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ORNL-4345.txt
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CENTRAL RESEARCH LIBRARY
DOCUMENT COLLECTION
2
.‘."\'”\ll' i
| |
R
0515
ORNL-4345
UC-70 — Waste Disposal and Processing
TEMPERATURE PROFILES WITHIN CYLINDERS
CONTAINING INTERNAL HEAT SOURCES AND
MATERIALS OF TEMPERATURE-DEPENDENT
THERMAL CONDUCTIVITIES. DESCRIPTION OF
FAST COMPUTER PROGRAMS AS APPLIED TO
SOLIDIFIED RADIOACTIVE WASTES
OAK RIDGE NATIONAL LABORATORY
operated by
UNION CARBIDE CORPORATION
for the
U.S. ATOMIC ENERGY COMMISSION
Printed in the United Stotes of America. Available from Clearinghouse for Federal
Scientific and Technical Information, National Bureau of Standards,
inia 22151
Printed Copy $3.00; Microfiche $0.65
U.S. Deportment of Commerce, Spri
Price:
LEGAL NOTICE
This report was prepared o5 an account of Government sponsored work. Neither the United States,
nor the Commission, nor any person acting on behalf of the Commission:
A. Mokes any warranty or representation, expressed or implied, with respect o the accuracy,
completeness, or usefulness of the information contained in this report, or that the use of
formation, apparatus, method, or process disclosed in report may not infri
privately owned rights; or
B. Assumes any liabi
any
= with respect fo the use of, or for damages resulting from the use of
formation, apparatus, method, or process disclosed in this report.
As used in the above, “‘person octing on behalf of the Commission'
includes any employee or
contractor of the Commission, or employee of such contractor, fo the extent that such employes
or contractor of the Commission, or employee of such contractor prepares, disseminate
provides access to, any information pursuant to his employment or contract with the Commi
or his employment with such contractor.
ORNL-4345
Contract No. W-7405-eng-26
- CHEMICAL TECHNOLOGY DIVISION
Chemical Development Section B
TEMPERATURE PROFILES WITHIN CYLINDERS CONTAINING INTERNAL HEAT
SOURCES AND MATERIALS OF TEMPERATURE-DEPENDENT THERMAL
CONDUCTIVITIES. DESCRIPTION OF FAST COMPUTER PROGRAMS
AS APPLIED TO SOLIDIFIED RADIOACTIVE WASTES
W. Davis, Jr.
JANUARY 1969
OAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee
operated by
UNION CARBIDE CORPORATION
for the
U. S. ATOMIC ENERGY COMMISSION
i W
e
iii
CONTENTS
Abstract . . . . . . L e e e
1. Introduction . . A
2. Methods of Solution . . . . .. .. .. L
3. Input Statements. . .. . ... .. ..
4. Execution Timesand Output ... ... ... ... ... .. ........
D, References . . . . v i i s e e e e e e e e e e e e e e e e e e e e e e e e
TEMPERATURE PROFILES WITHIN CYLINDERS CONTAINING INTERNAL HEAT
SOURCES AND MATERIALS OF TEMPERATURE-DEPENDENT THERMAL
CONDUCTIVITIES. DESCRIPTION OF FAST COMPUTER PROGRAMS
AS APPLIED TO SOLIDIFIED RADIOACTIVE WASTES
W. Davis, Jr.
ABSTRACT
The safety and economic aspects of producing and storing radio-
active wastes as solids, usually in vessels having a cylindrical geometry,
require that we know the maximum internal temperatures to be expected
as a result of decay heat. Such information is mandatory for materials
that have variable thermal conductivities under different storage con-
ditions. '
This report presents a computer program (STORE), which was written
to permit more rapid calculation of temperature profites within cylinders
containing homogeneously distributed heat sources and materials whose
thermal conductivities can be expressed as a tabular function of tem-
perature. A simplified version of this program was prepared for the cases
in which the thermal conductivity is constant or is a linear function of
temperature. Both of these programs have short execution times, typi-
cally from a few to 20 seconds on the IBM/360-75 as compared with the
five or more minutes required for the more accurate finite-difference
method. They are based on the assumption that the material density and,
therefore, the power density of the heat source are independent of tem-
perature; this assumption is, of course, contrary to physical reality.
However, in a test example involving a hypothetical vessel of glass con-
taining a large internal (fission product) heat source with a specific
power density of 0.2 cal sec™! cm™3 (i.e., 80,910 Btu hr~! £+73), the
temperature difference between the wall and the center of a 6-in.-diam
by 6.25-ft-long cylinder was overestimated by 36°C (an error of only
about 10%, as compared with the "exact"” value that is obtained by
solving finite-difference equations and compensating for the reduction
in the heat-source strength as the density decreases with increasing
temperature). Within the uncertainties inherent in thermal conductivity,
density, and heat capacity measurements of systems of interest in the
storage of solidified radioactive process wastes, the method and the
program presented in this report offer adequate accuracy and a large
time savings as compared with the more exact calculations. Also, they
are applicable to cylinders with any specified length/diameter ratio.
1. INTRODUCTION
On the basis of safety and economic studies, it appears rather probable that
high-]-4 and ini‘ermedicfe-level5 radioactive wastes which accumulate from the
processing of nuclear reactor fuels will be converted to solids for permanent storage
at some time — from 30 days to 30 years — after removal of the fuel from the reactor.
Essentially all of the beta energy, and more than half of the gamma energy, from the
fission products will usually be absorbed by the solid within which they are contained;
therefore, the temperature in the interior will be raised to a level higher than that of
the surface. The extent of this temperature elevation depends on the thermal conduc-
tivity of the solid, the diameter of the storage vessel (which is usually considered to
be a right circular cylinder), and the specific fission product power density. In cal-
culations it is assumed that the radioactive materials are distributed isotropically
throughout the cylinder; then, atlsfecdy state, the temperature, T{r, z), at any point
(r, z) is given by the equation
"
l_a_.!.Kré.I.‘+iiK-al +A=0, (n
rarL er azL z
where
K = temperafure-dependent thermal conductivity, cal cm-] sec °C ,
T = temperature, °C,
r = radial variable, cm,
z = vertical variable, cm,
A = sum of (fission product) power densities (i.e., absorbed power density),
-1 -3
cal sec cm
The quantity A is actually a function of temperature; that is, it decreases as
the temperature of the solid increases because the material specific volume increases
(the density decreases). However, the definition of A (above) points out that it is
the absorbed energy that is important. The fraction of the total gamma-ray energy
that is absorbed is a function of the dimensions of the cylinder.
3
Because of the temperature dependence of A, an "exact" solution of Eq. (1)
can be obtained only by use of finite-difference methods. However, in practical
cases the thermal conductivity is affected much more strongly by temperature than
the density is. For example, the effect of temperature on the thermal conductivity
6 °C-], while the coef-
of materials of interest is on the order of (50 to 200) x 10~
ficient of volume expansior. is in the range (1 to 5) x 10_6 °C-I. Thus, to a first
approximation, the quantity A may be assumed to be independent of temperature.
This report presents a computer program (STORE) which was written to provide
approximate solutions of Eq. (1) in terms of temperature as a function of spatial
. location within a cylinder of arbitrary length/diameter ratio. Such a program can
be very useful for evaluating the advantages, with respect to safety and economics,
of storirig radicactive waste materials because it can be executed much faster than
a program based on the more exact finite-difference method.
Additional reports, which are now being written, will illustrate the application
of this program, and the simpler programs.derived from it, to the calculation of
internal temperatures in cylinders containing intermediate-level waste solids that
are dispersed in an organic mairix (such as asphalt, polyethylene, or other plastics)
or high-level waste, existing as calcine or as solids that are dissolved, or dispersed,
in an inorganic matrix (such as a glass or a microcrystalline solid).
2. METHODS OF SOLUTION
It is convenient to express Eq. (1) in terms of a dimensionless temperature, v,
defined as
v = (T - TO)/TO 7 (2)
where To is a convenient reference temperature. In this report, To was chosen as
temperature of the surface of the cylinder. By combining Egs. (1) and (2), we
obtain
a3l af avi, AL
}--a-r—‘:Krsl-_-j""g-z-[K'—z'}"‘T;—o, (3)
with the boundary conditions
(4)
v=0atz=0hfor0=<r=a
v=0atr=afor0<z<h
where
a = radius of the cylinder, cm,
h = length of the cylinder, cm.
Using the notation of Carslaw and Joeger,7 we define
\4
S
O ) =g | Kdv, (5)
| o O
from which it follows that
&R W % Kg (®
On substituting the quantities of Eq. (6) into Eq. (3), we obtain
1 af @] 3fe@). A _ |
F"a'r_{r'TJ‘JrE\:'EE]JFKOTO'O' | 7
Here, Ko i< the thermal conductivity of the material at temperature T.- The boundary
conditions for ® are as follows:
®=0atz=0hfor0<r=a
~ (8)
®=0atr=afor 0£z<h
Further, it is convenient to define
x = r/a, Zéz/o, L=h/a. (9)
With these definitions, the solution of Eq. (7) is given by7
Z(L - 2Z) 442 - I (@m - 1) mx/t] sin[(2m - )T Z/1]
2 1_r3
U(x,Z) = , (10
ol (2m - 12T [(2m - 1) /2]
and
a2 |
®=Aa U/KOTO. (11)
After having calculated @, it is a straightforward process to calculate T. In
particular we have, from Egs. (2) and (5), |
T
LLLE (12)
T
o
1
KoTo
o
0=
For the case in which thermal conductivity can be expressed as a tabular function
of temperature, we first determine @ from Egs. (10) and (1 i) and then evaluate v and T
from Egs. (5) and (12). To do this, we must perform the numerical integration indicated
by Eq. (5) in order to calculate a fable of ® as a function of v. We then obtain v by
quadratic interpolation in this table with the known value of ®. Calculations are
performed by program STORE.
There are two special cases of considerable importance in radioactive waste
storage: the case in which thermal conductivity is a linear function of temperature
and the case in which the thermal conductivity is constant.
When thermal conductivity is a linear function of temperature, we require two
thermal conductivities at two temperatures, (T], K‘) and (T2, K2), from which we
obtain
K=Ko[1+b(T-To)] , (13)
. Ky Ty =T - Ky (T, -T)
o (T2 - T\]) .
Ky=-K
b = 2 ]
KO (T2 - T])
] v v bTov2 |
®=-K-o- JoKdv= Jo (l+bTov)dv=v+ 5 (14)
In this case, the value of ®is knownand vand T - T0 can be calculated from the
relationships
v=[/T+26T,8 - 11/bT, (15)
T-T=L/1T25T;a-1]/b. (16)
o
By replacing v by (T - To)/To and using the definition of K as a linear function of
temperature to eliminate b from Eq. (14), we obtain
- K+ K _
alx, 7) = (T To) o} _KAT (17)
! T K 2 TK '
o o co o
where the average thermal conductivity, K, at the two temperatures is
_ K+ Ko
K= 5 (18)
and we have defined
AT=T-T_. (19)
On replacing ®(x, Z) on the left-hand side of Eq. (17) by its value from Eq. (11),
we obtain
A= . | (20)
Equation (20) is quite similar to that for thermal diffusion in an infinite cylinder,
namely,
(21)
This equation may be derived in a manner analogous to that described, for example,
by McAdcms,S except that K would be a linear function of temperature rather than
a constant. Equation (21), while strictly applicable only to an infinite cylinder, is
frequently used and yiélds a rather accurate estimate of the specific power, A, as a
function of AT, providing that the cylinder is long (i.e., a cylinder with a length/
diameter ratio in excess of about 3 or 4).
Equations (20) and (21) imply that for an infinitely long cylinder we have
=1/4. (22)
The quantity U, calculated in program STORE, has values of 0.201, 0.245, 0.249,
and 0.250 for cylinders of Iengfh/dicme’rer-rcfios of 1, 2, 2.5, and 5, respectively.
Thus, as expected, Eq. (21) overestimates the AT for a given value of A, but to a
significant extent only if the length/diameter ratio is less than 2. In connection
with the mathematical reduction of Eq. (20) to Eq. (21), we note that, at the center
of a cylinder, namely at x =0 and Z = 4/2, Eq. (10) reduces to
N
L 32 sin L(2m - 1) 1/2] -
u(o, 4/2) = —8-1 -5 2 5 1 . (23)
w f=l (2m - DT [(2m - 1) /] ]
It can be shown that
Lim i sin LZm - 1 /2] Lg—z [1 --2—‘£] . (24)
m=1 (2m - 1) Io [(2m - 1) flc/h]J
£ — @
(a/h— =)
Thus, for a/h << 1, we would obtain Eq. (21) from Eq. (20).
The simplest calculations involve a constant thermal conductivity. In this
case, we solve for (T - To) by the procedure
T-T
o
T
o
o 2
=@=Aa U/KoTo : (25)
3. INPUT STATEMENTS
While program STORE properly accounts for the variation of thermal conductivity
with temperature, it does not describe the variation of A [see Eq. (1)] with temperature.
The power density decreases as the specific volume increases, that is, as the density
of the material decreases. Use of this program (or programs for the simpler relation-
ships between thermal condu;:fivi’ry and temperature) is advantageous because the -
execution time is significantly shorter than that of a program based on the more ac-
curate finite-difference equations.
The first three READ statements are:
READ 9001, NR, NZ, NOM
READ 9011, (RT), I=1, NR)
READ 9011, (Z(J), J=1, NZ)
9001 FORMAT (1615)
9011 FORMAT (16 F5.3)
Here
NR is the number of relative-radius units at which output data are to be printed.
The program is set for NR = 17 and for the values described below.
NZ is the number of vertical units at which output data are to be printed. A
reasonable number is in the range 20 to 45.
NOM is the number of terms to be used, instead of an infinite number, in evaluating
U(x, 2), Eq. (10). We have used as many as 100 and as few as 5; in general,
a value in the range 10 to 20 will be adequate.
R() are the NR radial positions at which data are printed. Output formats
carry headings of 0.0 (corresponding to the centerline of the cylinder),
0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.60, 0.7G,
0.80, 0.90, 0.95, and 1.0. Thus, the output tables will list temperatures,
and other values, for the centerline and for distances of 5, 10, 15, 20,
25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 95, and 100% of the distance from
the centerline to the wall. If these 17 values are not used, then certain
format statements will have to be changed.
Z(J) are the distances, in radius units, from the top or the bottom of the cylinder.
Convenient values are 0.0, 0.005, 0.01, 0.02, etc., corresponding to locations
on the top (or bottom) surface followed by distances of 0.005 radius, 0.01
radius, etc., into the cylinder from the top (or bottom). The last value of
Z(J)), namely Z(NZ), is used as the cylinder height/radius ratio. Thus,
if NZ = 25 and Z(25) = 25., then the length of the cylinder is 25 times that
of the radius.
Most of the values of Z(J) should correspond to relative heights of less
than one-half the cylinder height. About the fourth from the last value of
Z(J) should be the half-cylinder height [i.e., 12.5 in the above example .
The sub#equent two values should be on the order of 0.9 times Z(NZ) and
0.95 times Z(NZ), respectively. These values are used to check for passage
beyond the midplane of the cylinder; their judicial choice minimizes un-
necessary duplication in the calculation of temperatures beyond the mid-
plane. In the example above, it would be appropriate to use 30 or even
40 horizontal spacings for Z(J) < 12.5 followed by the four vcflues 12.5,
22.5, 24.0, and 25.0. |
it should be noted that the above READ statements are entered only once, re-
gcrdléss of how many calculational cases are to be performed. Thus, whether the
radius of the cylinder is subsequently set to 0.1, 5.0, 25.0, or 100.0 cm, all the
calculations will be performed at the centerline and, for example, 5%, 10%, efc., |
out toward the surface.
10
Program STORE uses thermal conductivity as a tabular function of temperature.
This table, which is read only once and is used for all cases to be calculated, is
entered as follows:
READ 9021, TP(NT), TK(NT)
9021 FORMAT (8E10.4)
One t‘emperafure, TP(NT), and its thermal conductivity, TK{NT), are punched on
a card, at 50°C intervals as the program is now written. DIMENSION allows a
maximum of 99 such pairs of values. At the end of the table, any negative floating-
point number, such as - 1.0, is entered in the TP(NT) field (columns 1 - 10) to
indicate the end of the table. The temperature interval 50.0°C was chosen because
it is cppropriofe’i& materials heated to temperatures in the range 0.0 to 3000°C or
higher. This interval is specifically involved in the integration shown in Eq. (5) for
evaluating @ as a function of v. Since it occurs in only a few statements, the program
could be modified fairly easily.
In addition to the READ statements already described, there is one READ state-
ment required for each case to be calculated, namely,
READ 2021, A, RAD, TINT, TO
Here
A is the internal (absorbed) power density in cal sec_] cm-3, as in Eq. (1).
Values in the range 0.1 to 0.8 are significant in connection with high-
level radioactive wastes from nuclear fuel processing.
RAD is the cylinder radius, cm.
TINT is the temperature interval at which temperature profiles are desired. For
example, if the temperature difference (T - To) is expected to be on the
order of 100°C, then TINT might be assumed to be 5. or 10., corresponding
to the radial and vertical locations of temperatures spaced 5 or 10°C
apart.
11
TO is the surface temperature, °C or °K, whichever is convenient.
TKO is the thermal conductivity of the material at To’ cal cm-] sec °C
4. EXECUTION TIMES AND OUTPUT
Each of the three programs (see Appendix A), namely, STORE, TKLIN (when K
is a linear function of T), and TKCON (when K is constant), has been executed many
times on the [BM/360-75 at ORNL to calculate temperature profiles in cylinders
containing high- or intermediate-level radicactive wastes generated by the processing
of nuclear fuels. Typically, the programs require 30 to 40 sec for compilation. De-
pending primarily on the number of temperatures at which profile data are desired,
TKCON and TKLIN require 1 to 10 sec per case. Program STORE has a longer
execution time, again depending (but less strongly) on the. number of temperatures
at which profile values are required; it generally requires 5 to 30 sec per case. For
comparison, a more accurate evaluation of Eq. (1), based on the use of finite-difference
equations, a grid of 30 radial divisions, 120 vertical divisions, and the near-minimum
of 500 iterations, requires on the order of 5 min with FORTRAN 1V, level H execution
under optimum timing. The time savings of TKCON, TKLIN, and STORE are thus
rather significant, while the loss of accuracy is not very great, as described below.
Output from STORE includes a table (Table 1) of NZ times NR values of U
[Eq. (10)], a table (Table 2) of temperatures above the surface temperature, (T - To),
and a table of ® values. Output also includes a table (Table 3) of distances, in
radius units, measured from the bottom or the top of the cylinder, where heating above
the surface temperature occurs by multiples of the quantity TINT. Finally, each
program outputs a table of the number of iterations required to obtain each multiple
of TINT at each radial location. The maximum number of such iterations in the
studies reported here was 20; usually, however, this number did not exceed 2. In
addition to the tables just mentioned, program STORE outputs a table of values for
T, K, ® XTA, XTB, and XTC [where T, K, and ® have been defined previously and
XTA, XTB, and XTC are the constants used to represent T as a quadratic function
HETGHT
(Z/A)
0.0
0.00%
0.010
0.015
0,020
0,025
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.120
0.140
0.160
0.180
0.200
0.250
0.300
0.350
0,400
‘0.450
0.500
0550
0.600
0.700
0.800
0.900
1.000
1.250
1.500
1.750
2.000
2.250
2.500
3.000
4.000
5,000
Table 1. Example of Output of the Dimensionless Quantity U [Eq. (10)] ot Grid Points
THC DIMENSIONLESS UlIesJ) FROM WHICH WE CALCULATE THETA(I.J)e VI(Is+J)+ AND
THE BESSEL FUNCTION SUMMATION.
VALUES OF THE RELATIVE DISTANCE {(R/A) FRUM THE CYLINDER AXIS ARE GIVEN
0.00
50
TERMS ARE USED IN
0.05 .
0.0
D+00}
0.205
0.N038
0.010
0.213
Q.016
0.021
0.025
0.030
0.035
0240
0.0644
0,049
0.0b57
0.066
Q074
0.081
0.089
0.1006
D121
0.135
0.147
0.15%
0.168
0.177
0.185
0.199
0.209
0.21R
0.225
0.236
D.242
04245
0.2647
0.248
J.2438
Ne247
D225
0.0
0.10
0.0
0.0U03"°
0.005
0,008
0.010
0.013
0.014
0.020
0.025
0.030
0.035
0.039
0.044
0.048
0.057
0.065
0.073
ND.081
0.088
0.135
0.120
0.134
0.146
0.158
0.167
0.176
0.184
N.197
0.208
0.216
0.223
0.234
0.240
0.243
0.245
0.246
0.246
0.245
0.223
0.0
0.15
0.0
0.003
0.20%
0.008
0.010
0.013
.0.015
0.020
0.025
0.030
0.035
0.039
0.044%
0.048
0.056
0.065
0.072
0.080
0,087
0.104
0.119
0.133
0.145
0.156
0.166
0.174
0.182
0.195
0.206
0.214
0.220
0.231
0.237
0.240
0.242
0.243
0.243
0.242
0.220
0.0
0.20
0.0
0.003
0.00%
0.008
0.010
0.013
0.015
0.020
0.025
0.030
0.034
0.039
04043
0.047
0.056
0.064%
0,072
0.079
0.086
L0.103
0.118
0.131
0.143
0.153
0.163
0.172
0.179
0.192
0.202
0.210
0.217
0.227
0.233
0.236
0.238
0.238
0.239
0.238
0.217
0.0
0.25
o.o
0.003
0.G05
0.008
0.010
0.013
0.015
0.020
0.02%
0.029
0.034
0.038
0.042
0.047
0.055
0.063
0.070
0.078
0.085
0.101
0.115
0.128
0.140
0.150
0.160
0.168
0.175
0.189
0.198
04206
0.212
0.222
0.227
0.231
0.232
0,233
0.233
0.232
0.212
0.0
0.30
0.C
0.003
0.005
0.007
0.010
0.012
0.015
0.919
0.02%
0.029
0.033
0.037
0.0642
0.046
0.054
0.061
0.069
0.076
0.083
0.099
0.113
0.125
0.137
0.147
0.156
0.164
0.171
0.183
0.192
0.200
0.206
0.216
0.221
0.224
0.225
0,226
0.226
0.225
0.206
0.0
0.35
0.0
0.002
0.005
0.007
0.010
0.012
0.014
0.019
0.023
0.028
0.032
0.036
0.040
0.045
0.052
0.060
0.067
0.074
0.081
0.096
0.109
0.122
0.133
0.142
0.151
0.159
0.165
0.177
0.186
0.193
0.199
0.208
0.213
0.216
0.217
0.218
0.218
0.217
0.199
0.0
0.40
0.0
0.002
0.005
0.007
0.009
0.012
0.014
0.018
0.023
0.027
0.031
0.035
0.039
0.043
0.051
0.058
0.065
0.072
0.078
0.093
0.106
0.117
0.128
0.137
0.145
0.153
0.159
0.170
0.178
0.185
0.190
0.199
0.204
0.207
0.208
0.209
0.209
0.208
0.190
0.0
0.45
0.0
0.002
0.005
0.007
0.009
0,011
0.013
0.018
0.022
0.026
0.030
0.034
0.038
0.042
0.049
0.056
0.063
0.069
D0.0T75
0.089
0.101
0.112
0.122
0.131
0.139
0.146
0.152
0.162
0.170
0.176
0.181
0.189
0.19%
0.196
0.198
0.198
0.198
0.198
D.181
0.0