A hollow cylinder has length $$L,$$ inner radius $${{r_1}}$$, outer radius $${{r_2}}$$, and thermal conductivity $$k.$$ The thermal resistance of the cylinder for radial conduction is

Consider a parallel-flow heat exchanger with area $${A_p}$$ and a counter-flow heat exchanger with area $${A_c}.$$ In both the heat exchangers, the hot stream flowing at $$1$$ $$kg/s$$ cools from $$80{}^ \circ C$$ to $$50{}^ \circ C$$. For the cold stream in both the heat exchangers, the flow rate and the inlet temperature are $$2$$ $$kg/s$$ and $$10{}^ \circ C$$, respectively. The hot and cold streams in both the heat exchangers are of the same fluid. Also, both the heat exchangers have the same overall heat transfer coefficient. The ratio $${A_c}/{A_p}$$ is ______________

Consider the radiation heat exchange inside an annulus between two very long concentric cylinders. The radius of the outer cylinder is $${R_0}$$ and that of the inner cylinder is $${R_i}$$ . The radiation view factor of the outer cylinder onto itself is

A project consists of $$14$$ activities, $$A$$ to $$N$$. The duration of these activities (in days) are shown in brackets on the network diagram. The latest finish time (in days) for node $$10$$ is _____________