Table of Contents¶

1 Buckingham $\pi$: Dimensional Analysis

Buckingham $\pi$: Dimensional Analysis¶

Mokbel Karam, and Prof. Tony Saad (www.tsaad.net)
Department of Chemical Engineering
University of Utah

In [1]:

from IPython.display import Image
from IPython.core.display import HTML 

In this Jupyter notebook we will execute the code presented in the paper.

Example 1: Pressure Drop in Pipe¶

In [2]:

from buckinghampy import BuckinghamPi

Pressure_Drop = BuckinghamPi()
Pressure_Drop.add_variable(name='{\\Delta}p',dimensions='M*L^(-1)*T^(-2)')     # pressure drop
Pressure_Drop.add_variable(name='R',dimensions='L')                            # length of the pipe
Pressure_Drop.add_variable(name='d',dimensions='L')                            # diameter of the pipe
Pressure_Drop.add_variable(name='\\mu',dimensions='M*L^(-1)*T^(-1)')           # viscosity
Pressure_Drop.add_variable(name='Q',dimensions='L^(3)*T^(-1)')                 # volumetic flow rate

Pressure_Drop.generate_pi_terms()
Pressure_Drop.print_all()

$$\text{Set }1: \quad\pi_1 = \frac{R}{d}\quad\pi_2 = \frac{Q \mu}{d^{3} {\Delta}p}\quad$$

$$\text{Set }2: \quad\pi_1 = \frac{R \sqrt[3]{{\Delta}p}}{\sqrt[3]{Q} \sqrt[3]{\mu}}\quad\pi_2 = \frac{d \sqrt[3]{{\Delta}p}}{\sqrt[3]{Q} \sqrt[3]{\mu}}\quad$$

$$\text{Set }3: \quad\pi_1 = \frac{d}{R}\quad\pi_2 = \frac{Q \mu}{R^{3} {\Delta}p}\quad$$

Example 2: Speed of Virus Infection¶

Using mass, length, time and temperature as fundamental dimensions:

In [3]:

from buckinghampy import BuckinghamPi

Virus_Infection = BuckinghamPi()
Virus_Infection.add_variable(name='V_{p}',dimensions='L*T^(-1)', non_repeating=True)   # virus spread rate
Virus_Infection.add_variable(name='P_{r}',dimensions='L')                              # precipitation
Virus_Infection.add_variable(name='{\\theta}',dimensions='C')                          # temperature
Virus_Infection.add_variable(name='C_{a}',dimensions='L^(3)/T')                        # airflow
Virus_Infection.add_variable(name='C_{e}',dimensions='T')                              # seasonal changes
Virus_Infection.add_variable(name='E_{fs}',dimensions='L^(-2)')                        # social structures
Virus_Infection.add_variable(name='H',dimensions='M*L^(-3)')                           # humidity

Virus_Infection.generate_pi_terms()
Virus_Infection.print_all()

$$\text{Set }1: \quad\pi_1 = \frac{P_{r}^{2} V_{p}}{C_{a}}\quad\pi_2 = \frac{C_{a} C_{e}}{P_{r}^{3}}\quad\pi_3 = E_{fs} P_{r}^{2}\quad$$

$$\text{Set }2: \quad\pi_1 = \frac{C_{e} V_{p}}{P_{r}}\quad\pi_2 = \frac{C_{a} C_{e}}{P_{r}^{3}}\quad\pi_3 = E_{fs} P_{r}^{2}\quad$$

$$\text{Set }3: \quad\pi_1 = \frac{C_{e}^{\frac{2}{3}} V_{p}}{\sqrt[3]{C_{a}}}\quad\pi_2 = \frac{P_{r}}{\sqrt[3]{C_{a}} \sqrt[3]{C_{e}}}\quad\pi_3 = C_{a}^{\frac{2}{3}} C_{e}^{\frac{2}{3}} E_{fs}\quad$$

$$\text{Set }4: \quad\pi_1 = \frac{V_{p}}{C_{a} E_{fs}}\quad\pi_2 = \sqrt{E_{fs}} P_{r}\quad\pi_3 = C_{a} C_{e} E_{fs}^{\frac{3}{2}}\quad$$

$$\text{Set }5: \quad\pi_1 = C_{e} \sqrt{E_{fs}} V_{p}\quad\pi_2 = \sqrt{E_{fs}} P_{r}\quad\pi_3 = C_{a} C_{e} E_{fs}^{\frac{3}{2}}\quad$$

Example 3: Economic Growth¶

In [4]:

from buckinghampy import BuckinghamPi

Economic_Growth = BuckinghamPi()
Economic_Growth.add_variable(name='P',dimensions='K', non_repeating=True)     # capital
Economic_Growth.add_variable(name='L',dimensions='Q/T')                       # labor per period of time
Economic_Growth.add_variable(name='{\\omega_{L}}',dimensions='K/Q')           # wages per labor
Economic_Growth.add_variable(name='Y',dimensions='K/T')                       # profit per period of time
Economic_Growth.add_variable(name='r',dimensions='1/T')                       # rental rate period of time
Economic_Growth.add_variable(name='{\\delta}',dimensions='1/T')               # depreciation rate

Economic_Growth.generate_pi_terms()
Economic_Growth.print_all()

$$\text{Set }1: \quad\pi_1 = \frac{P r}{L {\omega_{L}}}\quad\pi_2 = \frac{Y}{L {\omega_{L}}}\quad\pi_3 = \frac{{\delta}}{r}\quad$$

$$\text{Set }2: \quad\pi_1 = \frac{P {\delta}}{L {\omega_{L}}}\quad\pi_2 = \frac{Y}{L {\omega_{L}}}\quad\pi_3 = \frac{r}{{\delta}}\quad$$

$$\text{Set }3: \quad\pi_1 = \frac{P r}{Y}\quad\pi_2 = \frac{L {\omega_{L}}}{Y}\quad\pi_3 = \frac{{\delta}}{r}\quad$$

$$\text{Set }4: \quad\pi_1 = \frac{P {\delta}}{Y}\quad\pi_2 = \frac{L {\omega_{L}}}{Y}\quad\pi_3 = \frac{r}{{\delta}}\quad$$

Example 4: Pressure Inside a Bubble¶

Using mass, length and time as fundamental physical dimensions:

In [5]:

from buckinghampy import BuckinghamPi

Pressure_In_Bubble = BuckinghamPi()
Pressure_In_Bubble.add_variable(name='{\\Delta}p',dimensions='M*L^(-1)*T^(-2)')  # pressure 
Pressure_In_Bubble.add_variable(name='R',dimensions='L')                         # diameter
Pressure_In_Bubble.add_variable(name='\\sigma',dimensions='M*T^(-2)')            # surface tension
try:
    Pressure_In_Bubble.generate_pi_terms()
    Pressure_In_Bubble.print_all()
except Exception as e:
    print(e)

The number of variables has to be greater than the number of physical dimensions.

Using force and length as fundamental physical dimensions:

In [6]:

from buckinghampy import BuckinghamPi

Pressure_In_Bubble = BuckinghamPi()
Pressure_In_Bubble.add_variable(name='{\\Delta}p',dimensions='F*L^(-2)')       # pressure 
Pressure_In_Bubble.add_variable(name='R',dimensions='L')                       # diameter
Pressure_In_Bubble.add_variable(name='\\sigma',dimensions='F*L^(-1)')          # surface tension

Pressure_In_Bubble.generate_pi_terms()
Pressure_In_Bubble.print_all()

$$\text{Set }1: \quad\pi_1 = \frac{\sigma}{R {\Delta}p}\quad$$