Nusselt Number Correlations

Why This Matters

The Nusselt number is your bridge between fluid mechanics and heat transfer—it tells you how much better convection is at moving heat compared to pure conduction. When you're designing heat exchangers, cooling systems, or analyzing industrial processes, you need to predict convective heat transfer coefficients without running expensive experiments. That's exactly what these correlations do: they connect dimensionless groups you can calculate ($Re$, $Pr$, $Ra$) to the Nusselt number, which gives you $h$.

You're being tested on knowing which correlation applies to which physical situation—forced vs. natural convection, internal vs. external flow, laminar vs. turbulent regimes. The exam won't just ask you to plug numbers into equations; it'll test whether you understand the underlying physics: Why does the Prandtl number exponent differ? When does viscosity variation matter? What makes natural convection fundamentally different from forced convection? Don't just memorize formulas—know what flow regime and geometry each correlation assumes.

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Forced Convection: Internal Pipe Flow

Internal pipe flow correlations dominate heat exchanger design and process engineering. The key physics here involves boundary layer development within a confined geometry, where the flow regime (laminar, transitional, or turbulent) dramatically affects heat transfer rates.

Dittus-Boelter Correlation

• Turbulent flow in smooth circular pipes with $Re > 10{,}000$—the most widely used correlation for fully developed turbulent internal flow
• Correlation: $Nu = 0.023 \, Re^{0.8} \, Pr^n$ where $n = 0.4$ for heating and $n = 0.3$ for cooling (the exponent changes based on heat flow direction)
• Assumes constant properties—best for moderate temperature differences where viscosity doesn't vary significantly across the flow

Sieder-Tate Correlation

• Accounts for viscosity variation between bulk fluid and wall temperature—critical for oils and other viscous fluids
• Correlation: $Nu = 0.027 \, Re^{0.8} \, Pr^{1/3} \left(\frac{\mu_b}{\mu_w}\right)^{0.14}$ where the viscosity ratio corrects for property gradients
• Use when temperature differences are large—the viscosity correction factor captures how fluid near the wall behaves differently than the bulk

Gnielinski Correlation

• Covers transitional and turbulent flow ($3{,}000 < Re < 5 \times 10^6$)—fills the gap where Dittus-Boelter fails
• Correlation: $Nu = \frac{(f/8)(Re - 1000)Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}$ where $f$ is the Darcy friction factor
• Most accurate general-purpose correlation—preferred in modern engineering calculations for its wider validity range

Petukhov Correlation

• Friction-factor-based approach for turbulent flow in smooth tubes—explicitly links momentum and heat transfer
• Correlation structure mirrors Gnielinski—both derive from similar theoretical foundations connecting wall shear to heat flux
• Valid for $10^4 < Re < 5 \times 10^6$ and $0.5 < Pr < 2000$—excellent for liquid metals to viscous oils

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Forced Convection: External Flow

External flow correlations apply when fluid moves over a surface rather than through a channel. The physics shifts to boundary layer growth without confinement, where upstream conditions and geometry shape the heat transfer.

Flat Plate Correlation (Laminar)

• Laminar boundary layer flow over a flat plate—foundational correlation derived from exact boundary layer solutions
• Correlation: $Nu_x = 0.332 \, Re_x^{1/2} \, Pr^{1/3}$ gives local values; average $\overline{Nu}_L = 0.664 \, Re_L^{1/2} \, Pr^{1/3}$
• Valid for $Re_x < 5 \times 10^5$ and $Pr > 0.6$—the square-root dependence on $Re$ reflects laminar boundary layer physics

Hilpert Correlation

• Single cylinder in crossflow—uses the form $Nu = C \, Re^m \, Pr^{1/3}$ where $C$ and $m$ depend on Reynolds number range
• Accounts for flow separation around the cylinder—wake formation behind the cylinder complicates heat transfer patterns
• Empirical constants vary with $Re$—you'll typically be given the appropriate $C$ and $m$ values for the flow regime

Zhukauskas Correlation

• Cylinder banks in crossflow—essential for shell-and-tube heat exchanger design
• Correlation: $Nu = C \, Re^m \, Pr^{0.36} \left(\frac{Pr}{Pr_w}\right)^{0.25}$ with constants depending on arrangement (staggered vs. inline) and row number
• Staggered arrangements enhance mixing—turbulence generated by upstream cylinders increases heat transfer in downstream rows

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Natural Convection Correlations

Natural convection occurs when buoyancy forces drive fluid motion due to density differences from temperature gradients. The Rayleigh number ($Ra = Gr \cdot Pr$) replaces Reynolds number as the key parameter, combining buoyancy and viscous effects.

Churchill-Chu Correlation (Vertical Plates)

• Covers laminar through turbulent natural convection on vertical surfaces—a single correlation spanning $10^{-1} < Ra < 10^{12}$
• Correlation: $\overline{Nu} = \left[0.825 + \frac{0.387 \, Ra^{1/6}}{(1 + (0.492/Pr)^{9/16})^{8/27}}\right]^2$ handles the full range elegantly
• Transition occurs near $Ra \approx 10^9$—below this, flow is laminar; above, turbulent plumes dominate

McAdams Correlation (Horizontal Plates)

• Orientation matters critically—hot surface facing up or cold surface facing down promotes convection; opposite orientations suppress it
• Laminar correlation: $Nu = 0.54 \, Ra^{1/4}$ for $10^4 < Ra < 10^7$ (hot side up); turbulent: $Nu = 0.15 \, Ra^{1/3}$ for $10^7 < Ra < 10^{11}$
• Simpler geometry, simpler physics—horizontal plates avoid the complexity of vertical boundary layer development

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Analogy Methods

Analogy methods exploit the mathematical similarity between momentum, heat, and mass transfer to estimate one from another. These approaches are powerful when direct correlations aren't available.

Colburn Analogy

• Relates heat transfer to friction via the Colburn j-factor: $j_H = St \cdot Pr^{2/3} = \frac{f}{2}$ where $St = Nu/(Re \cdot Pr)$
• Works best for turbulent flow over flat plates and inside tubes—the analogy breaks down when pressure gradients or property variations are significant
• Practical application: if you know the friction factor from fluid mechanics, you can estimate $Nu$ without a dedicated heat transfer correlation

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Quick Reference Table

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Self-Check Questions

1. You're analyzing flow in a pipe at $Re = 5{,}000$. Why is Dittus-Boelter inappropriate here, and which correlation should you use instead?

2. Both Sieder-Tate and Dittus-Boelter apply to turbulent pipe flow. Under what physical conditions would Sieder-Tate give significantly different (and more accurate) results?

3. Compare and contrast natural convection correlations for vertical plates (Churchill-Chu) and horizontal plates (McAdams). How does orientation fundamentally change the flow physics?

4. An FRQ asks you to estimate heat transfer from a tube bank in a shell-and-tube heat exchanger. Which correlation applies, and what geometric factors affect the correlation constants?

5. The Colburn analogy relates heat transfer to momentum transfer. What dimensionless groups does it connect, and under what conditions does the analogy become unreliable?