Nusselt Correlation Derivation

Nusselt Correlation

N u = \frac{h L}{k} = C \cdot R e^{m} \cdot P r^{n}

Reynolds and Prandtl

R e = \frac{V L}{ν}, P r = \frac{ν ρ C_{p}}{k}

Plugging them in

\frac{h L}{k} = C {(\frac{V L}{ν})}^{m} {(\frac{ν ρ C_{p}}{k})}^{n}

Rearranging

h = \frac{k}{L} \cdot C \cdot \frac{V^{m} L^{m}}{ν^{m}} \cdot \frac{ν^{n} ρ^{n} C_{p}^{n}}{k^{n}}

Simplified Expression

h = C \cdot L^{m - 1} \cdot V^{m} \cdot ν^{n - m} \cdot ρ^{n} \cdot C_{p}^{n} \cdot k^{1 - n}

Note how the heat transfer coefficient is related to Velocity to the
{ #m}
power. From what I've seen m is generally about 0.6-0.7 (or lower.() With knowledge of the fan affinity laws V is proportional to P^3 . With the m coefficent of 0.7 my best guess is that doubling the cooling ability needs 16x the amount of power. (later on I actually calculate this correlation from experimental testing)