INTRODUCTION
FluidizedBed technology has been successfully applied for cases of solid fuels as energy sources. The various advantages of fluidized beds, either bubbling, circulating, or entrained over more conventional combustion and gasification techniques are listed elsewhere (^{Basu, 2016}; ^{Kunii and Levenspiel, 1991}; ^{Kunii and Levenspiel, 1997}; ^{Geldart, 1986}; ^{de SouzaSantos, 1987}; ^{de SouzaSantos, 2010}). Fluidized beds have been particularly useful when dealing with highmoisture biomasses, municipal solid residues, highash coals and other low heatingvalue fuels (^{de SouzaSantos, 2010};, ^{Muskala et al., 2011}).
During the years, many models have been developed to allow predictions of the behavior of existing units as well and optimizations of future ones. The list of those is too extensive, but many are described or listed in reviews (^{GomezBarea and Leckner, 2010}; ^{Philippsen et al., 2015}; ^{Alagoz, 2006}; ^{Saraiva et al., 1993}). Nonetheless, no model has been found in the literature including the alternative of liquid fuel feeding into fluidized beds.
The present development
An existing simulation program (www.csfmb.com) has been validated (^{de SouzaSantos, 1987}, ^{1989}, ^{1994a}, ^{1994b}, ^{2007}, ^{2008}, ^{2009}, ^{2010}; ^{Rabi and de SouzaSantos, 2003}, ^{2004}, ^{2008}; ^{Krzywanski et al., 2016}; ^{Englebrecht et al., 2011}) and used in studies involving various possibilities and consuming many fuels, including lowquality ones (^{Muskala et al., 2011}; ^{de SouzaSantos, 1989}, ^{1994a}, ^{1994b}, ^{2007}, ^{2008}, ^{2009}, ^{2010}, ^{2015}; ^{Rabi and de SouzaSantos, 2003}, ^{2004}, ^{2008}; ^{Krzywanski et al., 2016}; ^{Englebrecht et al., 2011}; ^{de SouzaSantos and Chaves, 2012a}, ^{2012b}, ^{2012c}; ^{de SouzaSantos and Ceribeli, 2012}, ^{2013}; ^{de SouzaSantos and Beninca, 2014}; ^{de SouzaSantos and Lima, 2015}; ^{de SouzaSantos et al., 2015}).
In many situations, fluidized beds require inert solids, such as sand or alumina, as temperature regulator and to allow more stable fluidization (^{Basu, 2016}; ^{Kunii and Levenspiel, 1991}; ^{Kunii and Levenspiel, 1997}; ^{Geldart, 1986}; ^{de SouzaSantos, 1987}; ^{de SouzaSantos, 2010}). Ash particles, detached from the feeding solid fuel, can replace part or even the total amount of that inert material.
In addition to solid fuels, liquids or slurries are also an important segment of the spectrum found in industry. Among them there are heavy or viscous oils, whose combustion in pulverized form is cumbersome. Furthermore, several works have explored the application of residual hydrocarbon to enrich low heating value solid fuels to allow their ignition, as well as improve the combustion or gasification processes (^{Breault, 2010}). Amid those, glycerol has shown particular interest since it is a byproduct of biodiesel production, which has increased substantially in many countries (^{Leonetia et al., 2012}; ^{Wei et al., 2011}; ^{Sricharoewnchaikul and Atong, 2012}; ^{Manara and Zabanioutou, 2016}; ^{Czernichowski, 2009}; ^{FossoKankeu et al., 2015}; ^{Maintinguer et al., 2015}; ^{Pagliaro and Rossi, 2010}). For instance, just the Brazilian output should reach 4.1 million m^{3} in 2016 (^{Barros, 2015}). Since glycerol represents around 11% of biodiesel production, its rate of generation would also increase fast in the coming years (^{Dasari et al., 2005}; ^{Schultz et al., 2014}).
An alternative for the utilization of those liquid fuels is their injection into fluidized bed boilers or gasifiers.
BASIC MODELS
The following possibilities can be visualized:

As the liquid fuel enters the fluidized bed, it immediately coats the inert solid particles. Then, the fuel goes through pyrolysis or devolatilization and the resulting coke layers remain on the inert particles. Then, those layers are attacked by gases. This model is called here CIP (Coated Inert Particle) and the situation is illustrated by Figure 1. The Appendix describes how it can be applied for each situation where the supporting solid has a particular basic shape, i.e., planar or flat, cylindrical, or spherical.

As the liquid fuel is sprayed into the bed, the drops go through pyrolysis before having the opportunity of meeting any inert particle. A hollow coke shell forms from each fuel drop after the volatiles escape from its interior through small holes in the respective shell. Then, the remaining shell of that hollow sphere of coke reacts with the gases. The average size among the spheres is assumed to be the same as the average drop size in the original sprayed fuel. This model is called here CSP (Coke Shell Particle) and the situation is illustrated by Figure 2.
Before describing the mathematical models, it is important to stress that the present work assumes isothermal conditions in the layer of fuel and inert particles. The local temperature of particles, as well as temperature, pressure, and composition of the surrounding gas, should be given by a comprehensive model for fluidizedbed equipment. The governing equations that constitute the mathematical model of fluidizedbed reactors can be found elsewhere (^{de SouzaSantos, 2010}). The only objective here is to present the equations proposed for the reaction rates between gases and the liquid fuel injected into the reactor with fluidized inert solid particles.
The treatment for a single heterogeneous reaction i, in which a chemical component j of the surrounding gas is consumed or produced, has been presented by ^{de SouzaSantos (2010)}. In any case, the fundamental equation for conservation of species leads to
The Laplacian operator is generalized as:
the coefficient p takes the following possible values: 0 for plane geometry, 1 for cylindrical, and 2 for spherical. Here:
The Thiele coefficient is given by:
On the above, it has been assumed that all reaction rates could be written as:
which is valid for most of the combustion and gasification reactions. For those, the reaction order n varies between 0 and 2. However, the main cases of carbonoxygen and carbonwater reactionswhich control most of the combustion and gasification processesfollow a first order behavior. A list of several reactions found in those processes as well their kinetic coefficients are presented elsewhere (^{de SouzaSantos, 2010}).
The basic equations to allow computation of the CIP and CSP models are present below.
Coated Inert Particle (CIP) Model
Figure 1 illustrates the CIP model. The inert particle, coated by a coke layer, may present three basic shapes: plane (in the case of chips), cylindrical (in the case of pellets or fibers), and spherical or almost spherical. As detailed in the Appendix, the solutions can be condensed into a single formula for the rate of consumption or production of component j by reaction i as
The sum that appears in the denominator represents the three resistances in series for mass transfers of chemical species j that depend on the inert particle original shape. Table 1 summarizes the various possibilities. The first parameter U_{1} represents the resistance to the mass transfer at the gas boundary layer, U_{2} the resistance at the ash layer, and the U_{3} the combined resistances of reacting gas mass transfer trough the reactive coke layer and its respective chemical reaction with the coke material. The various resistances for the basic geometries are summarized at Table 1. An example of deduction is shown in the Appendix.
Shape  Resistance  

U1  U2  U3 (*)  
Plate 



Cylinder 



Sphere 



^{*}Here:
Coke Shell Particle (CSP) Model
Figure 2 illustrates the situation for the CSP model. Here just the spherical shape is possible.
The first aspect to face is to estimate the radius of the internal space filled with gases.
The original mass of the droplet is given by
On the other hand, the mass of the coke before the formation of ash over it, is provided by
However,
Therefore,
The deduction for the resistances are presented in the Appendix. These resistances can be employed in Eq. (7) to compute the rate of consumption or production of gas component j by reaction i.
DISCUSSION
According to Equation 58 of the Appendix, in the case of CSP model, if no mass transfer at the internal cavity surface takes place (i.e., N_{2} = 0), the parameter w equals the variable a and the resistance forms reproduce the CIP case for spherical geometry.
Another interesting aspect of that case can be appreciated by noticing that the parameter g_{j} (Eq. 45 in the Appendix) is a function of the average concentration inside the cavity. At the beginning of cavity formation, the concentration would be equal to those obtained from the pyrolysis. After a while, the composition tends to equalize the concentration of the equilibrium atmosphere in which the particle is immersed. Therefore, for long exposure times, g_{j} would tend to zero. This situation reproduces the case where N_{2} = 0.
CONCLUSIONS
The possibility of using the fluidizedbed technique for combustion and gasification of liquid fuels is known. In those cases, the liquid is sprayed over or injected into the bed where inert solid is fluidized.
The adaptation of existing simulation models to such situations required methods to compute the kinetics for heterogeneous reactions. To allow that, two possible approaches are proposed here: the CIP (Coated inert Particle) Model and the CSP (CokeShell Particle) Model. The work introduces the analytical solutions for those models, as well as for cases of different basic shapes of inert particles present in the bed.
The present proposals are theoretical. Future publications might confirm the deductions made here and which model would better represent the process in each situation.