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O A heated fluid at temperature T (degrees above ambient temperature) flows in a pipe with fixed length and circular cross section with radius r. A layer of insulation, with thickness w

O A heated fluid at temperature T (degrees above ambient temperature) flows in a pipe with fixed length and circular cross section with radius r. A layer of insulation, with thickness w

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.4: Hyperbolas
Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
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2. Problem 4.30 in [Boyd-Vandenberghe]. Solve the problem on CVX for the following parameters: α₁ = .5, α₂ = .7, α3 = .9, α₁4 = .1; 03: Cmax = 100; Tmin= 5, Tmax = 20; Tmin= 10, Tmax 30; Wmin = 1, wmax = 5. (Note: You must use CVX in geometric programming mode.)
Transcribed Image Text:2. Problem 4.30 in [Boyd-Vandenberghe]. Solve the problem on CVX for the following parameters: α₁ = .5, α₂ = .7, α3 = .9, α₁4 = .1; 03: Cmax = 100; Tmin= 5, Tmax = 20; Tmin= 10, Tmax 30; Wmin = 1, wmax = 5. (Note: You must use CVX in geometric programming mode.)
4.30 A heated fluid at temperature T (degrees above ambient temperature) flows in a pipe with fixed length and circular cross section with radius r. A layer of insulation, with thickness w<r, surrounds the pipe to reduce heat loss through the pipe walls. The design variables in this problem are T, r, and w. The heat loss is (approximately) proportional to Tr/w, so over a fixed lifetime, the energy cost due to heat loss is given by a₁Tr/w. The cost of the pipe, which has a fixed wall thickness, is approximately proportional to the total material, i.e., it is given by a2r. The cost of the insulation is also approximately proportional to the total insulation material, i.e., a3rw (using w < r). The total cost is the sum of these three costs. The heat flow down the pipe is entirely due to the flow of the fluid, which has a fixed velocity, i.e., it is given by a Tr². The constants ai are all positive, as are the variables T, r, and w. Now the problem: maximize the total heat flow down the pipe, subject to an upper limit Cmax on total cost, and the constraints Tmin ≤ T ≤ Tmax, 1min ≤r≤rmax, Express this problem as a geometric program. Wmin w Wmax, w ≤ 0.1r.
Transcribed Image Text:4.30 A heated fluid at temperature T (degrees above ambient temperature) flows in a pipe with fixed length and circular cross section with radius r. A layer of insulation, with thickness w<r, surrounds the pipe to reduce heat loss through the pipe walls. The design variables in this problem are T, r, and w. The heat loss is (approximately) proportional to Tr/w, so over a fixed lifetime, the energy cost due to heat loss is given by a₁Tr/w. The cost of the pipe, which has a fixed wall thickness, is approximately proportional to the total material, i.e., it is given by a2r. The cost of the insulation is also approximately proportional to the total insulation material, i.e., a3rw (using w < r). The total cost is the sum of these three costs. The heat flow down the pipe is entirely due to the flow of the fluid, which has a fixed velocity, i.e., it is given by a Tr². The constants ai are all positive, as are the variables T, r, and w. Now the problem: maximize the total heat flow down the pipe, subject to an upper limit Cmax on total cost, and the constraints Tmin ≤ T ≤ Tmax, 1min ≤r≤rmax, Express this problem as a geometric program. Wmin w Wmax, w ≤ 0.1r.
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