Diffusion dimensionless time

FIG. 16 36 Dimensionless time-distance plot for the displacement chromatography of a binary mixture. The darker lines indicate self-sharpening boundaries and the thinner lines diffuse boundaries. Circled numerals indicate the root number. Concentration profiles are shown at intermediate dimensionless column lengths = 0.43 and = 0.765. The profiles remain unchanged for longer column lengths. 

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...

An analog expression can be found for parallel diffusion in Chapter 8. By integration of this expression over the entire sphere volume the fraction / (r) of initial gas remaining at dimensionless time r becomes... 

From the dimensionless time parameter (t) used in the solution, the diffusion coefficient (D) can be obtained from... 

Fig. 22. Variation of the dimensionless bubble radius and extraction efficiency with dimensionless time for diffusion-controlled bubble growth when the bubble population is constant. P, = 0.10, ifo = 0.10, /3 = 5.87.

A chronomal is a dimensionless parameter [symbohzed by I or /(a)] that is proportional to time. Chronomals are especially useful in deahng with diffusion, chemical reactions, and other related processes. One can chose to express the properties of such systems as t equal to Kb Ib, where Kb contains all the physical constants and has overall units of time, whereas Ib is a chronomal expressed in terms of the extent of reaction C In many respects, the chronomal can be regarded as dimensionless time. 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

The dimensionless time, t, for Sh to come within 100x% of the steady value indicates the duration of the unsteady state for Pe = 0, Tq.i == 31.8, and — 2.35. Diffusivities in gases are of order 10" times diffusivities in liquids hence, for particles with equal size and equal exposure, transient effects in a stagnant medium are much more significant in liquids. 

Self-diffusion coefficients were calculated by Carman-Haul equations (16-18). Examples of the percentage attainment of equilibrium with root time plots (Wt/Wa vs. /t) and of the dimensionless time plots (r vs. t) are shown in Figures 1 and 2, respectively. Further calculation (17, 18)... 

The solid lines are calculated rate curves drawn through experimental data for systems of different initial pH. The ordinate is the ratio of acid concentration, and the abscissa is the dimensionless time parameter. Data for two experimental runs are shown for each initial pH the corresponding diffusion coefficients are noted for each rate curve... 

The magnitude of Fo is often used as dimensionless time (to estimate how long diffusion has proceeded). The Schmidt number (Sc) is simply the ratio of the Pe to Re and is... 

Williamson and Adams6 presented data, shown in Figure 7, for the dimensionless centre temperature (Tc — Tsurf)/(7 0 — T rf) of variously shaped bodies as a function of dimensionless time Fo. Here Tc is the centre point temperature, T0 the initial body temperature and T surf the surface temperature. Fo is the Fourier number, which is given by Fo = at/82, where a is the thermal diffusivity, t is the time and S is the characteristic length for conduction, the distance of the centre point or centreline of the body to the nearest part of the surface. The data given in Figure 7 neglects the thermal resistance at the surface. 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

Fig. 2. Response of the TAP reactor to an inlet pulse of a gas that is irreversibly adsorbed (or reacted) with a dimensionless rate constant k. Tp is the dimensionless time, and F p is the dimensionless flow rate. The model takes into account the number of molecules in the pulse A p A, the effective Knudsen diffusion coefficient DeA, the number of surface sites, and the dimensions of the reactor (after 55). A, = 0, standard diffusion curve B, = 3 C, U = 10.

Here the porosity and the diffusivity vary with conversion of solid vs and v are the reactant and product molar volumes. A Thiele modulus < > and dimensionless time 9 can be defined, e.g., for a rate second-order in A and first-order in S ... 

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