Concentration Gradients and Diffusion

Figure 1 shows the schematic of a tubular reactor, of radius a and length L, where a — a/Lis the aspect ratio. Clearly, ifa>S> 1, or a <3C 1, a physical length scale separation exists in the reactor. This length scale separation could also be interpreted in terms of time scales. For example, a 1 implies that the time scale for radial diffusion is much smaller than that of either convection and axial diffusion, and concentration gradients in the transverse direction are small compared to that in the axial direction. 

The basic concept of diffusion refers to the net transport of material within a single phase in the absence of mixing (by mechanical means or by convection). Both experiment and theory have shown that diffusion can result from pressure gradients (pressure diffusion), temperature gradients (thermal diffusion), external force fields (forced diffusion), and concentration gradients. Only the last type is considered in this book that is, the discussion is limited to diffusion caused by the concentration difference between two points in a stagnant solution. This process, called molecular diffusion, is described by Pick s laws. His first law relates the flux of a chemical to the concentration gradient ... 

In the case of a solid catalyst operating in a liquid phase reaction system the problems of diffusion and concentration gradients can be particularly severe. Substrate diffusion can be further broken down into two steps, external diffusion and internal diffusion. The former is controlled by the flow of substrate molecules through the layer of molecules surrounding catalyst particles and is proportional to the concentration gradient in the bulk liquid, i.e. the difference in the concentrations of the substrate in the bulk medium and at the catalyst surface. The thickness of the external layer in a liquid medium is dependent on the flowing fluid and on the agitation within the reaction system typically it is 0.1-0.01 mm thick. Internal diffusion of substrate molecules is a complex process determined not only by the resistance to flow due to the... 

The mode of transport through a membrane may be passive, active, or facilitated. In passive transport, the membrane acts as a barrier, and permeation of the components is determined by their diffusivity and concentration gradient in the membrane. In facilitated transport along with the chemical potential gradient, mass transport is coupled to specific carrier components in the membrane. In active transport, driving force for transport is achieved by a chemical reaction in the membrane phase. 

Hydrogels such as the Q[6]-mediated alginate hydrogel beads discussed in Section 2.2.4 release their drug loads through a process of diffusion and concentration gradients unless some other mechanisms are built into the gel to modify release. This may be a direction for the future for certain drug applications. 

The rate of the chemical reaction is slow compared to internal and external diffusion, and concentration gradients both of the solid and gaseous reactant are negligible. 

When electrons are injected as minority carriers into a -type semiconductor they may diffuse, drift, or disappear. That is, their electrical behavior is determined by diffusion in concentration gradients, drift in electric fields (potential gradients), or disappearance through recombination with majority carrier holes. Thus, the transport behavior of minority carriers can be described by a continuity equation. To derive the p—n junction equation, steady-state is assumed, so that = 0, and a neutral region outside the depletion region is assumed, so that the electric field is zero. Under these circumstances,... 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

Fig. 20.1 Potential and concentration gradients in the electrolytic cell CU/CUSO4/CU. (a) The electrodes are unpolarised the potential dilference is the equilibrium potential and there is no concentration gradient in the diffusion layer. (f>) The electrodes are polarised Ep of the anode is now more positive than E. whilst E of the cathode is more negative and concentration gradients exist across the diffusion layer c, C), are the concentrations at the electrode...

In many practical mass transfer processes, unsteady state conditions prevail. Thus, in the example given in Section 10.1, a box is divided into two compartments each containing a different gas and the partition is removed. Molecular diffusion of the gases takes place and concentrations, and concentration gradients, change with time. If a bowl of liquid... 

In this approach, it is assumed that turbulence dies out at the interface and that a laminar layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by the random movement of the molecules, and the resistance to transfer becomes progressively smaller. For equimolecular counterdiffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Figure 10.5 by the full lines ABC and DEF. The basis of the theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distributions, and the thicknesses of the two films arc L and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases. In Figure 10.5, the scales are not necessarily the same on the two sides of the interface. 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

Pick s first law, 1855). The proportionality factor Dj is called the diffusion coefficient of the substance concerned (units cm%). In the diffusion of ions in solutions, Eq. (4.1) is obeyed only at low concentrations of these ions. At higher concentrations the proportionality between flux and concentration gradient is lost (i.e., coefficient D, ceases to be constant). 

In electrochemical cells we often find convective transport of reaction components toward (or away from) the electrode surface. In this case the balance equation describing the supply and escape of the components should be written in the general form (1.38). However, this equation needs further explanation. At any current density during current flow, the migration and diffusion fluxes (or field strength and concentration gradients) will spontaneously settle at values such that condition (4.14) is satisfied. The convective flux, on the other hand, depends on the arbitrary values selected for the flow velocity v and for the component concentrations (i.e., is determined by factors independent of the values selected for the current density). Hence, in the balance equation (1.38), it is not the total convective flux that should appear, only the part that corresponds to the true consumption of reactants from the flux or true product release into the flux. This fraction is defined as tfie difference between the fluxes away from and to the electrode ... 

If, as is normal, the solution is not stirred, then the conditions of laminar (uniform) diffusion characterising the above description will hold only for a short time. For longer periods, thermal and concentration gradients induce random convection processes and the resultant currents show sizeable fluctuations. 

Displacements of lattice members are determined by energy factors and concentration gradients. To a considerable extent, diffusion in solids is related to the existence of vacancies. The "concentration" of defects, N0, (sites of higher energy) can be expressed in terms of a Boltzmann distribution as... 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

Most solutes in soils are to some extent adsorbed on the soil solid only a small fraction is in the solution in the pores. However some adsorbed solutes, particularly exchangeable cations, can have considerable mobility on soil surfaces (see below), so it is important to consider the solid phase pathway as well as the solution. Because the diffusing solute passes rapidly between the solid and solution, the two pathways partly act in series. In such a heterogeneous medium as soil it is not realistic to account for the mobilities and concentration gradients of solutes in all the constituent parts. But if the soil volumes and reaction times... 

The transmembrane potential of cardiac cells is determined by the concentrations of several ions—chiefly sodium (Na+), potassium (K+), calcium (Ca2+), and chloride (Cl-)—on either side of the membrane and the permeability of the membrane to each ion. These water-soluble ions are unable to freely diffuse across the lipid cell membrane in response to their electrical and concentration gradients they require aqueous channels (specific pore-forming proteins) for such diffusion. Thus, ions move across cell membranes in response to their gradients only at specific times during the cardiac cycle when these ion channels are open. The movements of the ions produce currents that form the basis of the cardiac action potential. Individual channels are relatively ion-specific, and the flux of ions through them is... 

Equilibrium is reached when the driving force for the diffusion (the concentration gradient) is compensated for by the electric field (the potential gradient). Under these equilibrium conditions, there is an equilibrium net charge on each side of the junction and an equilibrium potential difference d< >e. This process is analogous to the way charge transfer across a nonpolarizable electrode/solution interface results in the establishment of an equilibrium potential difference across the interface. 

The coefficient C, related to the resistance to mass transfer between the two phases, becomes important when the flow rate is too high for equilibrium to be obtained. Local turbulence within the mobile phase and concentration gradients slow down the equilibrium process (Cs <=> Cm). The diffusion of solute between the phases is not instantaneous, hence the solute will be in a non-equilibrium process. 

Diffusion processes occur in all systems where concentration differences exist. Diffusion is the main mechanism which aids Ln the elimination of concentration gradients. Fick s first law of diffusion defines this phenomenon by correlating mass flow and concentration gradient. This law may be shown as... 

As diffusion proceeds, concentration and concentration gradient changes will take place as illustrated in Figure 2.4. To ensure that the broadening of the boundary is due to diffusion only, very accurate temperature control (to avoid convection currents) and freedom from mechanical vibration must be maintained. The avoidance of convection is a problem common to all kinetic methods of investigating colloidal systems. 

---