Water at 20^∘ C flows through a 5 -cm-diameter pipe that has...

Water at $20^{\circ} \mathrm{C}$ flows through a 5 -cm-diameter pipe that has a $180^{\circ}$ vertical bend, as in Fig. $\mathrm{P} 3.43 .$ The total length of pipe between flanges 1 and 2 is $75 \mathrm{cm}$. When the weight flow rate is $230 \mathrm{N} / \mathrm{s}, p_{1}=165 \mathrm{kPa}$ and $p_{2}=134 \mathrm{kPa}$. Neglecting pipe weight, determine the total force that the flanges must withstand for this flow.

Water at $20^{\circ} \mathrm{C}$ flows through a 5 -cm-diameter pipe that has
a $180^{\circ}$ vertical bend, as in Fig. $\mathrm{P} 3.43 .$ The total length of pipe between flanges 1 and 2 is $75 \mathrm{cm}$. When the weight flow rate is $230 \mathrm{N} / \mathrm{s}, p_{1}=165 \mathrm{kPa}$ and $p_{2}=134 \mathrm{kPa}$. Neglecting pipe weight, determine the total force that the flanges must withstand for this flow.

Key Concepts

Mass Flow Rate and Momentum Flux

The mass flow rate, defined as the weight or mass of the fluid passing through a cross-section of the pipe per unit time, is directly linked to the momentum flux entering or exiting the control volume. In high-speed flows or flows through bends, even a modest mass flow rate can result in significant momentum flux changes due to differences in velocity magnitude or direction. This concept is essential for quantifying the thrust or reaction forces generated by the fluid as it negotiates bends or changes in pipe geometry.

Pressure Forces on Fluid Boundaries

In fluid flow problems, pressure is the normal force exerted per unit area on the boundaries of the control volume. The pressure difference between different points along the flow leads to net forces acting on the fluid in the direction of decreasing pressure. In applications such as determining the forces on pipe flanges, these pressure forces, when integrated over the surface areas of the inlets and outlets, contribute significantly to the total force that the structural supports must withstand.

Control Volume Analysis

A control volume is an imaginary fixed region in space through which fluid flows, allowing the analysis of forces, momentum, and energy changes. By applying the conservation laws over this control volume, one can balance the inlet and outlet conditions, as well as any body forces acting on the fluid. This method is especially useful in complex geometries, like a 180° vertical bend in a pipe, where it simplifies the analysis of forces exerted on the pipe flanges due to the momentum changes and pressure variations.

Conservation of Linear Momentum

This principle states that the net force acting on a control volume is equal to the rate of change of momentum flux through that control volume. When fluid flows through a bend or change in direction, its momentum changes direction, and the resulting force required to cause that change is determined by applying this conservation law. The analysis involves accounting for both the momentum entering and leaving the control volume and equating their difference to the sum of external forces (such as pressure forces and gravity if applicable).

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