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A method of evaluating local turbulent diffusivity of heat in a gridded fuel subassembly
Nuclear Engineering and Design 62 (1980) 14%153 © North-Holland Publishing Company
A METHOD OF EVALUATING LOCAL TURBULENT DIFFUSIVITY OF HEAT IN A GRIDDED FUEL SUBASSEMBLY Hisashi HISHIDA, Katsuhiro SAKAI, Toshihiko IWASE and Hideyuki ASAI Mitsubishi Atomic Power Industries, Inc., Nuclear Development Center, 1-297 Kitabukuro-Cho, Omiya City, Saitama Prefecture, Japan Received 9 August 1980 In evaluating the turbulent dittusivity of heat associated with the coolant flow past a grid spacer within an FBR fuel subassembly, a heat diffusion technique is usually employed. However, measurement of subchannei bulk coolant temperature using thermocouples usually involves difficulty due to a steep and non-linear temperature gradient in the subehannels adjacent to a heater pin. A series solution of the heat conduction equation for the coolant flow in subchannels past a grid spacer and a heated section of a dummy fuel pin was derived under a slug flow approximation where the boundary conditions on dummy fuel pins were satisfied by means of the point-matching technique. The solution may be utilized in analyzing the turbulent diffusivity of heat within subehannel coolant flow as a function of distance from a grid spacer based on the measured temperature distribution on the wall of dummy fuel pins, which may be obtained without affecting the subchannel coolant temperature. In an illustrative example, the turbulent diffusivity of heat was most exaggerated at about 50 mm beyond a grid spacer and was approximately five times larger than the corresponding diffusivity without a grid spacer.
1. Introduction
measured at the center may not always represent the bulk subchannel temperature [4]. In this paper, a method of evaluating turbulent diffusivity of heat within a gridded fuel subassembly as a function of distance from a grid spacer to the downstream direction is discussed based only on the axial variation of temperature field on the wall of dummy fuel pins behind a grid spacer. For the analysis of temperature field in relation to turbulent diffusivity of heat, a series solution of the threedimensional heat conduction equation is utilized where the necessary boundary conditions are satisfied partly by the point-matching technique on the wall of dummy fuel pins [5]. As is already clear, the proposed analysis does not rely on the experimental determination of the bulk temperature for the adjacent subchannels but on the coolant temperature on the wall of dummy fuel pins which is directly measurable without affecting the flow characteristics. In the following discussion, an upstream portion of the center pin is electrically heated in order to evaluate turbulent diffusivity of heat in the cen-
Evaluation of turbulent diffusivity associated with coolant flow within a gridded FBR fuel subassembly is of considerable importance in connection to the core hot spot analysis, since grid spacers generally facilitate subchannel coolant mixing resulting in the reduction of temperature peaking within a subassembly. The vortex formation which affects turbulent diffusivity in coolant flow behind a grid spacer depends strongly on the axial distance from the grid spacer and its shape [1, 2]. Experimental investigations on turbulent diffusivity of heat have been reported as mixing coefficients for different shapes of grid spacers [3] where mixing coefficients were evaluated based on the bulk coolant temperature measured at the center of the adjacent subchannels under consideration utilizing the heat diffusion technique. However, the coolant temperature gradient within subchannels adjacent to a heater pin is quite steep and not linear due to the geometric configuration of subchannels so that the temperature 147
H. Hishida et al. / Evaluating turbulent diffusivity of heat
148
tral subchannels, however the proposed method may easily be applied to the peripheral and the corner subehannels with slight modification of the boundary conditions if necessary. It should be noted that turbulent diffusivity within the central and the peripheral subchannels do not differ essentially from each other [6]. Experiments were performed using a water loop with a gridded 169 dummy fuel pin subassembly installed in the test section. An upstream portion of the center dummy fuel pin was electrically heated. At the end of the heated zone, a grid spacer was placed and the axial variation of coolant temperature distribution on the wall of dummy fuel pins past the grid spacer was measured by rotating dummy fuel pins with thermocouples embedded on the wall as illustrated in fig. 1. Since turbulent diffusivity of heat within a gfidded fuel subassembly is strongly dependent upon the shape of a grid spacer, only illustrative examples with specific conclusions are given in this paper.
Z
DUMMY !FLYEL PIN
i
i
,
II
i i
HEATED SECTION THERMOCOUPLE
©C)O0 Fig. 1. Schematicrepresentation of a dummyfuel pin subassemblyemployedin heat diffusionexperiments with the coordinate system. and -= kJ/~. We seek the solution of eq. (1) in a series form such that
2. M a t h e m a t i c a l f o r m u l a t i o n
2.1. Basic equations
T(r, ~p, z) = m=0 ~__0Umn(r, ~O)l)m.n(Z),
With the assumption that the slug flow approximation is valid for the subchannel coolant flow such that u(r, q~) = Uo, the heat conduction equation for the coolant flow past the heated section of a dummy fuel pin may be expressed in the cylindrical coordinate system illustrated in fig. 1 as follows:
(3)
then, Um,.(r, ~p) a n d Vm,.(z) satisfy the following differential equations respectively.
1 0 {r 0 r Or 7r um,.(r,
~O)}+102 Um,n(r,~o) r 2 tgcp2
+ aZ,.,.u,.,.(r, ~0) = O,
(4a)
- ~-~z~ Vm'"(z)+ t'CPcU°°V"'"
OT c [ 1 0 f OT ~p,z)} pCpuo Tz (r, ~p, z) = K~[ r-~r [ r 7 r (r,
"~ ~Tl2nI)m,n(Z) =
+ 1 02T, -~ -~'~-f~2 t r, ~p, z )
0,
(4b)
where am..'s are eigenvalues to be determined later based on the boundary conditions.
+ "02T" ¢,z)], ~-~'z2 (r,
(1)
2.2. Boundary conditions
where/¢c is defined in terms of the molecular conductivity k¢ and the turbulent diffusivity of heat en of the coolant as
i¢~ = k¢ + pCpe.
(2b)
(2a)
The coolant temperature distribution T(r, ~o, z) is subject to the following boundary conditions: (1) In a subassembly of regular hexagonal
H. Hishida et al. / Evaluating turbulentdiffusivity of heat configuration consisting of dummy fuel pins, T(r, tp, z) should satisfy the azimuthally symmetric boundary condition such that
0 T(r, tp, z)l
-Olm,nJ6ra+l(am,nro) 3F 6 m J6m(Olmnro)
(5)
~p=0
(2) The amount of heat to be transferredlfrom the running fluid to the surface of a dummy fuel pin is negligible, that is
n(r). grad T(r, ~p, z)l r=r~ = 0,
= 0.
(7)
•
•
ro
am,,, is the nth eigenvalue corresponding to the mth characteristic equation which is derived from the boundary condition (7) as
{-Otm,nJ6m+l(Otm,nR) + ~ - J6m(OtmmR)} - Fm.,'{-am,~Y6m+l(am.nR) 6m } + ~ Y6m(Otm,nR) = 0 f o r m = 0 , 1 . . . . , (13) and c,~,," in expression (11) is an arbitrary constant corresponding to the m - nth eigenfunction. It is clear from eqs. (12) and (13) that
Fm,O= 0 and am,o= 0 for m = 0, 1, 2 . . . . .
(4) The coolant temperature distribution should be finite and uniform for z--> ~, that is
z -...>oo) = ~ C p + Tin,
(8)
Vm,,'(Z) = C'," exp(-flm.nZ),
(15)
where /3m.n is defined such that
T(r, ~p, 0) = Texp(r, q~, 0).
T(r, tp, z)= Co,o+ m~=on~__1 gm,n(r)
(9)
p C Uo 2 1/2
~ra,,"
=~a2m,~.(2k~.e) ) [ ~
_pCpu 2kc"
According to the boundary condition (5),
Um.~(r,~p) should be of the following form: (10)
where gm,n(r) is determined by substituting expression (10) into eq. (4a) such that
and rm.n is derived in terms of the boundary condition (6) with r = r~ as follows:
(11)
(16)
Then, the solution T(r, q~,z) which satisfies boundary conditions (5), (7), (8) and (6) with r = r~ may be written in the following form: oo
× exp(flm,~z)cos 6mq~,
2.3. Solution
gm,n(r) = Cm,n{J6m(am,nr)- rm,nY6m(Otm,nr)}
(14)
The solution of eq. (4b) which is finite for z --->o0is as follows:
where Q, w, Cp and Tin are the rate of thermal output of a heating section, the mass flow rate, the specific heat and the inlet temperature of coolant, respectively. (5) The coolant temperature distribution at z = 0 past the heating section of the center dummy pin should be equal to that obtained by experimental measurement Te~p(r,~, 0), that is,
;Arn,n(r,tp) = g,~.n(r) cos 6mtp,
(12)
(6)
where r* is the position vector of a point on the periphery of dummy fuel pin i at z and n(r*) is a unit vector normal to the periphery at r*. (3) At a sufficient distance R from the center dummy pin with a heating section, it may practically be assumed that
T(r,
.
- a , . nY6m*l(' m,'ro) + 6 m Y6m(am.nro)
=0.
r=R
ro
r,.,= =
=~6
OT (r, tp, z)[ Or
149
(17)
where arbitrary constants c ' ~ ' s appeared in solution (i5) are included in Cm,,'S of solution (11). Boundary conditions (6) for r = r* with i = 1, 2 . . . . . and (9) may approximately be satisfied by means of the point-matching technique.
2.4. Determination of Cra.n In order for T(r, tp, z) to satisfy approximately boundary condition (6) with r = r*,
H. Hishida et al. / Evaluating turbulent diffusivity of heat
150
the azimuthal angle 0 -< ~0 < ~'/6 is subdivided into j divisions. Let the position vectors of the intersections of q~ = 6oj and the periphery of dummy fuel pin i be r_~j and r+~j as illustrated in fig. 2, then the distance r_~4 - ]r_ d and r+~j = Ir+i,il, and the unit vectors n(r i,j) and tl(r+ij ) normal to the periphery of dummy fuel pin i at r-~j and r+~,; respectively are expressed as follows:
n~_,j = -x-U,f,(i, j ) - U~f~(i, j),
(18)
for i--- 1, 2 or 3, r_+~j= r0[2yi cos q~j -+ {1 - (23,/)2 sin e ~oj}l/2],
(19a)
f,(i, j) = {1 - (236)2 sin 2 q~j},/2
(20a)
and
directions, respectively. Similar expressions may easily be derived for r_.ij and n_~ij with i being greater than 6, however the radial temperature gradient of coolant usually diminishes rapidly as the radial distance from the center fuel pin increases and boundary condition (6) with i being greater than 6 may be disregarded in connection with boundary condition (7). In a case that a line q~ = q~j coincides with a tangent to the periphery of a dummy fuel pin, it should be clear that r+i,j = r-i,y. With expressions (18), (20) and (21), boundary condition (6) is to be satisfied pointwise on the periphery of each off-centered dummy fuel pin by the following equation:
n(r).gradT(r,q~,z)
L(i,j) = 2i3, sin q~j, for i = 4, 5 or 6,
[
(21a)
(°)
q~,
r_.ij = ro 2(i 2 - 5i + 7) 1/2 y cos - ~ - ~oj ,
/71"
\11/2"1
+_ 1-4(i2-5i+7),2sm2~-~-~o,) i 1, (19b) ]1/2
f,(i,j)={1-4(i2-5i+7)y2sin2(6-~oi}l (20b) and
5
2
=0.
(22)
(23)
then, eqs. (22) with respect to the intersections r._~,i are solved first for c-'(°)m,,with n = 2, 3, . . . , N and m being fixed. Here, N - 1 is the total number of intersections under consideration where each intersection is sequentially numbered by p(+-i, j) = 17 2 . . . . . N - 1. It should be clear from definition (23) that c~3 = 1. The resulting simultaneous algebraic equations of order N - 1 may be written in a matrix equation form such that
M(°)C (°) = B (°), I
I-_T_f//., OT Ll"l)-oTr (r'q~'z)
(21b)
where 3' is the pitch to diameter ratio of an assembly under consideration. Ur and U~ are the unit vectors to the radial and the azimuthal
i= 0
=t
The above eq. (22) contains arbitrary constants Cm,'S and due to the exponential terms (15), c,..,'s are to be determined for each sufficiently narrow subinterval of the axial distance z i.e. Zk <--Z < Zk+l with z0 = 0 so that eq. (22) may be satisfied for any value of z. Let c~). stand for cm,. within a subinterval Zk < Z < Zk+l and define 6~!. for m > 0 and n > 0 such that E(I,) _-_.,.
f~(i,j)=2(i2-5i+7)'/2ysin(6-q~,) ,
i=4
r~r~ij
3
Fig. 2. Definitions for vectors r+i4, n(r+ij) and fuel pin n u m b e r i.
(24)
where M (°) _ (M~°,),) is a square matrix of order (o} N-landC = ( C . (o) ) a n d B (o) = ( b p(o)) a r e column vectors, whose elements may be expressed as follows:
H. Hishida
et al. I E v a l u a t i n g
N
N
coo+ X" c(,) ~
M(O) p,n = [_{6~mr J6,n(Ctm..r)- Ctm.J6rn+l(ct,..r • " )
151
turbulent diffusivity of heat
•
m'~=0
m,1 n = l
C"(') m,n
exp(/3,.,.z0
× {J6,.(a,.,.r)- F,...Y6,.(am,.r)} cos 6rn~p. (28) -
r.•.
6m Y6m(otto,r) -5 -
+ F,.,.ot,..,Y6,.+l(am,.r)}f,(P)
COS6mq~
6m + ~ {./6,,(a,.,.r) r
- F,.,,Y6,,(el,,,,r)}f~(p)sin6mq~],
(25)
at the pth intersection and by ) = - M ~ t .
(26)
Matrix eq. (24) is solved for every integral value of m whose upper bound may usually be set equal to N. Once g~!.'s have all been obtained for 0 <~r~,~,o~ are to be determ <- N and 2 -< n -< N , x(0) mined based on the discretely measured temperature distribution on the wall of the center or the surrounding dummy fuel pins at z = 0 in the following way. Let Tcw(rq, goq,0) be the measured temperature at point q located on the wall of a dummy fuel pin whose position vector is given by (ra, ~q, 0), then from boundary condition (9) the following relation should be held:
T~p(rq, q~q,0) = T(")(rq, ~Oq,0) N
= c,,,, + E
m~(I
N
Z n=l
× {J~,, (m,.,r) - Fro,, Y6,, (am,.rq)} cos6mq~q for any q,
(27)
where T ~k) (r, q~, z) represents the coolant temperature distribution T(r, ~o, z) in a flow region z k ~ z < Zk+ 1. W i t h t e m p e r a t u r e data at N + 1 different points, eq. (27) may be reduced to a system of simultaneous algebraic equations for , N and ~m,J -c0) with m = 0, 1, ~o),~~ are found without difficulty. Next, we consider the coolant temperature distribution at z = zt =- Az with a sufficiently small Az. Then we shall have
T(°)(r, ~, zl) = T°)(r, q~, zt)
Since T(°)(rq, ~0q,z~) should be equal to the discretely measured temperature distribution Texp(rq, ~pq, z~), ~(0) related to the turbulent difusivity of heat by definitions (2a) and (2b) in the flow region 0 - z -< z ~ may numerically be determined by adjusting the value so that the relation Tt°)(rq, Cq, z~) = T~xp(rq, ~%, Zl) should be best satisfied. According to boundary condition (9) and the continuity requirement of z(k) T(k)(r, ~p, z) and T(k+l)(r, q~, Z) at z = Zk+,, ~m,. and C_-(k+l) m.tl should be related as follows: (k+t) m,n = 6tk) m.n exp [3,,.,(Zk
-
Zk+l)
for 0 <- m < N and 2 --- n < N
(29)
and c~,/may be obtained by equating T
3. N u m e r i c a l results a n d d i s c u s s i o n
Computer program F L O W M I X was completed based on the analytical procedures mentioned in the preceding sections and was applied to the evaluation of turbulent diffusivity associated with coolant flow in a FBR fuel subassembly. Experimental investigation on the coolant temperature distribution in a gridded 169 dummy fuel pin subassembly was performed using a water loop. The diameter of a dummy fuel pin was 6.5 mm arrayed with the pitch to diameter ratio of 1.2I. The upstream section being 250 mm long of the center fuel pin was electrically heated. The thermal output and the heat flux were maintained at 1.85 kW and 8.65 cal/cm 2 s, respectively. A honeycomb type grid spacer with the height of 15 mm was located just behind the heated section. Thermocouples were embedded on the surface of
H. Hishida et al. / Evaluating turbulent diffusivity of heat
152
D I S T A N C E FROM T H E GRID SPACER 1.5
.< 1.0
A
20mm
0
80mm
R e = 1.4 × 104
gridded subassembly may easily be evaluated as a function of distance from a grid spacer. Nomenclature
(r, ~p, Z)
T(r, ~o, z)
cylindrical coordinates where the axial direction of a fuel subassembly is parallel to the z-direction calculated coolant temperature at
To~,(r, ~,, z)
measured coolant temperature at
kc kc
inlet coolant temperature thermal conductivity of coolant effective thermal conductivity of coolant defined by eq. (2a)
z 0.5 .<
(r,~, z) (r, q~,z)
b-
Fig. 3. Temperature distribution on the wall of fuel pin 1 and 20 mm and 80 mm behind the grid spacer.
kc/ fcc P
c~
dummy fuel pins 0, 1, 2 and 4 at 20 mm, 50 mm, and 80 mm behind the grid spacer and by rotating these dummy fuel pins with thermocouples, coolant temperature field on the wall of dummy fuel pins was measured. Detailed data are shown in ref. [7], however for an illustrative example, periphera ! coolant temperature fields on dummy fuel pin 1 at 20 mm and 80 mm behind the grid spacer are shown in fig. 3 for Re = 1.4 x 104. Since eH of coolant past a grid spacer is strongly dependent upon the shape of a grid spacer, only specific conclusion is shown in this paper with respect to the grid spacer under consideration at the flow velocity of around 4 m/s which corresponds to the Reynolds number of 1.4 x 104. (1) Coolant mixing is most facilitated in the flow region about 50 mm behind the grid spacer. (2) For Re = 1.4 × 104, e , = 1.72 cmZ/s, which is approximately five times as large as that for a bare rod bundle at 50 mm behind the grid spacer. (3) Based on the analytical procedure mentioned in this paper with experimentally measured coolant temperature fields only on the wall of dummy fuel pins past a grid spacer, turbulent diffusivity of coolant flow within a
w Uo EH
O Otm,n
ro Y r* r~ij
n(r) V~ J~ Y~ i
J
density of coolant specific heat of coolant mass flow rate of coolant bulk velocity of coolant turbulent diffusivity of heat thermal output rate of the heated section nth eigenvalue corresponding to the ruth characteristic eq. (13) radius of a dummy fuel pin pitch to diameter ratio position vector of a point on the periphery of dummy fuel pin i position vector of intersections of ~o = q~j and the periphery of dummy fuel pin i illustrated in fig. 2 unit normal vector to the periphery at r unit vector to the radial direction unit vector to the azimuthal direction Bessel function of the first kind of order m Bessel function of the second kind of order m subscript indicating dummy fuel pin number subscript indicating the ]th subdivision of azimuthal angle (0, zr/6)
H. Hishida et al. / Evaluating turbulent diffusivity of heat
p(±i,j)
indicates identification number for the intersection of ¢ = ~p~ and the periphery of dummy fuel pin i superscript indicating the kth axial location
References [1] R. Nijsing, KfK 2232 (1975) 109.
[2] H. Hoffmann, KfK 1843 (1973). [3] H. Hoffmann, F. Hofmann and K. Rehme, IWGFR/29 (1978) 82. [4] T. Ginsberg and D.M. France, Internat. Seminar on Heat Transfer in Liquid Metals, Trogie, Yugoslavia, Sept. 1971. [5] R,A. Axford, Nucl. Engrg. Des. 6 (1967) 25. [6] H. Hoffmann and E. Baumg/irtner, IAEA-SM-173/IV (1973). [7] H. Hishida et al., PNC report SJ206 73-02 (1972).
153
A METHOD OF EVALUATING LOCAL TURBULENT DIFFUSIVITY OF HEAT IN A GRIDDED FUEL SUBASSEMBLY Hisashi HISHIDA, Katsuhiro SAKAI, Toshihiko IWASE and Hideyuki ASAI Mitsubishi Atomic Power Industries, Inc., Nuclear Development Center, 1-297 Kitabukuro-Cho, Omiya City, Saitama Prefecture, Japan Received 9 August 1980 In evaluating the turbulent dittusivity of heat associated with the coolant flow past a grid spacer within an FBR fuel subassembly, a heat diffusion technique is usually employed. However, measurement of subchannei bulk coolant temperature using thermocouples usually involves difficulty due to a steep and non-linear temperature gradient in the subehannels adjacent to a heater pin. A series solution of the heat conduction equation for the coolant flow in subchannels past a grid spacer and a heated section of a dummy fuel pin was derived under a slug flow approximation where the boundary conditions on dummy fuel pins were satisfied by means of the point-matching technique. The solution may be utilized in analyzing the turbulent diffusivity of heat within subehannel coolant flow as a function of distance from a grid spacer based on the measured temperature distribution on the wall of dummy fuel pins, which may be obtained without affecting the subchannel coolant temperature. In an illustrative example, the turbulent diffusivity of heat was most exaggerated at about 50 mm beyond a grid spacer and was approximately five times larger than the corresponding diffusivity without a grid spacer.
1. Introduction
measured at the center may not always represent the bulk subchannel temperature [4]. In this paper, a method of evaluating turbulent diffusivity of heat within a gridded fuel subassembly as a function of distance from a grid spacer to the downstream direction is discussed based only on the axial variation of temperature field on the wall of dummy fuel pins behind a grid spacer. For the analysis of temperature field in relation to turbulent diffusivity of heat, a series solution of the threedimensional heat conduction equation is utilized where the necessary boundary conditions are satisfied partly by the point-matching technique on the wall of dummy fuel pins [5]. As is already clear, the proposed analysis does not rely on the experimental determination of the bulk temperature for the adjacent subchannels but on the coolant temperature on the wall of dummy fuel pins which is directly measurable without affecting the flow characteristics. In the following discussion, an upstream portion of the center pin is electrically heated in order to evaluate turbulent diffusivity of heat in the cen-
Evaluation of turbulent diffusivity associated with coolant flow within a gridded FBR fuel subassembly is of considerable importance in connection to the core hot spot analysis, since grid spacers generally facilitate subchannel coolant mixing resulting in the reduction of temperature peaking within a subassembly. The vortex formation which affects turbulent diffusivity in coolant flow behind a grid spacer depends strongly on the axial distance from the grid spacer and its shape [1, 2]. Experimental investigations on turbulent diffusivity of heat have been reported as mixing coefficients for different shapes of grid spacers [3] where mixing coefficients were evaluated based on the bulk coolant temperature measured at the center of the adjacent subchannels under consideration utilizing the heat diffusion technique. However, the coolant temperature gradient within subchannels adjacent to a heater pin is quite steep and not linear due to the geometric configuration of subchannels so that the temperature 147
H. Hishida et al. / Evaluating turbulent diffusivity of heat
148
tral subchannels, however the proposed method may easily be applied to the peripheral and the corner subehannels with slight modification of the boundary conditions if necessary. It should be noted that turbulent diffusivity within the central and the peripheral subchannels do not differ essentially from each other [6]. Experiments were performed using a water loop with a gridded 169 dummy fuel pin subassembly installed in the test section. An upstream portion of the center dummy fuel pin was electrically heated. At the end of the heated zone, a grid spacer was placed and the axial variation of coolant temperature distribution on the wall of dummy fuel pins past the grid spacer was measured by rotating dummy fuel pins with thermocouples embedded on the wall as illustrated in fig. 1. Since turbulent diffusivity of heat within a gfidded fuel subassembly is strongly dependent upon the shape of a grid spacer, only illustrative examples with specific conclusions are given in this paper.
Z
DUMMY !FLYEL PIN
i
i
,
II
i i
HEATED SECTION THERMOCOUPLE
©C)O0 Fig. 1. Schematicrepresentation of a dummyfuel pin subassemblyemployedin heat diffusionexperiments with the coordinate system. and -= kJ/~. We seek the solution of eq. (1) in a series form such that
2. M a t h e m a t i c a l f o r m u l a t i o n
2.1. Basic equations
T(r, ~p, z) = m=0 ~__0Umn(r, ~O)l)m.n(Z),
With the assumption that the slug flow approximation is valid for the subchannel coolant flow such that u(r, q~) = Uo, the heat conduction equation for the coolant flow past the heated section of a dummy fuel pin may be expressed in the cylindrical coordinate system illustrated in fig. 1 as follows:
(3)
then, Um,.(r, ~p) a n d Vm,.(z) satisfy the following differential equations respectively.
1 0 {r 0 r Or 7r um,.(r,
~O)}+102 Um,n(r,~o) r 2 tgcp2
+ aZ,.,.u,.,.(r, ~0) = O,
(4a)
- ~-~z~ Vm'"(z)+ t'CPcU°°V"'"
OT c [ 1 0 f OT ~p,z)} pCpuo Tz (r, ~p, z) = K~[ r-~r [ r 7 r (r,
"~ ~Tl2nI)m,n(Z) =
+ 1 02T, -~ -~'~-f~2 t r, ~p, z )
0,
(4b)
where am..'s are eigenvalues to be determined later based on the boundary conditions.
+ "02T" ¢,z)], ~-~'z2 (r,
(1)
2.2. Boundary conditions
where/¢c is defined in terms of the molecular conductivity k¢ and the turbulent diffusivity of heat en of the coolant as
i¢~ = k¢ + pCpe.
(2b)
(2a)
The coolant temperature distribution T(r, ~o, z) is subject to the following boundary conditions: (1) In a subassembly of regular hexagonal
H. Hishida et al. / Evaluating turbulentdiffusivity of heat configuration consisting of dummy fuel pins, T(r, tp, z) should satisfy the azimuthally symmetric boundary condition such that
0 T(r, tp, z)l
-Olm,nJ6ra+l(am,nro) 3F 6 m J6m(Olmnro)
(5)
~p=0
(2) The amount of heat to be transferredlfrom the running fluid to the surface of a dummy fuel pin is negligible, that is
n(r). grad T(r, ~p, z)l r=r~ = 0,
= 0.
(7)
•
•
ro
am,,, is the nth eigenvalue corresponding to the mth characteristic equation which is derived from the boundary condition (7) as
{-Otm,nJ6m+l(Otm,nR) + ~ - J6m(OtmmR)} - Fm.,'{-am,~Y6m+l(am.nR) 6m } + ~ Y6m(Otm,nR) = 0 f o r m = 0 , 1 . . . . , (13) and c,~,," in expression (11) is an arbitrary constant corresponding to the m - nth eigenfunction. It is clear from eqs. (12) and (13) that
Fm,O= 0 and am,o= 0 for m = 0, 1, 2 . . . . .
(4) The coolant temperature distribution should be finite and uniform for z--> ~, that is
z -...>oo) = ~ C p + Tin,
(8)
Vm,,'(Z) = C'," exp(-flm.nZ),
(15)
where /3m.n is defined such that
T(r, ~p, 0) = Texp(r, q~, 0).
T(r, tp, z)= Co,o+ m~=on~__1 gm,n(r)
(9)
p C Uo 2 1/2
~ra,,"
=~a2m,~.(2k~.e) ) [ ~
_pCpu 2kc"
According to the boundary condition (5),
Um.~(r,~p) should be of the following form: (10)
where gm,n(r) is determined by substituting expression (10) into eq. (4a) such that
and rm.n is derived in terms of the boundary condition (6) with r = r~ as follows:
(11)
(16)
Then, the solution T(r, q~,z) which satisfies boundary conditions (5), (7), (8) and (6) with r = r~ may be written in the following form: oo
× exp(flm,~z)cos 6mq~,
2.3. Solution
gm,n(r) = Cm,n{J6m(am,nr)- rm,nY6m(Otm,nr)}
(14)
The solution of eq. (4b) which is finite for z --->o0is as follows:
where Q, w, Cp and Tin are the rate of thermal output of a heating section, the mass flow rate, the specific heat and the inlet temperature of coolant, respectively. (5) The coolant temperature distribution at z = 0 past the heating section of the center dummy pin should be equal to that obtained by experimental measurement Te~p(r,~, 0), that is,
;Arn,n(r,tp) = g,~.n(r) cos 6mtp,
(12)
(6)
where r* is the position vector of a point on the periphery of dummy fuel pin i at z and n(r*) is a unit vector normal to the periphery at r*. (3) At a sufficient distance R from the center dummy pin with a heating section, it may practically be assumed that
T(r,
.
- a , . nY6m*l(' m,'ro) + 6 m Y6m(am.nro)
=0.
r=R
ro
r,.,= =
=~6
OT (r, tp, z)[ Or
149
(17)
where arbitrary constants c ' ~ ' s appeared in solution (i5) are included in Cm,,'S of solution (11). Boundary conditions (6) for r = r* with i = 1, 2 . . . . . and (9) may approximately be satisfied by means of the point-matching technique.
2.4. Determination of Cra.n In order for T(r, tp, z) to satisfy approximately boundary condition (6) with r = r*,
H. Hishida et al. / Evaluating turbulent diffusivity of heat
150
the azimuthal angle 0 -< ~0 < ~'/6 is subdivided into j divisions. Let the position vectors of the intersections of q~ = 6oj and the periphery of dummy fuel pin i be r_~j and r+~j as illustrated in fig. 2, then the distance r_~4 - ]r_ d and r+~j = Ir+i,il, and the unit vectors n(r i,j) and tl(r+ij ) normal to the periphery of dummy fuel pin i at r-~j and r+~,; respectively are expressed as follows:
n~_,j = -x-U,f,(i, j ) - U~f~(i, j),
(18)
for i--- 1, 2 or 3, r_+~j= r0[2yi cos q~j -+ {1 - (23,/)2 sin e ~oj}l/2],
(19a)
f,(i, j) = {1 - (236)2 sin 2 q~j},/2
(20a)
and
directions, respectively. Similar expressions may easily be derived for r_.ij and n_~ij with i being greater than 6, however the radial temperature gradient of coolant usually diminishes rapidly as the radial distance from the center fuel pin increases and boundary condition (6) with i being greater than 6 may be disregarded in connection with boundary condition (7). In a case that a line q~ = q~j coincides with a tangent to the periphery of a dummy fuel pin, it should be clear that r+i,j = r-i,y. With expressions (18), (20) and (21), boundary condition (6) is to be satisfied pointwise on the periphery of each off-centered dummy fuel pin by the following equation:
n(r).gradT(r,q~,z)
L(i,j) = 2i3, sin q~j, for i = 4, 5 or 6,
[
(21a)
(°)
q~,
r_.ij = ro 2(i 2 - 5i + 7) 1/2 y cos - ~ - ~oj ,
/71"
\11/2"1
+_ 1-4(i2-5i+7),2sm2~-~-~o,) i 1, (19b) ]1/2
f,(i,j)={1-4(i2-5i+7)y2sin2(6-~oi}l (20b) and
5
2
=0.
(22)
(23)
then, eqs. (22) with respect to the intersections r._~,i are solved first for c-'(°)m,,with n = 2, 3, . . . , N and m being fixed. Here, N - 1 is the total number of intersections under consideration where each intersection is sequentially numbered by p(+-i, j) = 17 2 . . . . . N - 1. It should be clear from definition (23) that c~3 = 1. The resulting simultaneous algebraic equations of order N - 1 may be written in a matrix equation form such that
M(°)C (°) = B (°), I
I-_T_f//., OT Ll"l)-oTr (r'q~'z)
(21b)
where 3' is the pitch to diameter ratio of an assembly under consideration. Ur and U~ are the unit vectors to the radial and the azimuthal
i= 0
=t
The above eq. (22) contains arbitrary constants Cm,'S and due to the exponential terms (15), c,..,'s are to be determined for each sufficiently narrow subinterval of the axial distance z i.e. Zk <--Z < Zk+l with z0 = 0 so that eq. (22) may be satisfied for any value of z. Let c~). stand for cm,. within a subinterval Zk < Z < Zk+l and define 6~!. for m > 0 and n > 0 such that E(I,) _-_.,.
f~(i,j)=2(i2-5i+7)'/2ysin(6-q~,) ,
i=4
r~r~ij
3
Fig. 2. Definitions for vectors r+i4, n(r+ij) and fuel pin n u m b e r i.
(24)
where M (°) _ (M~°,),) is a square matrix of order (o} N-landC = ( C . (o) ) a n d B (o) = ( b p(o)) a r e column vectors, whose elements may be expressed as follows:
H. Hishida
et al. I E v a l u a t i n g
N
N
coo+ X" c(,) ~
M(O) p,n = [_{6~mr J6,n(Ctm..r)- Ctm.J6rn+l(ct,..r • " )
151
turbulent diffusivity of heat
•
m'~=0
m,1 n = l
C"(') m,n
exp(/3,.,.z0
× {J6,.(a,.,.r)- F,...Y6,.(am,.r)} cos 6rn~p. (28) -
r.•.
6m Y6m(otto,r) -5 -
+ F,.,.ot,..,Y6,.+l(am,.r)}f,(P)
COS6mq~
6m + ~ {./6,,(a,.,.r) r
- F,.,,Y6,,(el,,,,r)}f~(p)sin6mq~],
(25)
at the pth intersection and by ) = - M ~ t .
(26)
Matrix eq. (24) is solved for every integral value of m whose upper bound may usually be set equal to N. Once g~!.'s have all been obtained for 0 <~r~,~,o~ are to be determ <- N and 2 -< n -< N , x(0) mined based on the discretely measured temperature distribution on the wall of the center or the surrounding dummy fuel pins at z = 0 in the following way. Let Tcw(rq, goq,0) be the measured temperature at point q located on the wall of a dummy fuel pin whose position vector is given by (ra, ~q, 0), then from boundary condition (9) the following relation should be held:
T~p(rq, q~q,0) = T(")(rq, ~Oq,0) N
= c,,,, + E
m~(I
N
Z n=l
× {J~,, (m,.,r) - Fro,, Y6,, (am,.rq)} cos6mq~q for any q,
(27)
where T ~k) (r, q~, z) represents the coolant temperature distribution T(r, ~o, z) in a flow region z k ~ z < Zk+ 1. W i t h t e m p e r a t u r e data at N + 1 different points, eq. (27) may be reduced to a system of simultaneous algebraic equations for , N and ~m,J -c0) with m = 0, 1, ~o),~~ are found without difficulty. Next, we consider the coolant temperature distribution at z = zt =- Az with a sufficiently small Az. Then we shall have
T(°)(r, ~, zl) = T°)(r, q~, zt)
Since T(°)(rq, ~0q,z~) should be equal to the discretely measured temperature distribution Texp(rq, ~pq, z~), ~(0) related to the turbulent difusivity of heat by definitions (2a) and (2b) in the flow region 0 - z -< z ~ may numerically be determined by adjusting the value so that the relation Tt°)(rq, Cq, z~) = T~xp(rq, ~%, Zl) should be best satisfied. According to boundary condition (9) and the continuity requirement of z(k) T(k)(r, ~p, z) and T(k+l)(r, q~, Z) at z = Zk+,, ~m,. and C_-(k+l) m.tl should be related as follows: (k+t) m,n = 6tk) m.n exp [3,,.,(Zk
-
Zk+l)
for 0 <- m < N and 2 --- n < N
(29)
and c~,/may be obtained by equating T
3. N u m e r i c a l results a n d d i s c u s s i o n
Computer program F L O W M I X was completed based on the analytical procedures mentioned in the preceding sections and was applied to the evaluation of turbulent diffusivity associated with coolant flow in a FBR fuel subassembly. Experimental investigation on the coolant temperature distribution in a gridded 169 dummy fuel pin subassembly was performed using a water loop. The diameter of a dummy fuel pin was 6.5 mm arrayed with the pitch to diameter ratio of 1.2I. The upstream section being 250 mm long of the center fuel pin was electrically heated. The thermal output and the heat flux were maintained at 1.85 kW and 8.65 cal/cm 2 s, respectively. A honeycomb type grid spacer with the height of 15 mm was located just behind the heated section. Thermocouples were embedded on the surface of
H. Hishida et al. / Evaluating turbulent diffusivity of heat
152
D I S T A N C E FROM T H E GRID SPACER 1.5
.< 1.0
A
20mm
0
80mm
R e = 1.4 × 104
gridded subassembly may easily be evaluated as a function of distance from a grid spacer. Nomenclature
(r, ~p, Z)
T(r, ~o, z)
cylindrical coordinates where the axial direction of a fuel subassembly is parallel to the z-direction calculated coolant temperature at
To~,(r, ~,, z)
measured coolant temperature at
kc kc
inlet coolant temperature thermal conductivity of coolant effective thermal conductivity of coolant defined by eq. (2a)
z 0.5 .<
(r,~, z) (r, q~,z)
b-
Fig. 3. Temperature distribution on the wall of fuel pin 1 and 20 mm and 80 mm behind the grid spacer.
kc/ fcc P
c~
dummy fuel pins 0, 1, 2 and 4 at 20 mm, 50 mm, and 80 mm behind the grid spacer and by rotating these dummy fuel pins with thermocouples, coolant temperature field on the wall of dummy fuel pins was measured. Detailed data are shown in ref. [7], however for an illustrative example, periphera ! coolant temperature fields on dummy fuel pin 1 at 20 mm and 80 mm behind the grid spacer are shown in fig. 3 for Re = 1.4 x 104. Since eH of coolant past a grid spacer is strongly dependent upon the shape of a grid spacer, only specific conclusion is shown in this paper with respect to the grid spacer under consideration at the flow velocity of around 4 m/s which corresponds to the Reynolds number of 1.4 x 104. (1) Coolant mixing is most facilitated in the flow region about 50 mm behind the grid spacer. (2) For Re = 1.4 × 104, e , = 1.72 cmZ/s, which is approximately five times as large as that for a bare rod bundle at 50 mm behind the grid spacer. (3) Based on the analytical procedure mentioned in this paper with experimentally measured coolant temperature fields only on the wall of dummy fuel pins past a grid spacer, turbulent diffusivity of coolant flow within a
w Uo EH
O Otm,n
ro Y r* r~ij
n(r) V~ J~ Y~ i
J
density of coolant specific heat of coolant mass flow rate of coolant bulk velocity of coolant turbulent diffusivity of heat thermal output rate of the heated section nth eigenvalue corresponding to the ruth characteristic eq. (13) radius of a dummy fuel pin pitch to diameter ratio position vector of a point on the periphery of dummy fuel pin i position vector of intersections of ~o = q~j and the periphery of dummy fuel pin i illustrated in fig. 2 unit normal vector to the periphery at r unit vector to the radial direction unit vector to the azimuthal direction Bessel function of the first kind of order m Bessel function of the second kind of order m subscript indicating dummy fuel pin number subscript indicating the ]th subdivision of azimuthal angle (0, zr/6)
H. Hishida et al. / Evaluating turbulent diffusivity of heat
p(±i,j)
indicates identification number for the intersection of ¢ = ~p~ and the periphery of dummy fuel pin i superscript indicating the kth axial location
References [1] R. Nijsing, KfK 2232 (1975) 109.
[2] H. Hoffmann, KfK 1843 (1973). [3] H. Hoffmann, F. Hofmann and K. Rehme, IWGFR/29 (1978) 82. [4] T. Ginsberg and D.M. France, Internat. Seminar on Heat Transfer in Liquid Metals, Trogie, Yugoslavia, Sept. 1971. [5] R,A. Axford, Nucl. Engrg. Des. 6 (1967) 25. [6] H. Hoffmann and E. Baumg/irtner, IAEA-SM-173/IV (1973). [7] H. Hishida et al., PNC report SJ206 73-02 (1972).
153