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First wall fusion blanket temperature variation — slab geometry
330
Nuclear Engineering and Design 48 (1978) 330 339 @ North-Holland Publishing C~mpany
FIRST WALL FUSION BLANKET TEMPERATURE VARIATION - SLAB GEOMETRY *
J.A. FILLO Department of Applied Science, Brookhaven National Laboratory, Upton, N Y 119 73, USA Received 11 November 1976
The first wall of a fusion blanket is approximated by a slab, with the surface facing the plasma subjected to an applied heat flux, while the rear surface is convectively cooled. The relevant parameters affecting the heat transfer during the early phases of heating as well as for large times are established. Analytical solutions for the temperature variation with time and space are derived. Numerical calculations for an aluminum and stainless steel slab are performed for a wall loading of 1 MW(th)/m2. Both helium and water cooling are considered.
1. Introduction
To obtain electrical power from the D - T fusion reaction, a standard thermal cycle, i.e. a steam or gas turbine cycle, must be used. Since the thermodynamic efficiency of such a cycle is governed by the operating temperature of the heat source, the problems of heat removal from a CTR are of considerable importance. The choice of the coolant and flow network impinges on such aspects as capital cost and power-plant efficiency, as well as tritium breeding, radioactivity, material and structural criteria, and the reactor lifetime. It is generally assumed that in the case of a commercial reactor it is desirable to maximize the wall loading, i.e. the ratio of the maximum rated power output of the reactor divided by the torus surface area facing the plasma, since the reactor size and capital costs are directly related. Even though such factors as radiation damage and plasma-stability restraints may strongly affect the allowable wall loading, the ability to cool the first wall, i.e. the first material surface facing the plasma, appears to be tile major factor involved in establishing the wall-loading limit. The coolant or coolants are introduced so as to absorb the energy of the neutrons released in the fusion reactions along with the energy (neutron + gamma generation) developed in the blanket with or without tritium breeding. In addition the coolant or coolants must absorb a certain percentage of the energy from the plasma, primarily in the form of radiation, which impinges on the blanket surface exposed to the plasma. The neutron and gamma energy results in volumetric heat generation within the blanket, while radiation to the surface is responsible for an imposed surface heat flux. As just noted, in the design of blankets and the subsequent problems of cooling and heat removal, the first wall looms as the critical region of concern. Consequently, it is singled out for separate study and discussion, since all of the emissions from a reacting plasma may intersect this wall. These include [1 ] ions and neutral particles, primary neutrons, X-rays (bremsstrahlung) and cyclotron radiation. In addition, scattered neutrons and gamma radiation generated in the blanket regions exterior to this first wall are also imposed on it. Finally, D - T gas surrounding the plasma may react chemically with the first wall materials. One result of these many interactions is the generation of appreciable heat, which can be up to 20% of the plasma output. Thus, provision must be included for adequate cooling. In most analyses the above effects are usually lumped together as a radiant flux. While the incident fluxes are absorbed in a small depth of any first wall material, the usual assumption is to treat the lumped radiant flux as if it were simply deposited on the surface facing the plasma. In heat transfer analyses it is an applied * Work performed under the auspices of the US Energy Research and Development Administration.
J.A. Fillo / Slab geometry of first wall fusion blanket
331
heat flux boundary condition. The phenomena of sputtering and blistering, the formation of gaseous reaction products, and material vaporization processes may represent the major sources of impurity in the plasma. Therefore the first wall can be a limiting component in reactor power output due to a large power generation within it; and it may also limit plasma performance through impurity levels. As already noted, the useful first wall lifetimes in a power reactor may be severely limited through radiation damage and surface erosion. The attendant frequent replacement could unduly add to operating costs and require an excessive plant down time for replacement. Thus, the design and material selection for this first wall is a vital element in the design of a fusion power reactor. For example, in a number of designs [1 ] the first wall is constructed as a liner attached to the remainder of the blanket to absorb the brunt of the thermal loading. In addition to the above concerns with respect to the first wall, it is necessary to consider additional effects which the first wall will experience as well as potential scenarios, such as a plasma excursion and energy dump to the wall which could be disastrous for the first wall and cause subsequent failure of the blanket as a whole. These all have a bearing on the design of the blanket, need for careful analysis, as well as heat removal and cooling aspects of the blanket. Assuming that the plasma does not undergo any excursions to the first wall under normal operation, the first wall will be subjected to thermal cycling. This is not necessarily a problem in itself but rather the temperature excursions per pulse to which the structure could be subjected. Cracks could occur in the first wall as a consequence of thermal stresses, the propagation of these cracks leading to the ultimate failure of the blanket. Since fusion reactor designs for the experimental power reactor (EPR) or any of its variations (TNS) are to be pulsed devices where the plasma 'on' period may range from 30 to 40 sec to possibly several minutes, with a downtime less than or of the order of a minute, from a heat transfer standpoint the nonsteady or transient and steady, periodic states are of interest. While thermal conduction aspects of blanket designs may be analyzed by finite difference or finite element methods, the present study considers a slab geometry (fig. 1) which has the advantage of being amenable to analytical analysis. The surface facing the plasma is subject to a pulsed heat flux while the rear face is convectively cooled to remove the heat generated in the 'blanket'. The objective of the study is to identify the relevant parameters affecting or governing the heat transfer during the early phases of heating as well as for long times, once the steady, periodic temperature profile is established. Since a closed form solution is derived, the solution should be of interest as a base case against which results from numerical codes may be checked or compared, as well as a relatively simple means of assessing time-dependent effects on first walls. This assumes, of course, that the actual design may be approximated by slab geometry. Lastly, as a basis of comparison, we have chosen aluminum and stainless steel A1 (or SAP), rather than stain-
/
h, vf
Soe-r x
TX
T t
TT Qo ttT
Fig. 1. One-dimensional slab geometry.
332
J.A. lffllo /Slab geometry of first wall fusion blanket
less steel, has been advocated by BNL [2,3] in its designs, primarily on the basis of less long-time radioactivity. In addition, the better thermal properties of AI are apparent in the results.
2. Basic formulation - one dimensional A one-dimensional slab geometry (fig. 1) is used to model tire first wall blanket region. Since the volumetric internal heat generation can be closely approximated by a decaying exponential function, the one-dimensional, time-dependent heat conduction equation can be written 02v
~v
K Ox2
~t
So
-
pC
e-VX4o(t)
(1)
subject to the following boundary and initial conditions: Ov
- k ~xx = Q°c)(t)'
- k OVox= h(v
x : 0;
-
Of),
X=l
and
v = Vo,
t = 0.
(2a-c)
Here v is the temperature while S o and 7 are parameters dependent on plasma conditions as well as blanket structural material, and are derived from the results of a neutronics analysis. Qo is the incident flux to the first wall and is assumed to be a certain percentage of the plasma output, usually of the order of 20%. The heat transfer coefficient, h, and mean fluid temperature, vf, are given, and cooling is continuous during the plasma 'off' period. Taking into account an 'on' and 'off', cooling behavior is straightforward. We assume that the blanket structure is at some uniform temperature, vo, at time t = 0. For definiteness we consider only the case of an applied rectangular wave form but other cases such as a ramp followed by a constant flux and heat source (in time) may be treated in the same way. The applied heat flux and source may be represented by the function [4] :
¢(t)=o,
t < o,
~b(t) = 1,
nT
~b(t) = 0,
nT+T1
n = 0 , 1 ..... n=0,1 .....
(3)
In other words the applied flux, Qo(~(t), represents a flux Qo which is 'on' for time T1 and 'off' for time T - T 1 and so on, with period T.
3. Superposition of solutions
The solution to eq. (1) may be represented as the superposition of two solutions. Let (4)
O=U+W,
where b2u
3u
K bx 2
Ot
(5)
--0,
Ou
~u
-k~-x=0,
x=0;
-k~x=h(u-uf),
x=l;
U=Vo,
t=0,
(6a-c)
and b2w
K ax 2
Ow
~t
-
So
pC
e-'~%(t),
(7)
J.A. Fillo / Slab geometry of first wall fusion blanket ~w _k~x=Qog~(t),
aw - k ~ - x =hw ,
x=0;
x=l;
w=0,
t=0.
333 (8a-c)
Eqs. ( 5 ) - ( 6 c ) result in the complementary solution with effectively homogeneous boundary conditions (after a simple transformation, s = u - vf) while eqs. ( 7 ) - ( 8 c ) constitute the particular solution.
4. Complementary solution The solution to eq. (5) may be found in Carslaw and Jaeger [4]. The dependent variable, s, is defined as (9)
S=U--Of
so that l
s= ~
(10)
Zn(x) e -K#n2t f Z n ( ~ ) f ( ~ ) d~,
r/=l
0
where (1 la)
f(x) = initial temperature = u o - of and
Z n ( x ) = [ 2( k2~2 +h 2) ] Ll(k2t3 2 + h 2) + khJ
'/2
cos/3nX,
(llb)
t3n, n = 1,2, ..., are the positive roots or eigenvalues of (llc)
sin /31 = k~-COS/3l. Evaluating eq. (10), we find ¢¢
2(a2n + Bi 2)
U -- Of = (O O -- Vf) n=lG O~2 + B i 2 + B i
sin an I c o s ( ~ ) l e
-(Ka2nt/12) ,
(12)
~n
where Bi = Biot modulus = lh/k,
(13)
an = ~nl •
Note that the eigenvalue equation, eq. (1 lc), becomes (14)
tan a n = Bi/a n.
5. Particular solution The solution to eq. (7) may be found by applying Duhamel's theorem [4] once the solution to the problem for time independent heat flux and source term is known. The solution to this problem is also found by superposition of solutions. The first problem we solve is eq. (7) for ~(t) -- 1. Let w = wl (x) + w2(x, t) so that d2wl _ K dx 2
SO pC e-'rx,
(15)
32w2 K ax2
3w2 _ 0 , a~-
(17)
J.A. Fillo / Slab geometry of first wall fusion blanket
334 dWl -k-~-=Qo,
x=O,
(16a)
-k Ow2=Oo'x
- -k~dWl -=hWl,
x=l,
(16b)
.3w2 . -K~-x =nw 2, x=l,
(17b)
W 2 = --W 1 (X),
(1 7C)
x=O,
t = 0.
(17a)
Physically, wx represents the steady-state temperature solution for a slab which is subjected to a constant heat flux at x = 0, while at x = l there is radiation (or convection) into a medium at zero temperature. In addition the rate of heat production is a function of position. In the case of w 2 we have a slab insulated at x = 0, initially heated non-uniformly. The solution for wl is straightforward and results in
w,
=
Q°lrlL + Bi(1-/)] +S°12 [1 + Bi(1-/) +(~--1)e-B-~ Bi- e_BX/ll. -~:xl kBBi
(18)
The general solution to eq. (17) is the same as (10) but where
Q°l (1 S°l: II+Bi(1-1)+(~-l)e-B-B~Be-BXlll" f(x)- ~-[l+Bi -/)] k~i
(19)
Evaluating the integral, we find for w 2 that 2 W2
- - -
/2
B
n=, - -
+ Bi:
ie -B 1 - ~
2Qol( 1 ]/
1
coso~,, +
(o~ +8:)o~ + - - U - ' , ~ : )
rcosi .X le_, o ,:, \ l 13
(20)
Xan2+Bi2+BiL
Combining eqs. (18) and (20), the particular solution is
_Qo,[ f 7)]
w-~.-
1 +Bi 1 2
-n=l
+
kBBi
l2
- -
X ~n2 + Bi 2 + Bi
I1 + Bi( 1 _ / ) + (~i
B ie -B 1-~
os
1)e_ B _(~i)e_BX/l 1
1
cOSan+
(ct2n+B2)ct2 n+
e -(K~2nt/12).
(21)
6. Solution for time independent heat flux and heat source
The steady, periodic temperature distribution is of primary interest in this study. On the other hand, the general solution found by combining eqs. (12) and (2 I) may also be of interest since it predicts the transient, or simply, non-steady temperature history to steady-state assuming normal plasma behavior and 'long' burn time. In other words true steady-state is assumed to occur. The combined solution also forms the basis for determining the steady, periodic temperature distribution.
J.A. Fillo / Slabgeometry of first wallfusion blanket
335
The resultant temperature, o, is simply Qol Ii + Bi( 1 - - / ) 1 + Sol2 k--ff-~l Il o - Vf=~-B-~-
+(Oo
-
of) ~
{2 sin an
n = 1
an
+
2Sol2 -----
k(o 0 -
of)
+ Bi ( 1 - l x) + ( ~ - - 1 ) e - B - B_Bi e-BX/I1 IB
( BBi) BI 1 ie -B I c°San + (Ctn 2 +B2)an2
2Qol ( 1 ) } k(oo-Vf) ~2n2
a2+Bi2I(~-)le-(~a~t/12) X a 2 + B i 2+Bi
(22)
cos
All of the significant parameters affecting the solution are apparent, specific conclusions being dependent on blanket material choice and coolant as well as wall loading. In addition to the Biot modulus, Bi = hl/k, the solution can be restated in terms of the Fourier number, Fo = Kt/l 2 , the Pomerantsev modulus, Po = Sol2~ [k(o o - of)], and the Kirpichev modulus [5 ], Ki = Qol/[k(oo- of)]. The physical significance of the Pomerantsev modulus is that it shows the ratio of the amount of heat evolved by the source per unit time in the volume l (a parallelepiped with a base of 1 cm 2 and a height/) to the maximum possible amount of heat transferred by conduction through a unit area per unit time over the distance l (subject to the assumption that the temperature distribution occurs by the linear law). The Kirpichev modulus is the ratio of the heat flux, Qo, through the slab to the maximum possible heat flux at l provided that the temperature gradient at this point is maximum and equal to (v0 -vf)/l.
7. Steady periodic temperature The solution to eq. (1), subject to (2)--(2c) follows from (22) and Duhamel's theorem: oo
n=l
l
+~
2
1
n=l pC an2 +B 2
a n2
IB (
an2 + Bi2 + Bi
ie - n 1 -
cOSan+
e -(~2t/12)
os
"J: t
So(r)e -I~2n(t-r)/z21 dr
+pclfQo(r) e-lK~"(t-~)/121 dt a 2 + B i 2 + B i os 0
.
(23)
The steady, periodic variation in temperature which exists after the transient involving the initial conditions has died away may be achieved in a number of ways. The method of Weber [4] (see Carslaw and Jaeger) is found to be an 'elegant' procedure, and is quite general, the advantage being that the resulting series is not slowly convergent for values of time which are of interest. This is borne out by calculations in a latter section. The results given in [4] are immediately available for use in any problem in which the solution for constant external condition is given as a series of exponentials of the form (-~/nt), and the solution for external conditions (as well as the heat source) represented by (3) is to be obtained by Duhamel's theorem. Rather than repeat the derivation, we simply state the results for the integrals for the 'on' and 'off' periods. For the 'on' period we let
t = rT + t',
(24)
J.A. Fillo/Slab geometryof first wallfusion blanket
336
where 0 < t' < T1, and r is large. Setting
(25) and 0i(r) = So(r), Qo(r),
for i = 1,2 respectively,
(26)
we have to evaluate integrals of the type
e -Tnt
t
f~(r) e7nr dr,
(27)
0
which becomes, for large values of r, e~n(T 1-t')
I
_ e~,n(T-t')7
i2
(28)
'
where i = 1,2 (¢1 -- So and ¢2 = Qo). Consequently, for large values of the time the solution to (23) is
v-of:2~-~-.2 l ~ [2So h 12=,t ~ }1Ll an2+Bi2 [1 X an2 + Bi2 + Bi -
'[- Bie-" (
__~l)cOSan+B]+_~_(_~nn) B 2Qo ] }
e(Ka2n(T'--t"/t2'_--e(Ka2n(T--t"/12'1 (an~_) 1
-
-
_] cos
e (~a2T/t2)
(29) .
Since oo
2 2 Bi) n=l an(an +B2)(an2 + Bi2 + 1 [l+Bi(1-/) B Bi
Bie-/~ 1 -
+(\BBi ] - Be- ( B ) e - - 1Bi
cos an+ -Bx
and
(
n=l -an-2 an2+Bi 2+Bi c ° s t i f - ] = B-i
+ Bi 1 -
cos
/ l]
(30)
(31)
'
an alternative form of eq. (29) is O--Of =
Q°l
k--~-I l + B i
(1
s°12 II+Bi(1-I)+(BB~il)e-B-(B-~B)e-BX/II
-/)1 + ~-~
~ [ 2S°12 [13 (1 ~ii) -n~_l ~kan2(an:~+B2 ) ie -B - . cos a n+
I-e(~a2n(T1-t')/t2)-e(Ka2n(T-t')/t2)7
COS(-~)
BI+2Q°II ~
a2n+Bi2
an2+Bi 2+Bi
(32)
J.A. Fillo / Slab geometry of first wall fusion blanket
337
Note that physically the first two terms are the steady-state solution, the last term representing the steady, periodic variation of temperature with respect to the 'true' steady state. In an analogous way for the 'off' period, we set t = r T + Tt + t",
(33)
where r again is large. For this case t eTn(T - T 1-t") _ e~,n(T - t") e - ' r n t / d p ( r ) e "rnr d'c -+ (h _e~nT ) 0 ~'n( 1
(34)
so that the temperature, for large values of time, becomes u-of=
~ 2S°12 1 n = t [ k ( ~ 2 ~n2 + B 2
ie-B
B 1--~-COSt~n +
+
2Qo l 1 k (~2
F_e(•(x2(T-Tl-t")/12)_e(Ua2(T-t")/12)][OtnX ~ XL
1 -e("'~'/'2)
(35)
jcos~--/-).
8. Discussion of results The steady, periodic temperature distributions (actually, u - o f ) for a ½ cm aluminum and stainless steel slab are shown in figs. 2, 3 and 4, respectively. Input for the calculations are to be found in table 1. The neutronics results for A1 and the heat transfer parameters are taken from a BNL blanket design [4]. In the case of stainless steel we assume the same neutronics as for aluminum, which is an approximation. An examination of terms in eq. (32) reveals that the dominant contribution to the temperature variation is from the surface heat flux, Qo, and this would not change even for an exact stainless steel neutronics calculation. Note that we have fixed the Biot number for the comparison between the two metals.
AI SLAB
. - - SURFACE TEMPERATURE - - - COOLANT WALL TEMPERATURE
80
.t., 70 u/ 60 n,-.
50 ~E
20 [-
40
20 I0
i
o
SURFACE TEMPERATURE
COOLANT WALL TEMPERATURE
I
50
o
--
---
Io
L
20
50 40 50 60 o IO zo 50 TIME,sac TIME, sac Fig. 2. Steady, periodic temperature histories. Wall loading: 1 MW(th)/m2; helium-cooled ~Bi = 0.1. Fig. 3. Steady, periodic temperature histories. Wall loading: 1 MW(th)/m2; water-cooled ~Bi = 1:31.
I
40
,50
6O
J.A. b]'llo / Slab geometry of first wall fusion blanket
338
SS SLAB --SURFACE TEMPERATURE --COOLANT W A L L TEMPERATURE
120 IIO I00
9O
p
8o
~6o bA ~ 5o w
~
40
/
/
\
5O
IO 0
I0
20
30
TIME,
40
50
60
sec
Fig. 4. Steady, periodic temperature histories. Wall loading: 1 M W ( t h ) / m 2 ; w a t e r - c o o l e d ~ B i = 1.31.
While the difference between helium and water cooling for the AI slab is significant, it should be noted that the maximum surface temperature with the cooling may be reduced to about one-half (~35°C) [3] of the results in fig. 2 by improving on the design, i.e. by increasing the surface area of the coolant passage, as was done in ref. [3]. This, of course, introduces a two-dimensional effect which cannot be handled by the current calculation. To reduce the surface temperature in the one-dimensional calculation implies increasing the heat transfer coefficient. The bulk of the temperature difference for the helium-cooled case is in the difference between the wall and coolant fluid. For both the helium and water-cooled cases the maximum difference across the A1 slab is ~6°C, while for stainless steel, as a consequence of lower thermal conductivity, the maximum difference is ~60°C. In all cases, as we approach the end of the plasma ,on' period, the temperature approaches the 'true' steady-state, approaching it much faster for the A1 slab with water-cooling. It might be noted that the temperatures for the water-cooled case are virtually the same as in ref. [3] since the geometry of the water coolant passages (width > height) are such that the one-dimensional approximation is valid.
Table 1 Heat transfer parameters for temperature calculations Aluminum k ( W / c m °C) K (cm2/sec) Bi (dimensionless)
water helium
S O ( W / c m 3) B (dimensionless)
(2o (W/cm2) Eigenvalues: water -~Xl, = 0 . 9 4 6 6 , helium - a 1, = 0 . 3 1 1 1 .
1.75 1 1.31429 0.1 4.7 0.04272 20
Stainless steel 0.172 O.04938 1.31429 4.7 0.04272 20
J.A. Fillo /Slab geometry o f first wall fusion blanket
339
In the cases investigated, the infinite series in eqs. (32) and (35) reduce to a single term(only one eigenvalue is needed) and for the aluminum slab the exponential terms in brackets reduce to --Ka2 /12 t ' e
--K~2 /12 t '' or
e
Acknowledgements The author wishes to express his appreciation to Dr. J.R. Poweli, Fusion Technology Group Leader, for useful discussions, to Mr. Stan Majeski for preparation of the drawings, and to Ms. Carolyn Eterno for typing and prepara. tion of the manuscript.
Nomenclature B Bi C h k 1 Qo SO t t' t" T x o ¢~ t3
p r
= = = = = = = = = = = = = = = = = = = = =
dimensionless parameter, ",/l Biot m o d u l u s , hl/k Specific heat, Wsec/g °C heat transfer coefficient at cooled surface, W/cm 2 °C thermal conductivity, W/cm °C slab thickness, cm (fig. 1) applied surface heat flux, W/cm 2 (fig. 1) defined in eq. (1), W/cm 3 time, sec defined by eq. (24), sec defined by eq. (33), sec period o f applied heat flux Qo [eq. (3)] cartesian thickness coordinate, cm (fig. 1) temperature, °C defined by eqs. (13) and (3), dimensionless defined by eq. (11 c), c m - 1 defined by exponential in eq. (1), cm - 1 thermal diffusivity, c m 2 / s density, g/cm 3 temperature variable for integration [eqs. (23), ( 2 5 ) - ( 2 7 ) ] , sec function defined by eq. (3)
Subscripts f 0
= fluid = initial time.
References [ 1] [2] [3] [4] [5]
Fusion Reactor Design Studies, G A - A 1 3 4 3 0 (1975). J.R. Powell et al., BNL 19565 (1974). R. Benenati et al., Nucl. Eng. Des. 39 (1976) 165. H.S. Carslaw and J.C. Jaeger, C o n d u c t i o n of Heat in Solids (Oxford University Press, L o n d o n , 1959). A.V. Luikov, Analytical Heat Diffusion Theory (Academic Press, New York, 1968).
Nuclear Engineering and Design 48 (1978) 330 339 @ North-Holland Publishing C~mpany
FIRST WALL FUSION BLANKET TEMPERATURE VARIATION - SLAB GEOMETRY *
J.A. FILLO Department of Applied Science, Brookhaven National Laboratory, Upton, N Y 119 73, USA Received 11 November 1976
The first wall of a fusion blanket is approximated by a slab, with the surface facing the plasma subjected to an applied heat flux, while the rear surface is convectively cooled. The relevant parameters affecting the heat transfer during the early phases of heating as well as for large times are established. Analytical solutions for the temperature variation with time and space are derived. Numerical calculations for an aluminum and stainless steel slab are performed for a wall loading of 1 MW(th)/m2. Both helium and water cooling are considered.
1. Introduction
To obtain electrical power from the D - T fusion reaction, a standard thermal cycle, i.e. a steam or gas turbine cycle, must be used. Since the thermodynamic efficiency of such a cycle is governed by the operating temperature of the heat source, the problems of heat removal from a CTR are of considerable importance. The choice of the coolant and flow network impinges on such aspects as capital cost and power-plant efficiency, as well as tritium breeding, radioactivity, material and structural criteria, and the reactor lifetime. It is generally assumed that in the case of a commercial reactor it is desirable to maximize the wall loading, i.e. the ratio of the maximum rated power output of the reactor divided by the torus surface area facing the plasma, since the reactor size and capital costs are directly related. Even though such factors as radiation damage and plasma-stability restraints may strongly affect the allowable wall loading, the ability to cool the first wall, i.e. the first material surface facing the plasma, appears to be tile major factor involved in establishing the wall-loading limit. The coolant or coolants are introduced so as to absorb the energy of the neutrons released in the fusion reactions along with the energy (neutron + gamma generation) developed in the blanket with or without tritium breeding. In addition the coolant or coolants must absorb a certain percentage of the energy from the plasma, primarily in the form of radiation, which impinges on the blanket surface exposed to the plasma. The neutron and gamma energy results in volumetric heat generation within the blanket, while radiation to the surface is responsible for an imposed surface heat flux. As just noted, in the design of blankets and the subsequent problems of cooling and heat removal, the first wall looms as the critical region of concern. Consequently, it is singled out for separate study and discussion, since all of the emissions from a reacting plasma may intersect this wall. These include [1 ] ions and neutral particles, primary neutrons, X-rays (bremsstrahlung) and cyclotron radiation. In addition, scattered neutrons and gamma radiation generated in the blanket regions exterior to this first wall are also imposed on it. Finally, D - T gas surrounding the plasma may react chemically with the first wall materials. One result of these many interactions is the generation of appreciable heat, which can be up to 20% of the plasma output. Thus, provision must be included for adequate cooling. In most analyses the above effects are usually lumped together as a radiant flux. While the incident fluxes are absorbed in a small depth of any first wall material, the usual assumption is to treat the lumped radiant flux as if it were simply deposited on the surface facing the plasma. In heat transfer analyses it is an applied * Work performed under the auspices of the US Energy Research and Development Administration.
J.A. Fillo / Slab geometry of first wall fusion blanket
331
heat flux boundary condition. The phenomena of sputtering and blistering, the formation of gaseous reaction products, and material vaporization processes may represent the major sources of impurity in the plasma. Therefore the first wall can be a limiting component in reactor power output due to a large power generation within it; and it may also limit plasma performance through impurity levels. As already noted, the useful first wall lifetimes in a power reactor may be severely limited through radiation damage and surface erosion. The attendant frequent replacement could unduly add to operating costs and require an excessive plant down time for replacement. Thus, the design and material selection for this first wall is a vital element in the design of a fusion power reactor. For example, in a number of designs [1 ] the first wall is constructed as a liner attached to the remainder of the blanket to absorb the brunt of the thermal loading. In addition to the above concerns with respect to the first wall, it is necessary to consider additional effects which the first wall will experience as well as potential scenarios, such as a plasma excursion and energy dump to the wall which could be disastrous for the first wall and cause subsequent failure of the blanket as a whole. These all have a bearing on the design of the blanket, need for careful analysis, as well as heat removal and cooling aspects of the blanket. Assuming that the plasma does not undergo any excursions to the first wall under normal operation, the first wall will be subjected to thermal cycling. This is not necessarily a problem in itself but rather the temperature excursions per pulse to which the structure could be subjected. Cracks could occur in the first wall as a consequence of thermal stresses, the propagation of these cracks leading to the ultimate failure of the blanket. Since fusion reactor designs for the experimental power reactor (EPR) or any of its variations (TNS) are to be pulsed devices where the plasma 'on' period may range from 30 to 40 sec to possibly several minutes, with a downtime less than or of the order of a minute, from a heat transfer standpoint the nonsteady or transient and steady, periodic states are of interest. While thermal conduction aspects of blanket designs may be analyzed by finite difference or finite element methods, the present study considers a slab geometry (fig. 1) which has the advantage of being amenable to analytical analysis. The surface facing the plasma is subject to a pulsed heat flux while the rear face is convectively cooled to remove the heat generated in the 'blanket'. The objective of the study is to identify the relevant parameters affecting or governing the heat transfer during the early phases of heating as well as for long times, once the steady, periodic temperature profile is established. Since a closed form solution is derived, the solution should be of interest as a base case against which results from numerical codes may be checked or compared, as well as a relatively simple means of assessing time-dependent effects on first walls. This assumes, of course, that the actual design may be approximated by slab geometry. Lastly, as a basis of comparison, we have chosen aluminum and stainless steel A1 (or SAP), rather than stain-
/
h, vf
Soe-r x
TX
T t
TT Qo ttT
Fig. 1. One-dimensional slab geometry.
332
J.A. lffllo /Slab geometry of first wall fusion blanket
less steel, has been advocated by BNL [2,3] in its designs, primarily on the basis of less long-time radioactivity. In addition, the better thermal properties of AI are apparent in the results.
2. Basic formulation - one dimensional A one-dimensional slab geometry (fig. 1) is used to model tire first wall blanket region. Since the volumetric internal heat generation can be closely approximated by a decaying exponential function, the one-dimensional, time-dependent heat conduction equation can be written 02v
~v
K Ox2
~t
So
-
pC
e-VX4o(t)
(1)
subject to the following boundary and initial conditions: Ov
- k ~xx = Q°c)(t)'
- k OVox= h(v
x : 0;
-
Of),
X=l
and
v = Vo,
t = 0.
(2a-c)
Here v is the temperature while S o and 7 are parameters dependent on plasma conditions as well as blanket structural material, and are derived from the results of a neutronics analysis. Qo is the incident flux to the first wall and is assumed to be a certain percentage of the plasma output, usually of the order of 20%. The heat transfer coefficient, h, and mean fluid temperature, vf, are given, and cooling is continuous during the plasma 'off' period. Taking into account an 'on' and 'off', cooling behavior is straightforward. We assume that the blanket structure is at some uniform temperature, vo, at time t = 0. For definiteness we consider only the case of an applied rectangular wave form but other cases such as a ramp followed by a constant flux and heat source (in time) may be treated in the same way. The applied heat flux and source may be represented by the function [4] :
¢(t)=o,
t < o,
~b(t) = 1,
nT
~b(t) = 0,
nT+T1
n = 0 , 1 ..... n=0,1 .....
(3)
In other words the applied flux, Qo(~(t), represents a flux Qo which is 'on' for time T1 and 'off' for time T - T 1 and so on, with period T.
3. Superposition of solutions
The solution to eq. (1) may be represented as the superposition of two solutions. Let (4)
O=U+W,
where b2u
3u
K bx 2
Ot
(5)
--0,
Ou
~u
-k~-x=0,
x=0;
-k~x=h(u-uf),
x=l;
U=Vo,
t=0,
(6a-c)
and b2w
K ax 2
Ow
~t
-
So
pC
e-'~%(t),
(7)
J.A. Fillo / Slab geometry of first wall fusion blanket ~w _k~x=Qog~(t),
aw - k ~ - x =hw ,
x=0;
x=l;
w=0,
t=0.
333 (8a-c)
Eqs. ( 5 ) - ( 6 c ) result in the complementary solution with effectively homogeneous boundary conditions (after a simple transformation, s = u - vf) while eqs. ( 7 ) - ( 8 c ) constitute the particular solution.
4. Complementary solution The solution to eq. (5) may be found in Carslaw and Jaeger [4]. The dependent variable, s, is defined as (9)
S=U--Of
so that l
s= ~
(10)
Zn(x) e -K#n2t f Z n ( ~ ) f ( ~ ) d~,
r/=l
0
where (1 la)
f(x) = initial temperature = u o - of and
Z n ( x ) = [ 2( k2~2 +h 2) ] Ll(k2t3 2 + h 2) + khJ
'/2
cos/3nX,
(llb)
t3n, n = 1,2, ..., are the positive roots or eigenvalues of (llc)
sin /31 = k~-COS/3l. Evaluating eq. (10), we find ¢¢
2(a2n + Bi 2)
U -- Of = (O O -- Vf) n=lG O~2 + B i 2 + B i
sin an I c o s ( ~ ) l e
-(Ka2nt/12) ,
(12)
~n
where Bi = Biot modulus = lh/k,
(13)
an = ~nl •
Note that the eigenvalue equation, eq. (1 lc), becomes (14)
tan a n = Bi/a n.
5. Particular solution The solution to eq. (7) may be found by applying Duhamel's theorem [4] once the solution to the problem for time independent heat flux and source term is known. The solution to this problem is also found by superposition of solutions. The first problem we solve is eq. (7) for ~(t) -- 1. Let w = wl (x) + w2(x, t) so that d2wl _ K dx 2
SO pC e-'rx,
(15)
32w2 K ax2
3w2 _ 0 , a~-
(17)
J.A. Fillo / Slab geometry of first wall fusion blanket
334 dWl -k-~-=Qo,
x=O,
(16a)
-k Ow2=Oo'x
- -k~dWl -=hWl,
x=l,
(16b)
.3w2 . -K~-x =nw 2, x=l,
(17b)
W 2 = --W 1 (X),
(1 7C)
x=O,
t = 0.
(17a)
Physically, wx represents the steady-state temperature solution for a slab which is subjected to a constant heat flux at x = 0, while at x = l there is radiation (or convection) into a medium at zero temperature. In addition the rate of heat production is a function of position. In the case of w 2 we have a slab insulated at x = 0, initially heated non-uniformly. The solution for wl is straightforward and results in
w,
=
Q°lrlL + Bi(1-/)] +S°12 [1 + Bi(1-/) +(~--1)e-B-~ Bi- e_BX/ll. -~:xl kBBi
(18)
The general solution to eq. (17) is the same as (10) but where
Q°l (1 S°l: II+Bi(1-1)+(~-l)e-B-B~Be-BXlll" f(x)- ~-[l+Bi -/)] k~i
(19)
Evaluating the integral, we find for w 2 that 2 W2
- - -
/2
B
n=, - -
+ Bi:
ie -B 1 - ~
2Qol( 1 ]/
1
coso~,, +
(o~ +8:)o~ + - - U - ' , ~ : )
rcosi .X le_, o ,:, \ l 13
(20)
Xan2+Bi2+BiL
Combining eqs. (18) and (20), the particular solution is
_Qo,[ f 7)]
w-~.-
1 +Bi 1 2
-n=l
+
kBBi
l2
- -
X ~n2 + Bi 2 + Bi
I1 + Bi( 1 _ / ) + (~i
B ie -B 1-~
os
1)e_ B _(~i)e_BX/l 1
1
cOSan+
(ct2n+B2)ct2 n+
e -(K~2nt/12).
(21)
6. Solution for time independent heat flux and heat source
The steady, periodic temperature distribution is of primary interest in this study. On the other hand, the general solution found by combining eqs. (12) and (2 I) may also be of interest since it predicts the transient, or simply, non-steady temperature history to steady-state assuming normal plasma behavior and 'long' burn time. In other words true steady-state is assumed to occur. The combined solution also forms the basis for determining the steady, periodic temperature distribution.
J.A. Fillo / Slabgeometry of first wallfusion blanket
335
The resultant temperature, o, is simply Qol Ii + Bi( 1 - - / ) 1 + Sol2 k--ff-~l Il o - Vf=~-B-~-
+(Oo
-
of) ~
{2 sin an
n = 1
an
+
2Sol2 -----
k(o 0 -
of)
+ Bi ( 1 - l x) + ( ~ - - 1 ) e - B - B_Bi e-BX/I1 IB
( BBi) BI 1 ie -B I c°San + (Ctn 2 +B2)an2
2Qol ( 1 ) } k(oo-Vf) ~2n2
a2+Bi2I(~-)le-(~a~t/12) X a 2 + B i 2+Bi
(22)
cos
All of the significant parameters affecting the solution are apparent, specific conclusions being dependent on blanket material choice and coolant as well as wall loading. In addition to the Biot modulus, Bi = hl/k, the solution can be restated in terms of the Fourier number, Fo = Kt/l 2 , the Pomerantsev modulus, Po = Sol2~ [k(o o - of)], and the Kirpichev modulus [5 ], Ki = Qol/[k(oo- of)]. The physical significance of the Pomerantsev modulus is that it shows the ratio of the amount of heat evolved by the source per unit time in the volume l (a parallelepiped with a base of 1 cm 2 and a height/) to the maximum possible amount of heat transferred by conduction through a unit area per unit time over the distance l (subject to the assumption that the temperature distribution occurs by the linear law). The Kirpichev modulus is the ratio of the heat flux, Qo, through the slab to the maximum possible heat flux at l provided that the temperature gradient at this point is maximum and equal to (v0 -vf)/l.
7. Steady periodic temperature The solution to eq. (1), subject to (2)--(2c) follows from (22) and Duhamel's theorem: oo
n=l
l
+~
2
1
n=l pC an2 +B 2
a n2
IB (
an2 + Bi2 + Bi
ie - n 1 -
cOSan+
e -(~2t/12)
os
"J: t
So(r)e -I~2n(t-r)/z21 dr
+pclfQo(r) e-lK~"(t-~)/121 dt a 2 + B i 2 + B i os 0
.
(23)
The steady, periodic variation in temperature which exists after the transient involving the initial conditions has died away may be achieved in a number of ways. The method of Weber [4] (see Carslaw and Jaeger) is found to be an 'elegant' procedure, and is quite general, the advantage being that the resulting series is not slowly convergent for values of time which are of interest. This is borne out by calculations in a latter section. The results given in [4] are immediately available for use in any problem in which the solution for constant external condition is given as a series of exponentials of the form (-~/nt), and the solution for external conditions (as well as the heat source) represented by (3) is to be obtained by Duhamel's theorem. Rather than repeat the derivation, we simply state the results for the integrals for the 'on' and 'off' periods. For the 'on' period we let
t = rT + t',
(24)
J.A. Fillo/Slab geometryof first wallfusion blanket
336
where 0 < t' < T1, and r is large. Setting
(25) and 0i(r) = So(r), Qo(r),
for i = 1,2 respectively,
(26)
we have to evaluate integrals of the type
e -Tnt
t
f~(r) e7nr dr,
(27)
0
which becomes, for large values of r, e~n(T 1-t')
I
_ e~,n(T-t')7
i2
(28)
'
where i = 1,2 (¢1 -- So and ¢2 = Qo). Consequently, for large values of the time the solution to (23) is
v-of:2~-~-.2 l ~ [2So h 12=,t ~ }1Ll an2+Bi2 [1 X an2 + Bi2 + Bi -
'[- Bie-" (
__~l)cOSan+B]+_~_(_~nn) B 2Qo ] }
e(Ka2n(T'--t"/t2'_--e(Ka2n(T--t"/12'1 (an~_) 1
-
-
_] cos
e (~a2T/t2)
(29) .
Since oo
2 2 Bi) n=l an(an +B2)(an2 + Bi2 + 1 [l+Bi(1-/) B Bi
Bie-/~ 1 -
+(\BBi ] - Be- ( B ) e - - 1Bi
cos an+ -Bx
and
(
n=l -an-2 an2+Bi 2+Bi c ° s t i f - ] = B-i
+ Bi 1 -
cos
/ l]
(30)
(31)
'
an alternative form of eq. (29) is O--Of =
Q°l
k--~-I l + B i
(1
s°12 II+Bi(1-I)+(BB~il)e-B-(B-~B)e-BX/II
-/)1 + ~-~
~ [ 2S°12 [13 (1 ~ii) -n~_l ~kan2(an:~+B2 ) ie -B - . cos a n+
I-e(~a2n(T1-t')/t2)-e(Ka2n(T-t')/t2)7
COS(-~)
BI+2Q°II ~
a2n+Bi2
an2+Bi 2+Bi
(32)
J.A. Fillo / Slab geometry of first wall fusion blanket
337
Note that physically the first two terms are the steady-state solution, the last term representing the steady, periodic variation of temperature with respect to the 'true' steady state. In an analogous way for the 'off' period, we set t = r T + Tt + t",
(33)
where r again is large. For this case t eTn(T - T 1-t") _ e~,n(T - t") e - ' r n t / d p ( r ) e "rnr d'c -+ (h _e~nT ) 0 ~'n( 1
(34)
so that the temperature, for large values of time, becomes u-of=
~ 2S°12 1 n = t [ k ( ~ 2 ~n2 + B 2
ie-B
B 1--~-COSt~n +
+
2Qo l 1 k (~2
F_e(•(x2(T-Tl-t")/12)_e(Ua2(T-t")/12)][OtnX ~ XL
1 -e("'~'/'2)
(35)
jcos~--/-).
8. Discussion of results The steady, periodic temperature distributions (actually, u - o f ) for a ½ cm aluminum and stainless steel slab are shown in figs. 2, 3 and 4, respectively. Input for the calculations are to be found in table 1. The neutronics results for A1 and the heat transfer parameters are taken from a BNL blanket design [4]. In the case of stainless steel we assume the same neutronics as for aluminum, which is an approximation. An examination of terms in eq. (32) reveals that the dominant contribution to the temperature variation is from the surface heat flux, Qo, and this would not change even for an exact stainless steel neutronics calculation. Note that we have fixed the Biot number for the comparison between the two metals.
AI SLAB
. - - SURFACE TEMPERATURE - - - COOLANT WALL TEMPERATURE
80
.t., 70 u/ 60 n,-.
50 ~E
20 [-
40
20 I0
i
o
SURFACE TEMPERATURE
COOLANT WALL TEMPERATURE
I
50
o
--
---
Io
L
20
50 40 50 60 o IO zo 50 TIME,sac TIME, sac Fig. 2. Steady, periodic temperature histories. Wall loading: 1 MW(th)/m2; helium-cooled ~Bi = 0.1. Fig. 3. Steady, periodic temperature histories. Wall loading: 1 MW(th)/m2; water-cooled ~Bi = 1:31.
I
40
,50
6O
J.A. b]'llo / Slab geometry of first wall fusion blanket
338
SS SLAB --SURFACE TEMPERATURE --COOLANT W A L L TEMPERATURE
120 IIO I00
9O
p
8o
~6o bA ~ 5o w
~
40
/
/
\
5O
IO 0
I0
20
30
TIME,
40
50
60
sec
Fig. 4. Steady, periodic temperature histories. Wall loading: 1 M W ( t h ) / m 2 ; w a t e r - c o o l e d ~ B i = 1.31.
While the difference between helium and water cooling for the AI slab is significant, it should be noted that the maximum surface temperature with the cooling may be reduced to about one-half (~35°C) [3] of the results in fig. 2 by improving on the design, i.e. by increasing the surface area of the coolant passage, as was done in ref. [3]. This, of course, introduces a two-dimensional effect which cannot be handled by the current calculation. To reduce the surface temperature in the one-dimensional calculation implies increasing the heat transfer coefficient. The bulk of the temperature difference for the helium-cooled case is in the difference between the wall and coolant fluid. For both the helium and water-cooled cases the maximum difference across the A1 slab is ~6°C, while for stainless steel, as a consequence of lower thermal conductivity, the maximum difference is ~60°C. In all cases, as we approach the end of the plasma ,on' period, the temperature approaches the 'true' steady-state, approaching it much faster for the A1 slab with water-cooling. It might be noted that the temperatures for the water-cooled case are virtually the same as in ref. [3] since the geometry of the water coolant passages (width > height) are such that the one-dimensional approximation is valid.
Table 1 Heat transfer parameters for temperature calculations Aluminum k ( W / c m °C) K (cm2/sec) Bi (dimensionless)
water helium
S O ( W / c m 3) B (dimensionless)
(2o (W/cm2) Eigenvalues: water -~Xl, = 0 . 9 4 6 6 , helium - a 1, = 0 . 3 1 1 1 .
1.75 1 1.31429 0.1 4.7 0.04272 20
Stainless steel 0.172 O.04938 1.31429 4.7 0.04272 20
J.A. Fillo /Slab geometry o f first wall fusion blanket
339
In the cases investigated, the infinite series in eqs. (32) and (35) reduce to a single term(only one eigenvalue is needed) and for the aluminum slab the exponential terms in brackets reduce to --Ka2 /12 t ' e
--K~2 /12 t '' or
e
Acknowledgements The author wishes to express his appreciation to Dr. J.R. Poweli, Fusion Technology Group Leader, for useful discussions, to Mr. Stan Majeski for preparation of the drawings, and to Ms. Carolyn Eterno for typing and prepara. tion of the manuscript.
Nomenclature B Bi C h k 1 Qo SO t t' t" T x o ¢~ t3
p r
= = = = = = = = = = = = = = = = = = = = =
dimensionless parameter, ",/l Biot m o d u l u s , hl/k Specific heat, Wsec/g °C heat transfer coefficient at cooled surface, W/cm 2 °C thermal conductivity, W/cm °C slab thickness, cm (fig. 1) applied surface heat flux, W/cm 2 (fig. 1) defined in eq. (1), W/cm 3 time, sec defined by eq. (24), sec defined by eq. (33), sec period o f applied heat flux Qo [eq. (3)] cartesian thickness coordinate, cm (fig. 1) temperature, °C defined by eqs. (13) and (3), dimensionless defined by eq. (11 c), c m - 1 defined by exponential in eq. (1), cm - 1 thermal diffusivity, c m 2 / s density, g/cm 3 temperature variable for integration [eqs. (23), ( 2 5 ) - ( 2 7 ) ] , sec function defined by eq. (3)
Subscripts f 0
= fluid = initial time.
References [ 1] [2] [3] [4] [5]
Fusion Reactor Design Studies, G A - A 1 3 4 3 0 (1975). J.R. Powell et al., BNL 19565 (1974). R. Benenati et al., Nucl. Eng. Des. 39 (1976) 165. H.S. Carslaw and J.C. Jaeger, C o n d u c t i o n of Heat in Solids (Oxford University Press, L o n d o n , 1959). A.V. Luikov, Analytical Heat Diffusion Theory (Academic Press, New York, 1968).