Bulk flow correction factor

Some clarifications on the mass-transfer coefficients for use in the fiux expressions (3.1.139) are also in order. If the mass-transfer coefficient under convection was measured or correlated under particular conditions of A/r (= NiJ(N + Alsz) or A/az/(A/az + AIbz) ), then either of the btisic relations (3.1.126) or (3.1.134) should be used to determine the value of (T>ABC,/<5g) or (DjsCt/Si) as the case may be. These values are simply the mass-transfer coefficient for equimolar counterdiffusion in the presence of convection. For any other value of Ng, the value of k j, kcj or kg in equation (3.1.139) is obtained by using the bulk flow correction factor... 

One could also develop an analysis based on individual film coefficients, as we have done in equations (8.1.62(a)-d). Further, if the film coefficient/overall coefficient available is based on equimolar counterdiffusion, then a correction using the bulk flow correction factor has to be used for gas absorption/stripping therefore the expression for NTU will contain additional terms via... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

To assess the physical deviation between the average of products and the product of averages a momentum velocity correction factor can be defined by Cm = vz) / v1)a- By use of the Hagen-Poiseuille law (1.353) and the power law velocity profile (1.354) it follows that at steady state Cm has a value of about 0.95 for turbulent flow and 0.75 for laminar flow [55]. In practice a value of 1 is used in turbulent flow so that v1)a is simply replaced by the averaged bulk velocity vz) - On the other hand, for laminar flows a correction factor is needed. For more precise calculations a simplified (not averaged ) 2D model is often considered for ideal axisymmetric pipe flows [52, 69]. 

Pi is inlet pressure, microns pe is exit pressure, microns pm is mean pressure, microns n is viscosity at atmospheric pressure, poises Equation (3) reverts to Poiseuille s Law at sufficiently high pressures, where the second term in the large parenthetical factor becomes negligible compared to unity. As the pressure is decreased and intermolecular collisions become less frequent, a flow velocity profile is established where the forward velocity component near the wall becomes a finite value which increases with lower pressures. The reason for this condition is that at these pressures many molecules can stream from the bulk of flow, where the forward velocity is relatively high, to the wall, without suffering collisions with molecules having low forward velocities. Gas flow under such conditions is termed slip flow. Pressures corresponding to this type of flow are such that the second term of the correction factor in equation (3) is finite compared to unity. At lower pressures, where the mean free... 

This equation correctly predicts that C, surface - Q, bulk gas is positive for products and negative for reactants, because Ca. bulk gas > Ca, surface for reactant A. The kinetic rate law is evaluated on the external surface of the pellet, and the effectiveness factor is employed to predict the average rate of reactant consumption on the internal surface of the catalyst. When molar densities on the external surface are required in the rate law, (30-14) should be employed to re-express C,, surface in terms of bulk gas-phase conditions. This is advantageous because the plug-flow mass balance for species i is written in terms of C,-,buik gas-... 

The factor fj was measured experimentally (Carman, 1956) to fall between 0.8 and 1. Taking the average value of 0.9, the ratio of the Knudsen flux to the slip flux is about 1.4, meaning that the Knudsen flux under the molecular flow regime is higher than the flux predicted by the extra slip on the viscous flow. The reason for this is because the slip equation is an extension of the viscous flow, that is the fluid is still in its bulk state to induce viscosity. Therefore, it does not predict correctly the observed flux when Kn l, that is when the true molecular flow is dominant. 

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