A counterflow double-pipe heat exchanger cools 0.900 kg/s of a liquid process stream…

A counterflow double-pipe heat exchanger cools 0.900 kg/s of a liquid process stream from 150°C to a required 85.0°C using water entering at 20.0°C at 1.40 kg/s. Take the process-stream heat capacity as 2600 \(\mathrm{J\,kg^{-1}K^{-1}}\) and the water heat capacity as 4180 \(\mathrm{J\,kg^{-1}K^{-1}}\), both constant. The overall heat-transfer coefficient based on the available surface area is 310 \(\mathrm{W\,m^{-2}K^{-1}}\), and heat loss to the surroundings is negligible. Determine the water outlet temperature, verify numerically that counterflow terminal temperature differences remain positive, and calculate the required heat-transfer area using the log-mean temperature difference. The exchanger is assembled from identical 3.00 m modules, each providing 1.25 \(\mathrm{m^2}\) of area; determine the minimum integer number of modules required and the process-stream outlet temperature that this installed area would produce when both inlet temperatures and flow rates remain unchanged. For the installed-area calculation, use the counterflow effectiveness relation \(\varepsilon=[1-\exp(-NTU(1-C_r))]/[1-C_r\exp(-NTU(1-C_r))]\), with \(NTU=UA/C_{\min}\), \(C_r=C_{\min}/C_{\max}\), and \(Q=\varepsilon C_{\min}(T_{h,i}-T_{c,i})\).

First determine the required duty from the specified hot-stream temperature drop, then use it in the cold-stream energy balance.