Question

Most correlations for the convection heat transfer coefficient use the dimensionless Nusselt number, which is defined as (a) \(h / k\) (b) \(\mathrm{k} / \mathrm{h}\) (c) \(h L_{c} / k\) (d) \(k L_{c} / h\) (e) \(k / \rho c_{p}\)

Short answer

a) \(h / k\) b) \(k / h\) c) \(hL_c / k\) d) \(kL_c / h\) e) \(k / \rho c_{p}\) Answer: c) \(hL_c / k\)

Step-by-step solution

Recall the Nusselt Number definition

The Nusselt number (Nu) is a dimensionless number that represents the ratio of the convective heat transfer to the conductive heat transfer. It is generally defined as follows: $$ \nu = \frac{hL_c}{k} $$ where: - \(h\) is the convection heat transfer coefficient - \(L_c\) is a characteristic length - \(k\) is the thermal conductivity Now, let's match this definition with the given options.

Option (a)

\(h / k\) Comparing this option with the definition of the Nusselt Number we just mentioned, it doesn't match, as the characteristic length (L_c) is missing.

Option (b)

\(k / h\) This is simply the reciprocal of option (a), and it still lacks the characteristic length (L_c) term in the formula.

Option (c)

\(h L_{c} / k\) This option matches the definition we recalled earlier, and is the correct formula for the Nusselt Number: $$ \nu = \frac{hL_c}{k} $$

Option (d)

\(k L_{c} / h\) Just like option (b), it is the reciprocal of the correct formula (option c).

Option (e)

\(k / \rho c_{p}\) In this option, the terms \(\rho\) and \(c_{p}\) represent density and specific heat capacity, respectively. This formula does not comply with the Nusselt Number's definition.

Conclusion

Based on the analysis, the correct formula for the Nusselt number is given by option (c) \(h L_{c} / k\).