Water enters a circular tube whose walls are maintained at constant temperature at a specified flow rate and temperature. For fully developed turbulent flow, the Nusselt number can be determined from \(\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \mathrm{Pr}^{0.4}\). The correct temperature difference to use in Newton s law of cooling in this case is (a) The difference between the inlet and outlet water bulk temperature. (b) The difference between the inlet water bulk temperature and the tube wall temperature. (c) The log mean temperature difference. (d) The difference between the average water bulk temperature and the tube temperature. (e) None of the above.

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. In this case, the body is the fluid flowing through the circular tube, and the surrounding is the tube wall which is maintained at a constant temperature.

The Nusselt number is a dimensionless number that indicates the relative significance of conduction and convection in a system. In this exercise, the Nusselt number can be determined from the given formula: \(\mathrm{Nu}=0.023\mathrm{Re}^{0.8}\mathrm{Pr}^{0.4}\)

The Reynolds number is the dimensionless quantity that represents the ratio of inertial forces to viscous forces in a fluid, and it helps predict the onset of turbulence. The Prandtl number is the dimensionless quantity that represents the ratio of momentum diffusivity to thermal diffusivity. These two numbers are essential to determine the Nusselt number.

We will assess each option and determine which one is the correct temperature difference for Newton's law of cooling: (a) The difference between the inlet and outlet water bulk temperature. - This option does not consider the wall temperature and focuses only on the fluid's temperature changes, so it is not appropriate. (b) The difference between the inlet water bulk temperature and the tube wall temperature. - This option considers the temperature difference between the fluid and the tube wall but does not account for the change in temperature as the fluid flows through the tube. (c) The log mean temperature difference. - This option is appropriate because it accounts for the temperature variations as the fluid flows through the tube while also considering the temperature of the tube walls. (d) The difference between the average water bulk temperature and the tube temperature. - This option neither considers the inlet and outlet temperature differences nor the log mean temperature difference, so it is not appropriate. (e) None of the above. - We have found an appropriate option above, so this statement is not correct.

The correct temperature difference to use in Newton's law of cooling in this case is the log mean temperature difference (option c).