The flowrate between tank A and tank B shown in Fig. P8.1 15...

The flowrate between tank $A$ and tank $B$ shown in Fig. P8.1 15 is to be increased by $30 \%$ (i.e., from $Q$ to $1.30 Q$ ) by the addition of a second pipe (indicated by the dotted lines) running from node $C$ to tank $B$. If the elevation of the free surface in tank $A$ is 25 ft above that in $\operatorname{tank} B,$ determine the diameter, $D,$ of this new pipe. Neglect minor losses and assume that the friction factor for each pipe is 0.02

Key Concepts

Darcy-Weisbach Equation

The Darcy-Weisbach equation relates the head loss due to friction in a pipe to flow parameters such as velocity, pipe length, diameter, and the friction factor. It is a fundamental tool in hydraulic engineering for predicting how energy losses occur due to friction as a fluid flows through pipes. This equation is essential for calculating pressure drops and sizing pipe diameters to achieve a desired flow rate.

Hydraulic Head

Hydraulic head is a measure of the potential energy available to drive the flow of a fluid, typically expressed as an elevation difference or pressure head. In many fluid flow problems, the difference in hydraulic head between two points is the driving force that causes the fluid to move. Understanding hydraulic head is crucial when analyzing systems like interconnected tanks, where the head differential determines the flow rate.

Pipe Flow Rate and Diameter Relationship

In pipe flow, the flow rate is influenced by the diameter of the pipe, among other factors. The relationship shows that, for a given head loss and friction factor, larger diameters generally permit higher flow rates due to lower velocities and reduced frictional losses. This principle plays a key role in hydraulic system design when adjusting the size of the pipe to achieve a desired change in flowrate.

Friction Factor in Pipe Flow

The friction factor quantifies the resistance to flow due to friction between the fluid and the pipe walls. It depends on the nature of the flow (laminar or turbulent), the roughness of the pipe, and flow conditions. In many engineering problems, a constant friction factor might be assumed for simplicity, allowing for the use of the Darcy-Weisbach equation to determine parameters such as head loss or required pipe diameter. This concept is central to understanding fluid resistance in pipe systems.

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