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San Miguel, John Andre N. Date Performed: March 18, 2015

4ChE – A Date Submitted: April 17, 2015

Experiment 3

Efflux Time for a Tank with Different Exit Pipes

I. Introduction

Fluid flow in a pipe can be classified based on what is called the Reynolds

number. The Reynolds number is defined as the ratio of internal forces to viscous forces

and can be computed as a function of the pipe size and the average velocity, density and

viscosity of the fluid flowing, all of which is of equal importance [1][2]. The

dimensionless number can be calculated using the equation 1:

N ℜ=Internal FocesViscous Forces

=Dvρρ

(Equation 1)

Where: D = inside diameter of the pipe (m)

V = average velocity of the fluid (m/s)

Ρ = density of the fluid (kg/m3)

μ = viscosity of the fluid (Pa-s)

Based on the computed Reynolds number, the fluid flow can either be Laminar,

Transitional or Turbulent flow. Laminar Flow (NRe < 2100) is characterized by smooth

streamlines and ordered fluid motion. Turbulent flow (NRe >4000) is characterized by

velocity fluctuations and highly disordered fluid motion. When the fluid flow is not

under laminar or turbulent, it is said to be in Transitional state (2100 < NRe < 4000).

Laminar flow occurs when the fluid flowing is highly viscous such as oil and it flows in

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small pipes while turbulent flow occurs when the flow rate or velocity of the fluid

flowing is high [1][2].

Figure 1 shows the observations made from an experiment in determining the

type of flow. Osborne Reynolds (1842-1912), a British engineer and mathematician was

the one who conducted the experiment in distinguishing the difference between the

classifications. He injected dye steaks into the flow of a circular pipe and observed the

flow [2].

Figure 1: Observations of Reynolds’ Experiment

Figures 2 and 3 shows the diagram of the entry length along with its different

regions. The portion of a pipe where a fluid flowing enters at some point is called the

hydrodynamic entrance region. The region extends starting from the the fluid flow pipe

inlet up to the point where the velocity boundary layer merges at the centerline. The

length of this region is called the entry length (Le). The velocity boundary layer or simply

boundary layer is a hypothetical boundary surface where the effect of viscous shearing

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forces casued by fluid viscosity are felt. It also divides the fluid flow into two region, the

boundary layer region and the irrotational flow region. The boundary layer region is

where the changes in fluid velocity and the effects of visous forces are important. The

irrotational flow region (invisid core) is where fluid velocity is essentially constant rdially

and the effects of frictional forces are considered negligible. The boundary layer grows

in thickness in the entrance region and completely fills the pipe upon passing the entry

length. The region beyond the entrance region is called the hydrodynamically fully

developed region where the flow is said to be fully developed [2].

Figure 2: Simplified Pojection of Entry Length

Figure 3: Development of the velocity boundary layer in a pipe

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The entry length (Le) or the ength of the entrance region can be estimated and is a

function of the Reynolds number. For Laminar flow, the entry length can be estimated using

the formula: Le = 0.06D(NRe) where D is the dimensionless entrance length and NRe is the

Reynolds number. For turbulent flow, the formula for entry length is Le = 4.4D(NRe)1/6 [1].

The objectives of the experiments were to determine the efflux time needed to drain a

tank with a set of exit pipes with different lengths and diameters and to derive a

mathematical correlation between the efflux time and the pipe size and the tank diameter.

II. Methodology

A. Materials and Equipment

The apparatus used in the experiment were the Efflux Time apparatus,

pycnometer and the viscometer. The materials used were 1L beaker, stopwatch, taped

ruler, tape measure, ruler, and pails. The chemicals used that were of standard grade

were water and glycerol.

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Figure 4: Efflux Time Apparatus

Figure 5: Labelled Pipes 1-10

Tank

Movable Plate

Pipes

Level View Port

Pail

Water Inflow

Orifice of the Tank

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B. Preliminary

The density and viscosity of the reagents that were used were obtained. Using

the pycnometer, the density of the water and 50% glycerol-water mixture was

measured and using the viscometer, the viscosity of both reagents was measured. The

50% glycerol-water mixture were prepared already before the experiment.

The efflux time apparatus was properly inspected for any defects that can be a

source of experimental and calculation error and dimensions were measured. The tank

was ensured that it is properly mounted and was stable. The dimensions (height and

circumference) of the tank were measured using the tape measure. The pipes were

properly labelled from 1 to 10 to avoid confusion. The dimensions (height and inside

diameter) of each pipe was measured using the tape measure.

C. Experimental Proper

3.5L of water was prepared in one pail using the 1L beaker for volume

measurement. The orifice under the tank was covered with a hand and the water was

slowly poured at the top of the tank. The hand was removed and the time to empty the

tank was recorded using the stopwatch. The water flowing out of the tank was

contained in another pail. The height of the fall/flow was adjusted using the movable

plate of the efflux time apparatus. The labelled pipe 1 was properly mounted onto the

orifice of the tank. The stability and tightness of the setup was inspected. Using a hand,

the orifice of the exit pipe was covered and the tank was filled again with the same

water. After 1 minute, water was again allowed to flow out of the setup. Using the

stopwatch, the time was recorded for every 2cm height interval in the level view port

located beside the tank. The stopwatch used the lap mode to record consecutively the

Pipe Number Length (cm) Diameter (cm) Height Interval (cm) Velocity (cm/s) Time (s) Nre Le (cm)1 75.5650 0.6 2 0.1644 101.1933 11.2104 0.40362 75.5650 0.8 2 0.4984 33.3300 45.3045 2.17463 75.5650 1.1 2 1.1713 15.5667 146.3913 9.66184 75.5650 1.6 2 2.6964 6.4900 490.1816 47.05745 75.5650 1.8 2 3.5224 5.4667 720.3927 77.80246 57.4675 1.4 2 1.4350 17.6767 228.2633 19.17417 62.8650 1.4 2 1.6789 11.8467 267.0532 22.43258 88.2650 1.4 2 1.7464 10.6067 277.8024 23.33549 100.9650 1.4 2 1.8354 10.5800 291.9511 24.523910 113.6650 1.4 2 1.8652 11.2467 296.6894 24.9219

EXPERIMENTAL

Pipe Number Velocity (cm/s) Time (s) Nre Le (cm) D/d L/D1 0.1432 97.7801 9.7607 0.3514 28.11742 0.4652 30.0938 42.2856 2.0297 21.08803 1.6175 8.6554 202.1566 13.3423 15.33674 7.2402 1.9336 1316.2080 126.3560 10.54405 11.5975 1.2072 2371.8496 28.9228 9.37256 4.3636 3.2084 694.0980 58.3042 3.40647 4.3210 3.2400 687.3229 57.7351 3.72638 4.1888 3.3422 666.3006 55.9693 5.23199 4.1947 3.3376 667.2307 56.0474 5.9847

10 4.1146 3.4025 654.4929 54.9774 6.7375

THEORETICAL

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time. At the same time, the efflux time or the time to drain the tank alone was recorded.

The movable plate of the apparatus was adjusted accordingly to the height of the pipe.

This was done in triplicates and repeated for the rest of the labelled pipe (pipe 2-10).

Using the 50% glycerol-water mixture, the whole experimental run was again executed

using the same procedure. Finally, the workplace was cleaned to remove any spills.

III. Results and Discussion

Table 1: Dimensions and Experimental Data for Water only Set-up

Table 2: Theoretical Data and Ratios for Water only Set-up

Pipe Number Length (cm) Diameter (cm) Height Interval (cm) Velocity (cm/s) Time (s) Nre Le (cm)1 75.5650 0.6 2 0.1506 100.9833 2.1370 0.07692 75.5650 0.8 2 0.4356 34.4200 8.2424 0.39563 75.5650 1.1 2 1.0892 13.4933 28.3372 1.87034 75.5650 1.6 2 2.7821 5.4700 105.2768 10.10665 75.5650 1.8 2 3.3562 4.4033 142.8775 15.43086 57.4675 1.4 2 1.5031 10.5967 49.7697 4.18077 62.8650 1.4 2 1.5703 10.1567 51.9939 4.36758 88.2650 1.4 2 1.6437 9.8200 54.4255 4.57179 100.9650 1.4 2 1.8253 10.3133 60.4383 5.076810 113.6650 1.4 2 1.5448 11.0233 51.1483 4.2965

Pipe Number Velocity (cm/s) Time (s) Nre Le (cm) D/d L/D1 0.2980 46.9745 4.2292 0.1523 28.11742 0.9419 14.8630 17.8218 0.8554 21.08803 3.3669 4.1581 87.5921 5.7811 15.33674 15.0710 0.9289 570.2977 54.7486 10.54405 24.1408 0.5799 1027.6949 110.9910 9.37256 9.0830 1.5413 300.7446 25.2625 3.40647 8.9943 1.5565 297.8090 25.0160 3.72638 8.7192 1.6056 288.7003 24.2508 5.23199 8.6325 1.6218 285.8276 24.0095 5.9847

10 8.5647 1.6346 283.5842 23.8211 6.7375

THEORETICAL

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Table 3: Dimensions and Experimental Data for 50% Glycerol Solution Setup

Table 4: Theoretical Data and Ratios for 50% Glycerol Solution Set-up

Table 1 and Table 2 shows the all the data for the water only set-up which includes

experimental and theoretical values, dimensions and ratios of the dimensions. Table 3 and 4

shows the same set of data for the 50% glycerol solution set-up. Sample calculations are

presented in the appendix.

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Figure 6 and 7 shows the relation between the height interval and efflux time for the

water only set-up. Two graphs were made to separate the pipes with constant length (Pipe 1-5)

with the pipes with constant inside diameter (Pipe 6-10). As it can be seen in Figure 6, Pipe 1 has

the longest efflux time and pipe 5 has the fastest efflux time using the same height interval. This

can be attributed to the pipe diameter. Pipe 1 has the shortest diameter which is just 0.6cm

while Pipe 5 has the largest diameter which is 1.8cm. Thus, efflux time is directly proportional

with pipe diameter given the same height interval and total height. As the pipe diameter

increases, more liquid can pass though and evidently from the data, the velocity of the flowing

liquid increases as well. Figure 7 shows that as the pipe length increases, the total efflux time

decreases with the same height interval used in the level view port. But from the graph, which

contains experimental data, Pipe 10 has a longer efflux time compared to Pipe 9 which is shorter

than Pipe 10 in terms of length. This can be partly due to vorticity. Vorticity can cause the efflux

time to extend because of the gap that I creates hindering the flow of the fluid out of the tank.

Almost all Pipe set-up has a vorticity upon allowing the liquid to flow out and differs only in the

residence time and magnitude of the vorticity.

0 20 40 60 80 100 1200

2

4

6

8

10

12

14

16

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Height Interval vs Efflux Time

Pipe 1Pipe 2Pipe 3Pipe 4Pipe 5

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Figure 6: Plot of Height Interval (cm) vs Efflux Time (s) of Pipe 1-5 for Water only set-up

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

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Height Interval vs Efflux Time

Pipe 6Pipe 7Pipe 8Pipe 9Pipe 10

Figure 7: Plot of Height Interval (cm) vs Efflux Time (s) of Pipe 6-10 for Water only set-up

Figure 8 and 9 shows the relationship between the height interval and efflux time for

the 50% glycerol solution set-up. As seen in both figures, it also follows the same trend as that of

the water only set-up. The difference is efflux time. One would generalize that it would take

more time for the 50% glycerol solution to flow out of the tank than water only because it is

more viscous therefore less flow but from the data, it has less total efflux time in most pipe

numbers.

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0 20 40 60 80 100 1200

2

4

6

8

10

12

14

16

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Height Interval vs Efflux Time

Pipe 1Pipe 2Pipe 3Pipe 4Pipe 5

Figure 8: Plot of Height Interval (cm) vs Efflux Time (s) of Pipe 1-5

for 50% Glycerol solution set-up

0 2 4 6 8 10 120

2

4

6

8

10

12

14

16

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Height Interval vs Efflux Time

Pipe 6Pipe 7Pipe 8Pipe 9Pipe 10

Figure 9: Plot of Height Interval (cm) vs Efflux Time (s) of Pipe 6-10

For 50% Glycerol solution set-up

Figure 10 shows the relationship between entry length (Le) and the ratio of the tank

diameter (D) to the pipe diameter (d) for the water only set-up. Plotted data was based on Pipe

1-5 only since they differ in diameter. As seen from the graph, as the entry length decreases, the

ratio of D with d increases. Pipe 5 has the longest entry length with a value of 77.8024cm and

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has the lowest D/d ratio with a value of 9.3725. As seen in the definition of entry length,

because pipe 5 has the largest pipe diameter, there are a lot of spaces for water to flow.

Therefore, it will take more length for the flow to be fully developed. The relationship is inverse

but not proportional. Pipe 5 has thrice the diameter of Pipe 3 but does not have a D/d ratio

equal to 1/3 of Pipe 1. Figure 11 shows the relationship between entry length (L e) and the ratio

of the tank diameter (D) to the pipe diameter (d) for the 50% glycerol solution set-up. As shown,

it follows the same relationship or trend with the water only set-up. Pipe 1 has the largest D/d

ration with a value of 28.1174 and has the lowest entry length with a value of 0.0769cm. This

means that the solution attained fully developed flow immediately.

5.0 10.0 15.0 20.0 25.0 30.00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0Experimental Le vs D/d

Figure 10: Plot of Experimental Le vs D/d of Pipe 1-5 for Water only set-up

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5.0 10.0 15.0 20.0 25.0 30.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

Experimental Le vs D/d

Figure 11: Plot of Experimental Le vs D/d of Pipe 1-5 for 50% Glycerol solution set-up

Figure 12 shows the relationship between the experimental Entry length (Le) with the

ratio of the Pipe Length (L) and Tank Diameter (D) for the water only set-up. The plotted

data contains pipe 6-10 because of varying pipe length. This time, entry length has a direct

relationship with the L/D ratio. As the entry length increases, the L/D ratio also increases.

Again, the trend is not proportional when comparing the values. Figure 13 shows the same

trend but differs for Pipe 10 which showed an inverse relationship. The entry length is

decreased and the ratio increased compared to Pipe 9. Following the previous figures

containing ratios, the relationship could be inverse or direct depending on what the ratio

contains and where the constant value is placed.

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3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

5.0

10.0

15.0

20.0

25.0

30.0

Experimental Le vs L/D

Figure 12: Plot of Experimental Le vs L/D of Pipe 6-10 for Water only set-up

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

1.0

2.0

3.0

4.0

5.0

6.0

Experimental Le vs L/D

Figure 13: Plot of Experimental Le vs L/D of Pipe 6-10 for 50% Glycerol solution set-up

Figure 14 shows the relationship of efflux time to the ratio of the tank diameter (D) to

the pipe diameter (d) for the water only set-up. Both the theoretical and experimental values of

efflux time were plotted against the D/d ratio. Efflux time increases as the D/d ratio increases as

well. The experimental values are close with the computed theoretical values. Different

trendline equations were fitted and the exponential equations gave the highest R2 value for both

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the theoretical and experimental points. These equations could be used to estimate the efflux

time given the dimensions of the tank and the connecting pipe. Figure 15 which shows the same

relationship as Figure 14 but this time for 50% glycerol solution set-up has the same direct

trend. Exponential equation gave also the highest R2 value.

5.0 10.0 15.0 20.0 25.0 30.00.0

20.0

40.0

60.0

80.0

100.0

120.0f(x) = 0.173794564118716 exp( 0.233751164125676 x )R² = 0.979381391169498f(x) = 1.31045131298482 exp( 0.154936174459992 x )R² = 0.997862735663672

Efflux Time vs D/d

ExperimentalExponential (Experimental)TheoreticalExponential (Theoretical)

Figure 14: Plot of Efflux Time vs Ratio of Tank Diameter Pipe Diameter (D/d) for Water only set-up

5.0 10.0 15.0 20.0 25.0 30.00.0

20.0

40.0

60.0

80.0

100.0

120.0

f(x) = 0.0832805929512855 exp( 0.234229285257029 x )R² = 0.978528577772392

f(x) = 0.959362368896098 exp( 0.167476815100419 x )R² = 0.998025451035058

Efflux Time vs D/d

ExperimentalExponential (Experimental)TheoreticalExponential (Theoretical)

Figure 15: Plot of Efflux Time vs Ratio of Tank Diameter Pipe Diameter (D/d) for 50% Glycerol solution set-up

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Figure 16 shows the relationship of efflux time with the ratio of the pipe length (L)

and the tank diameter (D) for the water only set-up. The theoretical points shows a linear

increase while the experimental points shows a polynomial relationship. The figure also

shows the fitted equation that would give an acceptable R2 value. For the theoretical points,

the equation is linear with an R2 value of 0.954 and for the experimental points, the

equation is polynomial in 2nd degree with an R2 value of 0.7321. Figure 17 shows this time

the relationship for the 50% glycerol solution set-up. The data in the figure follows the same

trend for the theoretical points but has a different trend for the experimental points. The

experimental curve has a 2nd degree polynomial equation while the theoretical line has a

linear equation based on the acceptable R2 value. Compared with the polynomial equation

in Figure 16, the R2 value in Figure 17 is more acceptable. These equations can be used to

determine the efflux time given the length of the pipe and the tank diameter for design

purposes.

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

f(x) = 0.0544722392240489 x + 3.03284650841238R² = 0.954039877056093

f(x) = 1.2628635876142 x² − 14.0746665558721 x + 49.146052600682R² = 0.732104189334405

Efflux Time vs L/D

ExperimentalPolynomial (Experimental)TheoreticalLinear (Theoretical)

Figure 16: Plot of Efflux Time vs Ratio of Pipe Length to Tank Diameter (L/D)for water only set-up

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3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

2.0

4.0

6.0

8.0

10.0

12.0

f(x) = 0.0283360039681552 x + 1.44981015120989R² = 0.981114951214742

f(x) = 0.356480573356888 x² − 3.4579472587751 x + 18.1729322731288R² = 0.979537728460956

Efflux time vs L/D

ExperimentalPolynomial (Experimental)TheoreticalLinear (Theoretical)

Figure 17: Plot of Efflux Time vs Ratio of Pipe Length to Tank Diameter (L/D)for 50% Glycerol solution set-up

IV. Answers to Questions

1. Give practical applications of the principle of efflux time. What areas of chemical

engineering can we apply this concept?

Answer: A practical application of the efflux time is a cylindrical water tank. The tank

could

be just a simple house tank to large tanks as those seen in different industries. The

water would flow out of the tank to the connecting pipes to the shower or release

valves located in different parts of the house. In chemical engineering, the concept of

efflux time is incorporated by various industries such as chemical, food and

pharmaceutical industry. Process and Storage vessels in many industries appear in

different geometries and the time to drain these vessels of their content is known as the

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efflux time. Efflux time is important because it is part of the assessment in cases of

emergency, plant shutdown and other process considerations.

2. In cases where there is a desired efflux time, what design consideration must be

specified?

Answer: There were a lot of formulas or relationships that where derived that relates

efflux time to different parameters such as Reynolds’ number, diameter and length of

the pipe or the velocity and height of the flow stream. Based from those formulas or

relationships, considerations should be made on the working parameters for the process

such as those mentioned previously. Some other parameters could be viscosity and

density of the feed, area of the vessel, area of the connecting pipes and friction forces

present.

V. Conclusion and Recommendation

It is therefore concluded from the experiment that the total efflux time is a function

of the pipe dimensions used such as the pipe diameter and pipe length. Varying either the

diameter or length can either increase or decrease the total efflux time depending on their

relationship. Increasing the diameter will decrease efflux time as well as increasing the pipe

length. There were also other factors that could affect efflux time such as the presence of a

vortex which lengthens the total time. Mathematical correlations between the efflux time

and the ratio of the pipe size to tank diameter were derived based from the results to show

the relationship of efflux time and pipe dimensions which is important for operational

designs of tanks connected with pipes.

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For future and similar experiments concerning efflux time, it is recommended to use

other liquids especially highly viscous liquids to further prove the relationships derived from

this experiment. Also the use of different height interval and the use of a bigger volume

could help improve the experiment results.

VI. References

1. Shieh, J-C. (2007). Fundamentals of fluid mechanics chapter 8: pipe flow. Department of

Bio-Industrial Mechatronics Engineering.

2. LPS. Chapter 8: Flow in Pipes: Fluid Mechanics. pp. 322-328

3. Devi, a U., Singh, P. V. G., & Dharwal, S. J. (2011). A Review on Efflux Time 1, 9(1), 57–63.

4. Subbarao, C. V., Rao, P. S., Raju, G. M. J., & Prasad, V. S. R. K. (2012). Review on efflux

time. International Journal of Chemical Sciences, 10(3), 1255–1270.

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