drag reduction by polymer additives in gravity driven flo · 2 environmental pollution control...

INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010

© Copyright 2010 All rights reserved Integrated Publishing Association

RESEARCH ARTICLE ISSN 09764259

452

Drag reduction by polymer additives in gravity driven flow Ch.V. Subbarao 1 , King.P2, Bhaskara Sarma.C 3 , Prasad.V.S.R.K 4

1 Department of Chemical Engineering, MVGR College of Engineering, Chintalavalasa, Viziangaram, Andhra Pradesh. Email:[email protected]

2 Environmental Pollution Control Engineering Laboratory, Department of Chemical Engineering, AU College of Engineering, Andhra University, Visakhapatnam

3 Gayatri Vidya Parishad College of Engineering for women, Madhurawada, Visakhapatnam, Andhra Pradesh

4 Anil NeeruKonda Institutes of Technology and Sciences, Sangivalasa531162, Bheemunipatnam Mandal, Visakhapatnam, Andhra Pradesh

[email protected]

ABSTRACT

A mathematical equation was developed, based on macroscopic balances, to compute efflux time for the case of gravity draining of a Newtonian liquid from a large cylindrical tank through an exit pipe located at the centre of the bottom of the tank, the flow in the pipe line being turbulent. The equation was fine tuned with the experimental data and an empirical equation for friction factor was proposed. The proposed equation will be of use in arriving at the minimum time required for draining the tank. The effect of addition of watersoluble polymer, polyacrylamide, on drag reduction was investigated for the cases of partly laminar (when draining through the cylindrical tank), partly turbulent (while draining through the exit pipe), gravity driven once through flows and % reduction in efflux time was reported. The concentration of polymer on drag reduction was also established from the experimental data.

Keywords: Drag reduction, Efflux time, Friction factor, macroscopic balances, Turbulent flow, Water soluble polymer

Nomenclature

p A Area of pipe, m 2

t A Area of tank, m 2

R D

Radius of tank, m Diameter of pipe, m

f Friction factor, dimensionless Fr Froude number, dimensionless g Acceleration due to gravity .m/sec 2 gm Modified form of acceleration due to gravity ,m/sec 2 GV Gate Valve H Initial height of liquid in the tank, m

' H Final height of liquid in the tank, m h Height of liquid in the tank at any time, m L Length of exit pipe, m LI Level indicator




453

tot m Total mass of liquid in the tank, kg

2 1 ,P P Pressures at station 1 and station 2 respectively, N/m2

Re Reynolds number in the pipe, dimensionless

act t Experimental efflux time, s

eff t Efflux time, s

eq t Efflux time based on equation, s

2 1 ,V V Velocities at station 1 and 2 respectively, m/s

exp 2 V Experimental Average velocity, m/s

2 1 ,W W Mass flow rate at station 1 and 2 respectively, kg/s

2 1 ,Z Z Elevation at station 1 and 2 respectively , m L h Z Z + + = 2 1

µ Viscosity of liquid, kg/m.s

ρ Density of liquid, kg/m3

1. Introduction

Processing and storage equipment in the chemical and allied industries are designed and fabricated in a large variety of shapes. The time required to drain these vessels off their processed liquid contents is the efflux time and this is of crucial importance in many emergency situations besides productivity considerations as mentioned by Hart and Sommerfeld (1995). During draining, the fluid friction determines the extent of drag. Drag increases many fold when flow transforms from laminar to turbulent condition. This is the case during draining of a liquid from large cylindrical tank (where the flow is laminar) through an exit pipe (when the flow in the pipe is turbulent). The extent of reduction in the drag can be correlated to efflux time. One of the methods of reducing drag is the addition of small amount of polymers, which is likely to reduce the shear along the walls of the vessel or pipeline. Besides this, polymer additions greatly reduce heat losses in thermal systems Hoyt (1990).The addition of polymers in practice during the transportation of oil from offshore platforms to shore facilities would significantly reduce the drag thus showing considerable savings in the pumping costs as reported by Johnson et al. (1984). Sellin(1988) mentioned that polymer additions in sewage pipes and stormwater drains increase the flow rates such that they do not result in overflowing Increase in the range and coherence of water jets from firefighting hoses could be achieved by the addition of polymers. However Fabula(1971). mentioned that this has not been widely employed. Drag reduction technique on a torpedo by




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allowing a jet of seawaterpolymer solution from the torpedo nose has been in use in military applications as taken from Fabula et al (1980). Unthank et al (1980) reported that the addition of low concentrations of polymers might be capable of improving blood flow through stenotic vessels without altering flow through normal vessels. Drag reducing polymers were found to control corrosion as cited by Nelosn (2003).The Polymer additions were preferentially chosen to reduce drag in systems where hardware or pumping constraints do not allow an alternative mechanical solution.

1.1 Previous work

Analytical expressions for efflux time for annular and toroidal containers through restricted orifice were available in the literature Hart and Sommerfeld (1995). Modeling and experimentation of gravity draining of a Newtonian liquid from a cylindrical tank through restricted orifice was also reported by Jouse (2003). Efflux time data from tank with exit pipes of different lengths and fittings was reported in the Reynolds number range of 40,000 60,000 as mentioned by Vandogen and Roche.jr (1999). The maximum pipe length considered was one meter. The effect of pipes and fittings on actual pipe lengths was expressed in terms of corresponding equivalent lengths. However, there is a possibility of formation of pockets of air or vapour when intense turbulence conditions were prevailing as in the case cited Vandogen and Roche.jr (1999) . Simulation and experimental work was carried out for draining a Newtonian liquid from a cylindrical tank through an exit pipe at about a Reynolds number of 6,000 for a fixed pipe length was reported by Morrison (2001). The friction factor equation used was reported to be valid for Reynolds number >5000.

Present work deals with a study on the draining of a liquid from a cylindrical tank (where the flow is essentially laminar) through an exit pipe (when the flow is assumed to be turbulent) and the analysis is based on macroscopic balances, since Bird et al (2005) mentioned that macroscopic balances provide global description of large systems without much consideration to fluid dynamics within the system. Often they are useful for making an estimate of the order of magnitude of various quantities. They are also used to derive approximate relations which can be modified with the help of experimental data to compensate for terms about which sufficient information is not available. Since the efflux time is a function of drag and for the system under consideration, flow being partly laminar and partly turbulent, a brief review of laminar and turbulent drag reduction is presented here.

Drag reduction in laminar flow using hydrophobic surfaces fabricated by silicon wafers was carried out and pressure reductions up to 40% were reported Jia et al (2004). However, the development of such surfaces might invite some theoretical complications as cited by Troung(2001). Drag reduction was studied in laminar flow by Driels and Ayyash (1976).. using Polyox WSR301 (i.e. Poly Ethylene Oxide) dissolved in water The results were expressed in the form of resonance test where the frequency of forcing pressure was varied and the amplitude of oscillation of manometer liquid was measured. The reason for drag reduction in this case was attributed to pulsed flow by Daniel Thomas (1981) . Drag reduction in turbulent flows using complaint surfaces was also reported by Koji & Choi et al (1997). Ionic or nonionic surfactants were used for drag reduction, but at high concentrations of 2000 ppm as mentioned by Truang (2001) in his review article. Moreover, Aguilar et al (2006) mentioned that chemicals like copper hydroxide were added to the surfactant to




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reduce the viscosity to that of water without diminishing drag reducing ability.Even though drag reduction by microbubbles was the cheapest as reported by Lvovetal (2005), the control and injection of microbubbles posed technical challenges as mentioned by Truang (2001). Drag reduction by using Polymer additives require very little input in the form of energy. Most of the work reported by polymer additives referred to turbulent flow. Since the pioneering work of Toms as highlighted by Lumely (1977), on drag reduction using small amount of soluble polymer additives, many experimental works were reported using homogeneous polymer solutions by Toonder et al (1995) as well as heterogeneous polymer solutions Vleggar and Tels (1997). The above review reveals that in any draining operation where a liquid from a large cylindrical tank (the flow being laminar) is allowed to flow through exit pipe (when the flow being turbulent), the prevailing drag is to be necessarily reduced. Watersoluble polyacrylamide polymer was used in the present study to investigate whether drag reduction is significant enough to warrant the use of polymer additives. The scope of the present work includes

1. development of mathematical equation for efflux time 2. verification of efflux time with the experimental data and fine tuning of friction factor

and verifying the validity of finetuned equation with experimental data 3. study of tank draining pattern with and without Polymer addition 4. verification of efflux timings with polyacrylamide solutions and 5. calculation of percentage drag reduction with polyacrylamide solutions

2. Development of mathematical equation for efflux time

The cylindrical tank along with the exit pipe (Fig.1) is filled with a Newtonian liquid. The exit pipe was provided with a valve at the bottom. The fluid is assumed to leave the pipe (station 2) under turbulent conditions. The tank is provided with a level indicator (LI) to monitor the level in the tank. Using unsteady state mass balance

Rate of mass accumulated = Rate of mass in at station 1 –Rate of mass out at station 2

( ) 2 1 W W m dt d

tot − = (1)

For the present system, 0 1 = W (Since no liquid is added at the time of draining), Equation (1) becomes

( ) 2 W m dt d

tot − = (2)

ρ π h R m tot 2 = (3)

The mechanical energy balance equation between station1 and station2 can be written as




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Figure 1: Cylindrical tank along with exit pipe

( ) 2 / 4

2 2

2 2

2

2 2 2

1

2 1 1 V D L f gZ V P gZ V P

+ + + = + + ρ ρ

(4)

Since the inlet and outlet are open to atmosphere and liquid drains very slowly, 2 1 P P = and 0 1 = V .

Assuming variation in friction factor to be negligible, for any value of h and L, noting that L h Z Z + + = 2 1 , equation (4) becomes

( ) 2 2 2

4 1 V d

L f L h g

+

= + (5)

( ) ( ) D L f L h g V / 4 1 ) ( 2

2 + +

= (6)

p A V W ρ 2 2 = (7) Where ( ) 2 4 / D A p π =

Substituting 2 V & p A in equation (7)




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( ) ( ) 2 2 4 /

) / 4 1 ( 2 D

D fL L h g W π ρ

+ +

= (8)

Substituting the value of 2 W & tot m in equation (2)

( ) ( ) ( ) 2 2 4 /

/ 4 1 2 ) ( D

D fL L h g h R

dt d π ρ ρ π

+ +

− = (9)

for incompressible liquid , ρ is constant and hence

( ) ( ) ( ) 2 2 4 /

/ 4 1 2 ) ( D

D fL L h g h R

dt d π π

+ +

− = (10)

The above equation, on integration between the limits H and 0 (complete draining) yields

( ) ( ) ( ) 2 / 4 1

p

t eff A

A g

D fL L L H t + − + = (11)

Where 2 R A t π =

( ) ( ) ( )

+

− + = 2

2 / 4 1 2 p

t eff gA

A D fL L L H t (12)

The above equation can be written as

( ) ( ) L L H g

t m

eff − + = 2 (13)

Where m g modified form of acceleration due to gravity and Eq.(13) is modified form of

Torricelli’s equation is given as

m g hence is expressed as

= g g m

( )

+

2

/ 4 1

1

p

t

A A D fL

(14)

Where g g m is proportional to ( ) 2 Fr for the system of tank with exit pipe which is similar to

that defined for draining a liquid through a restricted orifice as reported by Jouse (2003).




458

A free falling particle travels downward with constant acceleration while a free falling surface in contrast decelerates continuously and this deceleration is given by m g .

Eq13 suggests that the plot of ( ( ) L L H − + ( ) vs efflux time is a straight line having a

slope of m g 2 .

3. Experimental procedure

3.1 Part A

The apparatus used for experimentation consisted of a stainless steel cylindrical tank of diameter 0.32 m (Fig.1) provided with a level indicator and a 4 x 103 m I.D mild steel pipe directly welded to the tank at the centre of the bottom of the tank. A Gate valve (GV) provided at the bottom most point served as the outlet for draining. Valve was closed and the tank was filled up to the mark and allowed to stabilize. The stopwatch was started immediately after the opening of the bottom valve. The drop in water level was read from the level indicator. The time was recorded for a known drop in the liquid level. The measurements were continued till the water level reaches to a desired value just 0.04 m above the tank bottom. The experimental efflux time was designated as tact. The experiments were repeated to check the consistency of data.

3.2 Part B

Experimental data on efflux time were also obtained with premixed polyacrylamide (PAM) solutions of three different concentrations of 10, 20 & 30 ppm. The stock solution of PAM was prepared by dissolving 1.6 g of polyacrylamide in 400 ml of water. A small quantity of isoproponal was added to the solution to serve as disinfectant. The solution was stirred for 4 hours and then allowed to hydrate for 24 hours. The clear solution without any nonhomogeneity was diluted suitably to prepare 30, 20 and 10 ppm solutions. Since all the solutions prepared were dilute, their density and viscosity were assumed to be equal to that of water as reported by Sichee Fore etal (2005). The premixed solutions were added to the cylindrical tank and pipe system and efflux times were obtained in the manner described above in PartA. The variables covered in these studies are compiled in table1.

Table 1: List of experiments performed in the absence and presence of polymer additives for At/Ap =6400

H, m L, m H, m L, m H, m L, m H, m L, m

0.32 0.28 0.24 0.20

1 0.32 0.28 0.24 0.20

0.75 0.32 0.28 0.24 0.20

0.5 0.32 0.28 0.24 0.20

0.25




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4. Results and Discussion

For a potential flow through exit pipe, the Eq (13) can be written in the format as

( ) ( ) ( ) L H L H g

t m

eff + − + = ' 2

where

( ) 2 / 1

p t

m

A A g g

= (15)

Even though it is theoretically possible to drain the liquid from the tank completely, it was found during experimentation that complete draining was not possible due to surface tension forces and hence it was felt necessary to modify the Eq.(13) as

( ) ( ) ( ) ( ) 2 * * / 4 1 * ' 2

2

+ + − + =

p

t eff A

A D L f L H L H t ` (16)

For steadystate fully developed turbulent flow the friction factor is given as by Mc.Cabe et al (1993)

32 . 0 Re 125 . 0 0014 . 0 + = f (17)

Re in the above equation is defined as µ

ρ exp 2 Re DV

= (18)

and ( ) act t D H H R V 2

2

exp 2 4 ' −

= (19)

While computing the Reynolds number, density and viscosity of water were taken as 1000 kg/m3 and 103 kg/m.sec respectively. Calculated Reynolds Number values were found to be in the range of 3400 – 4600 which reveals that the flow in the exit pipe was turbulent.

Substituting f, V2exp and Re in Eq. (16), the efflux time teq could be obtained. Plots of

( ( ) ( ) L H L H + − + ' vs. eff t (efflux time obtained from Eq. (16) and actual efflux time ( act t ) were shown in Fig.2 for 1m length of exit pipe.

In view of the deviation between theoretical and experimental values, an iterative technique was used to obtain the following fine tuned friction factor equation

25 . 0 Re 125 . 0 0014 . 0 + = f (20)

Eq.(21) takes into account the cumulative effect of contraction coefficient, fluid motion within the tank, unsteady state condition prevailing within the tank and the roughness of the exit pipe. Plots of data in accordance with Eq.(21) were shown in Fig 3 for the case of exit pipe of length 1m.




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Figure 2: Comparison of efflux time ( eq t ) and experimental values tact for 1m length pipe

Figure 3: Comparison of efflux time (teq) and experimental values (tact) for 1 m length pipe.

The efflux time values teff ( teq and tact) obtained by incorporating ‘f’ values computed from Eqs.18 and 21 for all the cases of exit pipe lengths employed in these studies were compiled and shown in table2 and table3 respectively.




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Table 2: Comparison of efflux time for different pipe lengths (Without polymer addition) before finetuning

S.No L, m H, m (teq),s ( tact),s % error

1 0.32 1231 1715 39

2 0.28 1067 1485 39

3 0.24 900 1250 39

4

0.75

0.20 731 1040 42

5 0.32 1199 1558 30

6 0.28 1045 1367 31

7 0.24 886 1166 32 8

0.5

0.20 725 1060 46 9 0.32 1166 1600 37 10 0.28 1023 1400 37 11 0.24 874 1201 37 0.25

0.20 718 990 38

Table 3: Comparison of efflux time for different pipe lengths (without polymer addition) after finetuning

S.No L ,m H, m (teq),s ( tact),s % error

1 0.32 1668 1715 2.8 2 0.28 1460 1485 1.7 3 0.24 1248 1250 0.16 4

0.75

0.20 1034 1040 0.58 5 0.32 1504 1558 3.58 6 0.28 1310 1367 4.3 7 0.24 1110 1166 5.03 8

0.5

0.20 913 1060 16 9 0.32 1423 1600 12.43 10 0.28 1248 1400 12.17 11 0.24 1066 1201 12.61 0.25

0.20 876 990 12.99

The data revealed that in spite of finetuning of Eq.(20), a maximum deviation was observed for the cases of 0.5m exit pipe lengths with initial height of liquid level of 0.20 m and for all the initial heights of liquid levels with 0.25 m exit pipe length.




462

When the liquid is drained under potential flow conditions, the ratio 2 ) / /( 1 p t m A A g g

= is

proportional to Froude Number as reported by Jouse (2003). Therefore slow draining of liquid from the tank keeps the Froude Number constant. In case of flow with fluid friction the

Eq.(14), ( ) ( ) 2 / * / 4 1

1

p t

m

A A D fL g g

+ = is proportional to (Fr) 2 .The Froude Number remains

constant only when (1+4fL/D) is constant and this is dependent on the pipe length and friction factor. The data was presented in table4.

Table 4: 1+4fL/D for different pipe lengths

S.No H, m 1+4fL/D L, m 1 0.32, 0.28,0.24,0.20 17.6 1 2 0.32,0.28,0.24,0.20 13.5 0.75 3 0.32,0.28,0.24,0.20 9.31 0.5

Since, the efflux time was more than what is anticipated for a fully developed turbulent flow resulting in more drag, drag reduction techniques are to be explored. Drag reducing agents including Polyacrylamide (PAM) used in the present study have a tendency to increase the modified form of acceleration due to gravity gm and hence the Froude number.

4.1 Verification of efflux time for polymer solutions and development of friction factor equation for polymer solutions

The Friction Factor equation for the cases of draining solutions containing 10, 20 and 30 ppm PAM have been fine tuned on the lines similar to that for developing Eq.(21) and the following equations were obtained.

315 . 0 Re 125 . 0 0014 . 0 + = f (21)

31 . 0 Re 125 . 0 0014 . 0 + = f (22)

The limitations of friction factor equations (17), (20),(21) and (22) used for the calculation of efflux times for draining the tank in the presence of polymer PAM can be seen from the compiled data in Tables 5, 6 and 7.

Table 5: Comparison of efflux time for 10 ppm solution.

S.No H, m (teq),s (tact),s Error (%) Remarks

1 0.32 1187 1190 0.2348 2 0.28 1027 1030 0.279 3 0.24 868 900 3.67 4 0.20 702 740 5.29

L =1 m

5 0.32 1190 1300 9.17 L =0.75 m




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6 0.28 1030 1101 6.93 7 0.24 870 940 8.1 8 0.20 704 759 7.8 9 0.32 1163 1194 2.6 10 0.28 1017 1075 5.7 11 0.24 863 924 7.1 12 0.20 705 780 10.68

L =0.5 m

13 0.32 903 1330 47 14 0.28 782 1100 41 15 0.24 669 1033 54 16 0.20 545 853 56

L =0.25m

Table 6: Efflux time comparison for 20 ppm solution

S.No H ,m eq t ,s act t ,s Error (%) Remarks

1 0.32 1210 1220 0.75 2 0.28 1048 1060 1.14 3 0.24 884 912 3.2 4 0.20 717 760 6.01

L =1 m

5 0.32 1189 1325 11.40 6 0.28 1037 1135 9.45 7 0.24 878 960 9.3 8 0.20 718 780 8.6

L =0.75 m

9 0.32 1190 1270 6.74 10 0.28 1037 1120 7.9 11 0.24 878 945 7.6 12 0.20 725 820 21.33

L =0.5 m

13 0.32 918 1377 50 14 0.28 795 1140 44 15 0.24 678 1045 54 16 0.20 554 893 62

L =0.25 m

Eq.(17) was found to be valid for low concentration solutions of 10 ppm and the fine tuned Eq.(20) does not show any advantage over Eq.(17). For higher concentrations of PAM the fine tuned eqns. 21 and 22 have shown little advantage for the exit pipe lengths > 0.5 m for all initial heights. These equations were found to be valid for the case of exit pipe length of 0.25 m for all the initial liquid levels while for 0.5 m length no specific advantage could be drawn.




464

Table 7: Efflux time comparison for 30 ppm solutions

S.No H ,m ( eq t ),s ( act t ),s Error (%) Remarks

1 0.32 1242 1316 5.9 2 0.28 1068 1076 0.77 3 0.24 902 936 3.78 4 0.20 731 780 6.65

L =1 m

5 0.32 1237 1360 9.98 6 0.28 1073 1190 10.87 7 0.24 905 1002 10.71 8 0.20 733 810 10.53

L =0.75 m

9 0.32 1211 1295 6.95 10 0.28 1057 1160 9.7 11 0.24 894 970 8.5 12 0.20 733 840 22.7

L =0.5 m

13 0.32 1182 1405 18.88 14 0.28 1036 1229 18.66 15 0.24 889 1100 23.70 16 0.20 738 913 24.25

L =0.25 m

In conclusion, no generalized fine tune equation could be developed for all the cases of exit pipe lengths and heights of initial liquid levels in the tank containing PAM.

4.2 Tank Draining pattern with and without Polymer solution

The draining patterns of liquid from the tank for both the cases of with and without polymer solution were shown in Fig 4. The data on time required to reach desired level were plotted for three different concentrations of polymer solutions. The cross plot of data of Fig 4 is shown in Fig 4A which gives the effect of concentration on efflux time.

Figure 4: Study of draining pattern with and without Polymer solution




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0 200

400 600 800

1000 1200 1400

1600 1800

0 5 10 15 20 25 30 35

polymer concentration (ppm)

Efflu

x tim

e (s

ec)

Figure 4A: Influence of polymer concentration on efflux time

The plots reveal that draining was found to be effective for the lower concentrations of polymer, i.e. 10 ppm. Similar trends were observed for the experimental data at the other initial heights with different exit pipe lengths used in the present study. These results are in confirmity with the observations made by Dries and Ayyesha (1976).

4.3 Calculation of % Drag reduction

The % drag reduction for a given height of liquid level and length of the exit pipe is computed from the following equation as:

addition. polymer without /t 100 * addition) Polymer with addition Polymer without ( = reduction Drag % act (23)

Table8 gives drag reduction data for polymer solutions of three different concentrations. Drag reduction was found to be significant at low concentrations of PAM solutions. It was reduced up to 27% when 10ppm polymer solutions were used.




466

Table 8: Drag reduction for different pipe lengths and different initial heights of liquid in the tank

S.No Polymer Concentration (ppm)

% Drag reduction for H=0.32m

1 m 0.75m 0.5m 0.25m pipe 1 10 24.2 24.1 23.36 16.9 2 20 22.2 22.7 18.48 13.98 3 30 16.2 20.7 16.8 12.2

% Drag reduction for H= 0.28m 1 10 24.3 25.8 21.3 21.4 2 20 22 23.5 18.1 18.6 3 30 20.9 19.8 15.1 12.2

% Drag reduction for H= 0.24m 1 10 21.4 24.8 20.75 14 2 20 20.3 23.2 18.95 13 3 30 18.3 19.8 16.8 8.4

% Drag reduction for H=0.20m 1 10 21.3 27 22 13.8 2 20 19.1 25 16.98 9.8 3 30 17 22 14.15 7.7

5. Conclusions

Based on the theoretical and experimental analysis done on the drag reduction data, the following conclusions are drawn

• When a liquid is drained under potential flow conditions, slow draining of a liquid through an exit pipe keeps (Fr) 2 constant. The dimensionless group gm/g (Eq.(14)) does not vary with time as the free surface falls during the draining operation; subsequently the potential energy of the liquid gets converted to kinetic energy of exiting fluid and gm/g is proportional to (Fr) 2

• From the data on 1+4fL/D, it can be concluded that for a given pipe length, (1+4fL/D) remained constant and was independent of initial level of liquid in the tank.

• Whenever there is a transition from laminar to turbulent flow, drag reduction takes place. Lower the Polymer concentration, higher the drag reduction. Since the system is once through, the maximum drag reduction up to 27% only could be achieved.

• Addition of polymer solutions at 10 ppm concentration delayed the onset of turbulence in the exit pipe. The existence of such Reynolds number is a typical of drag reducing solutions and is characteristic of a specific solventpolymer system.

• Polymer additions were found to decrease friction in the flow systems in general, thus reducing drag.




467

Acknowledgements

The authors gratefully acknowledge the Principal and the Management of MVGR College of Engineering –Vizianagaram for providing the necessary infrastructural facilities for carrying out experiments. The authors also would like to thank Prof Ch.Durgaprasada Rao, Retd. Professor of Chemical Engineering, IIT, Chennai, India,for their useful discussions

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drag reduction by polymer additives in gravity driven flo · 2 environmental pollution control...

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