Velocity axial, streamline conditions

Under streamline conditions, the velocity at the axis will increase from a value u at the inlet to a value 2u where fully-developed flow exists, as shown in Figure 11.9, because the mean velocity of flow u in the pipe is half of the axial velocity, from equation 336. 

There are some special cases in FFF related to the two extreme limits of the cross-field driving forces. In the first case, the cross-field force is zero, and no transverse solute migration is caused by outer fields. However, because of the shear forces, transverse movements may occur even under conditions of laminar flow. This phenomenon is called the tubular pinch effect . In this case, these shear forces lead to axial separation of various solutes. Small [63] made use of this phenomenon and named it hydrodynamic chromatography (HC). If thin capillaries are used for flow transport, this technique is also called capillary hydrodynamic fractionation (CHDF). A simple interpretation of the ability to separate is that the centers of the solute particles cannot approach the channel walls closer than their lateral dimensions. This means that just by their size larger particles are located in streamlines of higher flow velocities than smaller ones and are eluted first (opposite to the solution sequence in the classical FFF mode). For details on hydrodynamic chromatography,see [64-66]. 

In a plug flow reactor all fluid elements move along parallel streamlines with equal velocity. The plug flow is the only mechanism for mass transport and there is no mixing between fluid elements. The reaction therefore only leads to a concentration gradient in the axial flow direction. For steady-state conditions, for which the term IV is zero the continuity equation is a first-order, ordinary differential equation with the axial coordinate as variable. For non-steady-state conditions the continuity equation is a partial differential equation with axial coordinate and time as variables. Narrow and long tubular reactors closely satisfy the conditions for plug flow when the viscosity of the fluid is-low. 

The calculation is carried out for an open-loop circuit, that is, a circuit in which the entrance conditions to the reactor core are constant in time. This implies that the external loop removes all the heat generated in the core. The core proper is a right circular cylinder of height h and radius po. All fluid particles and heat are transported around the circuit at a uniform velocity V, and it is assumed that thermal equilibrium and density changes occur instantaneously with variations in temperature. Moreover, it is assumed that no lateral exchange of heat occurs between adjacent fluid streamlines i.e., the reactor is taken to be a line from the standpoint of heat generation in the fuel. Thus all axial paths through the core are identical. 

However, in microchannels, compressibility and rarefaction change this flow pattern, and none of the above conditions hold good. Density changes in microchannel gaseous flows are appreciable, and the flow can no longer be assumed incompressible. The other effect is due to rarefaction. There is an increase in mean free path A due to a decrease in pressure in microchannels. Thus, the Knudsen number increases along a microchannel. As a consequence, the axial velocity varies with axial distance, lateral velocity component does not vanish, streamlines are not parallel, and pressure gradient is not constant. 

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