Transfer by molecular diffusion

Heat and mass transfer are analogous processes. Molecular diffusion in homogeneous materials or phases is similar to heat transfer. Convective diffusion or convection in homogeneous materials or phases corresponds to heat transfer by convection. Mass transfer at the phase boundary corresponds to heat conduction. Mass transfer between phases occurs like heat transfer in several chronological steps. The slowest step controls the rate of the entire process. Thus the mathematical descriptions of heat and mass transfer operations are analogous. Calculation methods and approaches to calculate the heat transfer coefficients may similarly be used to calculate mass transfer coefficients. (See Table 1-18 in Chapter 1.7.2 for the analogy of heat and mass transfer.) 

X X coordinate of the diffusion space Dj diffusion coefficient of component i in the diffusion space (m /h) 

Introducing the partial pressure of gases as measure of concentration, Pick s first law becomes 

Pick s first law describes equimolar diffusion, in which all components of the system may diffuse independent from each other. During thermal separation processes, matter is transported through phase boundaries. If a phase boundary is selectively permeable to one component, only one-directional diffusion is possible (an especially important case for absorption, adsorption, and drying). For one-directional diffusion, Stefan s law gives 

---

For mass transfer by molecular diffusion from a single sphere of diameter d to an infinite stationary medium, it can be shown that... 

Laminar sub-layer. 0 < y+ < 5, Newton s Law of Viscosity (4), describes the flow and gives u+ = y+ here laminar flow and transfer by molecular diffusion dominate. 

Transfer by molecular diffusion is discussed in Section 12.2 and the concept of the mixing length in Section 12.3.2. By analogy with kinetic theory, the eddy kinematic viscosity, E, is given by ... 

The data available on the molecular diffusion coefficient of organic vapors in air are meager, but they indicate (in accordance with approximate theory) an inverse proportionality to the square root of molecular weight. The rate of mass transfer by molecular diffusion will be proportional to the diffusion coefficient and to the SVC, itself proportional to vapor pressure times molecular weight (M). We should expect, therefore, under standard conditions of ventilation, that the rate of loss will be proportional to vapor pressure X The ratio of observed rate to... 

Several possibilities are summarized in Figure 14—30. In the trivial case (case homogeneous mixture, there will be no mass transfer by molecular diffusion or convection since there is no concentration gradient or bulk motion. The next case (case b) corresponds to the flow of a well-mixed fluid mixture tlvrough a pipe. Note that there is no concentration gradients and thas molecular diffusion in this case, and all species move at the bulk flow velocity of Vt The mixture in the thud case (case c) is stationary (17= 0) and thus it corresponds to ordinary molecular diffusion in sfationary mediums, which we discussed before. Note that the velocity of a species at a location in this... 

As explained in Section II,A, when a soluble gas is mixed with an insoluble gas, it must diffuse through the latter to reach the interface. It is usual to refer to a gas film resistance. This implies a stagnant film of gas across which the soluble gas is transferred by molecular diffusion from the bulk gas with partial pressure p to the interface where the partial pressure is p,. If the component B has negligible vapor pressure, the reaction will proceed only in the liquid phase. 

Mass transfer by molecular diffusion is a result of molecular collisions on the microscopic scale. Fick s law states that mass flux due to molecular diffusion is proportional to the gradient of concentration ... 

INTERPRETATION OF DIFFUSION EQUATIONS. Equation (21.16) is the basic equation for mass transfer in a nonturbulent fluid phase. It accounts for the amount of component A carried by the convective bulk flow of the fluid and the amount of A being transferred by molecular diffusion. The vector nature of the fluxes and concentration gradients must be understood, since these quantities are characterized by directions and magnitudes. As derived, the positive sense of the vectors is in the direction of increasing b, which may be either toward or away from the interface. As shown in Eq. (21.6), the sign of the gradient is opposite to the direction of the diffusion flux, since diffusion is in the direction of lower concentrations, or downhill, like the flow of heat down a temperature gradient. 

Ammonia, NH3, is being selectively removed from an air-NH3 mixture by absorption into water. In this steady-state process, ammonia is transferred by molecular diffusion through a stagnant gas layer 5 mm thick and then through a stagnant water layer 0.1 mm thick. The concentration of ammonia at the outer boundary of the gas layer is 3.42 mol% and the concentration at the lower boundary of the water layer is essentially zero. 

Pictures of bubbles and clouds have inspired some workers to develop reactor models based on the predicted behavior of individual bubbles [3,10]. In these models, the equations for gas interchange include a term for flow out of the bubble and a second term for mass transfer by molecular diffusion to the dense phase. In some models, the cloud is included as part of the bubble in others, diffusion from bubble to cloud and cloud to dense phase are treated as mass transfer steps in series. In these models, the mass transfer coefficient is assumed to vary with following the penetration theory, and the diffusion contribution is the major part of the predicted gas interchange rate. 

Monte Carlo simulation was carried out by Blauch and Saveant based on a percolation process, and Z>app was obtained as shown in Eq. (14-4) considering charge hopping and bounded motion of the redox center [14]. Bounded motion is a kind of local oscillation of redox molecules. In this model, charge transfer by molecular diffusion is not taken into account. 

The basic equation for mass transfer by molecular diffusion is Pick s law... 

Introduction. When a fluid is flowing in laminar flow and mass transfer by molecular diffusion is occurring, the equations are very similar to those for heat transfer by conduction in laminar flow. The phenomena of heat and mass transfer are not always completely analogous since in mass transfer several components may be diffusing. Also, the flux of mass perpendicular to the direction of the flow must be small so as not to distort the laminar velocity profile. 

Mass transfer by molecular diffusion resides at the other end of the spectrum. It yields the lowest possible mass transfer rate and sets an upper limit on time requirements. The solvent spill considered in Practice Problem 4.4, for example, requires several days to complete evaporation by diffusion into stagnant air. The same data applied to turbulent air flow at the same temperature yield an estimate of several minutes, lower by three orders of magnitude. The factor of 1000 can thus be viewed as separating the two extremes of diffusive and turbulent mass transfer. 

Since the fluid is stationary at the solid surface, the mass of species i is transferred by molecular diffusion normal to the surface and is expressed by the diffusion rate equation (Equation 1.78) as... 

The mass transfer by molecular diffusion from the surface is carried away... 

The velocity of falling absorbent is very low, and mass is transferred by molecular diffusion. The absorption rate is low so as to keep the density of absorbent remains unchanged. 

---