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"# 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."
]
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"metadata": {},
"source": [
"# Example 1: Pressure Inside a Bubble\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using mass, length and time as fundamental physical dimensions:"
]
},
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"name": "stdout",
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"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',units='M*L^(-1)*T^(-2)') # pressure \n",
"Pressure_In_Bubble.add_variable(name='R',units='L') # diameter\n",
"Pressure_In_Bubble.add_variable(name='\\\\sigma',units='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": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" sets Pi 1\n",
"------ --------------------------\n",
" 1 \\frac{\\sigma}{R {\\Delta}p}\n"
]
}
],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Pressure_In_Bubble = BuckinghamPi()\n",
"Pressure_In_Bubble.add_variable(name='{\\\\Delta}p',units='F*L^(-2)') # pressure \n",
"Pressure_In_Bubble.add_variable(name='R',units='L') # diameter\n",
"Pressure_In_Bubble.add_variable(name='\\\\sigma',units='F*L^(-1)') # surface tension\n",
"\n",
"Pressure_In_Bubble.generate_pi_terms()\n",
"Pressure_In_Bubble.print_all()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 2: Pressure Drop in a Pipe\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"scrolled": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" sets Pi 1 Pi 2\n",
"------ ------------------------------------------------------- -------------------------------------------------------\n",
" 1 \\frac{R}{d} \\frac{Q \\mu}{d^{3} {\\Delta}p}\n",
" 2 \\frac{R \\sqrt[3]{{\\Delta}p}}{\\sqrt[3]{Q} \\sqrt[3]{\\mu}} \\frac{d \\sqrt[3]{{\\Delta}p}}{\\sqrt[3]{Q} \\sqrt[3]{\\mu}}\n",
" 3 \\frac{d}{R} \\frac{Q \\mu}{R^{3} {\\Delta}p}\n"
]
}
],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Pressure_Drop = BuckinghamPi()\n",
"Pressure_Drop.add_variable(name='{\\\\Delta}p',units='M*L^(-1)*T^(-2)') # pressure drop\n",
"Pressure_Drop.add_variable(name='R',units='L') # length of the pipe\n",
"Pressure_Drop.add_variable(name='d',units='L') # diameter of the pipe\n",
"Pressure_Drop.add_variable(name='\\\\mu',units='M*L^(-1)*T^(-1)') # viscosity\n",
"Pressure_Drop.add_variable(name='Q',units='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 3: Economic Growth\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" sets Pi 1 Pi 2 Pi 3\n",
"------ --------------------------------- ------------------------ ------------------\n",
" 1 \\frac{P r}{L {\\omega_{L}}} \\frac{Y}{L {\\omega_{L}}} \\frac{{\\delta}}{r}\n",
" 2 \\frac{P {\\delta}}{L {\\omega_{L}}} \\frac{Y}{L {\\omega_{L}}} \\frac{r}{{\\delta}}\n",
" 3 \\frac{P r}{Y} \\frac{L {\\omega_{L}}}{Y} \\frac{{\\delta}}{r}\n",
" 4 \\frac{P {\\delta}}{Y} \\frac{L {\\omega_{L}}}{Y} \\frac{r}{{\\delta}}\n"
]
}
],
"source": [
"from buckinghampy import BuckinghamPi\n",
"\n",
"Economic_Growth = BuckinghamPi()\n",
"Economic_Growth.add_variable(name='P',units='K',explicit=True) # capital\n",
"Economic_Growth.add_variable(name='L',units='Q/T') # labor per period of time\n",
"Economic_Growth.add_variable(name='{\\\\omega_{L}}',units='K/Q') # wages per labor\n",
"Economic_Growth.add_variable(name='Y',units='K/T') # profit per period of time\n",
"Economic_Growth.add_variable(name='r',units='1/T') # rental rate period of time\n",
"Economic_Growth.add_variable(name='{\\\\delta}',units='1/T') # depreciation rate\n",
"\n",
"Economic_Growth.generate_pi_terms()\n",
"Economic_Growth.print_all()"
]
}
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