Question

Consider a laminar ideal gas flow over a flat plate, where the local Nusselt number can be expressed as $\mathrm{Nu}_{x}=0.332 \mathrm{Re}_{x}^{1 / 2} \operatorname{Pr}^{1 / 3}$. Using the expression for the local Nusselt number, show that it can be rewritten in terms of local convection heat transfer coefficient as \(h_{x}=C[V /(x T)]^{w}\), where \(C\) and \(m\) are constants.

Short answer

Question: Show that the given expression for the local Nusselt number, \(\mathrm{Nu}_{x}=0.332 \mathrm{Re}_{x}^{1 / 2} \operatorname{Pr}^{1 / 3}\), can be rewritten in terms of the local convection heat transfer coefficient as \(h_{x}=C[V /(x T)]^{w}\). Answer: We have shown that the given expression for the local Nusselt number can be rewritten in terms of the local convection heat transfer coefficient as \(h_{x}=C[V /(x T)]^{w}\) with \(C = 0.332 \left[\frac{k^\frac{4}{3} \rho^\frac{1}{2} C_p^\frac{1}{3}}{\mu^\frac{1}{6}}\right]\) and \(w = \frac{1}{6}\).

Step-by-step solution

Define Nusselt number and given expression

The Nusselt number is a dimensionless quantity that represents the ratio of convective heat transfer to conductive heat transfer. The given expression for local Nusselt number is: \(\mathrm{Nu}_{x}=0.332 \mathrm{Re}_{x}^{1 / 2} \operatorname{Pr}^{1 / 3}\)

Utilize definitions of Reynolds number and Prandtl number

The Reynolds number (Re) is a dimensionless quantity representing the ratio of inertial forces to viscous forces in a fluid flow. It is defined as: \(\mathrm{Re}_{x}=\frac{\rho V x}{\mu}\) The Prandtl number (Pr) is a dimensionless quantity comparing the relative thickness of momentum and thermal boundary layers. It is defined as: \(\operatorname{Pr}=\frac{\mu C_{p}}{k}\)

Isolate \(h_x\) (local convection heat transfer coefficient)

Now, we need to connect \(\mathrm{Nu}_{x}\) to the local convection heat transfer coefficient. The local Nusselt number can also be expressed as: \(\mathrm{Nu}_{x}=\frac{h_{x} x}{k}\) The equation now becomes: \(h_x x = 0.332 k\left(\frac{\rho V x}{\mu}\right)^\frac{1}{2} \left(\frac{\mu C_p}{k}\right)^\frac{1}{3}\)

Rewrite the expression to the desired format

Our goal is to rewrite the expression in the format \(h_{x}=C[V /(x T)]^{w}\). By isolating \(h_x\), we get: \(h_x = 0.332 \left[\frac{k^\frac{4}{3} \rho^\frac{1}{2} C_p^\frac{1}{3} V^\frac{1}{2}}{\mu^\frac{1}{6}}\right] x^\frac{1}{6}\) Compare this to the desired format \(h_{x}=C[V /(x T)]^{w}\): C = \(0.332 \left[\frac{k^\frac{4}{3} \rho^\frac{1}{2} C_p^\frac{1}{3}}{\mu^\frac{1}{6}}\right]\) w = \(\frac{1}{6}\) Now we have proved that the local Nusselt number can be rewritten in terms of local convection heat transfer coefficient as: \(h_{x}=C[V /(x T)]^{w}\)