Gas Flow Under Laminar Viscous Conditions in Excel Formulas

Laminar flow is a type of fluid flow where the fluid moves in smooth and parallel layers, without mixing or turbulence. Viscosity is a measure of how much a fluid resists flowing or deforming due to internal friction. Gas flow under laminar viscous conditions means that the gas is flowing in a pipe or a tube with a constant cross-section, and the flow is affected by the viscosity of the gas and the pressure difference between the two ends of the pipe.

One way to understand gas flow under laminar viscous conditions is to imagine a stack of thin cylindrical shells of gas inside the pipe, each with a different radius and velocity. The innermost shell has the highest velocity and the lowest radius, while the outermost shell has the lowest velocity and the highest radius. The shells slide past each other with some friction, which is proportional to the viscosity of the gas and the difference in velocity between adjacent shells. The friction causes the shells to lose some kinetic energy and momentum, which results in a pressure drop along the pipe. The pressure drop is also proportional to the length of the pipe and the inverse of the fourth power of the radius of the pipe.

To summarize, gas flow under laminar viscous conditions is a smooth and steady flow that depends on the viscosity of the gas, the pressure difference between the two ends of the pipe, the length of the pipe, and the radius of the pipe. The higher the viscosity, the higher the pressure drop. The higher the pressure difference, the higher the flow rate. The longer the pipe, the higher the pressure drop. The smaller the radius, the higher the pressure drop and the flow rate.

Basic Theory

Laminar flow occurs when a fluid moves in parallel layers, with little to no mixing between the layers. In gas flow, viscosity plays a significant role, and under laminar conditions, the Hagen-Poiseuille equation can be used to describe the flow rate (Q) through a cylindrical conduit:

$Q = \frac{\pi \cdot r^4 \cdot (P_1 - P_2)}{8 \cdot \mu \cdot L}$

Where:

Q is the flow rate,
r is the radius of the conduit,
$P_1 - P_2$ is the pressure difference,
$\mu$ is the dynamic viscosity of the gas,
L is the length of the conduit.

Procedures in Excel

Step 1: Set Up Excel Table

Create a table in Excel with the following columns:

Diameter (D),
Radius (r),
Length (L),
Pressure Difference ( $P_1 - P_2$ ),
Dynamic Viscosity ( $\mu$ ),
Flow Rate (Q).

Step 2: Input Values

Enter the relevant values for the diameter, length, pressure difference, and dynamic viscosity in the corresponding columns.

Step 3: Excel Formula

In the “Flow Rate (Q)” column, use the Hagen-Poiseuille formula:

$Q = \frac{\pi \cdot (D/2)^4 \cdot (P_1 - P_2)}{8 \cdot \mu \cdot L}$

Step 4: Calculate Flow Rate

Apply the formula to calculate the flow rate for each row in the table.

Scenario and Calculation

Let’s consider a scenario with the following values:

Diameter (D): 0.02 m,
Length (L): 100 m,
Pressure Difference ( $P_1 - P_2$ ): 500 Pa,
Dynamic Viscosity ( $\mu$ ): 1.8e-5 Pa·s.

Using the Excel table and formula, we find the flow rate (Q).

Diameter (m)	Length (m)	Pressure Difference (Pa)	Dynamic Viscosity (Pa·s)	Flow Rate (m³/s)
0.02	100	500	1.8e-5	[Excel Formula]

Excel Result

After plugging in the values and applying the formula in Excel, the calculated flow rate will be displayed in the corresponding cell.

MATLAB Comparison

To solve the same problem in MATLAB, use the Hagen-Poiseuille equation and compare the results obtained from Excel with MATLAB calculations.

% MATLAB code for gas flow under laminar viscous conditions
D = 0.02;        % Diameter (m)
L = 100;         % Length (m)
P1_P2 = 500;     % Pressure Difference (Pa)
mu = 1.8e-5;     % Dynamic Viscosity (Pa·s)

Q_MATLAB = (pi * (D/2)^4 * P1_P2) / (8 * mu * L);

Compare the MATLAB result ( $Q_{\text{MATLAB}}$ ) with the Excel-calculated flow rate to ensure consistency.