Suppose that each edge e = (u, v) in a flow network g = (v, e) has not only an upper bound ce on the net flow from u to v, but also a lower bound be. that is, any flow f in the network must satisfy be ≤ f(e) ≤ ce, for all edges e. it may be the case that for a network no such feasible flow exists. for a cut (a, b), let us define c(a, b) = p e=(u, v),u∈a, v∈b ce, and b(a, b) = p e=(u, v),u∈a, v∈b be.
(a) prove that if f is a flow in the network, then the value v(f) of the flow satisfies v(f) ≤ c(a, b) − b(b, a) for any cut (a, b) of g.
(b) prove that the value of a maximum flow in the network, if it exists, is the minimum value of c(a, b) − b(b, a) over all cuts (a, b) of the network.

Amass of m 1.5 kg of steam is contained in a closed rigid container. initially the pressure and temperature of the steam are: p 1.5 mpa and t 240°c (superheated state), respectively. then the temperature drops to t2= 100°c as the result of heat transfer to the surroundings. determine: a) quality of the steam at the end of the process, b) heat transfer with the surroundings. for: p1.5 mpa and t 240°c: enthalpy of superheated vapour is 2900 kj/kg, specific volume of superheated vapour is 0. 1483 m/kg, while for t 100°c: enthalpy of saturated liquid water is 419kj/kg, specific volume of saturated liquid water is 0.001043m/kg, enthalpy of saturated vapour is 2676 kj/kg, specific volume of saturated vapour is 1.672 m/kg and pressure is 0.1 mpa.

Steel is coated with a thin layer of ceramic to protect against corrosion. what do you expect to happen to the coating when the temperature of the steel is increased significantly? explain.

Draw the engineering stress-strain curve for (a) bcc; (b) fcc metals and mark important points.