ENVS440/540, 2014 Vol 1:1-10 Pretend Journal of Environmental Modeling
Fred G. R. Watson
Division of Science & Environmental Policy, California State University Monterey Bay, Seaside, CA, USA.
Abstract
I used measurement and modeling of an experimental apparatus to explore the physical processes governing leakage from tanks through outlet pipeworks. I measured water level in a 6.6 L acrylic tank drained through a bulkhead by vinyl tubing with an inline ball valve. I modeled the system using Stella software with a single state variable, single parameter model, and various functional forms for the drainage term. Each version of the model was calibrated to fit the data by adjusting a single rate parameter until the observed and modeled water height time series matched best. I obtained a better fit using a square-root drainage function, as compared with linear and threshold-linear functions.
The square-root model is a satisfactory model for the classical concept of the ‘leaky bucket’; but other models may be more accurate.
Introduction
The ‘leaky bucket’ is a common concept underlying many hydrological models. A storage volume such as a unit of soil or a stream channel reach contains water, and the water ‘leaks’ from the volume at a rate that is proportion to the hydraulic head, H (m), in the volume.
Goals
My general goal was to understand the nature of real leaky buckets and the factors governing the rate at which they leak.
Postulate
I postulated that leakage, Q out
(m 3 s -1 ), is proportional to the head, since the head determines the pressure at the leakage orifice, which is the physical quantity that is expected to determine the outflow rate. Specifically,
I postulated that the leakage is according to one of three possible functions:
1.
A linear function:
Q out
= r H where r (m 3 s -1 m -1 ) is a rate parameter related to the size of the orifice through which the leak occurs. This function is perhaps the simplest choice.
2.
A linear function with a maximum:
Q out
= min( Q out,max
, r H ) where Q out,max
(m 3 s -1 ) is a parameter expressing the maximum possible outflow rate. This function extends the first one, but recognizes a simple upper limit perhaps due to friction in the orifice.
3.
A square root function:
Q out
= r H 0.5
where r (m 3 s -1 m -0.5
) is a rate parameter in terms of the square root of the height. This also recognizes a limit to how much the flow rate can increase as the height becomes large, but it does so according to a continuously sloping curve, with no hard upper limit. It is somewhat consistent with pipe flow theory
(Flowdude 1923).
Laboratory methods
I built a leaky bucket and measured its leakage in order to provide data with which to calibrate a model and evaluate which of the above three functional forms best describes a real physical system. The ‘bucket’ was a vertical acrylic cylinder 183 cm high, 70 mm in internal diameter, 6 mm in thickness, and closed at the lower end (Fig 1.). The side wall of the cylinder at the
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ENVS440/540, 2014 Vol 1:1-10 Pretend Journal of Environmental Modeling lower end was drilled to accommodate an improvised bulkhead fitting leading through 5 cm of vinyl tubing to a ball valve and then another 50 cm of vinyl tubing to a free-fall outflow into a waste storage bucket, with various poly-vinyl chloride hose-barb inter-connects in between the segments of the system. The internal diameter of the smallest orifice in the outflow system was ???
mm, and the ball valve was nominally “half inch”.
A single drainage experiment was run. The water level was initially filled close to the top, while the valve was closed. Then, the valve was opened and the water level was measured at equal time intervals of 3.0 seconds until the flow ceased. Water level measurements were made by eye using a centimeter scale was glued to the side of the tank, and the timing of these measurements was facilitated by a second observer calling out the equal time intervals while looking at a stop watch that measured seconds.
Modeling methods
A simulation model was implemented in the Stella software package (ISEE Systems 2007). The model had a single state variable representing the volume of water, V (m 3 ), in the tank. The head in the tank was computed from the geometry of the tank as:
𝐻 =
𝑉
𝐴 where A (m 2 ) was the measured internal cross-sectional area of the tank, assumed to be constant throughout the tank. A single outflow was simulated, according to the postulates described earlier. The model was run for 42 seconds of simulated time, at a time step of 0.25
seconds. Euler’s method was used to approximate the continuous-time reality using a series of discrete time
steps. The model structure is illustrated in Figure , and
all model variables are summarized in Table 1.
Model calibration
The model had one free parameter, r (m 2 s -1 ). The value of r was calibrated to achieve the best fit of the model predictions to the observed water height data.
Calibration was performed manually in Stella and
Microsoft Excel by adjusting parameter values and running the model in Stella, copying the results to
Excel, looking at a plot comparing predicted and
Figure 1. Experimental apparatus used to obtained observations of the outflow from a ‘leaky bucket’.
Figure 2. Structure of model for simulating leakage from a bucket. The box denotes a state variable representing material storage, the thick arrow denotes a material flow, the circles represent parameters and internal working variables, the thin arrows represent dependencies, and the cloud represents a material sink. observed water height over time, and repeating these steps until the agreement between the predicted and observed time series was qualitatively optimal.
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ENVS440/540, 2014 Vol 1:1-10 Pretend Journal of Environmental Modeling
Table 1. Summary of all variables in a model simulating leakage from a bucket.
Variable name Type
Water volume Material storage / state variable
Water height
Tank area
Internal working variable
Measured parameter
Outflow
Outflow rate
Material flow
Free parameter
Results
Symbol
V
H
A
Q out
R
(Initial) value
1.73 / A
Varied

(0.07/2) 2
Varied
0.002 additional parameters to account for non-full pipe flow conditions.
Units m 3 m m 2 m 3 s -1 m 3 s -1 m -1
A single time series of 14 observations of water height was successfully obtained. The apparatus remained
stable and no blockages occurred (Figure ). There was
some minor inaccuracy in the correspondence between nominal measurement times and the actual time at which the water height scale was read, which we expected would result in a small amount of unbiased and apparently random variation in the observations.
Each of the three outflow functions lead to a reasonable correspondence between observations and modeled water heights, but the square-root function clearly led to the most accurate predictions. The results from the square-root version of the model are shown in Figure 3.
References
ISEE Systems. 20XX. Stella software. Version 9.0.3.
Lebanon, NH, USA.
Flowdude, P. 1923. A study on pipe flow.
1:23-45.
Acknowledgements
J. Pipes ,
I am grateful to the students in ENVS 440 and ENVS
540 at CSUMB in Fall 2010 for being my lab slaves.
The final modeled outflow curve was slightly more curved than the observed outflow curve, indicating that the square-root function does not perfectly represent the observed system. The observed outflow was faster at lower water heights than was estimated by the model. This may have been due to a change in the pipe flow hydraulics as the flow state transitions from a fullpipe situation to a partially filled situation at least for some part of the outflow apparatus.
200
150
100
Modeled
Observed
Conclusions 50
I postulated that leakage from a bucket occurs in either linear, threshold-linear, or square-root proportion to the hydraulic head in the bucket. The square-root postulate was best supported. Specifically, I found that leakage from a bucket through tubing with an inline valve was well approximated by a one-parameter square-root function of the height of water (and thus the pressure) in the tank from which the leakage was occurring. The square-root model appeared to perform better than linear, or threshold-linear models. A superior model to the square-root model may exist, perhaps requiring
0
0 10 20 30
Time (s)
40 50
Figure 1. Comparison between observed and modeled water height during drainage of water from a ‘leaky bucket’, showing a good fit between model and data.
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