuber

2 Santosh Kuar and other researchers [6] developed equations for efflux tie for two exit pipe systes for ravity drainin of a Newtonian liquid throuh two exit pipe syste. The authors considered two exit pipes each of 4X10-3 dia.the authors also stated that durin drainin, Froude nuber reains constant. Subbarao and other researchers [7] also perfored dra reduction experients for two exit pipe syste. The extent of increase in Froude nuber is observed to be ore for two exit pipe systes copared to sinle exit pipe syste in the absence and presence of polyer solutions. The authors also reported the ratio of cross sectional area of tank to pipe as 1600 below which addition of polyer solutions does not brin reduction in efflux tie. Theoretical equations for efflux tie for sphere and cylinder are developed by Reddy and Subbarao [8] for the case of turbulent flow in the exit pipe. For drainin the sae volue of the liquid, the equations so derived are copared to find out which of the tanks considered drain faster and reported faster drainin tie for sphere copared to cylinder.. Subbarao [9] copared efflux ties between cylinder and cone and showed that the efflux tie for cone is less than that of a cylinder. Hence Froude nuber for cone is hiher than that of a cylinder. Gopal Sinh and other researchers [10] used Polyacryl aide and polythene oxide as polyers in their studies on efflux tie throuh exit pipe for both lainar and turbulent flow in the exit pipe and reported optiu concentrations for both lainar flow and turbulent flow in the exit pipe. They stated that the reduction in efflux tie and hence increase in Froude nuber is influenced by type of polyer used. All the above works increased Froude nuber by Addition of polyer solutions or Chanin the eoetry of the vessel or Increasin the cross sectional area by providin two exit pipes. However, cross sectional area can also be increased by providin a sinle exit pipe of larer diaeter. Present work considers efflux tie easureent by providin 4X10-3 and 6x10-3 dia sinle exit pipes and copares the extent of increase in Froude nuber when the exit pipe dia is increased. The scope of work includes Efflux tie easureent usin water as Newtonian fluid Verification of these experiental values with theoretical odel reported in the literature. Calculation of Froude nuber for both exit pipes considered. 2.0 Materials and ethods 2.1 Description of apparatus The scheatic diara of the apparatus and the equipent are shown in Fi 1. The equipent used consisted of known diaeter tank riidly placed on a steel structure. A ild steel pipe of known diaeter (d) is welded to the tank at the centre of the botto of the tank, served as an exit pipe. A ate valve (GV) provided at the botto (1), 10,

3 ost point of the exit pipe, served as control valve for drainin of liquid fro the tank. A transparent plastic tube (LI) provided to the tank served as level indicator durin drainin operation. Efflux ties are easured with a stop watch of 1 sec accuracy. The lists of experients perfored are shown in table-1. Table-1 : List of experients S.No Dia. of tank, Lenth of exit pipe, Initial heiht of liquid in the tank, ,0.75,0.5, ,0.20, ,0.75,0.5, , 0.32,0.26, ,0.75,0.5, ,0.40,0.34, Experiental Procedure Gate valve (GV) was closed and the tank was filled up to the ark and allowed to stabilize. The stopwatch was started iediately after the openin of the botto valve. The drop in water level was read fro the level indicator. The tie was recorded for a fall in the liquid level to a predeterined level of above the tank botto. The experiental efflux tie is desinated as tact. The experients are repeated and the easureents were taken to check the consistency of data. 3. Results and discussion 3.1 Verification of efflux tie The followin efflux tie equation developed by Subbarao et al (Subbarao et al, 2008) is used to verify the efflux tie data

4 t eff 2 H L H' L (1) teff is the efflux tie, is odified for of acceleration due to ravity iven by 1 4 f L d 1 K c At Ap 2, where f is the friction factor, L lenth of the exit pipe, d, diaeter of exit pipe, and A p are cross sectional area of tank and exit pipe respectively, K c is the contraction coefficient whose values is reported as 1.5 by Joye and Barret [2]. This value is used for verifyin the validity of the atheatical odel. The followin equations for friction factor for turbulent flow reported in the literature [11, 12] are used to calculate the friction factor At f 0.25 Re (2) f Re (3) To verify whether the flow is turbulent or not, Reynolds nuber is calculated as Re DV 2 exp / (4) V 2exp is obtained usin the experientally easured data as V 2exp D H H / d t 4 act (5) Where D is the dia. of tank, d is the dia. of exit pipe, H is the initial heiht of liquid in the tank, H is the final heiht of liquid in the tank. The density and viscosity of water as a liquid in the present study are assued to be equal to 1000 k/ 3 and 0.001k/.sec respectively. The Reynolds nubers for all the cases considered is calculated and found to be in turbulent flow only. By substitutin V 2exp, K c, f in eq.1 to ives t eff. The plot of H L H' L vs t eff is shown in Fi.1 for a 0.3 dia. tank, dia exit pipe and 1 exit pipe lenth. The efflux tie obtained usin eq.2 and eq.3 is represented as teq.2 and teq.3 respectively while actual efflux tie is shown as tact. (1), 10,

5 Efflux tie, sec e-περιοδικό Επιστήμης & Τεχνολογίας tact teq.2 teq.3 Fi.1 : Coparison of efflux tie (dia.of tank=0.3, dia.of exit pipe=0.004 and lenth of exit pipe=1). Maxiu deviation of 23% is observed between experiental efflux tie and teq values. However, the Eq.2 does slihtly better for coputin efflux tie copared to Eq.3. The trend for other exit pipe lenths and other diaeters of tanks ad exit pipes is shown in the followin tables (table-2 to table-7) Table-2 : Efflux tie coparison for Dia. of tank =0.30,exit pipe dia=4x10-3 S.No Initial heiht of liquid in the tank, t act, sec teq.2 ( sec)/ teq.3 (sec) % Error Rearks / /26 Exit pipe lenth / /23 = / / /525 19/ / /24 Exit pipe lenth / /20 = /796 15/ /544 14/ / /20 Exit pipe lenth / / /853 7/ /592 4/5 =

6 Table-3 : Efflux tie coparison for Dia. of tank =0.34, exit pipe dia=4x10-3 S.No Initial heiht of liquid in the tank, tact, sec teq.2,sec /teq.3,sec % Error Rearks / /24 Exit pipe lenth / /17 = / / /751 21/ / /20 Exit pipe lenth / /21 = / / /762 19/ / /22 Exit pipe lenth / /22 = / / /785 21/ / /17 Exit pipe lenth / / / / /829 16/17 =0.25 Table-4: Efflux tie coparison for Dia. of tank =0.37, exit pipe dia=4x10-3 S.No Initial heiht of liquid in the tank, t act, sec teq.2, sec / teq.3,sec % Error Rearks / /26 Exit pipe lenth = / / / / /771 24/ / /19 Exit pipe lenth / /19 = / / /764 13/ /21 Exit pipe lenth /15 = / / / /25 Exit pipe lenth / / / / /683 22/23 =0.25 (1), 10,

7 Table-5 : Efflux tie coparison for Dia. of tank =0.30,exit pipe dia=6x10-3 S.No Initial heiht of liquid in the tank, t act, sec teq.2,sec /teq.3,sec % Error Rearks /441 11/11 Exit pipe lenth = /358 12/ /272 10/ /183 4/ /455 12/12 Exit pipe lenth /370 12/12 = /283 14/ /193 18/ /475 5/5 Exit pipe lenth /389 7/7 = /299 8/ /205 11/ /523 1/1 Exit pipe lenth / = /337 1/ /234 1/2 Table-6 : Efflux tie coparison for Dia. of tank =0.34, exit pipe dia = 6X10-3 S.No Initial heiht of liquid in the tank, t act, sec teq.2, sec/teq.3 % Error Rearks /587 5/5 Exit pipe lenth = /448 5/ /341 5/ /230 4/ /600 1/2 Exit pipe lenth = / /351 3/ /238 1/ /625 2/2 Exit pipe lenth = /482 2/ /370 3/ /253 3/ /675 1/1 Exit pipe lenth = / / /409 2/ /283 2/

8 / e-περιοδικό Επιστήμης & Τεχνολογίας S.No Initial heiht of liquid in the tank, t act, sec teq.2, sec /teq.3,sec % Error Rearks /856 6/6 Exit pipe lenth = /696 5/ /529 2/ / / /3 Exit pipe lenth = /712 3/ / / /371 1/ /901 0 Exit pipe lenth = / / / / / / / /0.6 Exit pipe lenth = /799 0/ /624 1/ /436 3/3 Table-7 : Efflux tie coparison for Dia. of tank =0.37, exit pipe dia=6x10-3 It can be seen fro the tables (Table-2 to table-7) that the deviation is very less when the exit pipe dia. is 6X Variation of Froude nuber with initial heiht of liquid in the tank: Froude nuber is defined by Subbarao and other researchers as L 1 4 f K d 2 At Ap c where is proportional to (Fr) 2 The followin plot shows variation of Froude nuber with initial heiht of liquid in the tank for 0.30 dia. tank while keepin the lenth of the exit pipe at 1. [3] y = 3E-11x + 2E-09 R² = 0,2819 Initial heiht of liquid in the tank, Fi.2 : Variation of / with initial heiht of liquid in the tank. (1), 10,

9 / e-περιοδικό Επιστήμης & Τεχνολογίας The plot suests that Froude nuber is independent of initial heiht of liquid in the tank. However, when the lenth of exit pipe is increased, variation of Froude nuber with lenth of exit pipe is shown below. y = -2E-09x + 4E-09 R² = 0,8928 Lenth of exit pipe, Fi.3 : Variation of efflux tie with lenth of exit pipe It can be seen fro the plot that as the lenth of exit pipe increases, Froude nuber decreases. The variation of for different dia. of tanks and exit pipes are shown in the followin tables (tables-8 and table-9). Table- 8 : Variation of S.No Initial heiht of liquid in the tank, for different dia.of tanks. / X10 9 Lenth of exit pipe, Rearks ,0.26,0.20, Dia. of storae tank = ,0.26,0.20, and Dia. of exit pipe ,0.26,0.20, =4X10-3, ,0.26,0.20, ,0.32,0.26, Dia. of storae tank =0.34 and Dia. of exit pipe =4X ,0.32,0.26, ,0.32,0.26, ,0.32,0.26, ,0.40,0.34, Dia. of storae tank = ,0.40,0.34, ,0.40,0.34, ,0.40,0.34, and Dia. of exit pipe =4X

10 Table- 9 : Variation of S.No Initial heiht of liquid in the tank, for different dia.of tanks. / X10 8 Lenth of exit pipe, Rearks ,0.26,0.20, Dia. of storae tank ,0.26,0.20, =0.30 and Dia. of exit ,0.26,0.20, pipe =6X10-3, ,0.26,0.20, ,0.32,0.26, Dia. of storae tank =0.34 and Dia. of exit pipe =6X ,0.32,0.26, ,0.32,0.26, ,0.32,0.26, ,0.40,0.34, Dia. of storae tank ,0.40,0.34, ,0.40,0.34, ,0.40,0.34, =0.37 and Dia. of exit pipe =6X10-3 It can be seen fro the tables (Tables- 8 and 9) that Froude nuber is independent of dia. of storae tank and depends only on the dia and lenth of the exit pipe. It can also be seen that as the dia. of exit pipe increases, Froude nuber also increases. 4. Conclusions Soe of the conclusions of the above study are a. The difference in experiental values and theoretical values of efflux tie usin a contraction coefficient of 1.5 leads to a axiu deviation of 25% for 4X10-3 dia exit pipe where is the axiu deviation is 11% in case of 6X10-3 dia exit pipe. b. Froude nuber reains constant and is independent of initial heiht of liquid in the tank. It is also indepedent of dia. of storae vessel when drained by exit pipe of sae dia. c. Froude nuber increases when the lenth of exit pipe is decreased. d. Froude nuber also increased with increased dia. of exit pipe. 5. Acknowledeents The authors are thankful to the Principal Dr. K.V.L.Raju and the anaeent of MVGR Collee of Enineerin, Vizianaara for providin the necessary infrastructural facilities. The authors are also thankful to the head of the departent of Cheical Enineerin, Andhra University for providin the necessary support. The authors would like to thank Professor Ch. Dura Prasada Rao, Retired Professor, IIT-Chennai and presently workin as Professor at MVGR Collee of Enineerin, Vizianaara, C.Bhaskara Sara of Gayatri Vidya Parishad Collee of Enineerin and Prof.V.S.R..K.Prasad of ANITS-Visakahpatna for their useful discussions. (1), 10,

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