One of the most important characteristics of the mineral dressing production process is the degree of useful component recovery by ores grinding.
At present there exist no technical facilities which permit to perform an automatical estimation of this parameter.
The method of automatical ultrasonic control of the useful component (mineral) recovery degree during ores grinding has been worked out at the Krivoy Rog Technical University.
Let the material being under investigation contain the particles, minimum dimension of which is r_{1} and upper limit size is r_{2}.
If the particles distribution according to their dimensions is expressed by the f(R) function and initial sample for the screen analisys includes N particles, the sample for control of the useful component recovery degree will contain
 (1) 
particles. Let j
(x
) be a function of the particles distribution according to the useful component recovery degree x
physical content of which becomes evident, if
 (2) 
one would apprehend the number of particles having the recovery degree in the space from x
till x + dx.
Let's assume that the radiator of ultrasound and receiver are set on the cylindrical vessel with water at the level of Z. At the beginning a sample under investigation is being kept in the layer having thickness of Z_{1}(V_{s} = SZ_{1} is the volume of the layer mentioned, where S is the area of vessel's crosssection).
In the process of particles settling their concentration will be changed together with the changing of time and space because of the dependence of settling speed upon the particles density. Let's designate with letters n_{Rx}(Z,t) the R radius particles concentration with the useful component content x
at the depth of Z at the time moment t. The equation being complied with the quantity n_{Rx}(Z,t) looks like following:
 (3) 
where V_{R,x} is velocity of the particles settling with the corresponding parameters R and x
Initial condition of the equation (3) can be written as follows
 (4) 
where is the initial particles concentration in the layer of Z_{1} thichness;
St is step function, accepting two values  0 and 1, by positive and negative argument accordingly.
The solution of equation (1) with the initial condition (4) is quite simple
 (5) 
If one determines the concentration of all the particles faking in consideration their distribution according to their dimensions within the space from r_{1} to r_{2} with function f(R) and the recovery degree with function j
(R), its dependence upon the time at the Z depth will be expressed by the formula
 (6) 
where
h
 is water viscosity;
r
_{o}  is water density;
 is particle density with the recovery degree of x;
r
_{1} and r
_{2}  the density of useful component material and of dead rock.
One can estimate the particles distribution according to the useful component recovery degree on the basis of measuring temporal dependence of the ultrasonic oscillations amplitude decrease at the certain level (depth) Z. As the oscillations amplitude decrease depends on the concentration of solid phase particles , so one can write down, taking into account (6):
 (7) 
where A_{o} is the amplitude of ultrasonic oscillations in clear water; V_{3F} is effective volume under the control between receiver and radiator; s(R,v) is full crosssection of n
frequency ultrasound decreasing on the particle of R radius.
After the results of measuring of amplitude of oscillations it is necessary to determine
 (8) 
As the formula (8) shows, S_{v}(t) is dependent on j
(x
); this according to the temporal dependence of S_{v}(t) permits to find out the law of particles distribution according to the useful component recovery degree x
. For this sake one has first of all to exclude the dependence upon initial concentration of the solid phase particles.
This can be done after determining temporally integral volume of the formed signal S_{v}(t) i.e.
 (9) 
where integration is carried out within the space from t_{o} till t_{1}, i.e. within the temporal space, when the volume S_{v}(t) being measured is not equal to zero.
Now the relation of volumes S_{v}(t) and will not be dependent on the initial quantity of the material under control.
 (10) 
The method of obtaining information concerning the function of distribution of ground material particles according to the recovery degree j
(x
) will be analyzed on the more simple example, when all the particles have the same dimension. Now it is better to return to the expression (5) again supposing subsequently R to be a constant value. In these approximations the value will be determined by the expression
 (11) 
Taking into consideration that does not depend upon time and is a small quantity, the integrals according to the variable x
in the expression (11) can be represented as
 (12) 
It is known that by great frequencies (v>1 MGz) the crosssection of the sound decreasing does not depend upon the particles density, so
 (13) 
By integrating according to t in the formula (13) one has taken into consideration the fact the function j
(x
) was normalized for one.
Shall we integrate the value within the little time space D
t so we shall have
 (14) 
So, the distribution function j
(x
) can be estimated on measuring results of the value by formula
 (15) 
in this case .
Tests of the device for ultrasonic control of the minerals recovery degree by ores grinding have been carried out on the chaching table for different ores types characterized by certain laws of distribution according to the useful component recovery degree.
In the course of experiment the dependence of ultrasound oscillations amplitude upon the particles settling time of the controlled sample with different laws of distribution according to the useful component recovery degree was being determined.
The functions of particles distribution according to the useful component recovery degree j
(x
) after being renewed in accordance with formula (15) are shown in figures 1 and 2 (histograms). Particles dimensions variation was about 10 mkm (near the class 0,074 mm) which became the reason of the distribution function j
(x
) renewal error. As one can see from fig. 1 and 2 the errors level depends on the nature of initial particles distribution according to the recovery degree. If the initial values of distribution have no strongly pronounced peculiarities, so the renewal error is not great.
Fig 1: Distribution functions of particles according to the useful component recovery degree. 3 and 4  0, +  initial distributions, histograms  renewed distributions by D r = 10 mkm

Fig 2: Distribution functions of particles according to the useful component recovery degree. 1 and 2  0, +  initial distributions, histograms  renewed distributions by D
r = 10 mkm 
For versions 2,3,4 the medium quadratic deviation is equal to s
_{2}= 0,2; s
_{3}= 0,44 and s
_{4} = 0,36 accordingly.
For the version 1 having a strongly pronounced sharp dependence of initial function j
(x
) on x
the renewal error becomes appreciably greater and is equal to s
_{1}= 0,64 in medium quadratic deviation.
The worked out method for ultrasound control of the useful component recovery degree during ores grinding has found wide utilization in practice.