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Heat Transfer in a Critical Fluid under Microgravity Conditions - a Spacelab
Experiment
-de Bruijn, R.
Publication date
1999
Link to publication
Citation for published version (APA):
de Bruijn, R. (1999). Heat Transfer in a Critical Fluid under Microgravity Conditions - a
Spacelab Experiment -.
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Time independent
integral
In this appendix we show that when Th* = G(u/i), Tb(t) must be proportional to r in which case
the integral I [eq.(2.53)] is time independent. Subsequently, / is calculated as a function ot u . We start with a general function f(x) defined for x > 0 . T h e assumption is that
(C.1)
(C.2)
(C.3)
(C.4)
then
When /i is a continuous function then
h(x<*) = ( / i ( . v ) ) " , (C.6)
and in particular
101
f(y) %•)
This implies that
ƒ ( ! ) M f(y)
T h e n
f(xy) = g(x) • ƒ(>)= û y j • f(y)
When
*»-M-C - Time independent integral
h(ea) = (h(e)Y1-- (C.7)
For y = ea, cc = In v and, necessarily,
h(y) = ( / l ( f ) )l n> = (?'"''(<'))'"' = (eln.v)ln*W _ ylnM«). (C 8 )
It follows that
ƒ(*) = ƒ( I ) • h(x) = ƒ( 1 ) • *'"*<«> = /I • A-M . (C.9) When Th(t')-x/, Th* = (u/tf and / is easily calculated. In terms of the variable ysjl -u/t,
for which eq.(2.53) becomes
/ = J ( l - y2) V
0
we find for the value u the relation
;M = J ( 1 - / ) rfy = y( 1 - y-f\0 - j - 2 u / ( 1 - y T rfy
Consequently
With /0 = 1 and /1 / 2 = -r, /t / 2 for k is a positive integer can be calculated easily.
(C.10)
2njy2( 1 -y2f~'dy = 2 J - J(l - y2fdy + j( 1 - y2f~X dy
0 0 0
2^^-i-V- (C.11)
2u ,
