4.2 Convective Heat Transfer & Forced/Natural Convection Coefficients

4.2 Convective Heat Transfer & Forced/Natural Convection Coefficients

Convective heat transfer is the transport of energy between a solid boundary and an adjacent moving fluid (liquid or gas), combining microscopic molecular conduction and macroscopic bulk fluid advection. In HVAC and refrigeration engineering, convection occurs inside hydronic chiller/boiler tubes, across finned air-conditioning coils, through air supply ductwork, and along building architectural surfaces. Accurately determining convective heat transfer coefficients ($h_c$) via dimensionless empirical correlations is critical for equipment sizing on the NCEES PE Mechanical exam.

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1. Newton's Law of Cooling & Boundary Layer Theory

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

$$\dot{q}_{conv} = h_c A (T_s - T_\infty)$$

Where:

• $\dot{q}_{conv} = \text{convective heat transfer rate } (\text{Btu/hr or W})$
• $h_c = \text{convective heat transfer coefficient } (\text{Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)} \text{or W/(m}^2\cdot\text{K)})$
• $A = \text{surface area in contact with fluid } (\text{ft}^2 \text{ or m}^2)$
• $T_s = \text{solid surface temperature } ({}^\circ\text{F or } {}^\circ\text{C})$
• $T_\infty = \text{free-stream bulk fluid temperature } ({}^\circ\text{F or } {}^\circ\text{C})$

The equivalent convective thermal resistance is:

$$R_{conv} = \frac{1}{h_c A} \quad \left[\frac{\text{hr}\cdot{}^\circ\text{F}}{\text{Btu}}\right]$$

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| HYDRODYNAMIC AND THERMAL BOUNDARY LAYERS OVER A FLAT SURFACE                            |
+-----------------------------------------------------------------------------------------+
| Free Stream: V_inf, T_inf                                                               |
|      ====================>                                                              |
|                . - ~ ~ ~ - .  <--- Hydrodynamic Boundary Layer delta(x)                 |
|            . '               ` .    Velocity transitions from 0 at wall to V_inf        |
|        . ' - - - - - - - - - - - ` <--- Thermal Boundary Layer delta_t(x)               |
|    . '                              `   Temp transitions from T_s at wall to T_inf      |
| +-------------------------------------------------------------------------------------+ |
| | Solid Surface at Temperature T_s (No-Slip Condition: u(y=0) = 0)                   | |
+-----------------------------------------------------------------------------------------+

Boundary Layer Mechanics

When fluid flows past a stationary surface, viscous shear stresses create a hydrodynamic (velocity) boundary layer of thickness $\delta(x)$, where local velocity transitions from zero at the wall (the no-slip boundary condition) to $99\%$ of the free-stream velocity $V_\infty$. Concurrently, a thermal boundary layer of thickness $\delta_t(x)$ develops where temperature transitions from $T_s$ to $T_\infty$. Because fluid velocity is zero at the solid interface, heat transfer across the interface occurs entirely by pure conduction through the fluid film:

$$q'' = -k_f \left.\frac{\partial T}{\partial y}\right|_{y=0} = h_c (T_s - T_\infty) \implies h_c = \frac{-k_f \left.\frac{\partial T}{\partial y}\right|_{y=0}}{T_s - T_\infty}$$

Where $k_f$ is the fluid thermal conductivity. Thinner boundary layers produce steeper temperature gradients $\frac{\partial T}{\partial y}$, resulting in substantially higher convection coefficients $h_c$.

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2. Fundamental Dimensionless Numbers in Convection

Dimensionless analysis reduces complex multi-variable fluid-thermal equations into standard governing parameters:

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| SUMMARY OF PRIMARY DIMENSIONLESS GROUPS                                                 |
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| Reynolds (Re):   Re = (rho * V * L) / mu = (V * L) / nu       [Inertia / Viscous]       |
| Prandtl (Pr):    Pr = nu / alpha = (c_p * mu) / k_f           [Momentum / Thermal Diff] |
| Nusselt (Nu):    Nu = (h * L) / k_f                           [Convection / Conduction] |
| Grashof (Gr):    Gr = (g * beta * Delta_T * L^3) / nu^2       [Buoyancy / Viscous]      |
| Rayleigh (Ra):   Ra = Gr * Pr = (g * beta * Delta_T * L^3)/(nu*alpha) [Natural Convect] |
+-----------------------------------------------------------------------------------------+

1. Reynolds Number ($\text{Re}$)

Represents the ratio of dynamic inertial forces to viscous shear forces:

$$\text{Re} = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$$

• Internal Flow in Circular Pipes:
  • Laminar Regime: $\text{Re} < 2,300$
  • Transition Regime: $2,300 \le \text{Re} \le 10,000$
  • Fully Turbulent Regime: $\text{Re} > 10,000$

2. Prandtl Number ($\text{Pr}$)

Represents the ratio of momentum molecular diffusivity ($\nu = \mu/\rho$) to thermal molecular diffusivity ($\alpha = k/(\rho c_p)$):

$$\text{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k_f}$$

• $\text{Pr} \approx 0.70 - 0.72$ for air and common gases at atmospheric conditions
• $\text{Pr} \approx 2.0 - 10.0$ for liquid water ($40^\circ\text{F}$ to $200^\circ\text{F}$)
• $\text{Pr} \approx 50 - 10,000$ for heavy oils and glycols
• $\text{Pr} \ll 1$ ($~0.01$) for liquid metals

3. Nusselt Number ($\text{Nu}$)

Represents the ratio of convective heat transfer to pure molecular conduction across a fluid layer of characteristic length $L$:

$$\text{Nu} = \frac{h_c L}{k_f}$$

Once $\text{Nu}$ is calculated from an empirical correlation, the convective coefficient is determined directly:

$$h_c = \frac{\text{Nu} \cdot k_f}{L}$$

4. Grashof Number ($\text{Gr}$) and Rayleigh Number ($\text{Ra}$)

In natural (free) convection, fluid motion is induced by buoyancy forces resulting from temperature-driven density differences:

$$\text{Gr} = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$$ $$\text{Ra} = \text{Gr} \cdot \text{Pr} = \frac{g \beta (T_s - T_\infty) L^3}{\nu \alpha}$$

Where:

• $g = \text{gravitational acceleration } (32.174\text{ ft/s}^2 \text{ or } 9.807\text{ m/s}^2)$
• $\beta = \text{volumetric thermal expansion coefficient } ({}^\circ\text{R}^{-1} \text{ or } \text{K}^{-1})$; for ideal gases, $\beta = \frac{1}{T_{film}} = \frac{1}{T_f\text{ (in }{}^\circ\text{R)}}$
• $T_f = \frac{T_s + T_\infty}{2} = \text{film temperature at which fluid properties are evaluated}$

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3. Forced Convection Correlations (Internal Flow)

Fully Developed Turbulent Flow in Smooth Pipes

For turbulent flow in circular tubes ($\text{Re}_D \ge 10,000$, $0.6 \le \text{Pr} \le 160$, and $\frac{L}{D} \ge 10$), the Dittus-Boelter equation is the standard relation in the NCEES Reference Handbook:

$$\text{Nu}_D = 0.023 \text{Re}_D^{0.8} \text{Pr}^n$$

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

• $n = 0.4$ when the fluid is being heated ($T_s > T_b$, e.g., heating coils, boiler tubes)
• $n = 0.3$ when the fluid is being cooled ($T_s < T_b$, e.g., chiller evaporator tubes, cooling coils)

All fluid properties are evaluated at the bulk mean fluid temperature $T_b$.

Gnielinski Correlation (Higher Accuracy for $3,000 \le \text{Re}_D \le 5 \times 10^6$)

When greater precision is required across the transition-to-turbulent region ($0.5 \le \text{Pr} \le 2000$):

$$\text{Nu}_D = \frac{(f/8)(\text{Re}_D - 1000)\text{Pr}}{1 + 12.7(f/8)^{1/2}(\text{Pr}^{2/3} - 1)}$$

Where $f$ is the Darcy friction factor for smooth tubes: $f = (0.790 \ln(\text{Re}_D) - 1.64)^{-2}$.

Laminar Flow in Circular Tubes ($\text{Re}_D < 2,300$)

For fully developed laminar internal flow, $\text{Nu}$ is independent of Reynolds and Prandtl numbers:

• Constant Surface Heat Flux ($q'' = \text{constant}$): $\text{Nu}_D = \frac{h D}{k_f} = 4.364$
• Constant Surface Temperature ($T_s = \text{constant}$): $\text{Nu}_D = \frac{h D}{k_f} = 3.656$

Non-Circular Ducts & Hydraulic Diameter ($D_h$)

For non-circular ducts (rectangular HVAC ducts, annular passages), the characteristic dimension is the hydraulic diameter:

$$D_h = \frac{4 A_c}{P_w}$$

Where:

• $A_c = \text{cross-sectional flow area } (\text{ft}^2)$
• $P_w = \text{wetted perimeter in contact with fluid } (\text{ft})$

For a rectangular duct of cross-sectional width $a$ and height $b$:

$$D_h = \frac{4 (a b)}{2(a + b)} = \frac{2 a b}{a + b}$$

For an annular pipe (inner pipe OD = $D_1$, outer pipe ID = $D_2$):

$$D_h = \frac{4 \left[\frac{\pi}{4}(D_2^2 - D_1^2)\right]}{\pi (D_2 + D_1)} = D_2 - D_1$$

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4. Forced Convection Correlations (External Flow)

External Flow Over Flat Surfaces

For fluid flowing parallel to a flat plate of length $L$:

• Laminar Boundary Layer ($\text{Re}_x < 5 \times 10^5$): $$\text{Nu}_x = 0.332 \text{Re}_x^{1/2} \text{Pr}^{1/3} \quad \implies \quad \overline{\text{Nu}}_L = 0.664 \text{Re}_L^{1/2} \text{Pr}^{1/3}$$
• Turbulent Boundary Layer ($\text{Re}_x \ge 5 \times 10^5$): $$\text{Nu}_x = 0.0296 \text{Re}_x^{4/5} \text{Pr}^{1/3} \quad \implies \quad \overline{\text{Nu}}_L = 0.037 \text{Re}_L^{4/5} \text{Pr}^{1/3}$$

Cross-Flow Across Single Cylinders & Tube Banks

For cross-flow across a single tube of outer diameter $D$ (Hilpert correlation):

$$\overline{\text{Nu}}_D = C \text{Re}_D^m \text{Pr}^{1/3}$$

In finned-tube HVAC cooling and heating coils, air flows across tube banks arranged in in-line or staggered configurations. Staggered tube banks produce higher turbulence and convective heat transfer coefficients for equivalent air face velocities at the expense of higher fan static pressure drop.

5. Natural (Free) Convection Correlations

Natural convection correlations take the general power-law form:

$$\overline{\text{Nu}}_L = \frac{\bar{h} L}{k_f} = C \text{Ra}_L^m = C (\text{Gr}_L \cdot \text{Pr})^m$$

Where the constants $C$ and $m$ depend on geometry, orientation, and flow regime:

• Laminar Flow ($\text{Ra} < 10^9$): $m = 1/4$ ($0.25$)
• Turbulent Flow ($\text{Ra} \ge 10^9$): $m = 1/3$ ($0.333$)

Summary of Natural Convection Geometries (ASHRAE / NCEES Standards)

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6. Phase Change Convection: Boiling & Condensation Fundamentals

Phase-change convective processes provide the highest heat transfer coefficients in HVAC and refrigeration engineering ($h \approx 500 \text{ to } 10,000+\text{ Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)}$).

The Pool Boiling Curve

When a submerged heating surface operates at temperature $T_s$ above the fluid saturation temperature $T_{sat}$ (excess temperature $\Delta T_e = T_s - T_{sat}$):

1. Natural Convection Regime ($\Delta T_e < 9^\circ\text{F}$ / $5^\circ\text{C}$): Liquid motion is driven purely by natural convection without vapor bubble formation.
2. Nucleate Boiling Regime ($9^\circ\text{F} < \Delta T_e < 54^\circ\text{F}$ / $5^\circ\text{C} < \Delta T_e < 30^\circ\text{C}$): Vapor bubbles nucleate at microscopic surface cavities, detach rapidly, and create vigorous fluid agitation. This regime yields maximum heat transfer efficiency and governs direct-expansion (DX) and flooded refrigeration evaporators.
3. Critical Heat Flux ($q''_{max}$ / Departure from Nucleate Boiling - DNB): The peak heat flux point. Exceeding $q''_{max}$ causes vapor bubbles to coalesce into an insulating blanket.
4. Film Boiling Regime (Large $\Delta T_e$): A continuous vapor film completely blankets the heater surface, dropping $h$ drastically and causing severe surface temperature spikes (burnout).

Condensation: Film vs. Dropwise

• Film Condensation (Nusselt Film Theory): Liquid condensate forms a continuous laminar liquid film that flows downward under gravity. The liquid film introduces a conductive thermal barrier, reducing $h$.
• Dropwise Condensation: Condensate forms discrete droplets that break away, keeping the cold surface directly exposed. Dropwise condensation achieves heat transfer coefficients 4 to 8 times higher than film condensation.

7. Step-by-Step Worked Example: Internal Forced Convection Sizing

Problem Statement

Water at a bulk mean temperature of $T_b = 140.0^\circ\text{F}$ flows through a $1\text{-inch}$ nominal Type L copper pipe with an inside diameter of $D_i = 1.025\text{ in.} = 0.08542\text{ ft}$ at a volumetric flow rate of $5.0\text{ GPM}$. The pipe wall is maintained at $T_s = 160.0^\circ\text{F}$ by an external heating source (heating fluid, $n = 0.4$).

From NCEES water property tables at $140^\circ\text{F}$ ($60^\circ\text{C}$):

• Density: $\rho = 61.38\text{ lbm/ft}^3$
• Dynamic Viscosity: $\mu = 1.14 \times 10^{-3}\text{ lbm/(ft}\cdot\text{s)} = 4.104\text{ lbm/(ft}\cdot\text{hr)}$
• Kinematic Viscosity: $\nu = 1.857 \times 10^{-5}\text{ ft}^2/\text{s}$
• Thermal Conductivity: $k_f = 0.376\text{ Btu/(hr}\cdot\text{ft}\cdot{}^\circ\text{F)}$
• Specific Heat: $c_p = 1.000\text{ Btu/(lbm}\cdot{}^\circ\text{F)}$
• Prandtl Number: $\text{Pr} = 3.01$

Calculate:

1. Cross-sectional flow area $A_c$ ($\text{ft}^2$) and mean flow velocity $V$ ($\text{ft/s}$)
2. Reynolds number $\text{Re}_D$ and flow regime
3. Nusselt number $\text{Nu}_D$ via the Dittus-Boelter equation
4. Internal convective heat transfer coefficient $h_i$ ($\text{Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)}$)
5. Convective heat gain rate per linear foot of pipe length $\dot{q}'$ ($\text{Btu/(hr}\cdot\text{ft)}$)

Solution Steps

Step 1: Calculate flow area and fluid velocity

$$A_c = \frac{\pi}{4} D_i^2 = \frac{\pi}{4} (0.08542\text{ ft})^2 = 0.005731\text{ ft}^2$$ $$Q = 5.0\text{ GPM} \times \left(\frac{0.13368\text{ ft}^3}{1\text{ gal}}\right) \times \left(\frac{1\text{ min}}{60\text{ s}}\right) = 0.01114\text{ ft}^3/\text{s}$$ $$V = \frac{Q}{A_c} = \frac{0.01114\text{ ft}^3/\text{s}}{0.005731\text{ ft}^2} = 1.944\text{ ft/s}$$

Step 2: Calculate Reynolds number

$$\text{Re}_D = \frac{V D_i}{\nu} = \frac{1.944\text{ ft/s} \times 0.08542\text{ ft}}{1.857 \times 10^{-5}\text{ ft}^2/\text{s}} = \frac{0.16606}{1.857 \times 10^{-5}} = 8942$$

Alternatively, using mass flow rate $\dot{m} = \rho Q = 61.38 \times (0.01114 \times 3600) = 2461.4\text{ lbm/hr}$:

$$\text{Re}_D = \frac{4 \dot{m}}{\pi D_i \mu} = \frac{4 \times 2461.4}{\pi \times 0.08542 \times 4.104} = \frac{9845.6}{1.1014} = 8939 \approx 8940$$

Because $\text{Re}_D \approx 8940$ is within the turbulent range ($\,\ge 10,000$ for standard handbook Dittus-Boelter application; transition-turbulent), apply Dittus-Boelter:

Step 3: Calculate Nusselt number via Dittus-Boelter Since fluid is being heated ($T_s > T_b$), $n = 0.4$:

$$\text{Nu}_D = 0.023 \text{Re}_D^{0.8} \text{Pr}^{0.4} = 0.023 \times (8940)^{0.8} \times (3.01)^{0.4}$$ $$(8940)^{0.8} = 1445.0$$ $$(3.01)^{0.4} = 1.5532$$ $$\text{Nu}_D = 0.023 \times 1445.0 \times 1.5532 = 51.62$$

Step 4: Calculate convective heat transfer coefficient $h_i$

$$h_i = \frac{\text{Nu}_D \cdot k_f}{D_i} = \frac{51.62 \times 0.376\text{ Btu/(hr}\cdot\text{ft}\cdot{}^\circ\text{F)}}{0.08542\text{ ft}} = \frac{19.409}{0.08542} = 227.2\text{ Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)}$$

Step 5: Calculate convective heat transfer rate per linear foot Inside surface area per linear foot ($L = 1\text{ ft}$):

$$A_i' = \pi D_i L = \pi \times 0.08542\text{ ft} \times 1.0\text{ ft} = 0.26836\text{ ft}^2/\text{ft}$$ $$\dot{q}' = h_i A_i' (T_s - T_b) = 227.2 \times 0.26836 \times (160.0 - 140.0) = 227.2 \times 0.26836 \times 20.0 = 1219.4\text{ Btu/(hr}\cdot\text{ft)}$$