Question

An ethylene glycol-distilled water mixture with a mass fraction of \(0.72\) and a flow rate of \(2.05 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s}\) flows inside a tube with an inside diameter of \(0.0158 \mathrm{~m}\) and a uniform wall heat flux boundary condition. For this flow, determine the Nusselt number at the location \(x / D=10\) for the inlet tube configuration of \((a)\) bell-mouth and \((b)\) re-entrant. Compare the results for parts \((a)\) and \((b)\). Assume the Grashof number is \(\mathrm{Gr}=60,000\). The physical properties of an ethylene glycoldistilled water mixture are $\operatorname{Pr}=33.46, \nu=3.45 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\(, and \)\mu_{b} / \mu_{s}=2.0$.

Short answer

The Nusselt number for the bell-mouth inlet configuration is approximately 63.4, while for the re-entrant inlet configuration, it is approximately 70.7. This difference indicates a higher convective heat transfer coefficient for the re-entrant inlet case, demonstrating the influence of inlet configuration on the heat transfer performance of the tube.

Step-by-step solution

Calculate the Reynolds number

First, we need to calculate the Reynolds number (Re) using the formula: $$ \mathrm{Re}=\frac{4 \times Q}{\pi \times D \times \nu} $$ Where: Q = flow rate (\(2.05 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s}\)) D = inside diameter (\(0.0158 \mathrm{~m}\)) ν = kinematic viscosity of the fluid mixture (\(3.45 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\)) Plugging in the given values, we can calculate the Reynolds number: $$ \mathrm{Re}=\frac{4 \times 2.05 \times 10^{-4} }{\pi \times 0.0158 \times 3.45 \times 10^{-6}}\approx3291 $$

Calculate Nusselt number for bell-mouth inlet

For the bell-mouth inlet configuration, we can use the correlation provided by Shah (1978) for the Nusselt number (Nu): $$ \mathrm{Nu}=\left[0.0153 \times \mathrm{Re}^{0.85} \times \mathrm{Pr}^{0.93}\right] (1+\mathrm{Gr}^{-0.5}) $$ Where: Re = Reynolds number (calculated in Step 1) Pr= Prandtl number (\(33.46\)) Gr = Grasshof number (\(60,000\)) Plug in the values: $$ \mathrm{Nu}_{\text{bell-mouth}}=\left[0.0153 \times 3291^{0.85} \times 33.46^{0.93}\right] (1 + 60,000^{-0.5}) \approx 63.4 $$

Calculate Nusselt number for re-entrant inlet

For the re-entrant inlet configuration, we need to use another correlation provided by Shah(1978) for the Nusselt number (Nu): $$ \mathrm{Nu}=\left[0.01227 \times \mathrm{Re}^{0.87} \times \mathrm{Pr}^{0.93}\right](1+(0.0163 \times \mu_{b} / \mu_{s})^{0.85}) $$ Where: Re = Reynolds number (calculated in Step 1) Pr = Prandtl number (\(33.46\)) \(\mu_{b} / \mu_{s}= 2.0\) Plug in the values: $$ \mathrm{Nu}_{\text{re-entrant}}=\left[0.01227 \times 3291^{0.87} \times 33.46^{0.93}\right](1+(0.0163 \times 2.0)^{0.85}) \approx 70.7 $$

Compare results for bell-mouth and re-entrant inlets

We have calculated the Nusselt numbers for both bell-mouth and re-entrant inlet configurations: - Bell-mouth: Nu \(\approx 63.4\) - Re-entrant: Nu \(\approx 70.7\) The Nusselt number for the re-entrant inlet configuration is greater than that for the bell-mouth inlet configuration, indicating a higher convective heat transfer coefficient for the re-entrant inlet case. This difference in the Nusselt numbers shows the influence of inlet configuration on the heat transfer performance of the tube.