Forced convection: dimensionless numbers and correlations

Newton's law of cooling, the Re, Pr, Nu, St numbers, standard forced-convection correlations for tubes and flat plates, and how h scales with velocity and diameter.

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Why it matters

Almost every heat exchanger, jacketed reactor and cooler in a plant relies on a pump or fan driving fluid past a surface. The film coefficient h is what limits the duty, and it cannot be derived from first principles for real equipment — it comes from dimensionless correlations. Knowing which correlation applies, and how h scales with velocity and diameter, is what lets you size an exchanger or predict what happens when you change the flow.

Key ideas

Newton's law of cooling. Convective heat flux is written q″ = h·(T_s − T_f). This defines h; it is not a law of nature. h depends on the fluid properties, the velocity, the geometry and whether the flow is laminar or turbulent.

Why dimensionless numbers. Dimensional analysis (Buckingham π) reduces the many variables of forced convection to Nu = f(Re, Pr) and, for short tubes, L/D. Correlations measured with one fluid and size can then be used for others.

Reynolds number Re = ρ·u·L/μ — ratio of inertial to viscous forces; decides laminar or turbulent flow.

  • Pipe flow (L = inside diameter D): laminar below about 2100–2300, turbulent above about 4000 (correlations like Dittus–Boelter need Re > 10 000).
  • Flat plate (L = distance from the leading edge x): transition at about Re_x = 5 × 10⁵.

Prandtl number Pr = c_p·μ/k = ν/α — ratio of momentum diffusivity to thermal diffusivity; a fluid property. Liquid metals ≈ 0.01, gases ≈ 0.7, water ≈ 2–7 (falls with temperature), oils 100–10 000. For laminar boundary layers δ/δ_t ≈ Pr^(1/3): high-Pr fluids have thin thermal layers.

Nusselt number Nu = h·L/k_f — dimensionless h: the ratio of convective heat transfer to pure conduction across a fluid layer of thickness L. Note k_f is the fluid's conductivity (in the Biot number it is the solid's).

Stanton number St = Nu/(Re·Pr) = h/(ρ·c_p·u) and the Colburn j-factor j_H = St·Pr^(2/3). The Chilton–Colburn analogy j_H ≈ f/2 (Fanning f) links heat transfer to friction in turbulent flow; it is the basis of the heat–momentum–mass analogies.

Peclet number Pe = Re·Pr, and Graetz number Gz = (π/4)·Re·Pr·D/L, used for laminar developing flow in tubes.

Correlations to remember.

  • Turbulent flow in smooth tubes, fully developed (Dittus–Boelter): Nu = 0.023·Re^0.8·Pr^n with n = 0.4 when the fluid is heated, 0.3 when cooled; Re > 10 000, 0.7 < Pr < 160, L/D > 10; properties at the bulk mean temperature.
  • Sieder–Tate adds a viscosity correction (μ_b/μ_w)^0.14 for large property variations (viscous oils): Nu = 0.027·Re^0.8·Pr^(1/3)·(μ_b/μ_w)^0.14.
  • Laminar fully developed tube flow: Nu = 3.66 (constant wall temperature), 4.36 (constant heat flux) — independent of Re.
  • Laminar flat plate: local Nu_x = 0.332·Re_x^0.5·Pr^(1/3); average over length L: Nu_L = 0.664·Re_L^0.5·Pr^(1/3) (Pr ≥ 0.6).
  • Turbulent flat plate (average, from the leading edge): Nu_L = 0.037·Re_L^0.8·Pr^(1/3). Coefficients and exponents vary between textbooks; use the one your syllabus or data book gives and respect its range of validity.

Scaling. For turbulent tube flow h ∝ u^0.8·D^(−0.2) at fixed properties; for laminar flat plates h ∝ u^0.5·L^(−0.5).

Formulas

q = h·A·(T_s − T_f) Newton's law of cooling; h (W/m²·K), A (m²), T_s surface, T_f fluid temperature.

Re = ρ·u·D/μ, Pr = c_p·μ/k, Nu = h·D/k ρ (kg/m³), u mean velocity (m/s), D diameter (m), μ (Pa·s), c_p (J/kg·K), k fluid conductivity (W/m·K).

Nu = 0.023·Re^0.8·Pr^0.4 (heating), Nu = 0.023·Re^0.8·Pr^0.3 (cooling) Dittus–Boelter: turbulent, Re > 10⁴, 0.7 < Pr < 160, L/D > 10.

Nu = 0.027·Re^0.8·Pr^(1/3)·(μ_b/μ_w)^0.14 Sieder–Tate: turbulent, large viscosity variation; μ_w at wall temperature.

Nu = 3.66 (uniform T_w), Nu = 4.36 (uniform q″) Laminar, fully developed, circular tube.

Nu_x = 0.332·Re_x^0.5·Pr^(1/3), Nu_L = 0.664·Re_L^0.5·Pr^(1/3) Laminar flat plate, Re < 5 × 10⁵, properties at film temperature (T_s + T∞)/2.

St = h/(ρ·c_p·u) = Nu/(Re·Pr), j_H = St·Pr^(2/3) ≈ f/2 Stanton number; Colburn analogy.

Worked examples

Example 1 (standard). Water flows at 1.0 m/s through a 25 mm inside-diameter tube and is being heated; its bulk mean temperature is 40 °C. Properties at 40 °C (from a data book): ρ = 992 kg/m³, μ = 6.53 × 10⁻⁴ Pa·s, k = 0.631 W/m·K, Pr = 4.32. Find h.

  1. Re = ρ·u·D/μ = 992 × 1.0 × 0.025/6.53 × 10⁻⁴ = 37 980 → turbulent, Dittus–Boelter valid.
  2. Heating, so n = 0.4: Nu = 0.023·Re^0.8·Pr^0.4 = 0.023 × 37 980^0.8 × 4.32^0.4 = 190.4.
  3. h = Nu·k/D = 190.4 × 0.631/0.025 = 4805 W/m²·K.

Answer: h ≈ 4.8 kW/m²·K.

Example 2 (GATE level). (a) Air at 5 m/s flows along a flat plate 0.5 m long (ν = 1.6 × 10⁻⁵ m²/s, k = 0.0263 W/m·K, Pr = 0.707 at the film temperature). Find the average h and the heat lost from one side of a 0.5 m × 1 m plate at 60 K above the air. (b) In turbulent tube flow, by what factor does h change if the tube diameter is halved at the same mass flow rate?

(a)

  1. Re_L = u·L/ν = 5 × 0.5/1.6 × 10⁻⁵ = 1.5625 × 10⁵ < 5 × 10⁵ → laminar over the whole plate.
  2. Nu_L = 0.664·Re_L^0.5·Pr^(1/3) = 0.664 × 395.3 × 0.8908 = 233.8.
  3. h = Nu_L·k/L = 233.8 × 0.0263/0.5 = 12.30 W/m²·K.
  4. q = h·A·ΔT = 12.30 × 0.5 × 60 = 369 W. (b)
  5. Same mass flow: u ∝ 1/D², so u rises 4 times; h ∝ u^0.8·D^(−0.2).
  6. Ratio = 4^0.8 × (0.5)^(−0.2) = 3.031 × 1.149 = 3.48.

Answer: (a) h ≈ 12.3 W/m²·K, q ≈ 369 W; (b) h increases about 3.5 times (at the cost of roughly 2^4.8 ≈ 28 times the pressure drop per unit length in turbulent flow).

Common mistakes

  • Using the solid's k in Nu, or the fluid's k in Bi.
  • Using Dittus–Boelter in laminar or transitional flow, or forgetting n = 0.4 for heating and 0.3 for cooling.
  • Using the local flat-plate correlation (0.332) when the average (0.664) is wanted.
  • Evaluating properties at the wrong temperature: bulk mean for tubes, film temperature for external flow.
  • Using the flat-plate transition Re (5 × 10⁵) for a pipe or vice versa.
  • Forgetting that in laminar fully developed tube flow Nu is constant, so h ∝ 1/D and does not depend on velocity.

For GATE CH

Expect calculations of Re, Pr, Nu and h with a given correlation, ratio problems (how h changes when velocity, diameter or flow rate changes), identification of the right correlation from the regime, and conceptual questions on the meaning of Pr and the Colburn analogy. Practise writing h ∝ uᵃ·Dᵇ from a correlation before plugging numbers; most ratio questions then take one line.

Quick check

  1. Water Pr ≈ 5; air Pr ≈ 0.7. Which has the thicker thermal boundary layer relative to its velocity layer?
  2. In turbulent tube flow, by what factor does h rise if velocity doubles?
  3. Laminar fully developed flow, uniform wall temperature: Nu = ?
  4. Which temperature is used for flat-plate properties?
  5. Define the Stanton number.

Answers: 1. Air. 2. 2^0.8 ≈ 1.74. 3. 3.66. 4. The film temperature (T_s + T∞)/2. 5. St = h/(ρ·c_p·u) = Nu/(Re·Pr).

Interview questions

All Heat Transfer interview questions

Try answering each one aloud before you open it.

  1. 1.What is forced convection in the context of heat transfer?Concept

    Forced convection is a mechanism of heat transfer in which fluid motion is generated by an external source like a pump, fan, or a mixer. This is in contrast to natural convection, where fluid motion is caused by buoyancy forces that result from density variations due to temperature gradients in the fluid.

  2. 2.Explain the significance of the Reynolds number in forced convection.Concept

    The Reynolds number (Re) is a dimensionless number that helps predict flow patterns in different fluid flow situations. In forced convection, it is used to determine whether the flow is laminar or turbulent. A low Reynolds number indicates laminar flow, while a high Reynolds number indicates turbulent flow. This distinction is crucial because it affects the heat transfer rate.

  3. 3.What is the Nusselt number and why is it important in forced convection?Concept

    The Nusselt number (Nu) is a dimensionless number that represents the ratio of convective to conductive heat transfer across a boundary. In forced convection, it is used to quantify the enhancement of heat transfer through a fluid layer as a result of convection. A higher Nusselt number indicates more efficient convective heat transfer.

  4. 4.How does the Prandtl number influence heat transfer in forced convection?Concept

    The Prandtl number (Pr) is a dimensionless number that relates the momentum diffusivity (viscosity) to the thermal diffusivity of a fluid. It indicates the relative thickness of the velocity boundary layer to the thermal boundary layer. In forced convection, a higher Prandtl number means that the thermal boundary layer is thinner compared to the velocity boundary layer, which can affect the heat transfer rate.

  5. 5.Why is the Dittus-Boelter equation used in forced convection calculations, and when is it valid?Application

    Nu = 0.023·Re^0.8·Pr^n (n = 0.4 for heating the fluid, 0.3 for cooling) gives the film coefficient for fully developed turbulent flow in smooth tubes with one line of algebra, which is why it is the workhorse for tube-side h in exchanger design. It is valid for Re above about 10 000, 0.7 < Pr < 160, L/D > 10 and moderate wall-to-bulk temperature differences, with properties at the bulk mean temperature. For viscous liquids with large property variation, Sieder–Tate (with the (μ_b/μ_w)^0.14 correction) is preferred, and in laminar flow it must not be used at all.

  6. 6.What happens to the heat transfer rate if the flow changes from laminar to turbulent in forced convection?Application

    If the flow changes from laminar to turbulent in forced convection, the heat transfer rate generally increases. This is because turbulent flow enhances mixing and disrupts the thermal boundary layer, leading to a higher convective heat transfer coefficient and thus a higher heat transfer rate.

  7. 7.How would you determine if a flow is laminar or turbulent in a forced convection scenario?Application

    To determine if a flow is laminar or turbulent in forced convection, you calculate the Reynolds number. For flow inside a pipe, if the Reynolds number is less than 2300, the flow is considered laminar. If it is greater than 4000, the flow is considered turbulent. Values between 2300 and 4000 indicate a transitional flow regime.

  8. 8.Calculate the Reynolds number for water flowing at 0.5 m/s through a pipe with a diameter of 0.1 m. Assume the kinematic viscosity of water is 1.0 × 10⁻⁶ m²/s.Numerical

    Re = (velocity × diameter) / kinematic viscosity = (0.5 m/s × 0.1 m) / (1.0 × 10⁻⁶ m²/s) = 50,000. This indicates that the flow is turbulent.

  9. 9.Using the Dittus-Boelter equation, estimate the Nusselt number for water being cooled in a tube at Re = 10 000 and Pr = 7, in fully developed turbulent flow.Numerical

    For cooling, n = 0.3: Nu = 0.023 × 10 000^0.8 × 7^0.3 = 0.023 × 1584.9 × 1.793 ≈ 65.4. If the water were being heated (n = 0.4), Nu = 0.023 × 1584.9 × 2.178 ≈ 79.4. Re = 10 000 is the lower edge of the correlation's validity, so treat the result as an estimate.

  10. 10.Explain how the Grashof number is related to forced convection.Concept

    The Grashof number (Gr) is primarily used in natural convection to quantify the ratio of buoyancy to viscous forces. In forced convection, it is less relevant because the flow is driven by external means rather than buoyancy. However, in mixed convection scenarios where both forced and natural convection are present, the Grashof number can help determine the relative influence of natural convection.

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