Thermal analysis of a tokamak divertor plate after a sudden coolant dry-out

Fusion Engineering and Design 13 (1991) 339-361 339 North-Holland

Thermal analysis of a tokamak divertor plate coolant dry-out

Anton io Sofia a, . , Vito Renda b and Loris Papa b

a ETSII-DENIM, Instituto de Fusion Nuclear, Universidad Politecnica, 28006 Madrid, Spain b Applied Mechanics Division, JRC-lspra Establishment, 21020 Ispra (VA), Italy

Submitted 7 April 1990; accepted 25 June 1990 Handfing Editor: G. Casini

after a sudden

The divertor plate represents a critical component in the design of a tokamak fusion reactor. Within the frame of the International Tokamak Experimental Reactor (ITER), many proposals are underway. The thermal behaviour following a complete Loss of Coolant Accident of a pre-designed divertor plate is analyzed in this paper. The aim is to carry out a parametric study on the most critical parameter, i.e. the plasma shutdown time. Several load hypotheses are studied in order to contribute to a better understanding of the design requirements for the components. According with these results, a significative material damage is expected to occur when plasma burning is not efficiently stopped within some seconds after the LOCA detection.

1. Introduction

Accidental thermal transients are an important fea- ture to be assessed for future fusion tokamak reactors. During the last years, JRC-Ispra has contributed to these analyses in order to evaluate the effectiveness of passive heat removal mechanisms after under cooling accidents ('Loss of Flow Accident', LOFA and 'Loss of Coolant Accident', LOCA) [1,2,3]. An overall plant model was used in most of the cases, using as reference reactors NET-Double Null (DN) and NET-Shielding Blanket (SB). The influence of the divertor plates was ignored, assuming that no cooling capability was lost in this reactor part. Since the steady-state surface heat deposition on the divertor plates is about two orders of magnitude greater than on the first wall, new problems arise when designing this component, and materials with better thermal performance are needed. A detailed 2-D analysis simulating a sudden LOCA in the cooling tubes of the component is presented in this paper. Both steady-state and transient calculations have been done. The reference design corresponds to the Joint Research Centre proposal [4] for the ITER (International Toka- mak Reactor) divertor plate [5].

* JRC Grant Holder.

The main characteristics of the new CFC-block divertor concept are the use of the high conductivity carbon fiber composite graphite (CFC) as plasma-facing material, and the arrangement of the cooling circuit embedded in both the graphite matrix and the AISI-316 back plate. Steady-state thermomechanical calculations were recently carried out based upon this design [6,7], focusing on the CFC analysis.

The TZM-alloy coolant tubes have the function of supporting the graphite tiles, as well as the AISI-316 block, so that the contact between these components is not ensured. The thermal effect of this gap is an important issue to be analyzed.

The extremely high temperatures reached by the components makes the heat radiation to be a non negligible heat transmission mechanism. Moreover, only heat radiation to the opposite first wall and the inner vacuum vessel can be foreseen as cooling mechanisms, whose effectiveness will be analyzed.

2. Calculation hypothesis

The accident scenario considered here assumes as instantaneous and complete dry-out in the cooling tubes, since only one cooling circuit has been foreseen. From

0920-3796/91 /$03 .50 © 1991 - Elsevier Science Publishers B.V. (Nor th -Ho l l and )

340 A. Soria et al. / Thermal analysis of a tokamak divertor plate

time = 0 onwards, the tubes are supposed to be void, and only heat radiation is present within the pipes. This is a very pessimistic hypothesis, but it will certainly provide bounding values for the expected temperature overshoots in the component.

2.1. G e o m e t ~

The finite element discretization used for the analysis is shown in fig 1. Four materials are present within the model: CFC graphite in the upper zone, two pieces of AISI-316, a ceramic material connecting them, and the TZM-alloy cooling tubes. The vacuum vessel is represented by the lower mesh which acts as cold heat sink by means of radiation across the cavity EFGHIJKLMNOE, the line PQRSP being also a cavity. In all cases, the bottom line AB is assumed to have a constant temperature of 373 K. Two pieces of stainless steel are present: the coolant tubes are embedded in the first one, which also supports the CFC-graphite matrix. The second one is a double-T shaped rail which allows the fixing of the tiles, avoiding in this way vibrations produced by the high speed of the coolant in the tubes.

If a gap between the CFC and the AISI-316 steel is assumed, a third radiative cavity must be considered.

During the transient analysis, no coolant capability exists in the tubes, and heat diffuses by radiation also there.

The divertor section analyzed is 17 mm wide and 75 mm long (line GC). The tube diameter is 10 mm, and its thickness is 2 m.. The minimal thickness of CFC graphite between the tube and the plasma facing surface is 11 ram. Since the graphite is not continuous in the z-direction, this discretization is supposed to accurately represent the temperature field during the transient.

2.2 P lasma thermal load

The surface heat deposition due to charged particle collision on the line CD in nominal steady state conditions is 10 M W / m 2, which is a standard assumption for the heaviest loaded divertor point. The plasma extinction time after an accident greatly depends on the detection time. Lack of information on this issue makes it impossible to establish a realistic hypothesis. Four cases were considered here, none of them regarding the possibility of a power overshoot (fig, 2): (a) automatic plasma shutdown at time zero, (b) linear power decrease from the steady-state value to

zero in 5 s,

- - T Z M - a l i o y

. . . . . . . C F C g r a p h l t e

S R . . . . . C e r a m l o m a t e r i a l

P Q ...... Ceromic rneterlm[

. . . . A]S[ - 3 1 6

T Z M - a l l o y

- - A I S / - 3 1 6

o . H

A B

Fig. 1. ITER divertor plate. Finite element discretization.

A. Soria et al. / Thermal analysis of a tokamak divertor plate 341

(c) steady-state power continuation for 5 s, then linear power decrease to zero at time 10 s,

(d) plasma burning continuation. After the total plasma shutdown (cases (a), (b), and (c)), the graphite is assumed to radiate to a perfect black body kept at temperature T~o. For times 0 < t < 10 s, Too = 1300 K, for 10 s < t < 50 s, T~ is linearly inter- polated between 1300 K and 373 K, whereas T~o = 373 K for times t > 50 s. The effect of this radiation boundary condition could be neglected during the first seconds of the transient, since the heat flux leaving the graphite surface is irrelevant when compared with the nominal heat flux of 10 M W / m 2. The flux between a small surface with emissivity 0.95 at 1000 K surrounded by a black body at 373 K is q = 56699 W / m 2. Heat radiation, however, plays a significant role for large times. The above-described selection of the temperature T~o tries to simulate the temperature evolution of the well-cooled first wall in the plasma chamber.

2.3. Activation thermal load

The short-term thermal response of the component is mainly driven by the intense surface heat deposition on the plasma side. In spite of this, the volumetric heat

Table 1 Volumetric heat deposition rates ( W / m 3)

Material Steady- l s 10m 60m 2h 3h state

CFC graphite 9.0e6 0 0 0 0 0

Other 1.2e7 2.8e5 2.4e5 1.9e5 1.5e5 1.2e5

generation in the solid plays an certain role in the steady-state case, whereas it is much more important in the long-term time transient. The heat generation rates used here are listed in table 1. They come from data corresponding to the NET-Shielding Blanket reference reactor [8].

2.4. Material properties

The thermal parameters (k: thermal conductivity, p: density, Cp: specific heat) are reported in table 2 [10], as a function of temperature (K). It is important to point out that surface emissivity plays a critical role in this kind of calculations, the results being quite sensitive to this parameter. Nevertheless, it is very difficult to cor-

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rectly estimate a good value for emissivity, since it greatly depends on the surface polishing and treatment. For the calculations presented here, the graphite emissivity was assumed to be 0.95, and 0.5 for all other materials.

The CFC graphite exhibits an anisotropic thermal conductivity. The high-conductivity plane is assumed to be the calculation plane. Heat diffusion in the perpendicular direction is more difficult due to the material discontinuity of the graphite tiles. One could, however, arrange the graphite with the high conductivity in the plane given by the vertical direction in fig. 1 and the direction perpendicular to the calculation plane, in order to enhance the flattening of the temperature peak produced by the maximum in the heat flux in the poloidad direction. The effect of this should be negligible in the cases presented here, since the heat diffuses mainly in the vertical direction of fig. 1. When analyzing another type of transient, such as a single-tube LOCA with two independent circuits, the symmetry of the component could no longer by used to reduce the calculations. In this case, an accurate description of the anisotropic thermal conductivity would be necessary if the second choice is done.

42424 W / m 2 / K . However, two facts make reasonable an increase of this value: (a) In order to maximize the turbulence, a twisted tape

could be fixed in the internal part of the tube. Recent experiments [9] showed that the forced convection heat exchange coefficient could reach a value two times greater than the coefficient observed without the tape.

(b) A nucleate boiling is expected in the critical zone of the tube wall. This implies also a significant im- provement in the global heat exchange conditions.

In order to carry out an accurate modelization of the phenomenon, one should use the Dittus-Boelter equation with a modified Reynolds number to account for the presence of the twisted tape, and use a proper nucleate boiling heat transmission correlation (i.e. the Thorn formula) to account for bubble formation. When considering the accident scenario analyzed here, these problems are present only in steady-state conditions, so that a much simpler approach was done, assuming a uniform value of the heat exchange coefficient of 60000 W / m 2 / K in both pipes. The coolant temperature was assumed to be 60 ° C.

2.5. Coolant characteristics 3. Modelling

The water velocity in the coolant tubes is about 10 m / s , flowing in a highly turbulent regime. This condition is imposed to obtain an efficient heat removal, as well as to reach a maximum heat exchange coefficient between the tube wall and the water bulk temperature. The extremely large temperature gradient in the tube wall implies an adequate pressurization to avoid a generalized two-phase flow. A coolant pressure of 10 bar was assumed here. The evaluation of the heat exchange coefficient under these conditions is another important source of uncertainty.

If the well-known Di t tus-Boel ter correlation is applied to this case, a Nusselt number of 648.7 is ob- tained, corresponding to a heat exchange coefficient of

3.1. Introduction

A standard finite element approach has been used to model the heat flux through the conductive domain. A computer program, called T H E R M , has been written to study the coupling between radiation across internal cavities and heat conduction. The program has been recently validated at JRC-Ispra comparing its results with those of the P - T H E R M A L code [11].

The elemental conductance and consistent mass matrices are found and assembled, then boundary conditions are inserted, to give the discretized equation:

C ( r ) T + K ( T ) T = ~ ( T ) (1)

Table 2 Thermal properties

Material k (W / m / K )

CFC 292-0.199 T+4 .41e-5 T 2

TZM 136.3 - 0.033 T + 2.63e- 6 T 2 ASISI-316 8.749 + 0.01416 T Ceramic 77.4-0.0346 T (T _< 1000)

42.8( T > 1000)

pCp(J/m3/K)

1800 (578 - 1.399 T + 2.03e - 4 T 2) 10200 (243 - 0.0217 T + 4.le - 5 T e ) 7960 (406.4+0.1771 T) 3200 (556.0+0.446 T)


Equation (1) is nonlinear due to the dependence of C, K, g on the temperature vector T. The way in which radiative cavities are handled is quoted in the following.

3. 2. Heat diffusion through cavities

An internal radiation cavity is assumed to be made up of n gray, diffuse-reflecting surfaces and n nodes. The gas filling the cavity is supposed to be non-par- ticipating. The radiative heat balance leads to the following expression for the net flux leaving the surface i [12]:

qi = mi L A i , ° T J ~' (2) j=l

where A, is area, Tj is the temperature of surface j , o is the blackbody radiation constant, and A, j is related to the view-factor matrix F u by:

" (8~ j - % ) (3) A'I=

with:

~ij = ( X i j ) -1 (4)

and

8ij - (1 - q ) F i i (5) Xij = f'i

if the surface i is not adiabatic, and

Xij = 8ij - Fij (6)

if the surface i is adiabatic. Here, 8,j is the Kronecker diadic and c~ is the emissivity of surface i. Summation in eq. (2) is to be extended only to the non adiabatic surfaces of the cavity.

The assembling of the flux term (2) directly in the load vector g [13] is the usual approach done in most of the large finite-element general purpose codes. The flux term is calculated using the temperatures at the preceding iteration or the preceding time step. This obviously limits the time step size and frequently leads to oscilla- tory or even divergent results [14].

Instead of this, the assembling of (2) into the conductance matrix, treating the cavity as radiative element, gives better convergence [15]. By approximating the temperature for each surface j as the average of nodal temperatures TjA and TjB, eq. (2) takes the form:

qi = i-6 ° j = l

If an anticlockwise node numeration is used, and the identity between nodes 0 and 1 and nodes n and n - 1 is implicitly assumed, the element radiative conductance is, for cavity e:

Ai (Kread)ij = ~ - o ( A , j (Tj + Tj+I) 3 + Ai j_ I (T j_ 1 -l= Tj) 3)

(8)

if the surface following node i is not adiabatic, and:

(g red) i j = 0 (9)

if the surface following node i is adiabatic. When emissivities are not temperature-dependent,

the matrix A ij needs to be formed only once. Other- wise, one has to form and invert Xu at each iteration. The major drawback of this formulation is that one has to invert sparse, non-symmetric matrices strongly de- pending on the temperature. The view-factor matrix F,v can be easily computed using Hottel's cross string method when all surfaces see each other. If not, more elaborate methods are needed. For the calculations presented here, the FACET view-factor preprocessor [16] was used in the case of re-entrant cavities. This package of subroutines first identifies which surfaces are seen from surface i, then accurately integrates the solid angle between each couple of elements, by means of the double contour integration formula:

= f f A f f A ~ r 2 d A i d A j

= 2 ~ i ~ c ~ c j l n r d l i (10)

where: r is the distance between dA i and dAj , Bi is the angle between the normal to dA~ and r, C~ is the contour of Ai, and dl, is the elemental contour of line C v

3. 3. Radiation boundary condition

A boundary segment, with emissivity ~, radiating to a black, close body at temperature T~ losses a heat flux given by:

q = o ¢ ( T 4 - T ~ ) . (11)

This heat flux can be represented by means of a lin- earized heat exchange coefficient:

q = o¢ .hrad(T- Too) (12)


with:

hra d = (T 3 + T2Too + TT 2 + 7"3). (13)

Two approaches can be used to insert the radiation boundary condition: (a) form and assemble q as given in (12) into the load vector g at each iteration level, (b) calculate the radiation heat exchange coefficient krad, and use it as a nonlinear convective boundary condition.

Following the first approach, the nonlinearity is concentrated in g, and the solution often diverges. It seems better to use the second formulation, although it implies a new temperature-dependent term in the conductance matrix. This matrix needs to be differentiated with respect to the nodal temperatures when solving the nonlinear problem with Newton-Raphson procedure.

3.4. Time integration scheme

The generalized trapezoidal rule applied to the time step [t,t + At] approximates the temperature time de- rivative as:

Tn+l _ T n 2k = - At (14)

Equation (1) is then solved in a generalized mid-point t . = t. + aAt (0 < a < 1), making:

T '"+"~t = (1 - a ) r " + aT "+1 (15)

and solving the nonlinear system:

[Kt.+aAt + ~ C t . + a a t ] T ' . +azat

= g,,+~at + ~ , +eatT,,.t (16)

The scheme is unconditionally stable if 0.5 < a, and offers O(At 2) convergence if a = 0.5 (Crank-Nichol- son-Galerkin two point recurrence formula), which was the scheme used for the calculations presented in this paper.

3.5. Iterative strategies

Equation (16) must be solved iteratively. The diffi- culties of such iteration are directly related to the time step size. Large time steps make the solution from time t to time t + A t to be far out of the convergence radius of a given iterative method. The particular way in which the radiative heat flux has been treated makes the

system matrix to be not banded. Therefore, it is difficult to guarantee the diagonally-dominant character, and methods like non-linear Gauss-Seidel or succesive over- relaxation seem a priori to be penalised. The iterative methods implemented to analyze the problems presented in this paper were: (a) Newton-Raphson algorithm family, (b) Picard direct iteration method. Given a nonlinear system of equations:

M ( T ) T = F ( T ) . (17)

The Newton-Raphson algorithm family finds a better solution T k + 1 from T k by making:

r k + l = T k + [ J k ] 1Rk" (18)

Within this format, the residual vector is given by:

R k = F ( T k) - M ( T k ) T ~ (19)

The Jacobian matrix Ji~ is formed by the partial deriva- tive of row i in equation (17) with respect to nodal temperature ~. One can form an invert the Jacobian matrix at each iteration level (full Newton-Raphson method), keep it constant during a certain number of iterations (or even time steps) (modified Newton-Raph- son method), or find approximate secant Jacobian matrices from the preceding corrections (quasi-Newton methods) [17].

The Picard direct iteration method is based upon the formula:

T k + l = ( M ( T k ) ) 1F(T k ) . (20)

This method requires a matrix inversion per iteration. When dealing with a time transient problem with a large number of degrees of freedom, performing more than one matrix inversion per time step can be extremely expensive task. The convergence radius of all Newton-Raphson methods decreases as the number of equations increases. On the other hand, the convergence behaviour of the Picard method is often unstable. The preferred way to solve equation (17) is to proceed with a quasi-Newton method without refactofization of the Jacobian matrix until a significant change in the time step size makes the convergence difficult, then refresh the Jacobian. Following this Newton-Raphson based scheme, the direct iteration method is used only when a risk of divergence is detected. According to our experi- ence, when divergence appears it is mainly due to a small convergence radius, rather than a bad quasi-New- ton Jacobian approximation, i.e., the approximation to

A. Sofia et al. / Thermal analysis of a tokamak divertor plate 345

the Jacobian is good, but the Newton-Raphson method hardly converges.

It should be pointed out that the proposed method allows the use of variable time steps minimizing the number of matrix inversions, since it is less sensitive to the inverted Jacobian matrix quality. Taking into account the number of degrees of freedom of a particular case, one should decide about giving priority to the minimization of the number of matrix inversions (which leads to more iteration) or the reduction of the number of iterations (which implies more matrix inversions). In the calculations presented here, - 6 0 time steps were used to analize each transient, using variable time steps ranging from 0.005 s up to 500 s. Typically, only one matrix inversion each ten time steps was needed.

1800

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9OO

600

300

4. Analysis of the results

The main results of this research are presented in this section. The geometry with a gap between graphite and steel will be referred to as geometry A, geometry B corresponding to the geometry with thermal contact between these components. Thermal load hypothesis 1, 2, 3 and 4 correspond to cases a, b, c and d in figure 2, respectively.

4.1. Steady state results

The steady-state temperature fields are shown in fig. 3. Note that for geometry A, the second pipe does not contribute to the heat removal efficiently, due to the heat flux inversion in the gap. This effect is caused by the intense steady state volumetric heat deposition in the AISI-316 steel, and the poor radiation through the gap. The thermal insulation of the double-T shaped steel piece makes it reach a maximum temperature of - 750 K. A plot of the isotherm lines for both cases is presented in fig. 4. The maximum temperature in the graphite is 1690 K in both cases, which is far above the desired limit value of 1273 K stated in the ITER report [18], the thickness of the graphite matrix being slightly too large for the other hypothesis considered here. A more extensive parametric study considering several materials and geometries, as well as a larger coolant tube diameter (15 ram) is now under way.

The extremely high heat flux concentrated on the upper side of the most heated pipe produces a large temperature gradient between the tube wall and the coolant. Since the coolant pressure is limited to 10 bar due to mechanical considerations, it seems impossible to

Fig. 3. Steady state temperature fields.

avoid some boiling in this critical zone. The saturation temperature at p = 10 bar is 175 o C. Isotherm A in fig. 4 corresponds to 159°C. The size of the boiling zone can be estimated in 1 /3 of the wet diameter.

346

VAL. ISO.

A 4.32E+@2 B 6.28E+e2 C 8.23E+@2 D 1. @2E+@3 E 1.22E+@3 F 1,41 E+e3 G 1.61E+@3

A. Soria et al. / Thermal analysis of a tokamak divertor plate

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[ I

4.2. Transient results

The main results of the transient analysis are presented in this section. Temperature histories for 4 selected nodes and mountain-like representation of the temperature field at several times are plotted.

Figure 5 shows the position of the selected nodes, and its corresponding plot markers for geometries A and B ( x = hottest point in the graphite matrix, * = hottest T Z M point, ~ - - - p o i n t C in fig. 5 (stainless steel), [] = point D in fig. 5). The analysis was done covering times up to - 2 2 0 0 s, since different t ime constants are involved.

All cases exhibit a short-time transient driven by the plasma load shutdown. The temperatures of the conduc- tively-connected pieces are uniformed within - 5 rain after the beginning of the transient. From this point on, both heat radiation and volumetric heat deposition due to activation control the temperature transient. In prac-

tice, a variable time step size ranging from 0.005 s to 500 s was used in all cases. Load hypothesis 1

Figure 6 shows the evolution of the selected nodes for geometry A and load hypothesis 1, in two different t ime scales (up to 20 s, and up to 2200 s).

Figure 7 corresponds to geometry B and load hypothesis 1, with the same time scales. Note that in case A1, the AISI-316 steel pieces take - 300 s to reach a uniform temperature, while the CFC-graphite has reached a flat temperature distribution only - 1 0 s after the sudden dry-out. This effect is caused by the excellent CFC-graphi te thermal conductivity, which makes the temperature uniform in absence of any other significant thermal load. The double T-shaped AISI-316 piece takes more time to reach a quasisteady state due to the presence of the low-conductivity ceramic material and the small short-term heat removal efficiency of the heat radiation.


B

C.

/ I t~

I ] Fig. 5. Selected analyzed nodes.

Load hypothesis 2 Figures 8 and 9 are related to load hypothesis 2. The

temperature histories for the selected nodes are plotted with two time scales in fig. 8, corresponding to geometry A. No temperature overshoot is observed in the graphite, whereas the temperature in the upper TZM tube reaches - 1 2 0 0 K) in - 6 s. The corresponding plots for geometry B are presented in fig. 9. The short- term graphite behaviour is almost the same but the heat wave rapidly diffuses to the stainless steel back plate. The whole divertor reaches a uniform temperature - 7 0 0 K after 200 s. From this point on, it linearly decreases to - 630 K at t = 2200 s.

Load hypothesis 3 Figure 10 reports the temperature evolution for the

selected nodes in geometry A. The first 20 s of the transient are shown in fig. 10a. A considerable temperature overshoot is observed, reaching almost 2400 K in the CFC-graphite surface. A significant vaporization of

the material is expected. However, the radiation gap efficiently protects the back plate of possible melting. The graphite temperature becomes uniform after - 30 s, and then decreases (fig. 10b). On the other hand, the AISI-316 back plate reaches a temperature close to 700 K at time - 2200 s.

Figure 11 shows the temperature field at times t = 0.2 s, t = 5 s, t = 1 0 s , t = 2 4 s , and t = 8 4 s .

Figure 12 is related to load hypothesis 3 and geometry B. The temperature overshoot in the graphite armour is equivalent to the case A3 (fig. 12a), but in this case, the thermal contact between CFC-graphite and back plate makes it to reach a temperature of - 9 5 0 K in only 30 s. This temperature is close enough to the melting point to suppose that a considerable material damage will be produced. The divertor temperature becomes uniform - 8 0 0 K after - 3 0 0 s and then decreases by thermal radiation to 650 K at t = 2200 s.

Figure 13 shows the temperature field shape at times t = 0 . 2 s, t = 5 s, t = 1 0 s, t = 2 4 s, and t = 8 4 s. Concerning the TZM-alloy pipes, load hypothesis 3 does not represent still a fatal damage for the graphite- embedded one. The recrystallization temperature ( - 1800 K) is not surpassed, the maximum temperature reached by the component being -1600 K at time t = 10 s is far below the melting point (2893 K).

Load hypothesis 4 In this case, no plasma shutdown is assumed after

the sudden dry-out of the coolant tubes. This calculation can assess the time needed by the divertor to burn out. The maximum allowable temperatures for each material are summarized in table 3. The analysis has been carried out up to t = 85 s. The results are shown in figs. 14 and 15 corresponding to geometries A and B respectively. The results are very similar for both cases, although the case shown in fig. 15 is slightly less severe for the graphite due to the thermal contact between the stainless steel and the CFC-graphite matrix, case A corresponding to a more adiabatic graphite behaviour. After the analyzed period, the component can be supposed to be burn out. The TZM recrystallization temperature is reached within - 8 s in both cases, and the TZM melting is expected at time t = 50 s. The sublimation temperature for the CFC-graphite is reached at time t = 80 s in the case A, an it is expected to occur at time t = 90 s in the case B. The stainless steel will not have a short term failure in case A, protected by the radiative gap. However, figure 15 shows that the stainless steel melting temperature is also reached in time t = 80 s if there is a perfect thermal contact between the two pieces.

3 4 8 A. Soria et al. / Thermal analysis of a tokamak divertor plate

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Fig. 8. Temperature transients in selected nodes for load hypothesis 2. Geometry A.

A. Soria et al. / Thermal analysis of a tokamak divertor plate

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A. Soria et aL / Thermal ana~sis of a tokamak dioertor plate

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A. Soria et aL / Thermal analysis of a tokamak divertor plate 353

Long term effects Comparing figs. 6b, 7b, 8b, 9b, 10b, and 12b, several

conclusions can be made: (a) The decay heat in the back plate makes this

component reach greater temperatures than the graphite

armour, when a radiation gap is present between these two components. This temperature inversion occurs

- 1500 s after the beginning of the transient in all cases. (b) The long-term temperature of the components

does not depend very much on the severity of the

280(;

2400

2000

1600

120~

800

400

Fig. 11. Temperature field evolution for load hypothesis 3. Geometry A; (a, top) t = 0.2 s, (b, bottom) t = 5 s.

354 A. Soria et aL / Thermal analysis of a tokamak divertor plate

short-term transient, but its influence is not negligible. The approximate temperatures at the end of the analysed time period (t = 2200 s) for the CFC-graphite with the gap are 551 K, 590 K, and 630 K, for load hypothesis 1, 2, and 3 respectively. The corresponding final tempera-

28°° K ....

tures is the AISI-316 back plate are 607 K, 658 K, and 709 K. As expected, the final divertor temperature with thermal contact between the pieces is about the average of the preceding temperatures for each load hypothesis: 578 K, 633 K, and 692 K, respectively.

2400

. i t 2000

1600

1200

8O0

4OO

° K / . " / 28013 j J "

I / ....

24OO

2000

1600

12013

80O

40O

Fig. 11. Continued; (c, top) t = 10 s, (d, bottom) t = 24 s).

A. Soria et al. / Thermal analysis of a tokamak dioertor plate 355

2800

2400

2000

1600

1200

8OO

400

Fig. 11. Continued; (e) t = 84 s.

5. Conclusion

Several 2-D thermal analyses have been performed to assess the reliability of a tokamak divertor design (ITER concept) in the case of a complete loss of coolant accident (LOCA). Four different plasma load be- haviours have been assumed. A study on the thermal effects of the radiative gap between the components has been carried out. The main conclusions of the analysis can be summarized as follows: (a) From the steady state point of view, the radiative

gap is not desirable, since it causes a heat flux inversion towards the most loaded coolant pipe. The maximum temperature reached by the CFC graphite with the geometrical assumptions used in this analysis (11 mm thickness, 10 mm coolant tube inner diameter) is about 400 K above the threshold safe steady state value (1273 K). Further research on the material choice and arrangement will provide the optimal design of the component.

(b) From the thermal transient point of view, the gap will efficiently protect the AISI-316 divertor back plate from temperature overshoots due to plasma burning continuation during the first seconds of the transient.

(c) An effort should be done in order to clearly identify the plasma extinction time, as well as the correct

shape of the plasma shutdown curve. The severity of the accident greatly depends on this issue.

(d) Heat radiation on well-cooled parts of the reactor seems to be an efficient passive heat removal mechanism. The heat generated and stored on critical components rapidly diffuses to the heat sinks. Large-scale, long term time transients can be accurately studied with coarse meshes and long time steps, since they are mainly dominated by residual power and heat radiation.

The consideration of two independent cooling circuits, and the assessment of the consequences of a single-circuit LOCA are important areas of further research which would help the designing of an optimized component. Future work will consider the numerical simulation of the nucleate boiling heat exchange proper-

Table 3 Maximum allowable temperatures

Material Max. Failure mechanism temperature (K)

CFC-graphite 3500 Sublimation TZM-alloy 1800 Recrystallization TZM-aUoy 2890 Melting AISI-316 1600 Melting

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A. Soria et aL / Thermal analys~ of a tokamak divertor plate

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A. Sofia et al. / Thermal analysis of a tokamak divertor plate 357

2800

2400

2000

1600

1200

800

400

2800

2400

2000

1600

1200

800

40O

Fig. 13. Temperature field evolution for load hypothesis 3. Geometry B; (a, top) t = 0.2 s, (b, bottom) t = 5 s.

358 A. Soria et al. / Thermal analysis of a tokamak dioertor plate

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A. Soria et al. / Thermal analysis of a tokamak dioertor plate

Fig. 13. Continued; (e) t = 84 s.

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ties, in order to bet ter assess accident scenarios in which the coolant is supposed to fill the faulted tubes dur ing a cer tain time, as in the case of L O F A or part ial LOCA.

Acknowledgements

The first au thor gratefully acknowledges the Com- mission of the European Communi t i e s for the f inancial support . Technical discussions with Mr. Farfalet t i - Casali, Mr. Janssens, Mr. Scaff idi-Argent ina and Mr. Piazza have been sincerely appreciated.

References

[1] V. Renda, A. Soria and L. Papa, Thermal effects of residual power on plasma facing components of NET, Fusion Engrg. Des. 6 (1988) 269.

[2] F. Fenoglio, V. Renda, L. Papa and A. Sofia, Conditions for the passive removal of residual power from a fusion reactor, Fusion Engrg. Des. 11 (1989) 455.

[3] A. Sofia, V. Renda, L. Papa and F. Fenoglio, Progress in the studies of passive heat removal in the Next European

Torus under accident conditions, Fusion Technology 16 (1989) 474.

[4] F. Farfaletti-Casali, Personal communication. [5] A. Cardella, R. Santa and G. Vieider, The divertor system

design for ITER, ITER-1L-PC-89-E-11 (1989). [6] W. Janssen, Optimization design of a divertor element for

a tokamak-type fusion reactor, JRC-Ispra, Second Interim Report (1988).

[7] F. Scaffidi-Argentina, G. Piazza, M. Biggio, Y. Crutzen, F. Farfaletti-Casali and A. Cardella, Thermomechanical and electromagnetic analyses of a new divertor concept for the ITER reactor, JRC-Ispra, TN-I.89.74 (1989).

[8] C. Ponti, Activation calculation for NET-shielding blanket, JRC-Ispra, TN-I.88.73 (1988).

[9] A. Cardella, Personal communication. [10] E. Zolti, Material data for thermal and mechanical analy-

sis of plasma facing components, NET-IN-89-012 (1989). [11] F. Andritsos, Use of P-thermal code for modelling the

ITER divertor plate under LOCA conditions, JRC-Ispra, TN-I.90.25 (1990).

[12] E.M. Sparrow and R.D. Cess, Radiation Heat Transfer (Brooks/Cole Publishing Co., Belmont (CA), 1966.

[13] H. Wolf, L.D. Wills, R.M. Krudener and M.D. Almond, Thermal and stress analysis of composite nuclear fuel rods by numerical methods, Chapter 8 in: Numerical Methods in Heat Transfer, Vol. 1. Edited by R.W. Lewis, K.

A. Soria et aL / Thermal analysis of a tokamak divertor plate 361

Morgan, O.C. Zienkiewicz (John Wiley and Sons, NY, 1981).

[14] J.F. Stelzer and R. Welzel, Experiences in nonlinear analysis of temperature fields with finite elements, Int. J. Num. Meth. Ing. 24 (1987) 59.

[15] J.H. Chin and D.R. Frank, Engineering finite element analysis of conduction, convection and radiation, Chapter 10 in: Numerical Methods in Heat Transfer, Vol. 3, Edited by R.W. Lewis (John Wiley and Sons, NY, 1984).

[16] A.B. Shapiro, FACET: a radiation view factor computer

code for axisymmetric, 2D planar and 3D geometries with shadowing, Methods Development Group. Mech. Eng. Dept. Lawrence Livermore Laboratory (1983).

[17] A. Sofia and P. Pegon, On the performances of quasi-newton iterative methods to solve the nonlinear heat diffusion equation, Proc. 6th International conference on Numeri- cal Methods in Thermal Problems, edited by R.W. Lewis and K. Morgan (Pineridge Press, Swansea, U.K., 1989).

[18] ITER Conceptual Design Interim Report, International Atomic Energy Agency, Vienna, Austria (1989).

Documents

Thermal analysis of a tokamak divertor plate after a sudden coolant dry-out