Question

In a ventilation network, three airways with resistances of 4.0 Ns\( ^2 \)m\( ^{-8} \), 6.25 Ns\( ^2 \)m\( ^{-8} \) and 9.0 Ns\( ^2 \)m\( ^{-8} \) are connected in parallel.

The equivalent resistance of the network in Ns\( ^2 \)m\( ^{-8} \) is _________ (rounded off to 2 decimal places)}

Hint:

Always double-check inverse resistance sums in parallel combinations, especially when using decimal approximations.

Answer 1

The relationship between pressure drop (\( \Delta P \)), airflow quantity (\( Q \)), and resistance (\( R \)) in an airway follows \( \Delta P = R Q^2 \).

For resistances connected in parallel, the formula for the equivalent resistance (\( R_{eq} \)) is:
\[ \frac{1}{\sqrt{R_{eq}}} = \sum_{i=1}^{n} \frac{1}{\sqrt{R_i}} \]

In this case, with three airways:
\[ \frac{1}{\sqrt{R_{eq}}} = \frac{1}{\sqrt{R_1}} + \frac{1}{\sqrt{R_2}} + \frac{1}{\sqrt{R_3}} \]

Given resistances are:

\( R_1 = 4.0\ \text{Ns}^2\text{m}^{-8} \)
\( R_2 = 6.25\ \text{Ns}^2\text{m}^{-8} \)
\( R_3 = 9.0\ \text{Ns}^2\text{m}^{-8} \)

Substitute the values into the formula:
\[ \frac{1}{\sqrt{R_{eq}}} = \frac{1}{\sqrt{4.0}} + \frac{1}{\sqrt{6.25}} + \frac{1}{\sqrt{9.0}} \]

Calculate the square roots:

\( \sqrt{4.0} = 2.0 \)
\( \sqrt{6.25} = 2.5 \)
\( \sqrt{9.0} = 3.0 \)

Now substitute these back:
\[ \frac{1}{\sqrt{R_{eq}}} = \frac{1}{2.0} + \frac{1}{2.5} + \frac{1}{3.0} \]

Calculate the fractions:
\[ \frac{1}{\sqrt{R_{eq}}} = 0.5 + 0.4 + \frac{1}{3} \] \[ \frac{1}{\sqrt{R_{eq}}} \approx 0.5 + 0.4 + 0.3333... \] \[ \frac{1}{\sqrt{R_{eq}}} \approx 1.2333... \]

Solve for \( \sqrt{R_{eq}} \):
\[ \sqrt{R_{eq}} = \frac{1}{1.2333...} \approx 0.81081... \]

Finally, square the result to find \( R_{eq} \):
\[ R_{eq} = (0.81081...)^2 \approx 0.6574... \]

Rounding off to 2 decimal places:
\[ R_{eq} \approx 0.66\ \text{Ns}^2\text{m}^{-8} \]

Answer:

The equivalent resistance of the network in Ns\( ^2 \)m\( ^{-8} \) is 0.66 (rounded off to 2 decimal places).

The equivalent resistance is \( \boxed{0.66\ \text{Ns}^2\text{m}^{-8}} \).