Fluid Flow with Logger Pro

Contents

• BODY OF ARTICLE
• Streamline Fluid Flow
• Exponential Decay Experiment
• Secular Equilibrium Experiment
• Series Decay-Chain Experiment
• Further Analysis
• Conclusion
• REFERENCES
• About the Author
• FIGURES

Graphical analysis of experimental data that exhibit exponential behavior is typically postponed at many institutions until students are able to understand the theory underlying the concept of radioactive decay or of RC time constants in ac circuits. In 1960 Smithson and Pinkston¹ described a laboratory exercise that uses the flow of water from a vertical column through a long horizontal capillary tube as a source of data that models radioactive decay. Many institutions have used this experiment simply as an early introduction to exponential behavior without reference to radioactive decay or ac circuits. Greenslade² recently described a modification of this experiment to demonstrate the concept of secular equilibrium in radioactive decay. This paper presents results of similar experiments, but visual measurements are replaced in this work by data obtained with modern sensors interfaced to a computer. Experiments are described from simple exponential decay to an analogue of the complex interactions of three nuclides in a radioactive-series decay chain.

Laminar flow of a viscous fluid through a horizontal capillary tube is described by Poiseuille's equation,³ Q = (P₁ – P₂)r⁴/8L, where Q is the volume flow per unit time, r is the radius of the tube of length L, (P₁ – P₂) is the pressure difference between the two ends of the tube, and is the coefficient of viscosity of the fluid. When one end of the tube is placed at depth h below the surface of the open vertical column of fluid of density cross section A, the pressure difference is simply gh when the other end of the tube is open to atmosphere. By applying the continuity equation where the volume flow rate at the top of the vertical column is expressed as Av = Adh/dt, where v is the instantaneous speed, it is readily found that the height of the column is given by h = h₀exp(–t), where h₀ is the column height above the tube at a reference time of t = 0 and

A hollow coffee stirrer, used as the capillary tube, was inserted into the side of a 710-mL plastic water bottle and fastened with plumber's putty. This was the equipment suggested at a workshop attended by one of the authors (TAW) at the Summer 2001 AAPT Meeting in Rochester, N.Y. However, instead of visual measurements of the height of the fluid reservoir as a function of time, we suspended the bottle on a force probe connected to a Vernier⁴ interface and accumulated the data with the Vernier Logger Pro software. Instead of height, we measured the corresponding weight as a function of time and fit the data to

where W_1BG is the weight of the bottle and fluid remaining after the flow stops. The subscript 1 anticipates use of this equation in a later section. The original bottom of the plastic bottle was cut off, the bottle was inverted, and the tube was placed near the bottom of the uniform diameter portion of this reservoir. Another piece of stirrer was used at the open end to maintain a constant geometry of the flexible plastic when supporting the amount of filled fluid. A data-sampling rate of one point/s was used for all experiments.

Equation (2) is exactly analogous to radioactive decay with a constant background added. Figure 1(a) presents both data and a graph of Eq. (2) using the best-fit values for the adjustable parameters of W₁₀, ₁, and W_1BG. For instructional and analysis purposes the data were imported into a spreadsheet (Microsoft Excel was used in this study) and sliders were constructed for each of the three parameters so students may study how a variation of each parameter affects the mathematical fit to the data. Students are asked to first find the best visual fit by manipulating the sliders, and finally to use Excel's Solver⁵ add-in tool to obtain the final best fit shown in Fig. 1. Solver is an optimization tool that we used to minimize the sum of squares of differences between the data and the corresponding values obtained from Eq. (2). Figure 1(b) is the same data and fit shown on a log-linear scale but with the background term subtracted. The overlap of the two graphs demonstrates excellent agreement except for small differences at large times.

Figure 1a.

Figure 1b.

Using the Solver best-fit value of ₁ = 0.00642, a tube length L = 13.9 cm, a diameter of the reservoir = 6.81 cm, and known properties of the fluid (water was used in all experiments), the diameter of the stirrer was found to be 1.9 mm from Eq. (1), which is close enough for our purposes to a direct measurement of about 2.1 mm by using a folding micrometer. The hollow cross section of the stirrer was not uniform, but more closely resembled that of a cylindrical cavity with a thin solid rod attached to the inner surface. We don't know what effect this has on Poiseuille's equation. For quantitative purposes it would be best to use a uniform capillary tube, but this was not the intent of our experiment. However, we now simply use this experiment in various lab courses as a measurement of the effective inside diameter of the stirrer. In lab courses where statistical treatment of data is emphasized, students routinely determine the tube diameter to be within 0.1 mm of the directly measured quantity. Historically this experiment was used to measure the viscosity of various fluids.

The exponential time dependence of column height exactly models radioactive decay⁶ where the number of nuclei present at time t is given by N = N₀exp(–t). The decay constant, , represents the probability per unit time for a nucleus to decay and is related to the half-life, T_1/2, the time for half of the original number of nuclei to decay, by T_1/2 = ln 2. In the fluid flow experiment the value for may then also be determined from this relationship by measuring the time for some reference height of fluid to fall by a factor of two.

In a radioactive decay series in which a parent nuclide (subscript 1) decays into a daughter nuclide (subscript 2) and having decay constants ₁ and ₂, respectively, the number of daughter nuclei at time t is given by the well-known result⁷

where N₁₀ is the number of parent nuclei at t = 0, the time at which no daughter nuclei are present. For the readily available ¹³⁷Cs beta decay to an excited state of ¹³⁷Ba (respective half-lives of 30.2 y and 2.55 m), the much longer parent half-life means that ₂>>₁, so Eq. (3) may now be written as

where the final equilibrium value of N₂ is written as N_2f = N₁₀₁/₂.

A constant water flow obtained directly from a faucet was used to fill a leaking reservoir as a model of this radioactive decay of a long-lived parent to a short-lived daughter. This fluid flow experiment may now be modeled as Eq. (4) plus a constant amount of background radiation. The weight of the leaking reservoir as a function of time was again obtained with the Logger Pro software and the data were fit to the equation

where W_2f = W₁₀₁/₂ and W_2BG is the initial weight of the fluid and bottle before filling begins. Figure 2 presents both data and a graph of Eq. (5) using the best-fit values for the adjustable parameters of W_2f, ₂, and W_2BG. Sliders for these parameters were again employed for instructional purposes, and the final best-fit values were also obtained by the use of the Solver add-in. This fit is in excellent agreement with the data for this example of secular equilibrium, and we disagree with the statement by Greenslade² that the "analogue breaks down for large times." Although at this point we are only interested in a qualitative agreement between the data and the mathematical model, all best-fit adjustable parameter values are consistent with direct measurements of corresponding quantities.

Figure 2.

A single radioactive-series decay chain begins with an initial amount of only a parent nuclide, which decays to a second, which decays to a third, etc., each with its respective decay constant. The set of solutions for the time dependence of the amount of each member of the series is known as the Bateman equations.^8,9 An elegant derivation of the general solution for each series member has been recently presented by Pressyanov.¹⁰

For a series of three nuclides, the third of which is stable (₃ = 0), the time dependence for the third member is found to be

This third member is represented in our experiment by a third plastic bottle without an outlet. We model the time dependence of its weight as

where W_3BG is the weight of the empty third bottle. The time dependence for the weight of the second bottle is modeled by Eq. (3) and is written as

where W_2BG is the starting weight of the second bottle.

Figure 3 shows the experimental apparatus, where three plastic bottles were mounted on three separate force probes, each of which was connected to the same interface. The water-filled bottle #1 was then allowed to leak into bottle #2, which had a slightly larger decay constant. This translates into having a slightly larger stirrer diameter, as seen from Eq. (1). Bottle #2 then filled bottle #3. The time dependence for the weight of each bottle was measured using the Logger Pro software and the data were modeled by Eqs. (2), (8), and (7) for bottle #s 1, 2, and 3, respectively.

Figure 3.

Figure 4 presents both data and best-fit graphs for all three bottles. The overall qualitative agreement between data and best fit for all three bottles is much better than we had anticipated and demonstrates the practicality of applying Solver for such a complex situation. The best-fit values for the interrelated adjustable parameters W₁₀, ₁, W_1BG, ₂, W_2BG, and W_3BG were provided by minimizing the sum of squares of differences between data and model values for all three data sets and relevant equations combined.

Figure 4.

As shown by Lapp and Andrews,⁷ when the time derivative of Eq. (3) is set to zero, we find the time for N₂ to reach a maximum is given by t_m = [ln (₂/₁)]/(₂ – ₁). Although we haven't pursued this, the reader may wish to investigate what effect the inclusion of this condition in Solver may have on best-fit values.

If ₁ < ₂, as in our case studied above, then after a sufficiently large time, Eq. (3) reduces to

For three cases of interest, a) ₂ = 2₁, b) ₂ < 2₁, and c) ₂ > 2₁, it follows from Eq. (9) that after a large time a) N₂ = N₁, b) N₂ > N₁, and c) N₂ < N₁, respectively. From this analysis we understand what conditions must be met for whether or not the population of N₁ crosses below that of N₂ for large times. In our case studied above, the best-fit values result in ₂/₁ = 1.85, so the model equations (less background terms) show a crossover. However, this ratio is so close to the threshold value of two that, combined with the different constant background terms, within the experimental and analytical uncertainties involved in such a complex situation, it is not obvious how the actual long-term data compare, even after background terms are subtracted. We suggest that readers may be interested in similar investigations, but for ₂/₁ much different from the threshold value of two.

Textbook discussions of a radioactive-series decay chain appear outside the normal realm of undergraduate lab experiments. This is due to the complexity of the necessary equipment and especially to the lack of availability of a suitable decay chain. Naturally occurring chains might be used, such as the buildup of ²²²Rn from an initially pure sample of ²²⁶Ra, but this adds greatly to the experimental complexity.

In our fluid flow model with bottles attached to force probes and interfaced to a computer, we now have simple but very versatile equipment that may be adjusted to any reasonable type of condition. It is now easy for us to simulate the various classes of conditions of series decay chains that are normally presented in textbooks.

With such simple equipment we believe that more undergraduate students should now have the pleasure of deriving expressions for at least the first three populations of a decay chain. The integrals involved are rather straightforward and the algebra is not cumbersome at all. This derivation is worth the effort because of the rich variety of experiments that may now be easily performed.

Many instructors may wish to use this experiment as a basis for statistical treatment of data. However, we were mainly interested in the overall qualitative agreement between theory and experimental data, and not in how close some value comes to its directly measurable amount. After all, this fluid flow from plastic water bottles through coffee stirrers is not exactly nuclear physics!

References

1. J. R. Smithson and E. R. Pinkston, "Half-life of a water column as a laboratory exercise in exponential decay,"Am. J. Phys. 28, 740–742 (1960). first citation in article
   
2. Thomas B. GreensladeJr., "Simulated secular equilibrium," Phys. Teach. 40, 21–23 (Jan. 2002). first citation in article
   
3. Francis Weston Sears and Mark W. Zemansky, University Physics, 4th ed. (Addison-Wesley, Reading, MA, 1970), p. 207. first citation in article
   
4. Vernier Software, 13979 SW Millikan Way, Beaverton, OR 97005-2886; 503-277-2299; http://www.vernier.com. first citation in article
   
5. Scott Godsen, "Optimization analysis of projectile motion using spreadsheets," Phys. Teach. 40, 523–525 (Dec. 2002). first citation in article
   
6. Ralph E. Lapp and Howard L. Andrews, Nuclear Radiation Physics, 4th ed. (Prentice Hall, New Jersey, 1972), p. 176. first citation in article
   
7. See Ref. 6, p. 187. first citation in article
   
8. See Ref. 6, p. 190. first citation in article
   
9. Robley D. Evans, The Atomic Nucleus (McGraw-Hill, New York, 1955), Chap. 15. first citation in article
   
10. Dobromir S. Pressyanov, "Short solution of the radioactive decay chain equations," Am. J. Phys. 70, 444–445 (April 2002). [ISI] first citation in article
    

1. Mechanical Simulation of a Half-Life
   T. T. Grove et al., Phys. Teach. 46, 369 (2008)
2. Weight-controlled capillary viscometer
   Rafael M. Digilov et al., Am. J. Phys. 73, 1020 (2005)
3. Using the Stock Market to Teach Physics
   David A. Faux et al., Phys. Teach. 42, 488 (2004)

Stephen J. Fairman received a B.S. physics education degree from Edinboro University of Pennsylvania and is currently teaching physics at Clarion Area High School in Clarion, PA.

Joseph A. Johnson is attending Edinboro University of Pennsylvania in the B.S. physics education program.

Thomas A. Walkiewicz, professor of physics, has taught at Edinboro University since 1970. He received a B.S. degree from Xavier University (Ohio) and a Ph.D. from The Pennsylvania State University. He has collaborated on experimental nuclear physics research at Oak Ridge National Laboratory for more than 25 years and has served as a consultant to industry for studies of radiation effects in materials. He was co-founder and vice president of Edinboro Scientific Company, a certified radon testing lab, and has served on the TPT Editorial Board.Department of Physics and Technology, Edinboro University of Pennsylvania, Edinboro, PA 16444; walkiewicz@edinboro.edu

Full figure

Fig. 1a. Weight vs time (black dots) for simple exponential decay plus a constant background. Red dots are calculated from Eq. (2) for Solver best-fit values ofW₁₀ = 4.825, ₁ = 0.006422, and W_1BG = 1.467. First citation in article

Full figure

Fig. 1b. The same data and fit, but with the background term subtracted for both, shown on a log-linear scale. First citation in article

Full figure

Fig. 2. Weight vs time (black dots) for the secular equilibrium buildup analogue of a relatively short-lived daughter. Red dots are calculated from Eq. (5) for Solver best-fit values of W_2f = 1.274, ₂ = 0.01513, and W_2BG = 0.2091. First citation in article

Full figure

Fig. 3. Apparatus for the decay chain experiment consisted of three plastic bottles, each suspended from its own force probe. Coauthors (L to R) Joseph Johnson and Stephen Fairman. First citation in article

Full figure

Fig. 4. Weight vs time (black dots) for the three-member series decay chain. Red dots are calculated from Eqs. (2), (8), and (7) for member #1, #2, and #3, respectively, using Solver best-fit values of W₁₀ = 6.212, ₁ = 0.007285, W_1BG = 2.000, ₂ = 0.01348, W_2BG = 1.966, and W_3BG = 2.786. First citation in article

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