A solid sphere of radius$${r_1} = 20\,\,mm$$ is placed concentrically inside a hollow sphere of radius $${r_2} = 30\,\,mm$$ as shown in the figure.

The view factor $${F_{21}}$$ for radiation heat transfer is

A double-pipe counter-flow heat exchanger transfers heat between two water streams. Tube side water at $$19$$ liter/s is heated from $${10^ \circ }C$$ to $${38^ \circ }C$$. Shell side water at $$25$$ liter/s is entering at $${46^ \circ }C$$. Assume constant properties of water; density is $$1000\,kg/{m^3}$$ and specific heat is $$4186$$ $$J/kg.K.$$ The $$LMTD$$ (in $${}^ \circ C$$) is ____________.

A manufacturer can produce $$12000$$ bearings per day. The manufacturer received an order of $$8000$$ bearings per day from a customer. The cost of holding a bearing in stock is Rs. $$0.20$$ per month. Setup cost per production run in Rs.$$500.$$ Assuming $$300$$ working days in a year, the frequency of production run should be

Consider the given project network, where numbers along various activities represent the normal time. The free float on activity $$4$$-$$6$$ and the project duration, respectively, are