X. R. Wang^{(1)}, D. K. Sze^{(2)}, M. S. Tillack^{(1)},
and C. P. C. Wong^{(3)}

1) Fusion Energy Research Program

University of California, San Diego, CA 92093-0417

2) Argonne National Laboratory

9700 South Cass Avenue, Argonne, IL 60439

3) General Atomics, San Diego, CA 92186-9784

^{*}Work supported by USDOE contract DE-AC03-95ER-54299

**ABSTRACT**

In this work, the heat flux limits of conventional plasma-facing
components (PFC) were examined. The limits are based on maximum
allowable temperature and stress levels in the structures. The
substrate materials considered were V, SiC composite and HT-9. The use
of Cu also was considered. However, low temperature limits, activation
and very limited radiation damage life time, make the using of Cu in
a commercial power plant unattractive. With selected heat transfer
enhancement, the heat flux allowable is about 5.3 MW/m^{2} for
lithium-cooled V-alloy, 2.7 MW/m^{2} for helium-cooled SiC
composite, and 2.7 MW/m^{2} for helium/water-cooled HT-9. Compared
with the maximum heat flux attainable with Cu and cold water (13.4
MW/m^{2}), acceptable power plant materials place severe
restrictions on heat removal. The thermal conductivity of SiC
composite at 1000 C and after irradiation is a factor of
several lowered than the value we used. This indicates a need to
examine the heat transfer problems associated with PFC, in terms of
material development and enhancement in heat transfer. Physics
regimes which can provide low peak and average heat flux should be
pursued.

**INTRODUCTION**

Design requirements of commercial fusion power plant in-vessel
components are potentially more stringent than those of experimental
devices. The radiation life-time, high temperature performance and
low activation requirements, severely limit the structure materials
selection. Those materials with good characteristics of life-time,
temperature limit, and low activation, such as V and SiC composite,
are not the best materials for heat removal. For present-day and
near-term experimental devices, such as ITER, low temperature water
cooling with copper as heat sink allows designers to envision possible
solutions to removal heat up to 13.4 MW/m^{2}. Relatively thick
armor provides protection against plasma erosion and severe off-normal
events such as disruptions. For a Demo or commercial power plant, the
temperature limit of the divertor target, combined with the requirement
of low activation, makes the selection of the armor material difficult.

Thus, we may have to design a bare-walled system. In this work, we examine the allowable temperatures and stresses for various material combinations. Both 1-D scoping and 2-D finite element calculations are carried out to determine the values obtained in a circular pipe geometry. The 2-D analysis is carried out with realistic one-sided heating and a "high-performance" design with very thin walls and relatively unconstrained boundaries. An innovative engineering solution, the so-called "Liquid Target" design, has been examined. With a liquid metal plasma-facing material, the problems associated with heat transfer, thermal stress, and erosion can all be reduced. The problems associated with a liquid target divertor also will be discussed.

**I. ALLOWABLE MAXIMUM HEAT FLUX ON DIVERTOR TARGET**

Temperature, thermal stresses and pumping power are three important limitations which determine heat flux limits on high heat flux components. As a parametric study, selected design combinations of structural materials and coolants are examined. These are: lithium-cooled V-alloy, helium-cooled SiC composite, helium-cooled HT-9, water-cooled Cu and water-cooled HT-9. Simple circular coolant tubes are used for all cases. The heat flux from plasma radiation at the front of the tubes varies with cos(theta). Therefore, a critical location for thermal and stress analysis will be at theta=0 where the heat flux has a peak value. For this study, the following design constraints are to be satisfied:

- The peak temperature of materials should be less than the temperature limits listed in Table 1.
- The primary and secondary stresses must remain below
the limiting value prescribed by the ASME code,
sigma
_{primary}+ sigma_{secondary}< 3S_{mt}. The allowable thermal stresses are defined by 2S_{mt}. - The ratio of the pumping power to removed thermal power should be less than a few percent.

where, T_{c} is the coolant temperature at the location where peak heat
flux occurs; the maximum film temperature drop
Delta T_{f} is determined by q_{c}^{''}/h;
temperature drop across the tube wall DeltaT_{w} can be estimated by

where K_{w} is the thermal conductivity of the tube wall;
q_{c}^{''} is the heat flux at the interface of the inner wall and
coolant; h is the heat transfer coefficient; q^{''} is the heat flux
from plasma, and q_{w}^{'''} is the volumetric heating in the wall.

The Properties of Structure Materials

Materials V-alloy SiC Cu HT-9 k, W/m-K 30 15 370 28 E, GPa 120 362 125 160 nu 0.36 0.16 0.34 0.33 alpha,10^{-6}/C 10.5 4.4 16.6 12.5 sigma_{a}, MPa 230 190 120 160 T_{max},C 750 1100 400 550

The peak thermal stress at theta=0 is estimated by

where, alpha is thermal expansion coeffcient; E is Young's modulus; and nu is Poisson ratio. Table 2 shows the parameters of thermal hydraulics results. For the V-alloy/Li design option, an average Nusselt number of 7, which was calculated by Gardner [1] based on uniform heat flux and Hartmann number great than 100, is used for the thermal analysis. The coolant tube is assumed to have an electrically insulated coating to minimizing the MHD effect of liquid metal flowing in the magnetic field. For the water cooled Cu and HT-9 tubes, swirl flow is used in order to enhance the heat transfer coefficient. This will result in a 60% increase of heat transfer and critical heat flux (CHF) compared with straight flow [2]. The calculations used the correlations of reference [2] showing that the CHF ranges from 8 to 17 MW/m2 for water velocity of 5-10 m/s. Roughened surfaces are used inside helium-cooled tubes to enhance the heat transfer by a factor of 2. It also increases the friction factor by 3 for the pumping power calculation. The thermal and thermal stress results are given in Fig.1 and 2.

Fig. 1 The allowable heat flux determined by structure temperature varies with the plasma surface heat flux.

Fig. 2. The allowable heat flux determined by thermal stress varies with the plasma surface heat flux.

Fig. 1 shows the results from temperature limits of structural materials. Refer to thickeness of 1 mm wall materials, the maximum heat flux are 13.4 MW/m2 for water-cooled Cu, 5.3 MW/m2 for both lithium-cooled V-alloy and helium-cooled SiC, 3 MW/m2 for both water and helium-cooled HT-9 design options. Fig.2 shows the maximum allowable heat flux determined by thermal stress limitation. The maximum heat flux with five design combinations are found to be 25 MW/m2 for water-cooled Cu, 7.6 MW/m2 for lithium-cooled V-alloy, 2.7 MW/m2 for helium-cooled SiC composite and both water and helium-cooled HT-9. It can be seen that the limits of surface heat flux for both water-cooled Cu and lithium-cooled design options come from temperature of structural materials.

Thermal Hydraulic Design Parameters

Design Options V-alloy/Lithium SiC/Helium Cu/Water HT-9/Water HT-9/Helium Tin/Tout, C 300/450 350/650 50/100 215/350 250/450 D, cm 1.0 0.8 0.8 0.8 0.8 v, m/s 1.3 89 8 3 134 h, kW/m2-C 3.3{a} 2.6{b} 4.7{c} 1.9{c} 3.7{b} fp, % 0.01 3.1{b} 0.23 0.04 7.0{b}a. average Nu=7; b. includes roughened surface; c. twisted tape assumed to be used in inside of coolant tubes.

**II. LIQUID METAL HEAT TRANSFER**

As a 2-D example of considering non-uniform heat flux distribution, only lithium-cooled V-alloy tube was analyzed in detail by finite element code. The most important effects for liquid metal heat transfer are MHD effects on velocity, nonuniform heat flux distribution and volumetric heating generation in liquid metal[3]. When the heat flux on the tube varies along the circumference, the Nusselt number and the film temperature drop will be a function of theta. To simplify the calculation of the temperature distribution in the lithium coolant and the wall, fully-developed laminar flow is assumed. The velocity distribution is assumed to be flattened by a transverse magnetic field strength of at least 7 T. Therefore a uniform velocity distributed across the coolant tube is used in the calculations. The common convection equation is expressed by

Where, the first term represents the heat conducted out of
structure across surface s; q^{'''} is the volumetric heat generation rate in the
structure and coolant; v is the coolant velocity;
rho and C_{p} are density and heat capacity of the coolant.
To simply the calculation, both v_{x} and v_{y} are set equal to 0.
A finite element code AYER [4] was used to calculate the temperature
distribution in both the coolant and wall. The quadrangular element
type is used to mesh the 2-D cross section of tube and
coolant. Then, the heat transfer coefficient can be determined as a function
of theta.

An average Nu at the front of the tube is found to be 3.0 which includes the effects of uniform coolant velocity, nonuniform heat flux and volumetric heat generation in the coolant. At the back of tube, a much lower value of Nusselt number, which is equal to 0.5, is found. The results are quite consistent with the analytic solutions of Hasan [5]. It should be noted that the results are only suitable for the case of fully developed temperature. Taghavi and Tillack [3] had investigated the effects of nonuniform heat flux on Nu in entrance region. They indicated that the average Nu in the entrance region can be several times the fully developed Nu.

**III. NUMERICAL SOLUTION**

Thermal and structural analyses were performed using the ANSYS finite
element code to compare the results obtained by 1-D analytic solution.
Only the case of lithium-cooled V-alloy tube was done with the ANSYS
program. The thermal loadings include the surface heat flux on the
tube, volumetric heating of 10.0 MW/m^{3}. A coolant bulk
temperature of 375 C was used at the midway point between
the inlet and outlet of the coolant tube. Both cases of Nu=7 and 3 are checked
by using the same element model. The external constraints are that the nodes at the
bottom of tube are fixed in the x and one node is fixed in y directions. The
coolant pressure of 1.0 MPa was taken to calculate the
pressure stress.

The results of thermal and thermal stresses computed by ANSYS program
are entirely consistant with that of analytic solutions. For the
surface heat flux of 5 MW/m^{2} and thickness of 1 mm, the peak
temperature in the front of the tube is 712 C. It is below
the temperature limit of 750 C, and the primary stress, peak
thermal stress are 10, 98 MPa, respectively. These stresses fall well
below the maximum allowable thermal stress of 230 MPa and maximum
primary plus secondary stress of 345 MPa. For Nu=3, the constraints of
thermal design window mainly come from temperature limits. The thermal
stress limit can allow for heat flux as high as of 8 MW/m^{2}, but
the temperature limit of V-alloy can only allow for a heat flux of 3
MW/m^{2}.

**IV. HEAT TRANSFER LIMITATION ON LIQUID TARGET DIVERTOR**

A liquid target divertor is an attractive design to eliminate some critical issues associated with a solid target divertor. The problems associated with sputtering, radiation damage, and thermal stress on the structural wall are all eliminated. A key issue associated with the liquid target divertor is plasma contamination. As the surface temperature of the liquid target increases, so does its vapor pressure. The acceptable vapor pressure defines the upper limit of the coolant temperature, and also the heat removal capability.

The particle density allowable in the plasma is estimated to be
10^{18}/m^{3} [6]. At 400 C, the lithium vapor pressure
exceeds this density limit. Thus, the maximun allowable surface
temperature for a lithium target divertor is 400 C. The
lithium vapor pressure increases exponentially with
temperature. Therefore, relaxation of the allowable particle density
has only a minor effect on the allowable temperature.

Lithium is fed into divertor target at 200 C. The maximun allowable surface temperature rise is 200 C. The temperature rise on the surface can be written as the following form [7],

in which q^{''} is the surface heat flux, k is the thermal conductivity,
alpha is the thermal diffusivity, t is the residence time of lithium on the
target, and ierfc is the error function. This equation can be simplified
to the following form with proper material properties:

in which t = l/v, with l being the length of the divertor plate and
v being the velocity of lithium. From this equation, it can be
seen that any heat flux can be handled,
with the reduction of the target length, and with the increasing of
the coolant velocity. In a real engineering design, there is a
minimun divertor target plate legnth we can use, and a maximun velocity
achieavable. If the plate length is assumed to be 1 m, and the
velocity is assumed to be 1 m/s, the allowable heat flux is
1.8 MW/m^{2}. The maximun allowable heat flux for a
lithium target divertor is 2~3 MW/m^{2}.

If Ga is used as the divertor coolant, the limiting factor is not heat transfer, but corrosion. Figure 24 in reference [8] showed the corrosion rate of various material facing Ga. According to BCSS [9], the maximun allowable corrosion rate is 20 um/y. Thus, the interface temperature of Ga with the structural temperature on this figure is limited to to below 250 C.

**CONCLUSIONS**

1-D thermo-mechanical scoping ananyses were performed for
five material combinations. Refer to thcikness of 1 mm wall materials, the
heat flux allowable are about 5.3
MW/m^{2} for lithium-cooled V-alloy with Nu=7, and 2.7 MW/m^{2} for
helium-cooled SiC composite and both helium and water cooled HT-9 with
heat transfer enhancements. The combination of cold water cooled Cu
has the highest heat flux limst of 13.4 MW/m^{2}. One side heat
transfer calculation shows low limits for lithium-cooled combination
to 3 MW/m^{2}. Heat transfer enhancements will be needed to increse
these design limits.
Lithium liquid target divertor surface heat flux removal capablity
will be limited by liquid metal vapor pressure to 2~3 MW/m^{2} for
a velocity of 1 m/s and a flow length of 1 m. High performance is expected for
higher coolant velocity and short flow path. More detailed analysis will
be needed to fully evaluate the application of this design option.

**REFERENCES**

- R. A. Gardner, "Laminar Pipe Flow In A Transverse Magnetic Field With Heat Transfer," {\it Int. J. Heat Transfer}, Vol. 11, pp. 1076-1081. Pergamon Press 1968.
- J. A. Koski {\it et al.}, "Thermal-Hydraulic Design Issues and Analysis for the ITER Divertor," {\it Fusion Technology}, Vol. 19, 1729 (1991).
- K. Taghavi {\it et al.}, "Special Features of First Wall Heat Transfer in Liquid Metal Fusion Reactor Blankets," {\it Fusion Technology}, Vol. 12, 104(1987).
- R. G. Lawton, "The AYER Heat Conduction Computer Program", LA-5613-MS, (1974).
- M. Hasan, "Effects of Nonuniform Surface Heat Flux and Volumetric Heating on Blanket Design for Fusion Reactors," {\it Fusion Technology}, Vol. 16, 44(1989).
- J. N. Brooks, Argonne National Laboratory, Private communication.
- H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids" Oxford Univ. Press, London and NY (1959).
- P. R. Luebbers, W. F. Michaud and O. K. Chopra, "Campatibility of ITER Candidate Structural Materials with Static Gallium," Argonne Natioanl Laboratory report ANL-93/91 Dec. 1993.
- Dale Smith et. al., "Blanket Comparison and Selection Study Final Report" ANN/FPP-84-1, Sept. 1984.