Jökull - 01.12.1957, Blaðsíða 9

(15)
oí the layer eroded, the approximation consists
of solving the equation
aTyy = Tt (10)
with the initial ancl boundary conditions
t = 0 T = g (y + r)
8U ’ (II)
v = 0 T = 0 V '
The solution of this simple problem is (Car-
slaw (10) page 41).
T = g(y + r-erf^=) (12)
and the temperature gradient is
Ty = s(1+V:íÍt=exp(_y2/4at))' (13)
The surface gradient may be written
(Ty)y = 0 =8 (! +Ce W)’ (14)
ce W = r/v/aat
where Ce(t) is the correction factor at the time t.
This holds for t > > r/v only.
Let a new short period of the length At and
a total erosion r start at the time t = + after
the middle of the first period. If t^ >> At
the initial temperature distribution for the
second period is given approximately by equa-
tion (12). The temperature distribution at the
end of the second period can therefore be
found approximately by inserting the expres-
sion (12) into the general solution (6).
fros/on r
—*-j^/í-«— —•-//[-»—
/7ý. 2
The computation of the integral in (6) can
often be avoided by the fact that it is easy to
furnish adequately accurate estimates of the
upper and the lower limits to the cumulative
effects of the erosional periods.
The upper limit is easily derived from the
fact that the gradient Ty in equation (13) has
its maximum at the surface. The effect of the
first period of erosion is therefore overestimat-
ed if the expression (14)
T = gy(l+Ce,l (4))
is applied as the initial temperature distribu-
tion to the second period of erosion. The sub-
script e, 1 applies to the first period.
As the initial temperature gradient is assum-
ed constant, the effect of the second period of
erosion can also be expressed by a correction
factor Ce 2 (t — ti) where t > tj_. If t —- +
> At the correct expression for this factor is
given in the r.h.s. of equation (9). On the
other hand, if
t - tj > > At
the factor can be expressed in good approxima-
tion according to equation (14)
Ce, 2 (í-4) = r/yVa (t-4) (16)
The surface gradient at the time t, where t
is a time at or after the end of the second
period, is therefore subject to the inequality
(Ty) yL0 < g í1 + Ce, 1 (4)) (! + Ce, 2 (l - 4 ))
(17)
The lower limit to the cumulative effect is
obtained in the following way. According to
the general solution (6), the solution at the end
of the second periccl of erosion, that is, for
t = tj + At, may be written
T = f(yz t r(z, /,jje % G,(&r,y;z)</z (18)
where F (z, t^) represents the effect of the first
period of erosion at the beginning of the second
period.
The integral (18) consists of two parts. The
first part is due to the first term in the paren-
thesis and leads to a solution of the form (8)
and its contribution to the surface gradient can
therefore be written
g(l+Ce,2(At))- (19)
In computing the second part due to the
second term in the parenthesis, it is to be ob-
served that the factor exp (vz/2a) is always
greater than or equal to unity and this part
of the integral is therefore underestimated if
this factor is replaced by unity.
The function F (y, t) is a solution of the heat
conduction equation (10) and hence
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