Thermal analysis of the APT power coupler and similarities to superconducting magnet current leads

A detailed thermal analysis has been performed on the 210 kW, 700 MHz RF power coupler (PC) which transfers microwave energy from high power klystrons to the superconducting (SC) resonant cavities for the acceleration of protons. The work is part of the design for Accelerator Production of Tritium funded by the US Department of Energy. The PC is a co-axial design with the RF power transmitted in the annular region between two concentric cylinders. The PC provides a thermal connection from room temperature to superconducting niobium operating at 2.15 K. Heat transfer mechanisms considered are conduction, infra-red radiation, RF joule heating in normal and superconducting materials, and, forced and natural convection cooling. The objective of the thermal analysis is to minimize the required refrigeration power subject to manufacturability and reliability concerns. The problem is reminiscent of the optimization of superconducting magnet leads. The similarities and differences in the results between SC leads and PCs are discussed as well as the critical parameters in the PC optimization.


I Thermal Analysis of the APT Power Coupler and Similarities to Superconducting Magnet Current Leads
Joseph A. Waynert, Dave E. Daney, and F. Coyne Prenger Los Alamos National Laboratory, Los Alamos, NM Abstract-A detailed thermal analysis has been performed of the 210 kW, 700 MHz RF power coupler (PC) which transfers microwave energy from high power klystrons to the superconducting (SC) resonant cavities for the acceleration of protons. The work is part of the design for Accelerator Production of Tritium funded by the US Department of Energy. The PC is a co-axial design with the R F power transmitted in the annular region between two concentric cylinders. The PC provides a thermal connection from room temperature t o superconducting niobium operating at 2.15 IC Heat transfer mechanisms considered are conduction, infra-red radiation, RF joule heating in normal and superconducting materials, and, forced and natural convection cooling. The objective of the thermal anatysis is to minimize the required refrigeration power subject to manufacturability and reliability concerns. The problem is reminiscent of the optimization of superconducting magnet leads. The similarities and differences in the results between SC leads and PCs are discussed as well as the critical parameters in the PC optimization.

I. INTRO&CTION
Power couplers (PCs) used in superconducting (SC) acceleratorsare similar to SC current leads in that they both represent a connection from a power source to a low temperature, SC device [l]. Radio frequency (RF) PCs energize SC cavities which establish RF electric fields to accelerate ionized particles. The PCs used in the Accelerator Production of Tritium (APT) program are somewhat like a high temperature superconducting (HTS) hybrid lead. An HTS hybrid lead is typically thermally anchored at liquid helium temperature at the cold end, and has another thermal intercept at a temperature below its critical temperature. A normal conducting stage provides the connection to room temperature.
Co-axial PCs are common, and represent the design baseline for APT. Fig. 1 shows a typical co-axial PC. The main components are: the inner conductor, which operates near room temperature; outer conductor, which connects room temperature to near liquid helium &He) temperature; and possible thermal intercepts to reduce the heat load to LHe. The outer conductor transitions to niobium near its connection to the beam tube to reduce RF losses. Similar to SC lead design, we want to minimize the total refrigeration input power required to compensate for the heat leaks.
n. DESCRIPTION OF THERMAL MODEL Fig. 1 shows a cut-away view of the PC as modeled, with, for illustration only, a counterflow heat exchanger for cooling of the outer conductor. The total length of the PC is about 1006 mm.
The outer conductor is 2.4 mm thickness stainless steel ( S S ) with a 15 pm copper plating, residual resistivity ratio RRR=50, on the inside diameter (ID). The outer conductor is 152.4 mm ID at the 300 K end and tapers to 100 mm ID at the 2.15 K end. About 120 mm from the beam tube (cold) end, the outer conductor mates with a niobium nipple with RRR40. The inner conductor is copper, RRR=50, with a wall thickness of 1.5 mm and tapers from 66.7 mm OD to 43.5 mm. Within the inner conductor is another concentric SS tube, which provides coolant for the inner conductor. Helium gas enters the ID of the SS tube from the left of Fig.  1, passes to the right end of the PC, and returns through a 3.2 mm radial gap between the stainless and copper tubes.
The inner and outer conductors are joined by a 1.27 mm thick copper plate at the furthest most position to the left of Fig. 1. This plate is an electrical shorting plate of the RF quarter-wave stub.
The thermal model is axisymmetric, one-dimensional, and uses a finite difference approximation. There are over 200 nodes. Temperature dependent properties are used for: thermal conductivities of the S S , copper, and niobium; RF resistivity I . in copper (RRFk2, for 750 MHz) and niobium[2]; and specific heat and enthalpy of the helium coolant. Ma-& radiation heating is calculated assuming gray body, diffuse scattering with the emissivity of all materials being 0.3. Copper can have emissivity values from less than 0.05 to 0.9 depending on the condition of the surface. The value of 0.3 is considered conservative for the 40 year life of the accelerator.

III. RESULTS
The analysis naturally divides into separate consideration of the inner and outer conductors. This paper focuses on the superconducting and cryogenic aspects of the PC, i.e., the outer conductor. The reader is refemxi to [3] for a detailed discussion of the results on the inner conductor.

A. Resistive Stage
The resistive stage of the PC differs from a classical (dc or low fresuency ac) current lead in that most of its crosssectional thickness is mechanical structure which carries no current. The thermal performance of the PC is consequently degraded because this extra material conducts heat to the cold end, but does not contribute to the current carrying capacity. In fact, a thermally ideal PC would have a copper wall with a thickness of less than one skin depth, and no supporting structure.
We can compare the performance the PC with a classical conduction cooled current lead by comparing the heat conduction equations for the t Q cases. For a classical current lead, the one-dimension2steady state heat conduction equation is For the RF case, the heat generation per unit volume of the outer conductor is for the heat conduction equation. Here, t is the total wall thickness and is the surface resistance in ohmdsquare, and D, the tube diameter. Note that (2) and (4)  Combining (5) and (6) and integrating, gives *

(7)
for the minimum value of the heat leak per unit current, which occurs at &=O (dT/dx = 0 at x=O). In the above, Lo) is a pseudo-Lorentz number that is given by where k is the average thermal conductivity at position, x of the copper-stainless steel composite wall. The same result as For our PC, we calculate an average value of LO' of 11.6 X lo-' A2Q2/K2 for the temperature range 300 K to 35 K. This value of LO' compares to 2.45 X lo-* A2Q2/K2 for a classical metal and about 2.1 X lo8 A2Q2/K2 for copper (RRR=lOO) over the same temperature range. Thus, we calculate, from (7), for an rms current of 65 A (for our 210 kW PC), a heat leak of 23 W to a 35 K thermal intercept, compared to a heat leak of 3.1 W for a classical conductioncooled current led with the same current.
As discussed above, the large difference in the heat leaks is due to the necessary SS structure of the PC. The analytically evaluated heat leak of 23 W compares to a value of 23.2 W determined from the one-dimensional finite difference modela better agreement than might be expected. The agreement seems especially good considering that LO' is relatively constant down to 100 K, but increases rapidly at lower temperatures because R, becomes constant, mainly due to the anomolous skin effect. Thus, (7) becomes increasingly suspect at low temperatures.
Another insight on PC performance to be gained from (7) is the dependence of the heat leak on the low end temperature T,. For T, lower than about 100 K, the heat leak at T, is nearly constant. This conclusion is verified by our finite difference results.

B. Superconducting Stage
The contribution to the room temperature, T,, refrigeration input power, Pi, is dominated by heat loads, Qi, at temperature, Ti as given in eqt. 1 where E, is the refrigeration cycle efficiency relative to Carnot. Factors, sometimes quoted as the ratio, P/Q, yield 770 W/W at 2.15 K; 108 W/W at 9 K, and 17 W/W at 50 K, using efficiencies of 0.18, 0.3 and 0.3 respectively, and show the importance of reducing the low temperature heat loads.
One critical factor in reducing the 2.15 K heat load is to maintain the Nb superconducting. This reduces the RF heat load generated in the Nb by a factor of two or more. Maintaining the warm-end temperature of the Nb at the SS/Nb interface below the superconducting critical temperature (TJ of Nb, 9.2 K, is not sufficient to guarantee the Nb is SC along its entire length; there may be a standing normal zone, SNZ. To investigate the existence of the SNZ, we begin with the onedimensional heat diffusion equation with generation and, for simplicity, constant material properties. Heat transfer is by conduction or radiation ( w i w h e inner conductor). Heat generation is from RF joule heating and infra-red radiation exchange, mainly with the inner conductor. For simplicity, we assume the Nb, of length L, is divided into two regions, one with a resistive normal zone with T>T,, and one SC, with TeT,. Equations (10) and (11)  area for conduction: and T,, T,, and T3 are the femperatures at normalized positions &=O, 1 and 6,=1 respectively. In both (10) and (ll), 6 varies from 0 to 1. Note that q1 is from infra-red radiation, mainly from the inner conductor, and q, is radiation plus RF joule heating. Fig. 2 shows a representative temperature profile as a function of the nohalized position variable, C=xL based on (10) and (11). There is a discontinuity in the slope at the interface of regions 1 and 2 because k,, which is the average k between about 11 K and 9.2 K, is about 2.5 times k,, which is the average from 2 K to 9.2 K, in this example.
At the interface between regions 1 and 2, the heat fluxes must be equal, i.e. the heating equals the cooling, which results in the constraint, where 2p -(1-5)

42
(1 -5) 775 = Fig. 3 shows the heating, left hand side of (14), and cooling, right hand side of (14), characteristics graphically. Notice that, depending on q,, p, and q, there may be no SNZ, one allowed SNZ, or two, based on the number of intersections of the heating curve, d with cooling curves a, b, or c respectively. In the case of the two intersections, curve b, the smaller SNZ is not stable. A small perturbation could cause the zone to expand slightly, and the excess heating would drive the resistive portion to the larger SNZ, or a perturbation could cause the zone to shrink slightly, in which case, the excess cooling would cause the zone to disappear.
In general, we are interested in designing a PC that will not have SNZs.
The power rating and electromagnetic performance of the PC, along with the RRR of the niobium, will determine the RF heating contribution to q,. The infrared radiation contribution to q, and q, can be calculated by sophisticated computer codes or estimated from first principles. Then q = q,/q2 can be determined. We define a new term R is the design parameter which allows us to avoid SNZs. R depends on p, which can be adjusted mainly through the choice of Nb nipple length, L, or cross-sectional area, A. (1 1) can be recast by calculating the heat transfed , Q, at the low temperature, T, Q, is the heat load at 2.15 K, in our case. Because 0 has a 1/ L2 dependence, there is an optimum length L to minimize Q,, assuming (T, -T,) is fixed. The value of L may be chosen for other reasons though, such as to eliminate SNZs or, for mechanical or manufacturing reasons. Thus, if L is fixed, the heat load can only be reduced by lowering T,, decreasing A, decreasing k by choosing a lower RRR Nb, or reducing q2. The value of q2 can be reduced by lowering the emissivity of the materials or lowering the inner conductor temperature.
In a typical situation, with geometry, emissivities and inner conductor temperature established, Q, is determined only by T,, the temperature at the SS/Nb interface. The value of T, is usually established by some type of thermal intercept or heat exchanger placed near the SS/Nb interface. The heat exchanger needs to be as close as possible to the interface to stabilize the high temperature end of the Nb. A space of 0.04 m was initially selected to allow room for the flange connecting the resistive portion of the outer conductor to the Nb nipple. Fig. 3 shows the finite difference results. There are two possible steady state solutions; one has a normal zone in the Nb with correspondingly higher heat loads. As seen from the gradient at the SS/Nb interface, the SS is passing an additional heat load to the Nb which is deposited at 2.15 K. To avoid the normal zone, and minimize the heat passed from the resistive portion of the outer conductor, the heat exchanger should be incorporated into the connecting flange itself.

IV. SUMMARY
At first glance, an RF power coupler might not seem very similar to a cryogenic current lead, and depending on your opinion after reading this article, you may still feel that way. The resistive portion of the outer conductor of the PC, appears to be a poorlydesigned classical conduction cooled current lead, because of the need for the SS to support the copper film. The SC portion of the PC has many issues similar to HTC and conventional SC leads.