Question

An intake shaft has resistance of 0.05 Ns$^2$/m$^8$ up to depth 400 m. Airflow rate = 100 m$^3$/s, air density = 1.2 kg/m$^3$. Barometer reads 99.375 kPa at surface. Find the barometer reading at 400 m depth (kPa), rounded to two decimals.

Hint:

Total pressure at mine depth = surface pressure + hydrostatic air column pressure − fan/shaft resistance losses.

Answer 1

Correct Answer: 103.5

Step 1: Shaft pressure drop due to resistance. Mine resistance formula: \[ \Delta P = R Q^2 \] \[ R = 0.05,\quad Q = 100 \] \[ \Delta P = 0.05 \times (100)^2 = 500\ \text{Pa} \] \[ = 0.5\ \text{kPa} \] Step 2: Static pressure increase with depth. \[ \Delta P_{static} = \rho g h \] \[ = 1.2 \times 9.81 \times 400 = 4708.8\ \text{Pa} = 4.7088\ \text{kPa} \] Step 3: Total pressure at 400 m depth. \[ P_{depth} = 99.375 + 4.7088 - 0.5 \] \[ P_{depth} = 103.5838\ \text{kPa} \] Rounded: \[ \boxed{103.58\ \text{kPa}} \]