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Design and Optimization of Thermal Systems Episode 3 Part 2

Tham khảo tài liệu 'design and optimization of thermal systems episode 3 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Lagrange Multipliers 497 FIGURE 8.7 Variation of the heat transfer rate Q with dimension L of the power source in Example 8.5. which gives L -L 0.04 m 4 cm 25 The second derivative is given by d2Q 8.611 5L 9 1 -15L 7 4 dL2 I 8 8 At L 0.04 m the second derivative is calculated as 3008.25 a positive quantity indicating that Q is a minimum. Its value is obtained as Q 77.05W. Figure 8.7 shows a sketch of the variation of Q with L and the minimum value is indicated. The problem can also be solved as a constrained one with the objective function and the constraint written as Q 2L 10L3 2 AT5 4 and L AT 5.6 Therefore from Equation 8.9 the optimum is given by the equations AT 5 4 2 15L1 2 XAT 0 4 2L 10 L3 2 AT1 4 XL 0 LAT 5.6 These equations can be solved to yield the optimum as L 0.04 m AT 140 Q 77.05 W X -17.2 It can be shown that if the constraint is increased from 5.6 to 5.7 the heat transfer rate Q becomes 78.77 i.e. an increase of 1.72. This change can also be obtained from the sensitivity coefficient Sc. Here Sc -X 17.2 which gives the change in Q 498 Design and Optimization of Thermal Systems for a change of 1.0 in the constraint. Therefore for a change of 0.1 Q is expected to increase by 1.72. Example 8.6 For the solar energy system considered in Example 5.3 the cost U of the system is given by the expression U 35A 208V where A is the surface area of the collector and V is the volume of the storage tank. Find the conditions for which the cost is a minimum and compare the solution with that obtained in Example 5.3. Solution The objective function is U A V given by the preceding expression. A constraint arises from the energy balance considerations given in Example 5.3 as AI 290 -100 1 5833.3 I V Therefore A may be obtained in terms of V from this equation and substituted in the objective function to obtain an unconstrained problem as U 35 I ss33.3. I 208V Minimum I 290 - 100 V or U 2 941167 208V Minimum Therefore U may be differentiated with respect to V and the .