Question

Jakob (1949) suggests the following correlation be used for square tubes in a liquid crossflow situation: $$ \mathrm{Nu}=0.102 \mathrm{Re}^{0.625} \mathrm{Pr}^{1 / 3} $$ Water \((k=0.61 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \mathrm{Pr}=6)\) flows across a \(1-\mathrm{cm}-\) square tube with a Reynolds number of 10,000 . The convection heat transfer coefficient is (a) \(5.7 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\) (b) \(8.3 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\) (c) \(11.2 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\) (d) \(15.6 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\) (e) \(18.1 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\)

Short answer

Based on the given values and calculations, the convection heat transfer coefficient (h) should be approximately 4.54 kW/(m²∙K). However, none of the provided options match this result. It's possible that there is an error in the given options, or an incorrect version of Jakob's correlation is being used.

Step-by-step solution

Substitute given values into Jakob's correlation for the Nusselt number

Using the given values, we substitute Re = 10000 and Pr = 6 into Jakob's correlation, which is: $$ \mathrm{Nu} = 0.102 \times \mathrm{Re}^{0.625} \times \mathrm{Pr}^{1/3} $$ We get: $$ \mathrm{Nu} = 0.102 \times (10000)^{0.625} \times (6)^{1/3} $$

Calculate the Nusselt number value

Perform the necessary operations to calculate the Nusselt number: $$ \mathrm{Nu} \approx 74.154 $$

Use the Nusselt number definition relation to find the convection heat transfer coefficient

We know that the Nusselt number can be defined as the ratio of the convection heat transfer coefficient (h) to the thermal conductivity (k) multiplied by the characteristic length (L). In this case, L = 0.01m (1 cm), and k = 0.61 W/(m∙K). The formula for Nusselt number is: $$ \mathrm{Nu} = \frac{hL}{k} $$ Substitute the values: $$ 74.154 = \frac{h \times 0.01}{0.61} $$

Solve for the convection heat transfer coefficient

Rearrange the equation and solve for h: $$ h = \frac{74.154 \times 0.61}{0.01} \approx 4535.4 \, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K} $$ Notice that, by converting the answer to kW/(m²∙K), we get: $$ h \approx 4.54 \, \mathrm{kW} / \mathrm{m}^{2} \cdot \mathrm{K} $$ However, none of the provided options match this result. The correct answer should be 4.54 kW/(m²∙K). It's possible that there is an error in the given options, or an incorrect version of Jakob's correlation is being used.