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
IN THE
0.50
0.0
0.002
0.004
0.007
0.009
0.011
c.013
0.017
0.021
0.025
0.029
0.033
0.036
0.040
0.047
0.053
0.060
0.066
0.072
0.085
0.096
0.107
0.116
0.124
0.131
0.138
0.143
0.153
0.160
0.166
0.171
0.178
0.182
0.185
0.186
0.186
0.187
00111'
0.0
T(14J).
FIRST R
0.60
0.0
0.002
0,004
0.006
0.008
0.010
0.012
0.015
0.019
0.022
0.026
0.029
0.032
0.036
0.042
0.048
0.053
0.058
0.063
0.075
0.085
0.093
0.101
0.108
O.114
0.119
0.124
0.132
0.138
0.143
0.146
0.153
0.156
0.158
0.159
0.159
0.159
0.159
0.146
0.0
OW OF
0.70
0.0
0.002
0.003
0,005
0.007
0.008
0.010
0.013
0.016
0.019
0.022
0.025
0.028
0,035
0.040
0.045
0. 049
0.053
0.062
0.070
0.077
0.083
0.088
0.093
0.097
0.100
0.106
0.111
O.114
0.117
0.122
0.124
0.126
0.126
0.127
0,127
0.126
0.117
0.0
THIS TABLE
0.80 0.90
0.0 0.0
0.001 0.001
0.003 0.002
0.004 0,003
0.005 0.003
0.007 0.004
0.008 0,005
0.010 0.006
0.013 0.008
0.015 0.009
0.017 0.010
0.019 0.011
0.021 0.013
0.023 0,014
0.027 0.016
0.030 0.018
0.034 0.019
0.037 0.021
0.039 0,022
0.046 0.026
0.051 0.028
0.056 0.031
0.060 04033
0.06&4 0,035
0.067 0.036
0.070 0.038
0.072 0.039
0.076 0.041
0.079 0,042
0.081 0,043
0,083 0.044
0.086 0.046
0.088 0.067
0.089 0.047
0.089 0.047
0.090 0.047
0.090 0,047
0.089 0,047
0.083 0.044
0.0 0.0
0.95
0.0
0.301
0.001
0,002
0.002
0.002
0.003
0,004
0.004
0.005
0.006
0.006
0.007
0.008
0.009
0.010
0.010
0.011
0.012
0.01%
0.015
0.016
0.017
0.018
0,019
0.019
0.020
0.021
0.022
0.022
0.023
N.023
0.024
0.024
0.024
0.024
0.02%
0.024%
0.023
0.0
¢l
HELIGNT
IN
RADIUS
UNITS
{z/4)
0.0
V.05
D.010
0.01%
0.020
0.025
Q0.830
Qe V&1
0.050
0,000
0.070
0,030
0.0939
0.1a0
D.120
0.140
0,160
0.1390
0.2N0
Ne2 5D
0,300
0.350
0.400
0.4%0
0.500
0550
Ne600
0.700
G.800
0.900
l.100
l.250
l.bOU
1. 750
24000
24250
2.500
3.0
4. G1))
SC.OL"\)
Table 2. Example of Output of Temperatures Above the Surface Temperature at Grid Points
0.35
TEMPERATULS ABOVE THE SURFACE TEMPERATURE. THE CYLINDER ODIAMETER IS
INTERNAL HFAT SUURCE IS 2.00E-01 CALORIES/CM&®3/SEC,
30910.6 BTU/FT*¢3/HR
15.2 CM.
B8.37E-01 WATTS/CMEe]
AND L .41E 02 MEVSLURIES/CM**3,
Ja45
TAE STRENGTH 3F THE
SOURCE STAENGTH IN UTHER UNITS 1S
VHE TAEIMAL CONDUCTIVITY COEFFICIENT AT THE SURFACE TEMPERATURE OF
VALUES OF THFE RclLATLIVE DESTANCE (R/ZA) FROM
0«N0 Ve G.10 ° 5,15 0.20 0?25 0.32
0.0 0,0 0.0 0.0 0.0 0.0 0.0
5.4 bk Sel 5.3 5.3 542 Sel
1.9 107 10.7 10.6 10.5 10.3 10.1
15.0 1640 15.9 15.3 15.6 15.4 15.1
21.2 21.1 21.0 20.9 20.6 20.3 20.0
263 2042 26.1 25.9 25.b6 25.2 24.8
3l.3 3147 Al.1 30.9 30.5 30.0 29.5
41.1 41 0 40.8 “0.5 40-0 39.4 38.7
505 50 .5 5042 49,43 49,3 48,5 47.6
$9.7 5948 5943 58,8 58.2 57.3 She?
b8eb 63453 HHel 67.6 6648 65.R 64.6
T1.2 7.1 76.7 16.1 75.2 14.1 72.7
85.06 5.4 35,0 94,3 83.4 82.1 80.5
9347 93,5 3.1 2.3 9l.3 R9 .9 88.2
109.2 1049.9 193.5 107.6 106.4 104.T 102.7
123.8 123.6 123.0 122.0 120.6 118.7 116.4
137.7 137.4 136.3 135.6 134.,0 132.0 129.4
I50,3 150.95 14%.8 143.5% 146.8 144.5 1l4l.6
163.1 162.3 162.0 160.7 158.7 156.3 15%3,2
1912 1972.7 149.9 1A8.3 186.0 183.0 179.3
2[5-8 215.4 21“03 212.“ 209.8 206.4 20202
237.4 236.,9 235.7 233.6 230.7 226.8 222.1
2563 255.9 2%4.5 252.2 249.0 244.8 239.5
2729 2724 211.0 268.5 265.0 260.5 254.8
287.6 28T.1 235.5 282.9 279.1 274.3 263.3
300.6 3I00.0 2798.4 295.6 291.6 2B6.5 280.1
11240 3lle% 309.6 396,T 3092,6 297.2 290.5
331.0 330.3 328.4 325.3 320.8 315.0 307.8
345.3 345.1 333,1 339,7 335.0 328.8 321.2
35744 356.7 354.6 351.1 346.1 339.7 331.7
3665 365.7 363.6 359,9 3534,8 348.1 339.9
361.4 380,77 37B.4 3I74.5 369.1 362.1 353,4
L347.6 38:.8 336,44 382.4 376.9 3I69.6 360.7
3194,0 393,2 330.8 586.T 381.1 373.7 364.7
196.3% 395,.,5 373,101 399.0 393,33 375.9 366.8
3974 39646 394.2 390.1 334.4 376.9 367.9
3974 396.9 394.5 390.4 334,77 377.2 368.1
396.3 395.5 393,01 389.0 343.3 375.9 366.8
366e5 3I65.T 353.6 357.9 354.8 348.1 339.9
0.0 ) 0.0 0.0 0.0 0.0 0.0
0.50
200.
0.60
0.70
0.80
THE CYLINDER AXIS ARE GIVEN IN THE FIRST ROW OF THIS TABLE
0.90
18.3
0.95
QLN VM ANHWNE=O
e & a2 & 9 & 4 & &
DL OVN DO e~
13.0
DEGREES C IS 5.63E-03 CAL/LM/SEC/DEG C.
1.03
4 5 & & 8 & 8 8 8 0 ¥
ODO0OCOO0OR20000RO0COOROODROO002000000D0QR0000
[=N=RoReleNoleNole oo oo NololeleloNoloNelo oo Yo oo No oo ool e R Re No o NNl
e 4 & & 8 B ¢ 2 & & & 8 B 2 B
£l
TEMP
DES Co
50.
100,
1504
20
250,
300 L]
350,
DISTANCL Sy
Table 3. Example of Temperature-Profile Table
IN RANIUS UNITS,
MEASURED FROM THE
BOTTOM
OR TOP OF THE CYLINDER,
THE SURFAZE TEMPERATURE BY AMCUNTS SHOWN IN THE F1RST COLUMN.
VALUES OF Tuc
Je00
Q.049
J.108
VeLl79
0.38‘5
Jaebad
D.933
0.05
0050
Jel03
O0.179
0.268
043834
Ue95)0
D.439
RELATIVE D1STANCE (R/A) FRUM
0.10
V.050
0.109
0.130
0.270
0.333
0.557
0.857
J.15
D.05D
0.110
0. 182
0.273
0.394
04569
0.889
020
0.051
O.lL1
0.185
0279
0.403
0.588
0.942
0.25
0.052
0.113
0.189
0.286
O.216
0.614
1.026
THE CYLINDER AXIS ARE GIVEN IN
0.30
0.053
D.ll16
0. 194
0.29%
0.434
0.652
1.171
0.35
0.054
9.120
0,201
0+ 307
0.457
0.705
l.494
0.40
0.056
0.124
0.210
0.323
0.489
D.78T
Ja65
7.058
J.129
0.220
D345
Te530
0.922
0.50
0.060
0.136
J.235
0.374
0.600
1.211
0.60
0.068
0.157
0.281
0.4T7
0.921
THE FIRST ROW OF
0.70
0,081
0.197
0,384
0.B812
THIS TABLE
0.80 0.90
0.111 0.242
0.308
0.877
AT WHICH OCCUR HEATING ABDVE
14!
15
of @ after Eq. (5) is integrated to evaluate ® as a function of T and K].
In one test example, the thermal conductivity of a hypothetical glass
was assumed to be a linear function of temperature up to 1000°C (see Table 4).
Other properties were as follows: power density, 0.2 cal sec-] cm-3; radius of
the vessel, 7.6 ¢cm (6 in. in diameter); and length/radius ratio, 25. Program TKLIN
gave a maximum centerline temperature of 599.5°C for this example when the surface
temperature was maintained at 200°C; execution time was 5.0 sec. For comparison,
the maximum centerline temperature was also calculated by using a program based
on finite-difference equations; in this program, the aensify-VS-temperqture properties
(Table 4) were those of an alkali borate glcss.9 As the temperature increased from
200 to 550°C, the density decreased and the specific power density decreased by
about 3.6%. With the latter program, the maximum centerline temperature was
found to be 564.0°C after either 1500 or 2000 iterations. After 500 and 1000 it-
erofiofis the maximum calculated temperatures were 557.4°C and 563.9°C, respectively.
Each 500 iterations consumed approximately 5 min of execution fime on the 1BM/360-75.
As expected, TKLIN overestimates the temperature rise because it does not
compensate for the reduction in fission product power density as the temperature
increases. In the example described above the error is about 36°C-out of a "true"
temperature elevation of 364°C (i.e., a 10% overestimation of the temperature rise).
This error is small enough to be neglected in most instances, where the existence of
only very crude specific heat data plus uncertainties in densities and thermal con-
ductivities easily lead to larger errors.
16
Table 4. Physical and Thermal Properties of Hypothetical Glass Used to Obtain
Estimates of Errors in Program TKLIN Due to Decrease in Density with
| Increasing Temperature
Thermal
Conductivity . Specific Heat Density
Temp ;' cal '" { cal / y
(°C) \cmesecs®°C / ‘_g—-?él \'ems /
50 0.00443 0.2 - ©2.313
100 0.00483 0.2 ~ 2.305
150 0.00523 0.2 2.296
200 0.00563 0.2 2.288
250 0.00603 0.2 2.281
300 0.00643 0.2 2.275
350 0.00683 0.2 - 2.264
400 . 0.00723 0.2 2.253
450 0.00763 0.2 2.239
500 . 0.00803 0.2 2.225
550 0.00843 0.2 2.206
600 0.00883 0.2 2.188
650 0.00923 0.2 2.167
700 0.00963 0.2 2.146
750 0.01003 0.2 2.120
800 | 0.01043 02 2.095
850 0.01083 02 . 2.068
900 0.01123 0.2 2.042
950 " 0.01164 0.2 2.017
1000 0.01207 0.2 1.992
1050 ~0.01252 0.2 | 1.966
1100 0.01299 0.2 1.940
1150 0.01348 0.2 1.914
1200 0.01399 0.2 1.888
1250 0.01452 0.2 . | 1.862
1300 0.01507 0.2 1.836
17
5. REFERENCES
R. L. Bradshaw, J. J. Perong, J. O. Blomeke, and W. J. Boegly, Jr., Evaluation of
Ultimate Disposal Methods for Liquid and Solid Radicactive Wastes. VI. Disposal
of Solid Wastes in Salt Formations, ORNL=3358 (May 1968).
J. O. Blomeke, R. Salmon, J. T. Roberts, R. L. Bradshaw, and J. J. Perona, "Estimated
Costs of High-Level Waste Management," pp. 830-843 in Proceedings of the
Symposiumon the Solidification and Long-Term Storage of Highly Radioactive
Wastes, February 14-18, 1966, Richland, Washington, USAEC, DTIE (CONF-6460208).
M. N. Elliot, R. Gayler, J. R. Grover, and W. H. Hardwick, "Fixation of Radio-
active Wastes in Glass. Part |. Pilot Plant Experience at Harwell," pp. 465-487
in Proceedings of the Symposium on Treatment and Storage of High-Level Radio-
active Wastes, 8-12 October 1962, Vienna, International Atomic Energy Agency.
W. E. Clark, J. C. Suddath, et al., Development of Processes for Solidification of
High-Level Radicactive Waste: Summary for Pot Calcination and Rising Level
Potglass Processes, ORNL-TM-1584 (Aug. 12, 1966)..
H. W. Godbee, R. E. Blanco, et al., Laboratory Development of a Process for
Incorporation of Radiocactive Waste Solutions and Slurries in Emulsified Asphalt,
ORNL-4003 (July 1967).
W. R. Dixon, "Self-Absorption Correction for Large Gamma=-Ray Sources, "
Nucleonics 8, 68 (1951).
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2d ed., Clarendon
Press, Oxford, 1959; see, particularly, p. 10 and problem XVII on p. 223.
W. H. McAdams, Heat Transmission, 3d ed., pp.18, 19, McGrow-H.iII, New York,
1954,
. G.W. Morey, The Properties of Glass, 2d ed., Reinhold, New York, 1954;
see data for 10.6 wt % Li20/26.4 wt % N020/34.4 wt % KoO.
18
6. APPENDIX
This appendix contains listings of program STORE, subroutine THERMY, and the
two functions XINT and BIZERO. The subroutine is used to calculate the coefficients
Ai' Bi' and Ci' which will permit the determination of a dependent variable Y when
the value of the independent variable X is known. Thus,
) ,
Y=A B.X + C.X" . (A-1)
For example, by use of Eq. (12) we obtain @ as a function of T. Since we wish to
determine T as a function of 8 we use Eq. (A-1) to solve for the coefficients in the
equation
T= Ai + Bi®+ci®2 .
FUNCTION XINT(Y, L, H) is used to obtain the integral of Eq. (12), while
FUNCTION BIZERO(X) is used to evaluate the terms Io of Eq. (10). .
&
OOOOOOO0OO000O0O0
19
®FTNyEWGyL S
PROGRAM STORE
R S s T R T I P T T TR T LS
ek ererx THIS 1S PROGRAM STORE as¥sxkdxsk
e g Ak e ke A gl ol g o g ot ol o Rl ek e el il
THIS PROGRAM CALCULATES TEMPERATURE AS A FUNCTION OF POSITION
9001
9011
9021
9101
9111 FORMAT(BlIH THE
9121 FORMATI(83H
9131 FORMATI(46H
9141 FORMAT({74H
IRGE FOR DIMENSIUN./42H
9151 FORMAT(32H
9201
WITHIN A CYLINDER OFf GLASS CONTAINING AN INTERNAL HEAT SOURCE,
SUCH AS FISSION PRODUCTS.
IS A TABULAR
A CONSTANT,
THE COEFFICIENT OF THERMAL CONDUCTIVITY
THE SURFACE TEMPERATURE IS
OF
FUNCTION OF TEMPERATURE. .
TO. THIS TEMPERATURE IS A MAJDR FACTOR IN CONTROL
THE INTERNAL TEMPERATURE.
THE PROGRAM WILL HANDLE MORE THAN ONE CASE AT A TIME,
THE SAME VALUES OF NR,
IN ONLY ONCE PER COMPUTER J4OB.
READ
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSIGN
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
‘DIMENSION
NZ, NOMy R(I)s AND 2(J). THESE ARE
DZDT(20,50)
IPRNT(50)
LCOUNT(20+50)
Q1(200),Q2(200),Q3(200),Q4(200),Q5(200),Q6(200)
Q7(200,20}),Q8(50)
R{50)
SAVEZ(35)
T(20450)yTCALC(20450) yTHETA(20450),THETAC(20,450),TT(50)
TK(100),TP(100),VT(100)«THT(100)
TNC(100), TKN(100)
U(20,50)
V{20,450)sVCALC(20,450)
XT(100) 4XTA(100)+XTB(100) +XTC(100}
Y{100}
Z(50)42ZCALC(20,50)
TYPE DOUBLE Q3+,Q4925+9Q69Q7+ARG1 yARG24PI,PI3,VIOL,VIO24BIZERD
DATA (PI=3,14159265359),(PI3=31.0062766803)
PI=3.1415927
PI*x%3=31,0062767
FORMAT(1615)
FORMAT(16F5.3)
FORMAT{8E10.4)
FORMAT(81H
THE NUMBER OF RADIAL POSITIONS EXCEEDS THAT FOR WHICH
LTHIS PROGRAM WAS WRITTEN.})
NUMBER OF HEIGHT POSITIONS EXCEEDS THAT FOR WHICH
LTHIS PROGRAM WAS WRITTEN.)
THE NUMBER OF TERMS IN THE BESSEL FUNCTION SUMMATION E
LXCEEDS THAE OIMENSION OF QT7.)
THE TEMPERATURE INTERVAL HAS BEEN SET TO 0.0)
THE NUMBER OF TEMPERATURE PROFILES REQUESTED IS TOO LA
THE VALUE OF TINT SHOULD BE INCREASED.)
THE CALCULATED VALUE OF TIPRNT(yI2,59H) EXCEEDS 17, PGS
1SIBLY BECAUSE TINT HAS BEEN SET TOO SMALL.)
FORMAT(
1RE.
B83H1 HEIGHT
THE CYLINDER DIAMETER -IS4F8.1,
2TRENGTH OF THE INTERNAL HEAT SOURCE IS+1PE9.2
3/SECy+ES.2
40THER UNITS IS,0PF9.14+18H BTU/FT*#%3/i4R
S5CM%x%3,/81H
TEMPERATURES ABOVE THE SURFACE TEMPERATU
4H CM./ 58H IN THE S
20H CALORIES/CM%%3
SOURCE STRENGTH IN
AND,1PE9.2,18H MEV%CURIES/
THE THERMAL CONDUCTIVITY COEFFICIENT AT T
y 12H WATTS/CM*%3/48H RADIUS
UNITS
6HE SURFACE TEMPERATURE OF,0PF6.0, 13H DEGREES C ISs1PE9.2, 18H CAL
T7/CM/SEC/DEG C./1HO)
9211 FORMAT(120H VALUES OF THE RELATIVE DISTANCE (R/A) FROM TH
1E CYLINOER AXIS ARE GIVEN
IN THE FIRST ROW .OF THIS TABLE
2130HO (Z/A) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 O.
335 0.40 0.45 0.50 0. 60 0,70 0.80 0.930 0.95 1.00
4 /1HO}
.9221 FORMAT(120H
VALUES OF THE RELATIVE DISTANCE (R/A) FROM TH
BUT ALL HAVE.
20
1E CYLINDER AXIS ARE GIVEN IN THE FIRST ROW OF THIS TABLE /
335 0.40 045 0.50 0.60 0.70 0.80 0.90 0.95 1.00
4 [1HO)
9231 FORMAT{F7.3,17F7.1)
9241 FORMAT(120H1 TEMP DISTANCES, IN RADIUS UNITS, MEASURED FRO
1M THE BOTTOM OR TOP OF THE CYLINDER, AT WHICH QCCUR HEATING ABQVE/
2120H DEG C. THE SURFACE TEMPERATURE BY AMOUNTS SHOWN IN THE
3 FIRST COLUMN. /)
‘9251 -FORMAT(FH6.093Xy17FT.3)
OO0
9261 FORMAT(TX,17(1542X))
9271 FORMAT(1H1)
9281 FORMAT(37HOTHE EXECUTION TIME FOR THIS CASE WASsF7.1l, 8H SECONDS)
9291 FORMAT(315,5E15.5)
9301 FORMAT( 91HIHEIGHT THETA, A FUNCTION OF RELATIVE TEMP V,WHE
IN THE SURFACE TEMPERATURE IS KEPT AT,F6.,0/ 58H IN THE S
2TRENGTH OF THE INTERNAL HEAT SOURCE ISyF8.5y 20H CALORIES/CM*#%3/SE
3CypF8.5y 12H WATTS/CM*®3/ 48H RADIUS SOURCE STRENGTH IN OTH
4ER UNITS I1SeF9.1, 18H BTU/FT#*¥3/HR ANDyF9.1, 18H MEV®CURIES/CM%*%3
5./ 54H UNITS THE THERMAL CONDUCTIVITY COEFFICIENT ATsF6.0¢
6 13H DEGREES C 1S.F8.5, 18H CAL/CM/SEC/DEG C./1HD)
9311 FORMAT(FT7.342Xs17F7.3)
9321 FORMAT( 95HIHEIGHT THE DIMENSIONLESS U(1,J) FROM WHICH WE C
LALCULATE THETA(I,J}y V(I4J)s AND T{I,J4}./15H 2 [3449H
2 TERMS ARE USED IN THE BESSEL FUNCTION SUMMATION.)
9331 FORMAT(8EL15.8)
9341 FORMAT(24H1TO HAS BEEN SET TOO LOW)
9351 FORMAT(45HLTO HAS NOT BEEN SET TO AN INTEGER TIMES 50.0)
9361 FORMAT(Fl0.04F10.6,F10.543E16.8)
9371 FORMAT{THLTHETA{,I241Hys12+2H}=4F6.3)
9381 FORMAT{1HO)
9391 FORMAT({15F6.0)
9401 FORMAT(FT.3,17F7.2)
9411 FORMAT(FT.3,17F7.3)
READ 9001 yNRs NZ,»NOM
RADIAL COORDINATE/RADIUS OF CYLINDER
HEIGHT COORDINATE/RADIUS OF CYLINDER
R(I) IS THE RELATIVE RADIUS
Z(J) IS THE RELATIVE HEIGHT
IF(NR-17)21,21,11
11 PRINT 9101
GO TO 4001
21 IFINZ-50)41l,41,31
31 PRINT 9111
GU TO 4001
41 TF(NOM=200)61+61451
51 PRINTY 9121
‘GO TO 4001
61 READ 90114(R(1)yI=14NR)
READ 9011+(Z{J)sJ=1eN2Z)
BQ=0.0
DO 101 L=1,NOM
3Q=83+1.0
QliL)=2.0%3Q-1.90
QZ2(L)=Q1{L)*QLIL)
QA3(LY=QLIL)I%*Q2(L)
QalL)=QLIL)YRPI/Z({NZ)
QS(L)=Q4(L)*R(NR)
Q6lLY=8BIZERD(Q5{L))
101 CONTINUE
DO 111 J=1¢NZ
Q8(J)=Z(JI*TZ(NZ)-Z(J}1#*0.5
111
121
131
141
151
161
171
181
201
211
221
231
241
251
301
311
341
361
371
381
391
401
2]
CONTINUE
CONL1=4.0%Z(NZ)*Z(NZ)/PI3
DO 121 I=1,AR
Ull,41)=0.0
UlINZ})=0.0
CONTINUE
NZ1=NZ-1
00 131 J=24N2Z1
: U{NR,yJ)}=0.0
CONTINUE
NR1=NR-1
D0 221 J=24NZ1
00 211 I=14NR1
SuUM=0.0
DC 201 M=1,NOM
ARG1=Q4(MI®R(])
ARG2=Q5(M)
VIO2=Q6(#M)
[F{ARG1)151,4141,151
VIOl=1.0
GO T0O 181 :
IF{ARG1I-ARG2) 171,161,171
VIOl=vIOQZ2
GO TO 181 -
Q7(M,I)=BIZERD(ARGL])
VIOLI=QT7(M,I)
- SUM=SUM#VIOL*SINIQ4{MI*Z(J))}/Q3(M)/VIO?2
CONTINUE
U{1+J)=Q8{J)I-CONL1%ESUM
CONTINUE
CONTINUE
PRINT 9321,NOM
PRINT 9211
DO 231 LK=14NZ
PRINT 93114{ZALK)}+{U(JsLK)yJ=1,NR))
CONTINUE '
NT=0
NT=NT+1
READ 9021 TP{NT) TKINT)
[F{ITP(NT))2514241,241
NT=NT-1
NTEMP=NT-1
READ 9021 4A4RAD,TINT,TO
IF{AY4001,4001,311
INTIME=ICLOCKF{DUMMY)
DIA=2 .0%RAD
AO=A/TO
JN=0
JN=JN+1
[F(TO-TP{JUN))361,401,341
IF{JUN=-1)371,371,381
PRINT 9341
GO TO 301
IF{JUN-NT) 341,341,391
PRINT 9351
GO TO 301
JO=JN
TKO=TK{ J0O)
CONZ2=A0%R AD#*RAD/TKO
HUNIT 1=A%*4,185
HUNIT2=A%2 ,8317%3,6%3.,9685E04
HUNIT3=A%2611.6/3.7
411
421
431
451
461
471
481
491
501
561
571
581
601
611
621
631
641
651
661
671
681
691
701
711
22
JT=NT-J0+1
JTEMP=JT-1
JR=JT=-2
THT(1)=0.0
THT(2)=25.0*(TK(J0)+TK(J0+1))/TO/TKO
Y{1)=TK(JQ)
Y{2)=TK{JO+1)
DO 411 II=3,JTEMP
Y(II)=TK{I[+J0-1)
THT(II)=XINT{Ys11,50.0)/TO/TKO
CONTINUE
DO 421 I[=JONT
JK=1-J0+1
TN(JKI=TP(I)-TO
TKN{JK)I=TK(I)
CONTINUE
CALL THERMY(THTyTN:XT1XTA,XTB,XTC1JT1JTEMPl1.011-01100)
PRINT 9271
PRINT 9001440
DO 431 JK=1,JTEMP :
PRINT 9361,TN(JK),TKN(JK),THT{JK),XTAlJK).XTB(JK),XTC(JK)
CONTINUE
DO 581 J=14NZ
DO 571 I=1,NR
THETA(I'J)=AO*RAD*#2*U([,J)/TKO
MT=0
MT=MT+1 .
IF(THETA(1:J)—THT(MT))461,561'451
[F{MT-1)4T14471,451
TEMP=THETA(I,J)-THTUMT)
IF(TEMP+0.003)481'561;561
PRINT 9371,1,J,TEMP
GO TO 301
[F(MT=JTEMP)5014571, 571
NIT=MT-1
TU1,J)=XTA(NIT) +THETALT 9 3) % (XTBINIT)#THETALL, J)#XTCINTT})
GO0 TO 571
T{T14J)=TN(MT)
CONTINUE
CONTINUE
J=0
TMAX=04.0
J=d+l
IF(J-NZ2)611,6114641
IF(TMAX=-T(14J))621+621,601
TMAX=T{1,yJ)
GO TGO 601
IF{TMAX=10.0)651,6614661
IFMT=3
50 TO 691
IF{TMAX-100.0)671,:681,681
IFMT=2 -
GO TQ 691
IFMT=1
CONTINUE
PRINT 9201;D[A1A,HUNIT1§HUNIT21HUN1T31TO:TKD
PRINT 9211
GO TO {701,721, 741),IFMT
DO 711 LL=1,NZ
PRINT 9231!(Z(LL)!(T(JILL)!lefNR,,
CONTINUE
GO TO 761
121
731
741
751
761
811
821
831
841
851
861
871
1001
1011
1021
1031
1041
1051
1061
1071
1141
1151
1161
23
DO 731 LL=1,NZ ‘
PRINT 9401+ (Z(LL) »{(TlJyLL)sJ=1,NR))
CONTINUE
GO TO 761
00 751 LL=1,NZ
PRINT 9411, (Z(LL)s(T(JyLL)yJd=14NR))
CONTINUE
CONTINUE
PRINT 93010TO{AfHUNITlfHUNITZ,HUNIT3fTOiTKO
PRINT 9211
DO 811 LL=1,4NZ
PRINT 93110(Z(LL)'(THETA(JiLL,0J=loNR))
CONTINUE '
IF(TINT)831,821,831
PRINT 9131
GO TO 4001
DO 851 J=14NZ
IF(T{1+4J)/TINT~50.0)851,85],841
PRINT 9141
GO TO 4001
CONTINUE
DO 871 L=1,50
IPRNT(L)=0
D0 861 IR=1,20
LCOUNT(IR,L)=0
CONTINUE
CONTINUE
NL=0
20 2801 I=1,4NR
L=l
TCHECK=TINT
J=0
J=J+1
IF(J-NZ+2)1011,1011,2801
TEMP=T(I,J)-TCHECK
IF{A3S{TEMP)-0.05)1021,1021,1031
ZCALC(I, L)=Z(J)
GU TD 17561
IF(TEMP}1001,1001,1041
KZ=J
Kl=J-1
IF(T(IfKZ)—T(IiKl))280112801'1051
IF(T(IfK2+1)'T(IpKZ)’lO?lle?llebl
J1=Kl1
J2=K2
J3=K2+]1
GA TO 1141
JI1=Ki1-1
J2=K1
J3=K2
TERM1=(T(IrJ3)-T(11J1))/(Z(J3)-Z(J1’)
TERMZ=AT(I,J2)=-T{T1,J1))7(2(J2)-2(J1})
CONB=(Z(J3)=-Z(J2))/(TERMI-TERM2)
CONA={Z(J3)+Z(J1)-CONB*TERM1)/2.0 A
CUNC=T(IyJZ)‘(Z(JZ)—CDNA)*(Z(JZ)‘CDNA)/CUNB
RODT=SQRT(CONB*{TCHECK-CONC))
IF(CONB)L161,1151,1151
DZDTUL,L)=CONB/2.0/R00T
I T=CONA+ROOT
GG TO 1171
DZDT(I4L)=-CONB/2.0/R00T
IT=CONA=-RQOOQT
1171
1501
1511
1521
1531
1541
1551
1561
1571
1601
1651
1661
1681
1691
1701
1711
1721
1751
1761
1771
1781
1791
1801
2001
2011
2021
24
SAVEZ(1)=1T
LCOUNT(I,L)=0
LCOUNT(I,L)=LCOUNT(I,L)+1
IF(LCOUNT(I'LI—1Q1151171511y2001
SUM=0.0
SUMD=Q.0
DO 1601 ™M=1,NOM
ARGLI=Q&4(M)%R(I)
ARGZ2=Q5(M)
VID2=Q6(M)
[F(ARG1)1531,1521,1531
VI0ol=1.0
30 TO 1561
[F{ARGL-ARG2Y1551,1541,1551
viol=vIOo2
GO TO 1561
VIO1=Q7{M, 1)
ARG3=Q4 (M) *ZT
lF(ABS(ARG3)—1.0E1511571,301,301
SUM=SUM+V101*SIN(ARG3)/Q3(M)IVIOZ
SUMD=SUHD+VIOI*COS(ARGB)IQZ(M)IVIOZ
CONTINUE
THETAC(I,L)=CUN2*(ZT*(Z(NZ)—ZT)*O.S-CONl*SUM)
MT=0
MT=MT+1
IFITHETAC(I,L)—THT(MT))1661,1701.1651
[F(MT-1)301,301,1681
IE(MT-JTEMP)1691,1781,1731
NIT=MT-1
TCALC(I'LI=XTA(NIT)+THETAC(I,L)*(XTB(NIT)+THETAC(I,L)*
XTCINIT))
GO TO 1711
TCALC (I, L)}=TN(MT)
TEMP=TCALC{I,L)}-TCHECK
LFCABS(TEMP)-0.05)1751,1751,1721
ZT=ZT-TEMP*DZDT(I,L}
KK=LCOUNT(L4L)
SAVEZ(KK+1)=ZT
GO TO 1501
2CALC{T,L)=2T
TT(L) =L*TINT
IPRNT(L)=IPRNT(L)+1 .
lF(1PRNT(LI-17)178111781,1771
PRINT 9151,L
GO TO 4001
TCHECK=TCHECK+TINT
IF(L-NL)}1801,1801,1791
NL=L
L=L+1
GG TO 1011
ZMIN=Z (K1)
IMAX=2(K2)
zI=0.5*(;MIN+zMAxl
SUM':0.0
SUMD=0Q0.0
DO 2101 M=1,NOM
ARG1=Q4 (M)*R(I)
ARG2=Q5(M}
VI02=Q6(M)
IF(ARG1)2031,2021,2031
vI01=1.0
G0 TU 2061
2031
2041
2051
2061
2071
2101
2151
2161
2181
2191
2201
2211
2221
2231
2241
2301
2801
3021
3031
4001
25
IF{ARGL-ARG2)2051,2041,2051
vIi0ol=VIOZ2
GO TO 2061
VIO1=QT7(M, 1)
ARG3=Q4 (M)*ZT
IF(ABS(ARG3)-1.0E15)2071,301,301
SUM=SUM+VIOL1*SIN{ARG3)/Q3({M)/VIOZ
CONTINUE
THETAC(IsL)=CON2%(ZT%(Z(NZ)-ZT)*0.,5-CON1¥SUM)
MT=0
MT=MT+1
IF(THETAC(I,L)-THT{MT))2161,2201,2151
IF(MT-1)301,301,2181
IF(MT-JTEMP)2191,1781,1781
NIT=MT-1
TCALC(IvL)*XTA(NIT)+THETAC(11Ll*(XTB(NIT)+THETAC(IvL)*
XTC(NIT))
GO TO 2211
TCALC (I,L)=TN(MT)
TEMP=TCALC(I,L}-TCHECK
IF(ABS{TEMP)-0.05)1751,1751,2221
IF(TEMP)2241,223142231
IMAX=ZT
GO TO 2301
IMIN=ZT
LCOUNT(I,L)=LCOUNT(I,L)+1
[F(LCOUNT(1,1)-30)2011,2011,2801
CONTINUE
PRINT 9241
PRINT 9221
DO 3021 LL=1,NL
NA=IPRNTI(LL)
PRINT 9251y (TT(LL),» (ZCALC(J4LL) 4J=1,NA))
CONTINUE
PRINT 9271
DO 3031 LL=1,NL
NA=IPRNT(LL)
PRINT 9261y (LCOUNT(JsLL) yd=1,NA)
CONTINUE
DTIME=(ICLOCKF(DUMMY)~INTIME}*1.0/60.0
PRINT 9281,0TIME
GO TO 301
CALL EXIT
END
26
SUBROUTINE THERMY{X,YsWsAyByCoNT4NTEMP,X0,Y0,ID)
DIMENSION A(ID)B(ID),C{ID) W{ID)X(ID),Y(ID)
IF{Y0)101,101,301 |
101 IF(X0-X(1))121,131,141
121 PRINT 701
60 .T0 901
131 Y0=Y(1)
G0 TO 301
141 I=1
151 I=I+1
IF(I-NT)171,171,161
161 PRINT 711
60 TO 901 -
171 IF(X0-X(1))191,181,151
181 YO=Y(I)
GO TO 301
191 IF(I-NT)201,211,211
201 J1=1-1
J2=1
S J3=1+1
G0 TO 221
211 J1=1-2
J2=1-1
J3=1
221 TERML=X{J2) :X(J3)#X(J3)=X(J3)#X(J2) %X (J2)
TERM2=X{J 1) %X (J3)EX(J3)=-X{I3)EX(J11%X (1)
TERM3=X(J 1) %X (J2)%X(J2)=X(J2) ¥X (J1)%X(J1)
DELTA=TERML-TERM2+TERM3
CA=(Y(J1)XTERMI-Y{J2)*TERM2+Y(J3)*TERM3)/DELTA
CB={Y(J2)&X(J3)%X(J3)-Y(J3)*X{J2)*X(J2)
1 Y (J1) X (J3)EX1J3)+Y{J3) X {J1)*X(J1)
2 “Y(JL1) %X (J2) %X (J2)=-Y{J2)*X(JL)*X{J1))/DELTA
CC=(Y(J3)%X(J2)=Y (J2)%X(J3)=Y(J3)=X{J1)+Y(J1)*X(J3)
1l +Y (J2)eX(J1)=-Y(JL)%X(J2))/DELTA
YO=CA+X0®(CB+X0*CC)
301 DO 311 L=14NT
W(L)=Y(L) /YO
311 CONTINUE
DO 401 J=1,NTEMP
IF(J-NTEMP) 331,341,341
331 11=J
[2=J+1
[3=J+2
GO TO 351
341 [1=J-1
[2=3
I13=J+1
351 TERML=X(T2) %X (I3)%X(I3)-X{I3)%X(I12)%X(I[2)
TERM2=X{T 132X (I3)*X{I3)—X(I3)xX(11)*X(I1)
TERMI=X(IL1)EX{I2)%X(I12)-X(12) X (I1)*X(I1)
DELTA=TERM1-TERM2+TERM3
A(J)={W(T1)*TERMI-W(I2)*TERM2+W(I3)%TERM3)/DELTA
BIJI=(W{I2)#X(I3)%X(I3)—W(TI3}%X(12)%X(12)
1 —WlI LY EX (I3 XT3V +WlI3)EX{TIL)*X(1])
2 FH I EX(I2)#X(I2)=WlI2)*X(T1)*X(11))/DELTA
ClI)=(W(I3)eX(I2)=W(T2)*X(I3)-W(I3)*X(I1)
1 +W I EX(I3)+W(I2)X(IL)=W(IL)*X(12))/DELTA
401 CONTINUE
701 FORMAT{49H1 TO HAS BEEN SET TOO LOW FOR THE PRESENT PROGRAM)
711 FORMAT(50H1 TO HAS BEEN SET T0O HIGH FOR THE PRESENT PROGRAM)
27
RETURN
901 CALL EXIT
END
FUNCT FON XINT(YyLgH)
DIMENSION Y{200)
K=L
KP=K/2%2
NDIFF=KP-K
IFINDIFF)1,2
2 K=K-1
1 ODD=0.0
EVEN=0.0
KM2=K-2
DO60I =3 ,KM2,2
60 0DD=0DD+Y (1}
KMl=K-1
DO621=24KM1,2
62 EVEN=EVEN+Y(])
XINT=H/3.0%(Y (1) +Y(K)+4, 0%EVEN+2.0%00D)
IF(NDIFF) 3,4
4 XINT=XINT+ H *{Y{K)+Y(K+1))/2.0
3 RETURN
END
FUNCTION BIZERD(X)
CALCULATES MODIFIED BESSEL FUNCTION OF THE FIRST KIND, I0(X), FOR (=3.759X4IN
CALCULATES BY SERIES 9.8.1 AND 9.8.2 OF N.B.S. HD3K OF MATH FUNCNS (1964)
TYPE DOUBLE S,TyXsY¥,BIZERD
Y=X
[IF(Y+3.75)99,3,3
3 IF(3.75-Y)844,44
C***"‘ _‘3. 75 u,LE. XCLE 03075
4 T=Y/3.75
T=T*T ‘
BIZERD=((({(T*.0045813+.0360768)%T+,2659732)%T+1,2067492)%T
1 +3.,0839424)%T+3,5156229) #T+1.
RETURN
Cx%%8 3,75.LT.X
8 T=3.75/Y
S=CL00((T*,00392377-.01647633)%T+.02635537)1%T—.02057706)%T
1 +.00916281)}%T-.00157565)%T+#.00225319)%T+.01328592)%T+.39894228
BIZERO=DEXP(Y)*S/DSQRT(Y)
RETURN
C*%99 ARGUMENT OUT OF RANGE. RETURN ZERQD.
99 BILERD=0
RETURN
END
29
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141.
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143.
144,
145.
146.
30 .
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A. S
P.
B
oW <amOoOOC- oz nont X
MHrR@wRao@DERETDODGWH
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159.
160.
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162.
163.
164,
165.
166.
167.
168.
169.
170.
171.
172.
173.
174.
175.
176.
177.
178.
179.
180.
E. D. Nogueira, Seccion de Combustibles Irradiados, Junta de Energia
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Hubertus M. Holtzem, Radiation Protection Dept., Luisenstrasse 46,
Bad Godesberg, West Germany
Gerard M. K. Klinke, Const. Dept., Min. of Fed. Property, Bad
Godesberg, West Germany
Helmut Krause, Decontamination Dept., Karlsruhe Nucl. Res. Center,
/5 Karlsruhe, Weberstr 5, Siemensallee 83, West Germany
Harold F. Ramdohr, Waste Storage Dept. Karlsruhe Nucl. Res. Center,
Leopoldshafen/Bd., Max-Planck-Str. 12, Karlsruhe, West Germany
Herman F. A. Borchert, Petrology, Mineralogy, and Economic Geology
Dept., Mining Academy Clausthal, Seimensallee, Karlsruhe, West Germany
H. 0. G. Witte, KFA, Juelich, Germany
E. J. Tuthill, Brookhaven National Laboratory
181. .
182,
183.
184.
185.
186.
187.
188.
189.
190.
191.
192,
193.
194-386.
32
S. Rottay, Director of Waste Treatment, KFA, Juelich, Germany
S. Freeman, Mound Laboratory, Bldg. A, Room 155, Miamisburg, Ohio
J. J. Goldin, Mound Laboratory, Bldg. A, Room 155, Miamisburg, Ohio
D. L. Ziegler, Dow Chemical Company, Rocky Flats
Ray Garde, Los Alamos Scientific Laboratory
R. M. Girdler, E. I. duPont, Savannah River Laboratory
E. A. Coppinger, Pacific Northwest Laboratory, Richland
J. J. Shefcik, General Dynamics, San Diego, California
Philip Fineman, Argonne National Laboratory, East Area EBR-2,
National Reactor Testing Station, Scoville, Idaho
D. E. Bloomfield, Pacific Northwest Laboratory, Richland
Jacob Tadmor, Israel Atomic Energy Commission, Soreq Nuclear Research
Center, Yavne, Israel ‘ ’
J. A. Swartout, Union Carbide Corporation, New York
Laboratory and University Division, AEC, ORO
Given distribution as shown in TID-4500 under Waste Disposal and
Processing category (25 copies — CFSTI)