Question

A tube with a bell-mouth inlet configuration is subjected to uniform wall heat flux of \(3 \mathrm{~kW} / \mathrm{m}^{2}\). The tube has an inside diameter of \(0.0158 \mathrm{~m}(0.622 \mathrm{in})\) and a flow rate of $1.43 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s}(2.27 \mathrm{gpm})$. The liquid flowing inside the tube is an ethylene glycol-distilled water mixture with a mass fraction of \(2.27\). Determine the fully developed friction coefficient at a location along the tube where the Grashof number is Gr \(=16,600\). The physical properties of the ethylene glycol-distilled water mixture at the location of interest are Pr $=14.85, \nu=1.93 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\(, and \)\mu_{v} / \mu_{s}=1.07$.

Short answer

Question: Calculate the fully developed friction coefficient at the location where the Grashof number is given as 16,600. Answer: To calculate the fully developed friction coefficient, follow the steps below: 1. Calculate the Reynolds number and the effective viscosity ratio using the given information and provided equations. 2. Determine the Nusselt number using the Gnielinski correlation and the modified Blasius correlation for the friction factor. 3. Calculate the heat transfer coefficient using the Nusselt number and the thermal conductivity of the liquid. 4. Find the fully developed friction coefficient using the relation between the heat transfer coefficient and the friction factor.

Step-by-step solution

Calculate the Reynolds number and the effective viscosity ratio

First, we need to determine the Reynolds number (\(Re\)) and the effective viscosity ratio (\(\mu_\mathrm{eff} / \mu_\mathrm{w}\)) at the location of interest. The Reynolds number can be found using the formula: $$ Re = \frac{VD}{\nu}, $$ where \(V\) is the fluid velocity, \(D\) is the inside diameter of the tube, and \(\nu\) is the kinematic viscosity of the liquid. From the given information, we can find the flow rate (\(Q = 1.43 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s}\)) and the cross-sectional area of the tube: $$ A = \frac{\pi D^2}{4} = \frac{\pi (0.0158 \mathrm{~m})^2}{4}. $$ Now, we can calculate the fluid velocity: $$ V = \frac{Q}{A}. $$ With the velocity found, we can determine the Reynolds number. As for the effective viscosity ratio, it is given as: $$ \frac{\mu_\mathrm{eff}}{\mu_\mathrm{w}} = 1 + 0.86 Re^{-0.25} \left(\frac{\mu_\mathrm{v}}{\mu_\mathrm{s}} - 1 \right), $$ where \(\mu_\mathrm{v} / \mu_\mathrm{s}\) is provided in the problem statement.

Calculate the Nusselt number

Next, we have to calculate the Nusselt number (\(Nu\)) using the Gnielinski correlation: $$ Nu = \frac{(f/8)(Re - 1000) Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}, $$ where \(f\) is the friction factor, which can be determined using the modified Blasius correlation: $$ f = 0.078 Re^{-0.22}, $$ and \(Pr\) is the Prandtl number given in the problem statement.

Calculate the heat transfer coefficient

With the Nusselt number, we can calculate the heat transfer coefficient (\(h\)) using the formula: $$ h = \frac{Nu k}{D}, $$ where \(k\) is the thermal conductivity of the liquid. The thermal conductivity of the ethylene glycol-distilled water mixture can be found in the reference tables.

Calculate the fully developed friction coefficient

Finally, we can calculate the fully developed friction coefficient using the relation between heat transfer coefficient and friction factor: $$ \frac{h}{k} = \frac{(f/8)(Re - 1000) Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}. $$ We have all the required variables to find the fully developed friction coefficient at the location where the Grashof number is given as 16,600.