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.