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What is a Film Heat Transfer Coefficient?
When a fluid flows past a heated surface, heat does not jump instantly from the wall into the bulk fluid. Instead, a thin boundary layer — or thermal film — forms adjacent to the wall. Within this layer, heat moves primarily by conduction through a near-stagnant fluid. The film heat transfer coefficient, denoted h, quantifies how effectively heat crosses this boundary:
$$q = h \cdot A \cdot (T_w - T_f)$$where \(q\) is the heat flow (W), \(A\) is the surface area (m²), \(T_w\) is the wall temperature, and \(T_f\) is the bulk fluid temperature. A high \(h\) means the film resistance is small — the fluid grabs heat quickly. A low \(h\) means the film resists heat transfer significantly.
In the Overall U-Value equation, each film coefficient appears as a resistance \(1/h\). The lowest film coefficient on either side of the wall almost always dominates the total resistance — and therefore controls the required heat transfer area.
Newton's Law of Cooling — The Foundation
The film heat transfer coefficient is defined directly through Newton's Law of Cooling:
$$q = h \cdot A \cdot \Delta T$$Rearranging to express \(h\) as a measurable quantity:
$$h = \frac{q}{A \cdot (T_w - T_f)} \quad \left[\frac{W}{m^2 \cdot K}\right]$$In practice, \(h\) is not measured directly — it is calculated from dimensionless correlations that relate fluid dynamics to heat transfer through the Nusselt number.
From Nusselt Number to Film Coefficient
The Nusselt number (\(Nu\)) is the dimensionless ratio of convective to conductive heat transfer across the thermal boundary layer:
$$Nu = \frac{h \cdot D}{k_f}$$Rearranging to get \(h\):
$$h = \frac{Nu \cdot k_f}{D}$$where:
- \(Nu\) = Nusselt number (from a correlation, dimensionless)
- \(k_f\) = fluid thermal conductivity (W/m·K)
- \(D\) = characteristic length — pipe internal diameter for tube flow (m)
This is the central equation. To find \(h\), you need to determine \(Nu\) from the appropriate correlation for your flow regime and geometry.
Turbulent Pipe Flow — Dittus-Boelter Correlation
For fully developed turbulent flow inside a smooth tube (Re > 10,000), the most widely used correlation is Dittus-Boelter:
$$Nu = 0.023 \cdot Re^{0.8} \cdot Pr^n$$where \(n = 0.4\) for heating (fluid is being heated) and \(n = 0.3\) for cooling (fluid is being cooled).
Validity range: Re > 10,000 · 0.7 < Pr < 160 · L/D > 10
Dittus-Boelter is widely used in preliminary design but can be ±25% inaccurate near its validity boundary. Use the Gnielinski correlation for more accurate results, especially in the transition region (3,000 < Re < 10,000).
Turbulent Pipe Flow — Gnielinski Correlation (More Accurate)
The Gnielinski correlation (VDI Heat Atlas) is more accurate across a wider range:
$$Nu = \frac{(f/8)(Re - 1000) \cdot Pr}{1 + 12.7\sqrt{f/8}(Pr^{2/3} - 1)}$$where \(f\) is the Darcy friction factor from the Colebrook equation.
Validity range: 3,000 < Re < 5,000,000 · 0.5 < Pr < 2,000 · L/D > 10
For smooth tubes, the friction factor can be estimated with the Petukhov approximation:
$$f = (0.790 \ln Re - 1.64)^{-2}$$Laminar Pipe Flow
For laminar flow (Re < 2,300), the Nusselt number approaches a constant — independent of velocity — depending on the thermal boundary condition:
- Constant wall heat flux: \(Nu = 4.36\)
- Constant wall temperature: \(Nu = 3.66\)
This means that in laminar flow, increasing velocity does not improve \(h\) once the flow is fully developed. The only way to improve \(h\) in laminar flow is to reduce the tube diameter (increase \(k_f / D\)) or use enhanced surfaces.
Applying the Dittus-Boelter correlation to laminar or transition flow (Re < 10,000) gives wildly incorrect results. Always check Re before selecting a correlation.
Transition Flow (2,300 < Re < 10,000)
The transition region is unpredictable. Flow alternates between laminar and turbulent patches, making heat transfer difficult to correlate reliably. In practice:
- Use Gnielinski down to Re = 3,000 with increased uncertainty (±30%)
- Design to avoid Re = 2,300–10,000 where possible — adjust velocity by changing tube diameter or flow path
Shell-Side Film Coefficient
The shell-side film coefficient in a shell-and-tube exchanger is more complex because flow crosses and re-crosses the tube bundle due to baffles. Simple correlations (like Kern's method) use a shell-side equivalent diameter:
$$Nu_{shell} = 0.36 \cdot Re_{shell}^{0.55} \cdot Pr^{1/3} \cdot \left(\frac{\mu_f}{\mu_w}\right)^{0.14}$$The rigorous shell-side method is Bell-Delaware, which accounts for baffle geometry, leakage streams, and bypass effects — implemented in the full Heat Exchanger Rating tool.
Reynolds and Prandtl Numbers — What They Mean
The film coefficient depends on two key dimensionless numbers:
Reynolds Number (Re) — Flow Regime
$$Re = \frac{\rho \cdot v \cdot D}{\mu} = \frac{G \cdot D}{\mu}$$Re quantifies inertial forces relative to viscous forces. High Re → turbulent flow → good mixing → high \(h\). Low Re → laminar flow → poor mixing → low \(h\).
Prandtl Number (Pr) — Fluid Property
$$Pr = \frac{c_p \cdot \mu}{k_f} = \frac{\nu}{\alpha}$$Pr is the ratio of momentum diffusivity (\(\nu\)) to thermal diffusivity (\(\alpha\)). High Pr fluids (viscous oils, Pr = 100–1000) have a thick velocity boundary layer relative to the thermal layer — the thermal film is small and \(h\) is dominated by fluid properties rather than velocity.
| Fluid | Pr (typical) | Typical h (W/m²·K, turbulent pipe) |
|---|---|---|
| Air (1 bar) | 0.71 | 20 – 100 |
| Water (25°C) | 6.1 | 3,000 – 10,000 |
| Light hydrocarbon (e.g. naphtha) | 3 – 8 | 500 – 2,000 |
| Crude oil (30°C) | 100 – 1,000 | 100 – 500 |
| Liquid sodium (reactor coolant) | 0.005 – 0.01 | 10,000 – 100,000 |
Effect of Velocity on h
From the Dittus-Boelter correlation, \(Nu \propto Re^{0.8}\), and \(Re \propto v\), so:
$$h \propto v^{0.8}$$Doubling the velocity increases \(h\) by a factor of \(2^{0.8} = 1.74\) — a 74% improvement in film coefficient. This is the primary lever for improving tube-side heat transfer.
However, increasing velocity also increases pressure drop, which scales as \(\Delta P \propto v^2\). So there is always a trade-off between better heat transfer and higher pumping cost. Typical design velocities for liquid-side tube flow:
- Water: 1.0 – 3.0 m/s
- Light hydrocarbon: 0.5 – 2.0 m/s
- Viscous oil: 0.3 – 1.0 m/s
- Maximum to avoid erosion: per API RP 14E / Crane TP-410
Effect of Tube Diameter on h
For fixed mass flow, reducing tube diameter increases velocity and Re, which increases \(h\). Additionally, since \(h = Nu \cdot k / D\), even at constant \(Nu\), halving \(D\) doubles \(h\) (at the cost of more tubes and higher pressure drop).
Effect of Fluid Properties on h
The fluid's thermal conductivity \(k_f\), density \(\rho\), viscosity \(\mu\), and specific heat \(c_p\) all directly enter \(h\) through Re and Pr. For high-viscosity fluids (crude oil, heavy residue):
- High \(\mu\) → low Re → risk of laminar flow
- Low \(k_f\) → low \(h\) even at same Nu
- Wall viscosity correction \((\mu_f / \mu_w)^{0.14}\) becomes important when the wall temperature differs significantly from bulk
Worked Example — Tube-Side Film Coefficient
A crude oil cooler (tube side) has the following conditions:
- Fluid: light crude oil at 60°C
- Flow rate: 150,000 kg/h through 200 tubes of 19 mm ID
- Fluid properties at 60°C: \(\rho = 830\) kg/m³, \(\mu = 1.8 \times 10^{-3}\) Pa·s, \(k_f = 0.135\) W/m·K, \(c_p = 2100\) J/kg·K
Step 1: Mass velocity per tube
$$G = \frac{\dot{m}_{total}}{N_{tubes} \cdot A_{tube}} = \frac{150000/3600}{200 \cdot \pi/4 \cdot 0.019^2} = \frac{41.67}{200 \cdot 2.835 \times 10^{-4}} = 734 \text{ kg/m}^2\text{s}$$Step 2: Reynolds number
$$Re = \frac{G \cdot D}{\mu} = \frac{734 \times 0.019}{1.8 \times 10^{-3}} = 7{,}747$$Flow is in the transition region — use Gnielinski.
Step 3: Prandtl number
$$Pr = \frac{c_p \cdot \mu}{k_f} = \frac{2100 \times 1.8 \times 10^{-3}}{0.135} = 28.0$$Step 4: Friction factor (Petukhov)
$$f = (0.790 \ln 7747 - 1.64)^{-2} = (0.790 \times 8.955 - 1.64)^{-2} = (7.074 - 1.64)^{-2} = (5.434)^{-2} = 0.0339$$Step 5: Nusselt number (Gnielinski)
$$Nu = \frac{(0.0339/8)(7747 - 1000) \times 28.0}{1 + 12.7\sqrt{0.0339/8}(28.0^{2/3} - 1)}$$ $$Nu = \frac{0.004238 \times 6747 \times 28.0}{1 + 12.7 \times 0.06507 \times (9.22 - 1)} = \frac{800.5}{1 + 6.79} = \frac{800.5}{7.79} = 102.8$$Step 6: Film coefficient
$$h = \frac{Nu \cdot k_f}{D} = \frac{102.8 \times 0.135}{0.019} = 730 \text{ W/m}^2\text{·K}$$This moderate film coefficient — controlled by the transition Re and high Pr of crude oil — will be the bottleneck in the U-value calculation. Checking: \(1/h = 0.00137\) m²·K/W. Compare this with cooling water at \(h_o = 5000\): \(1/h_o = 0.0002\) — the crude oil film resistance is nearly 7× larger.
How to Improve a Low Film Coefficient
| Strategy | Mechanism | Typical Improvement |
|---|---|---|
| Increase velocity | Higher Re → thinner boundary layer | h ∝ v0.8 |
| Reduce tube diameter | Higher Re at same flow, higher k/D | Significant (2–4×) |
| Twisted-tape inserts | Swirl flow mixes boundary layer | 1.5 – 3× (laminar) |
| Corrugated / enhanced tubes | Wall turbulence, increased surface | 1.5 – 2× |
| Increase fluid temperature (preheat) | Reduces viscosity → higher Re | Depends on fluid |
| More tube passes | Same flow, higher velocity per pass | Direct improvement |
Condensing and Boiling Film Coefficients
Phase-change processes have dramatically different film coefficients:
- Film condensation (Nusselt): Condensate film grows along the tube — h decreases down the tube length. Typically 5,000 – 20,000 W/m²·K for steam.
- Dropwise condensation: Condensate does not wet surface — h can be 5–10× higher, but difficult to sustain in practice.
- Nucleate boiling: High h (2,000 – 40,000 W/m²·K) due to bubble agitation.
- Film boiling: Very low h — the vapour film insulates the surface (Leidenfrost effect). Dangerous in cooling systems.
Phase-change film coefficients are covered in the dedicated articles on Condensation Heat Transfer and Boiling Heat Transfer (coming in Month 2).
Common Engineering Mistakes
- Using Dittus-Boelter in transition flow (2,300 < Re < 10,000) — use Gnielinski instead.
- Ignoring the wall viscosity correction for high-viscosity liquids — the correction factor \((\mu_f/\mu_w)^{0.14}\) can shift h by 10–30%.
- Treating h as a material constant — h changes with temperature, velocity, and fouling state. Always recalculate for your specific conditions.
- Applying tube-side correlations to shell-side flow — shell-side flow is fundamentally different due to baffles. Use Bell-Delaware or Kern's method for shell-side.
- Assuming turbulent flow at low Re just to get a higher h — always verify Re before selecting a correlation.
Key Validity Ranges at a Glance
| Correlation | Re Range | Pr Range | Accuracy |
|---|---|---|---|
| Laminar constant Nu = 3.66 | < 2,300 | Any | Fully developed only |
| Dittus-Boelter | > 10,000 | 0.7 – 160 | ±25% |
| Gnielinski | 3,000 – 5,000,000 | 0.5 – 2,000 | ±10% |
| Sieder-Tate (viscous) | > 10,000 | 0.7 – 16,700 | ±20% |
When to Use the Full ChemProCal Tool
The U-Value estimator below lets you test the impact of film coefficients on the overall U. When you need to:
- Calculate h from first principles — from your actual fluid properties, velocity, and tube geometry
- Run a full shell-and-tube rating with Bell-Delaware shell-side correlation
- Evaluate tube passes, baffle spacing, and shell diameter together
→ Use the Heat Exchanger Rating full tool.
Related Articles
- Overall Heat Transfer Coefficient (U-Value): Complete Engineering Guide
- Heat Exchanger Sizing: LMTD & U-Values
- Dimensionless Numbers in Chemical Engineering
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