Question

Water enters a 5-mm-diameter and 13-m-long tube at \(45^{\circ} \mathrm{C}\) with a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\). The tube is maintained at a constant temperature of \(8^{\circ} \mathrm{C}\). The exit temperature of the water is (a) \(4.4^{\circ} \mathrm{C}\) (b) \(8.9^{\circ} \mathrm{C}\) (c) \(10.6^{\circ} \mathrm{C}\) (d) \(12.0^{\circ} \mathrm{C}\) (e) \(14.1^{\circ} \mathrm{C}\) (For water, use $k=0.607 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=6.14, \nu=0.894 \times\( \)10^{-6} \mathrm{~m}^{2} / \mathrm{s}, c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=997 \mathrm{~kg} / \mathrm{m}^{3}$.)

Short answer

#tag_title#Step 2: Determine the flow type#tag_content#Based on the calculated Reynolds number, we can determine the type of flow in the tube. Generally, if Re < 2000, the flow is considered laminar, and if Re > 4000, the flow is considered turbulent. For intermediate values of Re (2000 < Re < 4000), the flow is considered transitional but is typically treated as turbulent for practical purposes. #tag_title#Step 3: Calculate the Nusselt number#tag_content#The Nusselt number is a dimensionless quantity that represents the ratio of convective heat transfer to conductive heat transfer. Depending on the type of flow (laminar or turbulent), we will use different correlations to calculate the Nusselt number. The most common correlations are the Sieder-Tate equation for turbulent flow and the Graetz-Nusselt equation for laminar flow. #tag_title#Step 4: Calculate the heat transfer coefficient#tag_content#The heat transfer coefficient (h) can be calculated using the Nusselt number, thermal conductivity of water (k), and tube diameter (d) with the following formula: h = \(\frac{Nu \cdot k}{d}\) #tag_title#Step 5: Calculate the heat transfer rate#tag_content#Using the heat transfer coefficient, we can calculate the heat transfer rate (Q) using the following formula: Q = \(\)hA\(\Delta T\) where A is the surface area of the tube and \(\Delta T\) is the temperature difference between the tube wall and the water. The surface area of the tube can be calculated as A = \(2 \pi rL\), where r is the radius of the tube and L is the length of the tube. #tag_title#Step 6: Calculate the exit temperature of water#tag_content#Finally, we can calculate the exit temperature of the water using the heat transfer rate, mass flow rate of water (m), and specific heat capacity of water (Cp) using the following formula: \(\Delta T = \frac{Q}{m \cdot Cp}\) The exit temperature of the water can be calculated by adding the temperature change to the initial temperature.

Step-by-step solution

Calculate the Reynolds number

The Reynolds number is a dimensionless quantity that determines the type of flow in the tube, i.e., laminar or turbulent. We can calculate the Reynolds number using the formula: Re = \(\frac{\rho vd}{\nu}\) where \(\rho = 997 \ \mathrm{kg/m^3}\) is the density of water, \(v = 0.3 \ \mathrm{m/s}\) is the velocity, \(d = 0.005 \ \mathrm{m}\) is the diameter of the tube, and \(\nu = 0.894 \times 10^{-6} \ \mathrm{m^2/s}\) is the kinematic viscosity of water.