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SYSTEM DESIGN NOTE

SDN 0007.10 - Thermal Distribution System Parameters

 1. Introduction

 A discussion of the requirements for the connections to the cryocoolers was provided originally in SDN7.02. This document was written before many of the detailed parameters of the instrument were known; the present document presents a more precise determination of the parameters for the thermal distribution system ("copper straps") in GNIRS.

 The thermal distribution system can be thought of as a collection of copper straps with a certain average length and total cross-section connecting the cryocoolers to the bench. Heat removed from the bench must pass through these straps. In addition, the heat absorbed by the third, "active" shield also provides a load on the cryocoolers, but does not pass through the straps.

 Since the length of straps is determined by geometry, the remaining parameter is their total cross-section.

 2. Initial Calculations

 For the purpose of the calculations, all relevant quantities were assumed to be linear (or constant) with temperature. The following approximations were used:

 Power dissipation in coldheads (total of 4)

 P_ch = 8.33*Tc - 217               (power in watts, Tc in K)

 This formula is reasonably close to the head performance provided by the manufacturer in the range 38-62K.

 Power removed from bench

 P_b = (kA/L)(Tb-Tc) - P_rad

 where P_rad is the heat input from the active shield (taken as 120 W worst case)

Tb is the bench temperature

Tc is the coldhead temperature

k is the conductivity of copper

A is the thermal distribution system cross-section

L is the thermal distribution system length

 The conductivity of copper increases with decreasing temperature; it is not very linear but the approximation

 k = 8.5 - 0.04*T     W/(cm-K)

 is close for T between 60 and 100 K, and underestimates the conductivity below 60 K. Note that for purposes of the calculations, T was taken as the bench temperature, and this also tends to underestimate the true conductivity slightly. Also note that the active shield is assumed to be connected directly to the cryocoolers.

 The equations above can be solved for any bench temperature to determine coldhead temperature and power removed from the bench. Some typical values are tabulated below, for A= 70 cm² and L = 100 cm.

 Bench Heat Removal

The last entry indicates that the bench cannot actually get as cold as 40K; its minimum temperature is slightly below 41K. This also indicates that at this level of radiation input, the bench temperature control must inject about 55W of heat to maintain the bench at 60 K.

 The cooling curves were produced by iteration. For a given bench temperature, the heat removed from the bench was estimated as indicated. The heat capacity of the bench and mechanisms was determined by multiplying the cold mass (510 kg) by the specific heat of aluminum. This is also roughly a linear function of temperature, given by

 Cp = 0.0055*T - 0.135          (J/g-K)

 The bench temperature after an hour was calculated assuming constant heat removal (and heat capacity), and the heat removal, conductivity, and bench heat capacity were recalculated. This procedure is approximate, in that it somewhat overestimates cooling power (about 2% maximum).

 Cooling curves were calculated for three cross-section values for the distribution system, 50, 70, and 90 cm². These are plotted in Figure 1. Note the effects of the smaller cross-section: because a given gradient produces less heat flow, the structure cools more slowly, and the coldheads are actually colder for a give bench temperature. In the end, everything approached equilibrium, with both bench and coldheads at a temperature close to 40 K.

 Another feature worth noting is the approximately linear change of bench temperature with time. This is because the heat capacity of the bench decreases with temperature, so even though the coolers are removing less heat (power) as things cool, less heat removal is needed to produced a given temperature change.

 Figure 1: Cooling curves assuming shield is connected directly to cryocoolers. The top curve for any given temperature is the bench temperature and the bottom curve is the coldhead temperature.

 3. Corrected Calculations

 In practice, it is not possible to connect the active shield to the coldheads directly. Instead, the connection must be made at some point on the thermal distribution system. After the connection point, the heat flowing along the distribution system includes both heat being removed from the bench and the heat from the active shield.

 One can set up equations similar to those above to deal with the situation. For the purposes of the calculation, the heat from the active shield was assumed to enter the distribution system at a point 35 cm from the coldheads. The results are shown below.

 Because there is heat flow through the last part of the distribution system at all times, even when the bench has reached its minimum temperature, there will be a gradient along the distribution system and the bench will never reach the coldhead temperature. Instead, it will approach the temperature at the point where the shields attach (Tx in Figure 2). In order to provide some overhead for control (and modeling errors), one would like to keep Tx below about 50 K, which in turn implies a minimum cross-section for the thermal distribution system of just under 70 cm². (Although the calculations assumed constant cross-section, it would be reasonable to provide a somewhat greater cross-section for the
shared portion of the distribution system; the 70 cm² value applies to this section.)

Figure 2. Cooling curves where shield is connected partway along distribution system. Tx is the temperature at the attachment point.

 The curves in Figure 2 illustrate cooling curves for the same cross-section values as Figure 1. However, the bench cools more slowly. The results are compared in the table below.

 Cooling Times

 This clearly shows that the cross-section of 50 cm² is unsatisfactory, unless the attachment point for the shield can be moved very close to the coldheads. Once the cross-section increases to 70 cm², performance is much better. The gain going to 90 cm2 is less impressive.

 The bulk of the distribution system consists of 8 straps composed of layers of sheet copper. The individual cross-sections should then be >8.8 cm², or >1.4 sq in.

 4. Shield Connections

 In the calculations above, the temperature at the point where the active shield attaches is approximately 50 K in equilibrium. If the bench is regulated to 60 K, the temperature at the attachment point will also be close to 60K. But the temperature of the active shield itself will necessarily be higher.

 If we assume that the connection between the shield and the coldhead consists of 4 straps (2 to each coldhead assembly) of cross-section 6 cm² and length 20 cm, the conduction will be roughly 6.8 W/K for a temperature gradient of 18 K between the distribution system and the attachment point on the shield.

 If we assume that the shield itself is a cylinder, its dimensions would be approximately 100 cm diameter x 200 cm length x 0.125 cm wall thickness. The cross-section is roughly 39 cm².

 The conductivity for 1100 series aluminum is about 2.5 W/(cm-K) at 100 K and somewhat better at lower temperatures, so the conduction from one end to the middle is then ~0.98 W/K.

 If we assume that all the heat load on the shield comes in at the ends, then the temperature distribution is linear, with a difference between the ends and the middle of roughly 61 K. The end temperature would then be 60 + 18 + 61 K = 139 K.

 If we assume that the heat load is produced uniformly along the length of the shield, the temperature distribution is parabolic, and the temperature difference between ends and middle is reduced to 31 K. The end temperature is then 109 K. This assumption is probably closer to reality.

 An increase of the thickness of the shield in the center will help somewhat. If the thickness of the shield in the center 60 cm of its length is increased to ~0.3 cm, the temperature of the ends drops by about 10 K.

 If the average temperature of the shield is 110K, the power radiated into the cold structure is a little over 4 W. Even if the end temperature is as high as 140 W, the total power radiated will be less than 8 W. These figures are comparable to other heat inputs, and indicate that the design continues to be acceptable.

 5. Detector Connection

 The detector connection will need to be "tuned" in practice, but the initial connection should be close to what is needed. The detector power dissipation is about 2 W, including its heater for temperature control. If we assume a length for the strap of 100 cm, an allowable temperature drop of 10 K and an average temperature of 30 K,

 k=11 W/(cm-K)

 kA/L = 0.2 W/K

 so A~1.8 cm² (0.28 sq in.)

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