 in this chat room, I would like all mathematical equations to be displayed using the 'displaystyle' in LaTeX. 

Certainly! I will use the `\displaystyle` command in all future mathematical expressions.

### Pressure Drop Calculation:

Given:

- Duct area: \( A_d = 1.4 \times 1.4 = 1.96 \, \text{m}^2 \)
- Flow speed of air: \( v = 10.5 \, \text{m/s} \)
- Wire diameter: \( D = 0.4 \, \text{mm} = 0.4 \times 10^{-3} \, \text{m} \)
- Screen pitch: \( P = 1.7 \, \text{mm} = 1.7 \times 10^{-3} \, \text{m} \)
- Density of air: \( \rho = 1.225 \, \text{kg/m}^3 \)

The open area of the screen \( A_s \) is calculated as:

\[
\displaystyle A_s = \left(\frac{P - D}{P}\right)^2 = \left(\frac{1.7 \times 10^{-3} - 0.4 \times 10^{-3}}{1.7 \times 10^{-3}}\right)^2 = \left(\frac{1.3 \times 10^{-3}}{1.7 \times 10^{-3}}\right)^2 = \left(0.7647\right)^2 \approx 0.5856
\]

The pressure drop \( \Delta P \) across the screen is calculated using:

\[
\displaystyle \Delta P = C_d \cdot \frac{1}{2} \cdot \rho \cdot v^2 \cdot \left(\frac{A_s}{A_d}\right)^2
\]

Assuming a drag coefficient \( C_d \) of 1.2:

\[
\displaystyle \Delta P = 1.2 \cdot \frac{1}{2} \cdot 1.225 \cdot (10.5)^2 \cdot (0.5856)^2
\]

\[
\displaystyle \Delta P = 1.2 \cdot 0.6125 \cdot 110.25 \cdot 0.343
\]

\[
\displaystyle \Delta P \approx 28.28 \, \text{Pa}
\]

Thus, the total pressure drop across the aluminum screen is approximately:

\[
\boxed{\displaystyle 28.3 \, \text{Pa}}
\]

If you have any further questions or need additional details, feel free to ask.

### Follow-up Questions:

1. How does changing the wire diameter \( D \) affect the pressure drop \( \Delta P \)?
2. How would varying the air density \( \rho \) influence the pressure drop?
3. What is the relationship between the drag coefficient \( C_d \) and different screen materials?
4. How would increasing the pitch \( P \) impact the pressure drop?
5. What simplifications are typically made when using empirical equations for pressure drop across screens?

### Tip:
When dealing with flow and pressure calculations, always ensure that the units are consistent across all parameters to avoid errors.

Explore how to calculate pressure drop in fluid mechanics using the drag coefficient, screen area, and air density. This problem involves detailed calculations for pressure drop across an aluminum screen based on given parameters. Suitable for advanced high school students.

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