Simultaneous Diffusion and Convection

By extending our definition of diffusion to include the process of heat transfer by conduction, the examples that follow demonstrate how some problems involving diffusion and flow can be treated effectively with the mathematical tools previously discussed. For example, consider the problem involving heat transfer to a flowing fluid. This problem was solved [12] with the use of the confluent hypergeometric function [13]. 

An example of heat transfer to a laminar flow fluid in a circular tube, this, the so-called Graetz problem, involves the determination of the temperature profile in a fully developed laminar flow fluid inside a circular tube. 

The governing equation for the Graetz problem may be obtained from an energy balance in cylindrical coordinates. Alternatively, one can start with the equation of energy in terms of transport properties for Newtonian fluids of 

The above equation 7.47 can be further simplified if both the terms 

Therefore, the general solution to Equation 7.55 is y(w) = C2yi(w) + Ciy2(w) 

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Frisch, H. L. Gas permeation through membranes due to simultaneous diffusion and convection. J. Phys. Chcm. 60, 1177 (1956). 

Fig. 12.2. Steady-state solute concentration profiles in simultaneous diffusion and convection across a membrane of uniform properties. Numbers adjacent to profiles indicate values of the Peclet number whose sign depends upon the direction of the volumetric flux relative to the extemed solution...

The above considerations deal with a point on the membrane. Expansion of these concepts to the dialyzer as a whole requires that the feed- and dialysate-side solute concentrations at a point on the membrane be expressed in terms of known concentrations. Such an approach has been developed by Schindhelm et al., who, by making the assumptions of a zero inlet dialysate concentration and a linear dialysate concentration profile, developed the following expression for the dialysance of solutes greater than 3(X) daltons molecular weight in a countercurrent dialyzer in the presence of simultaneous diffusion and convection. [Note that the equations for X and Y given in Ref. 37 contain misprints the correct expressions are as given in Eqs. (21.1-22) and (21.1-23).]... 

Examples are provided from heat transfer mass transfer simultaneous diffusion and convection simultaneous diffusion and chemical reaction simultaneous diffusion, convection, and chemical reaction and viscous flow. 

Chapter 7 is dedicated entirely to worked-out examples taken from the chemical engineering research literature. This chapter relies on the mathematics of the previous six chapters to solve problems in heat transfer mass transfer simultaneous diffusion and convection simultaneous diffusion and chemical reaction simultaneous diffusion, convection, and chemical reaction and viscous flow. 

Scherer PW, Shendalman LH, Greene NM. Simultaneous diffusion and convection in single breath lung washout. Bull Math Biophys 1972 34 393-412. 

For interactions between packaging and product the above descriptions of both material transport processes by diffusion and convection as well as the simultaneous chemical reactions come into consideration. The general transport equation (7-10) is the starting point for solutions of all specific cases occurring in practice. Material loss through poorly sealed regions in the package can be considered as convection currents and/or treated as diffusion in the gas phase. 

Let us briefly digress and consider the situation that diffusion and convection occur simultaneously. Then the flux becomes... 

Generally, single root models rely on solute transport theory to determine the supply of nutrients to the root. Solutes in soil are assumed to be moved by the additive and simultaneous processes of diffusion and convection. The governing equation in radial coordinates is (Nye and Marriott 1969 Barber 1995)... 

B2. Instead of treating diffusion and convection terms as additive Eq. ri5-16al. what other approaches could be used to analyze simultaneous convection and diffusion ... 

In reality, as one moves away from the interface towards the bulk solution, the contribution of convection to transport increases while that of diffusion decreases. Rather than treating simultaneously transport by diffusion and convection, the Nernst model makes a clear separation between the two transport mechanisms a total absence of convection inside the Nernst diffusion layer (y < S), and an absence of diffusion outside the Nernst diffusion layer (y > S). The intensity of convection affects the flux at the electrode by fixing the thickness of the Nernst diffusion layer. For the remainder of this book, the Nernst diffusion layer will simply be called the diffusion layer. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

Consider the unit square domain of the sine flow, where the right side is filled with reactant A and the left side contains only B. The initial interface between the two components is a vertical line along the center and one edge of the box. At t = 0 convective mixing is turned on in the model simultaneously with diffusion and reaction. The species balance equation in a dimensionless form for the flow with reaction, diffusion, and convection is... 

The solute transfer from one bulk solution to the other during the establishment of equilibrium involves diffusion and convection in the bulk and adsorption - desorption processes across the interface. Both hydrodynamical and physicochemical effects are simultaneously involved. We will analyze them successively, separating them rather arbitrarily, then examine how they are coupled. 

At intermediate temperatures, both diffusion and convection will be important, because the processes take place simultaneously. To understand such intermediate cases, we must look at how mass transport works. 

Mathematical approaches used to describe micelle-facilitated dissolution include film equilibrium and reaction plane models. The film equilibrium model assumes simultaneous diffusive transport of the drug and micelle in equilibrium within a common stagnant film at the surface of the solid as shown in Figure 7. The reaction plane approach has also been applied to micelle-facilitated dissolution and has the advantage of including a convective component in the transport analysis. While both models adequately predict micelle-facilitated dissolution, the scientific community perceives the film equilibrium model to be more mathematically tractable, so this model has found greater use. 

The effectiveness of the internal O2 transport by diffusion or convection depends on the physical resistance to movement and on the O2 demand. The physical resistance is a function of the cross-sectional area for transport, the tortuosity of the pore space, and the path length. The O2 demand is a function of rates of respiration in root tissues and rates of loss of O2 to the soil where it is consumed in chemical and microbial reactions. The O2 budget of the root therefore depends on the simultaneous operation of several linked processes and these have been analysed by mathematical modelling (reviewed by Armstrong... 

Bakd et al. theoretically analyzed simultaneous gas flow and diffusion in Weibel s symmetric model. Th applied a time-varying flow with simultaneous longitudinal diffusion and concluded that convective mixing is much less important than mixing induced by molecular diffusion. 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

The processes of convection, axial diffusion, radial diffusion, and chemical reaction in the liquid and tissue layers all occur simultaneously. A rigorous approach requires solution of several simultaneous differential equations. To avoid this complexity in preliminary models, the transfer... 

Although convection, axial diffusion, and radial diffusion actually occur simultaneously, a multistep procedure was adopted in the finite-difference calculation. For each 5-cm increment in tidal volume and for each time increment At, the differential mass-balance equations were solved for convection, axial difihision, and radial diffusion in that order. This method may slightly underestimate the dosage for weakly soluble gases, because the concentration gradient in the airway may be decreased. 

Since the possible simultaneous presence of three phenomena— convection, diffusion, and reaction—complicate analyses, it can be helpful to eliminate one of the three formally by means of a transformation. This is achieved in the analysis of the counterflow flame [184] by writing the equations in a convection-free form that is, by finding a transformation to a new spatial variable such that the convective and diffusive terms coalesce into a... 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

Flow along uncharged surfaces has been considered in secs. I.6.4f and e. surface conduction in sec. I.6.6d and mixed transport phenomena, simultaneously involving electrical, mechanical and diffusion types of transport In sec. 1.6.7. Specifically the Nemst-Planck equation ((1.6.7.1 or 2]) is recalled, formulating ion fluxes caused by the sum-effect of diffusion, conduction and convection. 

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