Teaching Thermal Hydraulics & Numerical Methods: An Introductory Control Volume Primer Page: 47 of 77
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Appendix I
Equation (38), the discrete version of the advection equation, is used for illustrating the matrix
assembly and t he donor ing.
~df ~(8
N N L Nt ( f VA)t - VaN,i (f VA), + SNN (38)
dt ti
This equation can be compacted into the relation
df_
dfN aN,k (f VA)k+SN N (Al -1)
where k denotes an inlet or outlet link and:
aNk=-1,iffk=t
aNk =1,f k=(A-2)
aNk = 0, if f k not linked t o node N
The aN k is -1.0 f or f low into volume N, link k is a terminal link t and aN k is 1.0 f or f low out of N, link k
is an incident link I. The aN k as def ined in Al -1 and Al -2 are the dot product (cosine) of the f low or
velocity vector with the outward unit normal vector.
Iliniet Iloutiet
(f VA) N (f VA )
(f VA )2_ _ _
For the f igure above we can see that the cosine of the angle between the f lux vectors of (f A) and
(f VA and the area unit normal finiet on the inlet f ace is -1.0 or aN1 aN,2 = -1.0 . Conversely, the
cosine of the angle between (f VA) and foutiet is 1.0 or aN,3 = 1.0 . Using this inf formation in Al -1 we
get
d t aN 1 (f VA ) - aN 2 (f VA )2 - aN 3(f V A )3 + SNY (AI-2a)
and upon using the values f or t he aNk we get46
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Lucas, D. S. Teaching Thermal Hydraulics & Numerical Methods: An Introductory Control Volume Primer, article, October 1, 2004; [Idaho Falls, Idaho]. (https://digital.library.unt.edu/ark:/67531/metadc889821/m1/47/: accessed April 30, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.