In a double pipe exchange $(232mm),$ the cold fluid $(c_p=1k\frac{J}{kg*K},$ flow rate $500k\frac{g}{h})$ passes through the pipe and the hot fluid goes through the annular. The inlet and outlet temperatures of cold fluid are $20C$ and $80C,$ and the inlet and outlet temperatures of hot fluid are $150C$ and $90C,$ respectively. The $h_i($heat transfer coefficient inside a pipe$)$ is $700\frac{W}{m^2.C}$ and overall heat transfer coefficient $U_0($based on the outside surface of pipe$)$ is $300\frac{W}{m^2*C},$ respectively. If the heat loss is ignored and the conductivity of pipe wall $($steel$)$ is taken as $45\frac{W}{m.C},$ find:

$(1)$ heat transfer coefficient outside the pipe $h_o?$

$(2)$ the pipe length required for counter flow, in m$?$

$(3)$what is the pipe length required if the heating medium changes to saturated vapor $(140C)$ which condenses to saturated liquid and other conditions keep unchanged?

$(4)$ When the exchanger is used for a year, it is found that it cannot meet the need of production $($the outlet temperature of cold fluid cannot reach $80C),$ explain why?

Solution: