Steady state diffusional flow

Orr [52] developed an apparatus, which was commercially available from Micromeretics, called the Knudsen flow permeameter which was based on the following form of equation (1.61). 

The flowrate of helium passing through a packed bed of powder is measured together with the upstream pressure p and the pressure drop across the bed Ap. Rearranging equation (l.(M) gives the alternative form  

For coarse powder it is necessary to conect for the effect of the support plug and filter paper. 

Equation (1.66) may be further simplified if the time is recorded for the pressure to fall from some preset high pressure (p//) to a preset lower pressure (pj). 

Under standard operating conditions A = 5.005 cm2, y iqoO cm, PH = 40 torr and pi = 20 torr. 

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Use of Operating Curve Frequently, it is not possible to assume that = 0 as in Example 2, owing to diffusional resistance in the liquid phase or to the accumulation of solute in the hquid stream. When the back pressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-3, a steady-state material balance leads to the fohowing equivalent relations ... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

Industrial fixed-bed catalytic reactors have a wide range of different configurations. The configuration of the reactor itself may give rise to multiplicity of the steady states when other sources alone are not sufficient to produce the phenomenon. Most well known is the case of catalytic reactors where the gas phase is in plug flow and all diffusional resistances are negligible, while the reaction is exothermic and is countercurrently cooled. One typical example for this is the TVA type ammonia converter [38-40]. 

An -> ideal nonpolarizable electrode is one whose potential does not change as current flows in the cell. Much more useful in electrochemistry are the electrodes that change their potential in a wide potential window (in the absence of a - depolarizer) without the passage of significant current. They are called -> ideally polarized electrodes. Current-potential curves, particularly those obtained under steady-state conditions (see -> Tafel plot) are often called polarization curves. In the -> corrosion measurements the ratio of AE/AI in the polarization curve is called the polarization resistance. If during the -> electrode processes the overpotential is related to the -> diffusional transport of the depolarizer we talk about the concentration polarization. If the electrode process requires an -> activation energy, the appropriate overpotential and activation polarization appear. 

An LCVD system is somewhat similar to a gas flame in which the combustion rate and the gas flow rate establish a steady-state flame. In an LCVD system, the monomer flow rate and the polymer formation rate establish a steady-state polymer-forming luminous gas phase. This situation is expressed schematically in Figure 20.17, where (a) indicates the diffusional transport of the energy-carrying... 

Gas-solid reactions are carried out on a commercial basis using fixed-bed, moving-bed, and fluidized-bed reactors. The fixed-bed reactor is an unsteady-state system as reactive gas is fed on a continuous basis through the reactor that is packed with a finite quantity of solid reactant. The solid is depleted and breakthrough of the gas reactant occurs after a certain reaction time. In the moving-bed reactor, both solid and gas are fed on a continuous basis and overall operation is steady state. The fluidized-bed reactor, where small solid particles are fluidized by upward flow of gas, also operates in a steady-state manner. Diffusional reaction resistances are reduced because of the small solid particles while solid backmixing reduces solid concentration gradients and promotes isothermal operation. 

For a water column of such physical characteristics, the ratio K/U, the scale height, is approximately 900 m. For distances much greater than the scale height, the transport resulting from flow is much more important than the transport by eddy diffusional dispersal. Conversely, for distances much shorter than the scale height, the eddy diffusional mode is the main mechanism of transport. In such a water column, the time required for the concentration to attain a steady-state will be calculated for chemical species reacting by two different mechanisms ... 

The simplest treatments of convective systems are based on a diffusion layer approach. In this model, it is assumed that convection maintains the concentrations of all species uniform and equal to the bulk values beyond a certain distance from the electrode, 8. Within the layer 0 x < 5, no solution movement occurs, and mass transfer takes place by diffusion. Thus, the convection problem is converted to a diffusional one, in which the adjustable parameter 8 is introduced. This is basically the approach that was used in Chapter 1 to deal with the steady-state mass transport problem. However, it does not yield equations that show how currents are related to flow rates, rotation rates, solution viscosity, and electrode dimensions. Nor can it be employed for dual-electrode techniques or for predicting relative mass-transfer rates of different substances. A more rigorous approach begins with the convective-diffusion equation and the velocity profiles in the solution. They are solved either analytically or, more frequently, numerically. In most cases, only the steady-state solution is desired. 

The boundary condition at the membrane wall derives from conservation of solute flux applied across the membrane. Stated in words At steady state the bulk flow of solute toward the membrane minus the diffusional flux of the solute away from the membrane toward the bulk of the fluid must equal the solute permeation through the membrane. Mathematically, taking into account that the y coordinate is in the direction of the concentration gradient and opposite to the permeation velocity,... 

Now, consider a one-dimensional parallel flow of two phases either in co- or countercurrent flow, exchanging mass and heat with each other. Neglecting diffusional (or dispersion) terms, in steady state the balance equations become... 

Chapters 9 to 11 deal with the dynamic analysis of a single particle exposed to a constant bulk environment. The method of differential adsorption bed discussed in Chapter 11 is suitable for the application of the single particle analysis. A permeation method called the time lag method is useful for characterisation of diffusional flow, viscous flow and surface flow of pure gas through a single pellet (Chapter 12). The diffusion cell method either in steady state mode or transient mode is useful to characterize binary diffusional systems (Chapter 13). All these methods evolve around the analysis of a single particle and they complement each other in the characterization of diffusion and adsorption characteristics of a system. From the stand point of system set-up, the time lag and diffusion cell methods require a careful mounting of a particle or particles between two chambers and extreme care is exercised to avoid any gas by-passing the particle. 

We observe that only one parameter a governs the steady state behavior of ideal reactors provided the catalyst is working in pure diffusion regime.Moreover,in a plug flow reactor,there is a critical value for a (or in practical terms a critical heirht for the reactor) at which complete conversion is obtained,i.e., u=2. This is because for the diffusional regime a zero order reaction is equivalent to a 1/2 order reaction in the kinetic regime. 

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