Figure 2. Physical representation of the adjacent heat exchange microchannels in which contact time is maintained about constant but the length of each stage increases progressively to accommodate the higher flow rate in each stage.
The law of conservation of mass is stated as
in which ρ denotes the density and u denotes the velocity.
The law of conservation of energy in the solid phase is stated as
wherein λ is the thermal conductivity.
The law of conservation of energy in the gas phase is stated as
in which T is the temperature, Y is the mass fraction, M is the molar mass, M' is the mean molar mass, V is the diffusion velocity, h is the enthalpy, and DT and Dm are the thermal and molecular diffusivity.
The law of conservation of surface species is stated as
in which θ denotes the coverage, γ denotes the active site density, and  denotes the production rate.
The law of conservation of gas-phase species is stated as
The law of conservation of momentum is stated as
wherein μ denotes the dynamic viscosity.
3. Results and discussion
The hydrogen mole fraction contour map in the reactor is plotted in Figure 3 wherein the leading edge of the subsurface layer has a tapered shape that corresponds to the angle of cross-bars in the top-most layer so that fluids from the bulk flow path are not trapped beneath the top-most layer. A hydraulic diameter of a channel is defined as four times the cross-sectional area of the channel divided by the length of the channel's wetted perimeter. A manifold is a volume that distributes flow to two or more connecting channels. The entrance, or inlet, surface of a header manifold is defined as the surface in which marks a significant difference in header manifold geometry from the upstream channel. The exit, or outlet, surface of the footer manifold is defined as the surface which marks a significant difference in the footer manifold channel from the downstream channel. For rectangular channels and most other typical manifold geometries, the surface will be a plane; however, in some special cases such as hemi-circles at the interface between the manifold and connecting channels it will be a curved surface. The value of the Reynolds number describes the flow regime of the stream. While the dependence of the regime on Reynolds number is a function of channel cross-section shape and size. The porous supports may be stacked between a heat transfer wall and a sheet with apertures. Alternatively, the porous supports may be etched, cut or otherwise have active surface feature grooves placed within the sheets. The sheets may be stacked with non-porous sheets that serve as walls to form an assembly. An active catalyst layer or layers may be disposed upon the large pore support. A surface feature is a projection from, or a recess into, a microchannel wall that modify flow within the microchannel. If the area at the top of the features is the same or exceeds the area at the base of the feature, then the feature may be considered recessed. If the area at the base of the feature exceeds the area at the top of the feature, then it may be considered protruded. The surface features have a depth, a width, and a length for non-circular surface features. Surface features may include circles, oblong shapes, squares, rectangles, checks, chevrons, zig-zags, and the like, recessed into the wall of a main channel. The features increase surface area and create convective flow that brings fluids to a microchannel wall through advection rather than diffusion. Flow patterns may swirl, rotate, tumble and have other regular, irregular and or chaotic patterns, although the flow pattern is not required to be chaotic and, in some cases, may appear quite regular. The flow patterns are stable with time, although they may also undergo secondary transient rotations. The surface features are preferably at oblique angles, neither parallel nor perpendicular to the direction of net flow past a surface. Surface features may be orthogonal, that is at a 90-degree angle, to the direction of flow, but are preferably angled. The active surface features are further preferably defined by more than one angle along the width of the microchannel at least at one axial location. The two or more sides of the surface features may be physically connected or disconnected. The one or more angles along the width of the microchannel act to preferentially push and pull the fluid out of the straight laminar streamlines.