Forced Convection

Forced convection occurs when external mechanical means such as fans, pumps, or wind creates fluid motion over a surface or through a conduit. This mechanism dominates heat transfer in most HVAC applications including air handlers, fan coil units, heat exchangers, and all ducted air distribution systems.

Fundamental Governing Parameters

Three dimensionless numbers characterize forced convection heat transfer behavior:

Reynolds Number (Re)

The Reynolds number quantifies the ratio of inertial forces to viscous forces in the fluid flow:

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

Where:

• ρ = fluid density (kg/m³)
• V = fluid velocity (m/s)
• D = characteristic length (hydraulic diameter for internal flow, m)
• μ = dynamic viscosity (Pa·s)
• ν = kinematic viscosity (m²/s)

Flow transitions from laminar to turbulent at Re ≈ 2300 for internal pipe flow and Re ≈ 500,000 for external flow over flat plates.

Nusselt Number (Nu)

The Nusselt number represents the ratio of convective to conductive heat transfer:

$$Nu = \frac{h D}{k}$$

Where:

• h = convection heat transfer coefficient (W/m²·K)
• D = characteristic length (m)
• k = thermal conductivity of fluid (W/m·K)

The Nusselt number is the primary output of convection correlations and directly determines the convection coefficient.

Prandtl Number (Pr)

The Prandtl number relates momentum diffusivity to thermal diffusivity:

$$Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$$

Where:

• ν = kinematic viscosity (m²/s)
• α = thermal diffusivity (m²/s)
• c_p = specific heat at constant pressure (J/kg·K)

For air at standard conditions, Pr ≈ 0.71. For water at 20°C, Pr ≈ 7.0.

Internal Flow Correlations

Turbulent Flow in Pipes and Ducts

Dittus-Boelter Equation

The most widely used correlation for turbulent flow in smooth tubes with moderate temperature differences:

$$Nu = 0.023 Re^{0.8} Pr^n$$

Where:

• n = 0.4 for heating the fluid (T_surface > T_fluid)
• n = 0.3 for cooling the fluid (T_surface < T_fluid)

Valid for:

• Re > 10,000
• 0.7 < Pr < 160
• L/D > 10 (fully developed flow)

Sieder-Tate Equation

Provides improved accuracy for large temperature differences by accounting for viscosity variation:

$$Nu = 0.027 Re^{0.8} Pr^{1/3} \left(\frac{\mu}{\mu_s}\right)^{0.14}$$

Where:

• μ = dynamic viscosity at bulk fluid temperature (Pa·s)
• μ_s = dynamic viscosity at surface temperature (Pa·s)

Valid for:

• Re > 10,000
• 0.7 < Pr < 16,700
• L/D > 10

Gnielinski Equation

Modern correlation providing superior accuracy across wider Reynolds number range:

$$Nu = \frac{(f/8)(Re - 1000)Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}$$

Where f is the Darcy friction factor from the Moody diagram or Colebrook equation.

Valid for:

• 3000 < Re < 5×10⁶
• 0.5 < Pr < 2000

Laminar Flow in Pipes

For fully developed laminar flow with constant wall temperature:

$$Nu = 3.66$$

For fully developed laminar flow with constant heat flux:

$$Nu = 4.36$$

These constant values apply for Re < 2300 with L/D > 10.

Entrance Region Effects

In the developing flow region near the entrance, heat transfer coefficients are higher than fully developed values. The thermal entrance length is:

$$L_t = 0.05 Re \cdot Pr \cdot D$$

For turbulent flow, entrance effects are typically negligible beyond 10-20 diameters.

External Flow Correlations

Flow Over Flat Plates

Laminar Boundary Layer (Re < 500,000)

$$Nu_x = 0.332 Re_x^{0.5} Pr^{1/3}$$

Where Nu_x and Re_x are evaluated at distance x from the leading edge.

Average Nusselt number over length L:

$$\overline{Nu}_L = 0.664 Re_L^{0.5} Pr^{1/3}$$

Turbulent Boundary Layer (Re > 500,000)

$$\overline{Nu}_L = 0.037 Re_L^{0.8} Pr^{1/3}$$

Valid for 0.6 < Pr < 60.

Flow Across Tube Banks

For staggered and in-line tube arrangements common in finned coil heat exchangers:

$$Nu_D = C Re_D^m Pr^{0.36} \left(\frac{Pr}{Pr_s}\right)^{0.25}$$

Constants C and m depend on tube arrangement and Reynolds number range. Typical values for 10-row deep staggered arrangements with Re_D > 1000:

• C ≈ 0.27
• m ≈ 0.63

Flow Over Cylinders

For cross-flow over single cylinders (representative of single tubes):

$$Nu_D = C Re_D^m Pr^{1/3}$$

Valid for 0.7 < Pr < 500.

Convection Coefficient Calculation Procedure

1. Determine fluid properties at the bulk temperature (average of inlet and outlet):

   • Density ρ
   • Specific heat c_p
   • Thermal conductivity k
   • Dynamic viscosity μ

2. Calculate Reynolds number using appropriate characteristic length

3. Evaluate Prandtl number from fluid properties

4. Select appropriate correlation based on geometry and flow regime

5. Calculate Nusselt number from the correlation

6. Solve for convection coefficient:

$$h = \frac{Nu \cdot k}{D}$$

Typical Convection Coefficients for HVAC Applications

These values represent single-phase forced convection without phase change. Condensation and evaporation produce significantly higher coefficients.

Heat Exchanger Design Applications

For heat exchangers, convection coefficients on both fluid sides determine the overall heat transfer coefficient U:

$$\frac{1}{U} = \frac{1}{h_i} + R_{fouling} + \frac{t_{wall}}{k_{wall}} + \frac{1}{h_o}$$

Where:

• h_i = inside convection coefficient (W/m²·K)
• h_o = outside convection coefficient (W/m²·K)
• R_fouling = fouling resistance (m²·K/W)
• t_wall = wall thickness (m)
• k_wall = wall thermal conductivity (W/m·K)

The fluid side with lower convection coefficient controls overall performance. In air-to-water heat exchangers, the air side typically controls with h values 20-50 times lower than the water side.

Enhancement Techniques

Several methods increase forced convection heat transfer:

Velocity Increase

Since h ∝ V^0.8 for turbulent flow, increasing velocity from 2 to 4 m/s increases h by approximately 74%. Pressure drop increases with V², requiring careful fan or pump power analysis.

Turbulence Promotion

Surface roughness, turbulators, and twisted tape inserts increase turbulence and Nusselt number by 50-200% at the cost of 2-4 times higher pressure drop.

Extended Surfaces

Fins on the low-h side increase effective area. Air-side fins in HVAC coils provide 10-20 times more surface area than the base tube area.

Reduced Hydraulic Diameter

For constant velocity, reducing hydraulic diameter increases Reynolds number and convection coefficient. Microchannel heat exchangers exploit this principle.

Practical Considerations

Temperature-Dependent Properties

Fluid properties vary significantly with temperature. Evaluate properties at the film temperature for external flow:

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

For internal flow, use bulk temperature unless temperature difference exceeds 20°C, then apply the Sieder-Tate correction.

Developing Flow Regions

Standard correlations assume fully developed flow. Short tubes and ducts near inlets have higher local heat transfer coefficients. Account for entrance effects when L/D < 10.

Non-Circular Ducts

For rectangular or other non-circular ducts, use hydraulic diameter:

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

Where A_c is cross-sectional area and P is wetted perimeter.

Uncertainty in Correlations

Empirical correlations typically have uncertainty of ±15-25%. Design heat exchangers with appropriate safety factors to account for this variability and fouling accumulation over time.

Sections

Lh,lam / D = 0.05 Re