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In a $1-1$ pass shell and tube exchanger, steam is condensing in the shell at a temperature $(T_{s})$ of $135^{\circ}C$ and the cold fluid is heated from a temperature $(T_{1})$ of $20^{\circ}C$ to a temperature $(T_{2})$ of $90^{\circ}C$. The energy balance equation for this heat exchanger is

$$In\:\frac{T_{s}-T_{1}}{T_{s}-T_{2}}=\frac{UA}{mc}_{p}$$

where $U$ is the overall heat transfer coefficient, $A$ is the heat transfer area, $\dot{m}$ is the mass flow rate of the cold fluid and $c_{p}$ is its specific heat. Tube side fluid is in a turbulent flow and the heat transfer coefficient can be estimated from the following equation:

$$Nu=0.023\:\left ( Re \right )^{0.8}\left ( Pr \right )^{1/3}$$

where $Nu$ is the Nusselt number, $Re$ is the Reynolds number and $Pr$ is the Prandtl number. The condensing heat transfer coefficient in the shell side is significantly higher than the tube side heat transfer coefficient. The resistance of the wall to heat transfer is negligible. If only the mass flow rate of the cold fluid is doubled, what is the outlet temperature (in $^{\circ}C$) of the cold fluid at steady state?

  1. $80.2$
  2. $84.2$
  3. $87.4$
  4. $88.6$
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