Internal flow in pipes and ducts is crucial for heat transfer in many systems. As fluid moves through confined spaces, velocity and temperature profiles develop, affecting heat exchange. Understanding these processes helps engineers design efficient heating and cooling systems.

Factors like Reynolds number, friction, and surface roughness impact internal flow behavior. Correlations for laminar and turbulent flows in various geometries allow us to predict heat transfer rates. This knowledge is essential for optimizing heat exchangers and other thermal management devices.

Internal Flow Characteristics

Fluid Boundary and Velocity Profile Development

• Internal flow is characterized by the fluid being completely bounded by solid surfaces, such as in pipes or ducts (cylindrical pipes, rectangular ducts)
• The velocity profile of the fluid in internal flow develops over a certain length, known as the hydrodynamic entry length, before becoming fully developed
  • In the hydrodynamic entry region, the velocity profile changes from a uniform profile at the inlet to a fully developed profile downstream
  • The shape of the fully developed velocity profile depends on the Reynolds number (laminar or turbulent flow)

Thermal Boundary Layer Development and Nusselt Number

• The thermal boundary layer in internal flow also develops over a certain length, known as the thermal entry length, before reaching a fully developed state
  • In the thermal entry region, the temperature profile changes from a uniform profile at the inlet to a fully developed profile downstream
  • The shape of the fully developed temperature profile depends on the boundary conditions (constant heat flux or constant wall temperature)
• The Nusselt number (Nu) is a dimensionless parameter that represents the ratio of convective to conductive heat transfer in internal flows
  • Nu = (convective heat transfer coefficient × characteristic length) / thermal conductivity
  • Higher Nusselt numbers indicate more effective convective heat transfer compared to conductive heat transfer

Friction Factor and Surface Roughness Effects

• The friction factor (f) is a dimensionless parameter that represents the resistance to fluid flow in pipes and ducts, and it depends on the Reynolds number (Re) and the relative roughness of the surface
  • f = (pressure drop × diameter) / (density × velocity^2 × length)
  • The friction factor is higher for rough surfaces compared to smooth surfaces, leading to increased pressure drop and pumping power requirements
  • In laminar flow, the friction factor is a function of the Reynolds number only (f = 64/Re for circular pipes)
  • In turbulent flow, the friction factor depends on both the Reynolds number and the relative roughness (Moody diagram)

Entry Lengths for Internal Flows

Hydrodynamic Entry Length

• The hydrodynamic entry length (Lh) is the distance from the inlet of a pipe or duct to the point where the velocity profile becomes fully developed
• For laminar flow, the hydrodynamic entry length can be estimated using the equation: Lh/D ≈ 0.05 Re, where D is the diameter of the pipe or duct
  • Example: For a laminar flow with Re = 1000 in a pipe with D = 0.02 m, Lh ≈ 0.05 * 1000 * 0.02 = 1 m
• For turbulent flow, the hydrodynamic entry length is much shorter and can be estimated using the equation: Lh/D ≈ 10
  • Example: For a turbulent flow in a pipe with D = 0.02 m, Lh ≈ 10 0.02 = 0.2 m
• The hydrodynamic entry length is important for determining the pressure drop and pumping power requirements in internal flows

Thermal Entry Length

• The thermal entry length (Lt) is the distance from the inlet of a pipe or duct to the point where the thermal boundary layer becomes fully developed
• The thermal entry length depends on the Prandtl number (Pr) and can be estimated using the equation: Lt/D ≈ 0.05 * Re * Pr, for laminar flow
  • Example: For a laminar flow with Re = 1000 and Pr = 0.7 in a pipe with D = 0.02 m, Lt ≈ 0.05 * 1000 * 0.7 0.02 = 0.7 m
• In turbulent flow, the thermal entry length is shorter than the laminar case and can be estimated using the equation: Lt/D ≈ 10
  • Example: For a turbulent flow in a pipe with D = 0.02 m, Lt ≈ 10 0.02 = 0.2 m
• The thermal entry length is important for determining the heat transfer characteristics and the effectiveness of heat exchangers

Heat Transfer Coefficients in Internal Flows

Laminar Flow Correlations

• The Nusselt number for fully developed laminar flow in a circular pipe with constant heat flux can be calculated using the Sieder-Tate correlation: Nu = 1.86 * (Re * Pr * D/L)^(1/3) * (μ/μs)^0.14, where μ is the fluid viscosity at the bulk temperature and μs is the fluid viscosity at the surface temperature
  • This correlation accounts for the variation of fluid properties with temperature and is valid for 0.48 ≤ Pr ≤ 16,700 and Re * Pr * D/L ≥ 10
  • Example: For a laminar flow with Re = 1000, Pr = 0.7, D = 0.02 m, L = 1 m, and (μ/μs) = 0.8, Nu ≈ 1.86 * (1000 * 0.7 * 0.02/1)^(1/3) * 0.8^0.14 ≈ 4.64
• Other correlations for laminar flow in circular pipes include the Hausen correlation and the Leveque solution, which are applicable for different boundary conditions and flow regimes

Turbulent Flow Correlations

• For fully developed turbulent flow in a circular pipe, the Dittus-Boelter correlation can be used to calculate the Nusselt number: Nu = 0.023 * Re^(4/5) * Pr^n, where n = 0.4 for heating and n = 0.3 for cooling
  • This correlation is valid for 0.6 ≤ Pr ≤ 160, Re ≥ 10,000, and L/D ≥ 10
  • Example: For a turbulent flow with Re = 50,000, Pr = 0.7, and heating, Nu ≈ 0.023 * 50,000^(4/5) * 0.7^0.4 ≈ 153.5
• The Gnielinski correlation is a more accurate alternative to the Dittus-Boelter correlation for turbulent flow in circular pipes, valid for 3,000 ≤ Re ≤ 5 × 10^6 and 0.5 ≤ Pr ≤ 2,000
  • Nu = (f/8) * (Re - 1000) * Pr / (1 + 12.7 * (f/8)^(1/2) * (Pr^(2/3) - 1)), where f is the friction factor
  • Example: For a turbulent flow with Re = 50,000, Pr = 0.7, and f = 0.018, Nu ≈ (0.018/8) * (50,000 - 1000) * 0.7 / (1 + 12.7 * (0.018/8)^(1/2) * (0.7^(2/3) - 1)) ≈ 148.8
• The Petukhov correlation is another accurate correlation for turbulent flow in circular pipes, valid for 3,000 ≤ Re ≤ 5 × 10^6 and 0.5 ≤ Pr ≤ 200
  • Nu = (f/8) * Re * Pr / (1.07 + 12.7 * (f/8)^(1/2) * (Pr^(2/3) - 1))
  • Example: For a turbulent flow with Re = 50,000, Pr = 0.7, and f = 0.018, Nu ≈ (0.018/8) * 50,000 * 0.7 / (1.07 + 12.7 * (0.018/8)^(1/2) * (0.7^(2/3) - 1)) ≈ 153.2

Convective Heat Transfer in Non-Circular Ducts

Hydraulic Diameter and Its Application

• Non-circular ducts, such as rectangular or triangular cross-sections, are commonly encountered in heat exchangers and other industrial applications (plate-fin heat exchangers, compact heat exchangers)
• The hydraulic diameter (Dh) is used to characterize the dimensions of non-circular ducts, defined as Dh = 4A/P, where A is the cross-sectional area and P is the wetted perimeter
  • Example: For a rectangular duct with width a = 0.1 m and height b = 0.05 m, Dh = 4 * (0.1 * 0.05) / (2 (0.1 + 0.05)) ≈ 0.0667 m
• The correlations for circular pipes can be used for non-circular ducts by replacing the diameter (D) with the hydraulic diameter (Dh)
  • Example: The Dittus-Boelter correlation for turbulent flow in a rectangular duct: Nu = 0.023 * Re^(4/5) * Pr^n, where Re and Nu are based on the hydraulic diameter

Heat Transfer in Annuli

• Annuli, or concentric circular pipes, are another common configuration in heat exchangers (double-pipe heat exchangers, shell-and-tube heat exchangers)
• The hydraulic diameter for an annulus is defined as Dh = Do - Di, where Do is the outer diameter and Di is the inner diameter
  • Example: For an annulus with Do = 0.05 m and Di = 0.03 m, Dh = 0.05 - 0.03 = 0.02 m
• Correlations for heat transfer in annuli can be derived from the correlations for circular pipes by using the hydraulic diameter and appropriate modifications to account for the geometry
  • Example: The Dittus-Boelter correlation for turbulent flow in an annulus: Nu = 0.023 * Re^(4/5) * Pr^n (Di/Do)^0.45, where Re and Nu are based on the hydraulic diameter
• The heat transfer in annuli is influenced by the ratio of the inner to outer diameter (Di/Do) and the direction of heat transfer (heating or cooling of the inner or outer wall)