{
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"source": [
"# Table of Contents\n",
" <p><div class=\"lev1 toc-item\"><a href=\"#Buckingham-pi:--Dimensional-Analysis\" data-toc-modified-id=\"Buckingham-pi:--Dimensional-Analysis-1\"><span class=\"toc-item-num\">1 </span>Buckingham $\\pi$: Dimensional Analysis</a></div><div class=\"lev1 toc-item\"><a href=\"#Example-1:-Pressure-Drop-in-Pipe\" data-toc-modified-id=\"Example-1:-Pressure-Drop-in-Pipe-2\"><span class=\"toc-item-num\">2 </span>Example 1: Pressure Drop in Pipe</a></div><div class=\"lev1 toc-item\"><a href=\"#Example-2:-Speed-of-Virus-Infection\" data-toc-modified-id=\"Example-2:-Speed-of-Virus-Infection-3\"><span class=\"toc-item-num\">3 </span>Example 2: Speed of Virus Infection</a></div><div class=\"lev1 toc-item\"><a href=\"#Example-3:-Economic-Growth\" data-toc-modified-id=\"Example-3:-Economic-Growth-4\"><span class=\"toc-item-num\">4 </span>Example 3: Economic Growth</a></div><div class=\"lev1 toc-item\"><a href=\"#Example-4:-Pressure-Inside-a-Bubble\" data-toc-modified-id=\"Example-4:-Pressure-Inside-a-Bubble-5\"><span class=\"toc-item-num\">5 </span>Example 4: Pressure Inside a Bubble</a></div>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Buckingham $\\pi$: Dimensional Analysis\n",
"**Mokbel Karam, and Prof. Tony Saad (<a href='www.tsaad.net'>www.tsaad.net</a>) <br/>Department of Chemical Engineering <br/>University of Utah**\n",
"<hr/>"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from IPython.display import Image\n",
"from IPython.core.display import HTML "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this Jupyter notebook we will execute the code presented in the paper."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 1: Pressure Drop in Pipe\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\text{Set }1: \\quad\\pi_1 = \\frac{R}{d}\\quad\\pi_2 = \\frac{Q \\mu}{d^{3} {\\Delta}p}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/latex": [
"$$\\text{Set }2: \\quad\\pi_1 = \\frac{R \\sqrt[3]{{\\Delta}p}}{\\sqrt[3]{Q} \\sqrt[3]{\\mu}}\\quad\\pi_2 = \\frac{d \\sqrt[3]{{\\Delta}p}}{\\sqrt[3]{Q} \\sqrt[3]{\\mu}}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
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"data": {
"text/latex": [
"$$\\text{Set }3: \\quad\\pi_1 = \\frac{d}{R}\\quad\\pi_2 = \\frac{Q \\mu}{R^{3} {\\Delta}p}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/markdown": [
"---"
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"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
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],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Pressure_Drop = BuckinghamPi()\n",
"Pressure_Drop.add_variable(name='{\\\\Delta}p',dimensions='M*L^(-1)*T^(-2)') # pressure drop\n",
"Pressure_Drop.add_variable(name='R',dimensions='L') # length of the pipe\n",
"Pressure_Drop.add_variable(name='d',dimensions='L') # diameter of the pipe\n",
"Pressure_Drop.add_variable(name='\\\\mu',dimensions='M*L^(-1)*T^(-1)') # viscosity\n",
"Pressure_Drop.add_variable(name='Q',dimensions='L^(3)*T^(-1)') # volumetic flow rate\n",
"\n",
"Pressure_Drop.generate_pi_terms()\n",
"Pressure_Drop.print_all()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 2: Speed of Virus Infection\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using mass, length, time and temperature as fundamental dimensions:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\text{Set }1: \\quad\\pi_1 = \\frac{P_{r}^{2} V_{p}}{C_{a}}\\quad\\pi_2 = \\frac{C_{a} C_{e}}{P_{r}^{3}}\\quad\\pi_3 = E_{fs} P_{r}^{2}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\text{Set }2: \\quad\\pi_1 = \\frac{C_{e} V_{p}}{P_{r}}\\quad\\pi_2 = \\frac{C_{a} C_{e}}{P_{r}^{3}}\\quad\\pi_3 = E_{fs} P_{r}^{2}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/latex": [
"$$\\text{Set }3: \\quad\\pi_1 = \\frac{C_{e}^{\\frac{2}{3}} V_{p}}{\\sqrt[3]{C_{a}}}\\quad\\pi_2 = \\frac{P_{r}}{\\sqrt[3]{C_{a}} \\sqrt[3]{C_{e}}}\\quad\\pi_3 = C_{a}^{\\frac{2}{3}} C_{e}^{\\frac{2}{3}} E_{fs}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\text{Set }4: \\quad\\pi_1 = \\frac{V_{p}}{C_{a} E_{fs}}\\quad\\pi_2 = \\sqrt{E_{fs}} P_{r}\\quad\\pi_3 = C_{a} C_{e} E_{fs}^{\\frac{3}{2}}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\text{Set }5: \\quad\\pi_1 = C_{e} \\sqrt{E_{fs}} V_{p}\\quad\\pi_2 = \\sqrt{E_{fs}} P_{r}\\quad\\pi_3 = C_{a} C_{e} E_{fs}^{\\frac{3}{2}}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
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],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Virus_Infection = BuckinghamPi()\n",
"Virus_Infection.add_variable(name='V_{p}',dimensions='L*T^(-1)', non_repeating=True) # virus spread rate\n",
"Virus_Infection.add_variable(name='P_{r}',dimensions='L') # precipitation\n",
"Virus_Infection.add_variable(name='{\\\\theta}',dimensions='C') # temperature\n",
"Virus_Infection.add_variable(name='C_{a}',dimensions='L^(3)/T') # airflow\n",
"Virus_Infection.add_variable(name='C_{e}',dimensions='T') # seasonal changes\n",
"Virus_Infection.add_variable(name='E_{fs}',dimensions='L^(-2)') # social structures\n",
"Virus_Infection.add_variable(name='H',dimensions='M*L^(-3)') # humidity\n",
"\n",
"Virus_Infection.generate_pi_terms()\n",
"Virus_Infection.print_all()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 3: Economic Growth\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\text{Set }1: \\quad\\pi_1 = \\frac{P r}{L {\\omega_{L}}}\\quad\\pi_2 = \\frac{Y}{L {\\omega_{L}}}\\quad\\pi_3 = \\frac{{\\delta}}{r}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\text{Set }2: \\quad\\pi_1 = \\frac{P {\\delta}}{L {\\omega_{L}}}\\quad\\pi_2 = \\frac{Y}{L {\\omega_{L}}}\\quad\\pi_3 = \\frac{r}{{\\delta}}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
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},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\text{Set }3: \\quad\\pi_1 = \\frac{P r}{Y}\\quad\\pi_2 = \\frac{L {\\omega_{L}}}{Y}\\quad\\pi_3 = \\frac{{\\delta}}{r}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/latex": [
"$$\\text{Set }4: \\quad\\pi_1 = \\frac{P {\\delta}}{Y}\\quad\\pi_2 = \\frac{L {\\omega_{L}}}{Y}\\quad\\pi_3 = \\frac{r}{{\\delta}}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"---"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Economic_Growth = BuckinghamPi()\n",
"Economic_Growth.add_variable(name='P',dimensions='K', non_repeating=True) # capital\n",
"Economic_Growth.add_variable(name='L',dimensions='Q/T') # labor per period of time\n",
"Economic_Growth.add_variable(name='{\\\\omega_{L}}',dimensions='K/Q') # wages per labor\n",
"Economic_Growth.add_variable(name='Y',dimensions='K/T') # profit per period of time\n",
"Economic_Growth.add_variable(name='r',dimensions='1/T') # rental rate period of time\n",
"Economic_Growth.add_variable(name='{\\\\delta}',dimensions='1/T') # depreciation rate\n",
"\n",
"Economic_Growth.generate_pi_terms()\n",
"Economic_Growth.print_all()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 4: Pressure Inside a Bubble\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using mass, length and time as fundamental physical dimensions:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The number of variables has to be greater than the number of physical dimensions.\n"
]
}
],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Pressure_In_Bubble = BuckinghamPi()\n",
"Pressure_In_Bubble.add_variable(name='{\\\\Delta}p',dimensions='M*L^(-1)*T^(-2)') # pressure \n",
"Pressure_In_Bubble.add_variable(name='R',dimensions='L') # diameter\n",
"Pressure_In_Bubble.add_variable(name='\\\\sigma',dimensions='M*T^(-2)') # surface tension\n",
"try:\n",
" Pressure_In_Bubble.generate_pi_terms()\n",
" Pressure_In_Bubble.print_all()\n",
"except Exception as e:\n",
" print(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using force and length as fundamental physical dimensions:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\text{Set }1: \\quad\\pi_1 = \\frac{\\sigma}{R {\\Delta}p}\\quad$$"
],
"text/plain": [
"<IPython.core.display.Math object>"
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"text/plain": [
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]
},
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],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Pressure_In_Bubble = BuckinghamPi()\n",
"Pressure_In_Bubble.add_variable(name='{\\\\Delta}p',dimensions='F*L^(-2)') # pressure \n",
"Pressure_In_Bubble.add_variable(name='R',dimensions='L') # diameter\n",
"Pressure_In_Bubble.add_variable(name='\\\\sigma',dimensions='F*L^(-1)') # surface tension\n",
"\n",
"Pressure_In_Bubble.generate_pi_terms()\n",
"Pressure_In_Bubble.print_all()"
]
}
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