Diffusion unsteady

Steady-state heat transfer Unsteady-state heat transfer Convective heat transfer (heat transfer coefficient) Convective heat transfer (heat transfer coefficient) Radiative heat transfer (not analogous with other transfer processes) Steady-state molecular diffusion Unsteady-state molecular diffusion Convective mass transfer (mass transfer coefficients) Equilibrium staged operations (convective mass transfer using departure from equilibrium as a driving force) Mechanical separations (not analogous with other transfer processes) ... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Phenomena of propagation such as vibrations are characterized by equations of hyperbolic type which are essentially different in their properties from other classes such as those which describe equihb-rium (elhptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows ... 

Parabolic The heat equation 3T/3t = 3 T/3t -i- 3 T/3y represents noneqmlibrium or unsteady states of heat conduction and diffusion. 

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

Diffusion-controlled mass transfer is assumed when the vapor or liquid flow conforms to Tick s second law of diffusion. This is stated in the unsteady-state-diffusion equation using mass-transfer notation as... 

Matching the flow between the impeller and the diffuser is complex because the flow path changes from a rotating system into a stationary one. This complex, unsteady flow is strongly affected by the jet-wake of the flow leaving the impeller, as seen in Figure 6-29. The three-dimensional boundary layers, the secondary flows in the vaneless region, and the flow separation at the blades also affects the overall flow in the diffuser. 

Dawes, W., 1995, A Simulation of the Unsteady Interaction of a Centrifugal Impeller with its Vaned Diffuser Flows Analysis, ASME Journal of Turbomachinery, Vol. 117, pp. 213-222. 

Dead-end Pores Dead-end volumes cause dispersion in unsteady flow (concentration profiles ar> ing) because, as a solute-rich front passes the pore, transport oceurs by molecular diffusion into the pore. After the front has passed, this solute will diffuse back out, thus dispersing. 

The simplified equation (for the general equations, see Section IV, L) in the case of unsteady-state diffusion with a simultaneous chemical reaction in isothermal, incompressible dilute binary solutions with constant p and D and with coupled phenomena neglected is... 

In the film-penetration model (H19), it is assumed that the reactant A penetrates through the surface element by one-dimensional unsteady-state molecular diffusion. Convective transport is assumed to be insignificant. The diffusing stream of the reactant A is depleted along the path of diffusion by its reversible reaction with the reactant B, which is an existing component of the liquid surface element. If such a reaction can be represented as... 

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. 

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... 

Show that the concentration profile for unsteady-state diffusion into a bounded medium of thickness L. when the concentration at the interface is suddenly raised to a consiant value C, and kept constant at the initial value of C at the other boundary is ... 

The unsteady version of the convective diffusion equation is obtained just by adding a time derivative to the steady version. Equation (8.32) for the convective diffusion of mass becomes... 

Unsteady-State Diffusion Through a Porous Solid... 

Figure 4.2. Unsteady-state diffusion through a porous solid.

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

The simulation example DRY is based directly on the above treatment, whereas ENZDYN models the case of unsteady-state diffusion, when combined with chemical reaction. Unsteady-state heat conduction can be treated in an exactly analogous manner, though for cases of complex geometry, with multiple heat sources and sinks, the reader is referred to specialist texts, such as Carslaw and Jaeger (1959). 

Unsteady-State Heat Conduction and Diffusion in Spherical and Cylindrical Coordinates... 

Longitudinal diffusion can be analysed using the unsteady-state diffusion equation... 

In the previous section, we detailed diffusion equations and generalized mass balance equations. We now turn to their practical uses in the pharmaceutical sciences. Mass transport problems can be classified as steady or unsteady. In steady mass transport there is no change of concentration with time [3], characterized mathematically by... 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

E. Unsteady Diffusion into a Semi-Infinite Membrane... 

---