In a shell and tube heat exchanger, steam of known temperature $T_S(K)$ flowing around the tube is heating a liquid $L$ of density ${}_L(\frac{kg}{m^3})$ and constant velocity $v(\frac{m}{s})$ from temperature $T_{L1}$ to $T_{L2}.($K$)$ The temperature in the tube varies obviously with axial direction $z,$ radial direction $r$ and time $t.$ To simplify the problem, assume that there are no change in the radial direction. This assumption is valid if the radius is small and no large amount of heat is transferred. The heat transfer from the steam to the liquid depends on the heat transfer coefficient on the steam side $(h{t}_0)$ and on the transfer on the liquid side $h{t}_i.$ Also neglect the thermal capacity of the metal wall separating the steam and the liquid and assume that the exchange between the steam and liquid occurs with an overall heat transfer coefficient $U(\frac{W}{m^2}(K)).$ Also assume constant heat capacity for the liquid. Develop a distributed model for this heat exchanger.