Bernoulli, beer jugs & draining tanks

Summary

Highlights

• Bernoulli's equation is great for theoretical predictions but not perfect due to real-life variables like friction. 📉
• Experiments show the calculated drainage times are consistently faster than the actual times observed. ⌛
• The introduction of a discharge coefficient helps align theoretical and experimental data. 📝
• The video humorously compares the elegance of scientific equations to the 'improvised' nature of engineering calculations. 😂
• The exploration emphasizes the real-world application of equations and the necessity of making adjustments for accuracy. 🌍️

Key Takeaways

• Bernoulli's equation helps calculate tank drainage time, but real-life factors like friction and viscosity affect accuracy. ⚖️
• Using real-world experiments, the video highlights the variance between theoretical and actual drainage times. 🧪
• A discharge coefficient is necessary to correct theoretical predictions in practical scenarios. 📏
• Engineers adapt fundamental equations with additional factors to make them applicable in the real world. 🔧
• Understanding the discrepancies in predictive equations encourages better design and application. 💡

Overview

Chapters

• 00:00 - 00:30: Introduction and Overview The chapter introduces the application of Bernoulli's equation in calculating the drainage time of a tank with specific dimensions. It outlines the plan to derive the equation, acknowledging that the math might be complex but is necessary for understanding. The chapter also includes a practical test in a kitchen setting to verify the equation, hinting at its inaccuracies and the intention to analyze where it might fall short.
• 01:00 - 05:00: Theory and Bernoulli's Equation In this chapter, the focus is on understanding the discrepancies between theoretical predictions and experimental results. The speaker discusses the equations and methods they use to attempt to reconcile these differences, specifically mentioning Bernoulli's equation. The chapter aims to provide a detailed walkthrough of these methods and encourages readers to explore these calculations on their own using the provided resources.
• 05:00 - 09:00: Differential Equation and Predictions The chapter begins with a theoretical discussion on a cylindrical tank system defined by certain dimensions such as diameter and height, as well as a circular drain hole at its base. The focus is on understanding the dynamics of the system by considering two specific points: point A at the base of the tank and point B, aligned with the liquid level. Applying Bernoulli's equation to these points forms the crux of this chapter, as it explores the energy dynamics within the system. Bernoulli's principle, which states that the energy at any given point in the fluid remains constant, is used to make predictions about the behavior of the fluid as it drains from the tank.
• 09:00 - 16:00: Experimental Setup and Execution The chapter on 'Experimental Setup and Execution' delves into the principles of fluid dynamics, focusing on the constant nature of specific energy components: pressure, velocity, and height of the liquid. It begins by eliminating the pressures at two points in the system, as they are both subjected to atmospheric pressure. This simplification assists in focusing on other variables. Furthermore, the chapter sets the height of point A as the reference or zero height line, facilitating calculations and understanding of fluid movement and behavior within the defined system.
• 16:00 - 19:30: Analysis of Experimental Results The chapter 'Analysis of Experimental Results' discusses the assumption made regarding the velocity of the liquid level inside a tank during experiments. Initially, it was assumed that the velocity was negligible. Upon performing the experiments, this assumption was validated for the largest drain hole and fastest flow rate, where the velocity through the drain hole was measured at 1.9 meters per second, and the peak velocity inside the tank was only 0.015 meters per second. Thus, it was deemed reasonable to disregard the velocity term as it had minimal impact on the overall results.
• 19:30 - 25:00: Discussion on Engineering Adjustments In the chapter titled 'Discussion on Engineering Adjustments', the focus is on deriving an expression that relates the velocity of liquid through a drain hole to the height of the liquid in a tank. It is noted that this velocity is a function purely of the liquid's height. Furthermore, the velocity can be understood in terms of the flow rate, which is obtained by dividing the velocity by the drain area. Rearranging these elements leads to a concise expression for calculating the flow rate.

Bernoulli, beer jugs & draining tanks Transcription

• 00:00 - 00:30 a fairly simple application of bernoulli's equation is in calculating how much time it takes to drain a tank of a certain height and diameter through a drain hole of a certain diameter today we're going to derive that equation no hard feelings if you don't stick around for all the maths but it's there if you need it but more importantly we're then going to go test it in my kitchen we're then going to go look at whether the equation works spoiler alert it doesn't but we're going to look at how much we're off by and
• 00:30 - 01:00 look at some of the reasons why we're off and then we're going to look at what what it is that we do to fix these situations the equations that i use the sheet that i use to make my predictions as well as the experimental results all appear in the sheet in the description so you can go use it yourself and look what it is that i've done [Music] [Applause] [Music] [Applause] [Music] enjoy
• 01:00 - 01:30 we start with the theory take a cylindrical tank with diameter capital d and height h assume the tank as a circular drain hole with diameter small d in its base we define two points a at the base of the tank and b in line with the liquid level we then apply bernoulli's equation to these two points bernoulli basically states that at any point in this tank the energy contained
• 01:30 - 02:00 by the liquid by virtue of its pressure velocity and height is constant the first thing to note is that we can get rid of both pressures because the tank is open to atmospheric pressure both at the top and at the drain point since both are atmospheric pressure they will cancel each other out next we define the height of a as the zero height line so we can get
• 02:00 - 02:30 rid of another term then we assume the velocity at the liquid level inside the tank is negligible after i performed the experiments i actually checked this assumption for the biggest drain hole and the fastest flow rate for a velocity of 1.9 meters per second through the drain hole the level in the tank at its peak is moving at 0.015 meters per second so it's a reasonable assumption to get rid of this velocity term it's going to have so little impact
• 02:30 - 03:00 and so we get an expression that says the velocity through the drain hole is only a function of the height of the liquid in the tank the velocity through the drain hole can be expressed in terms of the flow rate if we divide it by the area of the drain and so we arrange everything and get a nice expression for the flow rate
• 03:00 - 03:30 through the drain as a function of two things the size of the drain and the level of liquid in the tank increasing either of these increases the flow rate through the drain and your intuition should tell you as much the way you get more liquid through the drain is with a bigger drain or with more water in the tank the problem we have is that as we drain the liquid the height in the tank is changing which means that this is a time dependent problem and we need a differential equation to solve it we start with a basic mass balance the
• 03:30 - 04:00 rate of accumulation in the tank equals the flow into the tank less the flow out of the tank since there is only flow out of the tank we can ignore the inlet term but it is very common to be given a problem where there is a constant flow rate into the tank too it doesn't complicate the problem too much if that is the case you just plug in a constant value for the in term we replace the accumulation term with the derivative of the remaining volume in the tank v
• 04:00 - 04:30 with respect to time and this is equal to the negative drainage rate because the volume of the tank is dropping over time we can then take our previous expression for flow rate which we derived from bernoulli and shove it in here i need to replace the height of fluid in the tank on the right hand side with a volume term because it needs to match the dv on the left hand side for me to be able to solve this the height of a cylinder is equal to its
• 04:30 - 05:00 volume over its cross-sectional area which is obviously a function of tank diameter so i get an expression for height and i put it back into the differential equation and after a little bit of work i get the following expression if we clear our heads a little bit of all the maths we did and take a step back this expression tells us that in order to know the volume of liquid
• 05:00 - 05:30 left in the tank after a certain amount of time i need to know the volume right at the start the amount of time that has passed the diameter of the drain hole and the diameter of the tank that's it any volume i calculate can also easily be converted into a height by dividing it by the cross-sectional area of the tank i of course consulted the internet to double check my working out i see it's more popular to show this equation in terms of the height of the
• 05:30 - 06:00 liquid in the tank rather than the remaining volume of the tank but here you have both note that there are subtle differences in the equation so be careful of them there's another good question that comes up and that is how does this equation look for a tank that doesn't have a simple open drain at the base because actually no tank has a simple open base at the at the bottom what happens if there's piping out of the tank and that piping has a certain length and we're draining
• 06:00 - 06:30 it somewhere else that piping offers additional pressure drop and that needs to be taken into account i haven't gone into it here because this would be an entire exercise by itself but the thing i would do is i'd go and generate the system curve for that piping just like i did in a previous video where i generated the system curve for two kilometer pipeline plot that pressure drop as a function of q fit a curve to it using an excel curve fitter as an example and that
• 06:30 - 07:00 equation describes the pressure at the base of the tank for a certain flow rate through the pipe so i have an expression i can go and plug into the pressure term that i previously cancelled out as being zero because it's not atmospheric the pressure now is whatever the pressure drops through the pipe is so this would take a little bit more integration a little bit more mass i actually don't know how challenging it would be but at least you have the theory if that
• 07:00 - 07:30 was a bit too quick go through it again yourself and prove that you can derive this yourself the point is we're going to be using this final expression to make predictions of our own experiment let's go test we start with this beard jug which i've marked at a 10 millimeter interval up to 180 millimeters the beard jug is actually slightly conical in shape and so is a little irregular and it isn't that easy to measure a diameter so i fill it up with water to a level of 180 millimeters and i weigh it
• 07:30 - 08:00 if i assume a specific gravity of 1 i know that the volume of water in the jug is roughly 1.8 liters and from the volume of a cylinder i calculate the diameter to be around 112 millimeters i measured the bottom which is a little bit narrower than what i've calculated and i measure the top which is a little bit wider it's okay i'm not going to sweat the small stuff now i've got drill bits from 2 up to 10
• 08:00 - 08:30 millimeters and i'm going to test each of them i start with the smallest hole of 2 millimeters because it should be pretty obvious why i don't start with the largest drill bit i do a quick check and i'm comfortable that we have a two millimeter drain now before i even start i have set up a spreadsheet to predict how the level and the jug will change over time so here i have a nice curve describing the level change if i use a two millimeter drain
• 08:30 - 09:00 and it looks like the jug will take around 10 minutes to empty let's go and see what it does i've added some food coloring to better be able to see what's happening so i'm not going to stuff around below a level of 10 millimeters because there'll always be something left in the jug but you can see that it actually took 14 minutes instead of the 10 minutes that i predicted to empty the jug
• 09:00 - 09:30 if we plot the experimental points from the video you can see the general shape is fine but the jug drained a lot slower than i predicted i repeated this experiment for three four and five millimeter drain holes and then again for 6 8 and 10 millimeter drain holes [Music] for all of these i've plotted both the
• 09:30 - 10:00 theoretical as well as the actual points and you can see that i'm consistently under predicting how long it takes to drain my beer jug [Music] here i'm showing the theoretical time it takes to reach a level of 10 millimeters depending on drain hole versus the actual time i measured to get to 10 millimeters the error in my predictions is about 50 to 80 percent
• 10:00 - 10:30 what's the reason for this well bernoulli's equation which is where we started with this is an expression for a continuous streamline of fluid and very importantly it assumes that there are no frictional losses and no viscosity effects in the flow can you see here that apart from the density of the fluid there's no mention of viscosity or friction even my final expression for remaining
• 10:30 - 11:00 volume in the tank says nothing about density or viscosity of the liquid obviously if i filled my beer jug with honey it would take a lifetime to drain and if i made the hole small enough it might not drain at all due to the surface tension of the fluid but bernoulli's equation would tell us that it would drain just at a very low rate so what do engineers do when the math doesn't work that's right baby we slap a factor in front of the equation to make it work in this instance we're going to give it the very official sounding name of a
• 11:00 - 11:30 discharge coefficient and define it as the ratio of actual to theoretical flow in other words we're just forcing this thing to work and i go back to all my maths earlier and every time i have a flow rate i put a discharge coefficient in front of that term if i now account for this in my spreadsheet i can see that the discharge coefficient is between 0.5 and 0.66
• 11:30 - 12:00 meaning the flow is between 58 and 66 of what i predict with bernoulli i didn't do any fancy curve fitting i just kept manually changing the cell until the curve looked good as you can see me doing here another example of where we use discharge coefficients is in the orifice equation for orifice type flow meters
• 12:00 - 12:30 most of the equation for the mass flow rate through an orifice comes from a derivation of bernoulli's equation similar to what we did at the beginning then in order to make it match experimental data you can see it also has a coefficient of discharge in the frontier another interesting thing to note is that depending on the extent to which the water was agitated after the food coloring was stirred sometimes i formed a vortex where air
• 12:30 - 13:00 was being sucked into the drain hole which effectively reduces the drainage capacity bernoulli can't account for that and i only noticed that this was happening halfway through the experiments look here how with a 10 millimeter drain hole i can either form a vortex or not depending on how much the water swirls around in the jug one can find a lot of discussions online about how especially in physics and maths some of the most important and groundbreaking equations that we use
• 13:00 - 13:30 have a certain beauty about them they aesthetically they look really appealing and they're beautiful in their simplicity and i always think about this whenever i see any any um engineering equations especially in chemical engineering where we've taken something with a fundamental physical foundation and we've slapped a lot of a lot of factors in front of it and we just go and screw everything up and while i was being sarcastic and
• 13:30 - 14:00 saying are we just slapping factors in front of everything we do that with love it's not as if we throw away the physical foundation for something we just correct the fact that real life happens and we still want to use these things and apply them because our other option would have been to drain my beer jug plot the points and use excel to fit some generic curve to it some polynomial but that polynomial is absolutely useless i i can't use it to extrapolate i can't
• 14:00 - 14:30 use it in a new design scenario so we do want to keep the foundation and as well as good as possible we want to explain why it is that there is variation another example of this is the ideal gas law pvnr equals nrt that looks very attractive but it doesn't work for real gases and a lot of equations of state go and build on that put factors in front of that and thermodynamics
• 14:30 - 15:00 studies look at modeling those parameters for specific for specific chemicals so we see this a lot and while we may be sarcastic about it it's a really effective approach and it's the basis of how we design plants you