(Solved): The design of the cooling jacket for the apparatus shown in the figure below calls for the use of ...

The design of the cooling jacket for the apparatus shown in the figure below calls for the use of a copper tube of inner diameter $$D=0.5\mathrm{cm}$$ and total length $$L=4\mathrm{m}$$. Through this tube, cooling water flows with a mean velocity $$U=10\mathrm{cm}/\mathrm{s}$$. The tube is soldered to the side of the apparatus, the temperature of which is $$30^{?}\mathrm{C}$$. Neglecting the thermal resistance due to pure conduction across the copper wall of the tube, we can model the tube wall as isothermal with the temperature $$T_{\mathrm{w}}=30^{?}\mathrm{C}$$. The water stream enters the tube with the mean temperature $$T_i=20^{?}\mathrm{C}$$. In the calculations proposed below, the physical properties of water can be evaluated at $$25^{?}\mathrm{C}$$. Assume also that the tube is sufficiently straight; that is, neglect the effect of bends. (a) Verify that the flow is hydrodynamically fully developed over most of the tube length. Determine the pressure drop between inlet and outlet of tube. (b) Also verify that the flow is thermally fully developed along most of the tube. (c) Calculate the mean outlet temperature of the water stream, and (d) the total heat transfer rate extracted by the stream from the tube wall (i.e., from the apparatus to which the tube is attached.