Sigma theory

Consider the smallest particle in the feed sludge that has to be separated, the cut point size. d. This particle has a density, ps, and settles in a liquor of density, pl. and viscosity, The feed slurry enters the decanter at a rate of Qf, at a pond radius, ri, at point X at time t=0. By the time the particle traverses the clarifying length of the centrifuge, L, in time t-tg, the particle must settle to a radius T2, at point Y, the bowl internal radius, if it is to be collected by the conveyor. The centrifuge rotates at a constant angular 

At time l-t, the particle has a radial velocity vand a constant axial velocity of v a. It is also assumed that the particle travels a negligible distance from the pond surface before it reaches its terminal velocity. 

The terms on the right-hand side of equation (4.31) consist of r... which is solely a function of the process material, and the remainder of the terms, that are solely functions of the centrifuge. These latter lerms are collectively known as. Sigma. 1], the clarification capacity of the centrifuge  

Sigma has units of area, which is consistent with non-centrifugal clarifying 

Kquation (4.32) is the equation for Sigma preferred today, and is particularly recommended when deep pond decanters (ri/r 0.65) are used. When Ambler first derived his formula, only shallow ponds (ri/r.j 0.75) were used, and lie used difi erent starting assumptions for his derivation. 

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This approach to describing centrifuge performance has become known as the sigma theory . It provides a means for comparing the performance of sedimentation centrifuges and for scaling up from laboratory and pilot scale tests see Ambler (1952) and Trowbridge (1962). 

This theory concerns clarification in cylindrical bowl centrifiiges, such as the tubular bowl machine. It was first published by Hebb and Smith [1948], and is generally known as Ambler s sigma theory [Ambler, 1952]. 

The foregoing sigma theory is derived on the assunq)tion of 100% collection of a particle of a critical diameter. It is common to find machines characterised on the basis of only 50% efficiency of collection, ie. 50% of particles captured within the machine and 50% allowed to enter the centrate.The sigma theory can be modified to take this into account as follows. 

If the two centrifuges are arranged in parallel, each one takes one half of the total flow, Q, and the performance is compared with that of one centrifuge taking the whole of the flow. In terms of cut size, the Sigma theory (see section 7.2.2) can be used to write for the same machine (same S value) ... 

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