Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 35

35
5. HEAT CONDUCTION IN A ROD.
We shall now consider the classical problem of heat
conduction in a rod of length b . At the outset, / = 0 ,
the temperature y is throughout 0, then at one of the ends,
x = 0 , the temperature is suddenly raised to T and kept
constant. Hence the heat spreads by conduction to the
other end, x = b . Our problem is to find the temperature
at this end, ybt, as a function of the time t. The heat dis-
sipation from the surface to the surroundings is considered
as negligible. We base our calculations on the differential
equation for linear heat conduction:
dy __ d2y
dt dx
(63)
where y is a functíon of x and t, or y = yxt • According
to our assumption is:
dV = 0
i.n
dt
i2 n
n d fJOt
dx
2 n
n > 0
(64)
Further at the end, x = b:
d2n+lybt
2n-\-\ '
0
(65)
Then the General Serial Relation (3) becomes, with due
regard to (63), (64) and (65):
PoUbt
= Po'Jbt
= q0T -fqjb
p2b dybt dxl pfi d Pbt + •
2! 4! dx4
, Pi h2 dybt 1 Pi( d2ybt
1 2 * y- dt \ « 1 dt2
__ot j qJ3 b ypt |
dx 3! dx3
dy
(66)