$(15$ points$)$ Water at an average temperature of $80C$ flows through a pipe with an inner diameter of $0\mathrm{.}0266$ m and an outer diameter of $0\mathrm{.}0334$ m at a velocity of $2\mathrm{.}0\frac{m}{s}.$ The water is heated by saturated steam at $108C$ surrounding the pipe. The heat transfer coefficient on the steam side is $10500\frac{W}{m^2}*K.$ Calculate the heat transfer rate $q$ when the length of the pipe is $0\mathrm{.}305$ m $.$

Given:

PrandtI number of water at $80C$:$N_{Pr}=2\mathrm{.}72$

Density of water: $=980k\frac{g}{m^3}$

Thermal conductivity of water: $k=0\mathrm{.}633\frac{W}{m}*K$

Viscosity of water: $=0\mathrm{.}000432Pa*s$

Viscosity of water at the pipe wall temperature: ${}_w=0\mathrm{.}000356Pa*s$

Thermal conductivity of the pipe material: $k_{\text{p}\text{i}\text{p}\text{e}}=45\frac{W}{m}*K$