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 ______________

A food processing company uses $$25,000$$ $$kg$$ of corn flour every year. The quantity-discount price of corn flour is provided in the table below:

The order processing charges are Rs. $$500$$/order. The handling plus carry-over charge on an annual basis is $$20\% $$ of the purchase price of the corn flour per $$kg$$. The optimal order quantity (in $$kg$$) 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 _____________