Tham khảo tài liệu 'design and optimization of thermal systems episode 2 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 272 Design and Optimization of Thermal Systems R 2 R 100 EPS R R R R p p p p required solution is e flow rate Pressure FIGURE The results by the Newton-Raphson method for Example . where the increments AR and AP are calculated from the equations r F V. r F Ì I F Ar 1 1 if Ap i -F Ri P OR . vP . V I V I in ì AR Ì A -G R P l oR 1 IP i V i The four partial derivatives in the above equations are calculated for the R and P values at the ith iteration using analytical differentiation of the functions F and G. The iterative process is continued until a convergence criterion of the form F2 G2 is satisfied. Figure shows the computer output for 10-4 and starting values of 2 and 100 for R and P respectively. The results obtained are very close to those obtained earlier by the successive substitution method. The program is simpler to write for the successive substitution method. However the Newton-Raphson method converges at a faster rate due to its second-order convergence. It usually converges if the initial guessed values are not too far from the solution. Nevertheless if divergence occurs the initial guessed values may be varied and iteration repeated until convergence is achieved. Dynamic Simulation of Lumped Systems Dynamic simulation of thermal systems is used for studying the system characteristics at start-up and shutdown for investigating the system response to changes in operating conditions and for design and evaluation of a control scheme. We are interested in ensuring that the system does not go beyond acceptable limits under such transient conditions. For instance at start-up the cooling system of a furnace may not be completely operational resulting in temperature rise beyond safe levels. This consideration is particularly important for electronic systems since their performance is very sensitive to the operating temperature see Figure b . .
