91Ó°ÊÓ

Given: Mass flow rate \(m = 2\,\mathrm{kg/s}\), Tube diameter \(D = 40\,\mathrm{mm}\), Initial temperature \(T_i = 25^{\circ}\mathrm{C}\), Final temperature \(T_f = 75^{\circ}\mathrm{C}\), Tube surface temperature \(T_s = 100^{\circ}\mathrm{C}\). First, determine the water properties at the average temperature (\(T_{avg} = \frac{T_i + T_f}{2} = 50^{\circ}\mathrm{C}\)): 1. Density (\(\rho\)) \(= 990\,\mathrm{kg/m^3}\) - from a water properties table. 2. Specific heat capacity (\(c_p\)) \(= 4187\,\mathrm{J/(kg\,K)}\) - from a water properties table. 3. Thermal conductivity (\(k\)) \(= 0.627\,\mathrm{W/(m\,K)}\) - from a water properties table.

We need to determine the heat transfer rate necessary to raise the water's temperature from \(25^{\circ}\mathrm{C}\) to \(75^{\circ}\mathrm{C}\). Using the formula \(Q = mc_p\Delta T\), we obtain the required heat transfer rate: \(Q = m\times c_p\times (T_f - T_i) = 2\,\mathrm{kg/s} \times 4187\,\mathrm{J/(kg\,K)}\times(75-25)\,\mathrm{K} = 418700\,\mathrm{W}\)

We need to determine the convective heat transfer coefficient (\(h\)) by considering the flow inside the tube as turbulent with a Reynolds number greater than 10000. Using the Dittus-Boelter equation, we can find h: \(h = 0.023\,Re^{0.8}Pr^{n} \frac{k}{D}\), where \(Re = \frac{4m}{\pi D \mu}\) is the Reynolds number, \(Pr = \frac{c_p\mu}{k}\) is the Prandtl number, and the kinematic viscosity (\(\mu\)) at \(50^{\circ}\mathrm{C}\) equals \(6.93\times10^{-4}\,\mathrm{Pa\cdot s}\). The constant 'n' is 0.4 for heating (fluid temperature is increasing).

Now, using Newton's law of cooling, we can find the required tube length: \(Q = hA\Delta T_{lm}\), where \(A=\pi D L\) is the surface area of the tube, \(L\) is the tube length, and \(\Delta T_{lm} = \frac{T_s - T_f - (T_s - T_i)}{ \ln \left(\frac{T_s - T_f}{T_s - T_i}\right)}\) is the log mean temperature difference. Rearranging this equation, we obtain the tube length: \(L = \frac{Q}{h\pi D\Delta T_{lm}}\). Calculate \(L\) using all the obtained values. (b) Required tube lengths for different diameters and flow rates: Perform steps 1 to 4, considering the new given flow rates and diameter values. Calculate the required tube length for each combination of flow rate and diameter by iterating over each value and following the same steps as in part (a). Then, represent the information graphically. (c) Pressure gradient:

For laminar flow, use the Hagen-Poiseuille equation: \(\Delta P = \frac{32\mu QL}{\pi D^4}\). For turbulent flow, use the Darcy-Weisbach equation: \(\Delta P = \frac{4fL\rho v^2}{2D}\), with the friction factor \(f\) given by the Blasius equation: \(f = 0.079\,Re^{-0.25}\), and \(v = \frac{4Q}{\pi D^2}\) being the flow velocity. By assuming the tube wall is smooth and considering the flow rates given in part (b), we calculate the pressure gradient \(\frac{\Delta P}{L}\) as a function of the tube diameter. Plot the results for all three flow rates.