Dimensional analysis of forced convection

Variables and dimensions

Fundamental dimensions: mass M, length L, time T, temperature $\theta$ (heat is expressed in mechanical units, so there are 4).

$n = 7$ variables, $m = 4$ fundamental dimensions, so the number of $\pi$ groups is $n - m = 3$.

Repeating variables

Choose $D$, $V$, $\mu$, $k$ (together they contain $L, T, M, \theta$ and are independent). Each non-repeating variable ($h$, $\rho$, $c_p$) forms one group.

Group 1 with $h$: $\pi_1 = h\,D^aV^b\mu^ck^d$. Equating exponents to zero:

• $M$: $1 + c + d = 0$
• $L$: $a + b - c + d = 0$
• $T$: $-3 - b - c - 3d = 0$
• $\theta$: $-1 - d = 0 \Rightarrow d = -1$

Then $c = 0$, $b = 0$ and $a = 1$, so $\pi_1 = \dfrac{hD}{k} = Nu$.

Group 2 with $\rho$: the same procedure gives $\pi_2 = \dfrac{\rho V D}{\mu} = Re$.

Group 3 with $c_p$: $\pi_3 = \dfrac{c_p\mu}{k} = Pr$.

Result

$$\pi_1 = f(\pi_2,\pi_3)\;\Rightarrow\; Nu = f(Re, Pr)$$

Experiments give the form $Nu = C\,Re^m Pr^n$, for example Dittus-Boelter $Nu = 0.023\,Re^{0.8}Pr^{n}$.

Physical meaning