2.1 POZOS MULTILATERALES
2.1.5 CLASIFICACIÓN DE LOS POZOS MULTILATERALES EN FUNCIÓN
There are two main paths for the conduction o f heat into the salt pill. The first is the support structure which, in modem ADRs, is almost always composed o f kevlar strands. The support structure has to
o f mechanical strength to thermal conductivity is used. Kevlar does not have a particularly low thermal conductivity, but its tensile strength is 10 to 50 times higher than nylon, for example, making it the ideal choice. Duband, Hui and Lange (1993), Gush (1991) and Roach (1992) have described various designs for kevlar supports. The second path is through the wiring for electrical components, such as resistance thermometers, which may be present on the cold stage. The electrical and thermal conductivities in metals are connected by the Wiedemann-Franz law
where k and cr are the thermal and electrical conductivities, T is the temperature, e the charge on an electron, and kB the Boltzmann constant. The heat load through a wire depends both on the thermal conductivity and the Joule heating due to the electrical resistance, and these are clearly opposing effects according to 2.60. The total heat load should therefore have a minimum, and finding this point is important for high power wiring such as the magnet current leads. However, it is impractical for the experimental wiring to the salt pill, where the currents are always low. Therefore constantan or manganin are usually used as their thermal conductivities are reasonably low and their electrical resistivities do not vary greatly with temperature. Copper, with its higher thermal conductivity, is totally unsuitable for wiring to components on the salt pill, but is sometimes used for high current devices on the base plate, such as the solenoid in a mechanical heat switch. Constantan was used for the salt pill wiring at MSSL, and so the model needed routines to find the heat load through kevlar and constantan components.
The heat flow through a bar o f cross sectional area A under a temperature gradient dT/dx is given by thermally isolate the pill whilst providing mechanical support, and so the material with the highest ratio
2.60
Q'=X(T)A^
dx
Material a (3 Kevlar 83 x 10'^ 2.317 Constantan 1 x 10'3 2.54
Table 2.1. Thermal conductivity data used in the thermal model.
where I(T ) is the thermal conductivity. Thus if the ends o f a solid bar o f uniform cross section and length 1 are at temperature T] and T2,
A \
Q ' = - \ h ( T ) d T .
2.621
7iIn order to avoid performing the integral every time the thermal power is calculated, 2.62 can be written in terms o f the thermal potential
T
6
= J / l( T ) d T ,
2.630
SO
g = j [ e ( T , ) - e ( T A -
2-M
The thermal conductivities o f both kevlar and constantan have been measured at low temperatures: Duband, Hui and Lange (1993) and Poulaert et al. (1985) report results for kevlar, and data for constantan can be found in White (1979) and Pobell (1992). The thermal potentials o f most materials can be fitted to a power law o f the form
0 ( T ) = — T f
2.65P
and the values o f
a
and P for constantan and kevlar used in the model are given in Table 2.1.The MSSL laboratory ADR also included another support structure component, known as the foot rod, which consisted o f a stiff kevlar/epoxy composite rod. The material used was Aramid/Vinylester rod, cat. No. V E 417920, from Goodfellow Cambridge Ltd. (Cambridge Science Park, Cambridge UK). The foot rod allowed the salt pill to be removed and replaced easily, as it was stiff and could be glued to a bolt. This bolt could be screwed into a tapped hole at the bottom end o f the magnet bore by rotating the
Thermometer stage Support disc Kevlar cords Support riii]
Base plate
Main magnet Top rod
Salt pill (shaded area is the salt).
Base disc Toot rod top
Kevlar/epoxy composite rod Magnet bore,,
Fig. 2.2. Diagram o f the salt pill and surrounding components in the MSSL laboratory ADR.
whole salt pill, and so access was required only to the top end o f the bore when the salt pill was removed and replaced. However, no published data were available on the low-temperature thermal conductivity o f the foot rod material. The thermal conductivity was estimated at MSSL (Hepburn, 1995). The thermal potential was fitted with the quadratic polynomial
0.00001
N s \ - Total heat leak
Foot rod - Kevlar cords . Constantan wires
<D
X
5. x 102
3 A 1Temperature o f salt pill (K)
Fig. 2.3. Contributions to the heat leak in the MSSL laboratory ADR for a bath temperature o f 4.2K.
0(T) = 3.71 x 10-3 T + 2.11 x 10 -3 T 2.
2.66
Fig. 2.2 shows the salt pill and surrounding components o f the MSSL laboratory ADR. The foot rod had a radius o f 1mm and was 80mm long. The kelvar cords were situated at the opposite end o f the pill. There were ten cords, each 15mm long and containing 120 fibres o f kevlar 29, each 9pm in diameter (Goodfellow Cambridge Ltd. cat. no. AR305744). The wiring to the salt pill is not shown, but a typical arrangement in the work described later was to have six 10mm long constantan wires, each 55pm in diameter, passing from the base plate to the cold stage. The contributions to the heat leak for each o f these components are shown in Fig. 2.3.
Gas. Abundance by volume in dry air (parts per million).
Melting point (K). Boiling point (K).
Nitrogen 780900 63.15 77.344
Oxygen 209500 54.35 90.188
Argon 9300 83.75 87.294
Carbon dioxide 300 194.67 (sublimes)
Neon 18 24.55 27.102 Helium 5.2 - 4.22 Methane 1.5 90.65 111.65 Krypton 1.14 115.85 119.65 Nitrogen dioxide 0.5 182.29 184.67 Hydrogen 0.5 13.95 20.28 Ozone 0.4 80.45 162.64 Xenon 0.086 161.25 165.05
Table 2.2. Abundances, melting and boiling points of atmospheric gases. Data from Kaye and Laby (1986).
2.2.2. Conduction through Gas
A common feature o f all low temperature systems is the use o f vacuum spaces to aid thermal isolation. The salt pill o f an ADR sits in an evacuated chamber, the walls o f which are maintained at the bath temperature, usually between 1.6 and 4.2K. A vacuum o f 10'8 Torr is relatively easy to obtain at room temperature, and this will be improved as the temperature is lowered. In fact, since there is very little helium in the atmosphere, and all other atmospheric gases have melting points well above 4.2K, the pressure will be extremely low at 4.2K. Data for atmospheric gases are given in Table 2.2.
For approximately parallel surfaces at temperatures T] and T2, the heat conducted by a low-pressure gas is given by White (1979) as