Engineering problems mathematics

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The mathematical treatment of engineering problems involves four basic steps ... 

Dimensional analysis is a useful tool for examining complex engineering problems by grouping process variables into sets that can be analyzed separately. If appropriate parameters are identified, the number of experiments needed for process design can be reduced, and the results can be described in simple mathematical expressions. In addition, the application of dimensional analysis may facilitate the scale-up for selected biotechnology unit operations. A detailed description of dimensional analysis is reviewed by Zlokarnik [18]. 

Mixed-integer nonlinear optimization problems of the form (1.1) are encountered in a variety of applications in all branches of engineering, applied mathematics, and operations research. These represent currently very important and active research areas, and a partial list includes ... 

In the previous chapters we have shown how to solve many types of problems that occur in Chemical and Biological Engineering through mathematical modeling, standard numerical methods, and MATLAB. 

In analyzing an engineering problem, we frequently set up a mathematical model that we believe will describe the system accurately. Such a model may be based on past experience, on intuition, or on a theory of the physical behavior of the system. 

Thomas K. Sherwood and C. E. Reed, Applied Mathematics in Chemical Engineering, McGraw-Hill, New York, 1939 William R. Marshall and Robert L. Pigford, The Application of Differential Equations to Chemical Engineering Problems, University of Delaware, Newark, 1947 A. B. Newman, Temperature Distribution in Internally Heated Cylinders, Trans. AlChE 24,44-53 (1930) T. B. Drew, Mathematical Attacks on Forced Convection Problems A Review, Trans. AlChE 26,26-79 (1931) Arvind Varma, Some Historical Notes on the Use of Mathematics in Chemical Engineering, pp. 353-387 in W. F. Furter, ed., A Century of Chemical Engineering [17]. 

Although many practical engineering problems involving momentum, heat and mass transport can be modelled and solved using the equations and procedures described in the preceding chapters, an important number of them can be solved only by relating a mathematical model to experimentally obtained data. 

In fact, it is probably fair to say that very few problems involving real momentum, heat, and mass flow can be solved by mathematical analysis alone. The solution to many practical problems is achieved using a combination of theoretical analysis and experimental data. Thus engineers working on chemical and biochemical engineering problems should be familiar with the experimental approach to these problems. They have to interpret and make use of the data obtained from others and have to be able to plan and execute the strictly necessary experiments in their ovm laboratories. In this chapter, we show some techniques and ideas which are important in the planning and execution of chemical and biochemical experimental research. The basic considerations of dimensional analysis and similitude theory are also used in order to help the engineer to understand and correlate the data that have been obtained by other researchers. 

Tien attempting to gel an analytical solution to a physical problem, there is always the tendency to oversimplify the problem to make the mathematical model sufficiently simple to warrant an analytical solution. Therefore, it is common practice to ignore any effects that cause mathematical complications such as nonlincarities in the differential equation or the boundary conditions. So it comes as no surprise that nonlinearities such as temperature dependence of tliernial conductivity and tlie radiation boundary conditions aie seldom considered in analytical solutions. A maihematical model intended for a numerical solution is likely to represent the actual problem belter. Therefore, the numerical solution of engineering problems has now become the norm rather than the exception even when analytical solutions are available. 

Uncertainty and disturbances can be described in terms of mathematical constraints defining a finite set of hounded regions for the allowable values of the uncertain parameters of the model and the parameters defining the disturbances. If uncertainty or disturbances were unbounded, it would not make sense to try to ensure satisfaction of performance requirements for all possible plant parameters and disturbances. If the uncertainty cannot be related mathematically to model parameters, the model cannot adequately predict the effect of uncertainty on performance. The simplest form of description arises when the model is developed so that the uncertainty and disturbances can be mapped to independent, bounded variations on model parameters. This last stage is not essential to the method, but it does fit many process engineering problems and allows particularly efficient optimization methods to be deployed. Some parameter variations are naturally bounded e.g.. feed properties and measurement errors should be bounded by the quality specification of the supplier. Other parameter variations require a mixture of judgment and experiment to define, e.g., kinetic parameters. 

Where sufficient data to enable a complete mathematical representation of the product-engineering problem exists, mathematical techniques exist for their solution. However, significant work is still needed to establish a comprehensive generic methodology to generate and systematically reduce the number of alternatives through heuristics, so that product engineering can be accomplished even in the absence of complete data. 

Recent work has established how to mathematically represent the generic product-engineering problem [19], and mathematical programming has been successfully applied to these problems. [20] Of course, these techniques can only be applied where sufficient data are available to enable a complete mathematical representation of the product-engineering problem. 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

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