Chapter 19. Reacting Solids

Reactions where a solid reacts with a gas are very common in chemical, metallurgical, environmental, and electronic processes. Combustion of a coal particle is an example in the energy production sector as well as the process of gasification of coal where coal is reacted with steam to produce a mixture of CO and H₂ (syngas). In metallurgical industries, the reduction of metal ores (e.g., NiO) with hydrogen to produce metals (e.g., Ni) is an important application where a gas–solid reaction is encountered. Direct reduction of iron ores with hydrogen is another example. Oxidation of silicon to form an insulating film is an important application in the semiconductor fabrication of MOS (metal oxide semiconductor) devices. Similarly, in the production of polysilicon for solar cells, the precursor production involves many steps of gas–solid reactions such as chlorination of silicon, the deposition reaction of silane on a solid substrate, hydro-chlorination of silicon, and so on. In the pollution treatment field, CO₂ removal is an important problem and CO₂ capture by reaction with calcined lime (CaO) is an example of reacting solids.

In all these cases and other similar processes mass transfer effects play an important role as these are heterogeneous systems. Thus external (gas film) diffusion and pore diffusion in the interior of the solid affect the rate at which the solids react at the gas–solid interface. The scope of this chapter is therefore to study mass transfer effects in these systems.

The starting reacting solid can be relatively non-porous or highly porous and the modeling of the systems for these cases takes different approaches. For non-porous systems the reaction occurs at a sharp gas–solid interface. External or film mass transfer followed by diffusion through the product layer (in some cases) and the surface reaction are the factors to be considered. Analysis of such systems is discussed first.

For porous systems, the reaction occurs on the interior surface of the solid and hence simultaneous reaction during pore diffusion plays a role. The mass transfer steps here are similar to those for a porous catalyst analyzed in Chapter 18. However, an important difference is that the solid properties change as the solids react. This changes the porosity and in turn the effective diffusivity in the solid. We consider the various ways to represent how the pore structure affects the rate of reaction.

Diffusional effects in scenarios of a solid reacting with a second solid are also briefly discussed.

A+ ν_BB → Products

One widely used model for this reaction is the shrinking core model. This model can be applied to the case where there is no solid product and also to a case where there is solid product formed. The analyses for the two cases are slightly different and therefore treated separately in the following subsections. In both cases the solid reactant B is assumed to be non-porous so that the reaction occurs at a sharp interface.

Rate of mass transfer from gas to the surface = k_m(C_Ab – C_As)

This is balanced by the surface reaction assuming a pseudo-steady state:

Rate of reaction per unit surface area of the solid = k_sC_As

A first-order surface reaction is considered here. The two rates are equal due to the pseudo-steady state assumption and hence C_As can be eliminated and a rate expression based on bulk gas concentration can be derived.

This rate equation combining the two transport effects in turn leads to an apparent or overall rate constant, which is defined as

Rate per unit area of B = k₀C_Ab

This overall rate constant can be shown to be

Note that this can also be expressed as k₀ = k_sk_m/(k_s + k_m). Also note that depending on the relative values of the mass transfer coefficient and the surface reaction constant two limiting steps can be identified:

• The process can be mass transfer limiting if k_m << k_s and in this case k₀ = k_m.

• The chemical kinetics can be rate limiting if k_s << k_m and in this case k₀ = k_s.

The rate of reaction can now be incorporated into the transient mass balance for the solid to study the conversion of the solid as a function of time, which is presented next. It is useful to note that this part of the model is similar to the model for sublimation of a solid considered in Section 3.2.

ddt([4/3]πR3CB0)=−4πR2νBk0CAb

where ν_B is the stoichiometric coefficient of B and C_B0 is the concentration of solid B, which is equal to solid density divided by the molecular weight of B, that is, ρ_B/M_B. The previous equation can be simplified to

dRdt=–νBCAbCB0k0

The variables can be separated and the solution for time needed to reach a particular radius R(t) can be formally written as

Here we have also used Equation 19.1 for k₀ in terms of k_s and k_m. We also note that k_m depends on R and hence the explicit dependency of this term on R is also noted in the second integral on the right-hand side. The two integrals on the right-hand side can be written separately and the time needed for the solid to shrink to any R is given as

where t_R is the first term on the right-hand side of Equation 19.2:

and t_M is the second term:

The first term on the right-hand side can be integrated directly since k_s is not a function of the radius. The result is usually expressed in terms of the conversion of the solid B, denoted as X_B. This is related to R as

XB=1–(RR0)3

Hence the integrated form of the Equation 19.4 is

In order to integrate the second term in Equation 19.2 the functional dependency of k_m on R should be taken into account. In accordance with the Ranz-Marshall equation shown in Chapter 3, the dependency of mass transfer coefficient on particle radius is as follows:

• For small particles and/or small gas velocity k_m ∝ R^–1.

• For large particles and/or large external gas flow rates k_m ∝ R^–0.5.

Hence the radius dependency may change with time. At the start if we have large particles the dependency is –0.5 and will keep changing with time as the solid radius keeps decreasing; it may switch to the small particle regime where the dependency is –1.

Consider the case for small particles with low gas flow; here the mass transfer coefficient is given as

km=DAR

Hence the integrated form of Equaton 19.5 is

If k_s << k_m right from the beginning, the process is under a kinetic regime and the film mass transfer resistance need not be accounted for.

A few remarks on the kinetics and mechanism of solid–gas reactions may be in order:

• The order of the reaction with respect to the solid concentration is often a fitted parameter rather than that result due to mechanistic considerations. This is unlike gas–gas reactions. The effect of concentration of a solid cannot be easily studied unlike a gaseous system since the activity of the solid reactant is one.

• The surface reaction can proceed in two ways. The first is an adsorption + surface reaction mechanism. Gas first adsorbs on the surface and the adsorbed species then reacts with B on the surface of the solid. Models similar to gas–solid catalytic reactions, that is, Langmuir-Hinshelwood type of models, have been proposed for these cases. The second is the direct attack of gas on individual sites of a solid that are the most reactive (active sites). Crystal defects and orientation of the surface will have a strong effect in this case.

• In some cases the reaction may proceed at only some preferential directions on the solid surface, the so called wormhole effect. Modeling and analysis of wormhole formation in reactive dissolution of carbonate rocks has been studied by Kalia and Balakotaiah (2007, 2009) and Fredd and Fogler (1998) and other references cited within these sources.

The concentration profiles when the product layer forms are shown in Figure 19.2. The ash layer is assumed to occupy the region from λ to R at any instant of time. The reaction plane is now located at λ, which moves inward with time.

The pseudo-steady state assumption is made for the solid. Let λ be the position of the ash layer at any instant of time as shown in Figure 19.2.

The reaction takes place due to three steps in series, noted in the following together with the corresponding resistance.

The first step is the gas film diffusion; the mole/s transported are given as

The resistance for this step is 1/(4πR²k_m).

The second step is ash layer diffusion with the following rate:

The resistance is (R – λ)/(4πR D_eA).

Finally the surface reaction takes place at a sharp interface located at λ:

The resistance is 1/(4πλ²k_s).

The overall resistance is the sum of these resistances, and the rate at which the gaseous species is consumed by reaction is

Here k₀ is the overall rate constant, defined as

The three terms on the right-hand side are due to the three resistances indicated earlier. The dimensionless location of reaction plane λ*, defined as λ/R, is used for the convenience of later algebra.

Now we are in position to calculate the rate at which the solid phase reactant is consumed by writing a transient mass balance for the solid reactant. The method is similar to the solid balance discussed in the previous section and therefore not shown in detail. The following equation can be derived for the change in the reaction plane location with time:

Using Equation 19.12 for k₀ and separating the variables the result for the time needed to achieve a given conversion can be expressed as a sum of three terms, each corresponding to one of the resistances:

Here t_m is the contribution of the mass transfer resistance:

Note that the external mass transfer coefficient does not change as a function of time now since the overall radius of the particle remains constant. Particle size may change slightly due to shrinkage or expansion, which can be accounted for by a volume balance. These details are not addressed here.

The t_D term in Equation 19.14 is the time needed if ash layer resistance were in control:

Finally t_R term is the time if the reaction were in control:

The final results are presented in terms of the conversion of the solid, defined as

X_B = 1 – (λ*)³

Students may wish to verify the following results for the three contributions to the total time needed to achieve a given conversion.

The following diagnostic criteria given by Levenspiel (1999) are useful for finding the controlling regime:

• The time for complete conversion is proportional to R² if ash diffusion is controlling.

• The time for complete conversion is proportional to R if surface reaction is controlling.

• The time for complete conversion is proportional to Rⁿ where n is between 1.5 to 2.0 if external mass transfer is controlling. This is because k_m is a function of the radius. In this case the conversion should also change if gas velocity is changed.

Hence experiments with particles of different initial solid radius are indicative of the prevailing control regime for the given experimental conditions. Note that the controlling regime may change during the course of the reaction. Thus in the initial stages the process may be reaction controlled since there is no significant ash layer. In the later stages the process may be ash diffusion controlled. These changes are not captured in the above simple criteria and parameter fitting with tools available in MATLAB or other similar softwares may have to be used.

The gas diffuses into the pores, is adsorbed on the reactant surface, and reacts with B at the surface. The adsorption and reaction are lumped together and a “pseudo-homogeneous” rate of reaction is defined on the volume of the solid+gas locally at any location r. Note that a product layer can form on the surface of the solid. This layer is assumed to porous and the resistance due to the gas diffusion through this product layer is usually neglected at the first go at modeling. More detailed models account for this effect (see for example Christman and Edgar, 1983). Also the pore size changes due to product formation, which in turn affects the internal diffisivity.

RA=–kCAnCBm

where n and m are the order of reaction with respect to A and B respectively. Here C_B is the local concentration of the solid reactant. The rate is defined as moles produced per unit total volume (pore + solid) per unit time.

The following remarks on the kinetic model shown above are useful:

• Note the negative sign for R_A since A is consumed rather than produced by reaction.

• The rates are often based on the mass of the solid R_A, m in which case the units will be in mol/kg sec rather than mol/m³ s. If so, we should multiply this by ρ_s to get R_A, the volume-based rate of reaction used previously.

• Rate can also be defined using the surface area of the solid, in which case the initial surface area per unit volume of the solid must be known.

• The order of reaction with respect to the solid reactant is often a fitted parameter rather than a mechanism-based value. Usually it has a value ranging from 0 to 1.

• The reactions often follow a two-step scheme with adsorption of A on the solid first followed by reaction with B at the surface. This is often modeled by a Langmuir kinetics of the following form:

  RA=–kCA1+KACACBm

where ∊_p is the internal pore volume of the solid.

The diffusion coefficient is assumed to be a constant for further discussion. This parameter may change in time due to a change in the pore structure and due to reaction of solid B. Variable diffusivity analysis is briefly presented in the Section 19.3.1.

The corresponding rate of reaction of B is then incorporated into a transient mass balance for B (over a differential volume) and provides an equation to calculate the B versus t profile at each radial location:

The concentration of B is defined based on the total volume of the pellet (pore + solid) and hence the solid holdup term does not appear on the left-hand side. The R_A is assumed to be given by a power law model: RA=−kCAnCBm. Observed features of the solution for reaction, which is first order in B (m = 1) is shown next followed by a zero-order reaction (m = 0).

τ = (νkC_Ab)t

This provides the following dimensionless equations to be solved for the gas phase and the solid phase:

and

Two parameters that arise in the dimensionless formulation are

This is a Thiele modulus parameter based on the initial concentration of B and

This is a capacity term representing the ratio of the reaction time constant to the time constant for internal diffusion.

The pseudo-steady state hypothesis is often used. The term containing κ in Equation 19.20 is then dropped. This is usually a good assumption for gas– solid systems since C_Ab is usually much smaller than C_Bi. Hence the accumulation in the pore phase is often small and this term can be dropped. The assumption may not be valid for liquid–solid reaction. For further discussion, we use the pseudo-steady state hypothesis.

An illustrative plot of the concentration profile of gas A and solid B within the pellet is shown in Figure 19.4.

The following discussion is useful to understand the qualitative features of the volume reaction model and to identify the regime of operation. Two cases are normally distinguished: low Thiele and high Thiele. For low values of the Thiele parameter the concentration profile of A is nearly uniform in the pellet. The reaction of the solid takes place uniformly all over the pellet. This is often referred to as a homogeneous reaction model or uniform reaction model.

where τ is the dimensionless time.

An illustrative result for a (1,1) order reaction simulated numerically using the a modified MATLAB PDEPE solver is shown in Figure 19.5. The time required for a given conversion is smaller if the Thiele modulus is small and the uniform reaction model holds.

An approximate analytical solution can be obtained using the concept of cumulative concentration; this is presented as an exercise problem.

In the initial stage the concentration of B is above zero everywhere. The model equations for the gas and the solid are

and

The gas concentration can be solved independently since the reaction is first order. An effectiveness factor for the drop in gas concentration is used in the solid balance. Upon integration this yields the following expression for the conversion of the solid as a function of time:

This solution is valid until the concentration of B remains finite at the surface.

In the later stages of the reaction a B depleted layer develops near the surface of the solid. The concentration of B becomes zero at a dimensionless location λ in the solid and the reaction can now take place only in the region from 0 to λ. The previous model equations now need a minor modification. Equation 19.25 has to be solved separately for the region of no B, which is from λ to 1. The rate term is set to zero in this region. The reaction zone is from λ to 0 and the rate term is retained for this region. Matching of the concentration and the flux of A on either side of λ has to be used as additional conditions. The solution for the concentration profile and the conversion of the solid with this modified model is not very complicated, but the details are not shown here. The following solution for conversion can be derived for this stage of the reaction:

where λ is given by the following equation:

The reaction zone thickness depends on the Thiele parameter. It is on the order of 0.039 for ϕ = 100. Thus the sharp interface model can be used as an approximation for such cases. Note that the sharp interface model can be viewed as a limiting case of the volume reaction model for large Thiele modulus values. Details leading to Equation 19.29 are shown in the paper by Ishida and Wen (1968).

ZV=V¯PV¯B=νpMP/[ρp(1−εp)]νbMB/ρB

This parameter is a measure of the ratio of the volume of product formed per unit volume of the reactant. If Z_V less than one, the pore volume increases, while the pore volume decreases if Z_V is greater than one.

The porosity variation can then be modeled as

εp=εp0+CB0V¯B(1−ZV)(1−CBCB0)

Correspondingly the diffusion coefficient is assumed to vary as

where n has a range of values from between 2 and 3 depending on the porosity and tortuosity of the solid. A value of 2 for n holds for the random pore model with a single size distribution of the pore volume. This variation of diffusivity is then incorporated into the gas phase mass balance equation and the diffusivity in Equation 19.18 is no longer treated as a constant.

One of the effects of change in porosity is that an incomplete conversion of the solid can occur if the pore mouth gets closed due to the product occupying a larger volume than the starting reactant. The maximum conversion of the solid is shown to be given by the following equation:

XB(max)=εp0(ZV−1)(1−εp0)

The initial solid porosity has to greater than (Z_V – 1)/Z_V if complete conversion of the solid has to occur. Data on hydrofluorination of UO₂ and sulfation of limestone shows these effects.

The model focuses attention on the whole pellet, which is modeled as a single pore of length R/(3∊0), which is supposed to be representative of changes taking place in the pellet. The structural changes are accounted for in a simple manner using some simple parameters. These are average pore radius, radius of the associated solid, effective pore length, effective diffusivity through the porous layer, and reaction rate constant. For further details and application of the model please refer to Ramachandran and Smith (1977) and the follow-up citations for that source.

Carbothermic reduction of ores (e.g., NiO + C) is one of the important examples in metallurgical industry. Cement production (tricalcium aluminate formation) is another large-scale application.

Model development leads to the following equation for the conversion as a function of time:

X = 1 – exp(–ktⁿ)

The value of n = 4 was assigned in the original Avrami model. The rate constant k was interpreted as

k=πN¯(G¯)3/3

where N¯ is the nucleation rate and G¯ is the growth rate of the nuclei.

The equation is also known as the Johnson-Mehl-Avrami-Kolmogorov, or JMAK, equation since it appears to have been first derived by Kolmogorov in 1937 but was popularized by Avrami in a series of articles in the Journal of Chemical Physics.

In this model the reaction initiates at points where the reactant B is in contact with A. An ash layer develops in the interior of A near the surface. Further reaction proceeds through the diffusion in the ash layer. If there are a large number of contact points the ash layer overlaps and grows over a region near the surface rather than as discrete islands. In such case the behavior of the system is close to the shrinking core model. On the other hand for a low number of contact points the reaction front consists of several moving fronts within the solid A. Thus the contact point model is able to cover the two limiting cases of reaction control and ash diffusion control depending on the value of the parameters such as the number of contact points and the rate of growth at these contact points.

19.2 When would you use the volume reaction model?

19.3 How does the time for complete conversion depend on the particle radius for the shrinking core model if the reaction is controlling?

19.4 How does the time for complete conversion depend on the particle radius for the shrinking core model if ash layer diffusion is controlling?

19.5 What is meant by the law of addition of time?

19.6 When can you expect wormhole formation or anisotropic etching?

19.7 If the starting solid is relatively porous what type of model would you use?

19.8 Why is the ratio of the molar volume of the product and the reactant important in modeling gas–solid reactions?

19.9 What are the main differences between a gas–solid reaction and a solid–solid reaction?

19.10 What are the common models useful for solid–solid reactions?

N₂(g) + CaC₂(s) → CaCN₂(s) + C(s)

Particles of 2 mm reacted in 3 hours while those of 4 mm reacted in 6 hours. Suggest a controlling regime and develop a rate equation.

19.2 Time scales in the shrinking core model. Spherical particles of a reacting solid of 4.8 mm are completely converted to products in 100 min. If the ash layer diffusion coefficient is 1 × 10^–6 m²/s and the surface reaction constant is 0.01 m/s, find the contribution of the ash later and reaction to the time needed for the conversion. Estimate the time for the reaction of a particle of 9.6 mm.

19.3 Reduction of magnetite with hydrogen. Consider reduction of magnetite ore with hydrogen, which is important in direct routes for iron production. The kinetic data is given as

k_s = 1930 exp(–E/R_gT)

with E = 100, 000 J/mol and k_s in m/s. The effective diffusion coefficient is 3 × 10^–6 m²/s in the ash layer. The reaction is carried out in pure hydrogen at 1 atm pressure and 600 K. The initial particle radius is 5 mm and the density of magnetite is 4600 kg/m³. Plot the conversion versus time data and indicate the operating controlling regime.

19.4 Langmuir kinetics. Topochemical reactions with Langmuir kinetics are often encountered when gas A is strongly adsorbed. Assume that the reaction takes place at a sharp interface and the rate of the surface reaction is then represented as kC_A(1 + K_AC_A). Develop a shrinking core model for this case.

19.5 Temperature effects on conversion. What is the effect of temperature on conversion? How does the controlling resistance change? The effect of temperature on MgO conversion with carbon was reported by Sohn and Han (2011). The rate increases first with temperature and then decreases with temperature above 975 K. Provide a suitable explanation to this anomalous effect.

19.6 Parameter estimation for the shrinking core model. Find the controlling mechanism for the following time verus conversion data:

Time, Hour	0.18	0.347	0.453	0.567	0.733
Conversion	0.45	0.68	0.80	0.95	0.98

19.7 Cumulative concentration method for conversion. The model equations for a (1,1) order reaction can be solved in an approximate analytical manner using the concept of cumulative concentration. This exercise walks you through the steps of this method. The analysis borrows the generalized Thiele modulus concept used on gas–solid catalytic reactions. The equations to be solved are repeated here for convenience. Using the pseudo-steady state is represented in slab geometry as

The solid phase balance is

Define a cumulative concentration variable, ψ, as

ψ(τ, ξ)=∫0τcAdτ

Show that the concentration of B can now be represented as c_B = exp(–ψ). Show that the diffusion-reaction model for the gas, Equation 19.31, can now be written as

and state the boundary conditions to be used. This has a resemblance to a diffusion-reaction problem; hence, show that the following expression for the generalized Thiele modulus can be derived:

Λ=ϕ(1−exp(−τ)2(τ+exp(−τ)−1

Correspondingly a “conversion” effectiveness factor can be defined:

ηs=tanhΛΛ

Show that the conversion of the solid at any time is given as the conversion in the absence of diffusion times the above effectiveness factor.

19.8 Effect of solid residence time in a fluid bed roaster. Fluid bed reactors are commonly used in metallurgy. The large-scale units are operated continuously. Since there is spread in residence time, the conversion of the solid depends on the E-curve. Show that the following expression can be derived for the conversion:

1−X¯B=∫0tf(1−XB)E(t)dt

Here t_f is the time needed for complete conversion for a single particle. If one assumes that the solids are well mixed but segregated, what is the expression for the E-curve? Use this in the previous equation and show that the following equation holds:

1−X¯B=∫0tf[1−XB(t)]exp(−t/t¯)t¯dt

Use the model to predict the exit conversion for roasting of pyrrhoites if the mean residence time is 60 min and the time for complete conversion for a single particle is 20 min. Assume that the time for complete conversion is proportional to R^1.5.