Consider a long cylindrical tube of inner and outer radii, $${r_i}$$ and $${r_0}$$ , respectively, length, $$L$$ and thermal conductivity, $$k.$$ Its inner and outer surfaces are maintained at $${T_i}$$ and $${T_0}$$ , respectively $$\left( {{T_i}\,\,\, > \,\,\,{T_0}} \right).$$ Assuming one-dimensional steady state heat conduction in the radial direction, the thermal resistance in the wall of the tube is

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 ____________.

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