Oil having a specific heat capacity of $c_p=1\mathrm{.}88*{10}^{3}Jk{g}^{-1}{K}^{-1}$ is cooled in a plate heat exchanger from $100$ to $50C.$ The heat carrier medium may only be heated up from $20$ to $40C.$ The mass flow of oil is $0\mathrm{.}277kg{s}^{-1}.$ The dynamic viscosity of oil is temperature independent and is $0\mathrm{.}005Pa*s,$ the heat conductivity of oil is $0\mathrm{.}15W{m}^{-1}{K}^{-1}.$

a$)$ Please calculate the mean logarithmic temperature difference for both, parallel and counter$-$current operation.

$($substitute value $24$ for parallel and $42K$ for counter$-$current operation$)$

b$)$ The Reynolds number in the oil$-$sided part of the heat exchanger is $2000.$ Determine the oil$-$sided heat transfer coefficient with the correlation $=0\mathrm{.}4*(Re*Pr{)}^{(\frac{1}{3})}$ and a hydraulic diameter of $10cm.$

$($substitute value: $42W{m}^{-2}{K}^{-1})$

c$)$ The partial heat transfer coefficient on the side of the heat carrier medium is assumed to be $=100W{m}^{-2}{K}^{-1}.$ Please calculate the overall heat transport coefficient when having a layer thickness of $3mm$ and $$ steel $=15W{m}^{-1}{K}^{-1}.$

$($substitute value $24W{m}^{-2}{K}^{-1})$

d$)$ Please calculate the required exchange areas of the heat exchanger in countercurrent mode.