For turbulent flow, the pressure loss can still be calculated using the Darcy-Weisbach equation, but we need to calculate the friction factor \( f \) more accurately, typically using the Colebrook equation or an approximation for turbulent flow. Since solving the Colebrook equation analytically is complex, I'll use the **approximation** by **Blasius** or **Prandtl** for turbulent flow in smooth pipes, which can be adjusted for rough pipes like those with an acoustic silencer.

### Approximation for Friction Factor in Turbulent Flow

The friction factor for turbulent flow in smooth pipes can be approximated by:

\[
f = \frac{0.079}{\text{Re}^{0.25}}
\]

Where:
- \( \text{Re} \) is the Reynolds number, which is calculated by:

\[
\text{Re} = \frac{\rho v D_h}{\mu}
\]

- \( \mu \) is the dynamic viscosity of air, typically \( \mu \approx 1.81 \times 10^{-5} \, \text{Pa} \cdot \text{s} \).

### Step 1: Calculate Reynolds Number (\( \text{Re} \))

Let's first calculate the Reynolds number to determine the flow regime and use it for calculating the friction factor.

Given:
- \( \rho = 1.225 \, \text{kg/m}^3 \) (air density)
- \( v = 10.5 \, \text{m/s} \) (air velocity)
- \( D_h = 1.4 \, \text{m} \) (hydraulic diameter)
- \( \mu = 1.81 \times 10^{-5} \, \text{Pa} \cdot \text{s} \) (dynamic viscosity)

\[
\text{Re} = \frac{1.225 \times 10.5 \times 1.4}{1.81 \times 10^{-5}}
\]

\[
\text{Re} \approx \frac{17.955}{1.81 \times 10^{-5}}
\]

\[
\text{Re} \approx 9.92 \times 10^5
\]

### Step 2: Calculate the Friction Factor (\( f \))

Since \( \text{Re} \approx 9.92 \times 10^5 \) indicates turbulent flow, we'll use the Blasius equation:

\[
f = \frac{0.079}{(9.92 \times 10^5)^{0.25}}
\]

\[
f \approx \frac{0.079}{(178.4)}
\]

\[
f \approx 0.000443
\]

Given that the walls are rough (acoustic silencer), the friction factor might be larger. Let's adjust it up to \( f \approx 0.02 \) for rough turbulent flow.

### Step 3: Calculate the Pressure Loss

Using the Darcy-Weisbach equation with the updated friction factor \( f \):

\[
\Delta P = 0.02 \cdot \frac{6.2}{1.4} \cdot \frac{1.225 \cdot (10.5)^2}{2}
\]

\[
\Delta P = 0.02 \cdot 4.429 \cdot 67.478
\]

\[
\Delta P \approx 5.985 \, \text{Pa}
\]

### Final Answer:
The pressure loss across the wind tunnel, assuming turbulent flow and a rough duct wall, is approximately **5.985 Pascals**.

Would you like more details or have any further questions?

Here are five related questions:
1. How does the Reynolds number affect the friction factor and pressure loss?
2. What changes in pressure loss would occur if the duct material were smoother?
3. How would a reduction in duct size affect the pressure loss?
4. What is the significance of the Colebrook equation in calculating friction factors for turbulent flow?
5. How does the temperature of the air affect the pressure loss in the duct?

**Tip:** For more accurate calculations in rough ducts, consider using the Colebrook equation or Moody chart, which take into account both the Reynolds number and the relative roughness of the duct.

Learn how to calculate pressure loss for turbulent flow in ducts using the Darcy-Weisbach equation and approximations like the Blasius equation. Includes step-by-step calculations for a specific scenario with air flow and duct dimensions.

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