Modelling tank drainage using a simple apparatus

A straightforward experimental set-up, requiring a two-liter bottle, a ruler and a stopwatch, is used to provide data appropriate for modelling with Torricelli's Law in the simplest case, and a more sophisticated differential equation when losses are taken into account and a pipe extension is considered. With only an exit hole included in the model, the governing equation can be integrated analytically to find the liquid height in the tank as a function of time. In the more complicated instance, the governing equation requires numerical integration. In both cases, a discharge coefficient can be employed as a fitting parameter. Excellent agreement between the experimental data and the theoretical modelling results has been obtained. An analytical solution with comparable accuracy to the numerical version is also possible for a pipe extension under the operating conditions considered here. Student reception of the experiment was generally positive, although some nuances were not fully grasped.


Introduction
In the midst of COVID, when remote learning was necessary, laboratory classes became a matter of particular concern.To handle the problem, several approaches were taken, such as videotaping experiments and asking students to analyse the data thus generated (Faulconer & Gruss, 2018;Radhamani et al., 2021;Shambare & Simuja, 2021).Another method was to prepare kits with the necessary equipment and send them by mail to students to assemble and to conduct experiments (Jayawardena et al., 2021;Larriba et al., 2021;MacKay, 2020).Obviating the need to gather and ship multiple, identical elements, which might be expensive or delicate, a third strategy was to design readily performed experiments with materials available at home.However, the do-it-yourself labs had to have sufficient difficulty to cover the usual subject matter in a junior/senior-level Unit Operations course and be suitable for theoretical analysis and propagation of error.
In addition to statistical uncertainty analysis, a Unit Operation laboratory entails experiments illustrating physicochemical principles, fluid mechanics, heat and mass transfer, separations and chemical reaction engineering.Technical report writing, presentation, standard laboratory practice and safety are also important elements of the course.Typically, students will have taken one year of calculus, ordinary differential equations, one year of calculus-based physics, three years of chemistry, and standard chemical engineering courses, such as material and energy balances, thermodynamics, computational methods, and design of experiments, as well as more advanced topics.However, the tankdraining laboratory requires only some calculus, differential equations and physics or fluid mechanics.
A general discussion of the at-home approach can be found elsewhere (Rother, 2021).Herein, the details of a particular exercise concerning the draining of a tank are presented.This example has an easily constructed apparatus and allows analysis of the resulting ordinary differential equations at levels from simple to more complicated.
Tank draining is a classic problem used across STEM disciplines and encountered in chemical engineering Unit Operations (Brown et al., 1953;Geankoplis, 1978).Varying degrees of intricacy have been applied in the analysis, including completely numerical methods employing microscopic study of the Navier-Stokes equations (Rathaur & Ghosh, 2022;Sakri, 2017) and different geometric configurations, such as square ducts (Hart & Sommerfeld, 1995;Koehler, 1984;Sommerfeld, 1992) and adding two immiscible liquids together (Rathaur & Ghosh, 2022).In this case, however, students were asked to employ an unsteady mass balance, together with extended Bernoulli's equation, to arrive at the rate of tank drainage.Subsequently, they needed to integrate the governing differential equation, analytically or numerically depending on complexity, to find the time dependence of the liquid height.
To guide the students and increase their interest, they were given a video from CACHE (Computer Aids for Chemical Engineering), which demonstrates that a pipe extension decreases the drainage time -a finding that seems to contradict the idea of pipe losses (Gupta, 2009).In addition, a second video was provided from the University of Louisiana Lafayette, where a more elaborate apparatus is used in their laboratory (CIVE434Fall2013, 2013).
Section 2 below contains the theoretical development to investigate tank drainage, while the experimental setup is described in Section 3. Next, experimental results are compared to those from the model in Section 4. Finally, student and instructor Assessment (Section 5) of the lesson is discussed, with Concluding Remarks following in Section 6.

Theory
The governing equation for the rate of change of the liquid height h with respect to time t can be derived by employing a transient mass balance in (1), in conjunction with Bernoulli's equation between points 1 and 2 in Figure 1(a).
In ( 1), the time rate of change of the mass of the system M sys depends on the difference between the rate of mass flowing into the system ṁin and the rate of mass leaving ṁout .Equation (1) assumes there is no creation or destruction of mass.A momentum balance on a system with losses can be written as an extended form of Bernoulli's equation (Bird et al., 2002;Wilkes, 2005): where u i is the average velocity at point 1 or 2 , P i is the pressure at point i, z i is the elevation at point i, ρ is the density of the fluid, and g = 981 cm/s 2 is the acceleration due to gravity.The last two terms on the left side of (2) account for frictional losses and minor losses due to fittings or sudden expansions and contractions, respectively, where u m is the average fluid velocity at the relevant point in the flow.The variable f F is the Fanning friction factor and K is a geometric factor to account for minor losses.The hydraulic mean diameter D e reduces to the pipe diameter d in the case of an extension with circular cross section.To use Bernoulli's equation, as written above, the fluid must be incompressible.Applying (2) to Figure 1(a) yields the exit velocity u 2 after substituting P 2 = P 1 = P atm , atmospheric pressure, and using the quasi-steady approximation at point 1 .That is, since A a, u 1 is approximately zero compared to u 2 , where A = π 4 D 2 and a = π 4 d 2 .Thus, the liquid at point 1 is descending slowly, and the level is nearly steady.Because the tank drains slowly, there is no appreciable friction with the walls from a relatively low-viscosity liquid, and, if the edges of the orifice at point 2 are round, there are no losses due to exit effects either.In this way, the dissipative terms on the left side of (2) can be omitted.Then, Torricelli's Law, u 2 = 2gh, which holds if there are no losses in the system, can be introduced into the mass balance (1) to yield the simplest differential equation for the rate of tank drainage.(See (3) below with C D = 1.)However, if the edges of the exit are sharp, as shown in Figure 1(a), the effluent narrows to the vena contracta point, and the decreased flow due to the smaller cross-sectional area should be taken into account.These losses are usually represented through a discharge coefficient C D , so that the rate of drainage is written as NB. Depending on the equation, a contraction coefficient C C sometimes appears in place of the discharge coefficient C D .In practice, the ratio of the two coefficients is so close to unity as to be virtually negligible (Lienhard (V) & (IV), ( 1984)).More discussion of tank draining coefficients is found in Appendix 1. Equation ( 3) can be integrated analytically to find the time-dependence of h: where As discussed above, for a tank with a rounded exit, the discharge coefficient 1984;Wilkes, 2005); however, values between 0.76 and 0.99 have been observed (Hicks & Slaton, 2014) For the purposes here, the discharge coefficient will serve as a to-bedetermined fitting parameter for the data.When a pipe extension of length L and diameter d is incorporated into the system, as shown in Figure 1(b), the mass balance on the liquid in the tank is unchanged, but the application of Bernoulli's equation is modified.The height difference between points 1 and 2 expands to and losses in (2) must be included, so that the rate of tank drainage becomes where C 3 varies with flowrate.As a result, because of the increased liquid head in (6) and the additional length L, the tank exit velocity and the Reynolds number (See (9) below) never go to zero, as they do with an exit hole only, and the tank will generally drain more quickly with an extension.The minor losses coefficient K from (2) can be represented as where β = d 2 /D 2 , and accounts for the rapid reduction in area from the tank to the pipe (Bird et al., 2002).The Fanning friction factor f F , which is part of the dissipative pipe-wall term, depends on the Reynolds number Re and is not constant.For the liquid in the pipe, the Reynolds number is In ( 9), u m = u 2 is the average velocity in the pipe, μ is the liquid dynamic viscosity, and ν = μ/ρ is the liquid kinematic viscosity.If the pipe is smooth, as is the case for most plastic, and the flow is turbulent (Re > 4000, or at least Re > 2300), the friction factor (Wilkes, 2005) can be expressed as It should be noted that since the Fanning friction factor depends on the velocity through the Reynolds number and the velocity depends on the Fanning friction factor through the denominator in ( 7), iterations are required to find u 2 and f F at every timestep.Normally, these amounted to about 4 or 5 successive substitutions for convergence at each interval in time.
Due to the coupled variables, ( 7) cannot be integrated analytically, and a numerical technique must be employed.It is possible, however, to use a characteristic or average value of the Reynolds number as an approximation, and then the differential equation has an exact solution: where C 3,avg and f F,avg are typical values of C 3 from ( 7) and of the Fanning friction factor from (10), respectively, and C 1 is defined in (5) above.

Experiments
The great fluid mechanician, G. I. Taylor, preferred experimental equipment that was not only simple and beautifully devised to illuminate the phenomenon being studied but was often distinctly 'low tech' (Turner, 1997).With this idea in mind, though perhaps more from necessity than any grand planning, the tank draining apparatus is shown in Figure 2. A two-liter bottle was chosen as the tank because of its reasonable size and ease of acquisition.
Although the cross-sectional area is not completely constant, the data fit the model well, as seen below, except for the very end of the draining.
In addition to the two-liter bottle, ruler and timer, recommended materials consisted of duct tape, a pair of scissors or knife, a container for the effluent, straws or tubes for the pipe extension, and a cell phone to take pictures of the apparatus and produce a labelled schematic.
Unit Operations lab is an open-ended course, so that students were given considerable freedom in their approach to a given problem.The objective for this experiment was simply, 'To quantitatively investigate the effects of relative orifice size, pipe length and frictional losses on the draining of a tank.' Groups of two, who were located at their own residences during COVID, conducted the experiments with some guidance, such as performing at least three runs for each set-up for random uncertainty purposes; considering different liquids or two immiscible liquids in the draining; and employing pipe extensions with nonzero roughness or different lengths and diameters.Procedure was largely left up to  the students, but most chose to let the liquid start to drop and used the 'lap' feature on their cellphones' stopwatch to record the time every centimeter or half-centimeter.Other timing devices might do, but the frames per second information might be needed if there is not independent clocking going on concurrently.At a minimum, groups were expected to obtain the data necessary to make the comparisons shown in Figures 3(a) and 4(a) below.

Results
Students generally used Excel for data manipulation, numerical integration and leastsquares fitting.Many implemented Euler's method to solve (7), although occasionally the Runge-Kutta method made an appearance.Because the least-squares fitting could become complicated (see Appendix 2), some visual determination of the best fit was allowed.Also, a review of numerical methods was provided in class.In order to calculate the Re in (9), a value of the kinematic viscosity ν was necessary.However, during COVID, student access to thermometers could not be assumed, depending on their location.Accordingly, the value ν = 0.01 cm 2 /s was permitted for tap water in analysis, but many were able to make a better estimate.In the figures below, propagation of uncertainty and error bars were expected but are not presented here.
In the absence of a pipe extension, analytical results from (4) are presented in Figure 3(a), together with the experimental data.With a discharge coefficient of C D = 1, the drainage rate is overpredicted, demonstrating the presence of losses.Using the discharge coefficient as a fitting coefficient, the least squares method leads to C D = 0.698.(See Appendix 2 for details.)The fit is excellent apart from the last two data points, where the changing cross-sectional area becomes important.
In Figure 3(b), the Reynolds number at the vena contracta, estimated from (A4) of Appendix 1, is shown for the analytical results from part (a), with C D = 0.698.To produce the figure, intermediate data from the numerical results for Figure 3(a) is plotted.The decrease in Re is linear with time, as can be illustrated from ( 4), where h = (H 1/2 − C 1 t) 2 , in conjunction with Torricelli's Law, where u 2 ∝ h 1/2 .Then, since Re ∝ u 2 = u m , the decrease in Re with time has a constant slope.This observation is important in the case of a pipe extension, because, if true, it permits use of an analytical solution there, as well.
The demarcation of the boundaries for laminar, transition and turbulent flow is also indicated in Figure 3(b) for pipe flow.There is evidence that the transitions for flow through an orifice occur at much smaller Re (Borutzky et al., 2002;Merritt, 1967;Wu et al., 2002), but the standard values are given here for reference.
With a pipe extension included, comparison of experiment and theory is made in Figure 4(a) for the time-dependence of the liquid height.For C D = 1, the numerical results from ( 7) again overestimate the rate of drainage.With a narrowing of the effluent from the pipe assumed, a discharge coefficient of C D = 0.781 yields reasonable agreement between experimental data and computational values.
In Figure 4(b), the Reynolds number remains above 4000, illustrating that the flow is turbulent.Thus, because of the pipe extension, the Reynolds number does not decreases to zero, in contrast to the behaviour for an orifice only.
An important feature of Figure 4(b) is that the Reynolds number from (A4) of Appendix 1 is linear in time, as was the case for an orifice only (Figure 3(b)).In a fashion similar to Figure 3(b), intermediate data from the numerical results for Figure 4(a) is plotted to produce the graph.This finding might be considered unexpected, because the denominator in (7), labelled C 3 , is not constant due to the changing Fanning friction factor f F and its dependence on the Reynolds number in (10).However, the same scaling arguments for Re vs. t can be made for a pipe extension as for an orifice -but only if C 3 is essentially constant.In this way, although the Reynolds number varies between 4150 and 6680, the Fanning friction range is much smaller, with a deviation of only about 12%.This trend continues in Figure 5, where the variable C 3 changes less than 2%.
Due to the small variation in the denominator of ( 7) while the tank empties, it is possible to use (11) to model the experimental data, with a characteristic value for f F , as shown in Figure 5.When C 3,avg = 1.482, the corresponding Reynolds number and Fanning friction factor are Re = 5220 and f F,avg = 0.00929.The agreement with the experimental data is nearly for the analytical solution and the numerical calculations; however, the analytical solution is easier to manipulate.As a final note, one should check that C 3 is approximately constant before quickly resorting to the analytical solution.For example, if the pipe extension has a finite roughness, such as in galvanised iron, or if the pipe length or diameter differ widely from the values here, the variations in f F and C 3 might be more significant, leading to a poorer comparison between experiment and the theory.

Assessment
Students generally considered the online, at-home laboratory a positive experience.In terms of specific comments on the tank-draining experiment, one common remark was that producing usable data was encouraging.It was also noted that this lab was interesting and related to process engineering.One suggestion for improvement was to provide more detailed requirements for the experiment.There were also mixed responses to being compelled to work in groups: Some liked groupwork; others did not.A final opinion, which leads into my instructor observations, was that the tank-draining investigation was 'slightly too easy.' From my perspective, this lab, in particular, worked very well.With at least modest success, the students over the two-year period were able to obtain some experimental data and model the results by analytical and numerical integration of the governing differential equations.Although it could be viable in a different setting, giving more exact directions is not really possible due to the nature of the class.Likewise, the workingin-teams component is a non-negotiable requirement.Even if more detailed directions were not an option, more direct one-on-one communication with the instructor, such as mandatory office hours or help sessions, might have helped the students feel more connected.
The most intriguing remark concerned the difficulty, or lack thereof, for the experiment.Although the student work was satisfactory, subtleties were often missed.Three examples illustrate the point: (i) For the pipe extension, students did not verify that the flow was turbulent by checking the Reynolds number range (Re > 4000, or at least Re > 2300).If the flow is not turbulent, then (10) does not hold for the Fanning friction factor.(ii) Students did not recognise that iterations are required for the velocity with a pipe extension in (A2).Fortunately, after the first few timesteps, the error becomes small.(iii) Students did not notice that the denominator C 3 can be represented as a constant for the observed experimental conditions, so that the analytical expression (11) can be used in place of numerical integration of ( 7) to find the liquid height as a function of time.
After all, though the fundamentals of the tank-draining problem have been understood for some time, research continues to be conducted in this area to this day.After some reflection, in contrast to the approach of G. I. Taylor, mentioned in Section 3 above, my tentative conclusion is that students might think a problem cannot be difficult, if the apparatus is not complex.While it is true that the lab is designed to be clear in terms of data acquisition, the analysis can become quite involved, depending on the amount of work one is willing to invest.

Concluding remarks
Differential equations for the liquid height as a function of time, obtained from a transient mass balance and augmented Bernoulli's principle, closely model the experimental data from a simple apparatus made from a two-liter bottle, both without and with a pipe extension.A discharge coefficient C D is determined by the least squares method to fit the lab measurements.Both analytical and numerically integrated solutions can be implemented.Even for the case of a pipe extension, an analytical solution, which is as accurate as numerical integration, can be found by realising that the total losses term varies by only 2% for the experimental conditions employed here.
Student feedback was favorable, including appreciation for being able to readily generate functional data for modelling.Although some problem features were not discerned, lab groups were able to adequately model orifice drainage with an analytical solution and pipe effluent by numerical treatment.Some student concerns, such as more explicit requirements, could not be easily addressed because of the traditional nature of Unit Operations Laboratory, but better communication in a remote learning environment would undoubtedly help.and L j = 0 without pipe extension L with pipe extension.(A8) Setting ∂E ∂C D = 0 leads to (A9) which must be solved numerically: where the leading constant 4C j /C D can be dropped.Equation (A9) holds only for the case of a simple exit orifice (4) and for the approximate solution with a pipe extension at constant C 3 (11).For the instance of a pipe extension and numerical integration of ( 7), the Fanning friction factor f F depends on C D .Thus, a more complicated partial derivative would be required.However, a modified form of (A5), which requires minimisation, still applies: where h num (t i ) is the numerically determined value of h at t i .The discharge coefficients found by least squares fitting for an exit orifice only are listed in Table A1.
Because the tank diameter is not constant at the bottom, it is not clear whether the final experimental data should be included in the fitting.Still, even incorporating all the data yields a relative error of about 8%.In a plot, like Figure 3(a), the curves for C D = 0.69799 and C D = 0.70353 are virtually indistinguishable.
The pipe-extension results for C D are given in Table A2.Finding the minimum of (A10) from ( 7) is computationally intensive, because for each value of C D , numerical integration is required over the complete domain of the experimental data.The analytical solution from (11) with C 3 = 1.482 is much easier to optimise with respect to C D through (A5) or (A9).However, both approaches, analytical and numerical, provide the same C D to within 0.23%.

Figure 1 .
Figure 1.Schematic for tank draining (a) without and (b) with a pipe extension.In (a) the vena contracta is indicated.

Figure 2 .
Figure 2. Labelled experimental apparatus (a) without and (b) with a pipe extension.The height H = 15 cm, tank diameter D = 10.65 cm, and orifice diameter d = 0.5 cm.The pipe length and diameter are L = 10.05 cm and d = 0.5 cm, respectively.

Figure 3 .
Figure 3.Comparison of experiment and theory with an exit orifice of diameter d = 0.5 cm.Results are determined from (4) with C D = 1 (part a) only) and 0.698 (parts a) and b)).The liquid is tap water at room temperature.(a) Liquid height versus time without a pipe extension.Experimental data (dashed line with filled circles) are graphed with numerical results (solid lines) and (b) Reynolds number Re as a function of time for the numerical results.

Figure 4 .
Figure 4. Comparison of experiment and theory with a pipe extension of length L = 10.05 cm and diameter d = 0.5 cm.Numerical results are determined from (7) with C D = 1 (part a) only) and 0.781 (parts a) and b)).The liquid is again room-temperature tap water.(a) Liquid height versus time with a pipe extension.Experimental data are graphed with numerical results, including changing Re and (b) Reynolds number Re (left y-axis) and Fanning friction factor f F (right y-axis) as a function of time for the numerical results.

Figure 5 .
Figure 5.A graph of the denominator C 3 from (11), on the right y-axis, versus time, illustrating near constant behaviour during drainage.Resulting comparison between experiment and theory (left y-axis) for the same conditions as Figure 4 with C D = 0.781, but using the analytical expression (11) with a constant C 3 = 1.482 and C D = 0.783.

Table A1 .
Discharge coefficients C D for an exit orifice only.

Table A2 .
Discharge coefficients C D with a pipe extension.