Methodology for Performance Evaluation of Dust Control Systems with an Application to Electrostatic Precipitators

Abstract

A general methodology is presented that enables rigorous estimation of the total collection efficiency and the size distribution of particles penetrating dust control systems. This methodology assumes lognormal inlet particle-size distributions and can be used with fractional efficiency formulations that predict, under such conditions, lognormal outlet particle-size distributions. Multimodal inlet particle distributions can be accommodated additively. This methodology is applied to Electrostatic Precipitator Systems (ESPs), with the Nobrega et al. (2004) model selected for predicting their fractional efficiencies. For ease of use, a graphical solution has been developed for the Nobrega et al. fractional efficiency relations, but its availability is not a prerequisite for the application of the general methodology. For the latter, the fractional efficiencies corresponding to three particle diameters need to be estimated and this can be done either graphically or numerically using the model of Nobrega et al. or any other fractional efficiency formulation of interest.

Fine particles emerge as the most important pollutant worldwide in terms of human health, creating thus the need for credible particle size-specific inventories. In line with the above, a generic and rigorous method, capable of producing size-specific emission estimates from uncontrolled and controlled sources, has been developed (Economopoulou and Economopoulos 2001). For controlled sources, this method relies on the development of easy to use models that predict the total efficiency and the lognormal size distribution of particles penetrating the control systems used. Such models have already been developed for dry cyclone separators (Economopoulou and Economopoulos 2002a, 2002b) and venturi scrubbers (Economopoulou and Harrison 2007a, 2007b). The present methodology extends the use of the inventory methodology to ESP-controlled sources and, in addition, it provides a generalized basis for covering other types of control systems with any fractional efficiency formulation considered appropriate.

LIST OF SYMBOLS

A	=	area of the precipitator collection plate, m2;
b	=	width of the electrostatic precipitator, m;
c	=	half the wire-to-wire spacing, m;
Cc	=	Cunningham slip-correction factor, dimensionless;
di	=	particle diameter belonging to ith size class, μm;
dm	=	mass median diameter of particles in the inlet stream, μm;
dp	=	mass median diameter of particles in the outlet stream, μm;
De	=	Deutsch number, dimensionless;
dp	=	particle diffusion coefficient, m2/sec;
eTPM	=	emission factor for total particulate matter, kg per source activity unit;
E	=	average electric field strength, V/m;
Ef	=	overall collection efficiency;
Ef i	=	fractional collection efficiency;
H	=	height of the collection plate, m;
Kn	=	Knudsen number, dimensionless;
L	=	length of the collection plate (or channel length), m;
Mi	=	cumulative fraction-less-than-size di ;
N 0	=	ion density, ions/m3;
P	=	overall penetration, fractional;
Pe	=	Peclet number, dimensionless;
Pg	=	gas pressure, atm;
Pi	=	penetration of a given diameter di , fractional;
Qg	=	volumetric gas flow rate, actual m3/s;
qS	=	particle saturation charge, coulomb;
s	=	distance between the discharge and collecting electrodes, m;
t	=	variate defined by Equation (5);
Tg	=	gas temperature, °K;
v 0	=	gas flow velocity, m/sec;
V	=	applied voltage, V;
x	=	proportionality constant defined by Equation (21);
y	=	proportionality constant defined by Equation (22);
w	=	particle migration velocity, m/sec;

Greek Letters
∈	=	particle dielectric constant, dimensionless;
ϵ0	=	permittivity of free space, equal to 8.85·10−12 F/m;
μ g	=	gas viscosity, kg/m-s;
π	=	dimensionless constant, 3.14159;
ρ g	=	gas density, kg/m3;
σ g	=	geometric standard deviation of the input lognormal distribution; and
σ p	=	geometric standard deviation of the output lognormal distribution.

INTRODUCTION

Many health studies throughout the world have associated respiratory and cardiovascular problems with exposure to airborne fine particle concentrations. It is estimated that every year almost 370,000 people die prematurely in Europe because of air pollution—notably fine particles and ground-level ozone—and the life expectancy is reduced by an average of nine months (CAFE 2005). In the most polluted areas, the loss of life expectancy may be up to two years, or even higher (Economopoulou and Economopoulos 2002c).

Most of the information currently available on health risks has been based on studies in which particulate matter with diameter less than 10 μm (PM₁₀) in the air has been measured. Particulate matter with aerodynamic diameter less than 5 or 6 μm (PM₅ or PM₆) has also been considered, due to its significant deposition in the pulmonary region (WHO 1987). Today, an increasing body of information on particulate matter with diameter less than 2.5 μm (PM_2.5) tends to indicate that PM_2.5 is a better predictor of some health effects than PM₁₀ due to its ability to penetrate deep into the lungs (e.g., Pope et al. 2002).

In view of the above, air quality standards for fine particulate matter are being introduced worldwide and an increasing demand for size-specific particle emission inventories is emerging, mainly for PM_2.5 and PM₁₀, but also for other fractions, such as PM₅, PM₆, and PM₁₅. These are generally regarded as capable of remaining airborne over reasonably long periods, potentially affecting the quality of ambient air several kilometers downwind from the sources (USEPA 1999) and hence they have long been considered in the regulatory process. PM₃₀ is also of environmental concern, as the fraction of particles with diameter greater than 30 μm has an atmospheric lifetime of only minutes, falling out rapidly in the neighborhood of the source, causing local surface contamination and soiling.

To address the need for size-specific particle emission inventories, a methodology has been developed, which can be used for compiling fast and credible PM_2.5, PM₆, PM₁₀, PM₁₅, PM₃₀, and/or TPM inventories for sources with lognormal particle size distributions (Economopoulou and Economopoulos 2001). For the application of this methodology on controlled sources, easy to use models need to be developed for predicting the overall efficiency and the size distribution parameters of particles penetrating each type of control system used. So far, the required models have been developed for dry cyclone separators (Economopoulou and Economopoulos 2002a, 2002b) and venturi scrubbers (Economopoulou and Harrison 2007a, 2007b).

In this article, a general methodology is presented that enables rigorous estimation of the total collection efficiency and the size distribution of particles penetrating dust control systems. This methodology assumes lognormal inlet particle distributions and can be used with fractional efficiency formulations that predict, under such conditions, lognormal outlet particle distributions. Multimodal inlet particle distributions can be accommodated additively. This methodology is applied to electrostatic precipitator systems (ESPs), with the Nobrega et al. (2004) model selected for predicting their fractional efficiencies. As such, the present methodology extends the use of the inventory methodology to ESP-controlled sources and, in addition, it provides a generalized basis for covering other types of control systems with any fractional efficiency formulation considered appropriate.

Electrostatic precipitation is an effective method for controlling fine particle emissions from smelters, cement plants, power stations, paper mills, solid waste incinerators, metallurgical processes, and other heavy industries. Collection forces are applied directly to the individual particles rather than indirectly to the entire gas stream, which leads to the low energy requirements and small resistance to gas flow characteristics of the electrical method. The combination of high-collection efficiencies (often up to 99.8%), moderate energy use, ability to treat large gas flows at high temperatures and to handle corrosive atmospheres and particles, accounts for the widespread use and varied applications of electrostatic precipitators. However, their use is restricted to gas streams not containing explosive substances, entrained droplets, or sticky materials. In addition, their complex construction results in increased capital cost.

Several types and configurations of ESPs are used in practice, the most common of which is the single-stage, wire-plate type used for industrial service (see Figure 1), and the two-stage type, mainly used for air cleaning and light industrial applications. The single-stage, wire-plate type is the subject of our analysis. The operating principles are essentially the same for all types and they include: the corona discharge; the electric charging of the suspended particles; the collection of the charged particles in an electric field; and the removal of the precipitated material to an outside receptacle.

FIG. 1 Design characteristics of a single-stage, multi-wire electrostatic precipitator.

In a single-stage wire-plate electrofilter, the particles are electrically charged as they pass through the gas space between electrodes and plates, which is filled with highly concentrated gas ions of negative charge. Two distinct particle charging mechanisms occur, the most important being the charging by ions driven to the particles by the force of the applied electric field; a secondary charging process occurs due to diffusion, which depends on the thermal energy of the ions, but not on the electric field. The field charging process is predominant for particles larger than about 1.0 micrometer (μm) in diameter while the diffusion process is dominant for particles smaller than about 0.1 μm (White 1951). Both mechanisms are important for particles in the intermediate range.

Once charged, the particles move toward the oppositely charged collection surface where they accumulate and are periodically removed into hoppers by the application of a mechanical impulse or vibration to the plates. Poor gas flow, excessive gas velocities, or poor rapping conditions may cause re-entrainment of particles from the collection surfaces. The particles released from the collection electrodes tend to carry a positive charge (for negative corona). These particles may not be recharged or can only be partially recharged, and thus may be carried out of the precipitator, causing a substantial decrease of the precipitator efficiency.

DEVELOPMENT OF THE GENERAL METHODOLOGY

Mathematical Relations for the Overall Efficiency

Aerosols comprise of a mixture of particles of different sizes. If the fractional efficiency of the collector can be defined by an equation such as Ef _i = Ef _i (d_i ), and the size distribution of particles can be expressed in terms of a cumulative fraction-less-than-size d_i function, M_i =M_i (d_i ), then the overall precipitator efficiency is estimated by integrating the fractional efficiency over the particle size distribution:

The overall penetration can then be expressed as

In the majority of industrial processes, the size of generated particles can be satisfactorily fitted through a lognormal distribution function, with parameters the mass median diameter d_m and the geometric standard deviation σ _g (e.g., Licht 1980). The former represents the general size-magnitude of the particle, while the latter represents the spread of the size range around the mass median diameter. The lognormal distribution function, which results from applying the Gaussian law of distribution to ln d_i , takes the mathematical form:

Introducing Equation (3) into Equation (2) and rearranging we obtain:

where the variate t is defined as:

Mathematical Relations for the Size Distribution of Penetrating Particles

Based on the assumption of a lognormal input particle size distribution, the mass fraction of penetrating particles belonging to this ith size class, P_i =P_i (d_i ), can be computed by the following equation:

In Equation (6) there are three interrelated functions: the size-distribution of particles in the input stream; the size-distribution of particles in the output stream; and the fractional efficiency of the control system. If any of the two functions are given then the third one is fixed and can be calculated.

In the case under consideration, the output particle size distribution is to be calculated from a specified collection process acting upon a specified input dust. This task is facilitated by the premise that a lognormal particle-size distribution of the input dust yields a lognormal particle size distribution of the output dust. Under this condition, the size distribution of penetrating particles can be defined by its mass median diameter d_p and the geometric standard deviation σ _p , and the fractional penetration, P_i , can be expressed as a function of these two parameters through the following relation:

Combination of Equations (7) and (6) yields:

Equation (8) relates key output parameters to known input parameters and is used in the analysis that follows for developing the sought methodology.

Analysis of Functional Relations

Equation (8) correlates the output lognormal particle-size distribution parameters (d_p , σ _p ) to the input lognormal particle-size distribution parameters (d_m , σ _g ) and to the precipitator overall efficiency Ef. For n unknown variables, n number of equations are required, therefore, we consider Equation (8), for three different particle diameter sizes d_i =d ₁, d_i =d ₂, and d_i =d ₃, respectively.

The values of d ₁, d ₂, and d ₃ are selected from the following relations, as functions of the input dust lognormal distribution parameters d_m and σ _g :

where x and y are intermediate, real constants that can be calculated from the selected values of d ₁ and d ₂, as discussed below, and are introduced in our analysis in order to facilitate the ensuing mathematical computations.

Introducing Equation (9) into (8), for d_i =d ₁; Equation (10) into (8), for d_i =d ₂; and Equation (11) into (8), for d_i =d ₃; we obtain the following set of equations:

After appropriate mathematical transformations and rearrangements (see Supplementary Material section), the above set of Equations (12)–(14) is solved as follows:

where

In order to calculate the parameters x and y used in Equations (15)–(17), three characteristic particle sizes, d ₁, d ₂, and d ₃ need to be defined through the following procedure:

1.	Select appropriate (see below) values for the diameters d 1 and d 2, and calculate the values of x and y from Equations (10) and (9), rearranged in the following form:
2.	Calculate the diameter value d 3 from Equation (11) as function of the diameter values d 1 and d 2.

The selection of the particle sizes d ₁ and d ₂ in step 1 depends on the control system under consideration, since the set of particle sizes d ₁, d ₂, and d ₃ need to be within the diameter range that affects most the performance of the control system under consideration.

The parameters λ₁, λ₂, and λ₃ used in Equations (15)–(17), can be calculated from Equations (18)–(20), as function of the above calculated constants x and y and the fractional penetrations (1 –Ef _i ]_{di = d1} ), (1 –Ef _i ]_{di = d2} ), and (1 –Ef _i ]_{di = d3} ). The latter can be computed from any theoretical fractional efficiency formulation of interest.

Equations (15)–(17) yield the sought output particle size distribution parameters d_p and σ _p and the overall efficiency Ef as function of the input particle size distribution parameters d_m and σ _g and the above-computed parameters x, y, λ₁, λ₂, and λ₃ and (1 –Ef]_{di = d2} ).

To apply the above methodology to an electrostatic precipitator system, the following are considered:

•	The typical particle size–collection efficiency curve of a properly sized and operated electrostatic precipitator, exhibits a minimum in the range of 0.1 to 1 micrometers, with a decreased efficiency zone usually extending from 0.1 to 5 μm.
•	Within the above size range, recommended diameter values are 5–6 μm for d 1, and 1.2–1.5 μm for d 2,
•	With the above values of d 1 and d 2, Equation (11) yields d 3 values in the range of 0.2–0.5 μm.

Based on the above, the set of particle sizes, d ₁, d ₂, and d ₃, is always within the diameter range that affects most the performance of a properly sized and operated ESP and therefore is suitable for estimating the sought output particle size distribution parameters d_p and σ _p and the overall efficiency Ef.

COMPUTER SIMULATION PROGRAM

The main objective of the computer program described in this section is to test the validity of the premise that lognormal input particle-size distributions result in lognormal output particle-size distributions, the latter predicted through the use of the selected fractional efficiency relations. If this is verified, the methodology described above can be applied using the selected fractional efficiency relations of the control system under consideration.

This program comprises a subroutine that solves the selected fractional efficiency relations. The predictions of this subroutine are normally tested and validated against relevant predictions reported in the literature, under identical design and operating conditions.

The program integrates the fractional efficiency predictions of the above subroutine over the entire range of the lognormal inlet particle size distribution in order to calculate the overall collection efficiency and the size distribution of penetrating particles. As the functions to be integrated are usually complex, and accurate predictions need to be produced, the numerical integration is performed over 800 particle diameter increments, with smaller increments in the region of small particle sizes.

With the above-described program, one can proceed to test the validity of the premise that a lognormal particle-size distribution in the input stream results in a lognormal particle size-distribution in the output stream.

As analytical proof of the above assumption cannot be produced in general, its validity needs to be numerically tested and verified (Economopoulou 2006). For this purpose, the above computer program is expanded to perform the following:

•	Reads the lognormal inlet particle-size distribution parameters dm and σ g and calculates the mass fraction of particles belonging to any given ith diameter-size class, di .
•	Calculates the cumulative weight fractions of penetrating particles for more than 800 particle diameters.
•	Fits optimally the normalized cumulative weight fractions of penetrating particles with a lognormal distribution function. As illustrated in the flow diagram of Figure 2, the computational procedure begins with a pair of assumed initial values for the parameters dp and σ p and through a numerical hill-climbing search algorithm (Rosenbrock 1960), new pairs of dp and σ p parameter values are computed until the best fit, which minimizes the error function (mean square deviation) is obtained. To ensure the most accurate calculation of the parameters dp and σ p , the best-fit procedure is repeated several times with different initial values of dp and σ p and the most successful trial is selected.
•	Tests the above over a wide range of inlet particle-size distributions and control system design and operating conditions so as to ensure the validity of the conclusions derived.

FIG. 2 Algorithm for generating the best-fit lognormal distribution parameters d_p and σ _p from cumulative weight fraction series computed through the use of the Nobrega et al. (2004) fractional efficiency model.

Figure 3 demonstrates how closely, in a typical case, the lognormal distribution matches the size distribution of penetrating particles. In this example the system simulated is an ESP and its fractional efficiencies are predicted by the Nobrega et al. (2004) formulation (see below).

FIG. 3 Normalized cumulative weight distributions of particles penetrating electrostatic precipitators, as predicted by the theoretical fractional efficiency model of Nobrega et al. (2004) and optimally fitted by lognormal distribution functions.

Through this procedure, the above-mentioned assumption has been found valid for all control systems and fractional efficiency prediction relations tested so far (Economopoulou and Economopoulos 2002a, 2002b; Economopoulou and Harrison 2007a, 2007b) and the same is found in the present work to be the case with ESPs and the Nobrega et al. (2004) fractional efficiency formulation.

APPLICATION OF METHODOLOGY TO ESPs

ESP Collection Efficiency Theories

As described above, particle collection in precipitators occurs by the electric forces acting upon particles. These forces cause the particles to move toward the collecting electrodes at migration velocities determined by the equilibrium between the electric forces and the drag forces resulting from the gas viscosity.

Particle collection theories depend on the form of the gas flow through the precipitator. One theoretical approach assumes that particles carried in the gas are moving in laminar flow through the precipitator. In such cases the particles move toward the collecting electrodes with migration velocities that can be calculated from the laws of classical mechanics and electrostatics. In another, more realistic approach, the gas is assumed to move in complex turbulent flow patterns influenced by the electric wind effects of the corona and, in heavy dust loadings, by the momentum transferred to the gas by the mass movements of the dust particles. In this case, the movement of a particle through the precipitator is extremely complex and not amenable to analytical solution. Nor will any two particles follow the same path. For small particles, which are of particular importance in electrostatic precipitation, the particle migration velocities are much less than the velocity of the gas through the precipitator. Their motion is determined primarily by the turbulent flow patterns of the gas and only secondarily by the electric forces acting on the particles. Particle collection occurs when the particle happens to be carried close to the collection surface and enters the laminar flow boundary layer where the electric forces act to move them to the collection surface.

The typical particle-size collection efficiency curve for a properly sized and operated precipitator shows a minimum in the range of 0.1 to 0.5 μm. The efficiency curve is shaped by the combined effect of two particle electrical charging mechanisms (owing to field charging and ion diffusion), neither of which is highly effective in this particle size range.

As increasingly stringent particle emission legislation is coming into force worldwide, much higher performance levels are expected from control technologies such as electrostatic precipitators and higher accuracies are required in the estimation of their collection efficiency. For this purpose, several theoretical formulations have been developed over the years for predicting the fractional collection efficiency of precipitators and optimizing their performance, such as those by Deutsch (1922), Cooperman (1971), Leonard et al. (1982), Zhibin and Guoquan (1994), and Nobrega et al. (2004). From these models, only few address the effects of particle accumulation at the discharging electrodes and at the collection plates, a phenomenon known to influence adversely the performance of ESP. Instead, most of the ESP models tend to focus on the effects of turbulent mixing and secondary wind in multi-wire, single-stage electrostatic precipitators.

Lind (1997) presents an interesting survey of the research work in this area and of the contribution of several researchers to the understanding and solution of the influence of turbulence on the precipitator performance. Based on this historical résumé, and on a literature review carried out in the context of the present study, a short summary of the research advancements in this field is presented in the remainder of this section.

The Deutsch (1922) theory for particle collection assumes complete mixing by turbulent flow (infinite transverse mixing coefficient or degree of turbulence), which redistributes the particles at each downstream location homogeneously over the precipitator duct. He considers the forces acting on spherical particles to be the electrical Coulomb and the Stokes' fluid drag forces. Apart from the boundary layer, the particles in the downstream direction have the mean velocity of the fluid, they are fully charged and they migrate in a homogeneous electrical field (Riehle 1997). In order to improve the assumption of infinite diffusivity of the Deutsch model, many researchers tried to develop finite diffusivity models by dealing with the convective-diffusion equation using various boundary conditions.

In 1959, Friedlander introduced the particle vorticity diffusion and the migration velocity through a two-dimensional differential equation that accounted for the particle concentration in both axial and transverse directions. In 1965, Cooperman used a similar model adding re-entrainment to advance the problem. Cochet (1961) and McDonald et al. (1977) considered the diffusion charge and the field charge by introducing the concept of particle mobility.

More recently, Cooperman (1971) developed a modification of the Deutsch theory, taking into account longitudinal turbulent mixing effects and re-entrainment. According to his theory, at very high longitudinal turbulence, the concentration gradient carries particles out the outlet so fast that they do not have enough time to reach the collection plate. Although this case appears realistic in very high mixing conditions, it is too extreme to be used for actual design (Kim et al. 2001). Another limitation of this model stems from the absence of a general method to estimate the re-entrainment factor and the particle diffusivity (Kim and Lee 1999).

Bernstein and Crowe (1979) calculated numerically the electric field and the gas flow in a wire-plate precipitator, and concluded that the interaction between electric field and gas flow has a mixed effect on the electric migration velocity in different regions of the precipitator.

Leonard et al. (1980) made a further contribution to the understanding and the solution of the influence of turbulence on ESP performance. Their work concluded that higher efficiencies are expected with a finite level of turbulence. This is demonstrated in a graph, in which the electrofilter penetration was plotted as function of the Deutsch number (, where w is the particle drift, v ₀ is the gas velocity, L is the channel length and s is the distance between the discharge and collective electrodes), using the Peclet number as a parameter (, where d_p is the turbulent mixing coefficient. Pe is ∞ for laminar and 0 for fully turbulent flow). From the calculation and measurement of particle concentration profiles, they concluded that higher concentrations are expected towards the collector wall. Considering the finite diffusivity, they developed a complicated two-dimensional model using the method of the separation of variables from the convective-diffusion equation (Leonard et al. 1982), assuming uniformity of velocity components of charged particles and of particle diffusivity.

Zhibin and Guoquan (1994) considered the non-uniform electric field and measured the collection efficiency of a single-stage ESP covering a wide particle-size range. Their collection efficiency model estimated the turbulent diffusion coefficient taking into account the flow-induced turbulence, the secondary flow, and the turbulence induced by the electric wind. In this model the non-uniform distribution of the electric field intensity is calculated as function of the particle weight concentration distribution along the plane transverse to the gas flow direction.

Bai et al. (1995) proposed a moment model, assuming a lognormal particle size distribution of the input dust. Describing the continuous evolution of the particle size distribution along the precipitator, this modeling approach provides information on average particle size distribution properties. However, it does not consider the turbulent diffusion process, which is an important parameter affecting the collection of very small particles in the electrostatic precipitator.

A modification of the Bai et al. model was developed by Kim et al. (2001) to predict the continuous evolution of the lognormal particle-size distribution along the precipitator length, under the effect of flow convection, electrostatic force, and particle diffusion mechanisms. However, as noted by the authors, the predictions of the model are limited to the ESPs designed and operated under the condition of low electro- and hydro-dynamic flow effects. More comprehensive data are needed for use in the validation of the model before its practical application.

Nobrega et al. (2004) evaluated the performance of wire-plate electrostatic precipitators over a wide particle size distribution and concluded that the fractional efficiency predictions from literature correlations do not match well the experimental measurements. They proposed a new correlation, assuming uniform air velocity profiles and giving an empirically based formula to estimate the diffusion coefficient as a function of particle size. The model is based on the assumption that the concentration of particles is smaller at the centre of the precipitator duct and increases towards the collection plates. According to the authors, this assumption seems quite reasonable, is in agreement with the experimental data obtained by Schmid and Umhauer (1998), and is visually observed in photographs taken from inside a wire-plate precipitator by Riehle and Loffler (1993). The Nobrega et al. model also estimates an approximate combined field and diffusion particle charging, according to Cochet (1961), who modified the field charging equation using the Knudsen number K_n to account for increased particle charging in the submicron particle range (Schmid and Vogel 2003). Unfortunately, the authors presented no detailed information on how they derived their solution to the initial convective diffusion equation, and this raises some questions about the physical significance of their model and its theoretical basis.

Fractional Efficiency Model of Nobrega et al. (2004)

The development of fractional efficiency models that yield accurate predictions over a wide range of design and operating conditions is a particularly demanding task in the case of electrostatic precipitators, due to complex interactions among the electric field (generated by the corona discharge), the ionic current, the flow field (usually turbulent), and the particle dynamics (as determined by the electric field, the particle charging, and the fluid flow).

Earlier models, such as these by Deutsch (1922), Cooperman (1971), and Leonard et al. (1982) have numerous citations in the literature; however, their predictions are repeatedly reported to give inconsistent and unsatisfactory results when compared to experimental data, especially in the fine particle diameter region. Recent modeling approaches are more realistic, taking into consideration more physically justified processes occurring in the electrostatic precipitator, and thus they seem more promising; however, little is published regarding their validation against experimental data and considerable complexity governs the differential equations involved.

Among the most recent models, that of Nobrega et al. (2004) is somewhat simpler and its predictions appear in satisfactory agreement with experimental measurements over a wide range of particle diameters, system design, and operating conditions (Economopoulou 2006). The experimental validation of the Nobrega et al. model included experimental efficiency data for large, high efficiency ESP systems used in a cement plant as well as additional data sets published in literature (Riehle and Loffler 1990; Riehle 1992; Kim and Lee 1999). Comparison of the model predictions with these published data from ESPs of different sizes and various operating conditions have yielded results better than the Deutsch (1922), Leonard et al. (1982), Cooperman (1971), and Kim et al. (2001) models, especially at high ESP collection efficiencies.

The Nobrega et al. fractional efficiency can be computed from the following relations:

The Deutsch and Peclet numbers are as defined in the section on collection efficiency theories.

where Ef _i is the fractional efficiency of particles by mass; d_i is the size of an individual particle of a given kind (that is of a particle having a certain shape and composition); De is the Deutsch number (dimensionless); Pe is the Peclet number (dimensionless); L is the length of the collecting plate (m); s is the distance between the discharge and collecting electrodes (m); v ₀ is the gas flow velocity (m/s); w is the migration velocity (m/s); μ _g is the gas absolute viscosity (kg/m-s); E is the average electric field strength, defined by E=V/s(V/m); ϵ₀ is the permittivity of free space (8.85·10⁻¹² F/m); ∈ the particle dielectric constant (dimensionless); and q_S is the particle saturation charge (coulomb), i.e., the maximum charge to be collected due to electrical field charging.

D_p is the diffusion coefficient (m²/sec), assumed to be inversely proportional to particle size, D_p = 2 · 10⁻⁷ d ⁻¹ _i , with the proportionality constant experimentally determined. The Cunningham slip correction factor C_c , defined by C_c = 1 +K_n [1.257 + 0.4 exp (–1.10/K_n )], is introduced in order to take into account cases where the particle size is small enough to be of the order of the mean free path of the gas molecules; under this condition, the gas no longer behaves as a continuous medium with respect to the particle, and its drag effect is reduced. The value of C_c depends upon the ratio of the mean free path of the gas molecules to the size of the particle; this ratio is expressed as the Knudsen Number, defined by

Introducing Equation (23) into (4), we obtain the following relation, which yields the overall penetration of particles from ESPs, based on the Nobrega et al. (2004) formulation:

Graphical Solution to the Nobrega et al. Fractional Efficiency Formulation

In order to use Equations (15)–(20), one needs to estimate the fractional penetrations (1 – Ef _i ]_{di = d1} ), (1 – Ef _i ]_{di = d2} ) and (1 –Ef _i ]_{di = d3} ) for the particle sizes d ₁, d ₂, and d ₃. To facilitate this task and enable a graphical prediction of these fractional efficiency values, Equation (23) is expressed in the following simple functional form:

where, the Deutsch number takes the form of Equation (28), and the Peclet number can be expressed in the form of Equation (29).

As shown in Equations (28) and (29), both the Deutsch and Peclet numbers are a function of the particle diameter d_i . Thus, the functional form of Equation (27) is amenable to graphical solution, and this is provided by the nomograph in Figure 4. This nomograph yields directly the fractional penetration value (1 – Ef _i ) as a function of parameters De = De(d_i ) and Pe = Pe(d_i ), for any given diameter size d_i .

FIG. 4 Nomographs for estimating the fractional penetration 1 – Ef _i as a function of the parameters Pe(d_i ) and De(d_i ), based on the Nobrega et al. (2004) theoretical formulation.

The graphical solution in Figure 4 is spread into two nomographs designed with different y-coordinate scales, so as to accommodate for a wide range of (1 – Ef _i ) values, enhance the resolution of the nomographs and facilitate their use.

The mathematical analysis, based on which the nomographs in Figure 4 are produced, entails no assumptions or approximations and therefore the graphical results can deliver the exact numerical solution of the Nobrega et al. (2004) fractional efficiency formulation. Careful manual use is required when using these nomographs to avoid errors and yield accurate results.

STEP-BY-STEP APPLICATION PROCEDURE

The developed methodology for estimating the overall efficiency and the size distribution of penetrating particles is rigorous, subject to the sole assumption of lognormal size distributions of the inlet and output particles. For the application of this approach, input data are required on the ESP design configuration (L, H, b, s, c), electric equipment (V, number of cells and fields), operating conditions (Q_g , T_g , P_g ) and particle properties (d_m , σ _g , ρ _p ,∈). Once these data are known, the sought parameters d_p and σ _p can be estimated through the following simple step-by-step procedure.

Step 1. Calculate ESP Parameters

For the given gas temperature, estimate the gas density ρ _g and viscosity μ _g using appropriate literature relations (e.g., Economopoulou and Harrison 2007a). Compute the gas flow velocity v ₀=Q_g /A.

Select two diameter values in the regions 5–6 μm for d ₁ and 1.2–1.5 μm for d ₂. From these values, calculate parameters x and y using Equations (21) and (22). Based on x, estimate the diameter size d ₃ using Equation (11).

Step 2. Calculate the Required Fractional Penetration Values

If the Nobrega et al. model is to be used, then, for each of the three diameter sizes, calculate C_c ; q_S from Equation (25); w from Equation (24); and, based on these parameters, De(d_i ) from Equation (28); and Pe(d_i ) from Equation (29). The use of an excel spreadsheet can facilitate these calculations. Obtain the values of the fractional penetrations (1 –Ef _i ]_{di = d1} ), (1 –Ef _i ]_{di = d2} ), and (1 –Ef _i ]_{di = d3} ), using the appropriate nomograph in Figure 4. The required fractional penetration values can be calculated using any other fractional efficiency formulation of preference.

Step 3. Estimate the ESP Output Dust and Overall Collection Efficiency

Compute the parameters λ₁, λ₂, and λ₃ from Equations (18)–(20).

Calculate the output dust distribution parameters d_p and σ _p and the precipitator overall efficiency Ef from Equations (15)–(17), respectively.

EXAMPLE

Estimate the overall efficiency and the size distribution of particles penetrating two single-stage precipitators operating in parallel for the control of flue gas emissions from a dry process Portland cement kiln. The gas flow rate is Q_g = 200 am³/s at temperature 150°C and atmospheric pressure. The particles in the gas stream have a dielectric constant ∈ = 6.14, density ρ _p = 2,500 kg/m³ and follow a lognormal distribution with parameters d_m = 18.92 μm and σ _g = 9.78. The technical data and the electrical equipment of each electrofilter are:

•	Number of cells (laterally to the gas flow): 2
•	Number of electrical fields (longitudinal to the gas flow): 2
•	Number of passages per bus section: 25
•	Active length of each bus section, L: 3.6 m
•	Active height of each bus section, H: 11 m
•	Active width of each bus section, b: 7.25 m
•	Wire-to-plate spacing, s: 0.145 m
•	Wire-to-wire spacing, 2c: 0.10 m
•	Applied voltage, V: 50 kV

See also the diagrammatic presentation in Figure 5.

FIG. 5 Schematic diagram of the electrostatic precipitator arrangement.

Solution

Step 1. Calculate ESP Parameters

The estimated gas density and gas viscosity at T_g = 150°C, are ρ _g = 0.8372 kg/m³ and μ _g = 0.2170·10⁻⁴, respectively (Economopoulou and Harrison 2007a). In each electrofilter the collection plate area is A = 11 · 7.2 = 79.2 m², the gas flow is Q _g = 100 am³/s flue gas and the gas velocity is v ₀=Q_g /(2bH)= 0.627 m/s.

We select d ₁= 5.5 μm, d ₂= 1.4 μm, and calculate x= 3.9286 and y= 0.3517 from Equations (21) and (22). For these values we calculate d ₃= 0.356 μm using Equation (11).

Step 2. Calculate the Required Fractional Penetration Values

For each of the three diameter sizes we compute the following model parameters:

Table

			q S (d i ), coul	w(d i ), m/s	De(d i )	Pe(d i )
i	d i (μm)	C c (d i )	Equation (25)	Equation (24)	Equation (28)	Equation (29)
1	5.50	1.0429	6.563·10^−16	0.2098	16.617	0.837
2	1.40	1.1682	4.253·10^−17	0.0598	4.739	0.0607
3	0.356	1.6880	2.755·10^−18	0.0220	1.743	0.006

From the bottom nomograph of Figure 4 we obtain (1 –Ef _i ]_{di = d1} ) = 0.0003 and from the top nomograph (1 –Ef _i ]_{di = d2} ) 0.0061, and (1 –Ef _i ]_{di = d3} ) = 0.0105.

Step 3. Estimate the ESP Output Dust and Overall Collection Efficiency

From Equations (18)–(20) we calculate λ₁= 2.507, λ₂= −0.322, and λ₃= 2.870. From Equations (15)–(17) we calculate D_p = 0.825 μm, σ _p = 2.034, and Ef = 99.95%. From the outlet dust distribution parameters we can estimate the size-specific weight fractions W ₁, W _2.5, and W ₁₀ to be 0.290, 0.772, and 0.996, respectively. The exact numerical solution of the Nobrega et al. model yields very similar results, with W ₁= 0.272, W _2.5= 0.775, W ₁₀= 0.995, and Ef = 99.86%.

SUMMARY AND CONCLUSIONS

Particulate control systems use various mechanisms to separate particles from the gas stream. As the effect of these mechanisms varies with the particle size, different particle sizes are collected with different degrees of effectiveness.

To obtain the overall and size-specific efficiencies of a control system, a suitable formulation, capable of predicting satisfactorily the fractional efficiency of the control system, needs to be selected from the literature, this formulation must be solved and its predictions must be integrated over the appropriate range of the lognormal cumulative weight distribution, under the specified design and operating conditions. The integral that results from this mathematical analysis is generally complex and not amenable to analytical solution. The above complexities hamper the use of such modeling methods in emission inventory programs.

To overcome this difficulty and enable planners to compile credible PM_2.5, PM₆, PM₁₀, PM₁₅, PM₃₀, and/or TPM inventories for sources with lognormal particle size distributions, a generic method has been developed (Economopoulou and Economopoulos 2001). For controlled sources, this entails the development of graphical models enabling estimation of the overall efficiency, Ef, and of the output particle size distribution parameters, d_p and σ _p , for each type of control system used.

The present work expands the application of the generic method to particle sources controlled by ESPs and provides direct, analytical expressions for estimating the required parameters Ef, d_p , and σ _p . The advantage of this methodology is that it dispenses with the need to develop and use graphical models specific for the control system of interest and the fractional efficiency formulation under consideration. As such, this method can be applied on any ESP fractional formulation of interest, provided the output dust size distribution generated by this formulation closely resembles a lognormal distribution. In this general case, the required Ef, d_p , and σ _p , parameters can be estimated from Equations (15) to (17) as described above, with the three fractional efficiency values Ef _i estimated from the theoretical formulation under consideration for the selected particle sizes d ₁, d ₂, and d ₃. Regardless of which theoretical model is used, in cases of very high overall efficiency values (Ef > 99.9%), careful selection of the three particle diameter values is required, within or even below the recommended particle size ranges, so as to avoid fractional efficiency values close to unity.

The methodology proposed is rigorous, subject only to the assumption of lognormal size distribution of the inlet and outlet particles. The choice of the three diameter values that are used to fit the output dust distribution to a lognormal distribution, introduces negligible errors when the three diameter values d ₁, d ₂, and d ₃ are selected within the above-specified size ranges. As such, the method reflects the variation of the collection efficiency with particle properties, key design parameters, and basic source operating conditions (gas flow rate, temperature, and pressure).

The use and merits of this methodology on ESPs is demonstrated using the fractional efficiency formulation of Nobrega et al. (2004) and nomographs are developed for fast and rigorous fractional efficiency estimations. The model of Nobrega et al. (2004) was selected on the basis of simplicity and good accuracy of results against experimental efficiency data sets published in literature (Economopoulou 2006).

Application of this approach can also be extended to multimodal input dust distributions. In these cases, the particle emissions from each source need to be defined by two or three sets of , d_m , and σ _g parameters, rather than by a single one. The diameter values at which the dust distribution peaks are found on a frequency density plot ( versus ln d_i ) are the values of the mass median diameter for each parent mode. From the values of observed on this plot, the respective values of the standard deviation of each mode may be calculated by using Equation (3). Finally, the relative proportion of each parent mode in the mixture may be estimated from the relative areas under each of the bell-curves on the frequency density plot. After defining the emission factor and size distribution parameters d_m and σ _g of each parent mode, the methodology can then be applied separately for each mode, and the computed size-specific emissions from each mode must be added in the end.

The present methodology could easily be adapted and extended to additional control systems of interest, on the premise that a lognormal inlet dust distribution yields a lognormal size-distribution of particles penetrating these control systems. The validity of this premise can only be numerically verified. However, this has been found to be valid for all control systems and fractional efficiency formulations considered so far (Economopoulou and Economopoulos 2002a, 2002b; Economopoulou and Harrison 2007a, 2007b). Under this assumption, the optimum particle-size ranges for selection of the three diameters d ₁, d ₂, and d ₃ must be determined, so as to yield accurate performance evaluation of the control system under consideration.