Unsteady-state mass transfer

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 

At position y, the fluxes N A and N B will be as given in Table 10.6. At a distance y + 8y from the origin, that is at the further boundary of the element, these fluxes will increase by the amounts shown in the lower part of Table 10.6. 

Equation 10.66 is referred to as Fick s Second Law. This also applies when up is small, corresponding to conditions where C, is always low. This equation can be solved for a number of important boundary conditions, and it should be compared with the corresponding equation for unsteady state heat transfer (equation 9.29). 

For the more general three-dimensional case where concentration gradients are changing in the x, y and z direcdons, these changes must be added to give  

In general, it is necessary to specify the physical constraints operating on the system in order to evaluate the bulk flow velocity uF. In gas absorption, there will be no overall 

---

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

The equations represented by (230) reduce to the familiar equations for the special ideal case of unsteady-state mass transfer without coupling in a binary system if we let... 

A deep pool of ethanol is suddenly exposed to an atmosphere of pure carbon dioxide and unsteady state mass transfer, governed by Fick s Law, takes place for 100 s. What proportion of the absorbed carbon dioxide will have accumulated in the 1 mm layer closest to the surface in this period ... 

Table IV includes theoretical transition times (C2, R14, SI7c) in laminar flow between parallel plates, following a concentration step at the wall (Leveque mass transfer). Clearly, in laminar flow (Re 100 or lower), transition times are comparable to those in laminar free convection. Here, however, the dependence on concentration (through the diffusivity) is weak. The dimensionless time variable in unsteady-state mass transfer of the Leveque type is...

Experimental data relative to unsteady-state mass transfer as a result of a concentration step at the electrode surface are not available. However, for a linear increase of the current to parallel-plate electrodes under laminar flow, Hickman (H3) found that steady-state limiting-current readings were obtained only if the time to reach the limiting current at the trailing edge of the plate (see Section IV,E), expressed in the dimensionless form of Eq. (18), is... 

Unsteady-state mass transfer caused by excessively fast current or potential ramps. This is especially likely to occur in measurements involving laminar flow past elongated surfaces and in free-convection studies, in which the establishment of secondary flow patterns may require long times. A compromise between the time sufficient to reach steady-state transport and the time necessary to avoid bulk depletion and surface roughening (in metal deposition) is required, and is found most reliably by preliminary experimentation. 

K. Unsteady-State Mass Transfer to a Rotating Disk... 

The basic equation for unsteady state mass transfer is ... 

The theory of the unsteady-state mass transfer (surface renewal theory) starts... 

A complete parametric study of the unsteady state mass transfer model clearly shows that tp, the pulse period, 4, the polarization, A, the aspect ratio, and DF, the duty factor have a profound effect on the evolution and the final shape of the deposit. Large polarization s and aspect ratios lead to deposition that is mass transfer controlled. This results in keyhole formation, as the concentration gradient inside a high aspect ratio trench is very large. On the other hand, when the deposition is kinetically controlled (i.e. for small values of polarization and aspect ratio) the gradient down the length of the trench is much smaller and deposition proceeds at nearly the bulk concentration. This leads to conformal deposition, as there is negligible variation in the deposition rate at the mouth and at the bottom of the trench. 

Ratio of the species diffusion rate to the rate of species storage Unsteady-state mass transfer... 

Answer For unsteady-state mass transfer across expanding gas-liquid interfaces via the penetration theory, x = and y =. Since there is no convective mass transfer parallel to the interface, z = 0. 

Brauer, H. Unsteady state mass transfer through the interface of spherical particles, InL J. Heat Mass Transfer 21 (1978) p. 445/453,455/465... 

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. 

In Equation 6.18, denotes the depth of penetration by an eddy in the neighboring mass transfer boundary layer. Harriott s analysis of his data indicated that both slip velocity and transient effects are important in determining the overall mass transfer process. Although Harriott could not provide actual slip velocity and unsteady-state mass transfer data, his conclusions based on effects of diffusivity and particle density on in stirred tanks are quite logical. In his conclusion, Harriott argued that the slip velocity theory is relatively easy to use for quantitative predictions as compared to the modified penetration theory. 

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. 

---