What flow rate (gpm) will be produced in a 75 -mmdiameter...

What flow rate (gpm) will be produced in a 75 -mmdiameter water pipe for which there is a pressure drop of 425 kPa over a 200 -m length? The pipe roughness is $2.5 \mathrm{mm}$. The water is at $0^{\circ} \mathrm{C}$

Key Concepts

Darcy-Weisbach Equation

This equation relates the pressure drop (or head loss) due to friction along a given length of pipe to the flow velocity, the characteristic diameter of the pipe, the friction factor, and the length of the pipe. It is a fundamental tool in fluid mechanics for determining how much pressure is lost as fluid moves through a conduit, making it essential when calculating flow rates under given pressure drops.

Friction Factor

The friction factor is a dimensionless number that quantifies the resistance to flow resulting from the roughness of the pipe’s internal surface and the flow’s Reynolds number. It is typically determined using empirical relationships like the Colebrook?White equation or by reference to the Moody chart, and it directly influences the magnitude of the pressure losses predicted by the Darcy-Weisbach equation.

Pipe Roughness

Pipe roughness represents the characteristic irregularities on the interior surface of a pipe. It plays a vital role in turbulent flow conditions because it increases frictional resistance, thereby affecting the friction factor and ultimately the pressure drop over a fixed length. The roughness is a crucial parameter when analyzing real-life flow conditions using theoretical models like the Darcy-Weisbach equation.

Pressure Drop and Head Loss

Pressure drop in a pipe system is the loss of pressure from one point to another due to frictional forces and turbulence within the fluid. This loss, often expressed as head loss in terms of fluid height, is directly related to the energy required to overcome friction. Knowing the pressure drop is essential for determining the flow rate through a pipe using energy balance equations.

Fluid Properties

Fluid properties, such as density and viscosity, are fundamental parameters in fluid flow analysis. They determine the behavior of the flow, including whether it is laminar or turbulent, and affect key calculations such as the Reynolds number and friction factor. Temperature-dependent properties, like those of water at specific temperatures, must be taken into account to ensure accurate flow predictions.

Unit Conversion

Unit conversion is an important step in solving engineering problems to ensure consistency and correctness in the final answer. Flow rates, pressure drops, lengths, and diameters may be provided in different units (e.g., SI units versus US customary units), so converting these correctly (for example, from cubic meters per second to gallons per minute) is crucial for obtaining a meaningful and accurate result.

Please give Ace some feedback