Question

In Figure 14-49, water flows through a horizontal pipe and then out into the atmosphere at a speed${\mathit{v}}_{\mathbf{1}}\mathbf{=}\mathbf{15}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$. The diameters of the left and right sections of the pipe are$\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathit{c}\mathit{m}$and $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathit{c}\mathit{m}$.

(a) What volume of water flows into the atmosphere during a $\mathbf{10}\mathbf{}\mathit{m}\mathit{i}\mathit{n}$period? In the left section of the pipe?

(b) What is the speed v2?

(c) What is the gauge pressure?

Short answer

1. The volume of water that flows into the atmosphere during 10 min period is$6.4m^3$.

2. The speed of water flowing into the pipe is $5.4m/s$.

3. The gauge pressure of water is $9.8\times {10}^{4}Pa$.

Step-by-step solution

Given data

1. The speed of water flows into the atmosphere, $v_o=15m/s$.

2. The diameter of the inlet ,$d_i=5.0cm$.

3. The diameter of the outlet,$d_o=3.0cm$.

4. The pipe is horizontal.

5. The pipe outlet is open to the atmosphere.

Determining the concept of volume rate of flow

Find the volume rate of flow per sec and the speed of water flowing into the pipe by using the continuity equation. Then, find the gauge pressure of water by using Bernoulli’s principle. According to Bernoulli’s equation, as the speed of a moving fluid increases, the pressure within the fluid decreases.

Formulae are as follows:

$p\forall +\frac{1}{2}p{g}^{2}h+\mathrm{co}ns\tan t$

$A_v=cons\tan t=R_v=Volumerateofflowpersec$

Where, p is pressure, v is velocity, h is height, g is the acceleration due to gravity, h is height, A is the area, R is the rate of flow, and $\rho$is density.

(a) Determining the volume of water that flows into the atmosphere during 10 min period

The volume rate of flow of water can be calculated by using the continuity equation.

So,

$$\begin{gathered} R_v=Volumerateofflowpersec=A_v \\ {A}_v=\frac{{\mathrm{\pi d}}_{\mathrm{o}}^2}{4}{v}_o \\ =\frac{\mathrm{\pi }{\left(3.0\times {10}^{-2}\mathrm{m}\right)}^2\times 15\mathrm{m}/\mathrm{s}}{4} \\ =1.1\times {10}^{-2}{m}^3/s \end{gathered}$$

Hence, the volume of water flowing into the atmosphere in 10 min is given by,

$$\begin{gathered} V=R_v\times 10\times 60 \\ =1.1\times {10}^{-2}{m}^3/s\times 10\times 60 \\ =6.36 \\ \approx 6.4m^3 \end{gathered}$$

Hence, the volume of water that flows into the atmosphere during 10 min period is localid="1657694542906" $6.4m^3$.

(b) Determining the speed of water flowing into the pipe

The flow of water obeys the continuity equation.

So,

$A_i{v}_i=A_o{v}_o$

Hence,

$$\begin{gathered} v_i=\frac{A_o{v}_o}{A_i} \\ =\frac{{\left(3.0\times {10}^{-2}{m}^2\right)}^2\times 15m/s}{{\left(5.0\times {10}^{-2}{m}^2\right)}^2} \\ =5.4m/s \end{gathered}$$

Hence, the speed of water flowing into the pipe is $5.4m/s$.

(c) Determining the gauge pressure of water

Now,

Gauge pressure = actual pressure – atmospheric pressure

Gauge pressure $\left(∆p\right)$can be calculated using Bernoulli’s principle.

$p_{i}v+\frac{1}{2}\rho g_i^{2}h+p_1=p_{o}v+\frac{1}{2}\rho g_o^{2}h+p_2$

Here,

$h_1=h_{2}and{p}_i=p_o$

Hence,

role="math" localid="1657694208416" $$\begin{gathered} p_{o}v=p_{i}v+\frac{1}{2}\left(g_i^2-g_o^2\right) \\ =1.01\times {10}^{5}Pa+\frac{1}{2}\left(1000kg/m^3\right)\left({\left(15m/s\right)}^2-{\left(5.4m/s\right)}^2\right) \\ =1.99\times {10}^{5}Pa \\ =1.97atm \end{gathered}$$

Using given information and simplifying, we get

$$\begin{gathered} ∆p=p_i-p_o \\ =\left(1.97-1.00\right)atm \\ =0.97atm \\ =9.8\times {10}^{4}Pa \end{gathered}$$

Hence, the gauge pressure of water is $9.8\times {10}^{4}Pa$.