Question

Darcy's equation is applicable for

• A. Laminar flow only
• B. Both laminar and turbulent flows
• C. Turbulent flow only
• D. Open channel flow only

Hint:

Darcy's equation is used for calculating frictional head loss in pipe flow.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the flow regime for which Darcy's Law (or Darcy's equation) is applicable. It's important to distinguish between Darcy's Law for flow through porous media and the Darcy-Weisbach equation for pipe friction. Given the context of hydrology and soil physics, "Darcy's equation" almost always refers to Darcy's Law for porous media flow.
Step 2: Detailed Explanation:
1. Darcy's Law describes the flow of a fluid through a porous medium (like groundwater flow through soil). It states that the discharge rate (Q) is proportional to the hydraulic gradient (i) and the cross-sectional area (A). \[ Q = KiA \] or, in terms of velocity (v = Q/A): \[ v = Ki \] where K is the hydraulic conductivity. 2. This law is an empirical one, derived from experiments by Henry Darcy. It is fundamentally a linear relationship between flow velocity and hydraulic gradient. 3. This linear relationship holds true only under conditions where viscous forces are dominant and inertial forces are negligible. This is the defining characteristic of laminar flow. 4. In porous media, the flow regime is typically characterized by the Reynolds number (Re). Darcy's Law is valid for low Reynolds numbers, typically when Re<1. This corresponds to laminar flow conditions. 5. At higher velocities (higher Re), the flow becomes non-linear and eventually turbulent, and Darcy's Law is no longer valid. Other equations, like the Forchheimer equation, are used to describe flow in this regime. 6. Darcy-Weisbach Equation: It is important not to confuse Darcy's Law with the Darcy-Weisbach equation, $h_f = f \frac{L}{D} \frac{V^2}{2g}$, which is used to calculate friction losses in pipe flow. The Darcy-Weisbach equation itself is applicable to both laminar and turbulent flows, but the friction factor 'f' is calculated differently for each regime. However, the question simply says "Darcy's equation," which in its original and most common sense refers to Darcy's Law for porous media. Step 3: Final Answer based on standard interpretation:
Darcy's Law (often called Darcy's equation) for flow in porous media is only valid for laminar flow conditions. Therefore, option (A) is the correct answer. The checkmark in the provided image is on option (B), which would be correct if the question referred to the Darcy-Weisbach equation. Given the ambiguity, and that Darcy's original work was on porous media, (A) is the more historically and fundamentally correct answer for "Darcy's equation". If this is an exam on pipe flow, (B) could be the intended answer. Let's assume the context is soil/groundwater. Revisiting with the provided key: The key marks (B) as correct. This implies the question is referring to the Darcy-Weisbach equation for pipe friction, not Darcy's Law for porous media. Let's write the solution for that interpretation. Solution assuming Darcy-Weisbach Equation:
The Darcy-Weisbach equation is a fundamental equation in pipe hydraulics used to calculate head loss due to friction. It is given by: \[ h_f = f \frac{L}{D} \frac{V^2}{2g} \] This equation is dimensionally consistent and is considered the most accurate formula for pipe friction. Its key feature is its universal applicability to different flow regimes through the use of the Darcy friction factor, 'f'. - For laminar flow (Re<2000), the friction factor is a simple function of the Reynolds number: $f = \frac{64}{Re}$. - For turbulent flow (Re>4000), the friction factor depends on both the Reynolds number and the relative roughness of the pipe, and is typically found using the Moody chart or empirical equations (like the Colebrook-White equation). Since the equation's structure is valid for all flow regimes (with the value of 'f' changing accordingly), the Darcy-Weisbach equation is applicable to both laminar and turbulent flows. Step 4: Final Answer (following the key):
The Darcy-Weisbach equation is applicable for both laminar and turbulent flows in pipes. Therefore, option (B) is the correct answer.