Present the Darcy-Weisbach equation for head loss and identify the required parameters.

Present the Darcy-Weisbach equation for head loss and identify the required parameters.

• A. h_f = f (L/D) (v^2 / (2g)); f is friction factor depending on Re and roughness.
• B. h_f = f (D/L) (v^2 / (2g))
• C. h_f = ρ v^2 / (2g)
• D. h_f = f L v / (2g)

Answer: (A)

The key idea here is that friction in a pipe converts some of the fluid’s energy into heat, and the Darcy-Weisbach equation links that friction loss to the pipe’s geometry, the flow, and gravity. The head loss due to friction is written as h_f = f (L/D) (v^2/(2g)). Here, h_f is the loss of hydraulic head (meters of fluid), f is the Darcy friction factor (dimensionless) that captures how rough the pipe is and what regime the flow is in, L is the pipe length, D is the pipe diameter, v is the average fluid velocity, and g is the acceleration due to gravity. The term v^2/(2g) is the velocity head, representing the energy per unit weight associated with the flow’s speed. The ratio L/D scales the energy loss with the pipe’s length relative to its diameter, so longer pipes or smaller diameters (for the same velocity) produce more friction loss. The friction factor f depends on the Reynolds number and the relative roughness ε/D and is typically obtained from the Moody chart or the Colebrook equation. Why the other forms don’t fit: using the reciprocal D/L would misrepresent how geometry influences losses. Replacing L/D with the velocity head alone ignores the pipe length and friction factor, yielding only the dynamic pressure energy rather than the actual friction loss. Dropping the squared velocity and the dimensional structure, as in f L v /(2g), breaks both the physical meaning and the units needed for a length-valued head loss.

The key idea here is that friction in a pipe converts some of the fluid’s energy into heat, and the Darcy-Weisbach equation links that friction loss to the pipe’s geometry, the flow, and gravity. The head loss due to friction is written as h_f = f (L/D) (v^2/(2g)). Here, h_f is the loss of hydraulic head (meters of fluid), f is the Darcy friction factor (dimensionless) that captures how rough the pipe is and what regime the flow is in, L is the pipe length, D is the pipe diameter, v is the average fluid velocity, and g is the acceleration due to gravity. The term v^2/(2g) is the velocity head, representing the energy per unit weight associated with the flow’s speed. The ratio L/D scales the energy loss with the pipe’s length relative to its diameter, so longer pipes or smaller diameters (for the same velocity) produce more friction loss. The friction factor f depends on the Reynolds number and the relative roughness ε/D and is typically obtained from the Moody chart or the Colebrook equation.

Why the other forms don’t fit: using the reciprocal D/L would misrepresent how geometry influences losses. Replacing L/D with the velocity head alone ignores the pipe length and friction factor, yielding only the dynamic pressure energy rather than the actual friction loss. Dropping the squared velocity and the dimensional structure, as in f L v /(2g), breaks both the physical meaning and the units needed for a length-valued head loss.