Heat is generated uniformly in a long solid cylindrical rod ( diameter $$ = 10\,\,mm$$) at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The thermal conductivity of the rod material is $$25$$ $$W/m.K.$$ Under steady state conditions, the temperature difference between the center and the surface of the rod is ________________ $${}^ \circ C.$$

Two black surfaces, $$AB$$ and $$BC,$$ of lengths $$5m$$ and $$6m,$$ respectively, are oriented as shown. Both surfaces extend infinitely into the third dimension. Given that view factor $${F_{12}} = 0.5,\,\,{T_1} = 800\,\,K,\,\,{T_2} = 600\,\,K,$$ $${T_{surrounding}} = 300\,\,K$$ and Stefan Boltzmann constant, $$\sigma = 5.67 \times {10^{ - 8}}\,\,W/\left( {{m^2}{K^4}} \right),$$ the heat transfer rate from Surface $$2$$ to the surrounding environment is ____________ $$kW.$$

Saturated steam at $${100^ \circ }C$$ condenses on the outside of a tube. Cold fluid enters the tube at $${20^ \circ }C$$ and exits at $${50^ \circ }C.$$ The value of the Log Mean Temperature Difference $$(LMTD)$$ is _________ $$^ \circ C.$$

Two models, $$P$$ and $$Q,$$ of a product earn profits of Rs. $$100$$ and Rs. $$80$$ per piece, respectively. Production times for $$P$$ and $$Q$$ are $$5$$ hours and $$3$$ hours, respectively, while the total production time available is $$150$$ hours. For a total batch size of $$40,$$ to maximize profit, the number of units of $$P$$ to be produced is ____________.