Probability terms

E(t) has the unit of inverse time (e.g., min ), and in probability terms, E(t) is a density rather than a distribution. It is a fundamental indicator of the flow and mixing pattern in a chemical reactor and its... 

Equation (9) is valid for all x and z and for all times such that t sTzr. Although there are some bizarre conditional probabilities, termed pseudo-Markovian, which obey the SCK equation for all x, z, 11 s r, which do... 

In order to develop the field of chemical engineering, especially concerning the phenomena that should be expressed by the probability terms, the establishment of a consistent viewpoint is indispensable. 

Information entropy offers a good possibility of becoming a consistent viewpoint to treat phenomena that must be expressed by the probability terms. By using information entropy, it will become possible to define the evaluation indices for mixing and separation operations/equipment, to estimate turbulent flow structure in a chemical equipment, to establish scale-up rules based on the turbulent flow structure, to present a general particle size probability density distribution, and to define the amount of anxiety/expectation. 

When the expression of E k) in an equipment is the same as that in another equipment, it can be said that the structure of the turbulent flow fields in both equipment is identical. Anyway, turbulent phenomena can be expressed as probability terms. In other words, a discussion of the structure of turbulence in a chemical equipment is possible by making use of information entropy. 

The author intended to show the usefulness of information entropy in chemical engineering although information entropy is not common in this field. Not many examples have been considered for explaining the usefulness of information entropy. Therefore, it is still a question whether information entropy can be applied in chemical engineering. However, the author expects that information entropy will be accepted in chemical engineering, since it is related to the probability terms that are widely used in chemical engineering. Since there are many phenomena that can be expressed by probability terms in chemical engineering, information entropy would be valid over a wide area. The author expects that the number of researchers interested in information entropy will increase. 

The authors rejected the normal thermal probability term, involving They... 

Further simplications can be achieved by using normalised rate coefficients, which are similar to the probability terms used in Kisliuk s statistical method of kinetic analysis, viz. 

Thus, since the sticking probability term is present, precursor state kinetics can be introduced into the desorption. Furthermore, high values of v could be predicted and it is both coverage and temperature dependent. If ft is not unity, as for the molecular adsorption case, then provided hW >kT... 

This expression takes into acpount the relative bonding strengths between coatse/fine particles and fine/fine particles by means of the probability term. With some re-arrangement equation (5.17) can be shown to be directly... 

The change in free energy, AG, takes into account the energy required as heat, the change in enthalpy, AH, and the ratio of the probabilities of the final and initial states, the change in entropy, AS. Thus, if the initial and final states do not differ in enthalpy, that is, AH is essentially zero, the probability term, TAS, determines whether the change occurs. On the other hand, if the probabilities of initial and final states are the same, that is, AS is essentially zero, the difference in enthalpy, AH, determines whether the reaction or process goes as written. 

Probability terms are dimensionless and are usually associated with safeguards. Hence the word probability in the context of risk studies usually refers to the probabUity of failure on demand as distinct from the likelihood of failure over a period of time. For example, with regard to P-IOIB, the predicted probability of the spare pump not starting on demand is 0.1... 

Probability terms are often combined with equipment failure rates to come up with a system failure rate. P-IOIA has a failure rate of 0.5 year the probability that P-lOlB will not start on demand at the time P-101 A fails is 0.1 therefore, the overall failure rate for the pump system becomes (0.5 0.1) year or once in 20 years. 

We know only that expressirai (78) behaves like a probability function, with values in the realm [0,1]. Given expressions (75), (78) and the general MichaeUs-Menten equation (74), we derive an expression for the imreacted probability term, PuNREACT([5 ](t)). As such, the expression ... 

In this equation, Gj is a geometric constant including solid-angle terms, Qif(A.o) the excitation probability, and ax( i) the total absorption for Xi as given in Eq. (2.34). T is the specimen thickness and p the density. If the weight fraction W is separated from the excitation probability term and all constant terms collected into one constant Kj, Eq. (10.3) becomes... 

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