On the Effect of Flowing Circular Entities Swarms ... - ACS Publications

The use of finite size conductance electrode strips for the identification of dispersed phase flows may lead to difficulties related to the failure of...
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On the Effect of Flowing Circular Entities Swarms on Strip Electrodes Conductance M. Kostoglou* Department of Chemical Technology, School of Chemistry, Aristotle University, Univ. Box 116, 541 24 Thessaloniki, Greece ABSTRACT: The use of finite size conductance electrode strips for the identification of dispersed phase flows may lead to difficulties related to the failure of homogeneous theories for the effective conductivity of dispersions in the particular geometry. In this context, a model problem is presented here: circular nonconducting entities dispersed in a conducting liquid flow at constant velocity over a nonconducting wall with strip electrodes flush mounted on it. The mathematical model is formulated in detail and is decomposed to electrostatic and to swarm generation subproblems. The electrostatic problem is simplified for dilute swarm of small, compared to the electrode width, entities for which an approximate analytical closed form solution is derived. A stochastic algorithm is developed for the generation of entity swarms. It is shown that a swarm generates fluctuations to the effective conductance measured by the electrodes. The strength of the fluctuations is related to the variations of the local electric current magnitude of the electrical field undisturbed by the entities at a distance equal to the average distance between entities. The present work is a first step to understand the interaction between entity swarms flowing over strip electrodes in order to explore the possibility of exploiting the fluctuating conductance signal for the characterization of the swarm further to a simple volume fraction determination. tion of what is the relation between these fluctuations and the features of the entities and if this relation can be exploited in order to get an estimation of these features by analyzing the measured conductance fluctuations. It is noted that in practice an alternating current is used but its frequency is high enough (usually above 10 kHz)14 that no capacitance effects are present so the electrical field problem is similar to that of direct current.1 In addition, the observed signal fluctuations have much lower frequency compared to the alternating current frequency so there is no interference between the two phenomena. In this general context, a fundamental stochastic physics problem is set up regarding the interaction between a set of strip electrodes and a swarm of circular nonconducting entities flowing with constant velocity over the plane of the electrodes. The structure of the present work is the following: At first, the complete mathematical problem is formulated. Next, the simplification of the electrostatic interaction problem for the case of dilute dispersions is described in detail. Then, approximating closed form solutions for the electrostatic problem are derived. In the subsequent section, stochastic algorithms for the generation of entity swarms are developed. Finally, several indicative results are presented and discussed concerning the electrostatic interaction function or effective conductance time traces.

1. INTRODUCTION The interaction between electric fields and electrical conductivity spatial profiles is extensively used in practice as a nonintrusive technique for the estimation of phase distribution in a domain.1 There are several tomographic techniques of this kind based on different types of the imposed field and different measured quantities. In all cases, the final step of the calculations includes the transformation of the estimated local effective conductivity to a phase volume fraction. This can be achieved through different expressions available in the extensive literature on the effective conductivity of porous medium in a homogeneous external field. This theoretical topic was initiated long ago by Rayleigh2 and Maxwell3 but is evergreen, suggesting either empirical correlations for unstructured media4 or formal mathematical results for structured media of several types.5 A particular phase identification technique is based on strip metallic electrodes flush mounted on the wall of a pipe6,7 Although this technique was originally developed for separated flows of two continuous phases (a continuous conducting liquid and a continuous nonconducting gas), like annular or stratified,8,9 recently, it has been employed also for the characterization of nonconducting dispersed flows in continuous conducting liquids10,11 (e,g. liquid−liquid dispersions or bubbly flows). When dealing with dilute dispersions, the distance between the dispersed phase entities may be comparable to the width of the strip electrodes. In this case, the electric field cannot be considered homogeneous and the existing correlations do not hold. For such occasions, the effective conductance does not depend only on an integral descriptor, such as the dispersed phase volume fraction, but also on the exact location of the entities with respect to the electrode. This may be the reason of the conductance fluctuations observed experimentally12,13 addressing the ques© 2012 American Chemical Society

2. PROBLEM FORMULATION A schematic of the problem at hand is shown in Figure 1. The problem is defined in two dimensions implying that the third Received: Revised: Accepted: Published: 5615

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V=1

for 1 ≤ x ≤ 1 + λ

V = −1

−(1 + λ) ≤ x ≤ −1

(3) (4)

The boundary conditions at the surface of the particles are ∂V =0 ∂n ⃗

on

Si , ∀ i

(5)

where n⃗ denotes the normal to the surface direction. The far field conditions are V=0

Figure 1. Schematic of the problem under consideration.

G=

1 2

−1

⎛

⎞

∫−(1+λ) ⎜⎝ ∂∂Vy ⎟⎠

dx y=0

(7)

The dimensional conductance γ (which is the actually measured quantity) is related to G through G = γ/(δσ) where σ is the specific electrical conductivity of the continuous liquid phase. There are many works in the literature considering the numerical solution of the Laplace equation in a domain with multiple disk-shaped (2-dimensional) or spherical (3-dimensional) nonconducting particles relevant to the electrical or thermal conductivity (governed by the same equations). The linearity of the governing equations permits the use of methods with analytical support (i.e., multipole expansions or singularity/Green functions method)15 or methods that reduce the dimensionality of the problem (boundary elements).16 Even these methods are time-consuming in case of a large number of particles due to the large number of unknowns (degrees of freedom required for fine discretization). It is noted that in the present application a few realizations of the system are not enough for the study of the problem as is the case for a truly steady problem. A large number of consequent configurations must be studied as the particles move with respect to the electrodes. A direct numerical attack to the problem is intractable, but fortunately, a great simplification can be made in the limit of dilute suspensions, i.e., small volume fraction of particles. The effect of particles on the conductance can be expanded to a power series in which the linear term expresses the influence of individual particles to the conductance and the quadratic term describes the interaction between pairs of particles.17 This means that the linear term can be constructed by decomposing the problem to a series of problems containing just one particle each. The linearity of the governing equations permits superposition of the fields. The number of particles in the semi-infinite domain is infinite but a truncation at number N can be made considering that particles very far from the electrodes do not influence the conductance. The potential field V is decomposed to N + 1 independent fields according to the relation V = V(0) + Σi N= 1(V(i) − V(0)), i.e., one field V(i) for each particle and the field V(0) in the absence of particles (undisturbed field). These fields obey the same governing equation and wall boundary conditions with the original field but the boundary condition (eq 5) holds only for the surface Si. For each field, the conductance G(i) can be computed from eq 7 and the total conductance can be found as G = G(0) + Σi =N 1(G(i) − G(0)) where G(0) is the conductance in the absence of particles. In this way, the stochastic and the deterministic parts of the problem can be separated. The contribution of each particle to the conductance depends only on its radius and on its position xo,yo. So, the knowledge of the

(1)

The voltage values on the electrodes can be arbitrary, provided that their difference is equal to 2. The pair −1 and 1 (for the left and right electrode, respectively) is used to achieve antisymmetry which greatly simplifies the problem). The boundary conditions at the electrode plane are ∂V = 0 for x < −(1 + λ), −1 < x < 1, x > 1 + λ ∂y

(6)

The instantaneous normalized conductance G is given by the relation

dimension is extended to infinity. The domain of the problem is a semi-infinite plane described by the Cartesian coordinates: x (parallel to the wall) and y (normal to the wall). The two strip electrodes are mounted on the wall (assumed to be a nonconducting plane). A voltage difference of 2Vo is applied to the electrodes, and the resulting electric current is measured. The ratio of the conductance of the conducting liquid/ nonconducting entities mixture to the conductance of the liquid alone is related to the volume fraction of the nonconducting entities. The existing literature on the subject is based on the homogeneity of the macroscopic electrical field; i.e., the discreteness of the entities can be ignored at the scale of the macroscopic field. In this case, the conductance depends only on the volume fraction and not on the exact position of the entities. However, in the present problem, the situation is different. The microscale of the problem is defined by the distance between two entities (which will be called particles in the following). This distance is comparable to the width of the electrodes which constitutes the smallest scale of the macroscopic field. The consequence is that the exact location of the particles influences the measured conductance. Thus, the constant velocity flow of the particles yields a propagating sequence of locations (shift) of the swarm in the direction of the flow which creates a transient effective conductance signal. The electron flow is much faster than the fluid and particles flow so the electric field can be considered to be described by a series of pseudosteady states which differs in geometry due to the propagation of the swarm location. The steady-state mathematical problem for a particular configuration which is described by the location of the surfaces of the particles Si (the index i refers to all the particles in the system) is presented below. It is noted that all length variables are normalized to the half of the distance between the closest edges of the electrodes (denoted by δ). The normalized electrode width is denoted as λ (the dimensional width is λδ as shown in Figure 1). The coordinate origin is defined at the middle point between the electrodes. The electrical potential is normalized by Vo. 2.1. The Electrostatic Problem. The governing equation for the potential field V is the Laplace equation ∂ 2V ∂ 2V + =0 ∂x 2 ∂y 2

far from the electrode (i.e., x , y → ∞)

(2) 5616

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function g(xo,yo,R) that corresponds to the quantity G/G(0) − 1 for a particle of radius R with its center at xo,yo is enough to allow direct construction of G for any configuration of N particles as:

The characteristic parameter that defines the accuracy of this approximation is the fractional variation of the undisturbed field gradient across the particle. This assumption does not introduce appreciable error since the governing parameter is very small at any practical situation of interest (i.e., particle diameter at least ten times smaller than the electrode width). In this way, the boundary condition on the particle surface is transformed to the following:

N

G = G(0)(1 +

∑ g(xoi , yoi , R i)) i=1

(8)

There are still computational difficulties at the single particle problem. The sensitivity of the conductance to the presence of a single particle and to its position is very small (it is the collective effect of the swarm that makes it appreciable). On the other hand, the numerical accuracy is inevitably small due to the rediscretization needed for each particle position and to the singularities that arise at the edges of the electrodes. It is wellknown that singularities appear at the lines separating regions with Dirichlet and Neumann boundary condition. The accuracy of any numerical technique is comparable to the required sensitivity of the solution so this problem cannot be solved directly. The way to proceed is to decompose the problem into two distinct parts: one for the electrical field undisturbed by the particle, containing the singularity which, however, does not need high sensitivity and another for the disturbance field from which the singularity has been removed and for which the accurate computation of the effect of particle on conductance is possible. Specifically, the disturbance field v for the single particle problem is introduced as v = V(i) − V(0) (the index i is omitted from field v for clarity of presentation). This field obeys the same equation and boundary conditions with V except that (∂v)/(∂n⃗) = −(∂V(0))/(∂n⃗) on Si and v = 0 on the electrodes. It is noted that so far only formal development of the mathematical problem has been made without any nonformal approximation. The focus is given at the first order expansion term of the effective conductance with respect to the particle volume fraction, and the resulting stiff single particle problem is replaced by one that can be easily treated with conventional numerical methods. In fact, the validity of the decomposition approach as a function of the dispersed phase volume fraction φ can be appraised by comparing the Bruggeman18 relation for the effective conductivity of the continuum phase conductivity ratio (i.e., (1 − φ)1.5) to a first order expansion based on individual particles effect5 (i.e., (1 − 1.5φ)). Although the above relations refer to 3-dimensional spherical particles, they can serve for the assessment of field decomposition for other geometries, too. A simple calculation shows that the field decomposition approach can be considered very accurate up to φ = 5% and less accurate (but acceptable) for φ between 5% and 10%. 2.2. Approximate Solution of the Electrostatic Problem. At this point, several approximations will be introduced to the exact single particle problem in order to make a first computational study of the process. The first approximation is that at the size scale of the particle the undisturbed field can be approximated by the first two terms of its Taylor expansion around the center of the particle. This means that in the particle region:

⎛ ∂V (0) ⎞ ∂v ⎟⎟ = −⎜⎜ ∂x ⎝ ∂x ⎠x , y

o o

(10)

The function g can be found in terms of the field v as: g=

−1

⎛

⎞

∂v ⎜ ⎟ (0) ∫−(1 +λ) ∂y ⎝ ⎠ 2G 1

dx (11)

y=0

The second approximation which is the most important is based on two assumptions: First, the unperturbed field can be considered as linear in the scale of the perturbations created by the particles. The perturbation field for the specific problem is reduced inversely proportional to the distance from the center of the particle so the characteristic scale is a few times (less than ten) of the particle diameter (see Appendix). The condition for the validity of the assumption is similar to the condition of validity of eqs 9 and 10 but stricter since the characteristic size on which the linearity of the unperturbed field must hold is about an order of magnitude larger. Second, the existence of walls has no effect on the perturbed field. The criterion for the validity of this assumption is the smallness of the ratio of particle radius to particle distance from the wall. The above assumptions are exact for point particles, and their accuracy decreases as particle size increases. According to these assumptions, the boundary conditions on v are replaced by a far field zero gradient condition. Despite ignoring the wall conditions, the existence of electrodes still dominates the problem through the field V(0). The problem is modified employing a rotation of the coordinate system axes in order to align y axis with the gradient of the field V(0) at xo,yo (let us call the new direction Y) and a transformation of the field v such that its gradient on the particle to be zero and the far field gradient to be equal to ⎡⎛ ∂V (0) ⎞2 ⎛ ∂V (0) ⎞2 ⎤1/2 ⎟⎟ ⎟⎟ ⎥ + ⎜⎜ D = ⎢⎜⎜ ⎣⎝ ∂x ⎠x , y ⎝ ∂y ⎠x , y ⎦ o o

o o

i.e., the new field is defined as: w = DX + v. The resulting unit problem for the single particle influence on the linear field has been addressed in several ways in the literature during the last 150 years and by several scientific disciplines, especially for spherical particles. A more formal treatment for the case of cubic lattices of spherical particles has been performed by Rayleigh.2 A general account can be found19 in the context of perturbation techniques (in particular, the socalled homogenization method). In this context, the linear part of the effective conductivity-volume fraction function is called the Clausius-Mossotti equation. The same method for general periodic structures is explained by Adler.5 The isolated particle problem was solved for spherical particles by Jeffrey20 using multipole expansion of the field and by Maxwell3 using Green functions. The application of the first approach to the case of

o o

o o

o o

on particle surface S

⎛ ∂V (0) ⎞ ⎟⎟ (x − xo) V (0) = V (0)(xo , yo ) + ⎜⎜ ⎝ ∂x ⎠x , y ⎛ ∂V (0) ⎞ ⎟⎟ (y − yo ) + ⎜⎜ ⎝ ∂y ⎠x , y

⎛ ∂V (0) ⎞ ∂v ⎟⎟ = −⎜⎜ ∂y ⎝ ∂y ⎠x , y

(9) 5617

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cylinders is shown by Deen21 in the context of effective thermal conductivity of fibrous media. In all the above works, the particles are electrically conducting. The case of nonconducting particles treated here simplify the analysis which is straightforward (based on the approach of Jeffrey20), leading to the result that the flux excess E due to the existence of the particle is proportional to its area (see the Appendix for the derivation), i.e., E = −2πR2σD. The next step is to go back from the isolated particle approximation to the complete problem. The physical requirement for an infinite number of infinitesimally small particles (homogeneous dispersion) is that the effective conductance must be equal to the effective conductivity, i.e., G/G(0)=1 − 2φ. The attempt to match this general requirement with the single particle flux excess, found above, leads to the following equation: G (0)

G

∞

= 1 − 2 ∑ Ai i=1

In order to complete the approximating treatment, the undisturbed electric field is needed. There are several techniques for the solution of this kind of problem based on numerical (finite element, finite differences) or semianalytical techniques (dual and triple series, integral equations).22,23 The complexity of the problem that precludes an analytical solution is the discontinuous boundary condition on the wall. It was proposed6 that, when the width of the electrodes is much smaller than the distance between the electrodes, the current density profile on the electrode can be approximated by a uniform one. In this way, the boundary conditions on the wall are everywhere of Neumann type and the problem admits an analytical solution. The solution domain is truncated in the x direction, and a zero current condition is assumed on the truncation planes. The zero current condition is a weak one and does not alter the real (extended to infinity) field as it would do the strong zero potential condition. The uniform current and uniform voltage electrode boundary conditions lead, in general, to different absolute values of conductance, but it is expected that the predicted sensitivities of the conductance to the existence of the particles are the same in the two cases. The domain is truncated symmetrically at x = C and x = −C. The no flux boundary conditions on this plane and the V(0) = 0 condition at y→∞ are fulfilled by the following solution of the Laplace equation:

D(xoi , yoi ) ∞

∞

∫0 ∫−∞ D(x , y)dydx

(12)

πRi2).

where Ai is the area of the particle i (equal to From the physical point of view, it is apparent that the effect of a particle in the conductance reduction increases as the current density of the undisturbed field in the particle location becomes larger. Comparing eqs 12 and 8 yields that the approximating result for the function g(x,y,R) is D(x , y)

2

g (x , y, R ) = −2πR

∞

∞

∫0 ∫−∞ D(x , y)dydx

∞

V (0) =

2

= −2πR f (x , y)

∑ α i sin(κ ix)e−κ y i

κi =

i=0

(2i + 1)π 2C

(14)

The unknown coefficients αi can be found by replacing the series in the boundary condition at y = 0 (q is a normalized current density):

(13)

Thus, the approximating solution procedure has led to a quadratic particle size dependence of the function g which can be eliminated by introducing a new function f independent from the particle size. The above equations refer to the semiinfinite domain, but any practical use requires a domain truncation to distances where the influence of particles is small. Equation 12 leads to the expected results in several asymptotic limits. For example, considering the usual case of a uniform electric field, eq 12 takes the well-known form G/G(0) = 1−2φ where φ is the area fraction of the particles. In another limit, the field is nonhomogeneous (like the one considered here) but the particles are all of the same size and appear in random locations and the distances between them are much smaller than the electrode width (homogeneous dispersion). In this case, the discreteness effect is absent and the expression G/G(0) = 1 − 2φ is still expected to hold. It can be shown that the sum in eq 12 corresponds to the Monte Carlo computation of an integral which in its continuous form (infinite number of particles) leads to the expected result G/G(0) = 1 − 2φ. So, eq 12 is compatible with known asymptotic results of the problem. In particular, it gives the correct values for a uniform electric field with any kind of dispersion and for a homogeneous dispersion in an arbitrary electric field.

∂V (0) =0 ∂y −C < x < −(1 + λ), −1 < x < 1, 1 + λ < x < C (15a)

−σ

−σ

∂V (0) =q ∂y

∂V (0) = −q ∂y

1≤x≤1+λ

−(1 + λ) ≤ x ≤ −1

(15b)

(15c)

and exploiting the orthogonality properties of the sine function results in αi =

2q C σκ i 2

[cos(κ i(1 + λ)) − cos(κ i)]

(16)

Replacing the equation of the undisturbed field in the definition of f(x,y), the parameters σ, q, and C are eliminated between the numerator and denominator and the outcome is

f (x , y ) ∞

=

∞

[(∑i = 0 κ i−1[cos(κ i(1 + λ)) − cos(κ i)]cos(κ ix)e−κ iy )2 +(∑i = 0 κi−1[cos(κ i(1 + λ)) − cos(κ i)]sin(κ ix)e−κ iy )2 ]1/2 Y

C

∞ κ −1[cos(κ i(1 + λ)) − cos(κ i)]cos(κ ix)e−κ iy )2 +(∑i = 0 κi−1[cos(κ i(1 + λ)) − cos(κ i)]sin(κ ix)e−κ iy )2 ]1/2 dydx ∫0 ∫−C [(∑∞ i=0 i

(17)

The domain has also been truncated at y = Y and at x = −C 5618

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input parameter equal to the average value of δt which determines the particle area fraction. Another time scale important for the system is the minimum time step at which a snapshot of the system must be captured in order to perform a conductance calculation. The computed conductance signal must be continuous so the time before successive snapshots must be small. The criterion for the selection of this maximum permitted time step tm is that the movement of a particle during this time must correspond to a small fraction, e.g., 0.1, of the electrode width. According to the above, the particle generation and motion procedure is the following: All the particles in the observation domain are kept in a list. The time δt required for the appearance of a new particle is stochastically computed as it has been shown. Then, n time steps δτ are performed where n = [δt/tm] + 1 (the brackets denote integer part) and δτ = δt/n. At each time step, the x-coordinate of all particles is updated by adding the value δτ. When the n-th time step is completed, a new particle with coordinates x = 0 and y = L*RND (i.e., randomly selected y location) is added to the list. The list is inspected and particles with x > 1 are removed. Then, the procedure continues by choosing randomly the next value of δt. The above procedure does not prevent overlapping between particles as RSA does. To inhibit overlapping, the following procedure is needed: every time a new particle is added to the list, the distance between this and the previous K particles that entered the list (K is an input parameter of the algorithm) must be computed and compared to the sum of the corresponding particle radii. If it is smaller, then the selected δt value must be canceled and a new value is chosen. In practice (for computational efficiency), the overlapping test and the new particle rejection/adoption procedure precedes, and the division of the time step to substeps for field computation follows. The required value of K for elimination of overlapping depends on the number and size of the particles, but in general, the larger the K, the smaller is the possibility for overlapping.

and x = C just for practical computation purposes. That means that a correct choice of C and Y renders f independent from their actual values. The only fundamental parameter influencing the function f is the electrodes geometry as it is expressed through the parameter λ. 2.3. Creation of Particles Spatial Distribution and Motion. In the previous section, it was shown how to estimate the effect of a spatial pattern of particles on the conductance recorded by the electrodes. In this section, it will be shown how a random pattern of particles can be created and move with constant velocity over the electrodes. An orthogonal observation domain is considered with a Cartesian coordinate system (x-along the wall, y-normal to wall) and the origin of axes at the bottom left corner. The length of the domain is normalized to unity and the width is denoted as L. The coordination system is different from the one used for the electrical field problem for convenience. It is advantageous to handle each subproblem using the appropriate coordinate system and to switch from one system to the other during the merging of the two subproblems. A realization of the simulation process consists of passing a random spatial particle distribution through the observation domain with velocity u (normalized by the length of the domain) for a time period To. The residence time of a particle in the observation domain is 1/u and the simulation period contains T = Tou particle residence times. The direct way to handle the problem is to create a particle pattern on a strip of length Tou and width L using conventional methods. The most straightforward way for this is the use of the random sequential adsorption procedure (RSA). The particles are thrown one by one on the strip at random locations (overlapping is not allowed) up to a required particle area fraction.24,25 Using this approach, the strip with the particles must be moved over the electrodes with velocity u. This approach is straightforward, but it has the disadvantage of needing handling of a number of particles much larger than the number of particles in the observation domain. The alternative way is to consider only the particles located in the observation domain (lets imagine that the observation domain is moving along the immobilized strip with particles with velocity −u). The challenge here is to account for the appearance of particles in the domain without knowing their position on the strip. The particles in the observation domain can be described by their position. In particular, the x coordinate for each particle evolves in time whereas its y coordinate remains fixed. The problem can be described in terms of a discrete population balance26 (in particular crystallization) where the internal particle variable is its x position value instead of the particle size as is the case in crystallization. The population balance includes nucleation, growth (with respect to x), and removal (for x > 1) terms. A particular Monte Carlo approach for the solution of such population balances employs the general approach of the quiescence interval between events (see refs 26−28 for Monte Carlo solution of population balances). This approach is based on the fundamental statistical law that the time intervals between uniformly random occurring processes follow exponential distribution.29 The only stochastic phenomenon here is the particle appearance at the plane x = 0 (nucleation). The time variable t is normalized using the residence time 1/u. The time interval δt between successive particle appearance follows an exponential distribution and can be computed from the relation δt = −1/p*ln(RND) where RND denotes a uniformly deviate random number between 0 and 1 and p is an

3. INDICATIVE RESULTS AND DISCUSSION Before presenting results, a discussion is made on the validity of the series of the approximations that led to an explicit solution of the conductance problem. It is stressed here that the final results are valid at a specific limit: point particles and dilute dispersion. The characterization of the dispersion as dilute is based on the area fraction of the dispersed phase and of the particles as point on the ratio between particle size and current variation in the domain (or to make it more quantitative on the ratio of particle size to the electrode width R/λ). In all cases presented here (and for most practical cases), this ratio is very small, so the proposed approach leads to accepted results. A descriptor of the accuracy of the approach for finite size particles is the ratio of the particle size to its distance from the wall. There may be an error in the conductance for particles at distances a few times R from the wall, but they are very rare and their total contribution to the signal is restricted for the purposes of the present work. The local single particle flux excess is an exact result for point particles. The transformation of the local excess to conductance excess by division with the total gradient is not a formal result, but it is just an ansantz designed in order for the final expression to fulfill the expected results (for effective conductance in uniform electrical field and effective conductance of a homogeneous dispersion in a nonuniform field). 5619

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calculation of the properties of the swarm must be done for the period 1 < t < T. In case of infinitely small particles, the possibilities of overlapping are negligible and even using K = 0 the number of particles N in the domain is proportional to the particle area fraction. The value of N (and correspondingly the particle area fraction) undergoes fluctuations in time due to the stochastic nature of the process. Typical evolution of (normalized by its time average Nave) N for 1 < t < 5 and for four values of the average particle number in the domain is shown in Figure 3.

Based on the afore developed theoretical background, some indicative results for the two subproblems of creating particle swarms and of computing electrostatic interactions and for the complete effective conductance problem will be presented and discussed here. A snapshot of particles with normalized radius R = 0.01 in a domain with L = 0.5 produced using the developed algorithm with p = 0.005 and K = 0 is shown in Figure 2a. As it

Figure 2. Typical snapshot of the particle swarm for R = 0.01, p = 0.005, and (a) K = 0 and (b) K = 5. Figure 3. Evolution of the ratio N/Nave for several values of Nave (R = 0.0001, K = 5).

is expected, in the absence of special treatment, overlapping between particles appears. The overlapping for fixed value of p increases with particle radius. A typical snapshot using the same parameters but with K = 5 is shown in Figure 2b. It is clear that for this particular case the value K = 5 is large enough to prevent any overlapping between particles. The absence of overlapping is mandatory in all the results that follow. Another problem is the overlapping between particles and wall appearing in Figure 2. Not only is this behavior unrealistic but also in practice the particles cannot approach infinitely close to the wall due to viscous hydrodynamic interactions or to the inertia hydrodynamic forces like the lift force.30 On this account, a particle-free area close to the wall must be introduced in the simulation. The relevant parameter is Ymin and designates the lower value of y coordinate at which the center of a particle can be located. The value Ymin = R simply prevents particle-wall overlapping. Values of Ymin larger than R account for a particle free wall area. It must be noted that the derivation of eq 17 is based on point particles. In the case of finite particle size, the condition that the equation must give the exact value for a homogeneous dispersion requires that the lower limit of the y-integral in eq 17 should be R instead of 0. The particle generation and motion algorithm starts in an empty domain and with a list with no particle entries. As the particles are generated stochastically and move in a deterministic way, the list is progressively filled. When a particle reaches the exit edge of the domain, it disappears from the list. The disappearance of particles starts at about t = 1. After this moment, equilibrium is established among those particles entering and those leaving the domain (and consequently the list). The period 0 < t < 1 is the time needed for the filling of the domain with particles so any

The average values Nave are approximate, and they are used to give an idea about the nature of the problem instead of the actual but less informative parameter p. The amplitude of the area fraction fluctuations decreases as the number of particles in the domain increases. It was found that this amplitude (as it is expressed by the standard deviation of the time traces) is proportional to Nave−1/2. This is in agreement with the convergence rate of Monte Carlo technique in numerical integration as the number of sampling points increases.31 The next issue concerns the construction of the electrostatic interaction function f. The first step is to decide the dimension C of the domain for the evaluation of the undisturbed field V(0). The effective conductance in the absence of particles normalized by its value for C→∞ is shown in Figure 4. The C→∞ computation needs tens of thousands terms in the sums (due to Gibbs effect32), so it is not appropriate for the repeated computations needed for the construction of f. The value G(0)inf is computed by increasing C up to convergence of G(0) to a constant value. The deviation from unity of the ratio shown in Figure 4 expresses the error introduced by the truncation of the computation domain at x = C for several electrode geometries (λ). As it is expected, wider electrodes require larger domains. In any case, the error introduced by the use of C = 6 instead of C→∞ is smaller than 1% even for the wider set of electrodes examined here. On the basis of the undisturbed field, the function f can be constructed. The shape of this function for two electrode geometries will be presented next. In particular, a relatively thin (λ = 0.1) and a relatively thick (λ = 0.5) set of electrodes (with respect to separating distance) will be examined. The two5620

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Figure 4. Effective conductance G(0) of the undisturbed problem for truncation at y = C normalized by its value for C→∞, G(0)inf.

dimensional function f(x,y) has a rather complex shape, so the better way to present it is to show its x-variation for several y values. The curves are included in two graphs in order to use different scales and to isolate specific features. The absolute values of f are not important since they depend on the nondimensionalization (in the absence of this, f has units of inverse length squared). The shape of the function is what is important in the figures presented here. The function f for λ = 0.1 is shown in Figure 5a,b. The coordination system is the one of the electrostatic problem. The function f is symmetric with respect to x = 0 plane (since V(0) is antisymmetric). A peak of the function appears at the neighborhood of the electrode. The closer to the wall, the higher and sharper is the peak. As y increases, f is getting smoother. It is interesting that outside of the electrode region in x direction f does not depend on the distance from the wall at least for relatively small distances. For large values of x, f goes to zero as it is expected. The xdependence of f at larger distances from the wall is shown in Figure 5b. For y > 1, f is not sensitive to the location of the electrode and it is a monotonically decreasing function of x. The f function for the case λ = 0.5 is shown in Figure 6. The general pattern is similar except that the high f region is wider now (following the shape of electrode) and its relative magnitude is smaller. In this case, it is shown more clearly that as y decreases the peak in the electrode neighborhood is separated to two peaks. These peaks correspond to the lines separating the electrode from the nonconducting wall leading to a singularity of the current density (or equivalently to the excessive reduction of the local voltage in the uniform current density case treated here). The last step before simulation realizations is the merging of the two subproblems. It is noted that the location of electrodes in the observation domain coordinate system is from (C − 1 − λ)/(2C) to (C − 1)/2C (first electrode) and (C + 1)/(2C) to (C + 1 + λ)/2C. The observation domain, in general, does not have to coincide with the domain used for the computation of the electrostatic interactions. Given that only macroscopically homogeneous swarms are of interest here and the focus is on the fluctuations of the effective conductance, it can be shown that the criterion for the selection of the observation domain is the variation of f in a distance equal to the average distance

Figure 5. Electrostatic interaction function f versus coordinate x for several values of coordinate y shown in the panels (λ = 0.1).

between particles. Regions where this quantity is small do not contribute to the conductance fluctuations. In the present case based on f distribution, the observational window is chosen to span for x between −3 and 3 and for y up to 3 (Y = 3, L = 0.5). Thus, the value of C used in the integral limits in eq 17 can be different than the value used for the integrand computation. The very large number of f computations (for each particle at each time step) needed at each realization renders the direct use of eq 17 intractable. A nonuniform tensor product grid, which is fine at small y values and at x values in the neighborhood of the electrode (i.e., in regions where f is steep) and coarse everywhere else, is constructed. The function f is computed in all nodes of this grid, and a matrix with the values is constructed. The f computations during a simulation run are performed by finding the appropriate location of each particle in the matrix and performing a bilinear interpolation among nodal values of f. Some typical traces and the effect of problem parameters on them are presented next. It is noted that the dimensional time is given as 2tCδ/U where U is the dimensional velocity of the particles. What is actually shown in the Figures 7−9 is the (multiplied by the observation domain area for normalization) average value of f for the particles in the domain, denoted as f m. This does not correspond to any directly measurable quantity, but it is a very important 5621

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theoretical result since it isolates fluctuations due to the nonuniformity of the electrical field from the conventional fluctuations of the particle number in the domain pointed out previously. This function is derived as follows: the instantaneous conductance correction induced by the particles is Θ=1−

G G(0)

N

= 2πR2 ∑ fi

(18)

i=1

Denoting as Γ = 2C(Y − R) the total area of the observation domain, it can be shown by simple algebra that Θ = 2φ(t )

N (t )

Γ N (t )

∑ fi

= 2φfm

(19)

i=1

where φ(t) and N(t) refer to the instantaneous area fraction and the number of particles in the observation domain. Hence, the instantaneous conductance correction is the product of the instantaneous particle fraction φ and the instantaneous function f m. This function takes the value 1 for a uniform undisturbed field. Any instantaneous deviation of f m from unity is due to the electrical field nonuniformity. The time average of f m has to do with the uniformity of the particle concentration field, and it is 1 for a spatially uniform particle concentration field. Let us denote by an overbar time averages. It holds that Θ = 2φfm = 2φ̅ where Fm =

N (t ) Γ fm = 2φ̅ N̅ N̅

Γ N̅

N (t )

∑ fi

= 2φ̅Fm

i=1

N (t )

∑ fi i=1

According to the above, Θ̅ = 2φ̅ F̅m. Time average of Fm for a spatially uniform dispersion corresponds to the Monte Carlo integration of a function with exact integral equal to one, so it is by definition one. It is noted that in all cases a particle free region with width equal to a particle radius will be considered. The influence of particle number Nave on the discreteness effect is shown in Figure 7 for R = 0.001 and λ = 0.5. In this case, the discreteness does not influence the average conductance value. The increase of the particle number in the domain decreases the amplitude of fluctuations (since the size scale responsible for the fluctuation is the average distance between particles which is inversely proportional to particle number) but does not seem to influence the frequency of fluctuations. The effect of the particle radius is shown in Figure 8 (particle number 500, λ = 0.5). As the particle size increases, the average value of the particle effect decreases due to the increase of the particle free area width (assumed to be equal to R). To explain it better, it is noted that the time average value of f m is 1 for a spatially uniform particle concentration field. This is not the case here because there is a free from particles zone with width R. As R increases, the width of the free zone increases, the spatial uniformity of particle concentration field is destroyed, and the average value of f m deviates from unity. The existence of free from particles zone is also the reason of the reduction of the fluctuation amplitude when the particles exceed a certain size. Some results for the narrow electrodes (λ = 0.1) are presented in Figure 9. The fluctuation amplitude is in general larger. This is due to the fact that the variation of the f curve in the electrode neighborhood is larger. In addition, the frequency

Figure 6. Electrostatic interaction function f versus coordinate x for several values of coordinate y shown in the panels (λ = 0.5).

Figure 7. Evolution of f m for R = 0.001, λ = 0.5, and several number Nave of particles in the observation domain. The curves are shifted for clarity of presentation. Reference horizontal lines denote the location of unity in the vertical axis. 5622

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Figure 10. Dependence between IT and R for several pairs of p and λ.

Figure 8. Evolution of f m for Nave = 500, λ = 0.5, and several values of particle radius R. The curves are shifted for clarity of presentation. Reference horizontal lines denote the location of unity in the vertical axis.

agrees with all the findings discussed before based on time series data. The effect of parameter p on If and IT is shown in Figure 11 for R = 0.001. The range shown corresponds to Nave

Figure 9. Evolution of f m for λ = 0.1 and several pairs of R and Nave. The curves are shifted for clarity of presentation. Reference horizontal lines denote the location of unity in the vertical axis.

Figure 11. Values of If and IT versus p for R = 0.001 and two values of λ.

of fluctuations appears to be higher than in the wide electrode case. In order to quantify the notion of fluctuations, we introduce the fluctuation intensity If which is the standard deviation of the stochastic quantity showed in previous figures divided by its average values. In addition to If, the intensity IT of fluctuation of residual conductance is presented. This is the practically measurable quantity (based on conductance fluctuations) and includes not only the fluctuations in the magnitude of electrostatic interaction but also the fluctuation of the particle number in the domain also. The computations are based on long simulation runs to eliminate the effect of the stochastic nature of the process. The effect of particles size on IT is shown in Figure 10 for two values of parameter p (controlling the particle number Nave) and two type of electrodes. This figure

between 2000 and 200. As it is expected, IT is in every case larger than If but it is found to be smaller than the sum of If and the particle number fluctuation intensity. Overall, it is clear that the fluctuations of the conductance trace are related to the particle number and size so there is a prospect to use the conductance fluctuations for the estimation of the particle size. In this view, the simulation procedure developed here is a necessary tool to reveal qualitative relations between conductance trace features and particle size. After the establishment of such a relation for the simplest particle pattern problem considered here, more complex and realistic cases such particle size distributions and position dependent particle velocities must be considered. 5623

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4. CONCLUSIONS In the present work, a model problem relevant to the measurement of features of dispersed flows using a specific type of conductance measurements is pointed out. The 2dimensional version of the problem is considered with the constant velocity flow of nonconducting circular particles dispersed in a conducting liquid. A mathematical analysis of the complete problem (which is insolvable with current computational means) leads to its decomposition to a series of simpler problems. An approximating analysis of the single particleelectrode electrostatic interaction subproblem leads to closed form relations for the complete problem treated here. Several indicative simulation runs confirm that the fluctuations of the conductance signal are related to the discrete nature of the nonconducting material. In particular, it seems that the extent of fluctuations is related to the degree of variation of the local current magnitude at a distance equal to the average distance between particles. The algorithm developed here can be employed to quantify theoretically the relation between conductance fluctuations and particle size setting the principles for a particle size measurement method.

E ⃗ = −σ

■

∂w =0 ∂r

at r = R

*E-mail: [email protected]. Tel: +30-2310997767. Fax: +30-2310997759. Notes

The authors declare no competing financial interest.

■

(A1)

(A2)

Solving the Laplace equation with separation of variables, employing the identity X = rcosθ, fulfilling the boundary conditions A1 and A2, and matching the internal (inside particle) and external (outside particle) field at r = R inside to the following form for w:

w = 2Dr cosθ

R