Approximations

The most common explicit correlations for estimation of the friction factor in rough and smooth pipes are reviewed in this paper. Comparison of any friction factor equation with the Colebrook's equation was expressed trough the mean relative error, the maximal positive error, the maximal negative error, correlation ratio and standard deviation. The statistical comparison of different equations was also carried out using the "Model selection criterion" and "Akaike Information Criterion". It was found that the equation of Zigrang and Sylvester provides the most accurate value of friction factor, and that Haaland's equation is most suitable for hand calculations.

It is generally claimed that infinite idealizations are required for explaining phase transitions within statistical mechanics (e.g., Batterman 2011). Nevertheless, Menon and Callender (2013) have outlined theoretical approaches that describe phase transitions without using the infinite limit. This paper closely investigates one of these approaches, which consists of studying the complex zeros of the partition function (Borrmann et al. 2000). Based on this theory, we argue for the plausibility for eliminating the infinite limit for studying phase transitions. We offer a new account for phase transitions in finite systems, and we argue for the use of the infinite limit as an approximation for studying phase transitions in large systems.

Keywords infinite idealizations · phase transitions · finite systems · ineliminability · approximation · zeros of partition function

Many information measures are suggested in the literature. Among these measures are the
Shannon H,(O) and the Awad A,(O) entropies. In this work we suggest a new entropy measure, B~(0),
which is based on the maximum likelihood function.
These three entropies were calculated from the gamma distribution and its normal approximation, the
binomial and its Poisson approximation, and the Poisson and its normal approximation. The relative
losses in these three entropies are used as a criterion for the appropriateness of the approximation.
Copyright © 1996 Elsevier Science Ltd.

Exact values of the best one-sided approximation by Haar polynomials have been obtained in the integral and uniform metrics for some classes of the functions of several variables defined using modulus of continuity ω(f ; t) and ω_ρi (f ; δ).

Tolerance analysis usually complements tolerance synthesis; however, analysis algorithms, especially the Monte Carlo method proves highly expensive. This study offers an alternative to the Monte Carlo method. The huge sample sizes are replaced by small orthogonal experiments. The mean and variance of critical design requirements are calculated via the two methods. Preliminary results indicate the need for larger size orthogonal arrays to conform to results by Monte Carlo simulation (MCS). Two new modifications are introduced and their effects on the optimum tolerances and process combinations are studied. Results indicate the potential of the algorithm for further future developments.

Surrogate modelling is becoming a necessary modelling tool in engineering analysis and synthesis. This is, especially, true when mathematical models are not available. The capability of response surface methodology is explored with single unconstrained/constrained linear and non-linear problems. Problems with special features or with highly non-linear nature are dealt with. A further development with multi-objective problems starts with bi-objective functions and continues to deal with higher number of objective functions. A simple multi-objective problem is taken to initiate the development. Orthogonal arrays and response surface models are combined and used to explore the sets of Pareto Frontiers. One important result is that various array sizes are able to explore the Pareto set at no or little cost change with comparable accuracy. Except for one array, the optimum solution is typical for all arrays employed. One development is using the first derivative of the two objective functions to obtain the starting point for model construction. Design of experiments and analysis of variance (ANOVA) are employed to verify the relative significance of variables. Results of ANOVA guide the modeller in the model building process. We attempt to achieve a 'local accurate model' rather than a 'global accurate model'. Computational experience with different classes of problems is given.

A strategy for adaptive control and energetic optimization of aerobic fermentors was implemented, with both air flow and agitation speed as manipulated variables. This strategy is separable in its components: control, optimization, estimation. We optimized parameter’s estimation (from the usual KLa correlation) using sinusoidal excitation of air flow and agitation speed. We have implemented parameter’s estimation trough recursive least squares algorithm with forgetting factor. We carried separate essays on control, optimization and estimation algorithms. We carried our essays using an original computational simulation environment, with noise and delay generating facilities for data sampling and filtering.

Our results show the convergence and robustness of the estimation algorithm used, improved with use of both forgetting factor and KLa dead-band facilities. Control algorithm used in our work compares favorably with PID using the integrated area criteria for deviation between oxygen molarity and critical molarity (set point). Optimization algorithm clearly reduces energetic consumption, respecting critical molarity. Integration of control, optimization and adaptive algorithms was implemented, but future work is needed for stability. Methods were defined and implemented for stability improvement. We have implemented data acquisition and computer manipulation of air flow and agitation speed for actual fermentors.