A thermal transport phenomena model for the decomposition of a CO2 sorbent particle under concentrated solar irradiation is used to evaluate four approximate engineering models for the surface radiative properties of the particle. The radiative property models are formulated by considering the solid-phase to be opaque or semi-transparent and the size of the surface features to be either smaller or larger than the incident irradiation wavelength. Time to complete decomposition of the particle and maximum surface temperature in simulations employing the four models differ by approximately 2% and 0.5%, respectively.
© 2015 Optical Society of America
Particulates undergoing chemical reactions are found in many industrial and scientific applications including thermal decomposition of natural carbonates, ignition of metallic or other solid particles, metallurgic refining, and atmospheric and aerosol sciences. Chemical sorbent particles are an important subset of reacting particulates. One such novel technology is the solar thermochemical capture of carbon dioxide (CO2) via the two-step calcium oxide looping cycle [1, 2]. The high-temperature capture of CO2 via solar calcium oxide looping promises high efficiencies and capture rates . Concentrated CO2 is a commodity with applications in the pharmaceutical, medical, environmental, food and beverage, petrochemical, oil recovery, metallurgy, and manufacturing industries [4, 5], as well as the production of synthetic transportation fuels . Physical characteristics of the surface of a reacting particle can be complex, particularly with regard to the interaction of particles with high-flux solar irradiation at the particle surface. Treatment of radiative heat transfer in particulate media and the determination of surface radiative properties vary widely with application and are important considerations in many engineering applications .
In two-step calcium oxide looping for CO2 capture, calcium carbonate (CaCO3) is thermochemically decomposed into concentrated CO2 and calcium oxide (CaO) sorbent in the endothermic, solar-driven calcination reaction,8]. The thermal transport model was found to be more sensitive to heat transfer related parameters than mass transfer related parameters, but the model sensitivity to surface radiative properties was not investigated. The surface of a sorbent particle for calcium oxide looping is porous, and the porosity ϕ changes with time due to changing composition. The composition of the solid phase is a mixture of dense CaCO3 and porous CaO, and changes due to chemical reaction.
In this work, four different phenomenological models for the surface radiative properties are presented and compared using the thermal transport model. Understanding the sensitivity of the thermal transport model to surface radiative properties is important to understanding the interaction between sorbent particles and high-flux solar radiation, which in turn, provides input to the design of the reacting medium and a solar reactor for calcium oxide looping CO2 capture.
The models for surface radiative properties are formulated by considering different solid phase opacity and feature size relative to wavelength. Development of the surface radiative property models is motivated by the suitability and potential of using concentrated solar energy as the process heat for driving heterogeneous thermochemical processes featuring particulate media.
2. Problem statement
A semi-transparent, porous, and non-uniform particle is subjected to high-flux solar irradiation in an idealized environment (Fig. 1). Within the particle, local temperature, solid and fluid phase species concentrations, porosity, and fluid conditions vary with location and in time. The solid phase is a mixture of CaCO3 and CaO. The fluid phase in the pore space is a mixture of air and CO2. The local, intrinsic chemical reaction rate depends on temperature, solid phase composition, CO2 concentration, and particle morphology. The solid phase composition changes due to chemistry, while the fluid phase composition is affected by chemistry, species diffusion due to concentration gradients, and bulk advection due to pressure gradients. Convective mass transfer occurs between the fluid in the particle pore spaces and the ambient fluid surrounding the particle. Heat transfer from the surface to locations within the particle undergoing chemical reaction is captured in the heat transfer model. Conduction is modeled in both the solid and fluid phases. Surface and volumetric radiative transfer are considered. The solid phase is radiatively participating while the fluid phase is radiatively non-participating. Convective heat transfer occurs between both the solid and fluid phases and the ambient surrounding fluid at the particle surface.
The particle is initially 3% porous with solid phase composition 100% CaCO3 and fluid phase composition at equilibrium with the ambient. The particle is subjected to high-flux diffuse solar irradiation uniformly distributed over the surface to approximate the conditions of a single, rapidly rotating particle exposed directly to concentrated solar radiation. Uniform versus nonuniform irradiation was investigated previously . Radiative interactions with other particles in a reactor for CO2 capture is neglected for simplicity. The focus of the present study is to elucidate the interactions between the particle irradiation, spectral surface properties, and intra-particle transport phenomena. Incident irradiation initially goes towards particle heating until the calcination reaction temperature is reached. Calcination then proceeds until the particle is completely decomposed, and particle heating resumes.
3. Thermal transport model
A benchmark numerical model of thermal transport phenomena has been developed and applied to the calcination of a sorbent particle . The model is based on volume-averaged conservation equations and consists of five coupled equations: four conservation of mass equations for the four species of the system (CaCO3, CaO, CO2, and air) and one conservation of energy equation. Constraint equations include a simplified conservation of momentum equation and the ideal gas equation of state.
In order to properly resolve time-dependent composition and temperature profiles within the particle during cycling, mass, momentum, and energy equations are solved. Single global conversion expressions have been developed for the calcination reaction [10, 11]. However, they have limited applicability as they are limited to specific experimental conditions and do not account for mass diffusion limitations and internal temperature gradients. Governing equations for the local solid phase composition, fluid phase composition, and temperature are included in the numerical model, because these quantities influence the local intrinsic kinetics of the calcination reaction [12, 13].
Sorbent particles considered for calcium oxide looping CO2 capture range in size from the micrometers  to millimeters . The complete calcination numerical model and the surface radiative property models have been formulated for large sorbent particles.
Conservation of mass—Solid phase
The volume-averaged conservation of mass equations for CaCO3 and CaO read, respectively:
Conservation of mass—Fluid phase
For the fluid phase, the volume-averaged conservation of mass equations for CO2 and air read, respectively:8].
Conservation of momentum
Darcy’s law simplification of the conservation of momentum equation is used to determine the fluid velocity in the pore space. Darcy’s law simplification is used for its simplicity and appropriateness, because fluid velocities are low enough to maintain laminar flow at all times and locations within the particle.
Conservation of energy
The solid and fluid phases are assumed to be at local thermal equilibrium, yielding one conservation of energy equation for the system,Eq. (8), are given in .
Boundary and initial conditions
Two boundary conditions—one at the particle surface and one at the particle center—are required for each of the five conservation equations. The solid phase species do not move in space so the mass fluxes at the center and surface are zero. The mass flux at the center of the particle for all species is zero due to symmetry. For the fluid species, the boundary condition at the surface is a mixed condition defined by advection and diffusion at the particle surface and convective mass transfer away from the particle surface. Thus,
The initial conditions are:Eq. (8) reads:
The governing equations are solved numerically in space and time by employing the 1-D finite volume method and the explicit Euler time-integration scheme. Material properties used for the system and numerical solution methods are given in . The particle has a radius rp = 2.5mm. The magnitude of the incident flux is q″solar = 1MW m−2. The baseline simulation parameter set is shown in Table 1. The model predicts transient heat and mass fluxes, temperature distributions, and extent of the calcination reaction within the particle. The extent of the calcination reaction, referred to as the reaction extent, describes the solid phase composition and is defined as X = 1 − NCaCO3/N0,CaCO3, where NCaCO3 and N0,CaCO3 are the local and initial amount of substance of CaCO3, respectively.
4. Surface radiative property models
Four models for surface radiative properties are formulated by considering (a) the surface of the solid phase to be either non-absorbing or absorbing and (b) the size of the surface features to be either smaller or larger than the incident wave length. If the surface features are predominantly smaller than the incident wavelength, the effective medium (EM) assumption is used. If the surface features are predominantly larger than the incident wavelength, the optically discrete medium (ODM) assumption is used. The models are outlined in Table 2.
The thermal transport model requires the effective absorptance αeff and emittance εeff of the particle surface in Eq. (16) to describe radiative heat transfer at the particle surface. These effective properties are a mixture of the individual properties of the two solid phase species and the fluid in the pore space. Refractive indices of the constituents of the particle surface are used to evaluate reflectance, which is then used to evaluate absorptance and emittance. The computational approaches for evaluating effective absorptance and emittance for models assuming EM and ODM are shown in Fig. 2. The computational approach for models assuming the solid phase to be opaque or transmitting is the same, therefore Fig. 2(a) applies to Models 1 and 3, assuming EM, and Fig. 2(b) applies to Models 2 and 4, assuming ODM. The fluid in the pore space is treated as perfectly transparent and non-refracting in all models.
In Models 1 and 3 shown in Fig. 2(a), refractive indices for each surface constituent are converted to dielectric functions, then combined using the local reaction extent and the porosity of the surface to yield an effective dielectric function. The effective dielectric function is converted to an effective refractive index, meff = neff + ikeff = f(ε′eff, ε″eff). The effective refractive index is used with the Fresnel equations to evaluate the effective spectral, directional–hemispherical reflectance . The dielectric function and refractive index are related by :Equations (22) and (23) are used for species and effective quantities. After evaluating the surface constituent dielectric functions, the effective dielectric function is:
In Models 2 and 4 shown in Fig. 2(b), refractive indices for CaCO3 and CaO are used directly in the Fresnel equations to evaluate the spectral, directional–hemispherical reflectance for CaCO3 and CaO separately. and are then combined using the local reaction extent and porosity of the surface to yield ,
In all models, is averaged over all directions to evaluate the spectral, hemispherical reflectance ρλ,eff,
The Fresnel equations are :
The real parts of the refractive index n for CaCO3 and CaO are taken from [18, 19] and , respectively. The imaginary parts of the refractive index k for CaCO3 and CaO are taken from [18, 20] and [19, 21], respectively. Refractive indices versus wavelength for CaCO3 and CaO are shown in Fig. 3. Mathematical expressions are used to approximate refractive index experimental data resulting in the discontinuity of curves shown in Fig. 3.
Effective absorptance of the particle surface varies with surface composition, while effective emittance may vary with both surface composition and temperature. Figure 4 shows αeff and εeff versus the surface reaction extent. For the non-absorbing models (Models 1 and 2), αeff = εeff because the real components of the refractive index for all species do not vary with wavelength and are therefore not a function of temperature. Model 1 predicts the highest values for αeff and εeff. The change in αeff and εeff with reaction extent is 6% and 4% for Models 1 and 2, respectively. For the absorbing models (Models 3 and 4), surface temperature and composition both influence εeff, but composition has the stronger influence. Model 4 predicts the lowest values for αeff and εeff. The change in αeff with reaction extent is 5% and 2% for Models 3 and 4, respectively. The change in εeff with reaction extent for T = 800K is 6% and 3% for Models 3 and 4, respectively.
Radiative property models are compared based on time to complete conversion of the particle when the overall reaction extent Xoverall = 1, time to complete conversion of the particle surface when the local reaction extent Xsurface = 1, and the maximum achieved surface temperature. Overall reaction extent and the local reaction extent at the particle surface are shown versus time in Fig. 5. The particle in the simulation employing Model 1 reaches complete conversion at t = 32.1s. Models 2 and 3 reach complete conversion at t = 32.4s and t = 32.3s, respectively. Model 4 reaches complete conversion in the longest amount of time, t = 32.6s. Time to complete particle conversion increases with decreasing surface radiative properties. Similarly, the particle surfaces in the simulations employing Model 1, 2, and 3 reach complete conversion at t = 3.4s, and Model 4 reaches complete conversion at t = 3.5s.
Surface temperature versus time is shown in Fig. 6. The maximum temperature of the particle occurs at the surface at end of the 40s simulation. The maximum surface temperature achieved using Models 1 and 2 is 1975K and 1972K, respectively. The maximum surface temperatures achieve using Models 3 and 4 are 1971K and 1965K, respectively.
Simulations employing EM models exhibit higher predicted effective surface radiative properties, but non-absorbing models yield the highest maximum surface temperatures. Model 4 yields the lowest effective properties resulting in the less radiation absorption, longest time to complete conversion, and lowest maximum surface temperature.
Four phenomenological models for the surface radiative properties of a reacting sorbent particle are applied to a 2.5mm radius particle of CaCO3 subjected to 1MW m−2 solar irradiation. Comparing time to complete particle conversion and maximum surface temperature from simulations employing the four models, the largest change in time to complete conversion between models is less than 2%, and the largest change in maximum achieved surface temperature between models is 0.5%.
Model 1 predicts the highest effective surface radiative properties, while Model 4 predicts the lowest. EM models predict higher effective properties than ODM models, and non-absorbing models predict higher effective properties than absorbning models. Models 1 and 4 show the upper and lower bounds for surface radiative properties for the system considered. Pore size diameters can be in the range ∼0.01μm to 0.5μm, and CaO grain diameters are ∼1μm, thus depending on the wavelength band, the sorbent particle surface will exhibit both optically discrete and effective medium behavior. Future work will include development of a spectral mixed surface property model. Refractive index data, SEM imaging, and other surface evaluating techniques should be used to support selection of the appropriate surface property model.
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