Page 84 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 71

ac a2c
=
Mass diffusion: - D -
at ax2
dT k a2T
Thermal conduction: - -- (2.9)
=
at pc, ax2
do d"o
2-D Vorticity transport: - - (2.10)
=
V
at ax2
where c,T, and ~i) are concentration, temperature, and the z-component of
vorticity in a 2-D flow, respectively, and their corresponding diffusivities are D,
a, and V. This equation is thoroughly studied in the undergraduate curriculum.
It has solutions by Fourier and Laplace transforms, and similarity solutions for
X
initial and boundary conditions that collapse on the variable 77 = -

doesn't leave much room for finite element methods - just another technique for
a tired old problem, right? Wrong. FEMLAB can still give this problem a boost
which is not commonly considered. FEMLAB solutions are well suited to non-
constant coefficients, i.e. transport properties that depend on the field variable.
For instance, for suitably low pressures and high temperatures, a gas must satisfy
the ideal gas law:
nM
p=-=- PM (2.11)
V RT
where R is the gas constant and M is the relative molecular mass of the species.
Under these conditions, it is rare to find a gas that has a constant heat capacity.
For instance, over a range of temperatures, the heat capacity of COz gas is well
approximated by a quadratic in temperature, ( in MJkg-mol"C), with T in "C:
c, = 36.11 + 0.04233 T - 2.887~10-~T* (2.12)

It follows that

k kR
-- - f (T)
pc, 36.11PM
(2.13)
(T + 273)
(T)= l.+ 1.172~10" T - 7.995~10-~T'