Forced convection: laminar and turbulent correlations

Boundary layers, Re, Pr, Nu and St, laminar and turbulent flat-plate correlations, laminar and Dittus–Boelter tube-flow correlations, and outlet temperature with the log-mean temperature difference, with worked numericals.

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Why it matters

Coolant flowing through engine water jackets and radiator tubes, ram air over the radiator and intercooler, oil through an oil cooler and air flowing over a battery pack are all forced convection. The convection coefficient h is what sizes every one of these components, and it comes from empirical correlations of the form Nu = f(Re, Pr). Choosing the right correlation for the geometry and flow regime is the core skill.

Key ideas

Newton's law of cooling. Q = h A (T_s − T_∞). Unlike k, h is not a fluid property; it depends on velocity, geometry, flow regime and fluid properties. Typical values: free convection of air 2–25, forced convection of air 25–250, forced water 100–20 000, boiling and condensation 2500–100 000 W/m²·K.

Boundary layers. At a surface the fluid velocity is zero (no slip), so heat crosses the wall layer by conduction: h = −k_f (∂T/∂y)wall/(T_s − T∞). The velocity boundary layer grows from the leading edge; the thermal boundary layer grows alongside it. Their relative thickness is set by the Prandtl number: δ/δ_t ≈ Pr^(1/3) for laminar flow.

Dimensionless groups.

  • Reynolds number Re = ρVL/μ = VL/ν: inertia to viscous forces; decides laminar or turbulent.
  • Prandtl number Pr = ν/α = μc_p/k: momentum to thermal diffusivity (air ≈ 0.7, water 2–7, engine oil 100s–1000s).
  • Nusselt number Nu = hL/k_f: convective enhancement over pure conduction through the fluid layer (k is the fluid's).
  • Stanton number St = Nu/(Re·Pr) = h/(ρVc_p).

Laminar and turbulent. Turbulent eddies carry hot and cold fluid across the boundary layer, giving much higher h (and friction). Transition:

  • Flat plate: critical Re_x ≈ 5 × 10⁵ (laminar upstream of x_c, turbulent after).
  • Internal flow in a pipe: laminar below Re_D ≈ 2300, turbulent above about 4000 (10 000 for most correlations), transitional between.

External flow over a flat plate. Local h is highest at the leading edge and falls as the boundary layer thickens (h_x ∝ x^(−1/2) laminar, x^(−1/5) turbulent). For a plate with a laminar leading section followed by turbulence, use the mixed-flow average correlation.

Internal flow. After an entrance length the profiles become fully developed. For laminar fully developed flow, Nu is constant: 3.66 for constant wall temperature, 4.36 for constant heat flux. For turbulent flow, use Dittus–Boelter (or Gnielinski for better accuracy). Properties are taken at the mean bulk temperature; for external flow, at the film temperature (T_s + T_∞)/2.

Bulk temperature change. As fluid flows through a heated tube its temperature approaches the wall temperature exponentially, so the driving temperature difference is the log-mean temperature difference.

Reynolds–Colburn analogy. For flows without form drag, St·Pr^(2/3) = C_f/2: heat transfer and skin friction are linked, so enhancing heat transfer usually costs pressure drop.

Cylinders and tube banks in cross-flow (radiator tubes, sensor probes) use correlations such as Nu = C Re^m Pr^(1/3) with C and m from tables in your data book.

Formulas

Q = h A (T_s − T_∞) — h: W/m²·K; A: m².

Re = V L / ν, Pr = μ c_p / k, Nu = h L / k — L: plate length or tube diameter D; ν: m²/s; k: fluid conductivity, W/m·K.

Nu_x = 0.332 Re_x^(1/2) Pr^(1/3), average Nu_L = 0.664 Re_L^(1/2) Pr^(1/3) — laminar flat plate, Re < 5 × 10⁵, Pr ≥ 0.6.

δ = 5 x / √Re_x — laminar velocity boundary-layer thickness.

Nu_x = 0.0296 Re_x^(4/5) Pr^(1/3) — local, turbulent flat plate.

Nu_L = (0.037 Re_L^(4/5) − 871) Pr^(1/3) — average, mixed laminar–turbulent plate with Re_cr = 5 × 10⁵ (drop the 871 if turbulent from the leading edge).

Nu_D = 3.66 (constant T_s) or 4.36 (constant q″) — laminar, fully developed pipe flow.

Nu_D = 0.023 Re_D^0.8 Pr^n — Dittus–Boelter: n = 0.4 when the fluid is heated, 0.3 when cooled; Re_D > 10⁴, 0.6 < Pr < 160, L/D > 10.

(T_s − T_out)/(T_s − T_in) = exp(−h P L / (ṁ c_p)) — constant wall temperature; P = πD.

ΔT_lm = (ΔT_in − ΔT_out) / ln(ΔT_in/ΔT_out); Q = h A_s ΔT_lm = ṁ c_p (T_out − T_in).

Worked examples

Example 1 (standard, flat plate). Air at 20 m/s flows along a 1 m long flat car roof panel. Film properties: ν = 1.6 × 10⁻⁵ m²/s, k = 0.0265 W/m·K, Pr = 0.71. Find the average h and the heat transferred per metre width if the roof is 30 K hotter than the air.

  1. Re_L = V L/ν = 20 × 1/(1.6 × 10⁻⁵) = 1.25 × 10⁶ > 5 × 10⁵, so the flow becomes turbulent at x_c = 5 × 10⁵ × 1.6 × 10⁻⁵/20 = 0.40 m. Use the mixed correlation.
  2. Re_L^0.8 = (1.25 × 10⁶)^0.8 = 75 427; Pr^(1/3) = 0.892.
  3. Nu_L = (0.037 × 75 427 − 871) × 0.892 = (2790.8 − 871) × 0.892 = 1713.
  4. h = Nu k/L = 1713 × 0.0265/1 = 45.4 W/m²·K.
  5. Q = h A ΔT = 45.4 × 1 × 30 = 1.36 kW per metre width.

(Treating the whole plate as laminar would give 17.6 W/m²·K, and as fully turbulent 66.0 W/m²·K; the choice of correlation matters.)

Example 2 (GATE level, tube flow). Water flows at 1 m/s through a 20 mm diameter, 3 m long tube whose wall is held at 80 °C. Water enters at 20 °C. Take properties at the mean bulk temperature: ρ = 992 kg/m³, c_p = 4179 J/kg·K, ν = 0.658 × 10⁻⁶ m²/s, k = 0.631 W/m·K, Pr = 4.32. Find h, the outlet temperature and the heat transfer.

  1. Re_D = V D/ν = 1 × 0.02/(0.658 × 10⁻⁶) = 30 395, turbulent.
  2. Dittus–Boelter with n = 0.4 (water heated): Nu = 0.023 Re^0.8 Pr^0.4 = 0.023 × 3857 × 1.796 = 159.3.
  3. h = Nu k/D = 159.3 × 0.631/0.02 = 5025 W/m²·K.
  4. ṁ = ρ V πD²/4 = 992 × 1 × 3.1416 × 10⁻⁴ = 0.3116 kg/s.
  5. hPL/(ṁc_p) = 5025 × (π × 0.02) × 3/(0.3116 × 4179) = 947.2/1302.2 = 0.7274.
  6. T_out = T_s − (T_s − T_in) e^(−0.7274) = 80 − 60 × 0.4832 = 51.0 °C.
  7. Q = ṁ c_p (T_out − T_in) = 0.3116 × 4179 × 31.0 = 40.4 kW. Check: ΔT_lm = (60 − 29)/ln(60/29) = 42.6 K; h A ΔT_lm = 5025 × 0.1885 × 42.6 = 40.4 kW. Consistent.

Common mistakes

  • Using the solid's k instead of the fluid's k to get h from Nu (h L/k_solid is the Biot number, not Nu).
  • Applying pipe-flow thresholds (2300) to a flat plate (5 × 10⁵), or vice versa.
  • Using the laminar 0.664 correlation beyond Re = 5 × 10⁵.
  • Picking n = 0.4 in Dittus–Boelter when the fluid is being cooled (use 0.3).
  • Using the arithmetic mean temperature difference for a tube with large temperature change; use ΔT_lm.
  • Evaluating properties at the wrong temperature (film temperature for external flow, bulk mean for internal flow).

For GATE ME

Expect calculations of Re, Nu and h from a given correlation, boundary-layer thickness and local vs average h on a flat plate, laminar fully developed Nu values, Dittus–Boelter tube flow with outlet temperature, and MCQs on the meaning of Pr, Nu and St and on how h scales with velocity (V^0.5 laminar, V^0.8 turbulent). GATE usually supplies the correlation; your job is to identify the regime and apply it carefully.

Quick check

  1. h = 50 W/m²·K, L = 0.2 m, k = 0.6 W/m·K. What is Nu?
  2. In turbulent tube flow, how does h change if velocity doubles?
  3. What is Nu for laminar, fully developed flow in a tube with constant heat flux?
  4. Air (ν = 1.5 × 10⁻⁵ m²/s) flows at 2 m/s in a 50 mm tube. Laminar or turbulent?

Answers: 1. 50 × 0.2/0.6 = 16.7. 2. It rises by 2^0.8 ≈ 1.74 times. 3. 4.36. 4. Re = 6667, turbulent.

Interview questions

All Thermodynamics and Heat Transfer interview questions

Try answering each one aloud before you open it.

  1. 1.What is forced convection and how does it differ from natural convection?Concept

    Forced convection is the process of heat transfer where fluid motion is generated by an external source like a pump or fan. In contrast, natural convection relies on buoyancy forces that arise from density differences due to temperature variations in the fluid. Forced convection typically results in higher heat transfer rates compared to natural convection because the fluid movement is more controlled and can be intensified.

  2. 2.Explain the difference between laminar and turbulent flow in the context of forced convection.Concept

    Laminar flow moves in smooth layers, so heat crosses the boundary layer mainly by molecular conduction; turbulent flow has eddies that carry fluid across the layer, which raises the heat transfer coefficient and the friction several-fold. The transition depends on geometry: in a circular pipe flow is laminar below Re_D ≈ 2300 and turbulent above about 4000, while on a flat plate the boundary layer turns turbulent at about Re_x ≈ 5 × 10⁵. Because the mechanisms differ, laminar and turbulent flows use different correlations, and h scales roughly as V^0.5 in laminar and V^0.8 in turbulent flow.

  3. 3.Why are different correlations used for calculating heat transfer coefficients in laminar and turbulent forced convection?Application

    Different correlations are used because the flow characteristics and heat transfer mechanisms differ significantly between laminar and turbulent flows. In laminar flow, heat transfer is primarily due to conduction across the fluid layers, while in turbulent flow, it is enhanced by the mixing of fluid particles. Therefore, the mathematical models and empirical correlations used to predict heat transfer coefficients must account for these differences to provide accurate results.

  4. 4.What happens to the heat transfer rate if the flow transitions from laminar to turbulent in a forced convection scenario?Application

    When the flow transitions from laminar to turbulent, the heat transfer rate generally increases. This is because turbulent flow enhances mixing within the fluid, which disrupts the thermal boundary layer and increases the convective heat transfer coefficient. As a result, the overall heat transfer rate is higher in turbulent flow compared to laminar flow under similar conditions.

  5. 5.Explain why the Reynolds number is important in determining the flow regime in forced convection.Concept

    The Reynolds number (Re) is a dimensionless quantity that helps predict the flow regime in forced convection. It is calculated as Re = ρ·v·L/μ, where ρ is the fluid density, v is the velocity, L is the characteristic length, and μ is the dynamic viscosity. A low Reynolds number indicates laminar flow, while a high Reynolds number indicates turbulent flow. Knowing the flow regime is crucial for selecting the appropriate heat transfer correlations and designing efficient thermal systems.

  6. 6.How does the Prandtl number affect heat transfer in forced convection?Concept

    The Prandtl number (Pr) is a dimensionless number that relates the momentum diffusivity (kinematic viscosity) to the thermal diffusivity of a fluid. It is defined as Pr = ν/α, where ν is the kinematic viscosity and α is the thermal diffusivity. In forced convection, the Prandtl number helps determine the relative thickness of the velocity and thermal boundary layers. A high Prandtl number indicates a thicker velocity boundary layer compared to the thermal boundary layer, affecting the heat transfer characteristics.

  7. 7.What is the significance of the Nusselt number in forced convection heat transfer?Concept

    The Nusselt number (Nu) is a dimensionless number that represents the ratio of convective to conductive heat transfer across a boundary. It is defined as Nu = h·L/k, where h is the convective heat transfer coefficient, L is the characteristic length, and k is the thermal conductivity of the fluid. In forced convection, the Nusselt number is used to quantify the enhancement of heat transfer due to convection compared to pure conduction, and it is essential for calculating the heat transfer coefficient.

  8. 8.Calculate the Reynolds number for air flowing at a velocity of 2 m/s through a pipe with a diameter of 0.05 m. Assume the kinematic viscosity of air is 1.5 × 10^-5 m²/s.Numerical

    To calculate the Reynolds number (Re), use the formula: Re = v·D/ν, where v is the velocity, D is the diameter, and ν is the kinematic viscosity. Substituting the given values: Re = (2 m/s)·(0.05 m)/(1.5 × 10^-5 m²/s) = 6666.67. Therefore, the Reynolds number is approximately 6667, indicating turbulent flow.

  9. 9.A fluid with a Prandtl number of 0.7 flows over a flat plate. If the Reynolds number is 5000, determine whether the flow is laminar or turbulent.Numerical

    The flow regime is determined by the Reynolds number. For flow over a flat plate, a Reynolds number less than 500,000 typically indicates laminar flow, while a Reynolds number greater than 500,000 indicates turbulent flow. Since the given Reynolds number is 5000, which is much less than 500,000, the flow is laminar.

  10. 10.Why is it important to consider both the Reynolds and Prandtl numbers when analyzing forced convection heat transfer?Application

    Both the Reynolds and Prandtl numbers are crucial in analyzing forced convection because they provide insights into the flow regime and the relative thickness of the velocity and thermal boundary layers. The Reynolds number helps determine whether the flow is laminar or turbulent, which affects the heat transfer mechanism. The Prandtl number indicates the relative rate of momentum and heat diffusion, influencing the temperature profile and heat transfer rate. Together, they help in selecting appropriate correlations for calculating heat transfer coefficients.

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