Energy balances cracking

Cracking reactions are endothermic the energy balance is obtained by the production of coke that deposits on the catalyst and that is burned in the regenerator. 

From what we have said already, we can write down an energy balance which must be met if the crack is to advance, and fast fracture is to occur. Suppose a crack of length fl in a material of thickness t advances by 8a, then we require that work done by loads > change of elastic energy + energy absorbed at the crack tip, i.e. 

Sometimes the failure occurs by propagation of a crack that starts at the top and travels downward until the interface is completely debonded. In this case, the fracture mechanics analysis using the energy balance approach has been applied [92] in which P, relates to specimen dimensions, elastic constants of fiber and matrix, initial crack length, and interfacial work of fracture (W,). 

Griffith derived a similar equation using an energy balance approach, equating stored energy with the energy required for crack propagation ... 

It is envisaged that the degradation of the frictional interface properties and the corresponding increase in the relative displacements eventually lead to debond crack growth once the debond criterion is satisfied. The debond criterion based on the energy balance theory given by Eq. (4,35) under monotonic loading can be rewritten as... 

The strength of a brittle solid is defined as the applied thrust force required to produce cracks in the sample under test, with uniform stress action. From the energy balance of Griffith (1980), we get the basic expression... 

Chapter 4.1 deals with an important industrial problem, the vapor-phase cracking of acetone. Here the material- and energy-balance design equations are developed. We advise the students to try and develop the design equations independently before consulting the book s derivations. Numerical solutions and MATLAB codes are developed and explained for this problem and sample results are given that need to be checked against those of the students codes. 

Griffith used an energy balance approach to predict the crack propagation conditions (see Williams, 1984). The driving force is the elastically stored energy in the notched samples, which can be used to create new surfaces. A parameter Gc, the critical elastic strain energy release rate [GIc in mode I], can be determined and expressed in J m-2. 

The energy balance considerations in Griffith s original concept were later refined by Orowan and Irwin to include the effects of plasticity and elasticity for applicability to metals (Orowan, 1952 Irwin, 1957). Metals fail by ductile fracture, where the crack growth occurs in the direction of the primary slip system. When the slip plane is inclined to the crack, atoms across the slip plane slide past one another, relieving the stress, which results in a zigzag crack path. This is illustrated in Figure 10.14. 

Usually the radius of curvature p at the sharp notch of the crack is determined by the atomic sizes and is very small. It is immediately evident that the stress concentration at the sharp notches of the microcracks can become extremely large due to the above stress intensity factor, and the fracture should start propagating from there. Although this analysis indicates clearly where the instabilities should occur, it is not sufficient to tell us when the instability does occur and the fracture propagation starts. This requires a detailed energy balance consideration. 

In fact, the first quantitative attempt to incorporate the dynamics of the crack into the Griffith energy balance concept was given by Mott (1948). He suggested that unlike the Griffith case of fracture initiation or nucleation... 

Material Balance and Heat Balance, Required heat was mainly supplied by incineration of char, and some amount of produced combustible gas was fed as auxiliary fuel to the regenerator, as the amount of char was not sufficient for continuous thermal cracking. The material balance around the reactors is shown in Table-Ill and heat balance in Table-IV. Radiation and convection loss in Table-IV is larger than that of usual incinerators because of the thin refractory. It can be decreased in case of commercial plants. Energy balance of the total plant is shown in Fig-3. 

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