A high velocity water jet of cross section area = 0.01 m2 and velocity = 35 m/s enters a pipe filled with stagnant water. The diameter of the pipe is 0.32 m. This high velocity water jet entrains additional water from the pipe and the total water leaves the pipe with a velocity 6m/s as shown in the figure. The flow rate of entrained water is _____________ liters/s (round off to two decimal places).

The correct answer is 132.55.

Step-by-Step Explanation:

To find the flow rate of the entrained water, we apply the principle of conservation of mass (or volume, since water is an incompressible fluid) for the system. The total volume flow rate of water leaving the pipe must be equal to the sum of the flow rates entering the pipe.

Let:
- $A_i$ be the cross-sectional area of the high-velocity water jet = 0.01 m²
- $V_i$ be the velocity of the water jet = 35 m/s
- $d_o$ be the outer diameter of the pipe = 0.32 m
- $V_o$ be the exit velocity of the total water leaving the pipe = 6 m/s
- $\dot{Q}$ be the volume flow rate of the entrained water (in m³/s)

The cross-sectional area of the exit pipe $A_o$ is calculated as:

$A_o=\frac{\pi d_o^2}{4}$

By the conservation of flow rate, the total outlet flow rate is the sum of the inlet jet flow rate and the entrained water flow rate:

$\dot{Q}+A_i{V}_i=A_o{V}_o$

Rearranging the equation to solve for the entrained flow rate $\dot{Q}$:

$\dot{Q}=(\frac{\pi d_o^2}{4}\times V_o)-A_i{V}_i$

Substituting the given values into the equation:

$\dot{Q}=\{\frac{\pi }{4}\times {\left(0.32\right)}^2\times 6\}-(0.01\times 35)$

Let's calculate the terms:
- Outlet flow rate: $A_o{V}_o\approx 0.0804248\times 6\approx 0.48255$ m³/s
- Inlet jet flow rate: $A_i{V}_i=0.01\times 35=0.35$ m³/s

Subtracting the inlet jet flow rate from the total outlet flow rate:

$\dot{Q}=0.48255-0.35=0.13255$ m³/s

To convert the volumetric flow rate from cubic meters per second (m³/s) to liters per second (liters/s), we use the conversion factor $1{\text{m}}^3=1000\text{liters}$:

$\dot{Q}=0.13255\times 1000=132.55$ liters/s

Thus, the flow rate of the entrained water is 132.55 liters/s.