Forced Convection in External and Internal Flows

Introduction

Forced convection is heat transfer between a surface and a moving fluid when the motion is produced by a fan, pump, compressor, or other mechanical device. It is central to the design of heat exchangers, electronic cooling systems, vehicle components, turbine blades, solar collectors, air-conditioning equipment, and process machinery.

The rate of convection is commonly represented by Newton’s law of cooling:

where q̇ is heat-transfer rate (W), h is the convection coefficient (W·m⁻²·K⁻¹), A_s is the exposed surface area (m²), T_s is surface temperature (K or °C), and T_∞ is the undisturbed fluid temperature. The coefficient h is not a fluid property alone; it depends on geometry, velocity, fluid properties, and flow regime.

This chapter develops methods for estimating h in external flows over surfaces and internal flows through ducts. It emphasizes Reynolds number, Prandtl number, Nusselt number, boundary layers, entrance effects, friction relations, and the distinction between constant-temperature and constant-flux boundary conditions.

Learning objectives

After studying this chapter, a student should be able to:

• distinguish external and internal forced-convection problems;
• classify flow as laminar or turbulent using Reynolds number;
• explain hydrodynamic and thermal boundary layers;
• select appropriate Nusselt-number correlations;
• analyze fully developed and developing flow in ducts;
• apply energy balances to constant-wall-temperature and constant-wall-heat-flux cases;
• estimate pressure loss and understand its relationship to heat transfer;
• identify assumptions, limitations, and uncertainty in engineering correlations.

1. Governing dimensionless groups

For a characteristic length L and mean fluid speed V, the Reynolds number is

Here ρ is density (kg·m⁻³), μ is dynamic viscosity (Pa·s), and ν = μ/ρ is kinematic viscosity (m²·s⁻¹). Reynolds number compares inertial and viscous effects. Low values generally indicate orderly laminar flow; high values indicate turbulent flow. The transition value depends on geometry and disturbances.

The Prandtl number is

where c_p is specific heat (J·kg⁻¹·K⁻¹), k is thermal conductivity (W·m⁻¹·K⁻¹), and α = k/(ρc_p) is thermal diffusivity (m²·s⁻¹). It compares momentum diffusion with thermal diffusion.

The Nusselt number is

It expresses convection relative to conduction across a layer of thickness comparable with L. A Nusselt number close to unity indicates weak enhancement over pure conduction; large values indicate strong convection.

For ducts, the hydraulic diameter is used:

where A_c is flow cross-sectional area (m²) and P_w is wetted perimeter (m). For a circular tube, D_h = D. For a rectangular duct of width a and height b, D_h = 2ab/(a+b).

The bulk or mixing-cup temperature is the temperature associated with the energy carried by the flowing fluid. It is denoted by T_m or T_b, and is generally different from the local velocity-weighted average temperature.

2. Boundary layers and external flow

When a fluid first encounters a stationary surface, the no-slip condition causes fluid velocity to be zero at the wall and to increase toward the free-stream value. The resulting hydrodynamic boundary layer has thickness δ. A thermal boundary layer develops when wall and fluid temperatures differ. Its thickness depends strongly on Pr.

For Pr > 1, momentum diffuses more rapidly than heat, so the thermal layer is thinner than the velocity layer. For Pr < 1, heat diffuses more rapidly and the thermal layer is thicker. Liquid metals have very small Prandtl numbers; oils commonly have large values.

For flow over a smooth flat plate, the local Reynolds number is

where x is distance from the leading edge and V_∞ is free-stream velocity. Boundary-layer flow is commonly laminar for Re_x below approximately 5 × 10⁵, although surface roughness, free-stream turbulence, pressure gradients, and leading-edge geometry shift transition.

For a smooth isothermal flat plate with nearly constant properties, useful local and average correlations are listed below.

The turbulent average relation is commonly used for approximately 0.6 ≤ Pr ≤ 60 and sufficiently large Reynolds number. The subtraction term accounts approximately for the laminar leading-edge portion. Correlations should not be extrapolated beyond their stated ranges.

The exponent on Pr may be modified in more advanced correlations, especially for large temperature differences or variable properties. For gases and moderate temperature differences, evaluating properties at a film temperature is often useful:

The appropriate property temperature depends on the correlation and fluid. For liquids, viscosity variation can be important and a viscosity correction may be required.

Local versus average coefficients

A local coefficient h_x varies along the surface because the boundary layer grows. The average coefficient over length L is

The total heat-transfer rate is more safely evaluated as

when T_s and T_∞ are uniform. Confusing a local coefficient with an average coefficient can produce substantial design errors.

3. External turbulent flow and practical surfaces

Turbulent boundary layers contain eddies that transport momentum and energy across the layer. They generally produce larger heat-transfer coefficients than laminar layers, but also greater skin friction and pressure drag. Surface roughness can enhance turbulence and heat transfer while increasing drag, vibration, and fouling.

For a surface with roughness, a smooth-wall correlation is not automatically valid. Relative roughness, ε/L or ε/D_h, must be considered. In design, the thermal benefit of roughness should be compared with pumping or aerodynamic power:

where Δp is pressure drop (Pa) and V̇ is volumetric flow rate (m³·s⁻¹). In rotating machinery and vehicles, increased drag may outweigh the heat-transfer gain.

4. Internal flow in ducts

In internal flow, fluid enters a duct with a developing velocity profile. Viscous effects spread inward from the wall until the velocity profile becomes fully developed. The hydrodynamic entrance length is approximately

and, as an engineering estimate for turbulent flow,

The laminar estimate is most useful for a circular duct with a uniform inlet profile. Entrance lengths vary with inlet conditions and geometry.

A thermal boundary layer also develops. The thermal entrance length is approximately

for laminar flow under commonly used assumptions. A fluid may therefore be hydrodynamically fully developed but thermally developing. When both velocity and temperature profiles no longer change shape in the flow direction, the flow is thermally fully developed.

For a circular duct, Reynolds number is

where V_m is bulk mean velocity. For noncircular passages, replace D by D_h. In a smooth circular tube, flow is normally regarded as laminar below about Re_D = 2300, transitional between approximately 2300 and 4000, and turbulent above approximately 4000. These boundaries are engineering conventions, not universal physical discontinuities.

5. Fully developed laminar flow

For fully developed laminar flow in a circular tube with constant wall temperature, constant properties, and negligible axial conduction, the Nusselt number is exactly

For constant wall heat flux, the exact result is

These values apply only to thermally fully developed flow in a circular tube. They should not be used for the entrance region or arbitrary duct shapes.

For a fully developed laminar flow in a circular tube, the friction factor is

where f_D is the Darcy friction factor. The Darcy factor must not be confused with the Fanning factor; f_D = 4f_F. Pressure drop for a duct of length L is calculated from Darcy–Weisbach:

Minor losses from entrances, exits, bends, valves, and fittings may also be important.

6. Turbulent internal flow

A widely used correlation for turbulent flow in smooth tubes is the Dittus–Boelter equation:

where n = 0.4 when the fluid is being heated and n = 0.3 when the fluid is being cooled. It is typically applied for fully developed turbulent flow, smooth surfaces, approximately constant properties, and ranges commonly stated as 0.7 ≤ Pr ≤ 160, Re_D ≥ 10,000. Exact limits depend on the reference and should be checked for the selected version.

A more broadly useful correlation is the Gnielinski relation:

It requires a Darcy friction factor and is generally applied to fully developed turbulent flow in smooth or mildly rough ducts. One smooth-tube approximation is the Blasius relation:

for smooth turbulent flow over its conventional range, approximately 4 × 10³ ≤ Re_D ≤ 10⁵. For greater Reynolds numbers or rough ducts, the Colebrook equation is commonly used:

where ε is absolute roughness (m). Because f_D appears on both sides, iteration or an explicit approximation is required.

7. Constant wall temperature and constant wall heat flux

Two idealized boundary conditions are especially important.

For constant wall temperature, T_s is fixed and the bulk-temperature energy balance is

where ṁ = ρV̇ is mass flow rate (kg·s⁻¹) and P is heated perimeter (m). If h, P, and c_p are constant, integration gives

The total heat transfer is

and also

where the logarithmic mean temperature difference is

For constant wall heat flux, q''_s is uniform. The bulk temperature increases linearly for constant properties:

The local wall-to-bulk temperature difference is related to the coefficient by

In fully developed flow, h is constant and therefore the wall-to-bulk temperature difference is constant, even though both wall and bulk temperatures increase along the duct.

8. Worked example: turbulent tube heating

Air flows through a smooth circular tube of diameter 0.020 m and length 2.0 m at a mean velocity of 8.0 m·s⁻¹. Its properties are ρ = 1.20 kg·m⁻³, μ = 1.81 × 10⁻⁵ Pa·s, k = 0.026 W·m⁻¹·K⁻¹, and Pr = 0.71. The wall is maintained at 80 °C, and the inlet bulk temperature is 20 °C. Estimate the heat-transfer coefficient and outlet bulk temperature using Dittus–Boelter.

First,

The flow is turbulent according to the stated engineering classification. For heating, n = 0.4:

Thus,

The mass flow rate is

The heated area is A_s = πDL ≈ 0.1257 m². Therefore,

Using the constant-wall-temperature relation,

The predicted heat transfer is

The result is an estimate. The tube entrance, property variation, and transitional proximity should be assessed before using it for a safety-critical design.

9. Applied case: cooling an electronic enclosure

Background

An illustrative design team must cool a row of power electronics inside a forced-air enclosure. The allowable component temperature is 75 °C, while the incoming air is 25 °C. A finned heat sink is exposed to a measured air speed, but the available fan has limited pressure capability.

Problem

Increasing air speed raises Re and generally increases Nu and h. However, pressure drop rises, fan power increases, and acoustic noise becomes unacceptable. A heat-transfer-only design could therefore fail at the system level.

Analysis and approach

The team first determines the hydraulic diameter of the heat-sink passages, computes Reynolds and Prandtl numbers using properties at a suitable mean temperature, and checks whether the flow is laminar, transitional, or turbulent. It then selects a correlation whose geometry and boundary condition match the passages. The predicted coefficient is used in

with fin efficiency included when the fin temperature is not uniform. In parallel, Darcy–Weisbach and minor-loss coefficients estimate pressure drop. Fan power is approximated by P_fan = ΔpV̇/η_f, where η_f is fan efficiency.

The design is iterated: passage size, fin spacing, flow rate, and surface area are varied until thermal, pressure, noise, manufacturing, and contamination requirements are simultaneously satisfied. Temperature measurements are used to validate the model rather than treating a correlation as an exact prediction.

Outcome and learning points

The preferred design is not necessarily the one with the largest h. It is the design that meets the temperature limit with acceptable fan power and reliability. The case illustrates four professional principles: match the correlation to the flow, account for pressure loss, include contact and spreading resistances where relevant, and validate with measurements.

10. Selection procedure and limitations

A sound solution procedure is:

1. Define geometry, flow rate, temperatures, and boundary condition.
2. Evaluate properties at a justified reference temperature.
3. Select a characteristic length or hydraulic diameter.
4. Calculate Re, Pr, and, where needed, roughness ratio.
5. Determine whether the flow is developing or fully developed.
6. Select a correlation within its validity range.
7. Calculate Nu and then h = Nu k/L_c.
8. Apply an energy balance to determine heat rate or outlet temperature.
9. Check pressure drop, friction factor, units, and physical plausibility.
10. Consider uncertainty, fouling, nonuniform wall temperature, and property variation.

Important limitations include transition-region uncertainty, entrance effects, nonuniform velocity profiles, buoyancy mixed with forced convection, compressibility at high gas speed, radiation, roughness, fouling, and three-dimensional separation. Correlations are empirical or semiempirical models; their uncertainty may exceed the arithmetic precision of the calculated answer. A result reported as 44.7 W·m⁻²·K⁻¹ should not imply that the physical coefficient is known to four significant figures.

Modern computational fluid dynamics can resolve local velocity and temperature fields, but it still requires appropriate turbulence and wall models, mesh validation, and boundary conditions. Machine-learning correlations can assist interpolation of validated experimental databases, but they should not replace dimensional analysis, conservation laws, or verification against independent data.

Chapter Summary

Forced convection depends on fluid motion, geometry, and thermal boundary conditions. Reynolds number classifies the relative importance of inertia and viscosity, while Prandtl number compares momentum and thermal diffusion. Nusselt number converts a dimensionless correlation into a convection coefficient.

External-flow heat transfer is governed by developing boundary layers; flat-plate correlations distinguish local from average coefficients and laminar from turbulent flow. Internal-flow analysis requires hydraulic diameter, bulk temperature, entrance-length considerations, and friction relations. Fully developed laminar circular-tube flow has exact values of Nu_D = 3.66 for constant wall temperature and Nu_D = 4.36 for constant wall heat flux. Turbulent tube flow is commonly estimated with Dittus–Boelter or Gnielinski correlations, subject to validity ranges.

Energy balances connect convection coefficients to outlet temperatures and heat rates. Constant-wall-temperature problems produce an exponential bulk-temperature variation, whereas constant-wall-flux problems produce a linear bulk-temperature variation for constant properties. Practical design must balance heat transfer against pressure drop, pumping power, noise, fouling, uncertainty, and validation requirements.

Key Terms

Reynolds number: Ratio of inertial to viscous effects in a flow.

Prandtl number: Ratio of momentum diffusivity to thermal diffusivity.

Nusselt number: Dimensionless measure of convective heat transfer relative to conduction.

Forced convection: Convection driven primarily by mechanical fluid motion.

External flow: Flow over a surface with an external free stream.

Internal flow: Flow confined within a duct, channel, or pipe.

Boundary layer: Thin region near a wall where velocity or temperature changes rapidly.

Hydrodynamic entrance length: Distance required for the velocity profile to become fully developed.

Thermal entrance length: Distance required for the temperature profile to become fully developed.

Bulk temperature: Energy-average temperature of flowing fluid.

Hydraulic diameter: Equivalent diameter based on flow area and wetted perimeter.

Fully developed flow: Flow whose nondimensional velocity or temperature profile no longer changes axially.

Darcy friction factor: Dimensionless factor used in the Darcy–Weisbach pressure-drop equation.

Constant wall temperature: Boundary condition in which wall temperature is prescribed and uniform.

Constant wall heat flux: Boundary condition in which heat flux at the wall is prescribed and uniform.

Film temperature: Approximate mean of surface and free-stream temperatures for property evaluation.

Dittus–Boelter correlation: Empirical relation for fully developed turbulent tube heat transfer.

Gnielinski correlation: Turbulent internal-flow heat-transfer relation involving friction factor.

Minor loss: Pressure loss caused by fittings, entrances, exits, bends, or area changes.

Thermal boundary layer: Region in which fluid temperature changes from wall to free-stream value.

Review Questions

Short Answer Questions

1. What physical effects are compared by Reynolds number and Prandtl number?
2. Distinguish local and average convection coefficients over a flat plate.
3. Why is hydraulic diameter used for noncircular ducts?
4. State the fully developed laminar Nusselt numbers for a circular tube with constant wall temperature and constant wall heat flux.
5. What is the difference between Darcy and Fanning friction factors?

Long Answer Questions

1. Explain the development of hydrodynamic and thermal boundary layers in external flow and discuss the role of Pr.
2. Derive the exponential bulk-temperature relation for fully developed internal flow with constant wall temperature.
3. Compare Dittus–Boelter and Gnielinski correlations, including their inputs, assumptions, and limitations.
4. Explain why turbulent heat transfer generally increases pressure loss and discuss the design trade-off.
5. Describe a systematic procedure for analyzing forced convection in a duct from geometry and flow data to heat rate and outlet temperature.

Analytical / Application Questions

1. Air flows through a smooth tube. Explain how you would determine whether the flow is fully developed, select a correlation, calculate h, and check the pressure drop.
2. A heat exchanger tube is operated with constant wall temperature. Analyze how doubling mass flow rate affects Reynolds number, heat-transfer coefficient, and outlet temperature, stating assumptions.
3. A designer proposes roughening a duct wall to increase heat transfer. Evaluate the thermal, hydraulic, maintenance, and uncertainty consequences before approving the design.

Multiple Choice Questions with Answers

Question 1

Which dimensionless number compares inertial and viscous effects?

A. Prandtl number

B. Reynolds number

C. Nusselt number

D. Biot number

Correct Answer: B. Reynolds number

Question 2

For fully developed laminar flow in a circular tube with constant wall temperature, the Nusselt number is:

A. 1.00

B. 3.66

C. 4.36

D. 7.54

Correct Answer: B. 3.66

Question 3

The hydraulic diameter is defined as:

A. A_c/P_w

B. P_w/A_c

C. 4A_c/P_w

D. 2A_c/P_w

Correct Answer: C. 4A_c/P_w

Question 4

For turbulent air heating in a smooth tube using Dittus–Boelter, the exponent on Prandtl number is normally:

A. 0.2

B. 0.3

C. 0.4

D. 0.8

Correct Answer: C. 0.4

Question 5

Which relation gives pressure drop in a straight duct using the Darcy friction factor?

A. Δp = hA_sΔT

B. Δp = f_D(L/D_h)(ρV_m²/2)

C. Δp = ṁc_pΔT

D. Δp = kAΔT/L

Correct Answer: B. Δp = f_D(L/D_h)(ρV_m²/2)

Question 6

For constant wall temperature and constant properties, the bulk-temperature difference from the wall varies along a duct as:

A. A constant linear increase

B. An exponential decay

C. A sinusoidal function

D. A discontinuous step

Correct Answer: B. An exponential decay

Question 7

If Prandtl number is greater than one, which statement is generally correct?

A. Thermal diffusivity exceeds momentum diffusivity

B. The thermal boundary layer is generally thinner than the velocity boundary layer

C. The flow must be turbulent

D. Convection is impossible

Correct Answer: B. The thermal boundary layer is generally thinner than the velocity boundary layer

Question 8

A circular tube has diameter 0.01 m, mean speed 2 m·s⁻¹, density 1000 kg·m⁻³, and viscosity 0.001 Pa·s. Its Reynolds number is approximately:

A. 20

B. 200

C. 2,000

D. 20,000

Correct Answer: C. 2,000

Question 9

For a uniform constant wall heat flux in fully developed internal flow, the bulk temperature generally:

A. Remains constant

B. Changes linearly with axial distance for constant properties

C. Changes exponentially toward zero

D. Equals the wall temperature everywhere

Correct Answer: B. Changes linearly with axial distance for constant properties

Question 10

Why should pressure drop be considered when increasing flow speed for cooling?

A. Higher speed always reduces fan power

B. Pressure drop and required fluid-power input generally increase

C. Pressure drop has no relation to velocity

D. Higher speed eliminates all thermal resistance

Correct Answer: B. Pressure drop and required fluid-power input generally increase