# At one point in a pipeline the water’s speed is 3.00 m/s and the gauge pressure

At one point in a pipeline the water’s speed is 3.00 m/s and the gauge pressure is $$\displaystyle{5.00}\times{10}^{{{4}}}{P}{a}$$. Find the gauge pressure at a second point in the line, 11.0 m lower than the first, if the pipe diameter at the second point is twice that at the first.

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Tuthornt

Step 1
The diameters of the two pipe cross-sections are $$\displaystyle{d}_{{{1}}}$$ and $$\displaystyle{d}_{{{2}}}={2}{d}_{{{1}}}$$. The area of the first is $$\displaystyle{A}_{{{1}}}=\pi{{d}_{{{1}}}^{{{2}}}}$$ and of the the second $$\displaystyle{A}_{{{2}}}=\pi{{d}_{{{2}}}^{{{2}}}}=\pi{4}{{d}_{{{1}}}^{{{2}}}}={4}{A}{1}$$
Step 2
Let's first determine the relation between the velocities based on the continuity equation $$\displaystyle{v}_{{{1}}}{A}_{{{1}}}={v}_{{{2}}}{A}_{{{2}}}$$ and the fact that $$\displaystyle{A}_{{{2}}}={4}{A}_{{{1}}}$$ from which we can write
$$\displaystyle{v}_{{{2}}}={\frac{{{v}_{{{1}}}{A}_{{{1}}}}}{{{A}_{{{2}}}}}}$$
$$v_{2}=\frac{v_{1}}{4}$$
Step 3
Since the atmospheric pressures are the same we can write Bernoulli's equation as
$$\displaystyle{p}_{{{1}}}+\rho{g}{y}_{{{1}}}+{\frac{{{1}}}{{{2}}}}\rho{{v}_{{{1}}}^{{{2}}}}={p}_{{{2}}}+\rho{g}{y}_{{{2}}}+{\frac{{{1}}}{{{2}}}}\rho{{v}_{{{2}}}^{{{2}}}}$$ where $$\displaystyle{p}_{{{1}}}$$ and $$\displaystyle{p}_{{{2}}}$$ are gauge pressures. We can now insert $$\displaystyle{v}_{{{2}}}=\frac{{v}_{{{1}}}}{{4}}$$ to get
$$\displaystyle{p}_{{{2}}}={p}_{{{1}}}+\rho{g}{\left({y}_{{{1}}}-{y}_{{{2}}}\right)}+{\frac{{{1}}}{{{2}}}}\rho{\left({{v}_{{{1}}}^{{{2}}}}-\frac{{{v}_{{{1}}}^{{{2}}}}}{{16}}\right)}$$
$$\displaystyle{p}_{{{2}}}={p}_{{{1}}}+\rho{g}{\left({y}_{{{1}}}-{y}_{{{2}}}\right)}+{\frac{{{15}}}{{{32}}}}\rho{{v}_{{{1}}}^{{{2}}}}$$
Step 4
At this point we can insert all the given values to get the following
$$\displaystyle{p}_{{{2}}}={5}\times{10}^{{{4}}}+{10}^{{{3}}}\times{9.81}\times{11}+{10}^{{{3}}}\times{\frac{{{15}}}{{{32}}}}\times{3}^{{{2}}}$$
$$\displaystyle{p}_{{{2}}}={162}{k}{P}{a}$$