Segmented self-siphon: Experiments and simulations S. Viridi, Novitrian, F. Masterika, W. Hidayat, and F. P. Zen Citation: AIP Conf. Proc. 1450, 190 (2012); doi: 10.1063/1.4724138 View online: http://dx.doi.org/10.1063/1.4724138 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1450&Issue=1 Published by the American Institute of Physics.
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Segmented Self-Siphon: Experiments and Simulations S. Viridi∗ , Novitrian† , F. Masterika∗∗ , W. Hidayat‡ and F. P. Zen‡ ∗
Nuclear Physics and Biophysics, Institut Teknologi Bandung, Bandung 40132, Indonesia,
[email protected] † Nuclear Physics and Biophysics, Institut Teknologi Bandung, Bandung 40132, Indonesia ∗∗ Master Program in Physics Teaching, Institut Teknologi Bandung, Bandung 40132, Indonesia ‡ Theoretical High Energy Physics and Instrumentation, Institut Teknologi Bandung, Bandung 40132, Indonesia Abstract. Observation of fluid flow in a segmented self-siphon has been conducted experimentally and a model using molecular dynamics method is used to study the phenomenon in simulation. Influence of number of parts used in vertical straight segments observed in the simulation is nearly similar to the result observed in experiment. Mismatch error between experiment and simulation results, in predicting whether water can flow in a configuration of segmented a self-siphon or not, has been found about 10.4 %. During the flow through the self-siphon water velocity decreases, especially in the semi-circle bends. Keywords: molecular dynamics, fluid dynamics, self-siphon PACS: 47.11.Mn, 47.11.-j
INTRODUCTION Siphon that can be primed by itself, which is known as self-siphon, is widely used nowadays. It can be found as part of wastewater treatment [1], as apparatus in stabilizing wastewater ponds [2], as a large system to convey water over a dam [3], in microfluidic technologies for nucleid acid extraction [4], and in application for subcritical supercritical flows [5]. There ara several mechanisms that can allow a siphon to be able to be primed by itself, such as air lower pressure [6], one-way valve [7], capilarity [8], and hidrostatic pressure [9, 10]. The last mechanism has been studied previously using experiment [11] and simulation [12], and it is also used to build a modul for educational purpose [13]. New experiment has been conducted in this work, as correction to the previous one [11], as it is guided by prediction of the simulation [12].
EXPERIMENT SETUP A self-siphon pipes with similar construction to the “hidrostatic pressure” type [9, 10] are used in the experiment. In order to study the influence of length in each segment, a segmented self-siphon is made [11, 13] as it is ilustrated in Figure 1. Seven segments, with number of 0–6, are defined to construct the self-siphon. Each segment has their own parts as given in Table 1. All segments and parts are made of glass pipe with inner diameter about 0.6 mm. Each small part, e.g. part number 11, 9, or 1, has length of 2 cm. The bending radius in segment 0, 2, and 4 is about 1 cm. Segment 2 and 4 has bending angle π and segment 0 has bending angle 34 π . A container filled with water is used as the source of fluid
that will flow through the self-siphon. Segment 1 – 5 are placed inside a container, while segment 0 is at the container side, and segment 6 is placed outside of container.
FIGURE 1. Segmented self-siphon consists of segments and parts with pipe inner diamater about 0.6 mm.
Before the segmented self-siphon is immersed in the water, inner side of the siphon pipe is dried using a hairdryer and the right end or the outlet is closed using a small rubber stopper. As it is immersed in the water the air inside the self-siphon prevent the water to enter. Observation begins as the stopper is released and the water begins to enter the self-siphon. It will be observed in which configuration a flow can occur. A configuration is determined by number of parts are used in segment 5, 3, and 1, which is labeled as (N5 N3 N1 ). Water surface position is always kept in same position, which is about at black horizontal line without number in Figure 1.
The 5th International Conference on Research and Education in Mathematics AIP Conf. Proc. 1450, 190-195 (2012); doi: 10.1063/1.4724138 © 2012 American Institute of Physics 978-0-7354-1049-7/$30.00
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TABLE 1. Segments and parts of the segmented self-siphon in experiment. Segment
Part
1 2 3 4 5 0, 6
11, 12, 13, 14 10 6, 7, 8, 9 5 1, 2, 3, 4 15
SIMULATION
FIGURE 3. Considered forces in: (a) vertical straight segments and (b) in bend segments.
Division using segments and parts of a segmented selfsiphon is defined a little bit difference in the simulation as shown in Figure 2.
In the both types of segments there is a friction force between head and the inner wall of glass pipe, which is defined as f = −8πη Δhv,
(2)
as it is modified from Poiseuille equation [14], with η and v are fluid viscosity and velocity, respectively. For the vertical type of segments there is no force in the x direction, which means that max = 0.
(3)
But there is net of force in the y direction, which gives may = −mg + ρ g(yw − y)A − f , FIGURE 2. Model of Segmented self-siphon consists of segments and parts.
A small amount of water that is located in the interface between water and air, which is labeled as head, is considered to be a particle that will be simulated using molecular dynamics (MD) method implementing Euler method. Motion of head is different in the vertical straight segments and in the bend segments. Both type of segments and considered forces are given in Figure 3. Head is considered to have form of a silinder with thickness Δh and area of A = 14 π D2 with D is inner diameter of siphon pipe. Mass of head is m and its density is ρ . It is assumed that form of the head remains the same as it moves through all types of segments. Then, a relation holds 1 m = ρπ D2 Δh. (1) 4 In the following paragraphs considered forces acted on the head in both types of segments will be described.
(4)
where yw is position of water surface. And in the bend segments there is a relation I α = −mgR sin θ + ρ gRA(yw − y) − f R,
(5)
is moment of inertia of head, R is radius with I = of bend segments, a and α are linier dan angular acceleration, and θ is angular position in the bend segments. Equation (5) is the net force (or torque) in angular direction. In the bend segment there is no net force in radial direction or mR2
maR = 0.
(6)
Equation (4) and (5) are the main equations that will be solved using previously mentioned MD method, with the relation of Equation (1) and (2). Both Equation (3) and (6) guarantee that the head moves only along the self-siphon pipe with the initial condition that velocities perpendicular to the pipe are zero, v0 = 0 and ω0 = 0. Euler method that accompanies MD method will give following relations vi (t + Δt) = vi (t) + ai Δt, i = x, y,
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(7)
ri (t + Δt) = ri (t) + vi Δt, i = x, y, ω (t + Δt) = ω (t) + α Δt, θ (t + Δt) = θ (t) + ω Δt.
(8) (9) (10)
The MD method will give ax (t) = 0, ay (t) = −g + ρ g[yw − y(t)]A − 8πη Δhv(t), aR (t) = 0, g sin θ (t) g[yw − y(t)] 8πη Δhv(t) + − . α (t) = − R RΔh mR
(11) (12) (13) (14)
And for bend segment with (x0 , y0 ) is center of circular arc, there are addition relations x(t) = x0 + R cos θ (t), y(t) = y0 + R sin θ (t), vx (t) = Rω (t) cos θ (t), vy (t) = Rω (t) sin θ (t).
(15) (16) (17) (18)
Equations (15)-(18) are used in transfering information of motion parameters between straight vertical segments and bend segments. Equations (7)-(18) are numerical equations that lead to the solution, i.e. the evolution of motion parameters (position, velocity, acceleration) with time. A pair of parametric equations x(s) and y(s) is also defined as a guideline in checking whether the numerical solutions do follow the path along self-siphon pipe. These equations are [11] ⎧ 0, ⎪ ⎪ ⎪ ⎪ s − L1 , ⎪ ⎪ ⎪ 2R2 , ⎪ ⎪ ⎨ s − L1 − L3 , x(s) = 2R2 + 2R4 , ⎪ ⎪ ⎪ ⎪ s − L1 − L3 − L5 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s − L1 − L3 − L5 −2R0 cos θ3 , ⎧ s, ⎪ ⎪ ⎪ L1 + [1 − (s − L1 ⎪ ⎪ ⎪ ⎪ ⎪ −R2 )2 ]1/2 , ⎪ ⎪ ⎪ −s − 2R2 , ⎪ ⎪ ⎪ ⎪ L1 − [1 − (s − L1 − 2R2 ⎪ ⎪ ⎨ −L3 − R4 )2 ]1/2 , y(s) = s − R2 − L1 − L3 − L5 , ⎪ ⎪ ⎪ ⎪ L5 + 2R4 + R2 + [1 − (s ⎪ ⎪ ⎪ ⎪ −L ⎪ 1 − 2R2 − L3 − R4 ⎪ ⎪ 2 1/2 , ⎪ −R ⎪ 0 sin θ3 ) ] ⎪ ⎪ ⎪ ⎪ ⎩ −s − R2 − L1 − L3 +2R0 (1 − sin θ3 ),
s1 ≤ s ≤ s 2 s2 ≤ s ≤ s3 s3 ≤ s ≤ s 4 s4 ≤ s ≤ s5 , (19) s5 ≤ s ≤ s6 s6 ≤ s ≤ s 7 s7 ≤ s ≤ s8 s1 ≤ s ≤ s 2 s2 ≤ s ≤ s3 s3 ≤ s ≤ s4 s4 ≤ s ≤ s5 , (20) s5 ≤ s ≤ s6 s6 ≤ s ≤ s7 s7 ≤ s ≤ s 8
where s1 = 0, s2 = s1 + L1 , s3 = s2 + 2R2 , s4 = s3 + L3 , s5 = s4 + 2R4 , s6 = s5 + L5 , s7 = s6 + 2R0 cos θ3 , s8 = s7 + L6 .
(21) (22) (23) (24) (25) (26) (27) (28)
For the vertical straight segment it can written as shown previously in Figure 2 that L1 = N1 w, L3 = N3 w, L5 = N5 w.
(29) (30) (31)
The variables N1 , N3 , and N5 will be used to identify a configuration of segmented self-siphon with convension (N5 N3 N1 ).
RESULTS AND DISCUSSION There are 125 conducted experiments from (000) until (444) configuration. The same number of simulations is also performed and compared to experiment results. Simulation parameters are Δh = 0.01, Δt = 10−7 , and η = 10−8 , with water as fluid. Four examples of configuration are illustrated in Figure 4. The results that we are interested in, are with which configuration, identified by (N5 N3 N1 ), there is flow throug the whole pipe of segmented self-siphon. Two typical examples, one for each condition, are given in Figure 5. Thin continue line in Figure 5 is drawn using parameteric equations in Equations (19) and (20), while the thick continue line is the results of simulation using Equations (7) and (18). When water can flow through the whole pipe of segmented self-siphon then all parts of thin line will be covered by the thick line as seen in Figure 5.b, when not then only several parts will be covered as in Figure 5.a. When water can flow the whole parts of segmented self-siphon, it is interesting to observe how fast the flow in every part of the siphon. It can be seen in Figure 6 that the flow is slower in the bend parts than in the vertical straight parts, since in this parts there is partial energy transformation from vy into vx and then back into vy . We can assume that during this transformation an ineffecient process occurs.
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(a)
FIGURE 6. Velocity in x and y direction through all parts of segmented self-siphon.
(b)
it can be concluded that [15] 0, N5 − N3 < 0 , ν= 1, N5 − N3 ≥ 1
(c)
with v = 0 means there is no flow and v = 1 means there is flow. Unfortunately, simulation results do not give the exact preditions as observed in the experiments. If mismatch error is defined as the difference of condition that is predicted by Equation (32) in simulation and experiment
(d)
FIGURE 4. Configuration examples of segmented selfsiphon: (a) (000), (b) (104), (c) (343), and (444), which are drawn using parameteric equations x(s) and y(s).
(a)
(32)
ε = |νsimulation − νexperiment |
(33)
then it can found that as N5 increasing the mismatch error also increasing linearly after N5 ≥ 1 as illustrated in Figure 8. As a whole it can be obtained that mismacth error is obtained about 10.4 %. This error can be addressed to the formulation of Equation (2) which is actualy the friction force formulation defined for bubble that rises in fluid. In our case it is a little bit different since the fluid flows through a pipe. Normally the friction force formulation is for an object that moves through the fluid. Adjustment of this parameter could be the countinuity of this study.
(b)
FIGURE 5. Illustrattion of typical path that can be follow by water when: (a) flow does not occur (441) and (b) flow occurs (421).
It is interesting to observe the whole results in simulations and experiments and then compare them. When water flows through the whole parts of segmentend selfsiphon, it will be indicated by filled square, but when not it will be indicated by empty square. Illustration of the whole configuration resutls is given in Figure 7. It can be seen that the experiment results shown more regular results than the simulation results. From the experimens
CONCLUSION A model using MD method implementing Euler method has been develop and it can be use to predict whether a segmented self-siphon configuration can flow water or not with mismatch error about 10.4 % as compared to the experiment results. This error seems dependent to the length of the fifth segment or value of N5 , whose part that is in first contact with water or where the inlet located. During the flow through the self-siphon water velocity is decreased as observed in the simulation results, which can be addressed to the friction and also the semi-circle bends of the pipe.
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(a)
(b)
FIGURE 8.
Mismatch error ε as it is influenced by N5 .
supporting this work. (c)
(d)
REFERENCES 1.
2.
(e)
(f)
3. 4. 5.
(g)
(h)
6. 7. 8. 9. 10.
(i)
11.
(j)
FIGURE 7. Configurations where flow can occur (filled square) and not (empty square) as observed in simulation (a, c, e, g, and i) and experiment (b, d, f, h, and j).
ACKNOWLEDGMENTS
12. 13.
Authors would like to thank Institut Teknologi Bandung Alumni Association Research Grant in 2010-2011 for
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