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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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Thermistor response
T o derive the response of the thermistor to changes in the bulk temperature of the fluid, the fol-lowing situation is considered:
T= Tn lor
T= T0 + a[l + ^ for (D.1)
T= T„ + at., for
This corresponds to a constant heat input between / = -tlu,ul and ; = rl:ctll with an isentropic
response in the bulk fluid. By taking the Fourier transform, the spectrum density 5(w) at angular frequency CO is obtained, which is:
S(co) = - A s i n ù ^ ' W ^ k o )
TCO)2 V 2 J V 2 ) (D.2)
where 5(A) is the Dirac Delta function. Fourier analysis is employed here because the solution of the heat flow problem in which the bulk temperature has a simple harmonic variation with time is known, as a standard problem in linear acoustics. Equation (D.2) shows that the Fourier spectrum is heavily weighted towards low frequencies (much as expected). If we assume now that the ther-mistor is a homogeneous sphere of radius R then the temperature is:
T0 + A}0(k,i,r)exp(iwi) (D.3)
with j0 is the spherical Bessel function of order zero, k:lt = J-ia/D,,,, Dlh is the thermal diffusiviry
of the thermistor material, /• is the radial coordinate and A is a constant. For i > R it is then:
T0 + 7"|exp(/(flf) + -exp{ ((tor -k'r)} (DA)
D - Thermistor response
where the first term is the initial temperature, the second is the Fourier component in the bulk, and the third is the thermal boundary layer at the surface of the thermistor (B is a constant). The prime denotes properties of the fluid. Note that the thermal boundary layer takes the form of an outward-going thermal wave which attenuates rapidly with increasing r because of the imaginary part of k'.
T h e two boundary conditions which determine the constants A and B are those requiring conti-nuity of temperature on and of heat flow through the surface r = R . T h e result is that:
AJo(kR) = y ^ y (D.5)
where
(XV kR \Ji(kR)
' = irkTTJÏTRJMkRJ
(D-
6)Notice that, if ƒ « 1 , the surface temperature of the thermistor follows almost exactly the bulk temperature of the fluid at angular frequency co. Now, as T -» 7", , the ratio X/V goes to zero and, since D' = DT—>0 the term kR/(] +ik'R) also vanishes. Furthermore, as co -> 0, both the term
kR/( 1 + ik'R) and the ratio of Bessel functions go to zero.
T h e net result of all this calculation is that, the surface temperature of the thermistor should fol-low that of the bulk fluid more and more closely as T —» Tc.
