Jökull - 01.12.2007, Síða 84

Thorsteinsson et al.
d = mean diameter of borehole drilled in time t
!z = depth increment drilled in time t
!T = T0 – T
U = heat transfer coefficient of hose (unit:
Js"1m"2K"1) = k/b
k = coefficient of thermal conductivity of hose mate-
rial (unit: Js"1m"1K"1)
A = total surface area of hose = 2"rhosez = "dhosez,
with dhose= hose diameter and z = depth
b = hose wall thickness
For ice melting at the bottom of the borehole we
obtain from energy balance:
micelice
t =
mwcpw(T"Tf)
t (1)
Referring to Figure 6 the mass of ice in a depth in-
crement !z of the borehole is mice = !iceVhole =
!iceShole!z = !ice"(d2/4)!z and insertion in (1) then
yields
!icelice"(d
2
4 )!z
t =
mwcpw(T"Tf)
t (2)
which after rearrangement yields
d2 = ( 4")(
mw
t )(
cpw
!icelice
)(T"Tfv ) (3)
where v = !z/t is the drilling rate; i.e. the down-
ward velocity of the drill tip. Inserting numbers for
cpw, !ice, lice and Tf and the known value of mw/t =
450 l/hr = 0.125 kg/s, we obtain the relation
d = 0.015[m3/2s"1/2K"1/2] #
!
T
v (4)
for the diameter of the borehole (in m), expressed as
a function of the drilling velocity (set by the operator)
and the temperature of the water emerging from the
drilling tip.
Following Taylor (1984), we may express the heat
loss through the hose per unit time (from surface to
drill stem) as
mwcpw!T
t = UA!Tm (5)
where
!Tm =
T0"T
ln(T0)"ln(T ) (6)
is the logarithmic mean of T and T0. Here we have
neglected a small variation of T f with depth down the
borehole. Insertion in (5) then yields
mcpw!T
t =
k
b "dhosez!T
ln(T0/T ) (7)
where we have used U = k/b and A = "dhosez. Rear-
ranging, we obtain the expression
ln(T0/T ) = Ckz (8)
where
C =
! dhoseb
m
t cpw
(9)
and thus the temperature T at the drill tip becomes
T = T0e"CkZ (10)
Inserting numbers in (9) we obtain C = 0.021 KsJ"1.
The coefficient of thermal conductivity of the hose,
made of synthetic rubber, is not known, so we have
tentatively assumed the value for rubber given in the
CRC Handbook of Chemistry and Physics: k = 0.16
Js"1m"1K"1. Equation (10) is used to calculate the
curve in Figure 4 and insertion of (10) in (4) then
yields the following relation for the borehole diameter
as a function of depth and drilling rate
d = 0.015[m3/2s"1/2K"1/2] #
!
T0
v e
" 12 CkZ (11)
which is used to calculate the curves in Figure 5.
82 JÖKULL No. 57