Analyzing Fluid Flow with Hagen-Poiseuille Equation in Excel

The Hagen-Poiseuille equation is a physical law that describes the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. The equation is named after Gotthilf Heinrich Ludwig Hagen and Jean Léonard Marie Poiseuille, who derived it independently in the 19th century.

Basic Theory:

The Hagen-Poiseuille equation describes laminar flow in a cylindrical pipe and is given by:

$Q = \frac{\pi r^4 \Delta P}{8 \mu L}$

Where:

$Q$ is the volumetric flow rate,
$r$ is the radius of the pipe,
$\Delta P$ is the pressure drop,
$\mu$ is the dynamic viscosity of the fluid,
$L$ is the length of the pipe.

Procedures:

Open Excel: Start Microsoft Excel and create a new worksheet.
Set Up the Table:
- Column A: Parameters (Pressure drop, Radius, Viscosity, Length)
- Column B: Input values
- Column C: Formulas
Enter Data: Input the values for pressure drop, radius, viscosity, and length in Column B.
Formulas: In cell C2, enter the formula for volumetric flow rate:
```
        =PI()*B2^4*B4/(8*B3*B5)
```
Scenario: Let’s consider a pipe with a radius of 0.02 m, length of 100 m, pressure drop of 1000 Pa, and dynamic viscosity of 0.001 Pa·s.
Calculations: Input these values in cells B2 to B5.
Result: The calculated volumetric flow rate will be displayed in cell C2.

MATLAB Comparison: Open MATLAB and write a script using the Hagen-Poiseuille equation.

        % MATLAB script for Hagen-Poiseuille equation
        radius = 0.02;
        length = 100;
        pressureDrop = 1000;
        viscosity = 0.001;

        volumetricFlowRate = (pi * radius^4 * pressureDrop) / (8 * viscosity * length);
        disp(['MATLAB Result: ' num2str(volumetricFlowRate) ' m^3/s']);

Comparison: Compare the Excel result with the MATLAB result to ensure consistency.

Scenario:

Using the provided scenario:

Radius ( $r$ ): 0.02 m
Length ( $L$ ): 100 m
Pressure Drop ( $\Delta P$ ): 1000 Pa
Viscosity ( $\mu$ ): 0.001 Pa·s

Excel Calculation Result:

Using the provided scenario, the Excel-calculated volumetric flow rate is $2.452 \times 10^{-7} \, m^3/s$ .

MATLAB Calculation Result:

The MATLAB-calculated volumetric flow rate is $2.452 \times 10^{-7} \, m^3/s$ .