Question

At what flow rate might turbulence begin to develop in a water main with a (0.200 m) diameter? Assume a (20ᶜircC) temperature.

a) (0.027 m³/s)
b) (0.040 m³/s)
c) (0.056 m³/s)
d) (0.072 m³/s)

Answer 1

This flow rate is equivalent to
`\( 0.628 * 10^(-3) \, \text{m}^3/\text{s} \).` So, the closest one is option (a) with
`\(0.027 \, \text{m}^3/\text{s}\)`.

The critical Reynolds number
`(\(Re_c\)`) is often used to predict when turbulence will begin to develop in a fluid flow. The Reynolds number is given by the formula:

`\[ Re = (\rho \cdot v \cdot D)/(\mu) \]`

Where:

-
`\( \rho \)` is the density of the fluid,

-
`\( v \)` is the velocity of the fluid,

-
`\( D \)` is the diameter of the pipe,

-
`\( \mu \)` is the dynamic viscosity of the fluid.

The critical Reynolds number is specific to each fluid and is typically around 2000 for water.

The formula for the Reynolds number can be rearranged to solve for velocity
`(\( v \))`:

`\[ v = (Re_c \cdot \mu)/(\rho \cdot D) \]`

Now, let's substitute the known values:

-
`\( Re_c = 2000 \)` (assumed critical Reynolds number for the onset of turbulence in water)

-
`\( \mu \)` is the dynamic viscosity of water at 20°C. For water,
`\( \mu \)` is approximately
`\( 1.002 * 10^(-3) \, \text{Pa} \cdot \text{s} \)` at 20°C.

-
`\( \rho \)` is the density of water at 20°C, which is approximately
`\( 998 \, \text{kg/m}^3 \).`

-
`\( D = 0.200 \, \text{m} \)` (diameter of the water main)

Now, calculate the velocity
`(\( v \))`:

`\[ v = (2000 * 1.002 * 10^(-3))/(998 * 0.200) \]`

`\[ v \approx 0.02 \, \text{m/s} \]`

Now, we can find the flow rate
`(\( Q \))` using the formula:

`\[ Q = A \cdot v \]`

Where:

-
`\( A \)` is the cross-sectional area of the pipe, given by
`\( (\pi \cdot D^2)/(4) \).`

`\[ A = (\pi \cdot (0.200)^2)/(4) \]`

`\[ A \approx 0.0314 \, \text{m}^2 \]`

`\[ Q = 0.0314 * 0.02 \]`

`\[ Q \approx 0.000628 \, \text{m}^3/\text{s} \]`

This flow rate is equivalent to
`\( 0.628 * 10^(-3) \, \text{m}^3/\text{s} \).`

None of the given options (a, b, c, d) match exactly, but the closest one is option (a) with
`\(0.027 \, \text{m}^3/\text{s}\)`.