Jökull

Ataaseq assigiiaat ilaat

Jökull - 01.12.1957, Qupperneq 9

Jökull - 01.12.1957, Qupperneq 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 7
Qupperneq 1
Qupperneq 2
Qupperneq 3
Qupperneq 4
Qupperneq 5
Qupperneq 6
Qupperneq 7
Qupperneq 8
Qupperneq 9
Qupperneq 10
Qupperneq 11
Qupperneq 12
Qupperneq 13
Qupperneq 14
Qupperneq 15
Qupperneq 16
Qupperneq 17
Qupperneq 18
Qupperneq 19
Qupperneq 20
Qupperneq 21
Qupperneq 22
Qupperneq 23
Qupperneq 24
Qupperneq 25
Qupperneq 26
Qupperneq 27
Qupperneq 28
Qupperneq 29
Qupperneq 30
Qupperneq 31
Qupperneq 32
Qupperneq 33
Qupperneq 34
Qupperneq 35
Qupperneq 36
Qupperneq 37
Qupperneq 38
Qupperneq 39
Qupperneq 40
Qupperneq 41
Qupperneq 42
Qupperneq 43
Qupperneq 44
Qupperneq 45
Qupperneq 46
Qupperneq 47
Qupperneq 48
Qupperneq 49
Qupperneq 50
Qupperneq 51
Qupperneq 52
Qupperneq 53
Qupperneq 54
Qupperneq 55
Qupperneq 56
Qupperneq 57
Qupperneq 58
Qupperneq 59
Qupperneq 60
Qupperneq 61
Qupperneq 62
Qupperneq 63
Qupperneq 64
Qupperneq 65
Qupperneq 66
Qupperneq 67
Qupperneq 68

x

Jökull

Direct Links

Hvis du vil linke til denne avis/magasin, skal du bruge disse links:

Link til denne avis/magasin: Jökull
https://timarit.is/publication/1155

Link til dette eksemplar:

Link til denne side:

Link til denne artikel:

Venligst ikke link direkte til billeder eller PDfs på Timarit.is, da sådanne webadresser kan ændres uden advarsel. Brug venligst de angivne webadresser for at linke til sitet.