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58
— PROCESSES, EQUIPMENT, AND APPARATUSES OF FOOD INDUSTIRY—
UDC 621.929.2/.9 DOI 10.12737/1558
DEVELOPMENT OF MATHEMATICAL MODELS OF CENTRIFUGAL MIXING UNITS OF NEW DESIGN FOR THE PRODUCTION OF DRY
COMBINED FOOD PRODUCTS
V. N. Ivanets and D. M. Borodulin
Kemerovo Institute of Food Science and Technology, bul’v. Stroitelei 47, Kemerovo, 650056 Russia phone: +7(3842) 39-68-42, e-mail: Borodulin_dmitri@list.ru
(Received January 17, 2013; Accepted in revised form February 25, 2013)
Abstract: A method of modeling the continuous process of the mixing of bulk materials on the basis of cybernetic analysis with some elements of automatic control theory (ACT) [6, 9] has been considered. In this case, a mixing unit (MU) is represented in the form of a dynamic system, which is characterized by the known topology of the motion of material flows and subjected to various external disturbances.
The two developed mathematical models allow us to determine the degree of the smoothening of input material flow fluctuations from volumetric dosers by the mixers incorporated into a MU. The obtained numerical values of smoothability indicate that it is reasonable to equip the studied mixers of new design with volumetric dosers. This allows us to meet the requirements to MUs from both the engineering and economical viewpoints.
Key words: centrifugal mixer, time-and-frequency analysis, bulk materials, combined food products, modeling, cybernetic analysis
INTRODUCTION
The contemporary state of the market of food industry equipment is characterized by a considerable increase in the demand for machines and apparatuses that allow the production of high-quality food products of increased nutritional value (enriched with vitamins and biologically essential components) at low expenditures. In particular, the population should have new combined food products that compensate the deficiency of different food components and micronutrients in its ration due to considerable ecological disturbances in different regions of Russia and other countries.
Since the content of many food additives in the major product is small (1% and lower), the key problem consists in their uniform distribution over the entire volume. Using the results of studies, it has been revealed that continuous centrifugal mixers (CCMs) [2, 5] characterized by a high intensity of mixing due to the targeted organization of the motion of thin disperse layers are most promising for the solution of this problem. Centrifugal mixers enable the production of good-quality mixtures at a component ratio of 1:100 [2]. However, a single CCM is usually insufficient at higher ratios. In this connection, we propose to incorporate two serially arranged centrifugal mixers with a good smoothability into a single MU. In this case, it is possible to use volumetric dosers with certain advantages (high material feed rate, small dimensions,
low cost and maintenance expenditures) for the preparation of mixtures with high ratios of mixed components. For this reason, the objective of our work is to compare the operational efficiencies of two centrifugal MU of new design (differ from each other by the set of equipment incorporated in them), in which it is possible to obtain dry combined food products with a high ratio of mixed components, using cybernetic analysis and some ACT elements [6, 7, 9].
When studying the operation of certain mixing equipment, we artificially imposed a disturbance of one or another kind onto the input feed flow and then analyzed its consequences at the output of an apparatus (plotted a response curve) [10]. The function determined from the given curve for the residence time distribution of particles in centrifugal mixers was used in combination with the accepted flow pattern of mixed materials in an apparatus to predict the process of mixing in it [1, 8].
A number of scientific works [1, 6, 8, 13, 15] are devoted to the problems of the modeling of mixing processes. In our work, we have detailed the questions of the creation of a MU mathematical model, which would allow us to match the time-and-frequency characteristics of CCMs and dosers incorporated into a MU in the interactive operational mode of a computer. As a result, this provides the possibility of decreasing the amplitude of fluctuations in the output material flow of a mixer and improving the quality of a ready mixture.
OBJECTS AND METHODS OF STUDY
In the first case, the object of the study aimed at implementing the method of the sequential dilution of a mixture is a mixing unit that incorporates a block of two spiral and one batch dosers (D, i=1, N) and two serially arranged CCMs. Spiral doser Di and batch doser D2 deliver initial mixture components trough summing element SE1 into CCM1. This results in the mixing of components at a ratio of 1 : 500. The obtained mixture enters SE2, into which the major component (incorporated into the mixture in a great amount) is simultaneously fed with spiral doser D3, and then into CCM2, where the components are finally mixed at a ratio of 1 : 20. As a result, the mixture with a ratio of mixed components of 1 : 1000 is obtained at the output of CCM2.
The general structural functional scheme of the studied mixing unit operating by the method of the sequential dilution of a mixture is shown in Fig. 1. The dosers form the signals of the mass flow rates of materials that have masses Qdi(t) and Qd2(t) and concentrations Xd1(t) and Xd2(t) and are fed into SE1, thereupon the summary flow with parameters Xdc1(t) and Qdc1(t) enters CCM1. The mixture that leaves the first mixer and has a weight QM1(t) and a concentration XM1(t) and the material flow with parameters Xd3(t) and Qd3(t) from spiral doser D3 are fed into SE2. As a result, the material mass QM1(t)+Qd3(t) with a concentration XM1(t)+ Xd3(t) enters CCM2, and a mixture with parameters QM2(t) and XM2(t) leaves it.
Fig. 1. Structural functional scheme of the studied mixing unit.
To perform the monitoring and control of the principal parameters of the continuous process of mixing, let us use the structural functional scheme implying the estimation of the impulse responses of dosers and the transfer functions of mixers that are incorporated into the MU [6, 9, 10]. The transfer function of a mixer is the ratio of the output signal y(S) to the input signal x(S), both are Laplace transformed, at zero initial conditions. The transfer function is governed only by CCM internal properties, represents a dimensionless function of complex variables, and is denoted as W(S)=y(S)/x(S).
From Fig. 1 it can be seen that the two-stage MU consists of the two blocks of dosers WDB1(S) and WDB2(S) that have certain impulse responses, form signals of different kinds, and operate in parallel for SE1 and SE2. The principal elements of the scheme are the CCMs of new design developed by us with a horizontal rotor in the form of three and one hollow truncated cones (Wcm1(S) and Wcm2(S)) [11, 12].
The MU output signal for the given scheme in the operator form (WMU(S)) is determined by the formula
Wmu (S) = WD»(S) X Wcm 1 (S) + Wdb2(S)]x Wcm2 (S), (1)
where WDB1,2(S) are the impulse responses of the block of dosers, WCM1,2(S) is the CCM transfer function, and S is an independent complex variable that stands for differentiation with respect to time.
Here, the first block of dosers consists of a spiral doser and a batch doser. When the spiral doser forms a signal, the feed of a component Xd1(t) fluctuates by a time-dependent sinusoidal law with an average value Xd01 and an amplitude Xdm1:
Хd 1(t) = Xdo1 + Xdm1 x smCoy),
(2)
Performing the Laplace transform of the given signal from the time-dependent form to the operator form, we obtain the following expression:
W1( S ) = Xf-
S
Xdm1 X úd
S2 +
(3)
where Xd01 is the constant flow rate of a component dosed with a spiral doser and Xdm1 and wd1 is the amplitude and frequency of fluctuations.
For the formation of a square-wave signal from the batch doser Xd2(t), let us use the Fourier tenth-order expansion [6], which is represented by the following function in the temporal region:
XAt) = AAt +■ 4:
2 k=11
• cos-
2kx
t + B,
. 2kn • sin----1
T
, (4)
The Laplace transform of this signal gives the following expression:
w2(s ) = +■ (
Ak 2 x S
2S k=i S2 +
Bk2 x úd S2 +
),
(5)
where cok2=2^k/Td2 is the angular fluctuation frequency corresponding to the kth harmonic of the Fourier expansion of a square-wave signal from the batch doser, Td2 is the period of its fluctuations, and A02, Ak2, and Bk2 are coefficients in the Fourier expansion of the signal.
2 Td2
A =— J X (t )dt
T
d2
2 Td2
A, =— { X(t)
T
'X cos
Bk 2 =
d2 0 2 Td2
T
J X(t)x sin
d2
T
V d2 У
dt , dt
(6)
Then, taking into account Eqs. (3) and (5), the summary signal WDB1(S) in the operator form will be
d 2 d 2
The second block incorporates a spiral doser. Its signal in the time-dependent and operator forms is
Хd3 (t) = Xdo3 + Xdm3 x sin(0d3* ) ,
W3(S ) = У03
Xdm3 x úd 3 .
" S2 + ffli
(8)
(9)
When forming a CCM mathematical model, it is necessary to characterize the dynamics of the displacement of a material inside it. Professor Yu. I. Makarov in his work [10] considered a CCM as a control element with pronounced low-frequency filter properties. He has proved that the continuous process of mixture preparation can be described by the models that incorporate the corresponding combinations of serial and parallel plug-flow and stirred-tank zones. For the quantitative analysis of the operation of a CCM, its dynamic characteristics are usually approximated by first- or second-order aperiodic elements [6, 9].
The first-order element has the following form:
WcM ( S ) =
K x e~ö T1'x S + Г
(10)
The second-order element is
WcM (S) =
K x e
T22 x S2 + T1 x S +1
(11)
where K is the transfer coefficient (K = 1), and T1 are time constants (for the first and second CCMs) that characterize the time interval, within which the concentration decreases from a maximum value to a nearly zero level, T2 is the time constant that characterizes the period of attaining the maximum change rate of the output concentration of a mixture from a mixer in the transition regime with an impulse dosing disturbance, and t is the delay period.
Substituting the impulse responses of all the blocks and the transfer functions of MU mixers (Eqs. (7), (9), (10), and (11)) into Eq. (1), we obtain
Wmu(S)=
+ Xdm1 x úd 1 + A02 +
+ Z (■
A,x S + B„x o
-)) x
2S
K x e~
T22 x S2 + T1 x S +1
X
d 03 + Y , S dm3 •
s2 + o2
T'x s +1:
(12)
general structural functional scheme of the studied MU (Fig. 1) into the block structural scheme, whose elements are specified in the form of transfer functions (Fig. 2). The block structural scheme differs from the previous scheme by that the output signals of the block of the first- and second-stage dosers are substituted by parallel virtual elements linked to the output of corresponding mixers.
Fig. 2. Block structural scheme of the mixing unit.
+
The obtained model describes the process of the mixing of bulk components in the case of the sequential dilution of a mixture.
Let us further consider a procedure in the space of MU model states. To accomplish this, let us convert the
The transfer functions describing the virtual elements are such that the signal that appears at the output of the mixers upon the synchronous fictitious control action u(t) onto their outputs in the form of a unit impulse function is equal to the summary action of
real dosing impulses. From the block structural scheme it can be seen that it has two inputs and one output.
Let us transform the obtained transfer functions (Eqs. (7), (9), and (12)) into the corresponding differential equations. By way of example, let us
x
consider the first summand of Eq. (7) d01 u(t), which
S
is the image of the function y (t) , i.e.,
-u(t) = Y ^ y. Multiplying both sides of the
X,
S
equation by S with consideration for S x Y1 ^ y, we
obtain the differential equation y = xdol x u(t).
Transforming the other elements (summands) in a similar way, we obtain the following system of differential equations:
dy1(t)
dt
= Xdo1 x U(t)
d2 y (t ) 2
-ZT- + O x у2 (t) = Xdm1 xúd1 x U(t) dt2 dy3 (t) A0
= -JL x u(t) dt 2
d 2У 4 (t )
dt2 d ^ У 5 (t )
dt2 d ^У6 (t)
dt2 d ^ У 7 (t )
dt2 d ^ У 8 (t)
dt2
+ 0^2 x У4 (t) = A x u(t)
+ úd2 x У5 (t) = B1 x0d2 x U(t)
+ 0 x y6(t) = A2 x U(t)
+ 40 x У7 (t) = 2B2 x 0d2 x U(t)
+ 9®d22 x У8 (t) = A3 x U (t)
(13)
d2 y (t) 2
+ ^°<Í2 x У9 (t) = 3B3 xúd2 x U (t)
dt2 d ^ У23 (t )
dt2
+ 100úd22 x yM(t) = Al0 x U(t)
+ 100ú22 x y23(t) = 10Bl0 x Od2 x u(t)
2 dt2 d 2y 25 (t )
dt
dt2 dy26 (t)
+ °d23 x У25 (t) = Xdm3 x O3 x U(t)
= Xd03 x U(t)
dt
T, + y(t) = Km(y24 (t) + У25 (t) + у26 (t))
dt
where y1(t), y2(t), y3(t), y4(t) .... y25(t) are the internal signals that characterize the operation of corresponding transfer functions in the elements of the block structural scheme. The sum of y1(t) and y2(t) is the signal formed by the spiral doser, and the sum ofy3(t),y4(t) ...y23(t) is the signal formed by the batch dosers in the first block. The signals y25(t) and y26(t) are formed by the spiral
doser of the second block, y24(t) corresponds to the output signal of the first-stage CCM, and y(t) corresponds to the output signal of the second-stage CCM or the MU as a whole.
To solve system (14), let us reduce the order of the differential equations via the substitution of variables.
(
y(t ) ^ ' Xl(t )
У2 (t ) X2(t )
У 2(t ) X3(t )
Уз^ ) X4(t )
Уи (t) = X2k-3(t)
Ук (0 X2k-2(t)
У26 (t) X49(t )
У(0 у V X50 (t)
(k = 4,25)’
(14)
Such a transformation allows us to write the system of the differential equations describing the behavior of the MU with a batch doser signal that has n Fourier expansion harmonics in the Cauchy normal form.
X(t) = Xd0lxu(t)
X2(t) = X3(t)
X3(t) = -®dl xX2(t) + Xdm x0dl xu(t) A0
X4(t) = -^° xu(t)
(15)
X4k+l(t) = X4k+2(t) ( = 1,n)
X4k+2(t) = -k2 x x X4k+1(t) + 4 x u(t)
X4k+3(t) = X4k+4(t) ,
X4k+4(t) = -kl x x X4k+3(t) + kBk x O x u(t)
X4n+5(t) = X4n+6(t)
Km 2n
X4n+6(t) = (X1(t) + X2(t) + X4(t) + EX2k+3(í))-
k=1
1T
-T2 x X4n+5(t) -X X4n+6(t)
X4n+7(t) = X4n+8(t)
X4n+8(t) = -®d23 x X4n+7(t) + Xdm3 x ®d3 x u(t)
X4n+9(t) = Xdm3 x u(t)
Km 1
X4n+10(t) = T (X4n+5(t) + X4n+7(t) + X4n+9(t))-TX X4n+10(t)
T1 T1
It should be noted that the output signal y(t) of the second CCM is related with the state variable y „+10 (t) according to Eq. (15) via the relationship y(t) = x4„+10(t), which is the output equation for the considered MU.
Obtained model (16) that contains information on the formation of flow signals in the blocks of dosers also allows us to trace their fluctuations in parallel (during a single calculation procedure) with the output
signal that has passed the first CCM and is received at the output of the second mixer.
Let us model a two-stage MU consisting of three spiral dosers and two CCMs in a similar manner. Its structural functional scheme is shown in Fig. 3.
Fig. 3. Structural functional scheme of the mixing unit.
The output signal of the MU, where the first block incorporates two spiral dosers, in the operator form (Wmu(S)) is represented by Eq. (1), and its impulse response is determined as
Wdbi( S ) =
X.
S
Xdml Xúd
s2 +
X
S
Xdm2 + úd
S2 + oi
, (16)
The second block incorporates a spiral doser, whose impulse response is represented by Eq. (9). The transfer functions of the mixers are expressed by Eqs. (10) and (11).
Substituting the impulse responses of all the MU blocks and apparatuses (Eqs. (16), (9), (10), and (11)) into Eq. (1), we obtain the following model for the process of the mixing of bulk materials:
Wmu ( S ) =
X„
S
X.
S
K X e -
X dml X 01 ,
S 2 + On
+ Xdm 2 + úd
S2 + oh у
X
T¡ X S2 + T X S +1 S
X dm 3 X úd
S2 +
K X e T'x S +1
(17)
Let us consider a procedure in the space of MU model states and, to accomplish this, transform the general structural functional scheme of the studied MU (Fig. 3) into the scalarized block structural scheme (Fig. 4).
Applying the above considered expressions, we write the following system of differential equations:
d^7T) +°i xyl(t) = Xdm x0l xu(t)
dy>(t)
dt
= Xd01 xu(t)
^Уз() Xy3(t) = Xm2 XO02 XU(t)
df
= X« x*t)
dt
+û° Xy5(t) = Xdm, X°d3 xu(t)
, (18)
dt2
dy6(t)
dt
= Xd03 xu(t)
T ^ +T ^ + y,(t)=Ш0+уМ+y,(t) +y,(0) +y(t)=K(y>(t)+y«(t)+y,(t))
To solve it, let us reduce the order of the differential equations.
I yl(t)^ іXl(t) "
y1(t ) X2(t )
У2 (t ) X3(t )
y3(t ) X4(t )
y3(t ) X5 (t )
У4 (t ) X6(t )
У5(t ) X7(t )
y5(t ) X8(t )
У6 (t ) X9(t )
У7 (t ) X10(t )
y7(t ) X11(t)
V y (t ) у V X12(t ) у
(19)
Using Eqs. (18) and (19) as a basis, we obtain the resulting equation system (in the Cauchy normal form) describing the behavior of the mixing unit:
+
+
+
+
x
+
+
x
+
x
Fig. 4. Block structural scheme of the mixing unit.
Xl(t ) = x¿t )
X2 (t ) = ®d 2 X X1(t ) + Xdm1 X ®1 X U (t )
X3(t) = Xd01 X U(t)
X4 (t ) = X5 (t )
X5 (t ) = ®d 2 X X4 (t ) + Xdm2 X ® 2 X U(t )
X6(t) = Xd 02 X U (t)
X7(t ) = Xs(t )
X8(t) = -®d23 X X7(t) + Xdm3 X ®d3 X U(t)
X9 (t) = Xd 03 X U (t)
X10(t ) = X11(t ) к
Xll(t ) = (Xl (t ) + X3 (t ) + X4 (t ) + X6 (t ))-
(20)
T
1T
- T2 x xl0(t) - T^ x Xll(t)
к 1
(t) = T (X7 (t) + X9 (t) + X10 (t))- T X X12 (t)
According to Eq. (19), the output equation for the considered MUs is
y(t ) = xu(t ),
(21)
The obtained models can be implemented via different mathematical software that provides the possibility of calculating the MU time-and-frequency characteristics using the known values of doser impulse responses and mixer transfer functions.
RESULTS AND DISCUSSION
The frequency method of determining the smoothening degree requires the knowledge of the frequency transfer function of a mixer that operates in a certain regime (rotor speed, internal and external recycle ratios, taper angle, etc.). The given studies were performed on the white flour-potassium iodide mixture.
To determine the smoothability of the two centrifugal MUs (the first of them is schematized in Fig. 1, and the scheme of the second MU is shown in Fig. 2), the transfer functions of the mixers incorporated in them were represented as W(ja) = j x Im(a) + Re(a). After
Re (a) and Im(a) were determined, we plotted the amplitude frequency characteristic
A(a) = -y] (Im2(®) + Re2(a))
The studied MUs contain two CCMs each and identical spiral dosers, whose frequencies will be used to estimate the smoothability. For this reason, the obtained amplitude frequency characteristics will be identical for both mixing units.
Hence, the amplitude frequency characteristics of the first-stage CCMs of the studied MUs are plotted in Fig. 5.
Fig. 5. Amplitude frequency characteristics A1, A2, and A3 of CCM rotor speeds for the three operational regimes [12] at 10, 12.5, and 15 s-1, respectively.
From Fig. 5 it can be seen that the MU operating in the third regime has the best smoothing characteristics. The CCM smoothability was estimated from the plots for the third operational regime at a specified operational frequency of dosers. For example, if a dosing signal with a frequency a = 4.02 5 1 is sent to
the input of a mixer (first doser signal), the length of the transfer function vector is R(a) = A(a) = 0.032. The smoothability of the first-stage centrifugal mixer was then determined as
S (4.02) =
1
1
R(4.02) 0.032
- = 31.25,
(22)
Hence, the centrifugal mixer smoothens feed flow fluctuations at the given frequency of input signals by 31.25 times.
Let us further consider the amplitude frequency characteristic for the second-stage CCM [11] of the studied MUs (Fig. 6).
Fig. 6. Amplitude frequency characteristics A1, A2, and A3 of CCM rotor speeds for the three operational regimes at 10, 12.5, and 15 s-1, respectively.
If a dosing signal with a frequency® = 4.02 5 1
(third doser signal) is sent to the input of the mixer (at n = 15 5 1), the length of the frequency transfer
function vector is R(a) = A(®) = 0.0123. The smoothability of the first-stage centrifugal mixer is further determined as
S (4.02) =
1
1
R(4.02) 0.0123
81.3,
(23)
Hence, the centrifugal mixer smoothens feed flow
fluctuations at the given frequency of input signals by 81.3 times.
The data for all the MU operational regimes are given in Table 1.
Table 1. Smoothability of the mixing units
CCM operational regimes (rotor speed, s-1) Input signal frequency, s-1
2.093 4.02
White flour potassium iodide mixture
First-stage CCM [1] 10 9.17 24.39
12.5 9.43 28.57
15 10.00 31.25
Second-stage CCM [4] 10 20.3б 45.87
12.5 24.44 91.46
15 19.23 81.3
MU 10 29.53 70.2б
12.5 33.87 120.03
15 39.53 112.55
Hence, it follows from the results of frequency analysis that the smoothability ( ) grows with an increase in the operational speeds of CCM rotors and the input signals formed by the dosers. Its considerable growth occurs upon the switch from the first CCM operational regime to the second regime at both stages. The highest value of ( ) for the studied MUs is observed at a CCM rotor speed of 12.5 s-1.
To determine the degree of the smoothening of real dosing station signals, we also performed the time analysis of the MUs.
Let us first perform the analysis of the first MU (Fig. 1) at a CCM rotor speed of 10 s-1. Let us determine the real signal of the MU first-stage doser block from Eq. (2) using, for example, the MathCAD software for the case when the major component (white flour) is dosed with a spiral doser and the key component (potassium iodide) is dosed with a batch doser. The concentration of potassium iodide in the flour was found potentiometrically on an Elis-131-1 ion selective electrode, with which the equilibrium concentration of iodine ions in a solution was determined. The measurements of pi were performed on an ANION-4100 ion conductivity meter. The iodide selective electrode was preliminary calibrated against standard potassium iodide solutions with a mass concentration of 2. 1.5, 1, 0.5, and 0.1 g/dm3 [3]. The obtained signal is shown in Fig. 7.
The amplitude of the input signal of the first block of dosers is
Yh
Xmax T^mm
. d 0 ~ 0
' 2
8.03б - 5.5б4 ' 2 '
1.23б, g/s. (24)
The obtained signal was further sent to the input of the MU first-stage CCM [12]. The response of the system to the input signal is shown in Fig. 8.
b)
Fig. 8. Response of the system to the input signal of the first block of dosers: (a) output signal of the first-stage centrifugal mixer, (b) ratio of the amplitudes of the input (----) and (----) output signals.
The analysis of the obtained plots allows us to determine the real degree of the smoothening of feed flow fluctuations for the first block of dosers and also the numerical values of the real transfer functions of the first-stage CCMs WCMI(S).
By way of example, let us calculate S (m) of a CCM. To accomplish this, let us calculate the amplitude of the mixer’s output signal by the formula
T^max T^mrn
vOUT _Xd 0 - Xd 0
Xdm = ~
Then we find
б.832 - б.7бб 2 '
0.033, g/s. (25)
X 0 033
R(a) = —^ = ----------= 0.00482,
Xdo б.799
(2б)
Thereupon we calculate the mixer’s smoothability as 11
S(a ) =
R(a) 0.00482
: 207.22,
(27)
Fig. 7. Signal of the first block of dosers (spiral and batch).
The CCM transfer function can be calculated from the ratio of the amplitudes of the input and output signals, and its numerical value is then equal to
The obtained signal was further sent to the input of the MU second-stage CCM [11]. The response of the system to the input signal is shown in Fig. 10.
W '' і
XOU
1(S) = dm
0.033
XdNm 1.235
= 0.027,
(28)
The smoothability of the mixer at rotor speeds of 12.5 and 15 s-1 was determined in a similar way. The obtained results were compiled in Table 2.
Table 2. Smoothability and transfer function of the firststage mixer
n, s-1 Х , g/s s Х ( *) ( ** )
10 1.23б 0.033 207.22 0.027
12.5 1.23б 0.031 222.9 0.025
15 1.23б 0.025 272.9 0.02
* is the fluctuation frequency created by a doser, s-1.
S is an independent complex variable that stands for differentiation with respect to time.
The results of the performed analysis indicate that the CCM [12] smoothens well input material flow fluctuations produced by the first block of volumetric dosers. The best result was obtained at a rotor speed of 15 s-1.
The signal from the second block of dosers was then superimposed to the output signal of the first-stage CCM, thus leading to an increase in its amplitude and the numerical value of its impulse response (signals form the first-stage CCM and the second-stage block of dosers). The graphical interpretation of the given signal is shown in Fig. 9.
Fig. 9. Summary signal from the first CCM and the second block of dosers.
The amplitude of this signal is equal to
Fig. 10. Response of the system to the input signal of the second-stage block of dosers and the first CCM.
The ratios of the amplitudes of the input and output signals for the second CCM [11] are plotted in Fig. 11.
Fig. 11. Magnified system response fragment for the steady-state operational regime. Ratio of the amplitudes of the input (---) and output (—) signals.
Since the second-stage CCM is the end element in the functional structural scheme (Fig. 1), its output signal y(t) may be considered as the output impulse of the entire studied MU, and the transfer function WCM2(S) becomes WMU(S).
For further analysis, let us calculate S (m) and
WCM2(S) of the CCM at n = 10 s 1 . To accomplish this, let us calculate the amplitude of the mixer’s output signal by the formula
T^max
vOUT _Xd0
Xdm =
Xmm d 0
12.294 -12.144 2
= 0.075, g/s. (31)
Y IN _ dm
T^max Ymm
Xd 0 -Xd 0
2
12.474 -12.098 2
= 0.188, g/s. (29)
To determine the impulse response of the first-stage CCM and the second-stage block of dosers, the amplitude calculated by Eq. (30) should be divided by X obtained by Eq. (26):
W
2 (S ) = ^ = 01883 = 5.371, (30)
YL X ,
0.033
Further, we find
R(a) = Xdm = °.°75 = 0.00б14, XdO 12.219
(32)
Then we determine the smoothability of the mixer as 1 1
S (®) =
R(ffl) 0.00б14
= 1б2.8б7,
(33)
The transfer function of the second-stage CCM (or the MU transfer function) will be
W
•• ATI
xOUl
2(S ) = -XN
G.G75
G.188
= G.399, (34)
The smoothability of the second-stage mixer and its transfer function at rotor speeds of 12.5 and 15 s-1 was determined in a similar way. The obtained results were compiled in Table 3.
The results of the performed analysis indicate that the second-stage CCM [11] slightly worse smoothens input material flow fluctuations in comparison with the first-stage CCM [12]. This is explained by that the rotor of the second mixer consists of a single cone, so mixed particles reside in the working zone of the mixer for a shorter time.
Let us further perform the analysis of the second MU, whose regime parameters are the same as for the first MU. The real signal of the first-stage block of MU spiral dosers was determined by Eq. (15) for the white flour potassium iodide feed. The obtained signal is plotted in Fig. 12.
Table 3. Smoothability and transfer function of the second-stage mixer
n, s-1 First mixer and second block of dosers Second mixer or mixing unit
Х , g/s С ) Х , g/s ( ,) С **) С С ))
1G G.188 5.37 G.G75 1б2.8б G.399
12.5 G.186 б. 1G3 G.11 111.2 G.59
15 G.182 3.15 G.1G9 111.б G.595
* is the fluctuation frequency created by a doser, s-1.
S is an independent complex variable that stands for differentiation
with respect to time.
1464+
1454
14435
H 14531
Ü 14226 14122
Щ 14018
13 913
О
У
Fig. 12. Signal of the first block of dosers (both are spiral).
The amplitude of the given signal is xdm = 0.516 g/s.
Sending the given signal to the input of the MU first-stage CCM [12], we obtain the system’s response shown in Fig. 13.
Fig. 13. Magnified system response fragment. Ratio of the
amplitudes of the input (__ __ ) and output (__ ) signals.
The output signal amplitude is x^ = Ü.Ü25 g/s.
The ratio of the amplitude to the average mass flow rate is R(w) = G.GG18.
Then the smoothability of the first-stage CCM is
S (a) = -
1
1
R(w) 0.0018
and its transfer function is
- = 554.07
(35)
W '' і
X°u
1(S ) = -Xrn
X л™
G.G25
G.516
= G.G49
(3б)
The parameters of the implementation of the mathematical model of the MU first stage for the operation of the CCM at rotor speeds of 12.5 and 15 s-1 are given in Table 4.
Table 4. Smoothability and transfer function of the firststage mixer
n, s-1 Х , g/s Х , g/s С *) С ** )
1G g.516 G.G25 554.67 G.G49
12.5 g.516 G.G25 591.13 0.046
15 G.516 G.G2 716.46 G.G38
* is the fluctuation frequency created by a doser, s-1.
** S is an independent complex variable that stands for differentiation with respect to time.
From Table 4 it can be seen that the CCM [12] has the highest smoothability at a rotor speed of 15 s-1.
Further, the output signals of the first-stage CCM and the second block of dosers superimpose over each other. The graphical interpretation of the summary signal is shown in Fig. 14.
Fig. 14. Summary signal from the first CCM and the second block of dosers.
The amplitude of the given signal and the CCM
transfer function are
Yn
Xd,
0.181 g/s and
2(S ) = 7.1, respectively.
W
n CM
The obtained signal was sent to the input of the MU second-stage CCM [11]. The response of the system to the input signal is shown in Fig. 15.
t. sec
a)
Table 5. Smoothability and transfer function of the second-stage mixer
n, s-01 First mixer and second block of dosers Second mixer or mixing unit
X , g/s ( ) s X ( *) ( ,,) ( ( ))
10 0.181 7.1 0.00889 2204 0.049
12.5 0.18 7.51 0.0088 2228 0.049
15 0.176 8.94 0.025 797.95 0.139
* is the fluctuation frequency created by a doser, s'1.
S is an independent complex variable that stands for differentiation with respect to time.
b)
Fig. 15. Response of the system to the input signal of the first block of dosers and the continuous centrifugal mixer: (a) output signal of the second-stage CCM, (b) magnified fragment of the ratio of the amplitudes of the input (__ __ ) and output (-----) signals.
The amplitude of the output signal of the second-stage mixer is
xr = ^d^3- - = 19624 -19606 = 0.00889, g/s. (37)
R(ffl) = = 000889 = 0.00045, (38)
Xd0 19.615
The smoothability of the second-stage CCM is
S (®) = —^ =-------------------------------1-= 2204, (39)
R(ffl) 0.00045
The transfer function of the second-stage CCM (or the MU) is
WCM2(S) = XNr = = °.°49, (40)
The results obtained at rotor speeds of 12.5 and
The results of the performed analysis indicate that the second-stage CCM [11] has the same numerical values of the transfer function as for the first-stage CCM [12]. The difference exists only between the values obtained at n = 15 s'1, thus confirming the fact that the mixture components reside in the working zone of a mixer for a minimum period of time. For this reason, the second-stage CCM has not enough time to smoothen input flow fluctuations to an adequate degree.
Let us further consider some parameters of the implementation of the mathematical model of the studied MUs on the sugar-millet, salt-semolina, and river sand-ferromagnetic powder mixtures from Table 6 and 7.
Hence, the operational frequency regimes of dosers and CCMs have been matched for the preparation of high-quality mixtures with a high ratio of mixed components on the basis of cybernetic approach with some ACT elements. Theoretical and experimental analyses have allowed us to determine the obtained result error, which does not exceed +10.56 %. Consequently, the represented models adequately describe the obtained experimental data.
The smoothabilities of mixers with respect to input material flow fluctuations have been determined using the frequency and time methods. Their numerical values lie within a range from 50 to 2230 times. The discrepancy between the results of time-and-frequency analyses in the case of obtaining the white flour-potassium iodide mixture at a CCM rotor speed of 12.5 s-1 is 8.1%. Hence, the use of these methods of analysis is absolutely allowable.
The implementation of the mathematical models of mixing units that operate by the principle of the sequential dilution of a mixture shows that the best smoothability is attained for the mixing of components at first- and second-stage CCM rotor speeds of 15 and 10 s-1, respectively.
It has been established that it is necessary to prolong the time of the residence of mixed components in the working zone by sequentially passing them through a greater number of cones to increase the smoothability of mixers.
15 s- are given in Table 5.
Table 6. Smoothability and transfer function of the first mixing unit
First CCM First CCM and second block of dosers Second CCM or mixing unit
n, s-1 X , g/s X , g/s ( *) ( ,,) X , g/s ( ) X , g/s ( *) ( ,,) ( ( ))
Sugar-millet mixture
10 1.35 0.042 287.9 0.031 0.363 8.69 0.029 775.7 0.081
12.5 1.35 0.028 430 0.021 0.354 12.65 0.292 77.35 0.825
15 1.35 0.03 404.8 0.022 0.355 11.94 0.071 322 0.2
Salt-semolina mixture
10 1.47 0.012 1478 0.0082 0.383 31.65 0.066 522.4 0.173
12.5 1.47 0.018 1012 0.012 0.386 21.83 0.037 942.7 0.095
15 1.47 0.012 1502 0.008 0.383 32.1 0.038 910.2 0.1
River sand-ferromagnetic powder mixture
10 1.23 0.04 220.97 0.033 0.262 6.55 0.022 772.5 0.083
12.5 1.23 0.03 294.14 0.024 0.255 8.47 0.023 740.1 0.089
15 1.23 0.027 332.36 0.022 0.252 9.48 0.023 742 0.09
* is the fluctuation frequency created by a doser, s'1.
S is an independent complex variable that stands for differentiation with respect to time.
Table 7. Smoothability and transfer function of the second mixing unit
First CCM First CCM and second block of dosers Second CCM or mixing unit
n, s-1 X , g/s s X ( *) ( ,,) X , g/s ( ) X , g/s ( *) ( ,,) ( ( ))
Sugar-millet mixture
10 0.65 0.031 514.1 0.048 0.353 11.21 0.028 965.3 0.079
12.5 0.65 0.031 761.2 0.033 0.348 16.35 0.017 1593 0.049
15 0.65 0.022 726 0.034 0.348 15.61 0.02 1357 0.057
Salt-semolina mixture
10 0.93 0.006 3451 0.0069 0.381 58.75 0.079 496.9 0.207
12.5 0.93 0.006 1868 0.013 0.385 32.15 0.042 926.2 0.11
15 0.93 0.008 2734 0.0087 0.382 46.71 0.044 892 0.115
River sand-ferromagnetic powder mixture
10 0.63 0.037 275.4 0.058 0.259 7.087 0.022 824.4 0.084
12.5 0.63 0.037 362.8 0.044 0.253 9.113 0.023 780.3 0.091
15 0.63 0.025 409.3 0.039 0.251 10.187 0.019 924.6 0.078
* m is the fluctuation frequency created by a doser, s'1.
** S is an independent complex variable that stands for differentiation with respect to time.
The developed mathematical models have allowed second MU that incorporates three spiral dosers has the
us to compare the operational efficiency of two highest smoothability.
centrifugal MUs. The analysis of results shows that the
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