Finite difference program for calculating hydride bed wall temperature profiles Page: 4 of 47
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WSRC-TR-92-501
DISCUSSION
Heat Conduction Difference Equations
Heat conduction equation for material "i", with constant thermal
conductivity, k;,
aT. 2T1 a T
Pi C, - k D2 T = k a T + {1LT,()
at Lra r jr1
where ni = 0, 1, or 2 for planar, cylindrical, or spherical
geometries, respectively.
For finite difference approximations, let Ti; be the temperature
of material i, at the jth spacial node and nth time step for
j = 0,...,M; and n = 0,...,N. The following finite difference
approximations were used to approximate the derivatives:
aT Tj j - Ti j (2)
at At
aTi Ti j.1 - Tij_1 (3)
ari- 2 Ari
a2T" Ti- 1 - 2 Tf + Tfi1 t.)
ari (Ar1)2
Using the Crank--Nicolson method with these finite difference
expressions, the differential equation becomes
-Tj- + [2 +2yi] T +[ - Tij+1
(5)
= [y,-8,] Ti j- + [2 -2Yi] T' j + [Y1+&1] Tf +1
where
a __k a=At , = = 1At=, (6)
S pi C Y (Ar)2 ' 2riAr)
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Klein, J. E. Finite difference program for calculating hydride bed wall temperature profiles, report, October 29, 1992; Aiken, South Carolina. (https://digital.library.unt.edu/ark:/67531/metadc1338613/m1/4/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.