2. Numerical methods
Microchannel technology is capable of high heat and mass transfer
coefficients between a bulk reaction fluid and the catalytic heat
exchange surface. Alternating channel parallel plate designs can be
applied to thermally coupling endothermic steam reforming with
combustion in neighboring channels. A convenient way to supply heat is
to couple the endothermic reaction with an exothermic combustion
reaction in the heat exchange channels. The simulated microreactor is a
parallel plate reactor with alternating combustion and reforming
channels separated by walls. The device is 60 millimeters long. The
combustion channel is 700 microns wide, the reforming channel is 700
microns wide, and the wall separating the two channels is 700 microns
thick. As a result, a two-dimensional representation of the system may
be reasonable because of the large aspect ratio. The alternating channel
configuration allows solving only half of each channel, due to symmetry,
plus the wall connecting them. At these length scales, the continuum
approximation is still valid. However, in contrast to large scale
devices, radical quenching becomes important [29, 30] unless the
materials are properly prepared. The commercial computational fluid
dynamics software Fluent is used to solve the steady-state continuity,
momentum, energy, and species conservation equations in the fluid phase
and the heat equation in the solid phase using a finite volume approach.
The reactor body may be constructed from a number of materials using a
range of techniques. Suitable materials include ceramics with a low
coefficient of thermal expansion which are readily extrudable. These
include, but are not limited to, mullite, cordierite, alumina, and
silica. Other materials include metals which may be extruded, welded,
brazed, or diffusion bonded to make such structures. Using metals, it is
sometimes useful to start with metal oxide powders, which are then
bonded and reduced to the metallic state. Suitable metals include
copper, aluminum, stainless steel, iron, titanium, and mixtures or
alloys thereof. The dividing walls of the parallel plate reactor must be
of sufficient strength to maintain the integrity of channels. The
minimum wall thickness therefore depends upon material of construction.
In the present study, the wall thickness is in the range of about 0.5
millimeter to 5 millimeters and more particularly in the range of about
0.5 millimeter to 2 millimeters. The wall will act as a thermal barrier
to heat transfer, however, as the wall is very thin its resistance is
small. Consequently, the two channels will operate with a similar
operating temperature. The inlets and outlets are arranged for a
co-current flow of the reactants. In an alternative case, the inlets and
outlets are arranged for a countercurrent flow of the reactants.
Adaptive meshing is a method of refining the mesh of a simulation based
on the solution. An adaptive meshing scheme is used for the
discretization of the differential equations. The computational mesh is
initialized with 200 axial nodes, 200 radial nodes for the combustion
channel and the wall sections, and 200 transverse nodes for the
reforming channel. This discretization translates into a total of about
72,000,000 nodes. This initial mesh is adapted and refined during a
calculation to increase the accuracy of the solution in regions of high
gradients. Specifically, additional nodes are introduced to refine the
mesh using the tools built in the computational software so that the
normalized gradients in temperature and species between adjacent cells
are lower than 10 with a negative exponent of 6. Adaptation is performed
if the solution has not converged after about 2,000,000 iterations or
when the residuals are around 10 with a negative exponent of 6. This
last threshold, while not optimized, is meant to strike a balance
between cost and probability for convergence. Specifically, mesh
refinement before achieving complete convergence reduces the
computational effort, but a too early refinement, namely in a few
iterations, may lead to refinement in wrong regions. After mesh
refinement, a total of 20,000,000-60,000,000 nodes are used. Such an
adaptive meshing strategy, starting with a relatively coarse initial
mesh followed by refinement in regions of large gradients, achieves an
adequate balance between accuracy and computational effort.
The relationships between the mixture components and the properties are
typically complex and unknown. In these cases, it would be advantageous
to develop predictive models that are capable of relating the mixture
components to the properties so that the properties of new mixtures can
be estimated. While there have been various attempts to develop
predictive models for chemical mixtures [31, 32], none have gained
widespread use. The fluid density is calculated using the ideal gas law.
The individual properties of various gaseous species, such as thermal
conductivity, are calculated using the kinetic theory of gases, whereas
the specific heats are determined as a function of temperature using
polynomial fits from the thermodynamic database available in Fluent.
Mixture properties, such as specific heat and thermal conductivity, are
calculated from pure component values based on the mass-fraction
weighted mixing law. Binary species diffusivities are determined using
the Chapman-Enskog equation and then are used to calculate the
multicomponent mixture diffusivities. For the solid wall, a constant
specific heat and an isotropic thermal conductivity are specified. Given
that material conductivity varies with temperature and more importantly
with the material chosen, computational fluid dynamics simulations are
carried out over a wide range of conductivities.
Boundary conditions are chosen and defined in order to represent the
behavior of a real physical system that is being simulated. All
particular solutions to differential equations rely on enforcing
boundary conditions. The boundary conditions in a problem define how a
solution to a differential equation behaves at the boundary of a system.
Symmetry boundary condition is applied at the centerline of both
channels, implying a zero normal velocity and zero normal gradients of
all variables. No-slip boundary condition is applied at each wall-fluid
interface. Danckwerts boundary conditions are implemented for the
species and temperatures at the inlets to better mimic experimental
conditions. Both gases enter the channels at room temperature with a
uniform, flat flow velocity. The reactor exits are held at a fixed
pressure of 8 atmospheres and the normal gradients of species and
temperature, with respect to the direction of the flow, are set to zero.
Overall, the device is adiabatic, and hence no heat losses occur through
the side walls. Radiation losses play a secondary effect on the
operation of the reactor and hence are negligible. Continuity in
temperature and heat flux is applied at the fluid-solid interfaces.
Neither heat-transfer nor mass-transfer correlations are employed since
detailed transport within the solid and fluid phases is explicitly
accounted for.
The full problem is solved via a segregated solver using an
under-relaxation method. Convergence of the solution is monitored
through the residuals of the governing equations and the square root of
the sum of the entries of the vector of successive iterations of the
solution. The solution is deemed converged when the residuals of the
equations as well as the square root of the sum of the entries of the
vector of successive iterations are less than around 10 with a negative
exponent of 6. The coupling of the heat equation in the wall and the
reacting flow equations makes the problem stiff due to the disparity in
thermal conductivity between the gases and the wall. Parallel processing
employing a message passing interface is used to speed up the most
demanding calculations. In most cases, multiple simulations are run
simultaneously. Typical simulation times vary from about several hours
for high wall thermal conductivity and lower flow rates to about a few
days for low wall thermal conductivity and higher flow rates on a single
processor depending on the stiffness of the problem. Natural parameter
continuation is employed to study the effect of various operating
parameters.