Understanding Friction Factor in Laminar Pipe Flow

When a fluid flows through a pipe, it experiences resistance due to viscosity, known as friction. This frictional resistance depends on several factors, including the flow regime (laminar or turbulent), the fluid properties, and the pipe's characteristics.

What is Laminar Flow?

Laminar flow is a flow regime characterized by smooth, orderly movement of fluid particles in layers or laminae. It typically occurs at low velocities and when the fluid has relatively high viscosity. In pipe flow, the transition from laminar to turbulent flow usually happens around a Reynolds number of 2300.

Key Concepts: Friction Factor and Reynolds Number

• Friction Factor (f): This is a dimensionless quantity used in the Darcy-Weisbach equation to calculate the pressure drop or head loss due to friction along a given length of pipe. It accounts for the frictional losses in the pipe.
• Reynolds Number (Re): This is a dimensionless number that indicates whether the fluid flow is laminar, turbulent, or transitional. It is defined as the ratio of inertial forces to viscous forces. For flow in a pipe, it is typically calculated as:

\[ \text{Re} = \frac{\rho v D}{\mu} \]

Where:

• \( \rho \) is the fluid density
• \( v \) is the average fluid velocity
• \( D \) is the pipe diameter
• \( \mu \) is the dynamic viscosity of the fluid

Relationship between Friction Factor and Reynolds Number for Laminar Flow

For fully developed laminar flow in a smooth, circular pipe, the Darcy friction factor (\(f\)) has a specific theoretical relationship with the Reynolds number (\(\text{Re}\)). This relationship is derived from the Hagen-Poiseuille equation and is given by:

\[ f = \frac{64}{\text{Re}} \]

This equation is valid for \( \text{Re} < 2300 \), which is the typical range for laminar flow in pipes.

Analyzing the Options

Let's examine how the formula \( f = \frac{64}{\text{Re}} \) relates to the given options:

• Option 1: Varies linearly with the Reynold's number
  This would mean \( f = a \cdot \text{Re} + b \), where \(a\) and \(b\) are constants. Our formula \( f = 64 / \text{Re} \) does not fit this form unless \( a=0 \), but then \(f\) would be constant, which is not the case. So, this option is incorrect.
• Option 2: Is exactly equal to the square of Reynold's number
  This would mean \( f = \text{Re}^2 \). Our formula is \( f = 64 / \text{Re} \). These are clearly different. So, this option is incorrect.
• Option 3: Is independent of Reynold's number
  This would mean \( f \) is a constant value, not dependent on \( \text{Re} \). Our formula \( f = 64 / \text{Re} \) shows that \( f \) explicitly depends on \( \text{Re} \). So, this option is incorrect. (Note: For fully turbulent flow in very rough pipes, \( f \) can become nearly independent of \( \text{Re} \) but depends on roughness).
• Option 4: Varies linearly with the inverse of Reynold's number
  The inverse of Reynolds number is \( 1/\text{Re} \). Our formula \( f = 64 \cdot \frac{1}{\text{Re}} \) shows that \( f \) is directly proportional to \( 1/\text{Re} \). This is a linear relationship where the 'slope' is 64 and the 'y-intercept' is 0 (if we plot \( f \) versus \( 1/\text{Re} \)). So, this option accurately describes the relationship for laminar flow.

Based on the theoretical relationship \( f = 64 / \text{Re} \), the friction factor for laminar flow through a pipe varies linearly with the inverse of the Reynolds number.

Revision Table: Key Terms in Pipe Flow

Additional Information: Beyond Laminar Flow

The relationship \( f = 64 / \text{Re} \) is specific to laminar flow in circular pipes. For other flow regimes and pipe conditions:

• Transitional Flow (2300 < Re < 4000): The flow characteristics are unpredictable, fluctuating between laminar and turbulent. The friction factor is not easily defined by a simple formula and depends on various factors like entrance conditions and pipe roughness.
• Turbulent Flow (Re > 4000): The relationship between \( f \) and \( \text{Re} \) becomes more complex. It depends on both the Reynolds number and the relative roughness of the pipe surface (\( \epsilon/D \)). This relationship is often represented graphically by the Moody chart or calculated using empirical equations like the Colebrook-White equation or simpler approximations like the Haaland equation.
• Darcy vs. Fanning Friction Factor: There are two common definitions of the friction factor. The Darcy-Weisbach friction factor (\( f \)) used here relates head loss \( h_L \) to velocity head \( v^2/(2g) \) by \( h_L = f \frac{L}{D} \frac{v^2}{2g} \). The Fanning friction factor (\( C_f \) or \( f_F \)) is sometimes used, where \( C_f = f/4 \). The formula for laminar flow using the Fanning friction factor would be \( C_f = 16 / \text{Re} \). The question implies the use of the Darcy friction factor.