Article pubs.acs.org/EF
Determination of Sticking Probability Based on the Critical Velocity Derived from a Visco-Elastoplastic Model to Characterize Ash Deposition in an Entrained Flow Gasifier LaTosha M. Gibson,†,‡ Lawrence J. Shadle,‡ and Sarma V. Pisupati*,†,‡ †
John and Willie Leone Family Department of Energy and Mineral Engineering and The EMS Energy Institute, The Pennsylvania State University, 110 Hosler Building, University Park, Pennsylvania 16802, United States ‡ National Energy Technology Laboratory, 3610 Collins Ferry Road, Morgantown, West Virginia 26507, United States ABSTRACT: The fluid and solid mechanics of particle wall collisions were investigated for entrained coal gasifier applications. Critical velocity was used to characterize the conditions required for the reacted coal particles to stick to the wall. The critical velocity was derived from a viscoelastic model. Based on this method, the sticking probability was determined for simulated char particles from different coal specific gravity and size fractions. In this method, particles that exceeded the critical velocity were predicted to adhere, while those below it were predicted to rebound. This study showed that the sticking efficiency based on the critical velocity was closer to the predictions based on the temperature of the critical viscosity and critical angle than the predictions based on the temperature of the critical viscosity alone. However, there was a significant difference in the predictions between the “rules based criterion” and the critical velocity methodology for the sticking efficiency calculations of the larger size fractions for higher specific gravity fractions (SG3 and SG4). Nevertheless, the critical velocity methodology in this work is the first attempt to address the influence of both the ash and the carbon on the particle properties responsible for the inertial behavior of char particles within an entrained flow gasifier. Provided that the remaining uncertainty of the measurements of the particle compressive yield strength and the modulus of elasticity versus temperature is addressed in correlation to the measured viscosity and surface tension, this method could be a practical alternative in determining sticking efficiency.
1. BACKGROUND Because of the inefficiencies in plant operations due to flyash, there is a need to control ash deposition and the handling of slag disposal. Excessive char deposition in convective coolers can lead to unplanned shutdowns. Ash deposition also leads to a reduction in heat transfer in the radiative (slagging) section resulting in an increased cost of electricity generation.1 Therefore, the overall objective of this research project was to characterize the impact and deposition behavior of coal particles based upon their specific gravities and size fractions. This objective would help determine the particles within the population that were responsible for contributions to flyash.2−6 Nevertheless, by employing the discrete phase model, a computational tool that represents the gas phase as a continuum and the particles as a discrete phase, the trajectories of particles can be determined through a Lagrangian characterization.3,7 For those particles that are predicted to impact the wall, the coefficient of restitution (COR) must be defined. COR, which is the ratio of the rebounding velocity to the impacting velocity, depends not only on the properties of the particles but also on the properties of the surface of impact.8 Particles with a COR of zero are predicted to stick, while those with a COR greater than zero are predicted to rebound. Previously experimental and analytical work has been performed to characterize ash and char deposition.9−14 However, such efforts have fallen short in addressing the probable variation in behavior that can occur when char impacts the refractory wall, partially due to variability in the carbon and particle temperature. Figure 1 illustrates the three modes of behavior of a char particle-adhesion, rebound, and © 2014 American Chemical Society
entrapment. The three scenarios of particle impact include: (1) particle impacting the refractory wall, (2) particle impacting a slag layer on the wall, and (3) particle impacting adhered particles onto the wall. Three empirical methods used to characterize ash deposition are viscosity models, slagging indices, and measurements for the temperature of the critical viscosity. Of the different empirical methods mentioned, the viscosity models have been widely used to determine the threshold for particle sticking.15 Among the viscosity models, the modified Urbain model has been widely used.16 Such models based on the acid to base ratio have been correlated with the ratio of network-forming cations to network-disrupting cations. The network disrupting cations are defined as the cations that discontinue the network chain of oxides due to a lack of available vacancies. Typically, the temperature of critical viscosity, (TCV, i.e., the temperature above which slag transforms from a glassy Newtonian phase to a crystalline non-Newtonian phase) is determined. This temperature is then used within the viscosity models to determine the critical viscosity. The probability of the particle sticking is then based on the ratio of the predicted particle viscosity at the temperature of impact to the critical viscosity. Therefore, the higher the ratio of the particle viscosity to the critical viscosity, the lower the likelihood of the particle sticking and vice versa. However, the main pitfall of relying on the viscosity models to predict particle sticking is the fact that they Received: April 17, 2014 Revised: June 27, 2014 Published: July 1, 2014 5307
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Figure 1. Three modes of behavior of particle impacting a slag layer.
predict the adhesion of char particles in order to determine the slag layer thickness on the wall of a combustor. However, the model of Lee et al.26 was based on the empirical results of Tabkoff and Malak to determine the COR, which was measured at room temperature.27 At high temperatures, char particles exhibit viscoelastic behavior, as shown through swelling studies.28 Other methods for determining particle sticking employed parameters such as the temperature of critical viscosity, Weber number, and the influence of carbon either through a critical carbon conversion or the critical contact angle due to carbon, θCr as a rules based criteria.29,30 The latest methodology based on determining a critical velocity was proposed by Ai and Kuhlman.31 This formula was based on the behavior of dry elastic particles, but was extrapolated to be equated with the sticking efficiency of empirical ash deposition measurements through the modulus of elasticity as a function of temperature. In the present work, the critical velocity has been derived from a linear viscoelastic model. This critical velocity was developed to incorporate the influence of carbon and the ash composition on the yield strength of char particles. These calculations are a continuation of previous work regarding measurements for the coefficient of restitution and application of a population model to a particle size distribution to partitions slag and flyash.8,30 This previous work assessed the applicability of model predictions to gasifier conditions in regards to irregular particle and surface geometry.8 The particle orientation (a result of the degree of equancy and rotational velocity components) resulted in a variance in the coefficient of restitution as a function of the rebound angle.8 For the population model, an algorithm based on carbon conversion, impact probability, and sticking probability was developed to partition slag and flyash.30 The sticking probability was calculated based on empirical information about the carbon’s influence on interfacial surface tension and the influence of the ash composition on the viscosity (indicative of the particle’s stiffness). Results suggest that the largest size fractions, predominantly consisting of carbon, contribute the most to the flyash. The range of capture efficiencies for the PSD is consistent with experimental data on deposition from Li et al.14
only take the effects temperature and ash composition into consideration. Char particles that have a significant amount of residual carbon may not have enough minerals located in the peripheral areas of the carbon matrix as suggested by Li et al.14 In addition, the residual carbon can alter the oxidation state of the mineral matter in the char, thereby reducing its viscosity and influencing the particle’s sticking behavior. Furthermore, the viscosity models are still approximate at best when applied to different coals and cannot be used for particle tracking purposes to predict the magnitude and direction of the particles that are predicted to rebound from the surface. In addition, these models do not account for the influence of char particles that have adhered to the refractory walls and subsequently cooled. Previous efforts to develop the modeling tools that characterize ash deposition include the efforts of Rushdi et al. and the Energy and Environmental Research Center (EERC).1,17 Rushdi et al. implemented a subroutine to predict the particle shift temperature to determine the particle viscosity. Slagging and fouling were assessed using an empirical sticking efficiency and the ash impaction rate. Both of these efforts were largely based on the characterization of ash behavior as compared to char; however, they did not address the physics behind the tendency of a particle to adhere or rebound. One such model to predict the adhesion of char particles has been proposed by Shmizu et al., in which the gasifier temperature is assumed to be higher than the melting point of the ash.18 This model also assumes that the char will be captured by the slag surface and will rebound from where the char particles have adhered. The probability for the char capture rate is based on the ratio of the surface area covered by the unreacted char to the total surface area. However, the basis for this probability is inconsistent with the experimental work of others who have shown the resistance to wettability between slag and carbon19−22 and presented empirical evidence that the probability of adhesion only increases as a function of carbon conversion.23 In this case of contact angle measurements, wettability is the tendency of one fluid to spread on or adhere to a solid surface in the presence of other immiscible fluids.24 Wang et al.25 have utilized a model developed by Lee et al.10 to 5308
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This work combines the effort of the two previous documented works for the application of a population model30 and measurement of COR values8 by extending it to include a theoretical particle sticking function derived from the COR. By using the two variables, (the plastic loss and damping ratio) a critical velocity is determined for the purpose of partitioning (slag/flyash) so that the two variables can both be implemented in a single expression for the COR to define the fate of particles impacting a surface in a CFD model.
2. THEORY 2.1. Viscoelastic Description of Coal Char Structure. In terms of stiffness and the range of deformation, solid materials can be classified as elastic, elasto-plastic, or viscoelastic. A material that undergoes an elastic deformation with non-time dependent plastic deformation is called elasto-plastic.32 A material that deforms elastically but exhibits time-dependent plastic deformation is visco-plastic. There are four basic mechanical models (linear elastic, linear viscous, Maxwell, and Voigt) that exist to describe the range of deformation in terms of the amount of strain as a result of applied stress. To characterize the structure of coal, a macromolecular model was used. According to this model, the macromolecular structure of coal comprises of two structures.33,34 The first order structure consists of an arrangement of vibrating material bodies which in coal are centers built by aromatic rings containing atoms connected by strong covalent bonds. At temperatures exceeding 573 K, coal exhibits significant irreversible compliance under a constant stress that results from thermolytic degradation.28 Under conditions of thermal degradation, viscoelastic parameters such as elastic moduli and coefficients will no longer be constants but become time and temperature dependent on the state of the macromolecular network. This behavior is further evidenced by the sudden decrease of the velocity of propagation of ultrasonic waves above a certain temperature documented to be 523 K.35 Two viscoelastic models that have been proposed to predict the creep compliance as a result of uniaxial stress are the standard linear solid (SLS) model shown schematically in Figure 2 and the four element (FE) “Burgers”
Figure 3. Schematic of the Four Element “Burger’s” Model.
σm = σD = σs2
(3)
εm = εD + εs2
(4)
Based on the relationship of all components, the strain rate is expressed in eq 5.36 ⎡ d σ (t ) ⎤ E σ (t ) EE d ε(t ) = (E1 + E2)−1⎢ + 2 − 1 2 ε (t ) ⎥ ⎣ dt ⎦ dt η η (5)
For the (FE) Model, the Voigt Element is in series with the spring and the dashpot element. Therefore, the strain components which are the elastic strain εe, the viscoelastic strain εv, viscoplastic strain εp, are summed for the total strain. εtot = εe + εv + εp (6) The applied stress for the strain components are in eqs 7−9. σ = Eeεe
(7)
σ = Ev εv + ηv εv̇
(8)
σ = ηpεṗ
(9)
Rearranging eqs 7−9 and combining with eq 6 the stress strain relationship becomes the following (eq 10): ε=
(10)
The (SLS) model and the (FE) model are considered to be most applicable since there is an agreement between the model and actual time-dependent material for a wide range of materials with low as well as high damping under both free and forced vibration.37 However, the viscoelastic model proposed by Yigit et al. follows the Maxwell model in the manner in which the applied stress relates to the resulting strain.38 Nevertheless, the plasticity of both the elasto-plastic and visco-plastic materials must be taken into consideration when interpreting nanoindentation measurements for the modulus of elasticity.32 Understanding the rheological properties of materials is key to selecting the appropriate wall impact model. 2.2. Viscoelastoplastic (VEP) Model. In regard to the particle wall impact model for viscoelastic or plastic impact, one of the earliest theories was proposed by Tabor.39 In his model, the rebound velocity is related to the impacting velocity through the material constants, κ and β, which is a function of work hardening. The parameter, β is related to the Meyer’s index, n, while the parameter κ is a function of β and the yield
Figure 2. Schematic of the Standard Linear Solid Model.
model shown in Figure 3. In the SLS Model, a Maxwell element is in parallel with a spring element with the subsequent stresses and strain of the overall system. Because the Maxwell element is in parallel while the stress components are in series, in this model, one can write: σtot = σm + σs1 (1) εtot = εm = εs1
⎛ ⎛ E ⎞⎞ σ σ⎜ σ + 1 − exp⎜⎜ − v t ⎟⎟⎟⎟ + t ⎜ Ee Ev ⎝ η η ⎝ v ⎠⎠ p
(2)
For the Maxwell component, the strain components are in series while the stress components are in parallel. 5309
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Figure 4. Schematic of the viscoelastoplastic model and stress−strain curve.
velocity. Thornton and Ying proposed an analytical model for the COR as a function of the normal incident velocity and the yield velocity based on the theory of contact mechanics.39 An alternative equation as a function of the impacting velocity, yield strength, effective modulus, and impacting velocity was derived based on an empirical solution to a finite element model simulation.40 This simulation was based upon an elastic sphere colliding against an elastic-perfectly plastic substrate. Based upon this simulation, the plastic deformation was dominant while deformation due to wave dissipation was negligible.40 Jackson et al. developed an implicit model based on three phases of a particle wall impact; elastic compression phase, elastoplastic compression phase, and restitution phase.41 Whereas Jackson’s model has to be solved numerically, Yigit developed a linearized model by combining the elastic and plastic phases for loading into one phase for loading.38 In the viscoelastoplastic (VEP) model, the damping forces are considered to be negligible since the plastic effects can be neglibly small at low velocities. Instead of using the model by Biryukov and Kandotsev as proposed by Kim and Dunn, the VEP model by Yigit et al. was chosen for the present work to model the plastic effects in order to incorporate the viscoelastic behavior.38 In terms of elastoplastic behavior, there are two phases of loading that exist through the approach in addition to the rebounding phases of a particle impacting a surface: Phase I: Hertzian elastic loading phase F = Khz 3/2
for
0 ≤ z ≤ zy
zy =
4 RE * 3
(14)
F = K y(z − z y ) + Khz y 3/2
for
zy ≤ z ≤ z m
(15)
In this phase of deformation, the particle has been deformed past the yield strength at the deformation of yield, zy prior to deforming at a maximum length denoted as zm. Here, Ky is the linear contact stiffness of the elastic plastic loading phase as described by Yigit et al. for the nonlinear viscoelastoplastic impact model and is a function of zy, the deformation where yielding or damage occurs. This relationship is shown below in eq 16: K y = 1.5Kh z y
(16)
Phase III: Hertzian elastic unloading phase F = Kh(z 3/2 − zm3/2 + zy3/2) + K y(zm − zy)
(17)
In this phase of deformation, the force is unloaded for both the elastic−plastic and the Hertzian elastic loading phases. In the VEP Model, the elastic−plastic loading and the Hertzian loading phases are combined into one loading phase (for deformation). By reducing the three phases of deformation for particle impact into two, the end result is a linearization of the particle impact process for viscoelastic behavior. Based on this linearization, the plastic loss factor (γ), which is based on the linear and Hertzian contact stiffness, can be equated as the coefficient of restitution for plastic impact. Thus,
(11)
(12)
⎛ Ky ⎞ ⎟ epl = γ 2 = 1 − ⎜⎜1 − Kh z m ⎟⎠ ⎝
Here, R is the radius and E* is the effective contact modulus. The effective contact modulus is given by eq 13. 1 − vp2 1 − vs2 1 + = E* Ep Es
E *2
Plastic deformation occurs when the particle deformation rate is higher and no longer proportional to the applied force as illustrated at point 4 of the stress strain curve shown in Figure 4.42 Phase II: Elastic-plastic loading phase
In this phase of deformation, the force present is the Hertzian compressive force, which is dependent upon the Hertzian Stiffness shown in eq 12.
Kh =
0.68Sy2π 2R
(18)
For large velocities, the equation for the viscoelastic model can be derived through the binomial theory where Kh is the Hertzian contact stiffness given by eq 19:38
(13)
⎛ 2 ⎞ ⎛ Ky ⎞ ⎛ Ky ⎞ γ = ⎜ ⎟ ⎜ ⎟ ⎜ 2⎟ ⎝ 3 ⎠ ⎝ Kh ⎠ ⎝ mv0 ⎠
1/4
The parameter z is the deformation of the particle and the parameter zy is the threshold amount of deformation when the particle begins to yield due to plastic deformation. The deformation where yielding or damage occurs, zy, is given as a function of the yield strength Sy in eq 14:
2
(19)
Other variables in the equation include mass, m, and the initial velocity, v0. 5310
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The parameter for the plastic loss factor, γ can be equated with the coefficient of restitution when damping is absent; in fact, it (the damping ratio) is assumed to be absent in order to determine the coefficient of restitution. Because the coefficient of restitution in the Yigit’s model is based upon the point of impact, the normal as well as the tangential component must be calculated based on the impact angle. For large velocities, the coefficient of restitution for the viscoelastoplastic model is derived through binomial theory.38 This is represented as 1/4 ⎛ 2 ⎞ ⎛ Ky ⎞ ⎛ Ky ⎞ ⎜ ⎟ ⎜ ⎟ γ = ⎜ ⎟ ⎝ 3 ⎠ ⎝ Kh ⎠ ⎝ mv02 ⎠
Table 2. Ultimate and Proximate Analysis of Particle Size Distribution According to Specific Gravity (% wt Dry)
carbon hydrogen oxygen nitrogen sulfur ash
2
a
BSG3
BSG4
2.60 g/cm3
86.3 6.1 2.6 1.6 1.0 2.3
77.7 5.7 0.8 1.4 2.04 12.4
32.2 2.2 0.6 0.4 9.4 55.2
6.8 0.3 0.4 0.2 32.1 60.2
Table 3. Ash Analysis of Particle Size Distribution According to Specific Gravity BSG1
BSG1PS0
SG2PS0
BSG3PS0
BSG4PS0
>2.60 g/cm3
2.60−1.60 g/cm3
1.60−1.30 g/cm3
