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 wr, 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., aзrw (using wr). 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 a4Tr². The constants a 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, Tmin <≤rmax, Wmin WWmax, w≤0.1r. Express this problem as a geometric program. Solution. The problem is maximize subject to a Tr² a Tw¹ + a2r+aзrwCmax Tmin STTmax Tmin Tmax Wmin wwmax w≤0.1r. This is equivalent to the GP minimize (1/04)T-17-2 subject to (a/Cmax)Tw¹ + (02/Cmax) + (α3/Cmax)rw≤1 (1/Tmax)T1, Tmin T≤1 (1/Tmax) 1, Tmin-1 <1 (1/wmax)w1, Wminw 10wr¹≤1. ≤1