Time constant for diffusion

However, even with the most advanced measuring and simulation tools, the most efficient methods are simple calculations that give an order-of-magnitude estimation of the influence of a phenomenon. Time constants for diffusion, heat conduction, and acceleration are very useful. For example, the time constant for diffusion Td = f/D is the time it takes to fill a cube of size I by diffusion, and the time for a particle to accelerate from zero velocity to approximately two-third of the velocity of the surrounding fluids is 118/j, where p[Pg.331]

Diffusion in liquids is very slow. Turbulent transport or very narrow channels are necessary for good contact between the phases. The droplets must also be very small to minimize transport hmitations within the drops. An estimation of the time constant for diffusion in a 1-mm drop is (f (10-3)2... 

According to Pohorecki and Baldyga (1995) the characteristic time constant for diffusion is given by... 

For large K, values, the uptake curve depends only upon the value of the parameter (1 representing the ratio of characteristic time constants for diffusion in the pores and in the subparticles. For small (1 values, diffusion in the subparticles is controlling and the solution coincides with Eq. (16-96) with r, replacing rp. For large (3 values, pore diffusion is controlling, and the solution coincides with Eq. (16-96) with ZpDp/iZp + pplQ replacing D . 

The liquid film has a varying thickness and is alternately exposed to the gas and to the liquid with different concentrations. However, the film damps the effect of varying concentration, and the concentration at the wall is almost constant. The time constant for diffusion in the liquid film is bj/2D = 0.1 sec. (Eq. 32), and the contact time for the gas bubble and the liquid slug is 0.02 sec. Thus the wall concentration will be almost constant, and the mass transfers directly from the gas bubble and through the liquid slug can be added using the same driving potential. 

The characteristic time for diffusion inside a cloud or rain drop follows from the mathematical solution of Eq. (1-19). Diffusion coefficients in the aqueous phase are of the order of DL= 1.8 x 10-9m2/s. The resulting time constants for the approach to uniform concentrations are shown in Fig. 8-10. For large drops the equilibration takes longer than the fall time over a distance of 100 m. Large drops, however, also develop an internal circulation due to frictional drag that enhances mass transport by mixing. The time constant for diffusion inside a water drop thus represents an upper limit to the true mixing time. 

Higher rate of hthium ion insertion /de-insertion because of short hthium ion transport distance. The characteristic time constant for diffusion, t, is given by, T = L /D where L is the diffusion length and D is the diffusion coefficient Thus, the time for interealahon decreases substantially as micron-sized powders are replaced by nanometer particles. 

In bidisperse porous adsorbents such as zeolite pellets there are two diffusion mechanisms the macropore diffusion with time constant Rp /Dp and the micropore diffusion with time constant rc /Dc. Bidisperse porous models for ZLC desorption curves have been recently developed by Brandani [28] and Silva and Rodrigues [29]. In bidisperse porous adsorbents, it is important to carry out experiments in pellets with different sizes but with the same crystal size (different Rp, same rc) or pellets with the same size but with different crystals (same Rp, different rc). If macropore diffusion is controlling, time constants for diffusion should depend directly on pellet size and should be insensitive to crystal size changes. If micropore diffusion controls the reverse is true. The influence of temperature is also important when macropore diffusion is dominant the apparent time constant of diffusion defined by Rp2(H-K)/Dp is temperature dependent in the same order of K (directly related to the heat of adsorption) which is determined independently from the isotherm. The type of purge gas is... 

The reciprocal of the apparent time constants for diffusion Dp /Rp2(l+K) of nC.5 and nCe (calculated from ZLC desorption curves) plotted versus 1/T are shown in Figure 4. A strongly temperature dependence of time constants exists which is of the order of heat of adsorption. Time constants range from 0.002 s at 473K up to 0.03 s at 573K in the system He/nC.5, and from 0.00035 s at 473K up to 0.0053 s at 573K in the system He/nCe. 

Ammonia (NH3) or ammonium (NH4+) can exist in both the fuel and air streams. The diffusion of ammonium is fast, therefore, the ammonium entering the fuel cell from either side can quickly diffuse to the other side causing the contamination effect on both sides. For instance, for a typical membrane with a thickness of 10 to 100 jim, the estimated characteristic time constant for diffusion is 1 to 100 sec [149]. Ammonia may affect the PEMFC performance in different ways (1) by the reduction of the ionic conductivity of the membrane, which in its ammonium form is a factor of 4 lower than in the protonated form [149-151] (2) by poisoning the cathode catalyst [151] and (3) by poisoning the anode catalyst [149]. Recently, fuel cell tests have shown that the reduced membrane conductivity is not the major reason for performance losses induced by ammonia [149,150]. The effect of ammonia on the HOR was found to be minor at current densities below 0.5 A cm", but would increase with increasing current densities. The current density did not exceed 1 A cm in the presence of ammonia [149]. 

From a qualitative analysis of the competition between reaction and diffusion in an isothermal pellet one easily recognizes that the parameter governing the steady state behavior of the pellet is the ratio between time constants for diffusion and reaction,i.e.,T(j/xr-If Tr the reaction rate is much slower than the diffusion rate the concentration profile inside the pellet is then almost flat and equal to the external surface concentration.The effectiveness factor is around unity. However,when the concentration inside the pe-... 

From experimental data ( , Re) it is now possible to calculate the permeability B and the true time constant for diffusion and hence the true effective diffusivity. 

However from experimental results at Re=200 we get a ratio of the apparent time constants for diffusion... 

The mixing in the microchannel takes place by diffusion and convection depending on the flow geometry and the operating conditions used [8,9]. The final mixing on the molecular scale, where the reaction takes place, occurs only by molecular diffusion. The time constant for diffusion in an elementary structure is... 

All variables must be scaled with a variable characteristic of the process. The characteristic velocity U may be the average inlet velocity. The characteristic length L may be the tube diameter, the reactor length, or the catalyst particle diameter, depending on the processes involved. Time may be scaled with residence time or the time constant for diffusion. 

Figure 5.3. Time constants for diffusion in a slab with different boundary conditions, (a) No flux at z = 1 (b) y(f, oo) = 0.

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