Civil Engineering Reference
In-Depth Information
FIGURE 6.20
Schematic representation of a finite distribution of subdomains providing implicit information
concerning the temperature dependence of material properties and of the coupling between energy transfer and fluid
convection.
the process of calculating a discrete temperature field
T
p
over an open solution domain via numerical
integration of Eq. (6.27) for a given set of upstream boundary conditions . Given an explicit function
(
T
), a discrete set of upstream boundary values
T
B
and specified top surface and midplane boundary
conditions, a unique discrete temperature field
T
p
is determined over an open solution domain according
to Eq. (6.27). Next, we consider that within this same open solution domain we arbitrarily define a set
of closed subdomains such as those shown schematically in
Fig. 6.12
.
Referring to this figure, we next
consider the process of calculating a discrete temperature field
T
p
over a given closed subdomain for a
given set of boundary conditions via an iterative procedure. Given an explicit function
(
T
)
and boundary
conditions over the entire surface of the closed domain, a unique discrete temperature field
T
p
is determined
according to Eq. (6.27). Thus it is seen that depending on the type of boundary conditions, Eq. (6.27)
can assume either the role of an elliptic or that of a parabolic solver. Considering again a given set of
closed surfaces such as are shown schematically in
Figure 6.12
,
and following the set of different types
of temperature-field designations defined in previous sections, we represent values of the temperature
field on the downstream portion of this surface by the quantity
T
E
. At this point, it is significant to
observe that if the temperature field
T
E
is interpreted as being part of the temperature field
T
p
determined
via Eq. (6.27), adopted as a parabolic solver, then it follows that the quantity
T
E
can be interpreted as
being defined by a function
T
E
T
E
(
T
B
,
,
V
B
).
(6.34)
Next, we adopt the condition that there is sufficient information concerning the temperature field
downstream (e.g., information that has been obtained from weld cross sections or thermocouple mea-
surements) that a specified closed solution domain can be partitioned into a set of distributed subdomains
For a given partitioning of the solution domain into a set of distributed subdomains (e.g.,
Fig. 6.20
)
,
we consider the process of determining a discrete temperature field
T
p
over a closed solution domain by
adopting Eq. (6.27) as an elliptic solver and the values of temperature on the subdomain surfaces as
constraints. We wish to show that for a distributed set of closed subdomains, the equality
6
6
1
6
w
p
1
--
---------
T
p
w
i
T
i
T
i
(6.35)
i
1
i
1
