Jökull - 01.12.1975, Síða 24

Fig. 2. As heat conduction
and diffusion follow the
same rules, a case of the
former is used here. Temp-
erature v (T in the main
text) in a slab — 1 < x < 1
with constant heat produc-
tion at the rate A0 heat units
per unit of volume, and zero
surface temperature. The
numbers on the curves are
values of k • t/12, where t is
the time. After Carslaw and
Jaeger, 1959, Fig. 20. The
figure corresponds to diffus-
ion of a daughter element
from an isotropic mineral
plate if surface concentra-
tion is kept at zero. We have
only to replace k by the re-
spective coefficient D of dif-
fusion, and v or T by the
concentration C of the
daughter element, whose rate
of production is A0. The
0-2
0-4
0-6
0'8
1-0
x/l
figure shows how the increase in C anywhere in the plate slows down with time, to reach a
limited maximum, Cmax, for infinite time. For the central plane that maximum corresponds
to vmax = A012/2K in the figure.
Mynd 2. Hitaleiðsla og efnisdreifing (diffusion) í fösta efni fylgja sama stœrðfrceðilögmáli. Þess
vegna er hœgt að nota þessa mynd af varmaleiðslu út úr plötu, sem i er jöfn og jafndreifð
varmaframleiðsla, A0 á rúm- og timaeiningu. Hitanum er liér haldið föstum á yfirborðinu, x/l
— 1, x = fjarlægð frá miðju plötunnar. Þetta dæmi svarar alveg til jafndreifðrar og jafnrar
framleiðslu dótturefnis i plötunni, ef magninu á rúmeiningu er haldið föstu við yfirborð. Vegna
tapsins til hliðanna er dreifing C á hverjum tima t eins og línurnar sýna. Magnið vex hcegar
og hœgar, vegna þess að tapið eykst, uns náð er vissu hámarki. í miðri plötu svarar það til hita-
hámarksins vmaw = A0l2/2K á myndinni| þar sem K er varmaleiðslustuðull. Þar sem hálf plötu-
þykktin er i öðru veldi, verður þelta hámark i v eða C að öðru jöfnu 100 sinnum hcerra í cm-
plötu en í mm-plötu. I örþunnum krystalhimnum getur dótturefni því miklu síður vaxið i réttu
hlutfalli við timann en í krystalþlötum í grófkrystölluðu bergi.
perature Tmax increases with the square of the
thickness of the slab. It is further clear, cf.
Fig. 2, that as the temperature gradient towards
the surface increases in a given case, the heat
loss increases, while the heat production within
the slab is constant. Flence, the rise of the
central temperature slows down with time. This
is shown by the temperature curves for certain
values of the quantity k • t/12. With increasing
time t, the curves for k • t/12 = 0.02 .... 0.1
.... 1.0 ..... co are seen to rise slower and
slower. k =------is the thermal diffusivity, the
p-c
dimension of which is cm2/sec, p = density of
the slab material, and c its specific heat. K ancl
K are numerically very similar for most rock
materials, but they clearly have different dimen-
sion.
The development shown in Fig. 2 is quite
independent of T0. For convenience we put
T0 = 0 above, but the full formula for Tmax is
22 JÖKULL 25. AR