A cylindrical drilling platform operates on an exoplanet where the surface gravitational field is 18.0ms−2. Beneath the platform lies a reservoir of liquid ammonia with density 650kgm−3. Engineers must design a pressure-rated pipe so that the gauge pressure at the pipe’s control valve, located at the bottom of the reservoir, does not exceed 4.50×105Pa.
The pipe contains the same liquid ammonia as the reservoir and is open to the planet’s atmosphere at the liquid surface. Atmospheric pressure on this planet is 9.5×104Pa.
(a) Derive an algebraic inequality that relates the maximum allowable depth dmax to the given parameters so that the gauge pressure limit is not exceeded.
(b) Compute the numerical value of dmax for the liquid ammonia.
(c) The engineers replace the ammonia with a denser drilling fluid of unknown density but keep the valve at depth dmax. The gauge pressure now rises to 6.30×105Pa. Determine the density of the new fluid.
(d) Discuss one practical implication of using a denser drilling fluid for the mechanical design of the pipe system.