Determining the Diameter of a Nozzle for a Given Flow Speed Using the Continuity Equation

The continuity equation in fluid dynamics is a fundamental principle that is widely used to understand and predict fluid behavior in various applications. This article will guide you through a practical example where we use the continuity equation to determine the optimal diameter of a nozzle when we want to achieve a specific flow speed. The example involves water flowing through a pipe and emerging from a nozzle at a higher velocity.

The Problem and the Continuity Equation

Consider a scenario where water is flowing through a pipe with a diameter of 2 cm at a speed of 1 m/s. We want to design a nozzle such that the water emerges from it at a speed of 20 m/s. To solve this, we will use the principle of conservation of mass, represented by the continuity equation:

Continuity equation:

(A_1 v_1 A_2 v_2)

A1: Cross-sectional area of the pipe A2: Cross-sectional area of the nozzle v1: Velocity of the water in the pipe v2: Velocity of the water in the nozzle

Step 1: Calculate the Cross-sectional Area of the Pipe

The diameter of the pipe (d_1) is 2 cm, which is 0.02 m. Using the formula for the area of a circle:

A1(frac{pi d_1^2}{4})

Substitute the given diameter:

A1(frac{pi 0.02^2}{4})(frac{pi 0.0004}{4})(frac{0.0004pi}{4})~0.00031416**(m^2)

Step 2: Set Up the Equation with Known Values

We know the following values:

V11 m/s (velocity in the pipe) V220 m/s (desired velocity in the nozzle)

Substituting these values into the continuity equation:

A1 v1A2 v2'

Substitute the known area of the pipe:

0.00031416(m^2)*1 m/s A2 * 20 m/s

Step 3: Solve for A2

Rearrange to solve for A2:

A2 (frac{0.00031416 text{m}^2*1 text{m/s}}{20 text{m/s}} frac{0.00031416}{20} approx 0.000015708 text{m}^2)

Step 4: Calculate the Diameter of the Nozzle

Now that we have the area of the nozzle (A2) we can use the area formula to find the diameter (d2) of the nozzle:

A2 (frac{pi d_2^2}{4})

Rearrange to solve for d2:

d2(^2) (frac{4A_2}{pi})

Substitute the known area of the nozzle:

d2(^2) (frac{4 cdot 0.000015708}{pi} approx frac{0.000062832}{3.14159} approx 0.000019999)

Take the square root:

d2 ~ sqrt{0.000019999} ~ 0.004472 m ~ 4.47 mm

Conclusion

The diameter of the nozzle should be approximately 4.47 mm to achieve a speed of 20 m/s for the water emerging from it.

Note: In the theoretical derivation provided in the given content, there seems to be a discrepancy in the final calculations and the provided solution may be incorrect or simplified differently. The method explained here follows the step-by-step application of the continuity equation to accurately determine the nozzle diameter.