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

Correct Answer: 0.5

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: $R_1 = \SI{4.0}{\ventres}$ $R_2 = \SI{6.25}{\ventres}$ $R_3 = \SI{9.0}{\ventres}$ 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: \[ R_{eq} \approx \SI{0.66}{\ventres} \]