NONCA TAL YTIC HlFTEROGENEOUS SOLID FLUID REACTION MOD€LS C. Y. WEN
The successful design of a reactor depends greatly on a knowledge of reliable rate data. It is important to make a careful study of interactions and to isolate physical effects from purely chemical processes
oncatalytic heterogeneous reactions include solid-
N fluid, liquid-liquid, and liquid-gas systems. Here, we consider the heterogeneous noncatalytic reaction
systems involving only solid-fluid reactions. The reaction systems concerned are of considerable industrial importance and are readily found in chemical and metallurgical industries. Examples are the reduction of iron oxide or the oxidation of iron by steam, the roasting and smelting of ores, the combustion of solid fuels and solid propellant, the gasification of coal or oil shale, the regeneration of catalysts, just to mention a few. Such heterogeneous reaction systems occurring in various types of reactors are usually extremely complex and analyses often fail to reveal the true mechanism because of the great number of variables involved. The successful design of a reactor depends greatly on a knowledge of reliable rate data. The rates of heterogeneous reactions vary manyfold depending on conditions under which the experiment is performed. Physical effects such as diffusion and heat transfer can result in an erroneous rate expression if they are not properly accounted for. The order of reaction, the activation energy, and the selectivity determined may be so misleading that, if used in scale-up, the result may be a disastrous plant operation. I t is therefore important to make a careful study of the interaction and to isolate physical effects from purely chemical processes. I n spite of the industrial importance of solid-fluid reactions, there have been relatively few studies available on chemical kinetics and transfer rates of mass and energy 34
INDUSTRIAL A N D ENGINEERING CHEMISTRY
in heterogeneous noncatalytic systems. Partly, this is due to the intricate relationship among the rates of chemical reactions and the rates of mass and energy transfer. Even for an isothermal reaction system, the over-all reaction rates are influenced not only by the rate of chemical reactions occurring in or at the surface of a solid, and by the mass transfer rates of fluids through the solid as well as across the fluid-film surrounding the solid, but also by factors such as solid reactivity of crystallite orientation, crystallite size, surface characteristics, impurities, etc. However, it seems evident that the solid-fluid reaction taking place on a surface must depend on its chemical adsorption of the fluid reactants. Many features of heterogeneous kinetics of solid-gas reactions have been explained (3, 5, 77) by this hypothesis and the application of Langmuir’s adsorption isotherm. A surface reaction of solid-fluid systems can be considered to consist of the following steps: (1) diffusion of the fluid reactants across the fluid film surrounding the solid, (2) diffusion of the fluid reactants through porous solid layer, (3) adsorption of the fluid reactants a t the solid reactant surface, (4) chemical reaction with the solid surface, (5) desorption of the fluid products from solid reaction surface, and (6) diffusion of the product away from the reaction surface through the porous solid media and through the fluid film surrounding the solid. Since these steps take place consecutively, if any one of the above steps is much slower than all others, that step becomes the rate-determining. However, sin-
CAD CAS CA CAC
Solid Reactant
Figure 1. Schematic diagram of concentrationprofile for the unreacted-core shrinking model
CAD CAS
Solid Reactant Solid Reactant
---.-_Fisure 2. Schematic diagram of concentration profile in the second sage
Figure 3. Schematic diagram of concentration profile in the first stage
gle-step rate-determining processes are the limiting cases; the majority of solid-fluid reactions are influenced simultaneously by more than one step. Thus, a rigorous treatment seems unattainable even for the solid of the simplest geometry. Besides, a great number of difficult problems exist in practical systems, such as the changing size and shape of the solid during the reaction and formation of a product around the solid reactant which may crack or ablate. I n addition, the complex velocity profile of the surrounding fluid makes the problems of mass and energy transfers to the solid reactant more difficult to analyze. Therefore, in the following, we shall first analyze some of the simplest cases of noncatalytic reactions between solids and fluids. Later, we shall extend these considerations to formulate a more general model which can be used in a wide variety of situations. For this purpose, it is advantageous to classify phenomenologically the various noncatalytic, solid-fluid reactions into a few typical cases. \Ye can then propose various conceptual models to represent these typical cases for analysis and design of reactors.
hydrates, and removal of crystalline water from crystalline compounds, etc.
- -+
CaCOa(s)
CaS04.2HzO(s)
Solid reactant Solid reactants
(A)
fluid products --t
-
fluid and solid products
(B)
fluid products
(C)
Fluid and solid reactants -+ solid products
(D)
Fluid and solid reactants
Fluid and solid reactants
-
W@(s)
4C(s)
Na2S04(s)
(E)
Some examples of the reaction type A are pyrolysis of carbonaceous materials, combustion of double-base propellants, and thermal decomposition of some organic or inorganic compound, especially explosives, such as, T\;H,NO~(S)---t KzO(g)
+ 2 HzO(g)
Under certain conditions, the solid reactant decomposes gradually from the outer surface to its center while giving off fluid products. At temperatures much higher than the decomposition temperature, the reaction may occur a t the surface as well as inside the solid. Many examples for the reaction type B can be cited from the pyrolysis and thermal decomposition of organic and inorganic solid materials. Some of the typical reactions are pyrolysis of carbonaceous materials, calcination of carbonates, dehydration of hydroxides and 36
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
+ 3/~H20(g)
HzO(g)
SaaS(s)
-
Cao(s)
3C(s) -+ CaCz(s)
4CO(g)
co(g)
+
CaOCL(s) CaCL(s) 1/202(g) Reactions of type C are numerous and of great industrial importance. Combustions and gasifications of carbonaceous materials and oxidation of other solid compounds are typical examples of this class of reactions. They are : Gasification of carbonaceous matter --L
2HzO(g) +
+ 2Hz(g)
C(S)
+
--
+ Hz(g) COz(g) + 2Hz(g) CO(g)
CH4(g)
Combustion of carbonaceous matter C(S) C(s)
+ 02(g)
+ COz(9)
COz(g) 2CO(g)
Oxidation of sulfur
+ Ozk)
S(S)
SOz(g)
Production of carbon disulfide
C(s)
+ Sdg)
CSz(g)
Halogenation of metal and metal oxide 2A1203(s) A1203(S)
fluid and solid products
CaS04.1/2HzO(s)
+ + + + +
h@(OWa(s)
C(S)
Although there are a number of diI-erse solid-fluid reaction systems, it is possible to group them pheiiomenologically in two ways: One grouping is based on the phase combination of the reactant and the product, and the other grouping is based on the manner by which a reaction progresses. Classification based on the phases in which various species appear. Noncatalytic, fluid-solid reactions may be represented by one of the following schemes.
+
CaO(s)
C(S) $. H D ( g ) Classification of Solid-Fluid Reactions
-
Carbonaceous materials(s) carbon, ash(s) hydrocarbons (g and l), C O z , CO, and H,O(g)
+ 6C1z(g)
4AlC13(g)
+
+ 30z(g)
+ 6HCl(g) + 2,41C13(g) + 3HzO(g)
+ Fe(s) + Zr-ICl(g> FeO(s) + 2HC1(g) ~ C U ( S ) 2HCl(g)
--
+ Hz(g) FeCMg) + H&) FeCMg) + HzO(g) 2CuCl(g)
-
Formation of sodium cyanide from sodium amide NaNHz(1)
+ C(s)
+ Hz(g)
SaCN(1)
Other examples of type C reactions are those between solids (metals) and ions in aqueous solutions Na(meta1)
+ H(ion)
iXazSO,(aq)
+
+ S(s)
Na(ion) 3
+ l/zHz(g)
NazSJ&(aq)
Reaction in ion-exchange resins can also be classified under this group. A typical example of type D reaction is nitrogenation of calcium carbide to produce cyanamide CaCz(s)
+ Nz(g)
-
CaCNZ(s)
+ C(s)
Other examples are the rusting reaction of metals, such as 2Fe(s)
+ 02(g)
-
2FeO(s)
and the chemisorptions of gas or liquid on solid adsorbents. Reactions of type E are a more general form of heterogeneous noncatalytic fluid-solid reactions in which reactant and product components exist in both fluid and solid phases. Calcination of sulfides to make oxides, reductions of metal oxides, and steam-iron process to produce hydrogen are the typical examples of this type of reaction : Calcination
+ 302(g) 4FeSz(s) + 1 1 0 d g ) 2ZnS(s)
-+
+
+ 2SOdg) 2Fez03(s) + 8SOdg) 2ZnO(s)
Reduction of metal oxides
+ 3CO(g) FezOs(s) + 3Hz(g) FezOo(s)
+
2Fe(s) f 3COdg)
+
2Fe(s)
4- 3HzO(g)
Combustion of carbonaceous materials,
+
-+
Carbonaceous materials(s) Os(g) C02, CO, and HzO(g)
ashes(s)
Classification according to the manner by which the reaction progresses. Heterogeneous noncatalytic fluidsolid reaction may be classified by the manner by which the chemical reaction occurs. This classification is therefore dependent on the conditions of the systems, such as the internal structure of the solid, the relative velocities of chemical reactions, and the diffusion of reactants and products as well as the geometry of the solid. I n the following, four different cases are considered. (1) HETEROGENEOUS REACTIONS.When the porosity of the unreacted solid is very small so that the solid is practically impervious to the fluid reactants, the reactions will occur a t the surface of the solid or a t the interface between unreacted solid and the porous product layer. Also, when the chemical reaction rate is very rapid and the diffusion is sufficiently slow, the zone of reaction is narrowly confined to the interface between the unreacted solid reactant and the product. Such a reaction may be considered a heterogeneous surface reaction. We shall employ the unreacted-core shrinking model as shown by Yagi and Kunii (8, 77) to analyze the reaction of this type. (2) HOMOGENEOUS REACTIONS.I n many cases, we can assume that the solid contains enough voidage to pass freely the fluid reactant and the fluid product (that is, the AUTHOR C. Y. W e n is Professor of Chemical Engineering at the West Virginia University, Morgantown, W. Va. Prof. W e n coauthored the absorption annual review several years prior to 7962. The author wishes to acknowledge Dr. E, Sada and Mr. S. C. Wang for their assistance in the preparation of this paper. Part of the work appearing in Afipendin. B was sponsored by the Institute of Gas Technology, Chicago, Ill.
diffusivity of each fluid component through solid is large), and that the solid reactants are distributed homogeneously throughout the solid phase. Then it may be reasonable to consider that the reactions between fluid and solid components are occurring homogeneously throughout the solid phase. For such cases, we shall employ the homogeneous model to analyze the reactions between solid and fluid. ACCOMPANYING PHASECHANGES OF (3) REACTIONS SOLIDCOMPONENTS OR EVOLUTION OF VOLATILES.I n some cases, prior to the chemical reactions, the phase change of the solid component takes place because of poor heat conduction in a n exothermic reaction system. The solid reactants simply melt or sublime before they can be brought into contact with Auid reactants. T h e phase change may proceed with pyrolysis or devolatilization which often associates with gasification and combustion of solid fossil fuels. I n such cases, a homogeneous vapor-phase reaction takes place either around the outside of the solid, or in the solid product layer formed around the unreacted solid. CASESBETWEEN CASES1, 2, AND (4) INTERMEDIATE 3. I n practical cases, the distribution of the solid reactant in the solid phase cannot be considered homogeneous in the molecular scale. However, solid reactants may be considered as a n ensemble of small lumps of reactant distributed throughout the solid phase. T h e reaction rates between each small lump of solid reactants and the fluid reactants that diffuse into the solid may be described by one of the cases mentioned above. T h e over-all reaction rate depends on the distribution of the small lumps of solid reactant in the solid, the structure of solid, the intrinsic reaction velocities, and the transport properties of fluid reactants in the solid. I n analyzing the noncatalytic heterogeneous fluidsolid reaction, we must first select a conceptual model which will give a reasonable illustration of the phenomenological results. We then set u p a group of equations based on the selected model to obtain a quantitative relationship between the various variables in the system-e.g., the intrinsic reaction rates, the mass transfer rates, etc. For example, if the heterogeneous reaction model (unreacted-core shrinking model) fits the system, from material balance of each component a set of partial differential equations can be obtained with moving boundary conditions describing the consumption of the solid reactant by the chemical reaction. O n the other hand, if the homogeneous reaction model is chosen for the system, one obtains a set of partial differential equations from the material balances of each reacting component in the solid phase. These equations contain the terms relating the mole change during the process. I n the following sections, examples of the analysis using these conceptual mo~delsare presented. We shall also present a general model which is applicable under a wide variety of conditions. I t will be shown that the unreacted-core shrinking model and the homogeneous model are the special cases of the general model. I n VOL. 6 0
NO.
9 SEPTEMBER 1968
37
addition, the concept of the effectiveness factor developed in catalytic reaction theory will be extended to solidfluid reactions based on these models. The effect of the diffusion, the instabilities of solid-gas reactions, and the transition of the rate-controlling regions will be discussed based on the effectiveness factor. Order of Solid-Fluid Reactions
In the following sections, we consider the simplest case of fluid-solid reaction system--i.e., a n isothermal equimolal noncatalytic reaction of the type : aA (fluid)
+ S(so1id) + fluid and/or solid products
(1)
When the rates of diffusion through fluid-film and porous solid are both very fast, the over-all rate of a solid-fluid reaction is solely controlled by the inherent chemical reactivity of the solid reactant. Based on the Langmuir adsorption isotherm, it is possible qualitatively to describe the mechanism of uncatalyzed heterogeneous solid-fluid reactions under this condition. However, depending on the mechanism, the resultant rate equation involves more than two arbitrary constants, some times as many as seven (5). I n selecting these constants for each mechanism, the curve representing the rate equation of the favored mechanism which best fits the experimental data is chosen. Because of the unavoidable intrinsic scatter of the experimental data, it is rather clear that in many instances little meaning can be attributed to the magnitude of the adsorption equilibrium constants, the frequency factor, the apparent activation energy, etc., obtained by fitting the data to such multiconstant equations. Often, the difference in fit may be so slight that it is very difficult to determine whether it is simply due to experimental error or truly due to the difference in mechanism. Furthermore, an alternative mechanism may fit the data equally well, necessitating an additional extensive experimental search for the correct mechanism. Although a correct niechanism will allow extrapolation to conditions not actually investigated, in view of the difficulty in obtaining a correct mechanism, there is no reason why simple rate equations which fit the data satisfactorily should not be used provided no extrapolation beyond the range investigated is allowed. For design purposes, the nth-order rate equation can fit the data satisfactorily when the surface phenomenon controls the rate of a solid-fluid reaction. Based on adsorption isotherms, the order of solid-gas reactions can be shown to vary from zero to two depending on whether the gas reactant is strongly adsorbed or weakly adsorbed. Experimental studies also indicate many solid-gas reactions to have the similar range of the order of reaction, 38
INDUSTRIAL A N D ENGINEERING CHEMISTRY
depending on conditions such as reactant reactivity, pressure, temperature, etc. (3, 7 7). Thus, the rate of reaction for fluid component A , rA, and for solid reactant S, r S , can be represented as TA
=
ars = -aksCsmCAn or -akLCSmCAn
where, the stoichiometric coefficient, a, is the number of moles of fluid component A that will react with 1 mole of solid reactant S, and m and n are the order of reaction with respect to solid reactant S, and fluid reactant A , respectively. k,, k , are the reaction rate constants based on the reacting surface area and the reacting solid volume, respectively. Depending on the particular case under consideration, the reaction may be regarded as surface reaction or volume reaction. Consequently the unit of or r S depends on the rate expression, namely, if k , is used the rate is moles per unit time per unit surface area whereas if k , is used the rate is moles per unit time per unit volume. The reverse reaction is not considered here. Unlike catalytic reaction on solid surface, most reactions with solid surface cannot be truly reversible, since the phenomena of solid deposition in the reverse reaction do not necessarily provide the solid product with a structure that is exactly the same as that of the initial solid reactant in the forward reaction. This is seen in a number of solid-gas reactions which have been known to exceed their normal thermodynamic equilibrium (70, 73-75). However, it is possible to provide an additional term to account for the equilibrium hindrance. Although this is particularly convenient for the first-order reactions, it may become very complicated for other order of reactions and therefore will not be dealt with here. I n the following sections, we first examine the various conceptual models and compare these models in light of the rate-controlling factors of the reaction. IYe then attempt to show how a suitable model may be selected for a specific actual reaction system. The cases considered here include types of reaction discussed as C, D , and E. The model for the types of reactions under A and B which are much simpler, can be formulated in a similar manner. Heterogeneous Reaction Model (Unreacted-CoreShrinking Model)
Let us consider a case in which a fluid reacts with a solid to produce a fluid product plus a porous solid product layer or ash layer according to Equation 1. For convenience, we may call the porous inert solid product layer as the ash layer. The unreacted solid is impervious to fluid A because it is densely packed. O n the other hand, the solid product layer is quite porous SO
that the reactant fluid A can diffuse in and product fluid can diffuse out. We consider a spherical particle having an initial radius R being reacted by A . At first, the reaction takes place a t the outside surface of the particle, but as the reaction proceeds, the surface of the reaction will move into the interior of the solid leaving behind inert ashes. During the process, the reaction surface moves inward forming an unreacted core which shrinks with time, but the external radius of the particle still remains the same, R, assuming of course no deformation of the ash layer that has been formed. I n order for the fluid A to reach the surface of the unreacted core, it must move through various layers of resistances in series; the fluid film around the surface of the particle, the porous ash layer, and then the reaction surface at the core. Figure 1 illustrates the resistances in series as well as the concentration profiles within the particle. Referring to Figure 1, we shall designate the symbols appearing in the following discussion as: CAo,CAS,and C,, are the concentrations of fluid reactant A in the bulk of fluid stream, a t the surface of the particle, and a t the surface of the core, respectively. R , re, and r are: radius of the particle, radius of the unreacted core, and radius a t any point from the center of the particle, respectively. A fundamental equation relating the material balance of component A can be written as
The boundary conditions for a spherical particle are : a t the solid particle surface, R
to a first-order reaction with respect to fluid reactant A and a zeroth-order reaction with respect to solid reactant
s. The solution of Equation 2 with the boundary conditions given by Equation 3 yields the concentration profile of A as a function of particle radius, r , and time, t. However, an analytical solution of these equations can not be obtained except for a few special cases. I t is = 0 and obtain therefore customary to let an approximate solution. This technique is commonly known as the pseudo-steady-state approximation which greatly reduces the mathematical difficulties. We shall apply this technique to solve the unreacted-core shrinking model. The accuracy of the pseudo-steady-state approximation is examined in Appendix A. As is shown, the pseudo-steady-state solution is a good approximation for most of the solid-gas reaction systems except for systems with extremely high pressure and very low solid reactant concentration. However, for solid-liquid systems unless the liquid reactant concentration is very low, the error due to this technique may become excessive. Therefore, we must apply this technique carefully to only those cases where the approximation is reasonably accurate. An example showing an application of the pseudo-steady-state analysis on a solid-gas reaction system is presented in Appendix B. Pseudo-steady-state solution of unreacted-coreshrinking spherical particle with constant radius, A. Based on the unreacted-core shrinking model and the pseudo-steady-state assumption, the material balance equation as given in Equation 2 becomes,
, 0
De,
(z+ - - p"> r
dr
R>r>r,
(4)
with the boundary conditions given in Equation 3, we can obtain the solution readily for the concentration profile of A as,
at the moving interface, r,, '
(3)
(5)
and
J
where kmA is the mass transfer coefficient of component A from the bulk of the fluid phase to the solid surface across the film, and Cso is the concentration of solid reactant in the unreacted core which is assumed to be constant for uniform particles. B is the voidage of the porous layer of the particle. The initial condition is rc = R a t t = 0. The reaction considered is equivalent
The time required for the particle to reduce the unreacted core from R to rc is given by
or in dimensionless form, VOL. 6 0
NO. 9 S E P T E M B E R 1 9 6 8
39
~.. ..-
%(J-
3 NSia
- 1) (1 - $cy)
(7)
or
where
Chemical reaction controlling
or
J
If N A is the moles of reactanr. A transferred per unit time per unit area, and the subscripts s, a, and c refer to the external surface of the particle, a point in the interior of the ash layer, and the surface of reaction core, respectively, then the amount of A reacting per unit time, M A , can be expressed as :
t/t* = 1 - (1 - xp3 I n determining the rate-controlling factor, experiments may be performed with different-size particles and the time required to achieve a given conversion is measured. If R1 and R z are the radii of the two particles which have the same conversion but have the corresponding re, then action times of t l and t ~respectively, Fluid film diffusion controlling :
M A = 4nR2NAs= 4 ~ 7 =~4nr:NA, 1 ~ ~ = constant ~ and Ash diffusion controlling :
At complete conversion when t = t*, r , = 0, Equation 6 becomes :
For the fluid to react a t the surface of the core, it must diffuse through the film and ash layer to the reacting surface. The three terms appearing in the denominator of Equation 8 represent the three resistances in series for the reaction considered. Mihen the fluid film resistance controls, kmA 0;
CA = 0
; CB-0
CIHM~SHI-~CHI AR
A H = heat of reaction
. bmr
I il: IDinwnr*nlorrl
Figure BI. I n situ hydrogm&&ion and rnnparotura projles
of shale indi&'ng
mncdntration
comes the rate-controlling factor as the values of 9, become large. The dotted line represents the onset of the second-stage reaction. Figure 8 shows the effect of film diffusion on vr. The film diffusion has more significant effect when & is large indicating that the film diffusion is particularly important when diffusion through the solid is the rate-controlling factor. In Figure 9 the effectiveness factor for the unreacted-core 48
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure B2. Regresses of reaction sutface, r, and solid siujace, S, ar (I funcrion of dimcnrionlcss rim. Tmpmalure at ihc rtaction surface, r, us. dimmionless time
shrinking model is shown when film diffusion is unimportant. Again, the figure indicates that the ash diffusion becomes the rate-controlling factor when & is large. Partides with reducing diameter. Let us briefly examine the effectiveness factor for the unreacted-core shrinking particle without the ash layer formation. Again assuming the mass transfer coefficient to be = D,/r., we have represented as, k,
UNDERGROUND OIL SHALE GASIFICATION
> 0;
2 = r(t),t
Z = S ( t ) ,t
TR = TR' = T,
> 0;
(11)
where
(2) (z)
D s ~
= PM
QM
(Z) G) (Z)
; -AR
=
-
pML
t=O.r=S=O Assuming the pseudo-steady-state, a common practice for solidgas reaction, Equations 1 through 5 can be written as, I)=-
;:
0 s -dZCB
dZ2
0 =
(0
