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Design and Optimization of Thermal Systems Episode 3 Part 1

Tham khảo tài liệu 'design and optimization of thermal systems episode 3 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 8 Lagrange Multipliers 8.1 INTRODUCTION TO CALCULUS METHODS We are all quite familiar from courses in mathematics with the determination of the maximum or minimum of a function by the use of calculus. If the function is continuous and differentiable its derivative becomes zero at the extremum. For a function y x this condition is written as dy 0 8.1 dx where x is the independent variable. The basis for this property may be explained in terms of the extrema shown in Figure 8.1. As the maximum at point A is approached the value of the function y x increases and just beyond this point it decreases resulting in zero gradient at A. Similarly the value of the function decreases up to the minimum at point B and increases beyond B giving a zero slope at B. In order to determine whether the point is a maximum or a minimum the second derivative is calculated. Since the slope goes from positive to negative through zero at the maximum the second derivative is negative. Similarly the slope increases at a minimum and thus the second derivative is positive. These conditions may be written as Keisler 1986 For a maximum 0 dx2 8.2 For a minimum 0 dx2 8.3 These conditions apply for nonlinear functions y x and therefore calculus methods are useful for thermal systems which are generally governed by nonlinear expressions. However both the function and its derivative must be continuous for the preceding analysis to apply. Thus by setting the gradient equal to zero the locations of the extrema may be obtained and the second derivative may then be used to determine the nature of each extremum. There are cases where both the first and the second derivatives are zero. This indicates an inflection point as sketched in Figure 8.1 c a saddle point or a flat curve as in a ridge or valley. It must be noted that the conditions just mentioned indicate only a local extremum. There may be several such local extrema in the given domain. Since our interest lies in the overall maximum or minimum in the