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

24 min Convection & Dimensionless Numbers

The Reynolds number Re and Prandtl number Pr are dimensionless groups that characterize fluid flow and heat transfer. Re = ρvL/μ compares inertial to viscous forces and determines whether a flow is laminar or turbulent, while Pr = μ cp/k compares momentum diffusivity to thermal diffusivity. Together they form the inputs to virtually every convective heat-transfer correlation, making them the most important parameters in the field.

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?

  1. $$Re = \frac{VD}{\nu} = \frac{2 \times 0.05}{1.0 \times 10^{-6}} = 100{,}000$$
  2. $Re = 100{,}000 \gg 2300$ → 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?

  1. Apply Newton's law of cooling:
  2. $q = hA(T_s - T_\infty)$.
  3. $$x_{\text{cr}} = \frac{Re_{\text{cr}} \cdot \nu}{V} = \frac{5 \times 10^5 \times 1.5 \times 10^{-5}}{10} = 0.75 \text{ m}$$
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 $Pr$.

  1. $$Pr = \frac{\mu c_p}{k} = \frac{1.002 \times 10^{-3} \times 4182}{0.598} = 7.01$$
  2. This matches the known value.
  3. 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

🌊 Reynolds & Prandtl Numbers
Reynolds number Re
Flow regime
Prandtl number Pr
Newton’s Law of Cooling and Convection Basics Nusselt Number and Forced Convection Correlations