Sutherland formula

In this last formula u should be expressed in m/hr. We shall take the thermal conductivity of the combustion products to be equal to the thermal conductivity of air and find its value according to the Sutherland formula with the constants A0 = 0.0192kcal/hr m deg, C = 125 deg (MacAdams). At high temperature... 

If the condition is not fulfilled, Sutherland s formula needs a refinement given by Eq. (10.68), which can be called the generalised Sutherland formula. 

Only in this region of Stokes numbers the use of the generalised Sutherland formula is justified. Under a more severe constraint. 

The mathematical treatment was first developed by Lord Rayleigh in 1879, and a more exact one by Bohr has been reviewed by Sutherland [103], who gives the formula... 

The curves for hydrocarbon vapors and natural gases in the chart at the upper right are taken from Maxwell, the curves for all other gases (except helium) in the chart are based upon Sutherland s formula, as folloows ... 

Sutherland has developed a formula for such a model which can be used to give a fair empirical fit over a temperature range of about 100 to 200°C for most gases. The formula may be written as a correction to the mean free path ... 

Substituting v for a potential velocity field, Sutherland s formula is obtained. When we substitute Stokes velocity field into Eq. (10.25) and use Eq. (10.22), we obtain... 

At larger Reynolds numbers, hydrodynamic and diffusion fields are characterised by the appearance of boundary layers which enable one to estimate a local value of the retardation coefficient, as given by Eq. (8.108). A joint application of Eq. (10.31) and (8.109) allows to describe the effect of DAL on the radial component of particle velocity at the instant it approaches the bubble surface. Since in the non-retarded state the collision efficiency E, is described by Sutherland s formula (10.11), the effect of DAL can be approximated by the expression,... 

Negative Effect of Inertia Forces on Flotation of Small Particles. Generalisation of Sutherland s Formula. Extension of Limits of Applicability of Microflotation Theory... 

Levin (1961) has shown that inertia deposition of particles below a critical size, which corresponds to a critical Stokes number St = 1/12, is impossible. Regarding a finite size of particles, the collision is characterised by Sutherland s formula (10.11). Comparison of the results obtained from Sutherland s relation and by Levin enables to conclude that in the region of small St < St the approximation of the material point, accepted by Levin and useful at fairly big St, becomes unsuitable for Stsmall Stokes numbers were studied by Dukhin (1982 1983b) for particles of finite size. Under these conditions inertia forces retard microflotation. 

The negative effect of the centrifugal force can be summarised by the negative effect of SRHI, which is an essential deviation from Sutherland s formula. A common action of these factors appear if the limit trajectory ends not at the equator but at 0 = 0,. Results of such common action are shown in Fig. 10.15 for a fixed bubble radius a = 0.04cm and for a number of critical film thicknesses H,. = h,. / a,. 

The effect of the specific density of a particle and its radius on the combined effect of centrifugal forces is shown in Fig. 10.16. It can be expected that Sutherland s formula describes the transport stage of the elementary act also at Stokes number close to the critical one. It becomes clear from the results in Fig. 10.16, that it is applicable only at 0, close to 90°. This condition is not fulfilled over a wide range of 0, and Ap. 

The condition of applicability of Sutherland s formula obtained from Eq. (10.71) results in... 

Fig. 11.6. Results of the calculation of angles 0( and Ogd-ua a function of St at X=l and the decrease of collision efficiency due to a combined action of reflection and centrifugal forces as compared with Sutherland s formula for Eg.

The theory developed substantially changes the notion of kinetics of flotation of smooth particles since the calculated efficiency is decreased by approximately one order of magnitude when compared with Sutherland s formula. However, the theory needs further development because the final Eq. (11.46) underestimates the value of Vpg and the centrifugal force due to... 

Collision efficiency calculated from Eq. (10.60) is several times smaller than that obtained by Sutherland s formula if a(, amounts to some ten microns and a, 2 several hundred microns. 

Integration of Eq. (10.28) along the cross-section of the hydrodynamic layer allows us to check whether within its limits the radial velocity component is proportional to the tangential derivative of the velocity distribution along the bubble surface, which differs slightly from the potential distribution. The effect of a boundary layer on the normal velocity component and on inertia-free deposition of particles should be therefore very small. The formula for the collision efficiency given by Mileva as an inertia-free approximation is thus VRc times less than the collision efficiency according to Sutherland, which is definitely erroneous. 

The error in deriving this formula is that an incomplete expression for the hydrodynamic field of a bubble was substituted into the equation for the liquid stream-line. Of course, it is impossible to obtain a correct result and, in particular, the limiting case of Sutherland s formula by dropping the highest degree term (potential velocity distribution). 

If the molecules are treated as rigid spheres attracting as an inverse power of the distance it was shown by Sutherland (Chapman and Cowling, loc. cit., pp. 182—184 and 218) that an approximate formula for 1] is... 

We observe that the calculated value of Na is not independent of temperature. We plot the calculated values of Na against T in fig. 1 and according to Sutherland s formula (5) we should obtain an approximate straight line. In fact we do not get a straight line, but the plot has a sufficiently small curvature to make extrapolation to r-i = 0 possible. We thus find... 

The interpretation of the observed vibration-rotation bands by the authors cited above is probably not entirely correct since in their classification some of the combination bands violate the rule of Dennison. More recently Sutherland (ico) and Herzberg and Spinks (89) have tried to avoid this difficulty. They represent the vibration levels by a formula of the type... 

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