Question

Air will demonstrate compressibility effect due to variation in pressure as it travels deep into and out of the mine. Assuming 5% as a mathematically significant effect, at what elevation above or below atmospheric datum will produce a measurable difference in air quantity and pressure loss?

Answer 1

Density of the air in the atmosphere can be written as,

This equation can be derived from ideal gas law where m is the mass of air molecule,P is the pressure,T is the temperature and K is the Boltzmann constant.

Again,

where, h is the height and g is the acceleration due to gravity . Putting equation 1 in equation 2 we get,

Integrating the above equation from P₀ to P we get,

Now let h and P₀ be the height and pressure at atmospheric datum. Let h1 and h2 be the height such that and   at which the pressure loss is 5% significant above and below the atmospheric datum respectively. Let P₁ and P₂ be the pressure at h1 and h₂ respectively.

For 5% significant changes, P₁ = P₀-0.05P₀ and P2 =P₀+0.05P₀

All the calculation above are done under the approximation that temperature is constant in the region of interest and equal to the temperature at datum. Thus delta x gives the elevation above and below the atmospheric datum where the changes in the pressure will be significant by 5%.