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LOFT
ORNL P
1480
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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
ORNL-D-1480
MASTER P
HEATING IN INFINITE CYLINDERS BY UNIFORM GAMMA SOURCES*
H. C. Claiborne and C. W. Nestor, Jr.
Oak Ridge National Laboratory
Oak Ridge, Tennessee
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2 1965
AJG
As an aid to the reactor designer concerned with gamma heating in
olamonto that may be approximated by long cylinders, we have integrated
the point kernel describing the transfer of gamma energy for the case 01?
sources distributed uniformly throughout a right circular cylinder of
infinite length. Since the coefficients were available?,2 for a third-
degree polynomial representation of the energy absorption buildup factor,
we chose this representation and were able to reduce the solution to a
.
rooulthas trau dhe
fachadw age
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single numerical integration involving each term in the buildup factor
expansion by placing the origin of the coordinate system on the field
point P (see Fig. 1). In addition, we obtained an analytical
expression for the contribution from the linear term in the expansion.
With the coordinate system as shown in Fig. 1, we express H(r), the energy
deposition resulting from a uniform isotropic monoenergetic source S (Mev/

sec.cm) as
a
H(r) = 4Sug J J
* or
Thila).. B (up) e la dz n dn da
n=0 2-0 40102
where
RELEASIED FOR ANNOUNCEMENT
IN MUCLEAR SCIENCE ABSTRACTS.
= linear energy absorption coefficient, cm-2,
H = total linear attenuation coefficient, cm-,
Mela) = VR2 ge sina a'- r cos a, the maximum value of 1 for a
particular a (see Fig. 1), cm,
Research sponsored by the U. 8. Atomic Energy Commission mader contract
with the Union Carbide Corporation.

nearnown
::.;.
TO THE AEC AND ITS-CCKTRÁCFCRICKS
..
to
-
povna + z2, the distance from source point to field point, cm,
mlup)", the energy absorption bul.ldup factor, dimensionle88.
Upon substituting the above expressions for B (up) and p into Eq. (1),
. sories
we obtain
of
myla)
H(r) - -
.
Inform
- VO2 + 22
om dz n dn da , (2)
e
v212-m az
um
-
where the z and n inside the integral are now expressed in mean free paths and
& are the coefficients of the polynomial representation of the buildup factor.
For convenience in the integration we now define
-
;
--
-
-
me tha + z2
(n)
w
dz,
žo (ne + 2232-1
and if we let
n² + z² = n²t² ,
we have
eent at
ine er der etter
<(n) = 4 * Rei
.vngis at
m
By comparison of Eq. (4) with integral representations of the modified
Bessel functions of the second kind,\,(x), we find that .
n
zkiewinne
Fo(0) = *12(1) = 5 x (t) at,
Fi(n) = K (n),
Fa(n) = nki(n),
F3(n) = 72k (n) + nki(n).
We then express the heating, H(r), in the form
u
H(s) - I mi ovo(s)
and using Egs. (5) through (8), we obtain3
Vo(x) = 1 - * * Kuzcumq) da,
*a(r) = 3 - * urme Kawma) da,
va(x) = Vo:(0) - test cura de K, Curma) da ,
· 63() - ve(r) - Boat (uno bo ku Cuma) a.
(13)
In addition, if we write an expression for 8(r) in the coordinate
system centered on the axis of the cylinder, we have
(r) K(1) de xi dr'
(14)
Using the addition theorem for K (n) and performing the e integration, we
Pind that
UR
"(t) = \,(0) Le PPI (rº) dx' + I,() , x5, (r”) dr". (15)
After integrating with respect to rl and using the Wronskian identity,
: . X (x) Iz(x) + Kj(x) I (x) = ; we have
*(T) = 1 - HR Kx (HR) I (ur).
(16)
At r = 0, the point of maximum energy deposition, Mala)-- R,and on
integrating Egs. (10) through (13), we have
with a similares medicini, Akan tentakelis kartu su de comedor
.
V. (0) - 1 . Kialur) ,
(17)
vi(0) - 1 - UR Kz (AR) ,
(18)
(19)
va(0) - 4260) - Suge K. (NR)
*360) – veco) - Sun 13 (ur).
(20)
The average energy deposition rate ħ is given by
R.
1- SU(=) = dr ,
H(r) r dr ,
(21)
which we express in the same form as Eq. (9), with the vi(r) replaced by
Tm. The function T. 18 just the collision probability given in analytical.
and tabulated form by Case et al.5 In addition we have by integrating
Eq. (16),
Ti = 1 - 211(UR) Kı(UR).
(22)
The results of numerical integration of Eqs. (10) through (13)
and numerical integration to produce the functions are shown in Fig. 2
af functions of cylinder radius. We were able to obtain excellent agreement
between our numerically integrated it and the tabulated values, and also
between the results of the numerical integration for vi(r) and Fi and
analytical solutions (10) and (13). Tabulated values of the vn(r) functions
and the tm are available on request, &s are the FORTRAN-63 programs used
in the calculations on the CDC 1604A.

References
1. CAPO, M. A., "Polynomial Approximation of Gamma-Ray Buildup Factors,
for a Point Isotropic Source," APEX-510 (1959).
2. TOBIAS, M. L., D. R. VONDY, and M. P. LIETZKE, "NIGHTMARE – An IBM 7090
Code for Calculation of Garima Heating in Cylindrical Geometry," USAEC
Report ORNL-3198, Oak Ridge National Laboratory (1962).
3. For a definition of the Ki, (x) functions, see, e.g., W. G. Bickley
and J. Nayler, A Short Table of the Functions Ki (x), from n = 1 to
n = 16, Phil. Mag. 7, 20, 343-347 (1935).
4. WATSON, G. N., Theory of Bessel Functions, 2nd Ed., p. 361, Cambridge
U. Press, London (1952).
5. CASE, K. M., F. de HOFFMAN, and G. PLACZEK, Introduction to the Theory
of Neutron Diffusion, Vol. I, pp. 31-35, Los Alamos Scientific Laboratory
Los Alamos (1953).
.,
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END
DATE FILMED
11/19 /65



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