Kamack equation

It is preferable, when using the Kamack equation, to smooth out the experimental data. This can be done manually by plotting Q against d on log-probability paper or using a smoothing equation during computer data collection. 

The attenuation of the x-ray beam is proportional to the mass concentration at the measurement radius that has to be converted to the size distribution using the Kamack equation. A size range of about 8 1 is covered in about an hour. 

This pipet (Figure 16) was designed by Allen and Svarovsky ) to operate with a reduced volume of suspension (150 ml) and a modified Kamack equation to reduce the measurement time down to about an hour. At 500 rpm the size range for quartz is approximately 8 lm to 0.8 lm and these sizes are halved if the speed is doubled. 

Concentration undersize D, is determined using Kamack s equation [Kamack, Br. J. Appl. Phys., 5, 1962-68 (1972)] ... 

Kamack offered the following solution to equation (8.12). If Q is plotted as a function of v, = rJSy with f = o9-t as parameter, a family of curves is obtained whose shape depends on the particle size distribution function. The boundary conditions are that Q - 1 when f = 0 for all r, (i.e. the suspension is initially homogeneous) and = 0 for r, = S when f>0 (i.e. the surface region is particle free as soon as the centrifuge bowl spins). Hence all the curves, except for f>0, pass through the point Q = 0, S, and they will all be asymptotic to the line / = 0, which has the equation Q = - Furthermore, from equation (8.12), the areas under the curves are each equal to F(r/., ). 

The Kamack treatment builds up the concentration gradients within the centrifuge for each of the measurement times. The Q j values (Table 8.5) may be determined using equation 8.19 and, in combination with they / values, generate the concentration gradients. 

The approximation due to Kamack can be modified, for the scanning mode of operation, by replacing the constant r/S) with the variable r/S) where r is the position of the source and detector at time t i.e. equation (8.4) becomes ... 

Kamack s equation is applied with the assumption that the attenuation is proportional to the product of the extinetion coefficient and the cross-sectional area of the particles in the beam i.e. ln(/(//) is replaced by the optical density D where ... 

This is solvable for fixed R by an equation developed by Kamack (, 43) variable surface radius (5) by a modified equation developed by Allen and Svarovsky ( ) and for variable R by an equation developed by Allen. 

Kamack solved the above equation for a fixed measurement radius (r) and this solution was later extended to a variable surface radius (S ). 

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