Question

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 $\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \operatorname{Pr}^{0.4}$. 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 temperatures. (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.

Short answer

(c) The log mean temperature difference.

Step-by-step solution

Option (a) - Inlet and Outlet Water Bulk Temperature Difference

This is the difference between the temperature of the water entering the tube and the temperature of the water after being in contact with the tube wall. This option doesn't take into consideration the temperature difference between the fluid and the wall at any specific location along the tube.

Option (b) - Inlet Water Bulk Temperature Difference and Tube Wall Temperature

This option represents the difference between the temperature of the water entering the tube and the constant temperature of the tube wall. This option only takes into account the starting point of fluid motion in the tube but doesn't account for the temperature difference along the tube.

Option (c) - Log Mean Temperature Difference

The log mean temperature difference is a value that is calculated considering the temperature difference between the water and tube wall at both the inlet and outlet and taking the logarithmic average of those values. This option takes into account the varying temperature difference along the length of the tube.

Option (d) - Average Water Bulk Temperature Difference and Tube Temperature

This option represents the average temperature of the water inside the tube subtracted from the constant tube wall temperature. This option does not specifically consider how the temperature difference varies along the tube length.

Option (e) - None of the Above

If none of the above options accurately describe the correct temperature difference for Newton's law of cooling for this specific situation.

Analyzing the Nusselt Number Formula

The Nusselt number, \(\mathrm{Nu}\), is a dimensionless number that represents the ratio of convective heat transfer to conductive heat transfer. For fully developed turbulent flow, we are given the formula \(\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \operatorname{Pr}^{0.4}\). The Nusselt number depends on the Reynolds number, \(\mathrm{Re}\), and the Prandtl number, \(\operatorname{Pr}\), which are both dimensionless numbers representing the fluid's flow properties. Newton's law of cooling states that the heat transfer rate, \(Q\), between a fluid and a surface is proportional to the temperature difference, \(\Delta T\), between the fluid and the surface: $$Q = hA\Delta T,$$ where \(h\) is the convective heat transfer coefficient and \(A\) is the surface area. The Nusselt number relates the convective heat transfer coefficient to the conductive heat transfer coefficient as follows: $$\mathrm{Nu} = \frac{hL}{k},$$ where \(L\) is the characteristic length (in this case, the tube diameter) and \(k\) is the thermal conductivity of the fluid. From the information given for the problem and the formula for the Nusselt number, it's clear that the correct temperature difference for Newton's law of cooling should take into account the changing temperature difference along the tube length. This leads us to choose the option that accounts for the variation in temperature difference along the tube.

Final Answer

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