Ind. Eng. Chem. Res. 1995,34, 3417-3425
3417
Role of Stress in Reaction Engineering Catalytic and Noncatalytic Reactions V. Hlavacek,* D. Orlicki, J. J. Thiart, and H. Rode Chemical Engineering Department, State University of New York at Buffalo, Buffalo, New York 14260
In this paper we address the development of mechanical stress in catalytic and noncatalytic reacting systems. The thermoelastic formulation is presented and discussed as it applies to several examples of practical importance. These include fracture of catalyst pellets due to thermal stress, application of stresses during rapid cooling of reacted samples, and stress development during noncatalytic reactions in powders or particles. Stresses develop as a result of temperature gradients due to heat effects of the reaction, and differences in expansion coefficients and equivalent volume of different phases. The examples cited in this review illustrate that stresses play an important role in the design of chemical reactors and may change the reaction characteristics, in both a qualitative and quantitative way. 1. Preface
It is indeed a great honor and privilege to have this opportunity to contribute to the issue honoring 35 years of Transport Phemmena. One of us (V. Hlavacek) can testify about the impact of the book on the quality of education in Eastern Europe in general, and former Czechoslovakia in particular. He benefited immensely from having this book available early in his technical career. In this contribution, we have chosen a subject relating to two areas which are an integral part of a reacting system: heat and mass transfer with chemical reaction, and mechanical stress distribution in solid materials. The concept of stress interaction is rather new for the chemical engineering community and the authors wish to honor, by this contribution, the important anniversary in chemical engineering. 2. Introduction
It has been recognized for over half a century that the growth of an oxide layer on a metal surface may induce stress, and that the stress may eventually result in cracks in the oxide film (F’illing and Bedworth, 1923). Evidently, from the corrosion point of view, the stress occurrence is of considerable practical importance. If the stresses in a film are sufficiently high, relief will occur by rupture of the film, or by plastic deformation of the oxide film itself or by plastic deformation of the metal core. In the absence of plastic deformation, fracture of the film may have a profound influence on the subsequent oxidation behavior. An oxide film of particular porosity acts as a diffision resistance between the reactive metallic surface and fluid environment. Fracture of the film permits direct access of the oxidizer to the metal, and the high initial reaction rate is restored. The effect of compression stress in oxide film can result in blistering of the film if film adhesion is weak or film rupture if film adhesion is strong. Evidently, the extent t o which blistering or cracking takes place depends on two factors: the magnitude and direction of the stress, and the mechanical properties of the oxide film. The work from corrosion-oriented literature did not penetrate systematically t o reaction engineering literature. Even though many contributions have addressed the generation of stress in gas-solid reactions, a
* Author to whom correspondence
should be addressed.
0888-5885/95/2634-3417$09.00/0
consistent and general model describing the development of stress in gas-solid systems and predictions when the film will lose its impervious nature is still lacking. Therefore, the goal of this paper is to provide a review and consistent treatment of stress development in different areas relevant to chemical reactions. Typical applications represent in catalytic engineering design of tubular chemical nonisothermal nonadiabatic reactors cracking of catalyst support due to thermoelastic stress; also, in noncatalytic engineering, cracking of solid rocket fuel and transition of deflagration to detonation, stress development in noncatalytic reactions occurring in particles or powders, and production of cubic TaN using combustion of tantalum powder in liquid nitrogen. Other applications are not discussed, due to space limitations. 3. Model Formulation The thermoelastic equations must be solved along with the typical transport equations used in chemical reaction engineering. Because of the space limitations we will discuss only the case of a very fast exothermic reaction followed by rapid cooling (cf. production of cubic TaN). This development is taken from Viljoen et al. (1993). The process described below represents a combustion of fine tantalum powder dispersed in liquid nitrogen. Typically, the reaction must be initiated with a high energy release igniter. Once ignited, the Ta powder will burn, and after the reaction is completed it is rapidly cooled by liquid nitrogen. This process can result in an excessive thermal stress and a cubic modification of TaN is obtained. The energy balance during the cooling stage of the reacted sample can be written as
subject to
aT -k,-=h(T-T) ar
-a_T ar
at r = R
- ~at r = ~
T = T , at t = O
(2)
where k, is the effective thermal conductivity of the 0 1995 American Chemical Society
3418 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
material, h is the heat transfer coefficient accounting for radiation and/or convective heat losses, is the preform thermal capacity, and is linear expansion coefficient. The thermoelastic equation can formally be written in terms of the displacement vector, u’(Nowacki, 1963), as
&
a2uf Q -= pV2uf+ ( I + p)V(V*u’)- (313.+ 2p)BVT at2
(3)
with the following strain-displacement equations, err
- auf, -%
and the following stress-strain relations,
where i, j = r, 4, z, and &j= 0 for i tj , 6g = 1 for i = j . Note that A and p are the Lam’e constants as defined in the Nomenclature. The definition of the thermoelastic problem is completed by defining the proper boundary conditions. However, these boundary conditions w i l l depend on the configuration under study. The model described above can be rendered dimensionless if a set of characteristic magnitudes is defined, i.e., t* = (gC&2)/k, as the characteristic time and the preform radius, R , as characteristic length. Thus,
with
(A
+ 2p)A* = A
( I + 2p)p* = p
a -e
a@
- ~at
Tic MoSiz TaN
1.25 1.00 1.18 2.24
20.53 73.01 4.96 745.32
The problem described above is a fully coupled transient problem. Fortunately, in certain circumstances the governing equations can be simplified, making the problem more tractable. The temperature and equilibrium equations are coupled by terms representing heat as a consequence of strain and stresses resulting from temperature gradients. In the energy balance, the coupling between temperature and stresses occurs through the last term on the right-hand side of eq 6. However, the contribution of this term can safely be considered negligible compared with the other terms (Boley and Weiner, 1960). Thus, in this work, the effect of strain rate on the temperature field will be neglected. This assumption decouples the energy balance from the thermoelastic equations, and they can be solved independently. For the purposes of this study it will also be assumed that the system has a large 1engtWdiameter ratio. This enables us to neglect end effects and thus permits the plane strain assumption (cf. Boley and Weiner, 1960). The displacement and temperature fields can be considered two-dimensional [i.e., flr,4)l and the longitudinal ( z ) dependent terms disappear in the equations above. We will therefore consider stresses in the r and 4 directions only. The hyperbolic nature of eq 7 adds to the complexity of the problem. An additional simplification to overcome this problem is to consider the stress field in a quasisteady state by neglecting the inertia term. This assumption is not always admissible, and it depends on the relative time scales for the transfer of energy (z,) and mechanical disturbance relaxation (tm)(Thiart et al., 1991). The inertia term becomes important only when temperature changes occur on the same time scale as the characteristic scale for dissipation of mechanical disturbances. In Table 1values for the different characteristic times of several solid-solid reaction systems are presented. By comparing these times, it is apparent that zm
