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

Two cylindrical shafts $$A$$ and $$B$$ at the same initial temperature are simultaneously placed in a furnace. The surfaces of the shafts remain at the furnace gas temperature at all times after they are introduced into the furnace. The temperature variation in the axial direction of the shafts can be assumed to be negligible. The data related to shafts $$A$$ and $$B$$ is given in the following Table.

The temperature at the center-line of the shaft $$A$$ reaches $${400^ \circ }C$$ after two hours. The time required (in hours) for the center-line of the shaft $$B$$ to attain the temperature of $${400^ \circ }C$$ 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

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 ______________