Training Convection & Dimensionless Numbers Reynolds and Prandtl Numbers
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Reynolds and Prandtl Numbers

24 min Convection & Dimensionless Numbers

Reynolds and Prandtl Numbers

Dimensionless numbers let us collapse complex fluid dynamics into universal relationships. The Reynolds number tells us the flow regime; the Prandtl number describes the fluid’s thermal behavior.

Reynolds Number

$$Re = \frac{\rho V L_c}{\mu} = \frac{V L_c}{\nu}$$

Ratio of inertial to viscous forces. $V$ = velocity, $L_c$ = characteristic length, $\mu$ = dynamic viscosity, $\nu = \mu/\rho$ = kinematic viscosity.

Internal flow (pipe): $L_c = D$ (diameter). Transition at $Re_D \approx 2300$.
External flow (flat plate): $L_c = x$ (distance from leading edge). Transition at $Re_x \approx 5 \times 10^5$.

Prandtl Number

$$Pr = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}$$

Ratio of momentum diffusivity to thermal diffusivity. $\alpha = k/(\rho c_p)$.

Liquid metals: $Pr \ll 1$ • Gases: $Pr \approx 0.7$ • Water: $Pr \approx 7$ • Oils: $Pr \gg 1$ (50–100,000)

Example 1 — Pipe Flow Regime

Water ($\nu = 1.0 \times 10^{-6}$ m²/s) flows at 2 m/s in a 5 cm pipe. Laminar or turbulent?

$$Re = \frac{VD}{\nu} = \frac{2 \times 0.05}{1.0 \times 10^{-6}} = 100{,}000$$

$Re = 100{,}000 \gg 2300$ → turbulent.

Example 2 — Air over a Plate

Air ($\nu = 1.5 \times 10^{-5}$ m²/s) flows at 10 m/s over a flat plate. At what distance $x$ does transition occur?

$$x_{\text{cr}} = \frac{Re_{\text{cr}} \cdot \nu}{V} = \frac{5 \times 10^5 \times 1.5 \times 10^{-5}}{10} = 0.75 \text{ m}$$

Example 3 — Prandtl Number of Water

Water at 20°C: $\mu = 1.002 \times 10^{-3}$ Pa·s, $c_p = 4182$ J/(kg·K), $k = 0.598$ W/(m·K). Find $Pr$.

$$Pr = \frac{\mu c_p}{k} = \frac{1.002 \times 10^{-3} \times 4182}{0.598} = 7.01$$

This matches the known value. Water’s velocity boundary layer is ~7× thinner than its thermal boundary layer.

Practice Problems

1. Oil ($\nu = 5 \times 10^{-4}$) at 0.5 m/s in a 10 cm pipe. $Re$? Laminar or turbulent?
2. Air at 30 m/s, $\nu = 1.6 \times 10^{-5}$ m²/s. $Re$ at $x = 1$ m?
3. Find $Pr$ for air: $\mu = 1.85 \times 10^{-5}$, $c_p = 1005$, $k = 0.026$.
4. Liquid mercury: $Pr = 0.025$. Is its thermal BL thicker or thinner than the velocity BL?
5. Water velocity in a 2 cm pipe for $Re = 2300$? ($\nu = 10^{-6}$)
6. At what speed does air transition to turbulent in a 15 cm duct? ($\nu = 1.5 \times 10^{-5}$)
Show Answer Key

1. $Re = 0.5 \times 0.1 / (5 \times 10^{-4}) = 100$. Laminar.

2. $Re = 30 \times 1 / (1.6 \times 10^{-5}) = 1{,}875{,}000$. Turbulent ($> 5 \times 10^5$).

3. $Pr = 1.85 \times 10^{-5} \times 1005 / 0.026 = 0.715$

4. Thicker — heat diffuses faster than momentum when $Pr < 1$.

5. $V = Re \cdot \nu / D = 2300 \times 10^{-6} / 0.02 = 0.115$ m/s

6. $V = 2300 \times 1.5 \times 10^{-5} / 0.15 = 0.23$ m/s

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