Diffusion Fourier number

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... 

Kato et al. have used electric stepwise heating of a thin metal layer to measure thermal diffusivity of molten salts. The ratio of the temperatures with time and 2/i at distance x below tlie heating plate was evaluated as a function of the Fourier number ... 

Fo being the Fourier number and d the diameter of the disk. The mass transfer coefficient k can be considered as interpolating between the steady-state convective diffusion at large times (t - oo) and unsteady-state diffusion at short times (t — 0 and v = 0). The constants A and B of Eq. (147) follow from the solutions for these two limiting cases. For these two limiting cases... 

A conservative estimate of the disc s mass transfer performance may be obtained from the Nusselt model, assuming that there is no film mixing as it proceeds to the edge of the disc. For unsteady diffusion into a finite stagnant slab, the plot shown in Figure 8 from (8) gives the relative concentration distribution within the slab at various times, with a zero initial concentration and a surface concentration C0 imposed at time t = 0. The parameter on the curves is the Fourier number, Fo, where... 

The Fourier number (Fo) is an estimate of the temporal extent of diffusion 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. 

For some moist materials, the Lu number increases with the moisture content following a slow linear dependence. From Eq. (6.184), we can appreciate that, for Lu>l, the propagation velocity of the mass transfer potential is greater than the propagation velocity of the temperature field potential. The value of the diffusive filtration number of moisture LUp is normally much higher than one. The total internal pressure relaxation of the vapour-gas mixture in a capillary porous body is 2-3 orders of magnitude higher than the relaxation of the temperature field. The relationships between Fourier numbers may be expressed in terms of Lu and LUpi... 

Dividing by /cA/2Ax and using the definitions of thermal diffusivity a - k/pcp and the dimensionless mesh Fourier number T = aAtfAx gives... 

The quantities h, f,. e, and do not change v/ith time, and thus we do not need to use the superscript / for them. Also, the energy balance expressions are simplified using the definitions of thermal diffusivity a = A/pCp and the dimensionless mesh Fourier number t = aUtiP, where Ax = Ay - I. 

Obviously, the assumption of no lateral exchange between the subchannels is, strictly spoken, only (approximately) valid for a relative pitch of 1 however, for higher relative pitches it might be valid if the ratio of the residence time in the reactor to the time constant for lateral difhision/dispersion is much smaller than 1. For lateral diffusion/dispersion over a distance equal to the pitch (s) or the width of the reactor (d ), these ratios or Fourier numbers are, respectively. 

Owing to the nature of thin films generated by both spinning disks and spinning cones, transport by diffusion can become a significant method of transport of both heat and mass within the film. In the case of heat transfer this is especially true and will be the dominant mode of transport within the film. The equations for transport by diffusion have been studied and solved for many simple systems. One of the most comprehensive studies of this process was given by Crank in which the equations for diffusion are examined for several generalized systems. In the solution of the diffusion equation one parameter is of paramount importance that is, the Fourier number, Fo. This can be expressed as... 

For the estimation of the vapor-phase mass transfer coefficients we use Eqs. 12.2.31-12.2.33 and use 0 2 as the reference diffusivity. The Fourier number for the jetting zone is calculated as... 

Fourier number (Fo) - A dimensionless quantity used in fluid mechanics, defined by Fo = atIF, where a is thermal diffusivity, t is time, and / is length. [2]... 

Diffusive mixing efficiencies can be assessed using the Fourier number, which is defined by... 

A pseudo-diffusion approach [29] to model mass transfer in osmotic dewatering showed that effective diffusivity was not a unique function of Fourier number, as it would be expected in pure diffusion. The developed model had no predictive ability. On the other hand, experiments done on the frozen apple tissue showed that mass transfer was only diffusive [30]. 

Moreover, the mixing in the liquid-liquid system can be characterised by dimensionless numbers, such as, Sherwood number (Sh), which is the ratio of convective mass transfer to the molecular diffusion, and Schmidt niunber (Sc), which is the ratio of the viscous diffusion rate to the molecular diffusion. In addition to these, the Fourier number (Fo) can also give an idea about the dynamics of diffusive transport process. 

At constant fluid dynamic residence time, the RTD becomes narrower with decreasing chaimel dimension and thus decreasing diffusion time. The ratio of fluid dynamic residence time to radial diffusion time describes the Fourier number, Fo, which is a measure of the intensity of the radial molecular diffusion and thus radial mixing in den chaimels [12] ... 

Residence time distribution can be an important issue in the selection process. Microreactors usually operate at Reynolds numbers lower than 200. In this regime, laminar flow prevails and mass transfer is dominated by molecular diffusion. An injected substance in the channel will dissipate caused by the flow profile in the channel. Hence the input signal will be broadened until it reaches the exit of the channel (Figure 3.2). The extent of such a distribution depends on the channel design. In microchannels the mixing process can then be described by the Fourier number (no axial diffusion, dominating radial diffusion D ). A high Fourier Po number leads to a narrow residence time distribution ... 

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