ISSN 00405795, Theoretical Foundations of Chemical Engineering, 2014, Vol. 48, No. 6, pp. 848–854. © Pleiades Publishing, Ltd., 2014. Original Russian Text © S.A. Ivanova, V.A. Pavsky, 2014, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2014, Vol. 48, No. 6, pp. 701–708.
Stochastic Modeling of Protein Solution Foaming Process S. A. Ivanova and V. A. Pavsky Kemerovo Institute of Food Science and Technology email:
[email protected] Received July 5, 2012
Abstract—A stochastic model studying the process of formation and destruction of protein gasliquid dis perse system (foam) is proposed. The model allows one to describe the process state at each time moment of the first cycle. Mathematical expectation, variance, and other moments of random values, which characterize the number of bubbles per unit of volume, and the function that describes the rate of foam destruction, are proposed as a basis to calculate the efficiency parameters of the foaming process. The model could be useful for the imitation modeling of the foaming process. Keywords: proteinbased foams, stability, stochastic model, probability, random values moments, differential equations DOI: 10.1134/S0040579514040198
INTRODUCTION Gasliquid disperse systems (foams) are freely dis perse systems comprised of a gaseous disperse phase and a liquid dispersion medium. In both liquid and solid forms, the foam is widely used in various areas of industry (oil and gas, food, metallurgy, firefighting, etc.). Foams include highconcentration gasliquid disperse systems, which are characterized by the fact that the volume of the disperse phase exceeds the vol ume available for the free densest packing of the spher ical particles, which results in honeycomblike cellu larfilm structure [1, 2]. The bubbles are separated by a thin interlayers of disperse medium. The durability of liquid layers is provided by the presence of surfactants, which include both natural and synthetic compounds. The mechanism of the process is complicated due to the joint effect of multiple physicochemical, physi cotechnical, and other factors. Regularities, which cause the formation of disperse systems, significantly depend on the conditions of physicotechnical and technological process, and foaming and destruction of the obtained gasliquid layer occur simultaneously in the process of foam generation [3–13]. In many ways, these peculiarities complicate the mathematical description of the foaming process [3–5, 14–16]. The main characteristics of the foam are multiplic ity (the ratio of the foam volume to the solution vol ume), dispersion (the size of air bubbles), and stability (period of time from formation to partial or total destruction) [3, 4, 14, 17]. Among the mentioned and other, less common, characteristics, the stability of the
foam is an indicator that can be applied to any foam, independent of its application. It is known that foams based on protein solutions have high stability, and an increase in the protein con centration leads to an increase in foaming properties of the system as a whole [3, 18, 19]. The aim of the present work is to create a stochastic model that would systematically describe the foaming processes in protein solutions and determine the dura tion of formation process of the foam with the speci fied quality. THEORETICAL ANALYSIS Study of the foaming process. In order to study the effect of duration of the gassing process of the protein solution, a stochastic model that describes the effi ciency of the process on the average was used. Mathe matical expectation (average value) M i (t) of the ran dom value, which characterizes the number of parti cles (bubbles) of the disperse phase of the foam in the moment of time t under condition, that at the initial moment of time, their number was M i (0) = i, and dis persion Di (t ), Di(0) = 0, t ∈ [0, ∞), i = 0, 1, 2, … were considered as a basis for calculating the efficiency parameters. If the process of bubble formation depends mainly on the composition of the foamed solution (foamer) and intensity of the mechanical action, the destruc tion occurs under the effect of both internal and exter nal forces. Thus, the whole foaming process, which is a technological stream, can be considered as a dynamic system of streams or a mass service system,
848
STOCHASTIC MODELING OF PROTEIN SOLUTION FOAMING PROCESS
and the study can be conducted using mass service and random processes theory [20–23]. The operation of the foam generator (generation of the foam bubbles) will be assumed to be an unquench able source of the requirements characterized by parameter α. The destruction of the bubbles will be considered to be a service of requirements, character ized by parameter β. Determination of the duration of gassing process of protein solution for obtaining of the foam with specified quality. The studies have shown that the process of bubble formation can be considered as a Poisson pro cess [23], and their destruction can only be considered Poisson process in the first approximation. As it turned out, the process of protein foam destruction for t > 0 is described more efficiently by the function
A1 (1) exp(−(t − a)2 b12 ), b1 where A1, B, a, and b1 are nonnegative numerical parameters, the values of which can be obtained from statistical data, at B > 0, y(t ) > 0, ∀ t ∈ [0, ∞). In fact, it is known that the dynamics of the foam destruction can be conditionally divided into three parts, i.e., initial (insignificant destruction, when destruction factors have minimal effect on the foam; a gradual increase of the destruction rate occurs), active (denoted by considerable increase of the destruction rate, up to the maximum, the greatest impact of each destruction factor, including in combination), and decaying (decrease of the destruction rate) phases [4, 5, 23]. As the stability of the foam depends on the destruction rate, parameter β = y(t ) can be considered its characteristic in our model. Moreover, it was found that the boundaries that divide the phases of the foam destruction are determined by the critical points of the function y(t), which can be obtained by the methods of differential calculus. For the convenience of applying the function y(t) in engineering calculations, we should note, that it is reminiscent of the density of the normal distribution, which is tabulated in normalized form, thus, further, at A1 = A π , b1 = 2b, instead of y(t), we will write y(t) = B +
β(t ) = B + A
( 21πb exp(−(t − a)
2
)
(2b 2 )) .
Let us proceed to investigating the foaming pro cess. Mathematical model. Suppose we have a mass ser vice system that receives service requirements. The number of requirements arriving in time t is repre sented by the Poisson process Vk (t) =
(αt) −αt e , k! k
849
where α is the intensity determined as an average number of bubbles generated per unit of time, t ∈ [0, ∞), k = 0, 1, 2, .... The requirement received by the system begins to be served immediately. Service time is a random num ber η distributed exponentially as follows:
P(η < t) = 1 − e −βt , where β = 1 t mid is the intensity of service and t mid is the average requirement service time determined as a mean time of bubble life. A serviced requirement leaves the system. Mathe matical expectation M i (t) and dispersion Di (t ) must be calculated. Suppose that Pk (i, t ) is a probability of the fact that, at the moment of time t ∈ [0, ∞), the system contains k requirements and the condition that, at the initial moment of time t = 0, their number was i. The proposed stochastic model is described by a random process of birth and death [24] and is formal ized in the form of system of differential equations as follows:
⎧P '(i, t ) = −α P (i, t ) + β P (i, t ), 0 1 ⎪ 0 ⎪ ⎨Pk'(i, t ) = −(α + kβ)Pk (i, t ) + α Pk −1(i, t ) ⎪+ (k + 1)β Pk +1(i, t ), ⎪ ⎩ with the normalization condition
(2)
∞
∑ P (i, t) = 1, ∀ t ∈ [0, ∞) k
k =0
and initial conditions Pi (i,0) = 1, Pk (i,0) = 0, ∀k ≠ i. Let us introduce the function ∞
F (i, z, t) =
∑ z P (i, t), k
k
(3)
F (i,1, t) = 1, ∀ t ∈ [0, ∞) . (4)
k =0
For any fixed t, we can recurrently express the moments of random values of any order beginning from the first through the generating function F (i, z, t ) and obtain the system of differential equations directly for them [21, 22, 24]. In particular, considering (2)– (4), for M i (t ) and Di (t), we obtain the system of two equations in the following form [25]:
⎧ d M (t) + β M (t ) = α, i ⎪dt i ⎪⎪ d ⎨ Qi (t) + 2β Qi (t) = 2αM i (t), ⎪dt ⎪Qi (t) = Di (t) + M i2(t) − M i (t), ⎪⎩ the initial conditions of which are M i (0) = i, Di (0) = 0.
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 48
No. 6
2014
(5)
(6)
850
IVANOVA, PAVSKY
M(t) × 10–4 15 12 9 1 2 3 4
6 3
0
3
6
9
12
15 t, min
Fig. 1. Number of bubbles per unit of volume at the moments of time t = 3m, m = 1, 2, 3, 4, 5: 1. v = 1750 rev/min; 2. 2000; 3. 2500; 4. 3000 rev/min.
The solution of the system (5) with initial condi tions (6) has the following form [25]
⎧M (t) = α (1 − e −βt ) + ie −βt , ⎪⎪ i β ⎨ α ⎪Di (t) = (1 − e −βt ) + ie −βt (1 − e −βt ). β ⎩⎪
(7)
If stationary mode is achieved fairly quickly, the following formulas could be used for the expressanal ysis
M = lim M i (t ) = α β , D = lim Di (t ) = α β . t →∞
(8)
t →∞
Let us proceed to the study of the foam destruction process. In formulas (7), parameter (intensity) α is determined as the average number of bubbles formed in units of time, and parameter β can be interpreted as the average number of bubbles destructed in the same time unit. Subsequently, formulas (7) can provide an average estimate of the efficiency of the foaming pro cess at any moment in time. Upon a detailed analysis we should note that parameter β describes the foam destruction process from the moment of its formation and, therefore, generally depends on time; i.e., β = β(t), where β(t) satisfies (1). Considering the physi cal meaning of this parameter, functional dependence that defines it will allow one to study the foam destruc tion process in the limit of the first oscillatory cycle. Let us use the methods of differential calculus. Calcu lating the derivative β(t) and assuming it equal 0, we will find the maximum time t0, before which foaming should be stopped, i.e., without waiting for the highest foam destruction rate. Obtained critical time allows us not only to use formulas (7) efficiently (in which the value of parameter β is obtained from statistical data or at known function β(t), the theory of average integra
tion interval [0, t 0]) is used), but also to estimate the error of applying formulas (8). We should note that the derivation of the differen tial equation directly for the numerical characteristics is stable to the form of equation, i.e., it is independent on the fact that parameters α, β are constant or are functions of time, which cannot be guaranteed for the equations of system (2). This is why the stochastic model allows one to plan the foaming process of the first cycle based on the more simple dependences of the form (7), with lower calculation complications, as well as to determine the vicinity of the critical time t0, beginning from which the quantity of the foam lowers significantly. The solu tion of the system (5) with variable parameter β = β(t) can be found by numerical methods using appropriate software, e.g., Mathematica 5.2, which does not pos sess the abilities to study the tendencies of the process and to approach the investigation of the process sys tematically, but allows one to obtain numerical results for averages at β = β(t) and compare them with analyt ical solutions, where β(t) = β.. EXPERIMENTAL The quality of the gasliquid disperse system depends on the technology of its formation. In order to verify the extent of compliance of the proposed model to the studied process, the experimental stud ies, which consist of gas saturation of protein solution (skim milk protein concentrate, protein mass fraction of 4.4%) by rotary pulsation gassing device at rotor revolution rate v ranging from 1750 to 3000 rev/min; the filling factor of the working chamber was 0.3, the value of the gap between rotor and stator was 0.1 mm; temperature of the processed solution was 13 ± 2°С. Values of the parameters α and β, which can be deter mined by the statistical data, are required to apply the formulas (7). The main technical operation parameters of most foaming devices are the frequency of the working body revolution, filling factor of the working chamber, the duration of processing, and temperature. It is known that the increase in the rate of the revolution of the working body of the foaming device v leads to the intensification of the foamgeneration process (decreasing the time of the solution processing) [4, 5, 9, 18, 19]; however, in this case, the intensive mechan ical impact on the formed system is the reason for its destruction. The values v = 1750, 2000, 2500, 3000 rev/min were studied; corresponding parameters α, β will be denoted as αv , βv . Figure 1 shows the results of observations in the moment of time t = 3m min, m = 1, 2, 3, 4, 5 depending on the number of rotor revolu tions. The number of generated bubbles per unit time (value of parameter α) was determined for the used equipment and solution after freezing the samples in a
JOURNAL OF THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 48
No. 6
2014
STOCHASTIC MODELING OF PROTEIN SOLUTION FOAMING PROCESS β(t) 1.5
β(t) 1.50 1 2 3 4
1.0
1 2 3 4
1.00
0.50
0.5
0 0
851
3
6
9
12
3
6
15 t, min
Fig. 2. Changes of values of βv depending on time t = 3m: 1 – v : 1. v = 1750 rev/min; 2. 2000; 3. 2500; 4. 3000 rev/min.
nitrogen atmosphere with subsequent microscopic examination in the transmitted light (AxioVert.A1 microscope with AxioCamERc5 camera with a photodocumentation block), and number of bubbles was counted with digital images using appropriate software (comparison of the results of automatic and manual counting of the bubbles in the frozen samples showed that results of the former were underestimated by 18% in average). Assuming that destruction is almost absent in the first minute of generation, the α v = 25000, 33250, 495000, 63700 1/min. The use of both the mean value and mean square deviation allowed us to determine the value ranges of this parameter for each of these foaming processes. The following ranges of α v parameter value were obtained: 11500–38400, 12500–53900, 13200–77800, 22000–105000 1/min for ν = 1750, 2000, 2500, 3000, respectively, in which the value of this interval lies in the repeated test series. The direct determination of the number of the bub bles destructed under the effect of various factors is difficult, and thus the values of parameter β were picked. In fixed moments of time, the differences between generated number of bubbles and factual one were noted. Then, we calculated the ratio relative to the factually generated stream of bubbles consid ered per unit of time, for example, β 2500(3) = 49500 × 3 − 146324 : 3 = 0.00496 1/min. 146324 The values of parameter βv at time moments t = 3m are shown in Fig. 2. Analysis allows us to claim that, after ~12 min of the process, a second oscillatory cycle begins, which is characterized by an increase in the rate of foam destruction, which is followed by the decay period. Thus, the study of the foaming process for more than 12 min is impractical. Moreover, for all
9
12 t, min
Fig. 3. Changes of function β(t ) depending on time t: 1. values determined by the experimental data; 2. A = 7.50, B = 0.01, a = 6.90, b = 2.10; 3. A = 7.20, B = 0.01, a = 7.00, b = 2.00; 4. A = 6.90, B = 0.01, a = 7.10, b = 1.80.
presented variants, the highest values correspond to t = 6 min. Since βv is a function, then in reality, this moment of time will be in the vicinity of the men tioned point on the time axis. This information can be clarified in the process of investigation. At this stage, we will assume the interval (0, 12) with a center in point t = 6. We should note that, in reality, complete symmetry is improbable. In order to estimate the deviation from the symmetry point, if the need arises, we can use the third central moment to calculate the asymmetry coefficient. The corresponding formula is determined by the solution of the differential equation, additional to the system (5) obtained by the same way as the equation for the mathematical expectation and dispersion. The values of βv were approximated by the function β(t), the form of which, due to the presence of the spe cial points (extrema, bend points, etc.), allowed us to track the moments of time important to the process of formation and destruction of foam (explanation of physicochemical basis of which are provided in [3–5, 18, 19]). Figure 3 shows several variants of the approx imation of the function β2500 = β(t) of the form (1), the error varies from 15 to 20%, respectively. Upon con sideration of the lesser time interval, the accuracy increases almost by two times. Since the highest value of function β(t), which characterizes the rate of foam bubbles destruction, is assumed in the interval of 6– 8 min, then upon further studies, we will analyze the segment [0, 9). Let us determine the moment of time t0 (located in the interval from 2 to 4 min, judging by Fig. 3), beginning from which the intensive increase of the foam destruction rate occurs.
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 48
No. 6
2014
852
IVANOVA, PAVSKY
M(t) × 10–4 16 12 8 4
0
3
t0
6
t1
lowest value; obviously, it is unacceptable to achieve this value in the process of foam generation. 1 The calculation of the derivative function β(t) 2 allows one to determine the the argument value that 3 equals a hours. The second derivative of the function 4 β(t) equals zero at two points a ± b, which define the extrema of the function β'(t) and are the rates of change of function β(t). The third derivative of the function equals zero for t = a, t = a ± 3b. Thus, point t = a defines the maximum of function β(t), the bend ing point of function β'(t), and extremum β ''(t); points t = a ± b are the bending points of β(t) and extremum 9 t, min of function β'(t); t = a ± 3b are the bending points of β'(t) and extrema β ''(t).
Fig. 4. Dependence of number of bubbles on time t upon rotor pulsation processing of the protein solution: 1. exper imental data; 2. A = 7.50, B = 0.01, a = 6.90, b = 2.10; 3. A = 7.20, B = 0.01, a = 7.00, b = 2.00; 4. A = 6.90, B = 0.01, a = 7.10, b = 1.80.
As can be seen from Fig. 3, both statistical and empirical curves assume highest value at time moment t0, which is located approximately inside interval [6, 7]. Thus, it is impractical to perform the foam genera tion processes more than for 7 min, since it appears that the highest total effect of all destruction factors occurs in the vicinity of this moment of time; this is confirmed by Fig. 4. The solution of the system (5) with variable parameter β = β(t) in the form (1) is pre sented in Fig. 4. The obtained results for various values of the parameters of function β(t) describe the experi mental values with various degrees of accuracy, the value of which decreases with an increase in the time interval, wherein each of them describes the general character of the process in the interval up to 9 h quali tatively and with significantly accuracy. Thus, after studying the effect of function β(t) on the foam destruction dynamic, we will study the process of foam generation as a whole. Further, based on the obtained data, we will formulate the limits by the time of effect on the processes mass. The application of the differen tial calculus on the analysis of the function M (t ) from the first equation of system (5) allows (since β(t) ≠ 0) one to determine its maximum value, which is achieved at some point t0 from the equality
M (t 0 ) = lim α > 0. t →t 0 β(t ) At a constant value of the parameter α, function M (t ) achieves the highest value for this argument
value, at which the value of function β(t) is lowest in the interval [0, 7). It can be seen from Fig. 4 that the time interval in which the foaming process should be terminated is [3, t 0]. Analogously, at point t1 of the maximum of function β(t), function M (t ) assumes the
RESULTS AND DISCUSSION Considering the physical meaning of the function β(t), we obtain that the parameter a determines the moment of time, in which a maximum of foam destruction is achieved; a ± b is the moment of time, in which intensive change of destruction rate begins, namely a – b is the intensive increase; a + b is the intensive decrease of the destruction rate; and t = a ± 3b are the moments of stability of acceleration of foam destruction rate, beginning from which the intensive increase of acceleration (t = a − 3b) or intensive deceleration (t = a + 3b) occur. In this case, the time interval [a − 3b, a − b] where the moment of the end of the foaming process should belong clearly stands out, and the value t = a − 3b is obviously ideal. For the considered function β(t) (A = 7.50, B = 0.01, a = 6.90, b = 2.10; A = 7.20, B = 0.01, a = 7.00, b = 2.00; A = 6.90, B = 0.01, a = 7.10, b = 1.80, Fig. 3), these time intervals are [3.26, 4.80], [3.54, 5.00], [3.98, 5.30], and the ideal moments of time for the ending of the process are 3.26, 3.54, and 3.98 min, respectively (we should note that, upon the description of the rate of destruction of the protein foam using function β(t), the error was 15, 18, and 20% in the interval of 0–9 min, respectively). We should note, that, even at the initial phase, according to the experi mental data, one could claim that the foaming process should apparently end approximately after 3 min; more accurate conclusions cannot be drawn due to the discreteness of the function, which represents the experimental values. Wherein, after 2 min, the process ending is undesirable, since, at this time, the rate of bubble formation significantly exceeds the rate of destruction, while at 4 min, it is too late, i.e., the pro cess of intensive destruction is already irreversible. Comparing the experimental values with the values of the function M (t ) (Fig. 4) determined by the equa tion of system (5) with variable parameter β(t) of the form (1), in intervals of 0–4.8, 5.0, and 5.3, the error
JOURNAL OF THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 48
No. 6
2014
STOCHASTIC MODELING OF PROTEIN SOLUTION FOAMING PROCESS M(t) × 10–4 20
M(t) × 10–4 24 1 2 3 4
16 12
20 16
853
1 2 3 4 5 6
8 12 4 8 0
3
6
9 t, min
Fig. 5. Dependence of number of bubbles on time t upon rotor pulsation processing of the protein solution, β(t ) = β: 1. experimental data; 2. A = 7.50, B = 0.01, a = 6.90, b = 2.10; 3. A = 7.20, B = 0.01, a = 7.00, b = 2.00; 4. A = 6.90, B = 0.01, a = 7.10, b = 1.80.
amounted to 8–2%, respectively, and in the interval [0, 4), it does not exceed 3%, which can be considered within statistical error. Let us consider the function M (t ), which is a solution of the system (7). Let us divide the interval of 0–4 by the parts with a step of 1 and determine the average value of parameter β on each one. We will approximate the experimental data on the number of foam bubbles in the unit volume by pieces of the function M (t ) with constant parameter β as follows [20]: T
β = 1 β(t )dt, T
∫ 0
where T is the considered interval. The results, shown in Fig. 5, allows to determine the approximation error in the interval [0, 4) up to 4%, respectively; increase of the length of the interval significantly increases the error up to 30% or more. Thus, to analyze the foaming process in the interval [0, 4), we can use the formulas obtained from the system (5) at β = β(t ) and simple formulas (7), where β is constant. Analogously, considering approximation by the function βv (t) at v = 1750, 2000, 3000 rev/min with error that does not exceed 18% in the interval of 0– 9 min, we obtain the intervals [3.45, 4.91]; [3.24, 4.76]; [2.36, 4.25]. Recommended moments of time for the ending are ~3.5, 3.3, and 2.4 min, respectively. Let us study the foaming process during t0 minutes. Preliminarily analysis of experimental data (Fig. 2) have shown that on this time interval, apart from the case that corresponds to foam generation of protein concentrate at 3000 rev/min of the rotor, average val ues of the parameter β(t) = β approximately coincide and equal to 0.010 1/min. Knowing the performance of the equipment used by the foaming of the protein
4
0
1
2
3 t, min
Fig. 6. Dependence of number of bubbles on time t upon rotor pulsation processing of the protein solution: 1. exper imental data; 2. M 0 (t ), α = 49 500 1/min, β = β(t ), A = 7.50, B = 0.01, a = 6.90, b = 2.10; 3. M 0 (t ), α = 49500 1/min, β = β(t ), A = 7.20, B = 0.01, a = 7.00, b = 2.00; 4. M 0 (t ), α = 49 500 1/min, β = β(t ), A = 6.90, B = 0.01, a = 7.10, b = 1.80; 5, 6. boundary M 0 (t ), α = 13 270, 77 800 1/min, β = 0.010 1/min.
solution (parameter α), we will use formulas (7) to describe the process and will obtain the range (M i (t) − Di (t); M i (t) + Di (t )) of the number of bubbles per unit of volume of foam (Fig. 6). We should note that, with a probability close to 1, this interval includes experimental values of the bubble number in unit of volume of the foam at repeated series of tests of rotor pulsation processing of the pro tein solution (skim milk protein concentrate, with ini tial content of dry substance mass fraction of 9.2%) at 1750, 2000, 2500 rev/min. The case corresponding to 3000 rev/min does not attract any interest, since excessive hydromechanical impact only leads to sig nificant destruction of the foam due to the rotation of the working body of the device. CONCLUSIONS In the present work, the first cycle of the formation and destruction processes of protein foam were stud ied in detail. The proposed model allowed us to quan titatively describe the foaming process in both average and specific states. It was shown that the application of the numerical methods is well approximated by ana lytical methods that provide simple formulas, which are convenient for engineering calculations. It was established that the duration of the formation of pro tein foam in a rotor pulsation device at specified tech nological parameters of the process should be limited
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 48
No. 6
2014
854
IVANOVA, PAVSKY
(in our case, Fig. 4, for example t0 ≈ 3.26, 3.54, 3.98 min, respectively) before the moment of the highest rate of foam destruction. ACKNOWLEDGMENTS The work was supported by the Council on grants of the President of the Russian Federation (grant No. NS2175.2012.9) and Russian Foundation for Basic Research (project No. 120700145). NOTATION A, A1, B, a, b, b1—numerical parameters; D—limiting dispersion of bubble number; Di (t)—dispersion of the random value at the moment of time t; F (i, z, t )—generating function; M—limiting average value of the bubble number; M i (t)—mathematical expectation of the random value at the moment of time t; P (t ), Pk (i, t ) —probability;
Qi (t) = Di (t) + M i2(t ) − M i (t); t—time, min; t j — jth moment of time; t mid —average time of requirement service, min; Vk (t)—Poisson process; z—complex number; α—intensity of the inflow of the requirements per unit of time, 1/min; β—intensity of the service of the requirements per unit of time, 1/min; β(t), βv (t)—function that describe the process of protein foam destruction; β'(t), β ''(t), β'''(t)—derivative of the function β(t); η—a random value; v —the rate of rotation of the rotor of rotor pulsa tion device, rev/min.
5. 6. 7.
8.
9. 10. 11.
12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
SUBSCRIPTS AND SUPERCRIPTS i—initial number of requirements (bubbles); j, m—sliding variable; k—number of requirements (bubbles); v —number of revolutions.
22.
23.
REFERENCES 1. Fridrikhsberg, N.N., Kurs kolloidnoi khimii (Colloid Chemistry), Leningrad: Khimiya, 1984. 2. Gel’fman, M.I., Kovalevich, O.V., and Yustratov, V.P., Kolloidnaya khimiya (Colloid Chemistry), St. Peters burg: Lan’, 2003. 3. Kruglyakov, P.M. and Ekserova, D.R., Pena i pennye plenki (Foam and Foam Films), Moscow: Khimiya, 1990. 4. Tikhomirov, V.K., Peny: Teoriya i praktika ikh polucheniya i razrusheniya (Foams: Theory and Prac
24. 25.
tice of Production and Breaking), Moscow: Khimiya, 1983. Kann, K.B., Kapillyarnaya gidrodinamika pen (Capillary Fluid Dynamics of Foams), Novosibirsk: Nauka, 1989. Bikerman, J., Foams. Berlin: Springer, 1973. Pisarev, A.V. and Ivanenko, A.Yu., Mathematical model of particlebubble flotation complex motion in vibration flotation, Theor. Found. Chem. Eng., 2009, vol. 43, p. 321. Trushin, A.M., Dmitriev, E.A., and Akimov, V.V., Mechanics of the formation of microbubbles in gas dis persion through the pores of microfiltration mem branes, Theor. Found. Chem. Eng., 2011, vol. 45, p. 26. Arunagiri, A., Regupathi, I., and Murugesan, T., Inter facial area and mass transfer coefficients in liquidgas ejec tors, Theor. Found. Chem. Eng., 2011, vol. 45, p. 818. Amanbaev, T.R. and Antony, S.J., Modeling of bubble rise in a liquid subject to surface mineralization, Theor. Found. Chem. Eng., 2011, vol. 45, p. 876. Meshalkin, V.P. and Boyarinov, Yu.G., SemiMark ovian models of the functioning of complex chemical engineering systems, Theor. Found. Chem. Eng., 2010, vol. 44, p. 186. Padohin, V.A. and Zueva, G.A., Stochastic models of the grinding of disperse materials, Theor. Found. Chem. Eng., 2009, vol. 43, p. 682. Akhmadiev, F.G. and Nazipov, I.T., Stochastic model ing of the kinetics of processing of heterogeneous sys tems, Theor. Found. Chem. Eng., 2013, vol. 47, p. 136. Schramm, L.L., Emulsions, Foams, and Suspensions: Fundamentals and Applications, Weinheim: Wiley VCH, 2005. Kempner, D., Polimernye peny i tekhnologii vspenivaniya (Polymer Foams and Foaming Technolo gies), St. Petersburg: Professiya, 2009. Vetoshkin, A.G., Hydromechanics of separation of gas liquid systems with foam structure, Theor. Found. Chem. Eng., 2004, vol. 38, p. 569. Merkin, A.P. and Taube, P.R., Neprochnoe chudo (Del icate Wonder), Moscow: Khimiya, 1983. Walstra, P., Physical Chemistry of Foods, New York: Marcel Dekker, 2003. Dickinson, E., Food Emulsions and Foams, London: The Royal Society of Chemistry, 1987. Saati, T.L., Elementy teorii massovogo obsluzhivaniya i ee prilozheniya (Elements of Mass Service Theory and Its Applications), Moscow: URSS, 2010. Pavskii, V.A., Pavskii, K.V., and Khoroshevskii, V.G., Calculation of the survivability of distributed comput ing systems and problem solution feasibility, Iskusstv. Intell., 2006, no. 4, p. 28. Khoroshevskii, V.G., Pavskii, V.A., and Pavskii, K.V., Calculation of the survivability of distributed comput ing systems, Vestn. TGU, Upravl. Vychisl. Tekh. Inf., 2011, no. 2, p. 81. Ivanova, S.A., Stokhasticheskie modeli tekhnolog icheskikh protsessov pererabotki dispersnykh sistem obez zhirennogo moloka (Stochastic Models of the Process ing of Nonfat Milk Dispersions), Kemerovo: KemTIPP, 2010. Feller, W., An Introduction to Probability Theory and Its Applications, New York: Wiley, 1968, vol. 1. Yustratov, V.P., Pavskii, V.A., Krasnova, T.A., and Ivanova, S.A., Mathematical modeling of electrodialy sis demineralization using a stochastic model, Theor. Found. Chem. Eng., 2005, vol. 39, p. 259.
Translated by A. Shokurov
JOURNAL OF THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 48
No. 6
2014
