Knudsen formula

T = ratio of measured diffusivity to that calculated from the Knudsen formula with a mean pore diameter. 

In order to quantify the matter transfers involved at the ice/pore interface, we can express the incoming matter flux j [mol m" s" ] on a point of the ice surface by using the Langmuir-Knudsen formula ... 

The validity of this relationship is expected to be limited to droplet radii not to small. In principle the surface tension is a decreasing function of the radius r for small clusters. Hence, the result of Eq. (3) are corrected by Tolmans formula (see Ref. [14]) with 6 = lO- o m, the intermolecular distance in the liquid, see Fig. 1 - right. If the droplet radius is small compared with the mean free path X, the molecular growth is described by the Hertz Knudsen formula... 

The initial and boundary conditions at the reactor entrance and exit are the same as eqn (1), as suggest by previous workers [2-4]. The fundamental difference from eqn (1) is in the use of a non-constant diflusivity, which makes the reactor equation non-linear. Curve C ws a simulated curve computed via equation (2) with Eb as an adjustable parameter (D was obtained from a separate experiment with an empty tube and had been verified as being in close agreement with the Knudsen formula). 

Recently, Sameshima(24) has measured the rates of flow of various simple gases through a compact unglazed earthenware plate. The rates of flow definitely did not obey the Knudsen formula t = k JM, where t denotes the time required for the effusion at constant pressure of a volume F of a gas of molecular weight M, and where fc is a constant. On the other hand, the law t = k M was accurately obeyed when the gases effused through a platinum plate with a single orifice. For the earthenware plate Sameshima found a formula t = to apply. If the wall was very thin n approached zero, and the simple behaviour of the perforated platinum plate was found. If the wall was thick n approached unity and the equation became t = kij (ij denotes viscosity). 

To derive Eq. (3-1), Knudsen and co-workers mainly used samples from the Baltic and from the Mediterranean and the Red Sea. The offset of 0.03 in Eq. (3-1) reflects that the salt composition especially in the Baltic Sea is not exactly constant (Millero and Kremling, 1976) and thus contradicts the basic assumption of constancy that led to Eq. (3-1). However, the proposed titration method also had advantages it could be performed in reasonable time onboard a ship, and for salinity measurements in open ocean areas where S is close to 35 %o, the error induced by the non-constancy is less than that from titration (0.02 %o in salinity). Therefore, the so-called Mohr-Knudsen titration method (Mohr, 1856 see Chapter 11) and the Knudsen formula (3-1) served oceanographers for more than 60 years to determine salinity from chlorinity. 

In the early 1960s, bench salinometers were developed that allowed measurement of the electrical conductivity of a seawater sample relative to that of a standard with high precision. Cox et al. (1967) had related chlorinity and conductivity ratios of seawater to standard seawater at temperatures higher than 10 °C and tabulated their results (UNESCO, 1966). Following their work, the responsible international oceanographic organizations adopted a redefinition of salinity (Wooster et al, 1969). Firstly, it was assumed that salinity was proportional to chlorinity, to be consistent with the assumed constancy of the ionic composition. The constant was chosen so that for 5=35 %o, both the Knudsen formula (3-1) and the new relationship... 

The problem to use the above Knudsen diffusion for the interpretation of the data is that the magnitude does not work for DWNT membranes i.e. the measured value of permeabihty is larger than the permeabihty ealeulated by the Knudsen formula by a factor of 10-10, and therefore it was called fast transport. The above assumptions (1) and (2) were also shown to be flawed by the molecular dynamics work even before the experiments were performed. The AT dependence may be therefore much more complicated than the Knudsen meeharhsm. 

From the classic Knudsen formula the value of the Knudsen diffusion transport coefficient, >i,K, can be computed [46, 66, 67] ... 

KNUDSEN J c, ANTANUSE H s, RisBO j and SKIBSTED L H (2002) Induction time and kinetics of crystallization of amorphous lactose, infant formula and whole milk powder as studied by isothermal differential scanning calorimetry, Milchwissenschaft, 57, 543-546. 

The Knudsen fusion and Langmuir free-evaporadon methods. In both of these cases the rate of vaporisation of a substance in vacuum is measured and the following formula is utilised... 

The base of the nitrate (II) has proved to be identical with Knudsen s base for which an incorrect formula was given by Knudsen [66], and identical with the product (H) of interaction of methyl nitrate with hexamine described by Hahn and Walter [67]. 

Effective diffusivity. Resistance to diffusion in a catalyst pore is due to collisions with other molecules and with the walls of the pore. The corresponding diffusivities are called bulk diffusivity and Knudsen diffusivity DK. Many data and correlations of the former type exist the latter is calculable from the following formula (Satterfield, 1970, p. 42) ... 

For conditions where Knudsen or molecular diffusion does not predominate, Scott and DULLIEN(II) obtained a relation for the effective diffusivity. The formula they obtained for a binary mixture of gases is ... 

To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures D y eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11], Finally, combining Equations (6.15-6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)) ... 

Typical values for p are between 0.3 and 0.6, and for tp between 2 and 5. So, a reasonable assumption for the effective diffusion De is that it is Vio of the diffusivity I). This diffusivity D can be calculated from the Knudsen (corresponding to collisions with the wall) and molecular diffusivity (intramolecular collisions). The molecular diffusivity was estimated at 10 5 m2/s, which is reasonable for the diffusion in gases. The Knudsen diffusivity depends on the pore diameter. The exact formulas for the molecular and Knudsen diffusion are given by Moulijn et at1. For zeolites, the determination of the diffusivity is more complicated. The microporous nature of zeolites strongly influences the diffusivity. Therefore, the diffusion... 

Knudsen number was varied by varying the pressure of the medium (He) for particles of almost constant size. Their experimental data are compared with different models in Tables II and III. Comparison of the experimental data with the upper bound, the lower bound (for a Hamaker constant of 10-12 erg), and the Fuchs interpolation formula is also shown in Fig. 8. Since the Knudsen number was varied by varying the pressure of the medium for particles of constant size, the upper and the lower bounds approach the limiting values for large Knudsen numbers. The experimental data of Wagner and Kerker (12) lie between the upper bound and the Fuchs for-... 

There are three basic types of gas flow turbulent, viscous, and molecular. The type of flow passing through a given system is dependent on both the mean free path (MFP) of the molecule(s) and the diameter of the container (tube) through which they are flowing. A useful formula when talking about MFP is the Knudsen number (Kn), defined in Eq. (7.6). 

In this formula the average heat-transfer coefficient is based on the arithmetic average of the inlet and outlet temperature differences, and all fluid properties are evaluated at the mean bulk temperature of the fluid, except /j., which is evaluated at the wall temperature. Equation (6-10) obviously cannot be used for extremely long tubes since it would yield a zero heat-transfer coefficient. A comparison by Knudsen and Katz [9, p. 377] of Eq. (6-10) with other relationships indicates that it is valid for... 

We see that the only molecular constant involved is the molecular weight M of the gas, so that a measurement of the rate of flow of a gas through an orifice can be used to calculate M il S, the area of the orifice, is known. This formula was first verified by Knudsen and has been applied since to the measurement of molecular weights of unknown gases. 

The effusion method originally suggested by Knudsen is essentially a method of pressure measurement which utilizes the fact that pressure is the effect of the bombardment of the walls of the containing vessel by the molecules. If a small part of the wall is replaced by a hole leading to an evacuated space, then the molecular shower will pass through the hole, and the rate at which molecules do this depends only on the mean component of velocity of the gas molecules and the number present, and may be calculated by kinetic theory to be apj 2nmkT) molecules per second, where a is the area of the hole, p is the pressure, m is the mass of a molecule, k is Boltzmann s constant, and T is the absolute temperature 2 19 In the derivation of this formula it is assumed that ... 

The effective diffusion coefficients. Die, obtained from the parameter tdjf, include contributions from the Knudsen diffusion mechanism and from the molecular diffusion mechanism. Because of the very low tracer concentrations the Bosanquet formula (3) is applicable. 

A similar method of determining vapour pressure depends on finding the weight of substance escaping through a small hole in a box into a vacuum, and applying the formulae of Knudsen ( 2.VII J) for the passage of a gas at very low pressure ... 

Kelly filter, 307,319, 323, 324 Kenics blender, 302 Knudsen diffiisivity, 564 Kremser-Brown formula, 398,399,466... 

Another study, [17], used helium as their working fluid and carried out the experiments in 51.25 X 1.33 micrometer microchannels. They showed that, as long as the Knudsen number is in the slip flow range, the Navier-Stokes equations are still applicable and the discontinuities at the boundaries need to be represented by the appropriate boundary conditions. They obtained the following formula for the mass flow rate including the slip effects... 

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