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Title DESIGN PROBLEMS ASSOCIATED WITH HIGH-CURRENT-DENSITY"" WATER-COOLED SEPTUM MAGNETS

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Author Green, Michael A.

Publication Date 1965-08-01

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DESIGN PROBLEMS ASSOCIATED WITH HIGH-CURRENT-DENSITY WATER-COOLEDSEPTUM MAGNETS

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This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. ; _ 1 ~0 International Symposium on Magnet J oiJI_!ech~~logy-Stan~~~~ ~~iv~ --~-~t. 1965 L______---- UCRL-16309

UNIVERSITY OF CALIFORNIA Lawrence Radiation Laboratory Berkeley, California AEC Contract No. W -7405 -eng-48

DESIGN PROBLEMS ASSOCIATED WITH HIGH-CURRENT-DENSITY WATER-COOLED SEPTUM MAGNETS

Michael A. Green

August 1965 -1-

DESIGN PROBLEMS ASSOCIATED WITH HIGH-CURRENT-DENSITY WATER-COOLED SEPTUM MAGNETS

Michael A. Green

Lawrence Radiation Laboratory Berkeley, California

As the energy of particle accelerators in­ Bo magnetic field (gauss), creases, the need for small high-field magnets h vertical aperture, increases. These magnets are most likely to be AB overall area of the bundle. found in an accelerator's extraction system. The advent of slow extraction at CERN and Brookhaven The heat generated per unit volume may be has shown the need for high-current-density mag­ calculated once the current density is found (see net design. Fig. 1), 2 I define high-current-density water-cooled Pc = (i/A) Res conductors as conductors in which the heat trans­ where P heat generation per unit c fer from the copper to the water across the ther­ volume (W/cmZ), mal boundary layer can no longer be ignored. I Res resistivity. have arbitrarily set a lower limit of 5000 A/cm2, in the copper, for high-current-density conduc­ The resistivity is a function of the temperature of tors. In general, the film heat transfer becomes the copper. The heat generation per unit volume more dominant as the length of the cooling circuit will increase with temperature. decreases. The generalized equation for conductive heat High-current-density conductors are most transfer, which includes transient effects and heat likely to be found in extraction septum magnets of source, may be represented by various types. The elimination of the magnetic field outside the magnet requires the total current V'ZT +PC =.:£_8T; flow to be restricted to the magnet gap itself. K K 8t For the extraction of high-energy particles, the when the system is in steady state the equation thickness of the conductor in the horizontal direc­ becomes Poisson's equation, tion may be severely limited. The current den­ p sity in this conductor is determined by its hori­ V' 2 T+~ 0 K ' zontal dimension. The vertical dimension of sep­ tum conductor has no effect on current density, where T temperature, because the total current in the conductor bundle K thermal conductivity, is directly proportional to the vertical aperture c specific heat, of the magnet. p density of the conductor, t time. The use of high-current-density conductors should be avoided whenever possible for economic The differential equation can be solved by reasons. The optimum current density for mini­ several well-known techniques. In general, the mum cost in a septum magnet occurs at current heat conduction to the water can be neglected, densities of less than 500 A/cm2. The high­ particularly with standard conductors. There are current-density magnet should be a minimum­ cases in which the conductive heat transfer to the field magnet with a small vertical aperture. In water tube surface should be considered. These an extraction system with more than one magnet, cases usually can be solved from simplified one the current density of each magnet conductor or two-dimensional equations.i should be considered in cost-optimizing the ex­ 2 traction system. Current densities of 5000 A/cm or more require that the convective heat transfer from the Heat Transfer from a copper to the water be investigated. Fortunately, High-Current-Density Conductor a large amount of experimental work has been done with convective heat transfer from copper The current density of a conductor bundle tube walls to water. The empirical techniques for may be defined as follows: calculation of a heat transfer are well documented. The calculation can be made with an accuracy of 1'/A _- ---.Ni 2 Oo/o. E AB The basic convective heat transfer equation Ni = _!Q. B h is 47T 0 where Ni = ampere -turn requirement, i/A = current density, where Qc heat transferred, E = packing factor (including water he heat -transfer coefficient, passages), A heat-transfer area, -2-

Tw = temperature of the heated surface, references 2 and 5. The accuracy of all these Tb = bulk temperature of the water. methods decreases as the temperature difference between the tube wall and the bulk fluid increases. The value of the heat-transfer coefficient is determined by a number of factors such as the For high-current-density conductors, the velocity of the water, the hydraulic diameter of convective heat transfer plays an important role the tube, and the viscosity, density, and thermal in determining the current capacity of a given conductivity of the water. These terms are gen­ conductor. The two factors which influence con­ erally lumped into two dimensionless numbers, vective heat transfer most for a given fluid flow the Prandtl and Reynolds numbers. are the heat-transfer area and the allowable tem­ perature difference between the wall and fluid The Reynolds number of the water at its (see Fig. 2}. bulk temperature and velocity determines whether the flow is laminar or turbulent. In most cases, The most important condition that affects the flow will be turbulent. The heat transfer co­ the cooling of a magnet conductor is the length of efficient can be found as follovys: the water circuit. For high-current-density con­ ductors the length of the cooling circuit becomes for laminar flow, Reb< 2300, even more important. In general, the conductor wall temperature should not be allowed to exceed K 1oo•c. The heat-removal rate for a conductor is h 0.332 Df 5 33 c Re~· Pr~· for turbulent flow, Reb> 5000, since the heat removal by the water equals the h = 0.023 Kf R 0.8 P 0.33 heat transferred by the water and the heat c D ef rf generated, where i 2 Re t = h A ( T T ) A c w- b VDpb c Reb j.Lb VDpf Ref ---, and flf cfflf and the maximum Prf Kf Tb = Tout 4AH Ac =conductor cross-sectional area, D t = length of conductor, p Tout= exit water temperature, Tin = inlet water temperature~ and Tf = 0.5 (Tb + Tw) As the current density increases, the water The following nomenclature is used: circuit length must be shortened, the hole size in­ creased, or the water velocity increased. The K = thermal conductivity of the fluid, temperature rise in the water per gram of con­ C = specific heat of the fluid, ductor for various water -flow rates and current fl =viscosity, densities is shown in Fig. 3. V =bulk water velocity, D =hydraulic diameter, Special Problems Associated with AH =cross-section area of the tube, High-Current-Density Magnets P =wetted perimeter, p density of the fluid, There are several problems which become Re Reynolds number, more important as the current density increases. Pr Prandtl number. These include short burnout time in the event of cooling failure, the pumping and handling of cool­ The meaning of the subscripts: ing water, and increased cavitation, erosion, cor­ rosion, and electrolysis. f = water conditions at a temper.a­ ture Tf The burnout time is a function of current b = bulk water conditions at Tb density and maximum allowable copper tempera­ w =tube wall conditions. ture. If there is no cooling water in the cooling channel, time to failure is given (see Fig. 4) by There are a number of other methods for t = 4.19 pc (Tm -Twm) calculating the convective heat transfer. I have chosen these equations because of their connec­ PC tion with basic boundary-layer theory.6 For P = (i/A)2 Res, other heat-transfer calculation methods, see c -3-

where t = burnout time, high-current-density magnets is far more impor­ p =density of the conductor, tant than it is for standard magnets. This is be­ c = specific heat of the conductor, cause the convective heat transfer is far more Tm = maximum allowable temperature, important than with standard magnets. Also, Twm = maximum copper temperature most high-current-density magnets have smaller during normal operation, cooling passages than the standard magnets. Pc =heat generation rate ( W /cm3). Cooling-water stoppages and partial stoppages will affect high-current-density magnet perform­ The burnout time calculated in the preced­ ance greatly. ing paragraph is affected by water in the conduc­ tor and the stored energy of the magnet. The Cavitation and erosion usually are not prob­ burnout times shown in Fig. 4 represent the max­ lems in conventional magnet designs. Cavitation imum time allowed for the magnet to be shut off can be avoided by not allowing the lowest pressure in case of cooling failure. If the conductor con­ in the system to fall below the vapor pressure of tains some cooling water, this time may be ex­ the water. Erosion can usually be avoided by tended because of the heat capacity of the water. keeping the magnet water clean. However, when the water in the conductor starts to boil, most of it will be forced out by the steam The use of dissimilar metals should be pressure generated from the boiling water. avoided as in conventional magnet cooling sys­ terns. The deplating problem is worse than with The stored energy of the magnet may influ­ conventional-type magnets, because currents in ence the failure time. If the amount of stored en­ the cooling water are higher and the conductor and ergy released in a section of uncooled conductor its water channel generally are smaller in size. during magnet shutdown approaches the maximum The copper removed from the conductor by de­ allowable heat energy storage in the copper, the plating generally becomes a sludge which must be failure time will approach zero. removed by the cooling water filtration system. Elevated temperatures of the copper result in the High-current-density magnets will require increased formation of deposits on the walls of the a fast-response temperature sensor on the con­ cooling tube. In general, most of the cooling sys­ ductor near the end of each cooling circuit. The tern problems can be solved if the purity of the temperature -sensing system should be fast enough cooling water is maintained. to shut the magnet off before the conductor reaches burnout temperature. Optimization of Current-Carrying Capacity in Standard Rectangular Conductor The increase in current density in a water­ cooled conductor results in changes in the hand­ The current-carrying capacity of a standard ling of the cooling water. The heat generated in conductor is a function of convective heat transfer the conductor is directly proportional to the cur­ and cooling circuit length. The maximum current­ rent density. Since the cooling water has only a carrying capacity for a square conductor when the limited heat capacity, the amount of cooling water cooling-circuit length is zero (i.e., the heat ca­ needed must increase with current density. pacity of the water is neglected) is There are three ways that the flow rate can be in­ 2 creased. They include increasing the velocity of Q i Re s l = h AD. T Q the cooling water, increasing the conductor hole gen = 4.19 Ac c c out' area, and inc rea sing the number of cooling cir­ cuits. where A = b2 - 1TD2 c 4 Increasing the velocity improves the heat A = 1rDl transfer, but only at the expense of pumping pow­ er and increased erosion. Increasing the hole 0.023 Re~· 8 Pr~· 33 K size increases the heat transfer but also results h for turbulent flow. c in increased pumping costs. Increasing the num­ D ber of water circuits does not increase the cost of If the film temperature drop is a maximum, pumping, but results in increased manifolding namely Tw.,;; 100°C, and Tb the inlet water tem­ costs. perature, then the Prandtl number will be a con­ stant if we assume that The optimum solution involves increasing 0 33 the velocity and hole size slightly as well as in­ then P r f · = 1 . 4 ' hence creasing the number of water circuits. The ex­ 0 8 act design would depend on the magnet under con­ 0.032 1rRef· KlC.Tc sideration. In general, increasing the number of water circuits results in the best solution. How­ Q =Q gen out ' ever, as an increased amount of convective heat 2 transfer per unit volume of conductor is required, i Res l c an increase in hole size and water velocity be­ comes increasingly advantageous.

The purity of the magnet cooling water for -4-

A = heat transfer area, v = fLIP = kinematic viscosity, D = tube diameter, 6T = the maximum allowable film c temperature drop (::::70•C), 6 T R = temperature rise in the water required to remove the heat from a conductor carrying a where the constant current ic, 6T B =film temperature drop when 0 8 0.135 1r v · K the conductor carries i, X= 6 T p. water temperature rise when Res v 0 ·8 the current carries i, maximum current that can be and 6T =constant :::qo•c . carried (very short sample), c current capacity when the The maximum current flow will occur when circuit length is considered, Tin = inlet water temperature, Tout = outlet water temperature, Tw =wall temperature(,; 100•C). therefore maximum current-carrying capac.ity for a square conductor will occur when The optimum water hole size is larger when the cooling circuit length is considered. The D=0.602b. lange r the water circuit, the larger the water hole. When we no longer neglect the heat capacity The solution presented here does not take into of the water, the current-carrying capacity of the consideration the increase in Prandtl number and conductor goes down. The film-temperature drop the decrease in Reynolds number as 6TB de­ must be decreased because of the rise in tem­ creases. Fortunately, the effect of the change in perature of cooling water as it passes down the 6TB is small. conductor. The sum of the film-temperature drop when the water circuit length is not taken into Figures 5 and 6 show the current optimiza­ consideration is tion for a 5 X 5 -rom square conductor. The current-carrying capacity for four of these con­ 6Tc = 6TB+ 6Tp. ductors is greater than for one 10X 10-mm square The required temperature rise in the cooling conductor. A square conductor will perform bet­ water to allow the film temperature drop tore­ ter than a rectangular conductor of the Bame main at 6Tc is cross section. Heat transfer from the hot end of the conductor to the cold end of the conductor may 2 4i Res l be neglected except for very short conductors.3 c 6TR = Achievement of Very High Current Densities

where 6TR is a linear function of l. Since the In order to achieve current densities in the sum of 6Tc and 6TR is greater than 6Tc, the range of 20 000 A/cm2 and above, nonstandard current-carrying capacity of the conductor must techniques must be used. There are three ways be decreased. If we define of achieving current densities in this range. They are: using high water velocities, using very short 6TR water circuit lengths, and increasing the ratio of y = -- 6Tc ' heat transfer area to conductor volume (see Fig. 2). we have the current-carrying capacity for a con­ ductor with a nonzero water circuit length, In the design of very-high-current-density conductors, a combination of the three ways would i c be used. There are two basic conductor types which can be used in the very-high-current den­ [y +i]t sity range. These are multichannel conductors and cross-cooled conductors. then Y6T 6TR c The multiple-channel conductor allows the 6Tp T Y+1 y+ 1 out - Tin' heat transfer area to be increased, which increas­ es the current-carrying capacity of the conductor. 6Tc If the water velocity in the cooling channels is 6T = -- = T - T greater than 600 em/ sec and the cooling circuit B y + 1 w out' length is 50 em, the conductor shown in Fig. 7 should carry currents in excess of 12 OOOA .with where b side dimension of the square conductor, current densities exceeding 20 000 A/cm2. I feel that a conductor of this type can be built a tech­ Ac copper area, J nique that is similar to the one used for the CERN -5-

multichannel conductor. References 1 .... If a septum magnet could be designed to be a Dusinberre, George M., Heat Transfer Calcula­ single -turn magnet, the cross -cooled conductor tions by Finite Differences (International Text­ would be attractive. The conductor has a large book Co., Scranton, Pennsylvania, 1961). surface-to-volume ratio as well as very short 2 water circuit lengths. High water velocities Giedt, Warren H., Principles of Engineering should be possible if the manifolding is properly Heat Transfer (Van Nostrand, Princeton, New designed. The conductor shown in Fig. 8 should Jersey, 1957). be able to carry currents in excess of 20 000 am­ 3 peres. The greatest design problems, as I see it, Green, Michael A., "Heat Transfer Properties associated with a eros s -cooled magnet is the dif­ of a 5 -Millimeter Septum," Lawrence Radiation ferential expansion between the heated conductor Laboratory Report UCID-262 7, Sept. 1964. and the cooling manifolds. 4 Green, Michael A., "Parameters for the 8-GeV Conclusions Injector Synchrotron Extraction System," Lawrence .Radiation Laboratory Report UCID- Many of the problems associated with high­ 2628, June 1965. current-density septa are only briefly mentioned. Fabrication techniques developed in recent years 5Kreith, Frank, Principles of Heat Transfer (In­ make the construction of high-current-density ternational Textbook Co., Scranton, Pennsylvania, magnets possible. The development of inorganic 1960). insulations is important for the construction of 6 thin, slow -extraction septa. schlichting, Hermann, Boundary Layer Theory (McGraw-Hill Book Co., New York, 1960). In general, the following design criteria should be used in the design and fabrication of high current density magnets:

1. Avoid the use of high-current-density conduc­ tor whenever it is economically possible.

2. Minimize the vertical aperture of a high­ current-density septum.

3. Do not allow the copper temperature at the tube wall to exceed the water boiling temperature, particularly for small cooling passages.

4. Optimize the conductor for its current­ carrying capacity, not for maximum current den­ sity.

5. Minimize the number of turns in the magnet. Insulation takes up valuable space. Avoid multi­ turn magnets in the very-high-current-density range (20 000 A/cm2 or more).

6. Avoid the use of nonradiation-resistant insu­ lations.

7. Use high-speed temperature sensors, which are essential at the hot end of each cooling cir­ cuit to prevent conductor burnout due to cooling failures.

8. Avoid many cooling system problems by maintaining the purity of the cooling water.

9. Investigate conductive heat transfer from the copper to the water when the current densities are very high or when the heat path passes through materials of low thermal conductivity.

High-current-density magnets should be de­ signed according to the basic laws of physics; ex­ perience is also very valuable when these mag­ nets are built. L -6-

F.IGURE LEGENDS

3 Fig. 1. Heat generated per em of the conductor as a function of current density. 3 Fig. 2. Connective heat-transfer area required per cm of conductor as a function of current density for various h .6. T. c 3 Fig. 3~ The water temperature rise in the cooling circuit per em of con- ductor as a function of current density for various water flow rates.

Fig. 4. The minimum time for conductor burnout as a function of current density for various failure temperatures.

Fig. 5. Maximum current flow verses hole diameter in a 5X5-mm water­ cooled conductor for various water velocities. Maximum wall temperature = 100°C; maximum water temperature = 30°C; water circuit length not considered.

Fig. 6. Maximum current flow as a function of hole diameter in a 5 X 5 -mm water -cooled conductor for various water circuit lengths. Water velocity= 450 em/sec; maximum wall temperature= 100°C; maxi­ mum water temperature = 30 o C.

Fig. 7. The multichannel conductor with the cooling water flowing length­ wise down the conductor.

Fig. 8. The multiple -channel cross -cooled conductor.

.J -7-

.... 0 +­ u :::l "0 c: 0 u -0 N E u 3:'

c Resistivity of -6 0 -2xl0 +- copper at 60°C - ohm/em ....0 (l) c (l) 0" +- 0 (l) ::r:

10~--~~~~~~--~-L-L~~~---L~-L~UU~ 10 3 104 Current density, i/a

MUB-7312

Fig. 1 -8-

100

Q) E ::3 -0 > '- 0 u -::3 "'0 c hc~T= 25W/cm2 0 u 10.0 2 hc~T=50 W/cm cr0- -::3 "'" E Q.'Q) u hc~T=IOOW/cm 2 -oN Q) E '- u 2 ·-::3 he ~T = 200 W/cm 0" - Q) '- c 1.0 Q) ~T=Tw -Tb c'- '- Q)

1/) -c c -'- c -Q) I 0.1 103 104 105 106 2 Current density, i/a ( A/c m )

MUB-7313

Fig. 2 _, -9-

" 10

lo.. 0 u -:::3 \:) c: 0 u

lo.. Q) u a. 0 Water flow rate a. 0 0 u tO 20 g I sec 0 - -0 g I sec 1'0 w ~ 'o g /sec lo.. g /sec Q) X a. (\j g I sec Q) I/) II

lo.. I/) Q) 0.1 Q) 0::: lo.. :::3 0 -lo.. Q) a. E Q)

lo.. -Q) -0 ~ 0.01 3 4 5 6 10 10 10 10

Current density, i/a (A/cm 2 )

MUB-7314

Fig. 3 -10-

.[

10 Maxi mum copper temperature= 100 oc

u Q) (J) Insulation failure

Q) "0 temperature Q) ;_E I... Q) "0 1.0 I I 0 oc - (J) Q) 120 °C I... c: ::J 0 150 u oc 0 200 oc 0 300 - -c: oc E >. ::J 0' E I... Q) ·-c: c: Q) ~ 0.1 "0 Q) I... 0 -(J)

Cur r e n t density, i I a ( A I c m2)

MUB-7315

.'

Fig. 4 -11-

4000

3 000

<[

c: -Q) ... ~ 2000 (.) V = 150 em I sec

I 000

Water circuit length = 0

Hole diameter (mm)

MUB-7316

Fig. 5 -12-

Cooling circuits 4000 length= 0 meters / ---~

<(

u 0 a. 0 u

+­ c Q),_ ,_ :::l 0

Cooling water velocity 450 em/sec

Hole diameter ( mm)

MUB-7317

•· Fig. 6 -13-

MUB-7318

Fig. 7 -14-

Water ehanne I 0.4 x 0.25 em •' r'r-7-"-7~~-.::----= 0.1 em .. --7--:7'----c::::::_...... __ 0. 4 em

2em 0.5 em

MU B-7319

Fig. 8 ..

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