Mass transfer equation large Peclet numbers

Numerical solution of the mass transfer equation begins at a small nonzero value of z = Zstart, uot at the inlet where Cp, x, y,z = 0) = Ca, miet for all values of x and y. This is achieved by invoking an asymptotically exact analytical solution for the molar density of reactant A from laminar mass transfer boundary layer theory in the limit of very large Schmidt and Peclet numbers. The boundary layer starting profile is valid under the following condition ... 

Step 5. Use your results from step 4 together with the following assumptions the mass transfer Peclet number is large, and the tube diameter is much smaller than the overall length of the reactor to simplify the mass transfer equation. 

Another issue related to the numerical solution of the continuum equations is that of false diffusion. False diffusion exists when fiows are oblique to the orthogonal mesh and is most severe for large Peclet numbers. It is not a consequence of upwinding, but rather the treatment of variables as locally one-dimensional. The influence of false diffusion can be minimized by adopting fine computation meshes. For binary solid-liquid systems, false diffusion is generally most severe in the species conservation equation since local mass transfer Peclet numbers are large and real Fickian diffusion is small. 

The equations also hold in a corresponding manner for mass transfer. This merely requires the Nusselt number to be replaced by the Sherwood number and the Prandtl by the Schmidt number. Prerequisite for the validity of the equations is however, a sufficiently large value for the Peclet number Pe = RePr A 500... 

We have already noted that mass transfer in a liquid is almost always characterized by large values of the Peclet number (the Peclet number for mass transfer involves the product of the Schmidt number and Reynolds number instead of the Prandtl number and Reynolds number) and that the dimensionless form of the convection-diffusion equation governing transport of a single solute through a solvent is still (9-7), with 6 now being a dimensionless solute concentration. For transfer of a solute from a bubble or drop into a liquid that previously contained no solute, the concentration 6 at large distances from the bubble or drop will satisfy the condition... 

Consider a liquid-phase plug-flow tubular reactor with irreversible nth-order endothermic chemical reaction. The reactive mixture is heated with a fluid that flows cocurrently in the annular region of a double-pipe configuration. The mass and heat transfer Peclet numbers are large for both fluids. All physical properties of both fluids are independent of temperature and conversion, and the inlet conditions at z = 0 are specified. What equations are required to investigate the phenomenon of parametric sensitivity in this system ... 

Order-of-magnitude analysis indicates that diffusion is neghgible relative to convective mass transfer in the primary flow direction within the concentration boundary layer at large values of the Peclet number. Typically, liquid-phase Schmidt numbers are at least 10 because momentum diffusivities (i.e., i/p) are on the order of 10 cm /s and the Stokes-Einstein equation predicts diffusion coefficients on the order of 10 cm /s. Hence, the Peclet number should be large for liquids even under slow-flow conditions. Now, the partial differential mass balance for Cji,(r,0) is simplified for axisynunetric flow (i.e., = 0), angu-... 

If there is only one chemical reaction on the internal catalytic surface, then vai = — 1 and subscript j is not required for all quantities that are specific to the yth chemical reaction. When the mass transfer Peclet number which accounts for interpellet axial dispersion in packed beds is large, residence-time distribution effects are insignificant and axial diffusion can be neglected in the plug-flow mass balance given by equation (22-11). Under these conditions, reactor performance can be predicted from a simplified one-dimensional model. The differential design equation is... 

Various models are available to calculate liquid side mass transfer coefficients kj. The value of this hydrodynamic parameter and the equations that apply to its calculation largely depend on bubble size and the constitution of the bubble surface. Fig. 6 presents some recent measurements on mass transfer from single bubbles (19) which demonstrate the above influences. The evaluated kL values are plotted as Sherwood numbers vs. Peclet numbers. Large circulating bubbles with mobile surface yield kj, values which approach the... 

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