Mass transfer unsteady-state solutions

Pales and Stroeve [31] investigated the effect of the continuous phase mass transfer resistance on solute extraction with double emulsion in a batch reactor. They presented an extension of the perturbation analysis technique to give a solution of the model equations of Ho et al. [29] taking external phase mass transfer resistance into account. Kim et al. [5] also developed an unsteady-state advancing reaction front model considering an additional thin outer liquid membrane layer and neglecting the continuous phase resistance. 

In the unsteady-state mass balances for all four components in the second CSTR, the input terms due to convective mass transfer are based on the unsteady-state solutions in the exit stream of the first CSTR [i.e., C,i(t)] ... 

In the theoretical treatment, the heat- and mass-transfer processes shown in Fig. 6 were considered. Simultaneous solution of the equations describing the behavior of the unsteady-state reaction system permits the temperature history of the propellant surface to be calculated from the instant of oxidizer propellant contact to the runaway reaction stage. 

The above model of settler flow behaviour, combined with entrainment backmixing was used by Aly (1972) to model the unsteady-state extraction of copper from aqueous solution, using Alamine 336 solvent. An identification procedure for the relevant flow parameters showed an excellent fit to the experimental data with very realistic entrainment backmixing factors, fL = fQ = 3.5 percent, the fraction of well-mixed flow in the settlers, (XX = ay = 5 percent and an overall mass transfer capacity coefficient, Ka = 25 s->. 

There are no solutions for transfer with the generality of the Hadamard-Rybczynski solution for fluid motion. If resistance within the particle is important, solute accumulation makes mass transfer a transient process. Only approximate solutions are available for this situation with internal and external mass transfer resistances included. The following sections consider the resistance in each phase separately, beginning with steady-state transfer in the continuous phase. Section B contains a brief discussion of unsteady mass transfer in the continuous phase under conditions of steady fluid motion. The resistance within the particle is then considered and methods for approximating the overall resistance are presented. Finally, the effect of surface-active agents on external and internal resistance is discussed. 

Fo being the Fourier number and d the diameter of the disk. The mass transfer coefficient k can be considered as interpolating between the steady-state convective diffusion at large times (t - oo) and unsteady-state diffusion at short times (t — 0 and v = 0). The constants A and B of Eq. (147) follow from the solutions for these two limiting cases. For these two limiting cases... 

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

The first is the penetration theory of Higbie (1935). If the liquid immediately adjacent to a rising bubble is assumed to rise with the bubble, i.e., the relative velocity between the bubble and the liquid is 0, the mass transfer conditions are those of unsteady-state molecular diffusion. The mathematical solution of this problem leads to... 

Multiphase reactors can be batch, fed batch, or continuous. Most of the design equations derived in this chapter are general and apply to any of these operating modes. They will be derived for unsteady operation. The unsteady material balances include the inventories in both phases and mass transfer between the phases so that steady-state solutions fonnd by the method of false transients will be true transients if the initial conditions are correct. Compare Section 10.6. 

INTERNAL AND EXTERNAL MASS-TRANSFER COEFFICIENTS. The overall coefficient depends on the external coefficient and on an effective internal coefficient Diffusion within the particle is actually an unsteady-state process, and the value of decreases with time, as solute molecules must penetrate further and further into the particle to reach adsorption sites. An average coefficient can be used to give an approximate fit to uptake data for spheres ... 

Alternative Approach in the Absence of Liquid-Phase Chemical Reaction. The previous scaling law for the dissolution of spherical solid particles in a surrounding quiescent liquid can be addressed by performing an unsteady-state macroscopic mass balance on the liquid solution, with volume Viiquid- The accumulation of species A is balanced by the rate of interphase mass transfer (MT) when no chemical reaction occurs. Hence,... 

Now, it is instructive to re-analyze the unsteady-state macroscopic mass balance on an isolated solid pellet of pure A with no chemical reaction. The rate of output due to interphase mass transfer from the solid particle to the liquid solution is expressed as the product of a liquid-phase mass transfer coefficient c, liquids a Concentration driving force (Ca, — Ca), and the surface area of one spherical pellet, 4nR. The unsteady-state mass balance on the solid yields an ordinary differential equation for the time dependence of the radius of the peUet. For example,... 

In its simplest form, the penetration theory assumes that a fluid of initial composition x% is brought into contact with an interface at a fixed composition jCa. i for a time t. For short contact times the composition far from the interface (z - oe) remains at xX. If bulk flow is neglected (dilute solution or tow transfer rates), solution of the unsteady-state diffusion equation provides an expression for the average mass transfer flux and coefficient for a contact time 6. 

Chapters 1,4, and 5 emphasized the fact that the rate of mass transfer in multiphase reactors depends on the type and size of the equipment used. The reactors dealt with in this and subsequent chapters are of the type in which the gas phase is dispersed in a continuous liquid phase. The various phases taking part in the overall reaction sequence experience chaotic, turbulent motion in time and space. Under such conditions, mass transfer mainly occurs by a mechanism in which different eddies that come to the interface deliver/receive the solute during their lifetime at the interface and return back to the bulk phase. This unsteady-state mass transfer process has been exhaustively discussed in several texts (Astarita 1967 Danckwerts 1970). In the following, the various approaches to predict mass transfer coefficients in different multiphase reactors are discussed along with the advantages/drawbacks of each approach. 

A widely used method for adsorption of solutes from liquid or gases employs a fixed bed of granular particles. The fluid to be treated is usually passed down through the packed bed at a constant flow rate. The situation is more complex than that for a simple stirred-tank batch process which reaches equilibrium. Mass-transfer resistances are important in the fixed-bed process and the process is unsteady state. The overall dynjimics of the system determines the efliciency of the process rather than just the equilibrium considerations. 

Rate of leaching when diffusion in solid controls. In the case where unsteady-state diffusion in the solid is the controlling resistance in the leaching of the solute by an external solvent, the following approximations can be used. If the average diffusivity Da eff of the solute A is approximately constant, then for extraction in a batch process, unsteady-state mass-transfer equations can be used as discussed in Section 7.1. If the particle is approximately spherical. Fig. 5.3-13 can be used. 

The physical and mathematical similarity of equations (10-19) through (10-21) mean that the extensive published solutions of Carslaw and Jaeger (2) for heat transfer and Crank (3) for mass transfer can be used interchangeably to deal with either unsteady state heat or mass transfer. 

We now turn to estimating the effects of mass transfer on the breakthrough curves. These effects are complicated because of the unsteady-state process and the nonlinear isotherms. This means that the analysis is more elaborate than for the separation processes discussed in earlier chapters. Because of this complexity, we consider only one approximate analysis we assume that the adsorption is irreversible. While the analysis is not especially valuable quantitatively, it does let us estimate how the quality of the separation will change with variables like solution flow and adsorbent particle size. 

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