By Rachel Walsh
Old Faithful is a geyser in Yellowstone National Park that
is powered by an instrusive pluton that is believed to be at a depth of 2.4 km
to 4.8 km below the surface. The water is heated, and pumped through a plumbing
system up through many geysers, one of which is old faithful. The height the
water can reach is up to 56m. The duration of each eruption is 1.5 to 5 min.
and the volume of water can reach up to 32 m^{3 }. I am going to
investigate the rate, velocity of water at the hole and the possible pressure
by the pluton.
Assumptions:

the plumbing is one vertical pipe of equal
diameter through all 3.6 km (average) of the earth to the pluton.

The eruption is at a constant rate.

The change of temperature is irrelevant

The pressure at the opening of the hole is
atmospheric
Radius of opening: 0.2m
Possible viscosity of brine:1.3 x 10^{3 }Pa *s
(Francke et al)
Density of brine: 1090 kg/ m^{3}
Rate:
Q=
v/t to
32 m^{3 }/ 300s = 0.11 m^{3}/s
Velocity of water:
pgh
= ½ pv^{2 }to (9.8 m/s)(56m) = ½ v^{2
}
v
= 23.4 m/s
Change in pressure from pluton to surface:
Q
= pR^{4}(deltaP)/8hL à 0.11 m^{3}/s = p(0.2m)^{4}(deltaP)/8(1.3x10^{3}
Pa*s)(3600m)
deltaP = 1100 Pa
Sources:
Francke,
Henning. "Density and Viscosity of Brine: An Overview from a Process
Engineers Perspective." Sci Verse. Cheme Der Erde  Geochemistry,
n.d. Web.
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