Page 21 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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INTRODUCTION
1.6 Boundary and Initial Conditions
The heat conduction equations, discussed in Section 1.5, will be complete for any prob-
lem only if the appropriate boundary and initial conditions are stated. With the necessary
boundary and initial conditions, a solution to the heat conduction equations is possible.
The boundary conditions for the conduction equation can be of two types or a combination
of these—the Dirichlet condition, in which the temperature on the boundaries is known
and/or the Neumann condition, in which the heat flux is imposed (see Figure 1.5):
Dirichlet condition

T = T 0 on T (1.46)
Neumann condition
∂T
q =−k = C on qf (1.47)
∂n
In Equations 1.46 and 1.47, T 0 is the prescribed temperature; the boundary surface; n is
the outward direction normal to the surface and C is the constant flux given. The insulated,
or adiabatic, condition can be obtained by substituting C = 0. The convective heat transfer
boundary condition also falls into the Neumann category and can be expressed as

∂T
−k = h(T w − T a ) on qc (1.48)
∂n
It should be observed that the heat conduction equation has second-order terms and
hence requires two boundary conditions. Since time appears as a first-order term, only one
initial value (i.e., at some instant of time all temperatures must be known) needs to be
specified for the entire body, that is,
(1.49)
T = T 0 all over the domain
at t = t 0
where t 0 is a reference time.
The constant, or variable temperature, conditions are generally easy to implement as
temperature is a scalar. However, the implementation of surface fluxes is not as straight-

Γ T Γ qf

Ω

Γ qc

Figure 1.5 Boundary conditions