Dimensional Analysis of Pipe Flow

Pressure Drops in a Pipe Flow

There is a reduction in pressure as a fluid travels through a pipe. This pressure loss occurs due to several factors, primarily friction between the fluid and the pipe walls, changes in elevation, and the presence of fittings and valves that create additional resistance.

When fluid flows through a pipe, it encounters frictional forces from the pipe's inner surface, which oppose the motion of the fluid. This frictional resistance is influenced by the pipe's length, diameter, roughness, and the fluid's velocity and viscosity. The longer and narrower the pipe, the greater the frictional resistance, leading to a higher pressure drop.

Additionally, changes in elevation within the piping system can affect the pressure. When fluid flows uphill, it loses pressure due to the work done against gravity. Conversely, when flowing downhill, the fluid gains pressure. However, the net effect on pressure drop depends on the overall elevation change in the system.

Obstructions such as bends, elbows, valves, and other fittings introduce localized resistance, further contributing to the pressure drop. These components disrupt the smooth flow of the fluid, causing turbulence and energy loss.

The flow regime, whether laminar or turbulent, also plays a crucial role. In laminar flow, which occurs at lower velocities, the fluid moves in smooth, orderly layers, and the pressure drop is primarily due to viscous forces. In turbulent flow, which occurs at higher velocities, the fluid motion is chaotic, resulting in higher frictional losses and a greater pressure drop.

To quantify the pressure drop, engineers often use the Darcy-Weisbach equation, which relates the pressure drop to the friction factor, pipe length, diameter, fluid density, and flow velocity. For water flow in pipes, the Hazen-Williams equation is another commonly used formula, particularly in civil engineering applications.

Understanding and calculating pressure drop is essential for designing efficient piping systems in various industries, including water supply, oil and gas, and HVAC systems. Properly accounting for pressure drop ensures that the system can deliver the required flow rates and pressures, maintaining operational efficiency and safety.

Pressure Drop Variables

• Mean velocity (V)
• Diameter (D)
• Pipe length (ℓ)
• Wall Roughness (ε)
• Viscosity (μ)
• Density (ρ)

Pressure Drop Equation

Δp = p₁ - p₂

Functional Form of Pressure Drop

Δp = F(V, D, ℓ, ε, μ, ρ)

Dimensional Analysis Equation

$$ \frac{Δp}{\frac{1}{2}V^2} = \phi\left(\frac{ρVD}{μ}, \frac{ℓ}{D}, \frac{ε}{D}\right) $$

This equation shows how the pressure drop can be expressed as a dimensionless function of other dimensionless groups.

All about Wall Roughness (ε)

What is Wall Roughness (ε)?

Wall Roughness (ε) is a measure of the irregularities or roughness present on the inner surface of a pipe. These irregularities can significantly affect the flow of fluid through the pipe, leading to increased friction and pressure drop.

How Wall Roughness (ε) is Computed

Wall roughness is typically quantified using the absolute roughness (ε), which is the average height of the irregularities on the pipe's inner surface. The value of ε is determined through empirical measurements and varies depending on the material and condition of the pipe.

Common Values of Absolute Roughness (ε)

• Commercial Steel: ε ≈ 0.045 mm
• Cast Iron: ε ≈ 0.26 mm
• Concrete: ε ≈ 0.3 - 3 mm
• Plastic (PVC): ε ≈ 0.0015 mm

Dimensionless Roughness (ε/D)

In fluid dynamics, the roughness is often normalized by the pipe diameter (D) to form a dimensionless roughness:

Dimensionless Roughness: ε/D

This dimensionless roughness is used in various empirical correlations and equations, such as the Moody chart, to determine the friction factor (f) for flow in pipes.

Example Calculation

Given:

• Pipe diameter (D) = 0.1 m
• Absolute roughness (ε) = 0.001 m

The dimensionless roughness is:

ε/D = 0.001 m / 0.1 m = 0.01

Significance in Fluid Flow

The roughness affects the flow regime (laminar, transitional, or turbulent) and the friction factor, which in turn influences the pressure drop. Higher roughness values generally lead to higher friction factors and greater pressure drops.