Hagen Poiseuille Equation Formula and Applications

The Hagen-Poiseuille equation is a fundamental formula that describes the flow of a viscous fluid through a cylindrical pipe or tube. It was formulated by Gotthilf Hagen and Jean Léonard Marie Poiseuille in the 19th century, and it remains a cornerstone of fluid dynamics.

Theoretically and practically it has been proved that when a fluid flows viscously in a pipe of cross-section then according to the Hagen Poiseuille equation. 

Hagen Poiseuille equation is much more useful for determining when the other terms are known.

ΔP = 32Lvμ/gcD²

Where, 

ΔP = Pressure drop Lb/sq ft.

L = Length of tube 

μ = Viscosity (Lb - mass/ft sec)

gc = 32.2 (Lb - mass)F/(Lb - force)sec² 

v = Velocity

D = Diameter of the tube 

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Friction Loss in Pipe

When there is liquid flow in a pipe, friction resistance arises due to the roughness of the surface of the pipe. As a result, some energy of the liquid will be lost, which is called friction head loss in pipes. It is represented by hf.

Friction loss in pipe formula 

hf = 4flv²/2gd  

Where 

hf = Friction head loss in pipe 

f = Friction factor or coefficient of friction

l = Length of pipe

d = Diameter of pipe

v = Velocity of fluids 

Effect of Roughness in pipe and tube 

Generally, every pipe and tube is rough. Pipes are made very rough as compared to tubes. Due to this roughness of the pipe, the head loss of the pipe is more than that of the smooth pipe. 

The height of roughness is represented by K, it is called the roughness parameter. Friction factor, Reynolds Number, and Relative roughness are formed on the basis of dimensional analysis. 

Relative Roughness = K/D 

Where

K = Height of single unit of roughness

D = Diameter of pipe 

Note:- (1) If the clear pipe is smoothened then the value of the friction factor decreases. 

(2) If the friction factor does not decrease on re-smoothing the pipe for a fixed Rn, it is said to be hydraulically smooth.

The Hagen-Poiseuille Equation Applications

The Hagen-Poiseuille equation finds numerous practical applications in various industries.

1. Plumbing and Water Systems

Engineers use this equation to design efficient water supply systems, ensuring smooth water flow through pipes of different diameters.

2. Medical Applications

Understanding blood flow in vessels is essential for medical professionals. The Hagen-Poiseuille equation helps in evaluating blood flow rate and identifying potential issues.

3. Industrial Processes

In industrial settings, this equation aids in optimizing fluid flow through pipelines, ensuring the smooth transportation of chemicals and other materials.

4. Engineering Design

Engineers rely on the equation to design pipelines, optimize heat exchangers, and create systems that involve fluid transportation.