$(30$ points$)$ see figure below $-$ During excavation of a creek, the hydraulic piston of a heavy machine rests in the water. During extended use, the piston generates significant heat, warranting analysis if a heat exchanger is needed. Model the piston as a cylinder, and use properties for water at the film temperature. Hints: For parts a$.$ and b$.,$ model as a cylinder in cross flow. For parts c$.$ and d$.,$ model as a vertical plate.

a$.$ For the case of flowing water, what is the heat transfer coefficient $\frac{W}{m^2}*(K)?$

b$.$ For the case of flowing water, what is the rate of heat transfer $[$W$]?$

c$.$ For the case of still water, what is the heat transfer coefficient $\frac{W}{m^2}*(K)?$

d$.$ For the case of still water, what is the rate of heat transfer $[$W$]?$

e$.$ For the combined case, i$.$e$.$ considering both forced and free convection, what is the heat transfer coefficient $\frac{W}{m^2}*(K)?$

f$.$ For the combined case, i$.$e$.$ considering both forced and free convection, what is the rate of heat transfer $[$W$]?$