Understanding Pressure Drop in Laminar Pipe Flow

The question asks for the formula for the pressure drop per unit length (ΔP/L) in a pipe carrying fluid under laminar flow conditions. This pressure drop is stated to depend on the fluid's velocity (V), its dynamic viscosity (μ), and the pipe's diameter (d).

Laminar flow in a circular pipe is characterized by smooth, orderly fluid motion in layers. The relationship between pressure drop and flow parameters in this regime is described by the Hagen-Poiseuille equation. This equation is derived based on the principles of fluid dynamics, considering the balance between the pressure force driving the flow and the viscous forces resisting it.

Derivation of Pressure Drop per Unit Length

The Hagen-Poiseuille equation for the volume flow rate (Q) through a pipe of radius R (or diameter d = 2R) is given by:

\(Q = \frac{\pi R^4 \Delta P}{8 \mu L}\)

Substituting \(R = d/2\):

\(Q = \frac{\pi (d/2)^4 \Delta P}{8 \mu L} = \frac{\pi d^4 \Delta P}{16 \cdot 8 \mu L} = \frac{\pi d^4 \Delta P}{128 \mu L}\)

The average velocity (V) of the fluid in the pipe is defined as the volume flow rate divided by the cross-sectional area (A) of the pipe. The area is \(A = \pi R^2 = \pi (d/2)^2 = \frac{\pi d^2}{4}\).

\(V = \frac{Q}{A} = \frac{Q}{\frac{\pi d^2}{4}}\)

We can express Q in terms of V and A:

\(Q = V \cdot A = V \cdot \frac{\pi d^2}{4}\)

Now, substitute this expression for Q back into the Hagen-Poiseuille equation:

\(V \cdot \frac{\pi d^2}{4} = \frac{\pi d^4 \Delta P}{128 \mu L}\)

We want to find the pressure drop per unit length, \(\frac{\Delta P}{L}\). Let's rearrange the equation:

\(\frac{\Delta P}{L} = \frac{V \cdot \frac{\pi d^2}{4} \cdot 128 \mu}{\pi d^4}\)

\(\frac{\Delta P}{L} = \frac{128 \pi \mu V d^2}{4 \pi d^4}\)

Now, cancel out common terms (\(\pi\), \(d^2\)) and simplify the constants:

\(\frac{\Delta P}{L} = \frac{128 \mu V}{4 d^2}\)

\(\frac{\Delta P}{L} = \frac{32 \mu V}{d^2}\)

This formula shows that the pressure drop per unit length in laminar flow is directly proportional to the fluid's dynamic viscosity (μ) and the average velocity (V), and inversely proportional to the square of the pipe's diameter (d).

Evaluating the Options

Let's compare our derived formula \(\frac{32 \mu V}{d^2}\) with the given options:

• Option 1: \(\frac{d^2}{32\mu V}\) - This is incorrect. It shows an inverse relationship with viscosity and velocity and a direct relationship with diameter squared, which contradicts the derivation.
• Option 2: \(\frac{32\ \mu VL}{\gamma d^2}\) - This includes the length L in the numerator, but the question asks for pressure drop per unit length (ΔP/L). Also, it includes \(\gamma\) (likely specific weight), which is not a factor in the direct pressure-viscosity-velocity-diameter relationship for \(\Delta P/L\).
• Option 3: \(\frac{32\ \mu V}{d^2}\) - This exactly matches our derived formula for \(\frac{\Delta P}{L}\).
• Option 4: \(\frac{8\ \mu V}{d^2}\) - This has the correct dependency on μ, V, and d, but the constant is 8 instead of 32. This value (8) is related to the shear stress or pressure drop for flow between parallel plates, or might arise from confusion with related formulas.

Based on the derivation from the Hagen-Poiseuille equation, option 3 is the correct formula for the pressure drop per unit length in laminar flow.

Revision Table: Key Formulas in Pipe Flow

Additional Information on Laminar Pipe Flow and Pressure Drop

The Hagen-Poiseuille equation and the derived formula for pressure drop per unit length \(\frac{\Delta P}{L} = \frac{32 \mu V}{d^2}\) are fundamental results for understanding viscous flow in pipes.

• Assumptions: This formula is valid for fully developed, steady, incompressible laminar flow in a horizontal circular pipe. It assumes no slip at the pipe wall and a constant viscosity.
• Velocity Profile: In fully developed laminar flow, the velocity profile across the pipe cross-section is parabolic, with zero velocity at the wall and maximum velocity at the center. The maximum velocity is twice the average velocity (\(V_{max} = 2V\)).
• Energy Losses: The pressure drop in laminar flow is entirely due to viscous shear stresses within the fluid and between the fluid and the pipe wall. These losses are often referred to as major losses or friction losses.
• Reynolds Number: Laminar flow typically occurs when the Reynolds number (\(Re = \frac{\rho V d}{\mu}\)) is below approximately 2000. Above this value, the flow usually transitions to turbulent flow, where the relationship between pressure drop and velocity becomes non-linear, often proportional to \(V^2\).
• Practical Applications: This formula is used in designing piping systems for low-viscosity fluids at low speeds or high-viscosity fluids where laminar conditions are desired or unavoidable (e.g., some lubrication systems, flow in small capillaries).

Understanding the factors affecting pressure drop is crucial for calculating pumping power requirements and analyzing fluid distribution networks.