To solve this problem, we can use the principles of continuity for fluid flow, which states that the mass flow rate must remain constant in a closed system. In this case, the equation of continuity can be expressed as:

$A_1 v_1 = A_2 v_2$

where:

• $A_1$ and $A_2$ are the cross-sectional areas of the pipes (before and after the diameter reduction),
• $v_1$ and $v_2$ are the average velocities of the fluid in the pipes (before and after the diameter reduction).

The cross-sectional area of a pipe is given by:

$A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}$

Let:

• $d_1 = 23 \, \text{cm} = 0.23 \, \text{m}$ (diameter of the larger pipe),
• $v_1 = 0.73 \, \text{m/s}$ (average velocity in the larger pipe),
• $v_2 = 1.66 \, \text{m/s}$ (average velocity in the smaller pipe),
• $d_2$ be the unknown diameter of the smaller pipe.

Step 1: Find the diameter of the smaller pipe

Using the continuity equation:

$A_1 v_1 = A_2 v_2$

Substituting the areas $A_1 = \frac{\pi d_1^2}{4}$ and $A_2 = \frac{\pi d_2^2}{4}$ into the equation, we get:

$\frac{\pi d_1^2}{4} v_1 = \frac{\pi d_2^2}{4} v_2$

Canceling out $\frac{\pi}{4}$ on both sides:

$d_1^2 v_1 = d_2^2 v_2$

Solving for $d_2$:

$d_2 = d_1 \sqrt{\frac{v_1}{v_2}}$

Substitute the values:

$d_2 = 0.23 \, \text{m} \times \sqrt{\frac{0.73 \, \text{m/s}}{1.66 \, \text{m/s}}}$

Let’s calculate $d_2$.

Step 2: Calculate the flow rate

The volumetric flow rate $Q$ is given by:

$Q = A_1 v_1 = A_2 v_2$

Since $A_1 = \frac{\pi d_1^2}{4}$, we can calculate:

$Q = \frac{\pi d_1^2}{4} v_1$

Substituting the known values for $d_1$ and $v_1$:

$Q = \frac{\pi (0.23)^2}{4} \times 0.73$

Let’s calculate the flow rate as well.### Results:

1. The diameter of the smaller pipe is approximately 0.153 m (or 15.25 cm).
2. The flow rate is approximately 30.33 liters per second.

Do you need any more details or clarifications?

Here are some related questions to expand your understanding:

1. How would the flow rate change if the diameter was reduced further?
2. What is the relationship between flow velocity and pressure in a pipe?
3. How does fluid viscosity affect the flow rate in a pipe?
4. What happens to the flow rate if the liquid is compressible?
5. How would you calculate the flow rate for a non-circular cross-section pipe?

Tip: The principle of continuity can also be applied in open channels, though it requires adjustments for non-enclosed flows.