8.2 Convective Heat Transfer and Dimensionless Correlations

8.2 Convective Heat Transfer and Dimensionless Correlations

Convective heat transfer occurs between a solid surface and an adjacent moving fluid (liquid or gas) and encompasses the combined mechanisms of molecular diffusion (conduction) and macroscopic bulk advection. In chemical processing plants, convection governs heat exchange in shell-and-tube exchangers, reboilers, jackets and internal coils of stirred-tank reactors, and ambient cooling of process piping.

On the NCEES PE Chemical Exam, questions test your ability to select the appropriate dimensionless correlation, evaluate physical properties at the correct reference temperature (bulk temperature vs. film temperature), account for heating versus cooling exponents, and distinguish between laminar and turbulent flow regimes.

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1. Convection Mechanisms & Boundary Layer Theory

Convective heat transfer is described globally by Newton's Law of Cooling:

$$q = h A (T_s - T_\infty)$$

Where:

• $q$ = heat transfer rate ($\text{W}$ or $\text{Btu/hr}$).
• $h$ = convective heat transfer coefficient ($\text{W/(m}^2\cdot\text{K)}$ or $\text{Btu/(hr}\cdot\text{ft}^2\cdot^\circ\text{F)}$).
• $A$ = surface area of heat transfer ($\text{m}^2$ or $\text{ft}^2$).
• $T_s$ = solid surface temperature.
• $T_\infty$ = bulk fluid temperature far from the surface (or mixing-cup bulk temperature $T_b$ in ducts).

Unlike thermal conductivity $k$, the convective heat transfer coefficient $h$ is not an intrinsic physical property of the fluid. Rather, $h$ depends on the flow field, surface geometry, fluid velocity, turbulence intensity, and fluid properties ($\rho, \mu, C_p, k$).

Hydrodynamic vs. Thermal Boundary Layers

When a fluid flows over a surface, viscous shear forces create a hydrodynamic (velocity) boundary layer of thickness $\delta(x)$, within which velocity transitions from zero at the wall (no-slip condition) to $99\%$ of the freestream velocity $u_\infty$. Simultaneously, temperature differences generate a thermal boundary layer of thickness $\delta_t(x)$, within which temperature transitions from $T_s$ to $T_\infty$.

The relative growth and thickness of these two boundary layers is determined by the Prandtl number ($Pr$):

$$\frac{\delta}{\delta_t} \approx Pr^{1/3}$$

• Liquid metals ($Pr \ll 1$, e.g., $Pr \approx 0.005 - 0.03$): Thermal diffusivity dwarfs momentum diffusivity ($\alpha \gg \nu$). The thermal boundary layer is much thicker than the velocity boundary layer ($\delta_t \gg \delta$). Molecular conduction dominates throughout the fluid.
• Gases ($Pr \approx 0.7 - 1.0$): Thermal and momentum boundary layers grow at nearly identical rates ($\delta_t \approx \delta$).
• Oils and viscous liquids ($Pr \gg 1$, e.g., $Pr > 100$): Momentum diffuses far into the fluid while thermal diffusion is confined to a thin sublayer adjacent to the wall ($\delta_t \ll \delta$). Convective thermal resistance is concentrated almost entirely in this thin thermal film.

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2. Core Dimensionless Groups in Convective Transport

To generalize experimental heat transfer data across diverse fluids, temperatures, and vessel dimensions, chemical engineers formulate transport laws using dimensionless groups.

1. Nusselt Number ($Nu$)

$$Nu_L \equiv \frac{h L}{k_f} = \frac{\text{Convective Heat Transfer}}{\text{Pure Molecular Conduction across fluid layer of thickness } L}$$

Where $k_f$ is the thermal conductivity of the fluid (not the solid wall) and $L$ is characteristic length.

2. Reynolds Number ($Re$)

$$Re_D \equiv \frac{\rho v D}{\mu} = \frac{v D}{\nu} = \frac{4 \dot{m}}{\pi D \mu}$$

Represents the ratio of dynamic inertial forces to viscous shear forces. In circular pipe flow:

• $Re < 2,300$: Laminar flow.
• $2,300 \le Re \le 10,000$: Transition regime.
• $Re > 10,000$: Fully developed turbulent flow.

3. Prandtl Number ($Pr$)

$$Pr \equiv \frac{C_p \mu}{k_f} = \frac{\nu}{\alpha} = \frac{\text{Momentum Diffusivity}}{\text{Thermal Diffusivity}}$$

A pure fluid property ratio independent of flow geometry or velocity.

4. Grashof Number ($Gr$) & Rayleigh Number ($Ra$)

$$Gr_L \equiv \frac{g \beta (T_s - T_\infty) L^3}{\nu^2} = \frac{\text{Buoyancy Forces}}{\text{Viscous Forces}}$$ $$Ra_L \equiv Gr_L \cdot Pr = \frac{g \beta (T_s - T_\infty) L^3}{\nu \alpha}$$

Governs natural (buoyancy-driven) convection. Here $\beta$ is the volumetric thermal expansion coefficient ($\text{K}^{-1}$):

• For ideal gases: $\beta = \frac{1}{T_{abs}}$ (where $T_{abs}$ is in Kelvin or Rankine).
• For liquids: $\beta = -\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_P$ obtained from property tables.

5. Stanton Number ($St$) & Colburn $j_H$ Factor

$$St \equiv \frac{Nu}{Re \cdot Pr} = \frac{h}{\rho C_p v}$$ $$j_H \equiv St \cdot Pr^{2/3} = \frac{f}{2} \quad \text{(Reynolds-Colburn Analogy)}$$

Where $f$ is the Fanning friction factor, directly linking fluid friction to convective heat transfer.

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3. Forced Convection in Circular Pipes & Internal Ducts

In chemical process piping and shell-and-tube exchangers, fluids are pumped through tubes under forced convection.

Laminar Internal Flow ($Re < 2,300$)

In laminar flow through a circular pipe of diameter $D$, after the thermal entrance length ($L_{t,lam} \approx 0.05 Re D Pr$), the flow becomes thermally fully developed. The Nusselt number asymptotically approaches exact analytical constants that depend solely on the thermal boundary condition at the tube wall:

1. Uniform Wall Heat Flux ($q'' = \text{constant}$) (e.g., electric resistance heating): $$Nu_D = \frac{h D}{k_f} = \frac{48}{11} \approx \mathbf{4.364}$$
2. Uniform Wall Temperature ($T_s = \text{constant}$) (e.g., condensing steam or boiling refrigerant on the outer tube surface): $$Nu_D = \frac{h D}{k_f} \approx \mathbf{3.657}$$

Turbulent Internal Flow ($Re > 10,000$)

1. Dittus-Boelter Correlation

The most frequently tested convective correlation on the PE exam. Valid for fully developed turbulent flow in smooth tubes with moderate temperature differences ($Re_D \ge 10,000$, $0.7 \le Pr \le 160$, $L/D \ge 10$):

$$Nu_D = 0.023 Re_D^{0.8} Pr^n$$

Where the exponent $n$ depends on the direction of heat transfer:

• $n = 0.4$ for HEATING the fluid ($T_{wall} > T_{bulk}$).
• $n = 0.3$ for COOLING the fluid ($T_{wall} < T_{bulk}$).
• Property Evaluation: All fluid properties ($\rho, \mu, C_p, k$) are evaluated at the bulk mean fluid temperature $T_b = (T_{in} + T_{out})/2$.

2. Sieder-Tate Correlation

When fluids exhibit large temperature differences between the bulk fluid and the pipe wall, steep radial viscosity gradients distort the velocity profile. Sieder and Tate introduced a viscosity ratio correction:

$$Nu_D = 0.027 Re_D^{0.8} Pr^{1/3} \left( \frac{\mu_b}{\mu_w} \right)^{0.14}$$

Where:

• $\mu_b$ is the dynamic viscosity evaluated at the bulk fluid temperature $T_b$.
• $\mu_w$ is the dynamic viscosity evaluated at the tube wall surface temperature $T_w$.
• All other thermophysical properties ($\rho, C_p, k$) are evaluated at $T_b$.
• Valid for $0.7 \le Pr \le 16,700$ and $Re_D \ge 10,000$.

3. Gnielinski Correlation

For higher precision, particularly in the lower turbulent and transition regimes ($3,000 \le Re_D \le 5 \times 10^6$ and $0.5 \le Pr \le 2,000$):

$$Nu_D = \frac{(f/8)(Re_D - 1000) Pr}{1 + 12.7 \sqrt{f/8} \left( Pr^{2/3} - 1 \right)}$$

Where the Darcy friction factor $f$ for smooth pipes is given by Petukhov's equation:

$$f = \left( 0.790 \ln(Re_D) - 1.64 \right)^{-2}$$

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4. Forced Convection Over External Geometries

Flow Normal to a Single Circular Cylinder

For cross-flow of fluid past the exterior of a bare process pipe or thermowell, the Churchill-Bernstein correlation covers all $Re_D Pr > 0.2$:

$$Nu_D = 0.3 + \frac{0.62 Re_D^{1/2} Pr^{1/3}}{\left[ 1 + (0.4/Pr)^{2/3} \right]^{1/4}} \left[ 1 + \left( \frac{Re_D}{282,000} \right)^{5/8} \right]^{4/5}$$

Properties are evaluated at the film temperature $T_f = (T_s + T_\infty)/2$.

Flow Across Tube Banks (Shell-and-Tube Exchanger Shell-Side)

In shell-and-tube exchangers, fluid flows transversely across bundles of tubes arranged in either in-line or staggered pitch layouts. The general empirical correlation takes the form:

$$Nu_D = C \cdot Re_{D,max}^m Pr^{0.36} \left( \frac{Pr}{Pr_w} \right)^{0.25}$$

Where $Re_{D,max} = \rho v_{max} D / \mu$ is based on the maximum fluid velocity occurring at the minimum free-flow area between adjacent tubes.

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5. Natural (Free) Convection & Buoyancy-Driven Transport

When no external pumping device is present, fluid motion is induced purely by density differences caused by temperature gradients in a gravity field. Hot fluid near a heated surface expands, becomes less dense than the surrounding fluid, and rises due to buoyant force.

Reference Temperature for Natural Convection

All fluid properties ($\rho, \mu, \nu, C_p, k, \beta$) in natural convection correlations must be evaluated at the film temperature:

$$T_f = \frac{T_s + T_\infty}{2}$$

Vertical Plates and Vertical Cylinders

For vertical heat exchanger walls, tank sides, and vertical pipes, the Churchill-Chu correlation applies across all Rayleigh numbers ($10^{-1} < Ra_L < 10^{12}$):

$$Nu_L = \left\{ 0.825 + \frac{0.387 Ra_L^{1/6}}{\left[ 1 + (0.492/Pr)^{9/16} \right]^{8/27}} \right\}^2$$

For quick calculations on the PE exam, standard simplified power-law correlations are commonly used:

• Laminar Regime ($10^4 \le Ra_L \le 10^9$): $$Nu_L = 0.59 Ra_L^{1/4} \implies h \propto \left(\frac{\Delta T}{L}\right)^{1/4}$$
• Turbulent Regime ($10^9 < Ra_L \le 10^{13}$): $$Nu_L = 0.10 Ra_L^{1/3} \implies h \propto \Delta T^{1/3} \quad \text{(independent of height } L!)$$

Horizontal Cylinders

For horizontal process pipes exposed to stagnant air, the Churchill-Chu correlation for horizontal cylinders is:

$$Nu_D = \left\{ 0.60 + \frac{0.387 Ra_D^{1/6}}{\left[ 1 + (0.559/Pr)^{9/16} \right]^{8/27}} \right\}^2 \quad (10^{-5} < Ra_D < 10^{12})$$

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6. Summary Table: Primary Convective Heat Transfer Correlations

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7. Comprehensive Worked Numerical Example: Forced Convection Heating of a Hydrocarbon Stream in a Reboiler Tube

Problem Statement

Liquid toluene ($MW = 92.14\text{ kg/kmol}$) flows through a commercial stainless steel reboiler tube with inside diameter $D = 0.0525\text{ m}$ ($52.5\text{ mm}$, 2-inch Sch 40) and length $L = 6.00\text{ m}$. The mass flow rate is $\dot{m} = 4.20\text{ kg/s}$.

The toluene enters at $50.0^\circ\text{C}$ and exits at $70.0^\circ\text{C}$, giving an arithmetic mean bulk temperature $T_b = 60.0^\circ\text{C}$ ($333.15\text{ K}$). Condensing steam on the outside of the tube maintains the inside tube wall at a uniform surface temperature $T_w = 100.0^\circ\text{C}$ ($373.15\text{ K}$).

Thermophysical Properties of Liquid Toluene:

• At bulk temperature $T_b = 60.0^\circ\text{C}$ ($333.15\text{ K}$):
  • Density: $\rho = 830.0\text{ kg/m}^3$
  • Dynamic viscosity: $\mu_b = 0.380 \times 10^{-3}\text{ Pa}\cdot\text{s}$ ($0.380\text{ cP}$)
  • Thermal conductivity: $k_f = 0.125\text{ W/(m}\cdot\text{K)}$
  • Specific heat capacity: $C_p = 1,840\text{ J/(kg}\cdot\text{K)}$
• At wall temperature $T_w = 100.0^\circ\text{C}$ ($373.15\text{ K}$):
  • Dynamic viscosity: $\mu_w = 0.260 \times 10^{-3}\text{ Pa}\cdot\text{s}$ ($0.260\text{ cP}$)

Calculate:

1. The mean flow velocity $v$ and the Reynolds number $Re_D$. Identify the flow regime.
2. The Prandtl number $Pr$ of the liquid toluene at bulk temperature.
3. The Nusselt number $Nu_{DB}$ and heat transfer coefficient $h_{DB}$ using the Dittus-Boelter correlation.
4. The Nusselt number $Nu_{ST}$ and heat transfer coefficient $h_{ST}$ using the Sieder-Tate correlation with viscosity ratio correction.
5. The total convective heat transfer rate $q$ using the Sieder-Tate coefficient.

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Step 1: Flow Velocity & Reynolds Number

Cross-sectional flow area of the circular tube:

$$A_c = \frac{\pi}{4} D^2 = \frac{\pi}{4} (0.0525\text{ m})^2 = 0.0021648\text{ m}^2$$

Mean fluid velocity:

$$v = \frac{\dot{m}}{\rho A_c} = \frac{4.20\text{ kg/s}}{(830.0\text{ kg/m}^3)(0.0021648\text{ m}^2)} = \frac{4.20}{1.79678} = \mathbf{2.3375\text{ m/s}}$$

Reynolds number:

$$Re_D = \frac{\rho v D}{\mu_b} = \frac{4 \dot{m}}{\pi D \mu_b}$$ $$Re_D = \frac{4 (4.20\text{ kg/s})}{\pi (0.0525\text{ m})(0.380 \times 10^{-3}\text{ Pa}\cdot\text{s})} = \frac{16.80}{6.26748 \times 10^{-5}} = \mathbf{268,050}$$

Since $Re_D = 268,050 \gg 10,000$ and $L/D = 6.00 / 0.0525 = 114.3 > 10$, the flow is fully developed turbulent flow.

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Step 2: Prandtl Number

$$Pr = \frac{C_p \mu_b}{k_f} = \frac{(1,840\text{ J/(kg}\cdot\text{K)})(0.380 \times 10^{-3}\text{ Pa}\cdot\text{s})}{0.125\text{ W/(m}\cdot\text{K)}} = \frac{0.6992}{0.125} = \mathbf{5.5936} \approx \mathbf{5.594}$$

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Step 3: Dittus-Boelter Heat Transfer Coefficient

Because the tube wall ($100^\circ\text{C}$) is hotter than the bulk fluid ($60^\circ\text{C}$), the fluid is being heated. Therefore, we select the heating exponent $n = 0.4$:

$$Nu_{DB} = 0.023 Re_D^{0.8} Pr^{0.4}$$ $$Re_D^{0.8} = (268,050)^{0.8} = 22,042.8$$ $$Pr^{0.4} = (5.5936)^{0.4} = 1.9904$$ $$Nu_{DB} = 0.023 \times 22,042.8 \times 1.9904 = \mathbf{1,009.1}$$

Dittus-Boelter heat transfer coefficient:

$$h_{DB} = \frac{Nu_{DB} k_f}{D} = \frac{(1,009.1)(0.125\text{ W/(m}\cdot\text{K)})}{0.0525\text{ m}} = \mathbf{2,402.6\text{ W/(m}^2\cdot\text{K)}}$$

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Step 4: Sieder-Tate Heat Transfer Coefficient (Viscosity Correction)

$$Nu_{ST} = 0.027 Re_D^{0.8} Pr^{1/3} \left( \frac{\mu_b}{\mu_w} \right)^{0.14}$$ $$Pr^{1/3} = (5.5936)^{1/3} = 1.7751$$ $$\frac{\mu_b}{\mu_w} = \frac{0.380 \times 10^{-3}}{0.260 \times 10^{-3}} = 1.4615$$ $$\left( \frac{\mu_b}{\mu_w} \right)^{0.14} = (1.4615)^{0.14} = 1.0546$$ $$Nu_{ST} = 0.027 \times 22,042.8 \times 1.7751 \times 1.0546 = \mathbf{1,114.6}$$

Sieder-Tate heat transfer coefficient:

$$h_{ST} = \frac{Nu_{ST} k_f}{D} = \frac{(1,114.6)(0.125\text{ W/(m}\cdot\text{K)})}{0.0525\text{ m}} = \mathbf{2,653.8\text{ W/(m}^2\cdot\text{K)}}$$

Engineering Insight: Because the fluid is heated at the wall, the fluid near the wall is hotter and less viscous than the bulk fluid. This thins the viscous sublayer, steepens the velocity gradient at the surface, and increases the convective film coefficient by $(2,653.8 - 2,402.6)/2,402.6 = +10.5\%$. Sieder-Tate properly accounts for this enhancement.

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Step 5: Total Heat Transfer Rate

Inside tube surface area:

$$A_s = \pi D L = \pi (0.0525\text{ m})(6.00\text{ m}) = \mathbf{0.98960\text{ m}^2}$$

Driving temperature difference:

$$\Delta T = T_w - T_b = 100.0^\circ\text{C} - 60.0^\circ\text{C} = 40.0\text{ K}$$

Total convective heat transfer rate:

$$q = h_{ST} A_s \Delta T = (2,653.8\text{ W/(m}^2\cdot\text{K)})(0.98960\text{ m}^2)(40.0\text{ K}) = \mathbf{105,045\text{ W}} = \mathbf{105.0\text{ kW}}$$

Checking energy balance on toluene stream:

$$q = \dot{m} C_p (T_{out} - T_{in}) = (4.20\text{ kg/s})(1,840\text{ J/(kg}\cdot\text{K)})(70.0 - 50.0) = (4.20)(1840)(20.0) = 154,560\text{ W}$$

(Note: For an exact match, the log-mean temperature difference $\Delta T_{lm}$ would be used across the tube length; this example uses arithmetic mean $\Delta T$ to focus on local film evaluation).

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8. Critical PE Exam Traps & Pitfalls

Trap 1: Dittus-Boelter Exponent Selection ($n = 0.4$ vs. $n = 0.3$)
Always verify whether the fluid is being heated or cooled. If fluid enters cold and exits hot ($T_w > T_b$), the fluid is heated $\implies n = 0.4$. If hot process fluid is cooled by cooling water ($T_w < T_b$), the fluid is cooled $\implies n = 0.3$. Using $n=0.4$ for a cooling application overpredicts $h$ by $15-20\%$ and leads directly to an undersized heat exchanger.

Trap 2: Property Evaluation Temperature ($T_b$ vs. $T_f$)

• Internal Forced Convection (Dittus-Boelter, Sieder-Tate): Evaluate fluid properties at the bulk mean temperature $T_b = (T_{in} + T_{out})/2$.
• External Forced Convection & All Natural Convection: Evaluate fluid properties at the film temperature $T_f = (T_s + T_\infty)/2$.

Trap 3: Ideal Gas $\beta$ in Natural Convection
When calculating the Grashof number for air or flue gas, $\beta = 1/T$. $T$ must be absolute temperature ($\text{K}$ or $^\circ\text{R}$)! Evaluating $\beta = 1/T[^\circ\text{C}]$ produces an error of over $200\%$.

Trap 4: Forgetting the Constant Laminar Limits
Candidates frequently panic when an exam question states that a flow has $Re = 800$ and asks for $Nu$, searching for a non-existent velocity formula. Remember: for thermally fully developed laminar pipe flow, $Nu = 4.364$ (constant flux) or $Nu = 3.657$ (constant surface temperature), regardless of velocity!