Page 57 - Process Modelling and Simulation With Finite Element Methods

44 Process Modelling and Simulation with Finite Element Methods

-- dCA - - k,CA + k2CB
dt

-- dCB - k,C,-
dt k2cB - k3CB + k4CC

-- dCC - k3CB - k,Cc
dt
It may surprise you, but because the above system is linear, it has a general,
analytic solution. Though general, it lends little insight into the dynamics of the
system. Plot the graph of concentrations versus time for the initial value
problem. Start with pure cA=l with parametric values kl= 1 Hz, k2 =O Hz, k3=2
Hz, b=3 Hz and plot the graph versus time of concentrations.

1.4 Method 3: Numerical Integration of Ordinary Differential Equations
In the previous section, numerical integration was treated by marching methods,
commonly referred to as “time-stepping,” although in the reactor design
application, it was clearly spatial integration. In marching methods, the
unknowns are found sequentially. The other common method for integration is to
approximate the ODE and solve simultaneously for the unknown dependent
variables at the grid points. With marching methods, all solutions must be initial
value problems (IVP). The number of initial conditions must match the order of
the ODE system. But for second order and higher systems, a second type of
boundary condition is possible - the boundary value problem (BVP), where in 1-
D, there are conditions at the initial and final points of the domain. Hence, these
are two point boundary value problems. Marching methods can laboriously treat
BVPs by shooting - artificially prescribing an IVP and guessing the initial
conditions that satisfy the actual BVP by trial and error. In higher dimensional
PDEs, a BVP specifies conditions on the boundaries of the domain.
One of the major advantages of the finite element method is that it naturally
solves two-point BVPs. As an example, the reaction and diffusion equation in
1-D is

(1.19)

where u is the concentration of the species, 3 is the diffusivity, L is the length of
the domain, R(u) is the disappearance rate by reaction, and x is the dimensionless
spatial coordinate. If the unknown function u(x) is approximated by discrete