Forced and Mixed Convection in Ducts

Internal forced convection describes heat transfer between a duct wall and a fluid that is driven through the duct by a fan, blower, or pump. Unlike external flow over a plate or cylinder, the fluid is confined: boundary layers growing from the walls eventually meet, and from that point on the flow is said to be fully developed. The convective heat transfer coefficient is not a property of the fluid alone. It depends on the flow regime, the fluid properties, and crucially on how far the flow has developed along the duct. A practicing engineer who ignores development effects will misjudge heat transfer in short ducts, in heat exchanger passages, and in the many short straight runs between fittings that make up a real duct system.

Consider air entering a straight duct with a nearly uniform velocity profile. At the wall the no-slip condition forces the fluid velocity to zero, so a velocity boundary layer begins to grow on each wall. Because the same mass flow must pass through every cross section, the fluid in the untouched core accelerates as the boundary layers thicken. This developing region ends when the boundary layers merge at the duct centerline. Beyond that point the velocity profile no longer changes with axial position, and the flow is hydrodynamically fully developed. In laminar flow that profile is the familiar parabola; in turbulent flow it is much flatter, with steep gradients confined to a thin layer near the wall.

The distance required for the profile to settle is the hydrodynamic entrance length. For laminar flow in a circular tube it grows in direct proportion to the Reynolds number:

This relationship carries a warning that surprises many newcomers. At a Reynolds number of 2000 in a 200 mm duct, the laminar entrance length is 20 m. Laminar duct flow is almost never fully developed in practical equipment, which is one reason laminar correlations must handle entrance effects explicitly. Turbulent flow behaves very differently: intense cross-stream mixing completes the profile development quickly, and the hydrodynamic entrance length is nearly independent of Reynolds number, on the order of ten diameters:

Since HVAC duct flow is turbulent in the great majority of cases, a straight duct run becomes hydrodynamically developed after roughly ten diameters, and the velocity profile can be treated as settled for pressure drop purposes beyond that point.

Thermal development follows the same logic but on its own length scale. When the wall temperature differs from the entering fluid temperature, a thermal boundary layer grows alongside the velocity boundary layer. Whether it develops faster or slower depends on the Prandtl number, which compares the diffusion of momentum to the diffusion of heat. For laminar flow the thermal entrance length is well approximated by scaling the hydrodynamic entrance length with the Prandtl number:

For gases with Pr near 0.7, the thermal entrance is somewhat shorter than the hydrodynamic entrance. For viscous oils with Prandtl numbers in the hundreds or thousands, the thermal entrance can be enormously longer, and the flow may be hydrodynamically developed while still thermally developing over the entire length of the equipment. In turbulent flow the vigorous mixing collapses both entrance lengths to roughly ten diameters, so the distinction matters mainly for laminar applications such as oil coolers, viscous process flows, and small liquid passages.

The Graetz number compresses the entrance question into a single dimensionless group. It can be read as the ratio of the time needed for heat to conduct across the duct radius to the time the fluid spends traveling the duct length:

A large Graetz number means a short duct in thermal terms: entrance effects dominate and the mean heat transfer coefficient is well above the fully developed value. A small Graetz number means a long duct where most of the surface sees fully developed conditions. The classic Hausen correlation for laminar flow with constant wall temperature is built directly on this idea:

As the duct length grows and Gz approaches zero, this expression recovers 3.66, the fully developed laminar Nusselt number for constant wall temperature in a circular tube. For short ducts the correction term raises the mean Nusselt number substantially, which is exactly the physics: thin developing boundary layers transfer heat far more effectively than the settled profile.

The practical message for duct design is that entrance effects almost always raise the heat transfer coefficient above the fully developed value, so using a fully developed correlation over an entire short duct underestimates heat transfer. In HVAC systems this matters in two ways. First, straight duct runs between elbows, takeoffs, and transitions are frequently only a few diameters long, so the flow is perpetually re-developing and the effective heat transfer exceeds textbook fully developed values. Second, many published correlations, including the widely used Dittus-Boelter correlation covered in Session 2, carry an explicit requirement such as L/D of at least 10, precisely because they were fitted to developed flow. Applying them to a two-diameter duct segment without correction is a misuse of the correlation, not a conservative simplification.

Key takeaways

• Internal flow develops from the entrance: boundary layers grow, merge, and then the profile stops changing, marking fully developed flow.
• Laminar hydrodynamic entrance length scales with Reynolds number (0.05 * Re_D * D); turbulent entrance length is roughly ten diameters regardless of Reynolds number.
• Thermal entrance length follows the hydrodynamic length scaled by the Prandtl number in laminar flow, so high-Pr fluids stay thermally developing over long distances.
• The Graetz number measures entrance importance: large Gz means entrance-dominated, small Gz means fully developed conditions prevail.
• Developing flow transfers heat more effectively than fully developed flow, so entrance corrections raise, not lower, the predicted coefficient.

Turbulent internal flow is the workhorse regime of HVAC engineering. The chaotic cross-stream motion of eddies carries heat from the wall deep into the fluid far more effectively than molecular conduction alone, so turbulent heat transfer coefficients are typically an order of magnitude above laminar values at the same mass flow. Because the governing equations cannot be solved in closed form for the mean heat transfer, engineers rely on empirical correlations built from the dimensionless groups that control the physics: the Reynolds number for the flow intensity, the Prandtl number for the fluid's thermal character, and the Nusselt number as the dimensionless heat transfer coefficient being predicted.

The most widely used of these is the Dittus-Boelter correlation, a simple power-law fit proposed in 1930 that has survived nearly a century of use because it is easy to evaluate and adequate for many design calculations:

Fluid properties are evaluated at the bulk mean temperature, the mass-averaged fluid temperature at the cross section of interest. The correlation is stated to be valid for Reynolds numbers of 10,000 and above, Prandtl numbers from 0.6 to 160, and length-to-diameter ratios of at least 10. These limits are not decorative. Below Re of 10,000 the flow sits in the transition region where the correlation was never fitted, and predictions there can be substantially off. The L/D requirement restates the entrance lesson of Session 1: the correlation describes developed turbulent flow, and short ducts need entrance corrections.

The exponent n captures a real physical asymmetry. When the wall is hotter than the fluid, the fluid near the wall is warmer and, for liquids, less viscous than the bulk; the near-wall layer flows more easily and heat transfer improves slightly. When the wall is cooler, the effect reverses. For gases like air, viscosity increases with temperature, so the direction of the effect flips in detail, but the correlation's fitted exponents still give reasonable results across the stated range. In HVAC work with air at Prandtl number near 0.71, many engineers simply use n = 0.4 for heating and n = 0.3 for cooling without further thought, which is acceptable for ordinary design temperature differences.

When better accuracy is needed, or when the Reynolds number falls in the transition region between about 3000 and 10,000, the Gnielinski correlation from 1976 is the better tool. Rather than a pure power-law fit, it is built on the analogy between momentum transfer and heat transfer, which gives it a sounder physical footing across a wider range:

Gnielinski stated validity for Reynolds numbers from 3000 to 5,000,000 and Prandtl numbers from 0.5 to 2000, a range that comfortably covers everything from the transition region to highly turbulent flow, and from gases to viscous liquids. The correlation needs the Darcy friction factor, and for smooth tubes the Petukhov relation from 1970 is the standard companion:

Real HVAC ducts are not hydraulically smooth. Galvanized steel, duct liner, and flexible duct all carry surface roughness that raises the friction factor above the smooth-tube value. The accepted practice is to compute the friction factor for the actual roughness, for example with the Haaland equation covered in Session 5, and insert that friction factor into the Gnielinski correlation. This keeps the heat transfer prediction consistent with the pressure drop calculation, which matters because both feed into equipment selection and energy estimates.

A further refinement concerns property variation across the boundary layer. Both correlations evaluate properties at the bulk mean temperature, which is exact only when the wall and fluid temperatures are close. With large temperature differences, the fluid near the wall has markedly different viscosity and conductivity than the bulk, and the prediction drifts. Standard practice applies a property-ratio correction, most famously in the Sieder-Tate form that scales the Nusselt number by a viscosity ratio raised to a fitted power for laminar flow; analogous property-ratio corrections are used for turbulent flow. A simpler and common alternative for gases is to evaluate properties at the film temperature, the average of wall and bulk temperatures. Whichever approach is chosen, the key discipline is consistency: pick one property-evaluation scheme and apply it to both the heat transfer and the pressure drop calculations.

Choosing between the two correlations is straightforward in practice. For quick hand calculations in fully turbulent flow with moderate temperature differences, Dittus-Boelter is fast and adequate. For the transition region, for high-accuracy requirements, for rough ducts where the friction factor is known independently, or for fluids far from air, Gnielinski is the better choice. The numerical comparison below shows the two agreeing within a few percent in fully turbulent air flow, which is why Dittus-Boelter remains acceptable there, while only Gnielinski remains valid as the Reynolds number drops toward transition.

Key takeaways

• Dittus-Boelter (Nu_D = 0.023 * Re_D^4/5 * Prⁿ) is a fast, adequate correlation for fully turbulent flow with Re_D of at least 10,000, Pr from 0.6 to 160, and L/D of at least 10.
• The exponent is n = 0.4 for wall heating and n = 0.3 for wall cooling, with properties evaluated at the bulk mean temperature.
• Gnielinski covers Re_D from 3000 to 5,000,000 and Pr from 0.5 to 2000, making it the right choice for transition flow and higher-accuracy work.
• Gnielinski needs the Darcy friction factor: use Petukhov for smooth tubes and a roughness-based value such as Haaland for real ducts.
• Large wall-to-fluid temperature differences call for a property-ratio correction or film-temperature property evaluation, applied consistently.

Nearly every correlation in Sessions 1 and 2 was developed for circular tubes, yet the ducts on a real job site are overwhelmingly rectangular, with round duct reserved mainly for branches, risers, and high-pressure systems. The bridge between the two geometries is the hydraulic diameter, a single length scale that lets circular-tube correlations serve rectangular, triangular, and other cross sections with surprising accuracy. Understanding where it comes from, and where it stops working, keeps duct heat transfer calculations both simple and honest.

The hydraulic diameter is defined as four times the flow cross-sectional area divided by the wetted perimeter:

The factor of four is not arbitrary. For a circular tube, A_c = pi * D² / 4 and P = pi * D, so 4 * A_c / P reduces exactly to D. The hydraulic diameter therefore reproduces the ordinary diameter for the geometry the correlations were built on, and extends the same scaling to anything else. Physically, the ratio A_c / P is proportional to the fluid volume per unit of wall surface area along the duct, which is the quantity that governs how wall friction and wall heat transfer scale with size. For a rectangular duct with sides a and b, the area is a * b and the perimeter is 2 * (a + b), giving D_h = 2 * a * b / (a + b). A 400 mm by 200 mm supply duct, the example worked in detail in Session 6, has a hydraulic diameter of about 267 mm.

In laminar flow, the cross-sectional shape genuinely matters. The fully developed Nusselt number depends on the geometry and on the thermal boundary condition, and its values for rectangular ducts of various aspect ratios are tabulated in standard references, notably the compilation by Shah and London. For constant wall temperature, the circular-tube value of 3.66 and the parallel-plate value of 7.54 bracket the rectangular-duct results: as a rectangular duct becomes flatter, its fully developed Nusselt number migrates from the circular value toward the parallel-plate limit. The constant-heat-flux values follow the same pattern at slightly higher levels. This shape sensitivity is a laminar-flow phenomenon, rooted in how the velocity profile fills the corners and how the thermal boundary condition couples to it.

In turbulent flow, the picture simplifies enormously, which is fortunate because HVAC duct flow is turbulent in essentially all practical cases. Experiments have repeatedly shown that circular-tube correlations predict turbulent heat transfer in non-circular ducts quite well when the Reynolds and Nusselt numbers are both based on the hydraulic diameter. The intense eddy mixing that dominates turbulent transport washes out the geometric details that matter so much in laminar flow. So the working procedure for a rectangular supply duct is direct: compute D_h, form Re based on D_h, check that it exceeds 10,000, and apply Dittus-Boelter or Gnielinski exactly as for a round tube. Session 6 demonstrates this end to end. The same approach serves the friction factor in pressure drop calculations, where the Reynolds number in the Haaland equation and the diameter in the Darcy-Weisbach equation are both replaced by their hydraulic-diameter counterparts.

The approximation does have limits worth knowing. For ducts of extreme aspect ratio, very flat wide ducts behave thermally like flow between parallel plates, and the proper length scale trends toward twice the plate spacing rather than the hydraulic diameter; in practice such geometries appear as duct liners, ceiling plenums, and double-skin facades rather than as supply ducts. Triangular and other exotic sections follow the hydraulic-diameter approach reasonably in turbulent flow but deserve caution in laminar flow, where tabulated shape-specific values should be consulted. Entrance effects, covered in Session 1, apply in the same way: the entrance length scales with the hydraulic diameter, so the ten-diameter rule for turbulent flow becomes ten hydraulic diameters.

One subtle point concerns the wetted perimeter in ducts with internal obstructions. Turning vanes, splitter dampers, and internal liner surfaces all add wetted perimeter without adding flow area, which reduces the hydraulic diameter and raises both the friction factor and the heat transfer coefficient relative to a clean duct of the same outer dimensions. For ordinary design work this refinement is usually folded into fitting loss coefficients rather than computed explicitly, but it explains why a duct crowded with internal devices behaves as a "smaller" duct than its sheet-metal dimensions suggest.

Key takeaways

• The hydraulic diameter D_h = 4 * A_c / P reduces to the ordinary diameter for a circular tube and is the standard length scale for non-circular ducts.
• For a rectangular duct with sides a and b, D_h = 2 * a * b / (a + b).
• In laminar flow the fully developed Nusselt number depends on aspect ratio, lying between the circular-tube value 3.66 and the parallel-plate value 7.54 for constant wall temperature, with exact values tabulated by Shah and London.
• In turbulent flow, circular-tube correlations applied with the hydraulic diameter predict rectangular-duct heat transfer well, which covers nearly all HVAC applications.
• Use the hydraulic diameter consistently in the Reynolds number, the Nusselt number, and the pressure drop equations, and scale turbulent entrance lengths as ten hydraulic diameters.

Everything so far has assumed that the flow is driven entirely by the fan and that density differences within the fluid play no role. That assumption holds when velocities are high and temperature differences are modest, which describes most supply ductwork. But when temperature differences grow or velocities fall, the fluid's own weight begins to matter: warm fluid is lighter and tends to rise, cool fluid is heavier and tends to sink. These buoyancy forces superimpose a natural-convection motion on the forced flow, and the combined regime is called mixed convection. An engineer who never checks for it will occasionally, and expensively, be wrong about heat transfer in large low-velocity ducts, in ducts exposed to strong solar heating, and in any passage where the fan barely moves the air.

The strength of buoyancy relative to viscous forces is measured by the Grashof number:

The quantity that decides the flow regime, though, is the Richardson number, which compares buoyancy directly against the inertia of the forced flow:

The interpretation is deliberately simple. When Ri is much less than 1, inertia dominates and buoyancy can be ignored: pure forced convection applies, and every correlation in Sessions 2 and 3 is valid as written. When Ri is much greater than 1, buoyancy dominates and the flow behaves as natural convection in a confined passage. When Ri is of order 1, the two mechanisms are comparable and the heat transfer can differ markedly, usually upward, from either pure regime. In practice, engineers often treat Ri below about 0.1 as safely forced and Ri above about 10 as safely natural, with the decade in between requiring mixed-convection treatment. These are judgment thresholds, not sharp physical boundaries.

Orientation decides how buoyancy interacts with the forced flow. In a vertical duct, buoyancy acts along the flow direction, and two distinct cases arise. In aiding flow, buoyancy pushes in the same direction as the forced flow: the classic case is upward flow with heated walls, where the warm near-wall fluid is light and accelerates upward. The boundary layer thins, the wall shear rises, and the heat transfer coefficient increases above the forced-convection value. In opposing flow, buoyancy pushes against the forced flow: downward flow with heated walls is the standard example. The warm near-wall fluid wants to rise while the bulk flow descends, which decelerates the near-wall region, thickens the thermal boundary layer, and impairs heat transfer. If the opposition is strong enough, the near-wall fluid can stagnate or reverse, producing unstable, oscillatory behavior that no steady correlation can capture. Cooling the walls reverses the roles: upward flow with cooled walls becomes the opposing case.

Horizontal ducts present a different and very common situation. Here buoyancy acts across the duct rather than along it. With heated walls, the warm fluid near the wall rises toward the top of the duct while cooler core fluid sinks, setting up a pair of counter-rotating longitudinal vortices superimposed on the axial flow. This secondary motion continuously sweeps fresh fluid against the wall and typically enhances heat transfer relative to pure forced convection, sometimes substantially. With strong heating and low flow rates the stratification can become pronounced: a hot layer pools at the top of the duct while the bottom stays cool, and the circumferentially averaged heat transfer coefficient becomes a blunt instrument for describing a highly nonuniform reality. The top-to-bottom temperature asymmetry also matters mechanically, since it drives differential thermal expansion in large ducts.

For mainstream HVAC design, a quick Richardson number estimate usually settles the matter. Take a supply duct at 6 m/s with a 10 K wall-to-air temperature difference: the Grashof number is modest, the Reynolds number is large, and Ri comes out on the order of 0.002, as computed in the Session 6 worked example. Buoyancy is utterly negligible, and the analysis can proceed as pure forced convection with full confidence. Mixed convection earns attention in specific situations: large cross-section ducts at low velocity, where Re is small and D is large (note the D cubed in the Grashof number); rooftop or exposed ducts under strong solar heating, where the wall temperature can soar far above the air temperature; cold supply ducts running through hot ceiling plenums at part-load flow; and systems that rely partly on stack effect or natural ventilation, where the "forced" velocity itself is buoyancy-driven and the regime is mixed by construction.

When the Richardson number indicates that mixed convection matters, the honest approach is to consult mixed-convection correlations for the appropriate orientation and aiding or opposing condition, several of which appear in the standard heat transfer references. These correlations typically present the mixed Nusselt number as a correction to the forced-convection value, rising above it for aiding flow and horizontal ducts, and dipping below it for opposing flow. The correction is worth applying because the effect is not small: aiding and secondary-flow enhancement can raise heat transfer by tens of percent, while strong opposing flow can impair it by a similar margin. What the correlations cannot do is describe the unstable reversed-flow regime, which must be avoided by design rather than predicted precisely.

Key takeaways

• Mixed convection arises when buoyancy forces are comparable to the inertia of the forced flow, measured by the Richardson number Ri = Gr_D / Re_D².
• Ri much less than 1 means pure forced convection; Ri much greater than 1 means buoyancy-dominated flow; Ri near 1 requires mixed-convection treatment.
• In vertical ducts, buoyancy aiding the flow enhances heat transfer, while buoyancy opposing the flow impairs it and can cause instability or reversal.
• In horizontal ducts, buoyancy drives longitudinal secondary vortices and possible thermal stratification, usually enhancing the average heat transfer.
• Typical HVAC supply ducts have Ri on the order of 0.001, so buoyancy is negligible; check Ri explicitly for large low-velocity ducts, solar-heated ducts, and part-load conditions.

Heat transfer determines what a duct does to the air thermally; pressure drop determines what the fan must do to the air mechanically. The two are coupled through the same velocity field and the same friction factor, which is why a course on duct convection must treat pressure drop as a first-class topic. Undersize the pressure analysis and the fan cannot deliver design airflow; oversize it and the client pays for fan energy and equipment for the life of the building. The calculation has three parts: friction along straight duct, losses in fittings, and the conversion of total pressure into fan power.

Friction loss in a straight duct section follows the Darcy-Weisbach equation, the central relation of duct and pipe flow:

The group rho * V² / 2 is the velocity pressure, the kinetic energy per unit volume of the moving air, and the friction factor f is the dimensionless measure of how effectively the wall converts that kinetic energy into heat. In laminar flow the friction factor is known exactly, f = 64 / Re_D, a result that follows directly from the parabolic velocity profile. In turbulent flow the friction factor depends on both the Reynolds number and the relative roughness of the wall, and it is read from the Moody chart or, more conveniently today, computed from an explicit approximation.

The Haaland equation, published in 1983, gives the Darcy friction factor explicitly without the iteration that the original Colebrook relation requires:

For new galvanized steel duct, the roughness height epsilon is commonly taken as 0.15 mm in design practice, a value that represents commercial steel with its seams, joints, and slight surface irregularities. Relative roughness epsilon / D then sets how far the friction factor sits above the smooth-tube line: in large ducts the relative roughness is small and the wall behaves as nearly smooth, while in small ducts the same absolute roughness is proportionally larger and friction rises. Duct liner, internally insulated duct, and especially flexible duct carry much higher effective roughness, and flexible duct additionally suffers when it is compressed, sagging, or bent, since every bend in a flexible run acts partly as a fitting. Manufacturer data or handbook values should be used for these materials rather than the bare-steel roughness.

Straight-duct friction is only half the story, and in many real systems it is the smaller half. Every elbow, transition, takeoff, damper, and grille disturbs the flow and dissipates mechanical energy, and these fitting losses are expressed through dimensionless loss coefficients:

Loss coefficients are tabulated in standard design references such as ASHRAE Fundamentals and SMACNA duct design guidance, which devote extensive tables to them because fittings dominate so many systems. A few landmarks orient the intuition. A smooth radius elbow carries a small fraction of a velocity head, while a sharp mitered elbow without turning vanes can dissipate more than a full velocity head. A duct discharging directly into a room loses essentially one full velocity head, since the kinetic energy of the jet is dissipated in the space; this exit loss of K = 1.0 is a standard result. A sharp entrance from a plenum into a duct costs roughly half a velocity head. Dampers, fire dampers, and grilles each carry their own coefficients, usually obtained from manufacturer test data at the design flow. The critical discipline is consistency of the reference velocity: each K is defined against a particular velocity, normally the mean velocity in the duct at that fitting, and mixing reference velocities between fittings corrupts the total.

The total pressure drop the fan must overcome is the sum of all straight-duct friction terms and all fitting losses along the critical path, the run from the fan to the terminal with the highest resistance. In systems with many elbows, takeoffs, and dampers, the fitting total frequently exceeds the straight-duct friction by a large factor, which is why experienced designers obsess over fitting selection and duct routing rather than shaving a few pascals off straight runs. Once the total pressure is known, the air power required is the product of volume flow rate and pressure rise, divided by the overall fan efficiency:

The efficiency term deserves respect: it folds together the aerodynamic efficiency of the fan wheel, the motor efficiency, and any belt or drive losses, and it is properly taken from manufacturer selection data at the operating point rather than guessed. Two scaling laws, the fan affinity laws, govern how power responds to changes in flow and are among the most economically consequential relations in HVAC. At fixed system geometry, pressure drop varies with the square of the flow rate and fan power varies with the cube. Reducing airflow by 20 percent cuts fan power nearly in half. This cubic relationship is the entire economic logic behind variable-air-volume systems and variable-speed drives: trimming flow to match the actual load, rather than running constant volume against throttled dampers, converts directly into large energy savings.

For preliminary sizing, many designers work toward a straight-duct friction rate on the order of 1 Pa/m, then add fitting losses explicitly and check noise. This friction rate is a practice target that balances duct size against fan energy, not a code requirement, and it is adjusted up or down with project priorities: lower for quiet or energy-critical projects, higher where space is tight. The equal-friction sizing method extends the idea across a whole system by sizing each duct section for roughly the same pressure drop per meter, which tends to produce self-balancing layouts, with balancing dampers trimming the residual differences between branches.

Key takeaways

• Straight-duct friction follows Darcy-Weisbach: Delta p = f * (L / D) * (rho * V² / 2), with f = 64 / Re_D in laminar flow.
• The Haaland equation gives the turbulent Darcy friction factor explicitly from Reynolds number and relative roughness; commercial steel duct is commonly taken at epsilon = 0.15 mm.
• Fitting losses use Delta p = K * rho * V² / 2 with K values from ASHRAE, SMACNA, or manufacturer data, always referenced to a consistent velocity.
• Fitting losses frequently dominate straight-duct friction in real systems, so fitting selection and routing deserve the most design attention.
• Fan power is P = Q * Delta p / eta_fan, and the affinity laws (pressure with the square, power with the cube of flow) make flow reduction the most powerful energy lever available.

Theory becomes engineering judgment only when the numbers are worked by hand at least once. This session does exactly that for a completely ordinary piece of HVAC hardware: a rectangular galvanized steel supply duct. Every step uses the tools built in Sessions 1 through 5, and every result is checked against the kind of practice rules introduced in Session 7. Follow the arithmetic closely, because the pattern here, geometry to Reynolds number to friction factor to pressure drop to heat transfer coefficient to thermal consequence, is the same pattern behind every duct calculation you will ever perform.

The scenario is a straight horizontal supply duct, 400 mm by 200 mm in cross section, carrying air at 20 C with a mean velocity of 6 m/s. The duct is commercial galvanized steel. We want four answers: the pressure drop per meter of straight duct, the internal convection coefficient, the wall heat gain per meter for a 10 K temperature difference between the air and the surroundings, and a judgment on whether this duct is sensibly sized.

Now the interpretation, which is where the calculation earns its keep. First, the velocity: 6 m/s sits at the bottom of the typical 6 to 9 m/s range used for commercial main supply ducts, so it is entirely reasonable, quiet enough for most applications and not wasteful of duct material. Second, the pressure drop: 1.46 Pa/m for straight duct is acceptable and sits near the friction-rate targets designers commonly aim for, though slightly above a nominal 1 Pa/m target. Remember that this is straight duct only. In a real run, elbows, takeoffs, and dampers will add substantially, often doubling or tripling the total, so the system curve must include the fittings before a fan is selected.

Third, the heat gain: 245 W/m is a large number wearing an innocent expression. Over a 15 m run through an unconditioned ceiling plenum or attic, that is roughly 3.7 kW of unwanted heat picked up by the supply air, a meaningful fraction of the cooling capacity serving a typical zone, and it arrives as a direct penalty on both energy use and delivered air temperature. Because the estimate is an upper bound, the real figure is smaller, but not small enough to ignore. The conclusion is unambiguous: insulate supply ducts that run through unconditioned spaces, and where condensation is possible on cold duct surfaces, the insulation needs a proper vapor retarder as well. An uninsulated 400 by 200 duct at these conditions is not a minor oversight; it is a measurable load the mechanical system must carry for its entire life.

Finally, note what the calculation did not need: no CFD, no charts, no software. Six hand steps with standard correlations sized the duct thermally and hydraulically, checked the flow regime, and produced a defensible insulation decision. That is the working method this course is built to instill.

Key takeaways

• A 400 mm x 200 mm duct has D_h = 267 mm; at 6 m/s the Reynolds number is about 106,000, fully turbulent.
• The Haaland friction factor is 0.0179, giving a straight-duct pressure drop of 1.46 Pa/m, acceptable but only part of the system total.
• Dittus-Boelter gives Nu = 210 and h = 20.4 W/m²-K for air being warmed by the duct wall.
• The upper-bound heat gain of 245 W/m shows why ducts in unconditioned spaces must be insulated: a 15 m run approaches 3.7 kW of unwanted load.
• The Richardson number of 0.0025 confirms buoyancy is negligible, so the forced-convection analysis is complete and self-consistent.

The correlations of the previous sessions give exact answers to well-posed questions. Real design adds a harder problem: choosing the questions. Duct velocity, friction rate, aspect ratio, and insulation level are not dictated by physics alone; they are compromises between first cost, operating cost, space, noise, and comfort. The rules of thumb in this session are the distilled compromises of the industry, the ranges within which experienced designers work before refining with calculation. They come from design references such as ASHRAE Fundamentals and SMACNA duct design guidance, and they should be understood as typical practice ranges, not as code mandates. Codes govern fire, smoke, and life safety; velocity selection remains an engineering decision.

Velocity is the master variable because nearly everything scales with it. The commonly taught practice ranges run roughly as follows: main supply ducts in commercial systems, about 6 to 9 m/s; branch ducts serving individual zones, lower, typically in the 4 to 6 m/s region; risers and shafts, moderate values between the two; residential systems, lower still, reflecting smaller fans, shorter runs, and stricter noise expectations. Terminal devices, the grilles and diffusers that actually deliver air to rooms, operate far below duct velocities, and their selection follows manufacturer throw and noise data rather than any duct rule. These ranges exist because velocity is a three-way bargain. Higher velocity means smaller ducts, which saves sheet metal, hangers, and precious ceiling space. But pressure drop grows with the square of velocity and fan power with the cube, so each extra meter per second is bought with permanently higher energy bills. Lower velocity reverses the trade: cheap energy, expensive and bulky ducts.

Noise is the constraint that most often overrules the energy optimum. Air noise is generated wherever the flow is disturbed: at elbows, at dampers, at takeoffs, and especially at terminal devices discharging into occupied rooms. Room background noise is rated against NC, noise criteria, curves, and each space type carries an expectation: an open office tolerates more background sound than a conference room or a classroom. The design response is to keep velocities low near occupied spaces and at terminals, to prefer smooth radius fittings over abrupt ones in noise-sensitive paths, and to locate dampers and takeoffs away from diffusers so that the turbulence they generate has duct length in which to decay. A system that meets its pressure budget but whistles at every grille has failed just as surely as one whose fan cannot make the flow.

The pressure side of the design is managed through the friction rate, the pressure drop per meter of straight duct. Many designers begin sizing with a target on the order of 1 Pa/m for straight duct, then add fitting losses explicitly and check the noise implications. This target is a starting compromise, not a law: energy-critical or noise-critical projects size lower, space-constrained retrofits size higher. The equal-friction method systematizes the approach by sizing every section of the system for approximately the same friction rate, which produces layouts that are nearly self-balancing, with opposed-blade or splitter dampers trimming the residual differences between branches. A related classical method, static regain, sizes downstream sections so that the drop in velocity pressure at each expansion is converted back into static pressure, holding the available static pressure roughly constant along the run. Both methods then face the same reality check from Session 5: fittings often contribute more than the straight duct, so the fitting schedule deserves as much care as the duct sizes.

Fan energy ties the whole design together through the relation P = Q * Delta p / eta_fan

and the affinity laws that make flow reduction the most powerful efficiency measure available. A variable-air-volume system that trims flow to match the actual load exploits the cubic relationship between flow and power, which is why modern commercial systems rarely run constant volume. Duct leakage deserves mention in the same breath: every cubic meter per second that escapes through unsealed joints is flow the fan paid to move that never reaches the room, so sealing ducts to a proper leakage class is among the cheapest energy measures in the building. Insulation closes the thermal side: supply ducts in unconditioned spaces gain or lose heat along their length, as Session 6 quantified, and cold duct surfaces in humid environments condense, so insulation with a continuous vapor retarder is both an energy and a moisture-control decision.

Aspect ratio is a quieter but real design variable. A square duct carries a given flow with less sheet metal per unit of delivered air and lower pressure drop than a flat wide duct of the same area, because the flat duct has more wall surface per unit of flow area. Deep ceiling plenums therefore favor near-square ducts, and flat ducts should be reserved for cases where the available depth truly forces them. When flat ducts are unavoidable, the higher pressure drop and material cost should be carried explicitly in the design rather than discovered during balancing.

Balancing deserves a final word because it is where good designs are confirmed or quietly ruined. Dampers at each branch allow the airflows to be set to design values during commissioning, but dampers can only redistribute the flow the fan provides; they cannot create airflow the system cannot deliver. A design that depends on heavily throttled dampers to force balance is a design whose duct sizes were wrong: the throttling wastes fan energy permanently to correct a sizing error made once. Better to size the branches so that their natural pressure drops are close, leaving the dampers for fine trimming.

Key takeaways

• Typical practice velocities are about 6 to 9 m/s for commercial main supply ducts, lower for branches, and lower still for residential systems; these are design-practice ranges from references such as ASHRAE Fundamentals and SMACNA guidance, not code mandates.
• Velocity is a three-way trade between duct size and first cost, fan energy through the square and cube laws, and noise against NC criteria expectations.
• Size straight duct toward a friction rate on the order of 1 Pa/m as a starting target, then add fitting losses explicitly, since fittings often dominate the system total.
• Keep ducts near square where plenum depth allows; flat high-aspect-ratio ducts cost more material and pressure drop per unit of airflow.
• Seal ducts against leakage, insulate ducts in unconditioned spaces with a proper vapor retarder on cold surfaces, and size branches to balance naturally rather than by heavy damper throttling.