Fresh water flows horizontally from pipe section 1 of...

Fresh water flows horizontally from pipe section 1 of cross-sectional area $A_{1}$ into pipe section 2 of cross-sec tional area $A_{2}$. Figure $14-52$ gives a plot of the pressure difference $p_{2}-p_{1}$ versus the inverse area squared $A_{1}^{-2}$ that would be expected for a volume flow rate of a certain value if the water flow were laminar under all circumstances. The scale on the vertical axis is set by $\Delta p_{s}=300 \mathrm{kN} / \mathrm{m}^{2} .$ For the conditions of the figure, what are the values of (a) $A_{2}$ and (b) the volume flow rate? ${ }^{\infty} 69$ A liquid of density $900 \mathrm{~kg} / \mathrm{m}^{3}$ flows through a horizontal pipe that has a cross-sectional area of $1.90 \times 10^{-2} \mathrm{~m}^{2}$ in region $A$ and a cross-sectional area of $9.50 \times 10^{-2} \mathrm{~m}^{2}$ in region $B$. The pressure difference between the two regions is $7.20 \times 10^{3} \mathrm{~Pa}$. What are (a) the volume flow rate and (b) the mass flow rate?

Fresh water flows horizontally from pipe section 1 of cross-sectional area $A_{1}$ into pipe section 2 of cross-sec tional area $A_{2}$. Figure $14-52$ gives a plot of the pressure difference $p_{2}-p_{1}$ versus the inverse area squared $A_{1}^{-2}$ that would be expected for a volume flow rate of a certain value if the water flow were laminar under all circumstances. The scale on the vertical axis is set by $\Delta p_{s}=300 \mathrm{kN} / \mathrm{m}^{2} .$ For the conditions
of the figure, what are the values of
(a) $A_{2}$ and (b) the volume flow rate?
${ }^{\infty} 69$ A liquid of density $900 \mathrm{~kg} / \mathrm{m}^{3}$
flows through a horizontal pipe that has a cross-sectional area of $1.90 \times 10^{-2} \mathrm{~m}^{2}$ in region $A$ and a cross-sectional area of $9.50 \times 10^{-2} \mathrm{~m}^{2}$ in region $B$. The pressure difference between the two regions is $7.20 \times 10^{3} \mathrm{~Pa}$. What are (a) the volume flow rate and (b) the mass flow rate?

Key Concepts

Bernoulli's Equation

Bernoulli's equation is a statement of the conservation of energy for flowing fluids, relating pressure, kinetic energy, and gravitational potential energy along a streamline. It indicates that an increase in velocity (and hence kinetic energy) in a flowing fluid results in a decrease in pressure, assuming negligible viscous effects. This fundamental principle helps analyze how fluid speed and pressure change when the cross-sectional area of a pipe varies.

Continuity Equation

The continuity equation expresses the conservation of mass in steady, incompressible flow, stating that the product of the cross-sectional area and the local velocity remains constant along the flow. This relationship implies that if a fluid passes from a wide section of pipe into a narrow one, its velocity must increase to maintain the same volume flow rate, and vice versa.

Volume Flow Rate

Volume flow rate is the volume of fluid that passes through a cross-sectional area per unit time, calculated as the product of the area and the fluid velocity. This concept is central to analyzing fluid systems, as it connects the geometrical dimensions of the conduit with the dynamic behavior of the fluid under various pressure conditions.

Mass Flow Rate

Mass flow rate measures the mass of fluid passing through a cross-sectional area per unit time and is obtained by multiplying the volume flow rate by the fluid’s density. This concept is important for applications where the transport of physical mass is critical, and it ensures that conservation of mass principles are adhered to in fluid flow analyses.

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