{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# A battle between two bottles\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Author:Freek Pols
Time:10 minutes
Age group:14 - 18
Concepts:Bernoulli, fluid pressure
\n", "\n", "```{figure} demo41_figure2.jpg\n", ":width: 90%\n", ":align: center\n", "```\n", "\n", "## Introduction\n", "This demonstration was inspired by a question from the Dutch National Science Quiz: Which bottle empties faster, one with a long nozzle or one with a short nozzle? The demonstration is excellent for encouraging students to articulate their thoughts (why do you think so) or to explain Bernoulli's law.\n", "\n", "## Equipment\n", "* Two sturdy PET bottles without bottoms; \n", "* One bottle has a short nozzle/tube attached to the cap, the other a long nozzle; \n", "* A stand with two clamps; \n", "* A bucket or drip tray; \n", "* Water\n", "\n", "## Preparation\n", "* Remove the bottoms of the PET bottles. Secure the tubes in the caps using a glue gun.\n", "* Place the PET bottles in the stand as shown in the photo with the spouts facing downward.\n", "* Ask a student to keep the tubes closed with their fingers to prevent the water from flowing out (see {numref}`Figure {number} (left) `). Fill both bottles to the same height with water.\n", "* Position the bucket or drip tray beneath the bottles to keep the classroom dry!\n", "\n", "## Procedure\n", "Let students predict and explain which bottle they think will empty faster. Encourage them to explain their predictions thoroughly. Once enough ideas have been collected and no further ideas or explanations are forthcoming, have the student simultaneously release the tubes and let the bottles empty (see {numref}`Figure {number} (middle) `).\n", "\n", "The bottle with the longer tube empties first (see {numref}`Figure {number} (right) `).\n", "\n", "```{figure} demo41_figure1.png\n", "---\n", "width: 90%\n", "align: center\n", "name: demo41_fig1\n", "---\n", "The various stages of the demonstration.\\\n", "*left* The setup just before conducting the experiment.\\\n", "*middle:* Immediately after removing the fingers, the bottles start to empty.\\\n", "*right:* As less water remains in the bottle, the differences become clearer.\n", "```\n", "\n", "## Physics background\n", "The greater the height difference between the water surface and the outlet, the greater the water pressure at the outlet. Greater pressure results in a higher flow rate. For upper grades, you can use the Bernoulli equation, comparing the top of the water column and the point where the water exits the bottle (both are part of the same streamline). It is expressed as: $\\rho g\\Delta h = 1/2 \\rho v^2$.\n", "\n", "The pressure above and below the water column is equal (open to the air).\n", "\n", "The effect of friction is less significant than the effect of the water height.\n", "\n", "For a public event or lower grades, the following reasoning can also be used:\n", "The more fluid above you, the greater the pressure. This can be felt, for example, when diving in a pool, which may cause your ears to pop. Greater pressure also means a greater outflow speed.\n", "\n", "## Follow-up\n", "It is possible to simulate the process of emptying the bottles. Below we have made this python simulation. Not that the flow rate $Q = Av = A_{tube}\\sqrt{2gh}$ and the difference in height is $\\Delta V = Q\\Delta t = A_{top}\\Delta h$. Note that the emptying of the tube at the end is not included in the simulation." ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "g = 9.8 #m/s2\n", "A_top = 50 #cm2\n", "H_0 = 20 #cm\n", "\n", "A_tube = 1 #cm2\n", "L_tube_1 = 5 #cm\n", "L_tube_2 = 10 #cm\n", "\n", "h1 = []\n", "h1.append(H_0)\n", "h2 = []\n", "h2.append(H_0)\n", "t = []\n", "t.append(0)\n", "dt = 1 #s\n", "v1 = []\n", "v2 = []\n", "i = 0\n", "\n", "while min(h1[i],h2[i])>0:\n", " v1.append(np.sqrt(2*g*(h1[i]+L_tube_1)))\n", " v2.append(np.sqrt(2*g*(h2[i]+L_tube_2)))\n", " \n", " Q1 = A_tube * v1[i] * dt\n", " Q2 = A_tube * v2[i] * dt\n", "\n", " h1.append(h1[i] - Q1 / A_top * dt)\n", " h2.append(h2[i] - Q2 / A_top * dt)\n", " \n", " t.append(t[i] + dt)\n", " i += 1\n", "\n", "# Plotting the simulation\n", "plt.figure( )\n", "plt.plot(t, h1, 'k.', label='short tube')\n", "plt.plot(t, h2, 'r.', label='long tube')\n", "\n", "plt.xlabel('Time (s)')\n", "plt.ylabel('Height (cm)')\n", "plt.legend()\n", "plt.show()\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "base", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.14" } }, "nbformat": 4, "nbformat_minor": 2 }