For a multicomponent mixture containing any number of species, prove that M ¯ i = M + ( ∂ M ___ ∂ x i ) T, P − ∑ k x k ( ∂ M ___ ∂ x k ) T, P where the summation is over all species. Show that for a binary mixture this result reduces to Eqs. (10.15) and (10.16).

Athin walled concentric tube exchanger is used to cool engine oil from 160°c to 60°c with water that is available at 25°c acting as a coolant. the oil and water flow rates are each at 2 kg/s, and the diameter of the inner tube is 0.5 m and the corresponding value of the overall heat transfer coefficient is 250 w/m2. oc. how long must the heat exchanger be to accomplish the desired cooling? cpwater=4.187 kj/kg-candcpengine el=2.035 kj/kg·°c, oil . 120]