In a recent R&D note by Huang et al. (1984), an analytical solution for a packed bed reactor with a first-order reaction is derived. The solution is obtained by the Laplace-transform method and by inversion in the complex plane. Since the transform contains branch points a , so called second Bromwich contour (McLachlan, 1953) is used to obtain the solution in the time domain.Some criticism is also made of an earlier analytical solution of the same problem (but without chemical reaction) given by Rasniuson and Neretnieks (1980). We are supposed to have used the first Bromwich contour (McLachlan, 1953) in our derivation. Since the Laplace transform contains branch points, they contend, this procedure is not valid. This statement is correct. However, in our analytical inversion, neither the first nor second Bromwich contour is used. Instead, we integrate along the imaginary axis. This integration path follows direct application of the complex inversion integral for the Laplace transform. We would like to clarify our analytical inversion of the Laplace transform and discuss relative merits of various methods.Under very general conditions, the inversion integral of the Laplace transform is given (Churchill, 1972; Doetsch, 1976) by:

The two-dimensional theory of two simple generalizations of the Coulomb-Navier criterion for shear failure is developed. The first of these refers to a material with a single plane of weakness which has a different shear strength and coefficient of internal friction from the remainder of the material. In this case it is shown that failure may take place, according to circumstances, either in the plane of weakness or in planes cutting across it. The second criterion refers to a layered material whose shear strength varies continuously from a maximum in one direction to a minimum in the perpendicular direction. In this case it appears that, instead of the two directions of failure possible for an isotropic material, there is only one possible plane of failure which lies between the plane of minimum shear strength and the nearest to it of the two Coulomb-Navier planes. Numerical results are given for the case of uniaxial compression and experimental results are shown to be in reasonable agreement with them.

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The mathematical theory used in calculating temperatures of intrusions is reviewed, primarily from the point of view of finding temperatures in the country rock outside them. It is shown that the detailed behavior inside the intrusion, for example the mechanism of solidification and the possible effects of convection, becomes progressively less important as the distance from the contact increases, so that at distances of one-quarter of the thickness or more the simple theory of Lovering is adequate. The theory for sheets, stocks, laccoliths, and some irregularly shaped bodies is given, together with the effect of dissimilar thermal conductivities. The effects of latent heat, convection, and the circumstances of intrusion are discussed. Applications to metamorphism, rock magnetism, and argon loss caused by heating by intrusions are reviewed. NOTATIONT, temperature (excess of temperature over some arbitrary level, which is usually taken to be the initial temperature of country rock). T1 to T2, range of solidification of magma, T• To, initial temperature of magma. L, latent heat of magma. K, thermal conductivity of country rock. p, density of country rock. c, specific heat of country rock. • = K/pc, diffusivity of country rock. K•, p•, c•, •, thermal properties of solidified magma if different from those of country rock. ' ' thermal properties of liquid magma if different from those of K2', p2• C2 • country rock. c2 -c2' -•-L/(T• -T2) , equivalent specific heat of magma. To • -To + L/c, equivalen• initial •empera•ure. To, initial con•ac• •empera•ure. 2d, thickness of an intrusive sheee• (or diameter of a cylinder or sphere). t, •ime a•er intrusion. • -Kt/d 2, dimensionless time or Fourier number. x, distance from midplane of an intrusive shee•. • -x/d. Units are cgs, calorie, and øC unless otherwise stated. 443 444 J. C. JAEGER 1. INTRODUCTIONMany attempts have been made to calculate the cooling-history of igneous intrusions. The problem is a fairly definite one: a mass of magma at a known temperature and of a known shape is iniected into country rock at a known temperature; the subsequent variation of temperature at any point is to be calculated. Fortunately, most rocks and magma have much the same thermal properties, so that a simple approximation may be obtained by assuming these to be constant and equal; hence (neglecting the effects of latent heat, convection, and other complicating factors to be discussed later), the problem reduces to the conduction of heat in an infinite medium with a prescribed initial distribution of temperature, and the solution can be written down immediately. Ingersoll and Zobel [1913] gave solutions for intrusive and extrusive sheets and for a spherical laccolith. Lovering [1935] gave complete information for the sheet and laccolith in graphical form, stressing the fact that such numerical information for these bodies could be given in terms of two dimensionless parameters, so that special cases need not be considered, as had been done by many authors. A full review of this phase of ...

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Synopsis The similarities and differences between soil and rock mechanics are discussed with particular reference to the stability of slopes. The effects of constraints and of the stiffness of the system applying stress are of greater importance in rock mechanics. The criteria for failure of rocks are mostly empirical and lead to linear or power laws. Similar laws might be expected to hold for friction. While the Coulomb law is in general adequate for soils, it appears that the frictional behaviour of rocks is described better by a non-linear law and if a Coulomb law is used factors of safety are sensitive to the value adopted for the cohesion. The various methods for measuring friction are described and their limitations discussed. The process of wear and the area of contact between sliding surfaces are considered. It appears that in some cases residual values of friction are attained after small amounts of sliding. Gouge is built up during sliding and its behaviour appears to be time-dependent. At present, numerical values for friction and other parameters of jointed systems are uncertain and so simple formulae are still useful. A number of formulae for factors of safety for sliding on one or two plane surfaces are given. For the case of rock with closely spaced joints the use of soil mechanics theory for circular and other surfaces of sliding is reasonable. In this case, values of friction obtained from single joint surfaces or crushed rock are conservative since the interlocking of rock elements may cause a substantial increase in strength. Les similitudes et les différences entre la mécanique des sols et la mécanique des roches sont tuditudiées, avec réference particulière à la stabilité des pentes. Les effets des contraintes et de la rigidité du systéme appliquant les contraintes ont plus d'importance dans la mécanique des roches. Les critères pour la rupture des roches sont pour la plupart empiriques, et suggeèent des lois lintudiaires ou de puissance. On peut supposer des lois similaires pour le frottement. Alors que la loi de Coulomb est gènèralement correcte en ce qui concerne les sols, il semble que le frottement des roches serait plus correctement dtudi;fini par une loi non linéaire, et si l'on applique une loi de Coulomb les facteurs de sécurité sont affectés par la valeur adoptée pour la coh´esion. Les différentes methodes de mesure du frottement sont décrites, et leur champ d'application est étudié. On tient compte également du processus d'usure et de la superficie de contact entre les surfaces en glissement. 11 semble que dans certains cas on obtienne des valeurs résiduelles de frottement aprés des glissements de faible importance. Des cavités se produisent pendant le glissement et leur comportement semble dépendre d'un facteur temps. Actuellement, les valeurs numériques de glissement et les autres paramétres des systémes joints ne sont pas certains, et une formule aussi simple rend encore de grands services. On donne un certain nombre de formules pour les facteurs de sécurité du glissement pour une ou deux surfaces planes. Pour les roches avec des joints rapprochés on peut utiliser la théorie de mécanique des sols pour les surfaces de glissement circulaires et autres. Dans ce cas, les valeurs de frottement obtenues pour les surfaces à joint unique ou les roches écrasées sont plutôt en deçà de la vérité puisque l'enchevêtrement des roches peut résulter en unimportant accroissement de la solidité.

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