In Fig. 14-49, water flows through a horizontal pipe and...

In Fig. $14-49,$ water flows through a horizontal pipe and then out into the atmosphere at a speed $v_{1}=15 \mathrm{~m} / \mathrm{s}$. The diameters of the left and right sections of the pipe are $5.0 \mathrm{~cm}$ and $3.0 \mathrm{~cm} .$ (a) What volume of water flows into the atmosphere during a 10 min period? In the Ieft section of the pipe, what are (b) the speed $v_{2}$ and (c) the gauge pressure?

In Fig. $14-49,$ water flows through a horizontal pipe and then out into the atmosphere at a speed $v_{1}=15 \mathrm{~m} / \mathrm{s}$. The diameters of the left and right sections of the pipe are $5.0 \mathrm{~cm}$ and $3.0 \mathrm{~cm} .$
(a) What volume of water flows into the atmosphere during a 10 min period? In the Ieft section of the pipe, what are (b) the speed $v_{2}$ and (c) the gauge pressure?

Key Concepts

Continuity Equation

This principle states that for an incompressible fluid in steady flow, the mass or volume flow rate must remain constant throughout the flow field. It relates the product of the cross-sectional area and velocity at any two points along the streamline, ensuring that any change in the area of a pipe results in an inverse change in velocity.

Volumetric Flow Rate

Volumetric flow rate is defined as the volume of fluid passing through a given surface per unit time. It is calculated as the product of the velocity of the fluid and the cross-sectional area through which it flows, and is a key parameter in determining the overall transport of fluid in a system.

Bernoulli's Equation

Bernoulli's equation provides a relationship between pressure, velocity, and elevation in a flowing fluid. Under the assumptions of steady, incompressible, and non-viscous flow, it states that the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.

Gauge Pressure

Gauge pressure refers to the pressure measurement relative to ambient atmospheric pressure rather than absolute pressure. It is used to assess the pressure difference in a system, and is particularly useful when analyzing fluid systems where the absolute pressure is not as significant as the pressure difference driving the flow.

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