ISSN 0040-5795, Theoretical Foundations of Chemical Engineering, 2018, Vol. 52, No. 3, pp. 404–411. © Pleiades Publishing, Ltd., 2018. Original Russian Text © G.I. Kelbaliyev, S.R. Rasulov, G.R. Mustafayeva, 2018, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2018, Vol. 52, No. 3, pp. 352–360.
Viscosity of Structured Disperse Systems G. I. Kelbaliyeva, *, S. R. Rasulova, **, and G. R. Mustafayevab aNagiev
Institute of Catalysis and Inorganic Chemistry, Azerbaijan National Academy of Sciences, pr. Huseyn Javid 113, Baku, AZ1143 Azerbaijan b Azerbaijan State Oil and Industrial University, pr. Azadlig 16/21, Baku, AZ1002 Azerbaijan *e-mail:
[email protected] **e-mail:
[email protected] Received October 12, 2016
Abstract⎯A generalized viscosity equation has been proposed for structured disperse systems with their peculiar rheological properties. Some characteristic features of the viscosity curve have been revealed at the beginning of structuration. Different variants of viscosity equations for structured disperse systems have been considered, and their comparison with the experimental data has been performed. The possibilities of estimating the porosity and shear viscosity of structured media with a fixed layer of particles have been demonstrated. Keywords: disperse systems, aggregation stability, structuration, rheology, viscosity, coagulation, shear viscosity, porosity DOI: 10.1134/S0040579518020082
INTRODUCTION All disperse systems are characterized by aggregation and sedimentation (kinetic) stability, which in turn determine the structure of a disperse medium. The formation of structured disperse systems associated with the formation of aggregates of particles or clusters of aggregates is considered in a number of papers [1–4]. The structure of a disperse system is governed by the character of the mutual arrangement and orientation of particles and the existence of bonds between them. The fixation of the mutual arrangement of particles structures a system, which imparts certain physical properties to it in the long run. The interaction between particles with the formation of coagulation structures, which results from the coalescence of drops and bubbles or the adhesion of solid particles with a change in the principal rheological parameter, i.e., the viscosity of the entire system is observed in concentrated systems. The effect of structuration in disperse media is very efficiently illustrated by the viscosity curve (Fig. 1). The analysis of an experimental curve for the viscosity of a disperse system as a function of the concentration of particles shows that the entire region may be divided into three parts, where (a) region I (OA) is characterized by a monotonical change in the viscosity of a system at a low concentration of particles without the formation of coagulation structures, (b) region II (AC) corresponds to the beginning of the formation of coagulation structures and aggregates with a maximum deviation of the viscosity from its values without structura-
tion at point B, and (c) region III (CD) of intensive structuration until the creation of a framework at high system viscosities and low mobilities. It is noteworthy that, for different disperse systems, these regions appear at certain concentrations of particles and are determined by the nature of the system itself. In the coagulation structures of systems with the participation of the solid phase, region AC becomes narrower and, in some cases, may be not present at all. Moreover, this region allows damage into the structure, so region AC is characterized by thixotropic properties. In particular, it may be noted that the formation of aggregates from asphaltene particles occurs at low volumetric concentrations of 0.01 [5]. At high solidphase concentrations, a dense disordered structure with variable porosity and bulk (or shear) viscosity is formed from elastically bonded particles, and this also depends on the shear stress. The shear or bulk viscosities of a layer of particles are determined by their rearrangement under external deforming stresses. The presented curve of the viscosity as a function of the concentration of particles is typical for structured disperse systems. The objective of this study was to develop the models for calculating the viscosity of different rheological disperse systems with consideration for their structuration. STRUCTURATION OF DISPERSE SYSTEMS The coordinates and momenta of particles in aggregatively stable disperse systems are independent
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from each other, i.e., the position and motion direction and velocity of each particle do not depend on the position and velocity of other particles. This state is generally typical for diluted disperse systems, where there is no probability of interaction between particles and their collision and coagulation. The distance between particles in the absence of collision between them depending on the concentration of particles in volume can be approximately determined by the formula [6] l ≈ 80a 3 ρd Cm , where ρd is the density of particles, g/cm3, and Cm is the mass concentration of particles, g/m3. As follows from this formula, the distance between particles decreases with an increase in their concentration in a closed volume, and the probability of collision between them grows, which leads to the formation of coagulation structures. The further formation of aggregates together with aggregate and framework clusters leads to the creation of a continuous loose framework from mutually bonded particles with an appreciable increase in the viscosity of a system and a decrease in its fluidity and flow velocity. The destruction of bonds between particles and the structure may be provoked by external mechanical effects, i.e., an increase in the shear velocity due to growth in the velocity or pressure gradient, etc. Hydrodynamic forces stretch all of the bonds in aggregate to a critical length, after which the aggregate decomposes into smaller aggregates, and a further increase in the force leads to its complete destruction. In a quiescent state, the destructed bonds and the structural state of a system can completely recover, and this is characterized by the thixotropic properties of a system. It is noteworthy that, in some cases, a change in temperature may also affect the thixotropic properties. In particular, this concerns the appearance of physical phenomena (melting, vaporization), which change the phase state of a particle. In the case of oil, this primarily concerns wax particles, which can melt at a high temperature to change their phase state with the resulting destruction of certain structures. A decrease in temperature leads to the appearance of crystallization processes with the formation of solid wax particles able to form certain structures. Similar processes occur in oil-containing asphaltene particles, when it contains aromatic hydrocarbons (toluene, benzene), which destroy coagulation structures and aggregates when dissolving asphaltenes contained in oil. Turbulence in the flow of a disperse system produces a significant effect on the interaction between particles, as it increases the frequency of collisions by several times in comparison with a laminar flow. However, turbulence not only promotes the formation of a coagulation structures, but also destroys the formed aggregates and framework upon the distortion of equilibrium between the phenomena of coagulation and breakup. Flow turbulence deforms aggregates and clusters with the compression, stretching, and disruption of bonds between their particles, which results in
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η D
III II
I
C η0 – ηs
A
B
ϕ
0 Fig. 1. Regions of structuration in a disperse medium.
their destruction at high turbulent pulsations. The sizes of particles also affect the frequency of their collisions and, correspondingly, the creation of coagulation structures. In works [7, 8], the frequency of collision between drops in an isotropic turbulent flow is determined depending on the scale of turbulence as
(εa )
( )
13 ⎛ ⎞ ε exp ⎜ −C2 R2 t ⎟ , λ > λ0, a ⎝ ⎠ (1) 12 12 ⎛ ⎛ εR ⎞ ⎞ 2 ⎛ εR ⎞ ω ( a ) = C3ϕ0μ p ⎜ ⎟ exp ⎜ −C4 ⎜ ⎟ t ⎟ , λ < λ 0. ⎜ ⎝ νc ⎠ ⎝ νc ⎠ ⎟⎠ ⎝
ω ( a ) = C1μ pϕ0 2
13
R 2
(
For the fast coalescence of drops at t ! a2 εR
ε ω ( a ) ≈ C1μ2pϕ0 ⎛⎜ R2 ⎞⎟ , λ > λ0 ⎝a ⎠
)
13
,
13
(2)
or, at t ! ( νc εR ) , 12
12
⎛ε ⎞ 2 ω ( a ) ≈ C3μ pϕ0 ⎜ R ⎟ , λ < λ0, ⎝ νc ⎠ where μ2p is the degree of the entrainment of particles with a pulsing medium, μ2p → 1 for fine drops, and
(
)
μ2p → 0 for coarse drops, and λ 0 = ν3c εR is the Kolmogorov turbulence scale. These formulas were theoretically derived in papers [7, 8] using the equations of mass transfer in an isotropic turbulent flow, but experimentally confirmed in the works [9, 10]. As follows from these equations, the frequency of collisions between particles is inversely proportional to −2 3 their size, i.e., ω ~ a , and grows with an increase in the concentration of particles in the volume. As follows from Eq. (2), an increase in the viscosity of a medium and the size of particles decreases the fre-
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quency of collision between particles in an isotropic turbulent flow and, consequently, the probability of the formation of a coagulation structures.
At small r ! a, we have ∂ 2N ∂r 2 @ 1 ∂N ∂r , so 2r Eq. (3) is recast as
V ∂N = D ∂ N2 ∂r ∂r r = 0, N = N 0; r → ∞, N = N ∞. 2
In some cases, the collision of drops and bubbles promote their coalescence and growth, which distorts the aggregation and sedimentation (kinetic) stability of a disperse medium with the resulting qualitative transformation of its structure. The possibility of the formation of coagulation structures upon the collision between solid particles is also governed by many other factors, such as adhesion and cohesion, the surface properties of particles, their sizes, shape, propensity to adhesion, etc. When dry particles adhere to each other to form a structure, such a state is possible for every size of particles up to a certain critical flow velocity, after which it is destroyed. In most cases, coarsely disperse solid particles do not form coagulation structures, but can only create a dense packing of disordered structure at high concentrations with a minimum porosity and a rather high shear or bulk viscosity. To some extent, this fact may be explained by the existence of elastic collisions in the process of their interaction. When the surface of particles has a liquid layer of any substance, which improves the adhesion properties of their surface, the formation of particle aggregates with a loose structure is possible. Physical phenomena, such as the high-temperature sintering of particles and their crystallization from a liquid phase enable the formation of aggregates with a rather high density and hardness, and the physical processes of the melting and dissolution of particles can destroy the structure of any aggregates and framework. The formation of coagulation structures and aggregates imparts the character of Newtonian liquids with their typical rheological properties to disperse systems. COAGULATION OF FINELY DISPERSE PARTICLES IN A SHEAR FIELD Due to the chaotic motion of finely disperse particles and the existence of hydrodynamic forces, particles approach each other at a distance of one diameter and become bonded, which leads to fast coagulation with the formation of coagulation structures. These structures are formed by finely disperse particles due to their hydrodynamic displacement and Brownian diffusion. Let us consider the simplest equation for the flow of particles under the influence of convective and diffusion fields in the spherical form
(
)
2 V ∂N = D2 ∂ r ∂N . ∂r ∂r r ∂r
(3)
(4)
Taking into account that γ = ∂V ≈ V , we deter∂r a mine the shear flow velocity as V = aγ and, substituting the latter into Eq. (4) taking into consideration the dimensionless parameter z = r a , we find that
∂N = 1 ∂ N . (5) ∂z Pe ∂z 2 The solution of Eq. (5) with boundary conditions has the form 2
(
)
N − N∞ = exp −Pe r , N0 − N∞ a
(6)
6πa η where Pe = a γ = γ is the Peclet number or the D kBT dimensionless shear flow velocity that corresponds to 3 the dimensionless shear stress τr = a τ . kBT The Peclet number or the shear flow velocity for an isotropic turbulent flow will be as follows: 3
13
⎛ 2⎞ Pe ≈ α1 ⎜ a ⎟ ⎝ εR ⎠
γ, λ > λ0,
(7) 12 ⎛ νc ⎞ Pe ≈ α2 ⎜ ⎟ γ, λ < λ0. ⎝ εR ⎠ The frequency of collisions between finely disperse particles is determined as
ω = − 4 πa 2 D ∂ N ∂r r = a (8) 3 = 4πa ( N 0 − N ∞ ) γ. Using the experimental data [2], it is possible to derive the semiempirical dependence of the disperse medium viscosity on the Peclet number in the form
(
)
η − η∞ 12 = exp −mPe . η0 − η∞
(9)
Here, η0 is the viscosity of a nonstructured system, and η∞ is the viscosity of a system with a close packing of particles. A comparison of the results found by Eq. (9) and the experimental data [2] at ϕ = 0.45 and m = 1.4 is performed in Fig. 2. A system is characterized by a close packing of particles at high Peclet numbers, which correspond to complete structuration as follows from Eq. (6). The region of the decrease in viscosity in the case of shear
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flow is near the Peclet number Pe = 1, when hydrodynamic and Brownian forces are equal to each other and Pe > 1, it is possible that η → η∞ at high γ. VISCOSITY OF DISPERSE SYSTEMS Viscosity is one of the principal characteristics of rheological liquids as it determines their fluidity and depends on the shear stress η = τ γ and the concentration, sizes, and shape of particles in disperse systems. The simplest relationship between the viscosity and the content of particles in the case of a nonstructured laminar flow is the Einstein equation η ηc = 1 + 2.5ϕ , which is suitable for infinitely diluted media ( ϕ ≤ 0.01) with finely disperse spherical solid particles. A variety of empirical dependences of the effective viscosity of disperse systems on the content of particles in a flow can be encountered in the literature [1–3], and their citation could compose the entire bibliography. Many of these dependences take into account the concentration of particles for their close packing. To calculate the viscosity of suspensions, the authors [11] propose the Barnea–Mizrahi equation, which rather perfectly describes experimental data within a broad range of ϕ as follows:
⎡ kϕ ⎤ η (10) = exp ⎢ 1 ⎥ . ηс ⎣1 − k2ϕ⎦ Most of these formulas do not take into account structuration, which significantly deforms the viscosity curve. Based on an analysis of experimental studies, it follows that the viscosity in the structuration beginning region obeys the semiempirical equation
dη = −k η ( ϕ − ϕs ) , dϕ ϕ → ϕs , η = η0 − ηs .
(11)
Here, η0 is the viscosity of a nonstructured system, ηs is the maximum deviation of the system viscosity in the structural beginning region, and k is the coefficient that determines the existence of region AC (Fig. 1). The solution of this equation is
lnη = − k ( ϕ − ϕs ) + C , 2 where C is the integration constant determined from the initial conditions in the form C = ln ( η0 − ηs ) . The final solution of this equation is 2
η η0 − ηs 2 (12) = exp ⎡− k ( ϕ − ϕs ) ⎤ . ⎢⎣ 2 ⎥⎦ ηc ηc This expression determines the character of the viscosity–volumetric concentration curve in the region of its deformation and further structuration. The region of the beginning of structuration is characterized by thixotropic properties, as aggregates are
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η/ηc 25 20 15 10 5 0.001
0.01
0.1
1
10
100 Pe
Fig. 2. Viscosity of a disperse medium of polymethylmethacrylate particles of different size of 85 (dark points) and 141 μm (light points) versus Peclet number.
η, g/(cm s) 200
150
III
II
I 100 2 50 1
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35 ϕ
Fig. 3. Viscosity of a bituminous blend versus polyethylene content [12].
destroyed if the concentration of particles does not grow. The dependence of the viscosity of bituminous blends on their content of pure or waste polyethylene is plotted in Fig. 3 [12]. The presence of a certain concentration of polyethylene in bituminous blends improves their physical properties, makes them crackproof, and imparts thermal stability. The dashed lines in Fig. 3 correspond to viscosity values without structuration. In these curves, region II is assigned to the beginning of the structuration of polyethylene particles in the volume. The dependence of viscosity on the volumetric polyethylene content in bituminous blends at two temperatures is described by the equations
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( ) ( ) 2. T = 215°C, η = 440ϕ exp ( 0.044ϕ ( ϕ − 0.5) ) − A exp ( −640 ( ϕ − 0.2 ) ) , A = 16.
1. T = 220°C, η = 350ϕ exp 0.015ϕ ( ϕ − 0.5) − A exp −850 ( ϕ − 0.2 ) , A = 18,
The dependence of viscosity on the content of water drops in an oil emulsion is plotted in Fig. 4 [13] and described by the equation
⎡ 2.8ϕ ⎤ η = ( −11.6 + 0.6ϕ) exp ⎢ 2⎥ ⎣( ϕ − 100) ⎦ 2 − 7 exp ⎡⎣−0.01 ( ϕ − 60) ⎤⎦ . This formula can be used to calculate the relative viscosity of a disperse medium in the form of η ηc , where the dynamic viscosity of oil from the Muradkhanli field (Azerbaijan) is ηc ≈ 14 Pa s. The structuration and intensive viscosity growth regions can clearly be seen in these curves. On the whole, the generalized dependence of the viscosity on the volumetric content of particles can be written as
⎛ mϕ ⎞ η ηc = 1 + 2.5ϕ + A0ϕ exp ⎜⎜ 2⎟ ⎟ ⎝ ( ϕ − ϕ∞ ) ⎠ 2 − A1 exp k ( ϕ − ϕs ) .
(
(13)
)
The last term in this equation characterizes the beginning of structuration in a disperse system, i.e., the deformation of the viscosity curve in the structuration region. If A1 = 0 , region АС (Fig. 1) is absent as typical for disperse systems with a solid phase. The parameter ϕ∞ determines the condition for the formation of a close packing of particles, and coefficient A1 depends on the difference between the dynamic viscosities with and without structuration, i.e., A1 = ( η0 − ηs ) ηс . Parameter ϕs corresponds to the η, Pa s 80 70 60 50
III
II
I
40 30
η0 – ηs
20 10 0 20
30
40
50
60
70
80
90
Fig. 4. Oil viscosity versus water content [13].
100 ϕ, %
2
2
2
2
concentration of particles at the beginning of structuration. Because the region of the beginning of structuration is very small in some cases, it may be invisible in the viscosity curve. In this case, the last term in Eq. (6) may be omitted to simplify the viscosity equation. The viscosity of a disperse system is very sensitive to the sizes and shape of particles. Using Eqs. (1) and (13), it is possible to draw the important conclusion about that the effective viscosity of a disperse system in the case of slow coagulation between particles depends on the coagulation time. SHEAR VISCOSITY OF A LAYER OF SOLID PARTICLES The structuration of solid particles in a fixed layer and a shear flow is absolutely different in the degree of particle freedom in a flow. It is important to note that the shear flow (rearrangement) in a fixed layer is characterized by the shear or bulk viscosity, shear stress, and the change in the layer porosity in both radial and longitudinal directions. In chemical technology, fixed layers are used in catalytic fixed-bed reactors, adsorbers, packed columns, etc. The effective viscosity of a disperse system with a volumetric content of solid spherical particles ϕ without structuration ( A1 = 0) in flow can be determined by a formula similar to Eq. (13) [5, 8], i.e.,
⎛ mϕ ⎞ η = 1 + 2.5ϕ + 3 ϕ exp ⎜⎜ 2⎟ ⎟, ηc 4 ⎝ ( ϕ − ϕ∞ ) ⎠
(14)
where m = 2.2 + 0.033a, a is the size of particles, μm, and ϕ∞ = 0.74 is the concentration of particles in a maximally close packing. As follows from the experimental data and this formula, the effective viscosity of a disperse system strongly depends on the volumetric content and size of particles. Moreover, the effective viscosity grows with increasing particle size. In all appearances, a simple dense packing of particles is formed in this case instead of coagulation structures and aggregates. The experimental and calculated relative viscosities as functions of the volumetric content and size of particles in a suspension are compared in Fig. 5 [7]. The picture of structuration in a fixed layer of particles is slightly different, as the rearrangement of particles under external deforming stresses creates the conditions for the formation of denser packing, which has an effect on the porosity distribution and the diffusion coefficient. A fixed layer can be considered as a chain of elastically bonded particles spaced by a liquid (Fig. 6).
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If all sizes of spherical particles are assumed to be equal, their immediate contact may result in the formation of ordered isotropic structures. Let us select a chain of elastically bonded particles affected by an external force (pressure, upper layer, etc.) in a layer (Fig. 6). Every stable equilibrium corresponds to this type of system position, in which the potential energy E ( ξ) is minimal [14]. The deviation from this state under the influence of an external force with the resulting rearrangement of particles in the layer leads to the appearance of the force ( − ∂E ∂ξ) striving to return the system back to its equilibrium state. At small deviations from equilibrium, it is sufficient to confine the expansion of the difference E ( ξ) − E ( ξ0 ) in powers ( ξ − ξ0 ) to its second-order term taking into account that ∂E ∂ξ = 0 for an equilibrium state, i.e., 2 E ( ξ ) − E ( ξ0 ) ≈ k ( ξ − ξ0 ) , 2
(15)
where k = ∂ 2E ∂ ( ξ − ξ0 ) ξ=ξ is a positive coefficient. 0 Therefore, the force acting on the nth particle from the (n – 1)th particle can be determined from Eq. (15) as
Fn,n−1 = −k ( ξn − ξn−1 ) ,
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η/ηc 80 60
1 2 3 4
40 20
0
0.1
0.2
0.3
0.5 ϕ
Fig. 5. Viscosity of a disperse system versus volumetric content of solid spherical particles with different sizes of (1) 0.1, (2) 0.5, (3) 1, and (4) 1.5 μm.
V
(16)
where ξn is the deviation of the nth particle from its equilibrium state. The total force that acts on the nth particle from the (n – 1)th and (n + 1)th particles is equal to Fn,n−1 − Fn,n+1 . The friction force that acts on the particles in the layer is
FnT = −β ( ∂ξ ∂t ) ,
0.4
τc
(17)
where β is the coefficient of friction. Then, the displacement of a particle as a result of rearrangement in a layer under deforming shear stress is described as
d ξn dξ (18) + λ + κ0 ( −ξn−1 + 2ξn − ξn+1 ) = 0, 2 dt dt where λ = β m , κ0 = k m, and m is the mass of a spherical particle. The solution of Eq. (18) is 2
ξn = A exp ( − λ t 2) cos ( ωt − ϕn)
(19)
with an angular frequency ω = − 1 2 β m + 2 k m sin ( ϕ 2) . The coefficient κ is determined depending on the rate of deformation in the layer under particle shear stress as κ ~ τ ηs , and the phase shift between the vibrations of neighboring particles is proportional to the wavenumber k = 2π λ or ϕ = kx = 2πx λ , where x is the distance counted from the surface of a particle. The model
(
) ( ) ( )
y y cos 2π , a a y A = 0.22 + 0.38 exp −5 , a
ε − ε∞ = A exp −0.5
(20)
ξ
Fig. 6. Arrangement of a dense bed of solid particles in an apparatus.
constructed for the porosity in the radial direction of a layer by analogy with Eq. (19) is compared with the experimental data [2, 15] in Fig. 7. Curve 1 corresponds to the average layer porosity change and is described by a simple expression like ε = ε∞ + (1 − ε∞ ) exp ( −bt ) = 0.4 +0.6 exp ( −4.5y a)) . Substituting the variables dy = Vdy and ξn ≈ αεn into Eq. (18), we rearrange it as
d εn dε + λ V n + k0V 2 ( −εn−1 + 2εn − εn+1 ) = 0. 2 dt dt 2
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ε 1.0
0.8
0.6 1 0.4
0.2
0
1
2
3
4
5
6 y/a
Fig. 7. Change in the porosity of a layer in the radial direction.
Taking into account that λ V ~ γ and k0V 2 ~ γ 2 , we find
d 2ε n dε + γ n + γ 2 ( −εn−1 + 2εn − εn+1 ) = 0. 2 dt dt
(21)
Omitting the second derivative in Eq. (21) with consideration for γ = τs ηs , −εn−1 + 2εn – εn+1 = (εn − εn−1 ) – (εn+1 − εn ) ≈ ε − ε∞ , we recast this expression in the form
τ dε =− s. ηs (ε − ε∞ ) dt
(22)
Equation (22) is similar to the equation of packing in a layer of disperse particles [14] and depends on the shear direction and the viscosity. The particular solution of this equation at t = 0, ε = 1; t → ∞, ε → ε∞ is the expression for the average layer porosity
ε = ε∞ + (1 − ε∞ ) exp ( − τst ηs ) ,
(23)
which has the form of 1 in Fig. 7. The shear stress for a bed of particles in an apparatus is formed of two components, which characterize the pressure drop along the bed height and the weight of the upper layers of particles, i.e., τs = −ΔP + Δρz , where ΔP is the pressure drop along the bed height z, and Δρ is the difference between the densities of particles and a carrier
1
medium. Using Eq. (23), let us solve the inverse problem of estimating the shear viscosity in the form τst . ⎛ ε − ε∞ ⎞ ln ⎜ ⎟ ⎝ 1 − ε∞ ⎠ In turn, the bulk viscosity of a structured layer of particles is determined as [16] ηs = −
ηυ = 4 ηs 1 − ε . 3 ε The shear viscosity of a layer under deformation at ΔP = 1.0 Pa, t = 31.5 × 106 s (for a year) is nearly ηs ≈ 28.6 × 106 Pa s . The shear viscosity for the packing of mine rocks achieves a quite high value of nearly ~1023 Pa s [17].
CONCLUSIONS Highly concentrated disperse systems are classified as structured with typical rheological properties. The existence of various rheological models does not provide the possibility to select an unambiguous practical description of disperse systems, though this selection can be performed via the appropriate approximation of experimental data. It should be noted that the presented formulas of viscosity are not phenomenological expressions, but rather represent semiempirical equations describing experimental data. At the same time, they perfectly reflect the character of transformation in the viscosity curve upon structuration in a disperse
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medium, especially in the region of the beginning of structuration. NOTATION а size of particles, μm C1, C2, C3, C4 experimentally determined coefficients D F
k1, k2
molecular diffusion coefficient, m2/s force, N experimentally determined coefficients
kB M N
Boltzmann constant
N0
initial number of particles in the volume, m–3
N∞ r T t V
final number of particles in the volume, m–3 radius of a particle, μm temperature, °C time, s shear flow velocity, cm/s dimensionless coefficients
mass of a spherical particle, g number of particles in the volume, m–3
α1, α2 γ ε
ηc
medium deformation rate, s–1 porosity of a medium porosity of the maximally packed structure of a layer specific energy dissipation per unit volume, m2/s3 effective viscosity of a system, Pa s viscosity of a medium, Pa s
ηs
shear viscosity, Pa s
ηυ
bulk viscosity of a medium, Pa s
η∞
system viscosity corresponding to a close packing, Pa s degree of the entrainment of particles with a pulsing medium
ε∞ εR η
μp 2
νc ξ
ξ0 ρm τr τs ϕ ϕ0 ϕ∞ ϕs ω
kinematic viscosity, m2/s displacement of particles in a layer initial displacement of particles in a layer density of a medium, kg/m3 relaxation time, s shear stress, Pa volumetric concentration of particles initial volumetric concentration of particles volumetric concentration of particles for a close packing concentration of particles at the beginning of structuration
411
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frequency of collisions between particles, s–1 THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING
Translated by E. Glushachenkova Vol. 52
No. 3
2018
