Boundary conditions, Dankwert

Equations 6 and 7 are the well known Dankwerts boundary conditions (7). In Equation 5, Kj is the mass transfer coefficient around the gel, cuid cjn (t) the inlet solute concentration which is a function of analysis time. 

Typically, there are two types of boundaries in reacting flows. The first is a solid surface at which a reaction may be occurring, where the flow velocity is usually set to zero (the no-slip condition) and where either a temperature or a heat flux is specified or a balance between heat generated and lost is made. The second type of boundary is an inflow or outflow boundary. Generally, either the species concentration is specified or the Dankwerts boundary condition is used wherein a flux balance is made across the inflow boundary (64). The gas temperature and gas velocity profile are usually specified at an inflow boundary. At outflow boundaries, choices often become more difficult. If the outflow boundary is far away from the reaction zone, the species concentration gradient and temperature gradient in the direction of flow are often assumed to be zero. In addition, the outflow boundary condition on the momentum balance is usually that normal or shear stresses are also zero (64). 

The equations describing the solid phase are also valid at z=0 and z=L, but for the gas-phase the boundary conditions differ for each process step,. t the end of the reactor three different cases occur there can be a feed flowing inwards, the end can be closed or there can be an outlet flow. Each case is described with the appropriate Dankwerts conditions and the pressure ramp is imposed. The resulting equations are listed in table 1. 

Substitution of this mass balance into the differential equation for the column and application of the usual Dankwerts boundary condition at the column fluid entrance (z = 0) and zero change of concentration flux at the column exit (z = L) leads to a formal solution for the concentration ratio. When plug flow prevails and mass transfer resistance is small, cdco is given by... 

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